ThesisPDF Available

Building networks of sexual partners. Application for the study of the transmission dynamics of Human Papillomavirus (HPV)

Authors:

Abstract

Sexually transmitted diseases (STDs) have been a major public health threat for a long time in human history. Modern concerns about STD began with the pandemic of syphilis which spread over Europe in the early sixteenth century. The human papillomavirus (HPV) is the direct cause of more than half million new cases of cervical cancer, the second most common malignancy among women and a leading cause of cancer death worldwide. It also causes anogenital warts and other related diseases. In this work we have studied the transmission dynamics of HPV over a sexual contacts network. In order to predict the evolution of these kind of diseases, we need a reliable model of the underlying social network in which the infection spreads. We have built a lifetime sexual partners (LSP) network based on demographic data and surveys about sexual habits. Most of the modeling approaches to STD in general and HPV in particular, are done using classical models where the hypothesis of homogeneous mixing (everybody can transmit a disease to everybody) is assumed. However, homogeneous mixing is not a reasonable hypothesis and consequences of this assumption can be seen, for instance, in that the effects of vaccination schedules against HPV have been detected in Australia much sooner than what the classical models predicted. There is a debate concerning the vaccination of young men. Elbasha et al. found some evidences that the vaccination of boys could also be cost-effective. In our model we consider both heterosexual men, and men who have sex with men (MSM) populations and the connections among them letting us to study this matter. With our model simulate and carry out vaccination campaigns in order to figure out the best strategies. All these results can be useful for policy makers in Public Health to make appropriate decisions respect to HPV.
Universitat Polit`ecnica de Val`encia
Departament de Matem`atica Aplicada
PhD. THESIS
Building networks of sexual
partners. Application for the
study of the transmission
dynamics of Human
Papillomavirus (HPV)
Ph.D. CANDIDATE ADVISORS
Dr. Luis Acedo Rodr´ıguez
D. V´ıctor S´anchez Alonso Dr. Rafael Jacinto Villanueva Mic´o
Dr. Francisco Javier Villanueva Oller
Valencia - May 2019
Dr. Luis Acedo Rodr´ıguez and Dr. Rafael Jacinto Villanueva Mic´o, from
the Universitat Polit`ecnica de Val`encia and Dr. Francisco Javier Villanueva
Oller from the Universidad Rey Juan Carlos,
CERTIFY that the present thesis entitled Building networks of sexual
partners. Application for the study of the transmission dynamics of Human
Papillomavirus (HPV) has been performed under our supervision in the De-
partment of Applied Mathematics at the Universitat Polit`ecnica de Val`encia
by V´ıctor S´anchez Alonso. It constitutes his thesis dissertation to obtain the
PhD degree in Mathematics.
In compliance with the current legislation, we authorize the presentation
of this dissertation signing the present certificate.
Valencia, June 27, 2019
Luis Rafael Jacinto Francisco Javier
Acedo Rodr´ıguez Villanueva Mic´o Villanueva Oller
2
Abstract
Sexually transmitted diseases (STDs) have been a major public health threat
for a long time in human history. Modern concerns about STD began with
the pandemic of syphilis which spread over Europe in the early sixteenth
century.
The human papillomavirus (HPV) is the direct cause of more than half
million new cases of cervical cancer, the second most common malignancy
among women and a leading cause of cancer death worldwide. It also causes
anogenital warts and other related diseases.
In this work we have studied the transmission dynamics of HPV over a
sexual contacts network. In order to predict the evolution of these kind of
diseases, we need a reliable model of the underlying social network in which
the infection spreads. We have built a lifetime sexual partners (LSP) network
based on demographic data and surveys about sexual habits.
Most of the modeling approaches to STD in general and HPV in partic-
ular, are done using classical models where the hypothesis of homogeneous
mixing (everybody can transmit a disease to everybody) is assumed. How-
ever, homogeneous mixing is not a reasonable hypothesis and consequences
of this assumption can be seen, for instance, in that the effects of vaccination
schedules against HPV have been detected in Australia much sooner than
what the classical models predicted.
There is a debate concerning the vaccination of young men. Elbasha et
al. found some evidences that the vaccination of boys could also be cost-
effective. In our model we consider both heterosexual men, and men who
have sex with men (MSM) populations and the connections among them
letting us to study this matter. With our model simulate and carry out
vaccination campaigns in order to figure out the best strategies. All these
results can be useful for policy makers in Public Health to make appropriate
decisions respect to HPV.
3
Resumen en Castellano
Desde tiempos inmemorables en la historia de la humanidad las enfermedades
de transmisi´on sexual (ETSs) han sido una gran amenaza para la salud
ublica. Las preocupaciones comienzan en la edad moderna con pandemias
tales como la s´ıfilis, cuya propagaci´on ocurre en Europa a comienzos del siglo
XVI.
El virus de papiloma humano (VPH) es la causa directa de m´as de medio
mill´on de casos nuevos de c´ancer de cuello de ´utero, el segundo m´as ma-
ligno entre mujeres y una de las principales causas de muerte por c´ancer en
todo el mundo. Adem´as causa verrugas anogenitales y otras enfermedades
relacionadas.
En este trabajo estudiamos el contagio del VPH en una red de contactos
sexuales. Para predecir la evoluci´on de este tipo de enfermedades, necesi-
tamos un modelo fiable de la red social subyacente sobre el que la infecci´on
prolifera. Hemos construido una red de parejas sexuales durante toda la vida
basada en datos demogr´aficos y encuestas sobre h´abitos sexuales.
La mayor´ıa de los enfoques para modelizar ETSs por lo general y del VPH
en particular, se hacen usando modelos cl´asicos donde la hip´otesis de mezcla
homog´enea (todo el mundo puede transmitir a todo el mundo) es asumida
de manera impl´ıcita. Sin embargo, la mezcla homog´enea no es una hip´otesis
razonable y las consecuencias de estas suposiciones se ven de hecho, en que
los efectos de los calendarios de vacunaci´on contra el VPH se detectan en
Australia mucho antes de lo que los modelos cl´asicos predijeron.
Hay un debate sobre la conveniencia de la vacunaci´on de los ni˜nos. El-
basha et al. encontraron evidencias de que la vacunaci´on en ni˜nos podr´ıa lle-
gar a ser coste-efectiva. En nuestro modelo consideramos poblaciones tanto
de hombres que solo tienen relaciones con mujeres y que las tienen entre
ellos, permiti´endonos sacar conclusiones al respecto. Con nuestro modelo
simulamos y llevamos a cabo campa˜nas de vacunaci´on de modo que pode-
mos sacar conclusiones atendiendo a las mejores estrategias. Estos resultados
pueden ayudar a los responsables de Salud P´ublica a tomar decisiones apropi-
adas con respecto al VPH.
4
Resum en Valenci`a
Des de temps inmemorables en la hist`oria de la humanitat les malalties de
transmissi´o sexual (MTSs) han sigut una gran amena¸ca per a la salut p´ublica.
Les preocupacions comencen en l’edat moderna amb pand`emies com ara la
s´ıfilis, la propagaci´o de la qual ocorre a Europa al comen¸cament del segle
XVI.
El virus de papilloma hum`a (VPH) ´es el causant directe de m´es de mig
mili´o de casos nous de c`ancer de coll d’´uter, el segon mes maligne entre dones
i una de les principals causes de mort per c`ancer en tot el m´on. A m´es causa
berrugues anogenitales i altres malalties relacionades.
En este treball estudiem la din`amica de transmissi´o del VPH en una
xarxa de contactes sexuals. Per a predir l’evoluci´o d’este tipus de malalties,
necessitem un model fiable de la xarxa social subjacent sobre la qual la
infecci´o prolifera. Hem construt un xarxa de parelles sexuals durant tota la
vida basada en dades demogr`afiques i enquestes sobre h`abits sexuals.
La majoria dels enfocaments per a modelizar MTSs generalment i del
VPH en particular, es fan usant models cl`assics on la hip`otesi de mescla
homog`enia (tot el m´on pot transmetre a tot el m´on) ´es assumida de manera
impl´ıcita. No obstant aix`o la mescla homog`enia no ´es una hip`otesi raonable
i les conseq¨u`encies d’estes suposicions es veuen de fet, en que els efectes dels
calendaris de vacunaci´o contra el VPH es detecten a Austr`alia molt abans
del que els models cl`assics van predir.
Hi ha un debat en el que referix a la vacunaci´o dels xiquets. Elbasha
et al. van trobar evid`encies que la vacunaci´o en xiquets podria arribar a
ser cost-efectiva. En el nostre model considerem poblacions tant d’h`omens
que tenen realcions soles amb dones i els que tamb´e tenen relacions amb
homes i les connexions existents entre ells ens permeten traure conclusions
sobre este aspecte. Amb el nostre model podem simular diverses campanyes
de vacunaci´o de manera que podem traure conclusions atenent a les millors
estrat`egies. Estos resultats poden ajudar als responsables de Salut P´ublica
a pendre decissions apropiades respecte al VPH.
5
Contents
1 Introduction 8
1.1 The revelation of the HPV .................... 8
1.2 Vaccines and mathematical models ............... 10
1.3 Our proposal ........................... 14
2 Building LSP networks and describing the transmission dy-
namics of HPV on these networks 17
2.1 Origin of the data ......................... 17
2.2 Network model .......................... 18
2.3 Semi-Random Construction ................... 19
3 Calibrating the Sexual Contacts Network 23
3.1 Description of the model ..................... 23
3.2 Description of our resources ................... 26
3.2.1 Computers ......................... 26
3.2.2 Distributed computing environment ........... 26
3.2.3 Random Particle Swarm Optimization (rPSO) ..... 26
3.2.4 Fitness function and some features included in the rPSO 29
3.3 Selecting an optimal number of particles to calibrate the HPV
large network computational model with rPSO ......... 30
3.4 Improving the exploration of the rPSO ............. 31
3.5 Calibrating the model ...................... 32
3.6 Selecting the 30 realizations that best capture the data uncer-
tainty ............................... 33
3.7 Parameters returned by the calibration ............. 38
4 Including new features into the model: Dynamic LSP net-
work and vaccination campaigns 39
4.1 Introducing age and LSP dynamics into the LSP model . . . . 39
4.2 When should we start the vaccination campaign? ....... 41
4.3 Introducing vaccination ...................... 43
6
4.4 Introducing vaccination loss protection ............. 44
4.5 Introducing variations in the vaccination coverage ....... 44
5 Model validation: The Australian case 45
5.1 How to measure the decline ................... 46
5.2 Does our model return similar values to those in [1]? ..... 46
5.3 Study of the herd immunity effect over HPV LR infection . . . 50
5.4 Effect of the reduction of the vaccine effectiveness in the catch-
up vaccination ........................... 50
5.5 Discussion ............................. 53
6 Simulations 56
6.1 Decline of HPV LR infections and cases of genital warts in the
long-term in Spain ........................ 57
6.2 Decline of HPV oncogenic HR infections in the long-term in
Spain ................................ 59
6.3 What would happen in Spain if, after 20 years of the vaccina-
tion, the effect of the vaccine disappear completely? ...... 61
6.4 Simulation of the effect of the tourism on the contagion of
HPV in Spain ........................... 62
6.5 Simulation of different vaccination strategies in girls and boys
with the aim to eradicate the cancer diseases associated HPV
oncogenic HR ........................... 67
6.6 Quantifying the delay to recover normal levels of decline due
to a drop in the vaccination coverage .............. 71
6.7 How is the decline of HPV affected if the average number of
LSPs increases significantly? ................... 77
6.8 How does the decline of HPV changes if the percentage of
MSMs increases significantly? .................. 79
7 Conclusions and limitations 82
7
Chapter 1
Introduction
1.1 The revelation of the HPV
One of the biggest scientific discoveries in the past 30 years was the connec-
tion between HPV infection of the cervix and cervical cancer. This achieve-
ment resulted from the original seminal findings by Harald zur Hausen and
his group, they found that human papillomavirus genotype 16 can be de-
tected in cervical cancer tissue.
The finding was followed by epidemiologists, molecular biologists, vacci-
nologists, and clinicians culminating in 2006 with the development of effective
prophylactic vaccines for HPV, which have the means to prevent 70-80% of
cervical cancers [3]. Zur Hausen was awarded the Nobel Prize in Physiology
or Medicine in 2008, in recognition of his discovery.
With more than 600 million cases worldwide, including 20 million in the
United States, HPV is the most common STD, according to the Centers for
Disease Control and Prevention (CDC) and the World Health Organization
(WHO). More than 40 HPV types can infect the genital areas of men and
women, including the skin of the penis, vulva, and anus, and the linings of
the vagina, cervix, and rectum.
Most people who become infected with HPV do not know they have it.
Usually, the body’s immune system gets rid of the HPV infection naturally
within two years. This is true of both high risk (HR) and low risk (LR)
types. By age 50, at least 4 out of every 5 women will have been infected
with HPV at one point in their lives. HPV is also very common in men, and
often has no symptoms.
HPV types are often referred to as LR wart causing or HR cancer causing
[5], based on whether they put a person at risk for cancer. The types of
HPV that can cause genital warts (GW) are not the same as the types
8
Figure 1.1: Graphic illustrating the genomic organization of a typical mucosal
high-risk HPV. The genome contains early (E) and late regions (L), which
relate to the positions of the genes within the genome and their timing of
expression during the viral life cycle. The early (E) region carries a number
of genes which function at the level of viral replication and transcription,
i.e., E2, E1, E6, and E7. E2 encodes a protein which has an auxiliary role in
viral replication and also functions at the level of transcriptional regulation
of the viral early genes. The E6 and E7 genes encode the major transforming
proteins of the oncogenic HPVs. The late region (L) encodes viral structural
proteins, with L1 being the major capsid protein and L2 being the minor
capsid protein, adopted from [2].
that can cause cancer. Persistent HPV infections with genotypes 16 and
18 are responsible for about 70% of all cervical cancer, with 40-85% of other
anogenital cancers: anal, penile, vaginal, and vulvar cancer, and also 16-28%
of the head and neck cancers [3]. HPV is a cause of other non malignant
diseases, to mention genotypes 6 and 11 cause about 90% of anogenital warts
[6].
It is estimated that about 1 million new cases of GW are reported each
year and the cost of treatment is increased by the tendency of these warts to
recur after initial clearance. The cost of the treatment of GW was estimated
to exceed $ 3.8 billion in the U.S. in 1997 [8]. In Spain there were 35,000
cases in women in 2007 with an overall annual cost of e47 millions [9].
9
Figure 1.2: Harald zur Hausen (born 11 March 1936) is a German virologist
and professor emeritus. He has done research on cancer of the cervix, where
he discovered the role of papilloma viruses, for which he received the Nobel
Prize in Physiology or Medicine 2008. Adopted from [4]
1.2 Vaccines and mathematical models
Since the release of the first vaccines in 2006, nowadays there are three avail-
able: a quadrivalent (including HPV genotypes 16, 18, 6 and 11) and a
bivalent vaccine (including genotypes 16 and 18) and a nonavalent (includ-
ing genotypes 6, 11, 16, 18, 31, 33, 45, 52 and 58). All vaccines are efficacious
to prevent against precancerous lesions in the cervix, vulva or vagina; in ad-
dition, the quadrivalent and nonavalent prevent precancerous anal lesions,
anal cancer and anogenital warts.
According to the Advisory Committee on Immunization Practices (ACIP)
from the Centers for Disease Control (CDC) and Prevention, new recommen-
dations are given for use of a 2-dose schedule for girls and boys who initiate
the vaccination series at ages 9 through 14 years. Three doses remain rec-
ommended for persons who initiate the vaccination series at ages 15 through
26 years and for immunocompromised persons.
The HPV vaccine induces a herd immunity effect in genital warts when
a large number of the population is vaccinated. This aspect should be
taken into account when devising new vaccine strategies, like vaccination at
older ages or male vaccination. Numerous cost-effectiveness studies of HPV-
vaccination have been published in other countries. However, few studies
include the prophylactic effect of all HPV-associated diseases, or the impact
10
Figure 1.3: Age-adjusted prevalence of cervical human papillomavirus DNA
in sexually active women aged 15-69 years. Data are from IARC Prevalence
Surveys, 1990-2012, adopted from [7].
of vaccinating men.
In Spain HPV vaccine is given to adolescent girls as part of the national
immunization programme, and is recommended at different age groups in
different Autonomous Communities. In the Region of Valencia, Spain, the
vaccine is administered to <15 girls. Similar vaccination strategies of this
kind were modeled by Elbasha et al. [10,11] by means of a compartmental
model with 17 age groups for each gender. This model focuses mainly on the
development of cervical intraepithelial neoplasia (CIN) and its progression
from CIN1 to CIN3. According to these authors vaccination must be imple-
mented for adolescent girls aged between 12 and 14 years. Elbasha et al. also
found some evidences that the vaccination of boys could also be cost-effective
[10]. By vaccinating girls alone a 83% reduction in the incidence of GW is
expected but this reduction is increased to 97% if boys are also vaccinated.
It is important to develop mathematical models with good predictive
capacities. Some models have shown that the female vaccination program
has some herd immunity and the impact of implementing the vaccination in
males may not be cost effective. Besides, there is no economic analysis of the
11
nonavalent program in Spain, and it is important from the decision making
perspective.
Random network mathematical models may simulate the interactions and
propagation of all these viruses through sexual contacts among a population
of more than half million people (including heterosexual and homosexual
populations).
Before proceeding with the definition of our model, it is convenient to give
a brief and general perspective on the emergent field of network research for
the readers not familiar with these techniques and their application in epi-
demiology. A network is, basically, a model which derives from the abstract
mathematical concept of a graph composed by a set of points (the so-called
nodes) connected among them by some lines or edges (known as “links” in
the case of networks).
Figure 1.4: Example Network of 1,000 nodes, MSM in yellow, women in red
and men in blue.
There are several types of networks of interest for the applied sciences.
If we classify them according to the degree of a node, i.e., to the number
of links for a given node (or, more properly, to the distribution of these
number of links), we have two main categories in the literature: (i) random
networks, in which the links or edges occur with a fixed probability and the
statistical distribution of this number of links follows a Poisson’s law; (ii)
scale-free networks whose distribution of degrees follows a power-law with
12
an algebraic tail of the form P(k)'1/kγwith 2 < γ < 3. This means
that the nodes with very large degrees are more likely to appear in scale-free
networks than those in random networks. Random networks have been used
in epidemiology [12] and also as an elementary model of the brain [13]. On the
other hand, scale-free networks have been successfully applied to the Internet
and biological networks in which some nodes with a very large number of links
are determinant in the control of the dynamics (see [14]) and (iii) small-world
networks referring to the average length of a path connecting two typical
nodes in the network. Small-world networks are an important paradigm in
the science of networks. It was found that some sparse networks may present
the small-world phenomenon, i.e., there are short paths connecting every
pair of nodes through the links with other nodes. A mechanism to generate
these networks was discovered by Watts and Strogatz [15]. Many networks
in sociology exhibit this small-world property [16,17,18].
STD are more likely to produce large-scale infections than other transmis-
sible diseases, such as respiratory transmitted diseases, because the efficacy
of sexual contacts for the infection is large and the infectious agent has long
latency periods as in the case of HIV or HPV. Moreover, neither the carrier
nor his/her partner are aware of their exposure. For example, it has been
estimated that around 4050% of contacts are capable of transmitting HPV
[19].
In order to predict the evolution of these diseases we need a reliable model
of the underlying social network in which the infection builds up. Individuals
who change partners or have several partners simultaneously, are the hubs
favoring the spread of STD. The distribution of degrees of the nodes in the
network and the average chemical path from an infected individual to a
susceptible one, are important parameters controlling the final extension of a
new STD in a population and the speed at which it spreads. However, most
models are based on some assumptions, which could not be valid for certain
populations. Some studies claimed that the web of human sexual contacts is
a scale-free network characterized by a power-law distribution for the number
of individuals with a certain degree of connectivity, k:P(k)1/kαwith a
value of αin the range 2 < α < 3, and slightly smaller for males than for
females [17]. Although P(k) provides some valuable information about the
network, it is not a sufficient prescription on how to build it for a given
population size. Moreover, a power-law distribution of contacts could not be
representative of some populations, or could vary from country to country.
For example, it has been found that a densely connected core appears without
the need of a high connectivity degree in the Jefferson High School’s network
[18].
Even with the high prevalence of STDs there are few studies devoted to
13
ascertain the structure of sexual networks and its role in disease transmission.
Most studies are restricted to small communities such as the Jefferson High
Schools project [18] or that of Likoma Island [20].
Some field studies have ascertained the structure of moderate size real
networks of sexual contacts. In 2004, Bearman et al. published the results
for a set of 800 adolescents in a mid sized town of the United States [18].
They showed that the structure of this network is a big cluster with a ring
and extended filaments which contained most of the adolescents implying
that, potentially, the infection of an individual could spread to the whole
of the population, given sufficient time and infectivity. A similar study was
performed in 2007 at the Likoma Island in Malawi with the idea of predict-
ing and explaining the expansion of HIV in sub-Saharan populations [20].
That study disclosed that the sexual network contained many cycles, in con-
trast with the single cycle at Jefferson High School. For that reason, it was
speculated that superimposed cycles could be the explanation of the high
prevalence of HIV infection in small populations of Africa.
Some recent studies reveal that the evolution of partnerships is also an
important factor in the transmission of STDs. In particular, they pointed out
that the following items should be considered: (i) the cumulative distribution
of the lifetime number of partners, (ii) the distribution of partnership dura-
tion, (iii) the distribution of gap lengths between partnerships, and (iv) the
number of recent partners. A method for building up networks considering
these items has been developed by Schmid and Kretzschmar [21]. However,
this information is not available in most surveys, and we therefore face the
problem of developing reasonable models for STDs in many countries where
information about sexual behaviour is scarce. For example, in the case of
Spain, there is only available data about the number of sexual contacts in
a lifetime from surveys. This is sufficient for building a sexual network for
the transmission of HPV or other diseases with lifetime consequences and
progression. In these cases, the important fact is whether the individual has
had a contact with risk of infection. The remaining aspects of the network
such as the duration of partnership and the time intervals among them can
be incorporated effectively into a probability of transmission parameter.
1.3 Our proposal
In recent studies [1,22] a decrease on the number of infected persons and
the number of persons with GW is already reported for Australia after two
years of administering vaccinations to young girls. It showed both direct
and indirect prevention in males. These results were more impressive than
14
the predictions of the continuous models. New vaccination schedules, spe-
cially vaccination in boys, should take into account the herd immunity effect
vaccination in girls (in mainly heterosexual societies).
A Bayesian model for HPV vaccination was then proposed by Bogaards
et al. [23] and focused on the herd immunity effect of the female vaccination
on the male population in a static picture. A dynamic understanding for the
short and long term effects of vaccination policies is, however, still necessary
and even more so with HPV vaccines because their benefit to the whole
population is to be observed in the time span of several decades.
In this dissertation we propose a model based upon a network instead
of traditional continuous models. We show how to build a network model
for sexual contacts from the usual statistical data in surveys concerning the
number of partners in a lifetime. We consider both heterosexual men and men
who have sex with men (MSM) populations and the connections among them.
We perform simulations over this network substrate on the HPV infections
by different genogroups including both LR and HR infections. We will be
able to determine with higher accuracy the effect of vaccination in a short
and large periods of time, this is, the herd immunity effect. In particular, we
show that for the case of Australia the strategy of a vaccination for 12 13
year-old girls plus catch-up lead to a considerable reduction in the number of
cases of infection by HPV 6 and/or 11 (which are the main cause of GW). For
women in the 14 26 age-group we obtain a decrease of 59% after 3.64.6
years and 39% in men after 3 3.75 years. These results agree with the
conclusions of the study by Ali et al. [1].
In the following chapters we will explain in detail:
the building of the network model for sexual contacts. This is a technical-
computational chapter;
how the calibration of the model is carried out. This is a technical-
computational chapter;
model validation simulating the Australian scenario and obtaining sim-
ilar results;
the study of the decline of warts with the current vaccination campaign
in Spain: vaccination of girls with a coverage of 70%;
the study of the decline of oncogenic HPV with the current vaccination
campaign in Spain;
the study of what would happen if the effect of the vaccine disappear
suddenly after 20 years;
15
the study to determine if the tourism in Spain has a significant effect
on the HPV infection;
the study of the decline of oncogenic HPV if we vaccinate boys and
girls with a high coverage;
the study of how long the decline is recovered after a drop in the cov-
erage;
the study of how the decline of HPV is affected when the average
number of LSPs increases significantly;
the study of how the decline of HPV is affected when the number of
MSMs increases significantly.
16
Chapter 2
Building LSP networks and
describing the transmission
dynamics of HPV on these
networks
2.1 Origin of the data
Building a social network requires demographic data, for this model we have
used data from the region of Valencia (Spain) that was collected from the
Valencian Institute of Statistics (2013) [24], from this set of data we are
interested in the distribution of males and females along with their age. The
second set of data for this model has to do with sexual habits, this is the
LSP for an individual, which was obtained from the Health and Sexual Habits
Survey of 2003 [25], and summarized in Tables 2.1 and 2.2.
MEN
Age 0 LSP 1 LSP 2 LSP 3 4 LSP 5 9 LSP 10+ LSP
14–29 0.107 0.207 0.131 0.225 0.168 0.162
30–39 0.027 0.225 0.128 0.21 0.17 0.24
40–65 0.019 0.268 0.14 0.193 0.163 0.217
Table 2.1: Proportion of men per number of lifetime sexual partners (LSP)
per age group. Note that the sum of the rows are 1.
17
WOMEN
Age 0 LSP 1 LSP 2 LSP 3 4 LSP 5 9 LSP 10+ LSP
14–29 0.138 0.43 0.186 0.158 0.056 0.032
30–39 0.029 0.501 0.168 0.177 0.077 0.048
40–65 0.017 0.652 0.138 0.118 0.039 0.036
Table 2.2: Proportion of women per number of lifetime sexual partners (LSP)
per age group. Note that the sum of the rows are 1.
Some features of the distribution of contacts were: (i) the percentage of
males and females with no partners is very similar in each age-group; (ii) the
proportion of women with a single partner is, approximately, two times larger
than men with only one partner; and (iii) the percentage of men with two or
more partners is always larger than that of women except for women in the
age-groups 14–29, and 30–39 in the case of two partners. The asymmetry
in the behaviour of males and females should be taken into account in the
construction of the network.
2.2 Network model
In this dissertation, we use the random network model as a basis to simulate
the network of sexual contacts among individuals, but, in this model, the
average number of connections depend upon the age-group as deduced from
Tables 2.1 and 2.2. A basic property of the network we are going to discuss is
that the total number of LSP for the male population (M) must coincide with
the total number of LSP for the female population (F). This is so because
(in a purely heterosexual network) every link starting on a male must end in
a female and vice versa. In mathematical terms:
M
X
i=1
LSPi=
F
X
j=1
LSPj.(2.1)
About the estimation of sexual partners, there are some approaches to
the number of LSP in males and females [26,27] that are difficult to match.
Generally speaking, males tend to overestimate the number of their sexual
partners and females tend to underestimate it. Therefore, we considered the
average LSP male value, hkim, and calibrated the network so that results were
consistent with data of Tables 2.1 and 2.2, and estimated that the number
of LSP in males in Spain was at least 4.5. Networks with 100,000, 250,000,
18
500,000 and 750,000 have been used during the present study. It required a
substantial computational power.
2.3 Semi-Random Construction
From Table 2.1 (proportion of male LSP aged 14–29), we have the follow-
ing list: (0.107,0.314,0.445,0.67,0.838,1) for the accumulated proportion of
males less than or equal to a given LSP number. Now, we randomly generate
a number rbetween 0 and 1 and assign the number of contacts to every male
node in the 14 29 age group, in the network as follows:
r0.107 say that the corresponding male does not have an LSP,
0.107 < r 0.314 say that the corresponding male has one LSP,
0.314 < r 0.445 say that the corresponding male has two LSPs,
0.445 < r 0.67 say that the corresponding male has three or four LSPs
uniformly distributed.
0.67 < r <0.838 say that the corresponding male has five to nine LSPs
uniformly distributed.
0.838 < r 1 say that the corresponding male has 10 or more LSPs.
Every node in the network is labeled by its gender and age randomly as-
signed according to the population histogram. The assignment of the number
of bonds, as another label of the node, is not so straightforward since we must
guarantee that the condition in Equation (2.1) is verified. In order to ful-
fill this condition, we take advantage of the uncertainty of statistics reports
concerning individuals with 10 or more LSPs.
Starting with the males, we assign the number of LSPs up to nine partners
and, for 10 or more partners proceed as follows: let imax be the number of
males with nine or less partners. The unassigned males should be Mimax
and the number of bonds that should be distributed among them is Mhkim
Pimax
i=1 LSPi. By Euclidean division, this quantity can be expressed as
(Mimax)nm+rm, where nm10. In our procedure, we assign a random
number of bonds, uniformly distributed, in the interval [10,2nm10] to
every male with 10 or more LSPs, i.e., to the Mimax unassigned males.
Now, we denote as pmthe total number of bonds of the male population.
We must take into account, as expressed in Equation (2.1), that the total
number of bonds of the female population should be the same. To impose
19
that condition, we proceed as follows: (i) assign the number of bonds to the
females with nine or less partners following the statistical data in Tables 2.1
and 2.2; (ii) the sum of all female LSPs in this group of jmax members will
be denoted by sf. Then, nf=Fjmax is the number of females with
10 or more LSPs; (iii) the number of bonds starting in the males and still
unassigned to a female is pmsf=qfnf+rf, where 0 rf< nfand
nf10; (iv) we assign qf+ 1 bonds to rffemales still unassigned and qf
bonds to the rest of nfrffemales. The steps of this algorithm are also
enumerated in the flow diagram in Figure 2.1.
Notice that, for men and women with more than 10 LSPs, we assign their
LSP in the most equitable way, assuming that all of them have, more or less,
the same number of LSPs. Thus, we have a lot of hubs with a low number
of contacts instead of a few hubs with a lot of contacts.
From the point of view of STD transmission, the latter situation leads to a
faster transmission if the hub is infected, and, also, if the hub is vaccinated,
the transmission is cut faster. Therefore, due to the lack of data about
people with 10 or more LSPs, we make the decision of being conservative in
the transmission of the disease and in the effect of the vaccination campaigns.
Notice that this procedure implies that the condition in Equation (2.1) is
verified. After this procedure, we have obtained the following lists:
AgeM ale [i] is the age of the i-th male, i= 1, . . . , M,
AgeF emale [i] is the age of the i-th female, i= 1, . . . , F ,
kM ale [i] is the number of LSP for the i-th male, i= 1, . . . , M,
kF emale [i] is the number of LSP for the i-th female, i= 1, . . . , F .
These lists will be used to perform the connections of males and females
and build the network. Note that, in Tables 2.1 and 2.2, there are more
females than males with few LSPs (comparing male and female percentages).
It implies that there will be few women with a very large number of LSPs.
This fact suggests us to start the assignment procedure with women with
the largest LSPs. Otherwise, it would be possible that, when we have to
assign LSPs of men to a female with a large number of LSPs, there will not
be enough men with free sexual partners to be assigned and, for this female,
it would be impossible to satisfy the condition that the degree of each node
was the number of its LSP.
20
Assign LSP values to
males with 9 or less
LSP
Determine imax :
Number of males with 9
or less LSP
Remaining males: M-
imax
Remaining number of
bonds: N= M <k>m +



Perform euclidean
division:
N=(M-imax)nm+rm
Assign a random LSP in
the interval [10,2 nm-10]
to the remaining males
Assign LSP values to
the jmax females with 9
or less LSP
sf=



Number of females with
10 or more LSP nf=F-
jmax
Total bonds for the
males: pm=


Number of bonds
unassigned to females:
pm-sf=qf nf+rf
Assign qf+1 bonds to rf
females and qf bonds to
the remaining nf-rf
Figure 2.1: Flow diagram for the algorithm corresponding to the assignment
of a number of LSPs to every male and female in the network.
21
The assignment of partners was carried out by considering a principle of
psychological similarity [28] or assortativity. Hence, we are going to define
a weight function assuming that: women with few LSPs usually match men
with few LSPs; people with four or more LSPs use to join with people with
four or more LSPs; and couples where one of them has a large number of
LSPs and the other few LSPs will be uncommon. Then, for the woman iand
the man j, we define the following weight function:
π(i, j) =
|kF emale[i]kMale[j]|kF emale[i], kM ale[j]4
0kF emale[i], kMale[j]>4
100 otherwise
,
+|AgeF emale[i]AgeMale[j]1.8|.
(2.2)
The combined weight function, which takes into account the age differ-
ence of the partners, |AgeF emal e[i]AgeM ale[j]1.8|is defined in this
way because some studies show that the average age difference among the
members of a couple in Spain is 1.8 years [29].
The estimated percentage of men who have sex with men (MSM) is 3.88%
[25]. The situation for the MSM population is different of the one shown in
Tables 2.1 and 2.2, because the average number of sexual partners is esti-
mated in 39 regardless of age, but this number increases with age with a
peak of 59 in the 40-49 age-group [30]. A difficulty arises because we have
little information about the number of sexual contacts of women who have
sex with women (WSW) subpopulation. In a personal communication by
Dr. Mireia D´ıaz from the Catalan Institute of Oncology (IDIBELL) we were
informed that HPV hardly spreads among WSW, and almost all MSM, some-
times in their lives, had a woman partner. Consequently, we have simulated
these connections by assigning a contact to every man in the MSM subpop-
ulation with woman with 5 partners or more. This is done according to the
assortativity principle, that is, people use to join with people with similar
habits.
The assignment of links is then performed by the Greedy Randomized
Adaptive Search Procedure (GRASP) algorithm [31,32].
22
Chapter 3
Calibrating the Sexual
Contacts Network
The calibration of this kind of random computational models is an open
problem and several issues about fitting computational models to data with
uncertainty have to be addressed. For instance,
the fact that, for the same set of parameters, different realizations
usually return different outputs, and consequently, one realization may
fulfill the fitting requirements and another realization does not,
the determination of an appropriate measure of goodness-of-fit,
to find model parameters agreeing the values appearing in the medical
literature in a reliable and reproducible way,
adaptation of the optimization algorithms to these above issues and
the available resources,
the determination of the best parallel implementation of the optimiza-
tion algorithms, in terms of quality of the solutions and computational
efficiency of the program.
In the remainder of this section, we are going to describe thoroughly the
procedure we propose to approach the above issues.
3.1 Description of the model
As we explained in Chapter 2, we described how to build lifetime sexual
partners (LSP) networks. These networks are built with the main aim to
23
reproduce the demography in Spain [24], and data about the LSP, for men
and women, and for age groups 18 29, 30 39 and 40 65, presented in a
survey about sexual habits in Spain [25] and collected in Tables 2.1 and 2.2.
Then, over the LSP network, we describe the dynamics of the HPV [33,
34]. We divide the nodes into the age groups 14 17, 18 29, 30 39,
4065. For the implementation of the simulation of the transmission of HPV
among the individuals in the network, a standard epidemiological model with
susceptible and infected states and three types of infections (HR, LR and co-
infection) was carried out. The epidemiological model is defined by a set of
parameters, hence, it is necessary to include 11 model parameters that will
address the transmission dynamics of HPV on the LSP networks:
1. Average number of men LSP, needed for network LSP building.
2. We need some probabilities to determine if a sexual partner is going to
produce a contagion to another partner in a given time stage. These
parameters are different for each age group: 14–17, 18–29, 30–39 and
40–65. Notice that this means that the probability of contagion de-
pends upon the age group of the members of the relationship. More-
over, the probability of connection of these members in the network is
also age-dependent as proposed in Equation (2.2). The values of these
probabilities are determined in the process of the model calibration.
3. Average time an individual infected by a high (low) risk HPV clears
the infection and recovers (2 parameters).
4. Probability that a woman (man) infected of high (low) risk HPV trans-
mits it to his/her partner in a sexual intercourse (4 parameters).
Most of the above parameters have been studied in the literature and
there are some estimations we have to take into account:
The average number of men LSP: it is around 8 [30]. For this parameter,
our search will be in the interval [7,10].
The time for clearing the infections due to HPV HR, for both men and
women: in [10], the authors say that the mean duration of HPV 16/18
infection is 1.2 years; in [35], the mean duration of HPV 16 is 12.19
months (7.16 18.17); in [36] the duration clearance varies, in average,
between 6.5 months to 11.8 months. Thus, we considered the interval
[0.8,1.2] years.
24
The time for clearing the infections due to HPV LR, for both men and
women: in [10], the mean duration of HPV 6/11 infection is 0.7 years;
in [35], the mean duration of HPV infection is 7.52 months (6.808.61)
for any HPV; in [36] the duration clearance varies, in average, between
6.2 months to 11.7 months. Thus, we considered the interval [0.5,1]
years.
In [10], the authors estimated the probability of HPV infection trans-
mission per partnership and by type and, as in [9], this probability is
higher for transmission from males to females (0.8) than that for trans-
mission from females to males (0.7). Given that these data come from
estimations (not surveys) and after some runs of our model, we are go-
ing to be more flexible and consider the probability interval [0.2,0.6] for
LR transmission and [0.5,1.0] for HR transmission, given that, women
transmit less than men.
Simulations are executed by generating a network and carrying out a
large number of epidemic evolution time-steps starting with a number of
individuals infected by both HPV types as given by the CLEOPATRE study
[9]. After the warm-up period, we obtain a stable situation and we can
proceed with the calibration by comparing the model predictions with real
data and deducing the most probable values of the set of parameters.
We have used a calibration procedure using the Particle Swarm Opti-
mization (PSO) algorithm [37]. The prevalence data for each age group is
listed in Table 3.1.
Women HR-Infected LR-Infected
18–29 y.o. 24.10%,[22.1%,26.16%] 6.36%,[5.24%,7.54%]
30–39 y.o. 11.01%,[7.55%,14.47%] 1.26%,[0.31%,2.52%]
40–64 y.o. 5.97%,[4.65%,7.37%] 2.37%,[1.49%,3.25%]
18–64 y.o. 16.23%,[14.96%,17.53%] 4.41%,[3.71%,5.13%]
Table 3.1: Prevalence of HR- and LR-infected women per age groups from
the CLEOPATRE study [9]. Co-infections are included in both HR- and
LR-infected, mean and 95% confidence intervals.
Note that the network building and the transmission parameters involve
randomness and uncertainty due to the random processes used in the network
building and the transmission dynamics of the HPV. This fact is going to be
taken into account in the calibration and simulation.
25
3.2 Description of our resources
3.2.1 Computers
All the realizations are going to be executed on two computers with 64 cores
on 8 Xeon Sandy Bridge E5-4620 running at 2,2 Ghz, with 16 MB of cache
memory and 512 GB RAM memory1. The operating system is Ubuntu Server
16.04 LTS.
3.2.2 Distributed computing environment
We also have deployed a distributed computing environment called Sisifo.
Sisifo is a client-server based system designed to allow a problem to be solved
using distributed computation. Sisifo is able to assign tasks to a set of
personal computers (PCs), wait for the tasks to be completed and collect
the results for further analysis. Sisifo is made with simplicity as main aim,
giving as a result a system that requires almost no maintenance, needs very
little configuration time, and can be deployed in just a couple of hours.
The Sisifo Server keeps listening for request of the clients. The Server has
stored one or more executors, a set of problems to be solved in the Problem
files folder, and the solutions sent in the Result files folder.
The Sisifo Client is a program stored in one or several PCs that connects
to the server, and asks for a work packet. This work packet is composed of two
elements: a text file containing the model parameter values and the simulator
executable file. The Client, once the work packet is received, executes a
realization using the model parameters stored in the text file. When the
realization finishes, a solution file is generated, returned to the server and
dropped in the Results files folder. More details about how Sisifo works can
be found in [38].
In our case, the Sisifo clients are going to be located in each one of the
64 cores of the Sandy Bridge computers. The Sisifo server is located in a
regular PC with MS-Windows 7.
3.2.3 Random Particle Swarm Optimization (rPSO)
Using Python3 [39] and Mathematica [40], we have implemented an asyn-
chronous version of rPSO adapted to the Sisifo computing environment. To
do that, first, we recall the random PSO (rPSO) algorithm appearing in [37]
applied to the optimization of a function F.
1Both computers are not exactly the same. There are some minor hardware differences.
26
Step 1. Initialization.
Initialize Nparticles p1, . . . , pNchosen randomly in the parameter
space.
Initialize randomly their velocities v1, . . . , vN.
Evaluate the fitness of all the particles F(p1), . . . , F(pN).
Define the individual best fitness as pbest
i=pi,i= 1, . . . , N and
the global best fitness pbest
global as the pbest
iwhich fitness is optimum.
Step 2. Modify the particle velocities based on the previous individual best
and global best positions:
vi=ωvi+ψ1(pbest
ipi) + ψ2(pbest
global pi), i = 1, . . . N,
where ωis a random value in [1
4,3
4], ψ1is the exploitation rate and ψ2
is the exploration rate, both randomly generated in each iteration as a
number in the interval [0,1.5].
Step 3. Update the particle locations: pi=pi+vi, i = 1,...N.
Step 4. Evaluate the fitness of all the particles F(p1), . . . , F (pN).
Step 5. Update the individual best fitness pbest
i,i= 1, . . . , N and the global
best fitness pbest
global . Go to Step 2.
The above algorithm can be adapted to Sisifo computing environment
if, using the Sisifo Server, the computation of the fitness of the particles is
distributed among the Sisifo Clients.
However, in a typical PSO procedure, including rPSO, the set of particles
is updated once the fitnesses of all the particles have been calculated. This
means that, until all the fitnesses have not been evaluated and Step 4 is not
completely finished, the particles cannot be updated and new evaluations
cannot be performed. Then, in every iteration of rPSO, scenarios where
some Sisifo clients have finished their evaluations and are idle while other
Sisifo clients are still performing their evaluations will be common. In these
scenarios, we have an under-use of the Sisifo system.
In order to avoid the under-use system drawback, we propose the im-
plementation of an asynchronous version of rPSO in such a way that when
the fitness of a particle has been evaluated (Step 4), this particle is updated
(Steps 5, 2 and 3) without waiting for the evaluation of the remainder par-
ticles, considering the current existing global best and its individual best
27
particles. This way, we modify rPSO algorithm parallelizing Steps 2, 3, 4, 5
and sharing the updates of the global best particle.
We show in Figure 3.1 how we set the asynchronous rPSO in the Sisifo
environment. As we can see, the rPSO procedure generate the problems
and write them into the Problem files folder to be processed by the Sisifo
Server. Once the evaluation has been carried out and the solution file is in
the Result files folder, rPSO reads the content of the solution file, analyses it
and calculates the fitness. With this information, rPSO is able (Steps 2 and
3) to generate a new problem file.
Sisifo clients
Sisifo
Server
Problem
files
Result
files
rPSO
algorithm
Figure 3.1: Introduction of the asynchronous rPSO algorithm in the Sisifo
environment. rPSO manages the generation of new problems and put them
in the Problem files folder and the reading and processing of the solution files
located in the Result files folder.
When the procedure starts, with the initialization of the particles (Step
1) we create their corresponding problem files in the Problem files folder.
The Sisifo Server detects new problem files and distribute them among idle
Sisifo Clients. These clients carry out the realizations. When a Sisifo client
ends its task, a results file is generated and sent to the Sisifo Server that
drops it on the Result files folder. Every time a new results file appears in the
Results files folder, the asynchronous rPSO, in Step 4, reads the data from
the results file and calculates the fitness. Then, updates the velocity taking
into account the current existing global best and its individual best particles
(Step 2), updates the particle (Step 3) and with the new model parameters
creates a new problem file in the Problem files folder. And so on.
28
3.2.4 Fitness function and some features included in
the rPSO
There are some features included in our version of the asynchronous rPSO
algorithm that we have to describe.
1. In the Step 3, we include the possibility to discard the new particle
and generate a new one randomly with 10% probability. If it is not dis-
carded, with 10% probability we apply to the new particle a mutation.
2. The fitness evaluation is made as follows:
we start the model and after a warm-up time of 400 simulated
months, we get a stabilization of the model output;
we take the model output from month 401 to 500 for the subpop-
ulations in the data of Table 3.1, i.e., percentage of women HR-
and LR-infected per group ages 18 29 and 18 64;
then, for each subpopulation, we calculate the maximum and the
minimum of these portions of the model output and we see if these
maximum and minimum are inside the corresponding data 95%
confidence interval;
if this happens, the fitness is zero;
otherwise, the fitness is the sum of the distances from these max-
ima and minima to the corresponding 95% confidence interval of
the data.
3. The definition of the fitness function may provide the same fitness
values for different particles. Therefore, we are going to store all the
particles with the same best fitness and, in Step 2, we chose pbest
global
randomly among the stored particles.
4. As we mentioned before, physicians around the world have published
estimations of most of the model parameters [30,10,35,36]. It is
clear that we have to respect their estimation in our fitting procedure
avoiding that some model parameters overpass the estimation intervals.
Furthermore, finding model parameters in the range of the estimations,
gives credibility to our model. Therefore, when a model parameter
is less than 1% closer to an extreme of the interval provided by the
physician estimations, we discard this value and it is substituted by a
random value inside the interval.
29
It is worth to note that the above points 1, 3 and 4 will allow us to
explore extensively the whole space of parameters avoiding the accumulation
of particles close to the borders.
Remark 3.2.1 At this point, we want to remark that to evaluate the fitness
function, we do not need the model output for all the subpopulations. Only
is necessary the output corresponding to HR- and LR-infected women in the
group ages 18 29 and 18 64. Then, when a realization is carried out, we
retrieve and process from the results file the data corresponding to the above
subpopulations from the month 401 to 500, write them in a row and this row
is stored as the result of the realization to be used later.
Also, in order to compare properly the model output row with the data, we
build the data vector mean, percentile 2.5and percentile 97.5, repeating 100
times the 4values of each we have in the Table 3.1 and write them in a row.
3.3 Selecting an optimal number of particles
to calibrate the HPV large network com-
putational model with rPSO
As we have mentioned above, on previous implementations of the parallel
rPSO, we realized that we can not affirm that the higher the number of
particles, the better the quality of the solution. Moreover, we also detected
that, due to our asynchronous parallel implementation, sometimes the use of
more processors does not mean lower execution times. We would need more
experiments to detect the correct cause of the delays. However, in this case
is more practical to investigate what is the optimal combination of particles
to achieve the best quality with the lowest number of processors.
To perform this test, we are going to build HPV network models with
50,000 nodes. We execute 5 repetitions of the same problem with different
number of particles. Times are shown in seconds. Thus, the question is,
what is the quality of the solutions for different number of particles? Table 3.2
shows relevant information regarding the quality of the solutions for different
number of particles and 5 executions of each configuration. Each execution
carried out 1200 particle evaluations. We measure the quality of the solutions
on Table 3.2 by counting the number of fitness values equal to zero in the
5 executions (Total # 0s) and on average (Avg #0s). In addition, we also
recorded the iteration at which the first zero appear on average (Avg First),
from the last population of particles Fitness of the best, worst and average,
averaged over 5 executions (Best at end,Worst at end and Avg at end).
30
And, regarding executions time we show on Table 3.2 information about the
average total execution time for 1200 evaluations (Total Time), the worst
execution time, for one particle (Worst t) and the best execution time for
one particle (Best t). Red color indicates the worst configuration, bold letters
indicate the best and blue color the second best configuration.
# Total Avg Avg Best Worst Avg Total
particles #0s #0s First at end at end at end Time Worst tBest t
25 18 3.60 531.00 0.015486 0.429090 0.122573 8422.00 148.60 107.00
30 28 5.60 594.00 0.003785 0.445587 0.131193 6908.00 186.60 112.00
35 8 1.60 834.80 0.010211 0.576688 0.149215 7105.00 250.60 119.60
40 16 3.20 692.80 0.003332 0.524417 0.132628 8808.60 389.40 144.60
45 14 2.80 790.40 0.006458 0.639775 0.149670 10004.00 502.00 177.20
50 11 2.20 796.00 0.005168 0.658020 0.139089 11124.40 648.40 220.00
55 2 0.40 1171.20 0.004330 0.518894 0.137088 11905.80 775.00 207.40
60 12 2.40 765.00 0.002153 0.546259 0.132464 13787.80 857.00 245.40
64 5 1.00 985.40 0.002763 0.551228 0.140977 18770.00 1041.20 379.20
Table 3.2: Analysis of the quality and execution time of different rPSO con-
figurations. Results of 1200 evaluations for different number of particles on
the rPSO process (# Part.).
As we can see, 25 and 30 particles are the preferred configurations, since
we obtained the higher number of zeros in total and on average and the
lower executions times. Since the results of 25 particles show a very good
execution with 7 zeros and also a very good execution time, we can conclude
that the configuration with 30 particles should be selected bearing in mind
both quality and execution time. Results on total number of 0s and total
time were statically significant with pvalue of 0.1 after an ANOVA analysis.
3.4 Improving the exploration of the rPSO
As we explained on Section 3.2.4, we established a procedure with respect
to the intervals for the parameter values proposed by physicians. Some al-
gorithms such as differential evolution [41] allows to overpass these limits in
the search of a good combination of parameters. However, this is not a good
idea when implementing our parallel rPSO. First, remember that our parallel
version is asynchronous and the updating of the particles is made once every
particle is evaluated. Allowing particles with parameters close to the limits
trespass them, could lead to a chaotic search. Second, parameters out the
defined bounds have not a medical meaning. Therefore, when a model pa-
rameter is less than 1% closer to an extreme of the interval provided by the
estimations, we discard this value and it is substituted by a random value
inside the interval. This allows also to made a deeper and more efficient
exploration of the search space.
Figure 3.2 shows the exploration performed in some of the parameters
of the model (duration and contagion parameters). Figures represent the
31
values of the parameters during the 20 executions of rPSO algorithm with
500 realizations each one. X axis represents the realization number and Y
axis the value of each parameter. We can see that we find points on most of
the search space.
Time of clearance of Time of clearance of
HPV high risk HPV low risk
Women transmission probability Men transmission probability
of HPV low risk of HPV high risk
Figure 3.2: Upper figures: Time of clearance of HPV high risk and low risk,
both for men and women. Lower figures: on the left, probability to transmit
low risk HPV if the infected is a woman; on the right, probability to transmit
high risk HPV if the infected is a man. Red dashed lines correspond to the
bounds of each parameter. We can see that most of the search space is
explored in all the cases.
3.5 Calibrating the model
Now, we are going to perform the calibration. We use the Sisifo environment
with the modified rPSO and 30 particles. The HPV network models will have
32
100,000 nodes. In order to guarantee the reproducibility of the realizations,
we are going to include in the problem files the seeds for the generation of
the random numbers during the calibration process.
We assume that, initially, data of prevalence from Table 3.1 are not only
for women but also for men. Then, we label women and men as infected
randomly following these prevalence data. Also, we start executing the re-
alization and the first 400 months are used as a warm-up period to stabilize
the distribution of infected men and women. Thus, the goal is to calibrate
the model parameters in such a way that the model output related to women
HR and LR prevalence minimize the fitting function defined in Section 3.2.4.
We have performed 20 different calibrations using rPSO, each one with
around 500 realizations and 30 particles. A total of 10,100 realizations of
the model were performed with an equivalent sequential total computation
time of 161 days. Computer time, however, is highly reduced as we can see
in Figure 3.3 due to the benefits provided by Sisifo distributed architecture.
Figure 3.3: Histogram of the computation time of each one of the 10,100
realizations of the model using networks of 100,000 nodes. The average time
is 1374.5 seconds, around 23 minutes.
3.6 Selecting the 30 realizations that best cap-
ture the data uncertainty
Our goal, now, is to find a procedure to select 30 among the 10,100 real-
izations of the model in such a way that the means and the 95% confidence
intervals of these 30 realizations be as much close as possible of the corre-
sponding means and the 95% confidence intervals of the data in Table 3.1.
33
We decided to select 30 because, as we saw in Table 3.2, the total computa-
tion time is the best for 30 particles executing in parallel in the Sandy Bridge
computers.
Nevertheless, it would be interesting to reduce the number of eligible
realizations to much less than 10,100. In Figure 3.4 we can see the 100
realizations with error less than 0.01, that is, the realizations that almost
lie inside the 95% confidence intervals of the data, represented by the blue
horizontal dashed lines.
Figure 3.4: Drawing of the 100 model realizations with error less than 0.01
from month 400 to 499.
If we select 30 among these 100, it is clear that some percentiles of the
model will be far from the corresponding percentiles of the data, for instance,
the lower parts of the left figures. Therefore, we need to consider realizations
with errors greater than 0.01 without forgetting the objective to reduce the
number of eligible realizations.
Thus, we check the number of possible realizations depending on their
error. Then, there are 2 realizations with error 0, 100 realizations with error
less than 0.01, 392 with error less than 0.025, 1282 with error less than 0.05
and 2607 with error less than 0.075. In Figure 3.5, we draw the 1282 model
realizations of with error less than 0.05. Note that the realizations cover
completely the 95% confidence intervals of the data.
In order to determine the best 30 realizations that capture the mean
and the 95% confidence intervals of the data in Table 3.1, we are going to
introduce an adapted version of the rPSO algorithm before. Let Ebe the
34
Figure 3.5: Drawing of the 1282 model realizations with error less than 0.05
from month 400 to 499. These realizations cover the 95% confidence intervals
of the data, represented by the blue horizontal dashed lines.
realizations considered with card(E) = Mits number. For instance, for E
the set of realizations with error less than 0.01, M= 100. Also, for Ethe set
of realizations with error less that 0.05, M= 1282. Then E(i) is the ith
realization of the set E. We define the following fitness function F:
INPUT: Set of 30 indexes I={i1, . . . , i30}, 1 ijM,j= 1,...,30.
Step 1. Select the realizations E(i1), . . . , E (i30) and calculate the mean, per-
centile 2.5 and percentile 97.5 of them.
Step 2. Calculate the Root Mean Square Error (RMSE) of the difference be-
tween the mean, percentile 2.5 and percentile 97.5 of the 30 realizations
and the data (see Remark 3.2.1), and sum them up.
Now, the adapted version of rPSO to select the best 30 realizations is
Step 1. Initialization.
We have a set Ewith Mrealizations.
Initialize Nindex subsets S1, . . . , SNwith 30 elements of the set
{1, . . . , M}(particles) chosen randomly without repetition.
Evaluate the fitness of all the particles F(S1), . . . , F(SN).
35
Define the individual best fitness as Sbest
i=Si,i= 1, . . . , N and
the global best fitness Sbest
global as the Sbest
iwhich fitness is the min-
imum.
Step 2. For i= 1, . . . , N, build the new set Pi=SiSbest
iSglobal , that is,
joining the current particle, its individual best and the global best. Af-
ter removing index repetitions, the new particle Siconsists of a random
selection without repetition of 30 indexes of Pi.
Step 3. Evaluate the fitness of all the new particles F(S1), . . . , F (SN).
Step 4. Update the individual best fitnesses Sbest
i,i= 1, . . . , N and the
global best fitness Sbest
global . Go to Step 2.
Here, we also consider 10% of randomness and 10% of mutation when
updating new particles. In this case, mutation consists of changing some
of the indexes in the current particle by other randomly chosen indexes,
avoiding repetitions.
The above algorithm, in the tests we have executed, last around 20 min-
utes for 1 million of evaluations of the fitness function in the Sandy Bridge
computer, returning accurate solutions.
We have performed executions for 30, 40, 50 and 60 particles, being E
the realization sets with errors less than 0.025, 0.05 and 0.075. The lowest
error has been 0.1166 for the following realizations
85,109,460,474,475,476,485,493,496,497,523,531,
542,543,563,600,635,637,650,687,715,729,730,
842,974,987,1058,1060,1080,1238,
(3.1)
among the 1282 of the set of realizations with error less than 0.05. In
Figure 3.6 we draw the selected realizations and in Figure 3.7 we can see
the graphical result of the calibration and how resemble the means and 95%
confidence intervals.
3.7 Parameters returned by the calibration
The mean and the 95% confidence interval of the model parameters corre-
sponding to the 30 selected realizations from (3.1) are given in the Table 3.3.
Note that the obtained model parameters are in accordance to the medical
parameters appearing in the literature and collected in Section 3.1.
The last four parameters in Table 3.3, as we defined in page 24, are the
global probabilities that determine if the existence of a LSP implies sexual
36
Figure 3.6: Drawing of the 30 selected model realizations from month 400
to 499. These realizations are the ones whose means and 95% confidence
intervals resemble the most the data in Table 3.1, represented by the blue
horizontal dashed lines.
Figure 3.7: Means and 95% confidence intervals of the 30 selected realizations
(in red) compared to the means and 95% confidence intervals of the data
(blue).
37
Model parameter Mean 95% confidence interval
Average LSP men 8.63 [7.15,9.86]
Average time clearing HR HPV 1.08 [0.88,1.19]
Average time clearing LR HPV 0.60 [0.52,0.82]
Probability a woman transmits LR 0.23 [0.21,0.29]
Probability a man transmits LR 0.28 [0.22,0.37]
Probability a woman transmits HR 0.81 [0.68,0.95]
Probability a man transmits HR 0.91 [0.74,0.97]
Frequency, 14-17 years old 0.1098 [0.0485,0.1542]
Frequency, 18-29 years old 0.0776 [0.0568,0.0981]
Frequency, 30-39 years old 0.0620 [0.0024,0.0935]
Frequency, 40-65 years old 0.0190 [0.0046,0.0553]
Table 3.3: Mean and the 95% confidence interval of the model parameters
corresponding to the 30 selected realizations from (3.1). The last four pa-
rameters measure the global frequency of the sexual intercourses per age
group.
intercourses in the current time step per age group 1417, 1829, 3039 and
40 65. As we mentioned, these parameters measure the global frequency
of the sexual intercourses per age group. We can see that, in average, the
frequency in the older group is much less than the younger ones. In general,
there is high uncertainty in these parameters.
In the following, we are going to use the sets of parameters corresponding
to the realizations (3.1) (and their corresponding seeds) to perform simula-
tions.
38
Chapter 4
Including new features into the
model: Dynamic LSP network
and vaccination campaigns
4.1 Introducing age and LSP dynamics into
the LSP model
The computational model presented so far is static, that is, once we have
assigned the LSPs, they do not change over the time and the nodes do not
increase their age. We modulate the possibility of a contagion introducing
global probabilities that determine if the existence of a LSP implies sexual
intercourses in a given time step per age group 14 17, 18 29, 30 39 and
40 65. Also, note that the available data about LSPs come from 2003 and
the data of prevalence appears in a study (CLEOPATRA [9]) conducted in
2007 2008. The sexual behavior has changed in the last 10 15 years and
we would like to include it in our model in some way. Nevertheless, newer
data about LSPs and prevalence in Spain are not available.
Thus, in order to introduce age and LSP assignation dynamics in a simple
way, we are going to move the available data over the time. Figure 4.1 shows
the constant age distribution that does not vary over the time.
Then, we evolve the ages of the nodes over the time. As long as the nodes
do not change the age group, there is nothing to do. However, how do we
recycle the nodes that turn 64 years old? And how do we transform the
nodes in the first age group 14 29 when they turn 30?
To answer the first question, we propose to
preserve its sex in order to maintain the proportion males/females,
39
14-29 30-39 40-65
Time
Figure 4.1: Static LSP network. The ages remain constant over the time.
The time only affects the HPV contagion/clearing dynamics.
assign 14 years old to the node, erase all its LSPs and transform it in
susceptible,
assign new LSPs according to the LSPs of the age group 14 29 fol-
lowing the assortativity property.
To answer the second question, we propose to
preserve its sex and the infectious state,
erase all its LSPs and assign new LSPs according to the LSPs of the
age group 30 39 following the assortativity property.
This way, as time goes on, we will have the LSP structure by ages as we
can see in the Figure 4.2.
14-29 30-64
Time
Figure 4.2: Dynamic LSP network. The ages evolve over the time. The age
group 40-65 and their sexual behavior disappear when people grow.
40
If the node is MSM (male who have sex with males), we perform the
above procedure taking into account the division of age groups given in [30],
14 19, 20 24, 25 29, 30 39, 40 49, 50 59 and 60 64+.
Taking into account that the global number of LSP in the age groups
40 64 is less that in the age group 30 39, as the time goes on, the global
number of LSP in the whole network will increase and the transmission of
HPV will also increase. However, the proposed approach may be considered
as very conservative because the sexual behavior change in the last 10
15 years seems to result in an increase of the sexual intercourses greater
than the number that we can approach with the proposed dynamic network.
Nevertheless, the lack of data do not allow us to quantify the mentioned
change.
4.2 When should we start the vaccination cam-
paign?
When a realization executes, we use 500 months to stabilize the static net-
work, then, we change to a dynamic network. Thus, a key point arises and
it is to decide in which time instant starts the vaccination schedule. We
have performed a simulation with the selected 30 sets of parameters with
2300 months (around 191.6 years) where the network turns dynamic from
the month 500. In the Figure 4.3 we can see the levels of prevalence pre-
dicted by the model for 18-64 years old men and women for HR and LR, over
the next years.
The Figure 4.3 shows the mean and 95% confidence intervals of the 30 sim-
ulations for the prevalence of men and women for HR and LR. The horizontal
axis indicates the month and the vertical axis the percentage of infected. As
it can be seen, there are oscillation in the evolution of the prevalence.
The vertical lines correspond to:
magenta: month 500, the static network turns dynamic;
blue: months 980, 1580 and 2180, point out the peaks in the oscillations
in the means and the percentiles. Between them, there are 600 months
(50 years), the time of a complete generation in the model;
green: months 1280 and 1880, point out the valleys. Between the
valleys there are also 50 years, and 25 years between a peak and a
valley.
In some test executions with vaccination, we have observed that, if we
start the vaccination schedule when the prevalence is in the decreasing part
41
Figure 4.3: Mean and 95% confidence intervals of the prevalence for 18-64
men and women for HR and LR. Vertical dashed lines indicate milestones of
interest. Note the oscillations of the prevalence levels. The blue and green
vertical lines correspond to the peaks and valleys of the oscillations. The
magenta line, points the month 500 when dynamic network starts.
(towards a valley) of the oscillation, the herd immunity appears very much
sooner than when we start the vaccination schedule in the increasing part
(towards a peak).
If we take into account that in the paper [1], the authors report extraor-
dinary results in Australia, where two years after the vaccine was introduced,
the proportion of genital warts diagnosed declined by a 59% in vaccine el-
igible young women aged 12–26 years in 2007, and by 39% in men of the
same age, we conjecture that, if there are oscillations in the prevalence of
HPV over the time, they have taken advantage of a decreasing part of an
oscillation.
Recall that our goal is to determine the appropriate month where the
vaccination schedule starts, trying to save computations and favoring the
apparition of the herd immunity effect as soon as possible, in order to mini-
mize the opposite effect when the oscillation is in the increasing trend if we
have vaccinated a large enough number of individuals.
The oscillations in all the cases, men, women, HR and LR, are very simi-
lar, as we can see in the Figure 4.3, except, maybe in some upper percentiles.
Also, there are similarities from the first peak, in valleys and peaks. There-
fore, in order to take the maximum advantage of the decreasing trends and
to save computation, we are going to select the earliest peak, the month 980,
42
as the starting point for the vaccination.
4.3 Introducing vaccination
For our simulations, we are going to consider the vaccine GARDASIL9 [42].
GARDASIL9 is a vaccine indicated in girls and women 9 through 45 years
of age for the prevention of cervical, vulvar, vaginal, and anal cancer caused
by HPV types 16, 18, 31, 33, 45, 52, and 58, genital warts caused by HPV
types 6 and 11, and precancerous or dysplastic lesions caused by above HPV
types. It is also indicated in boys and men 9 through 45 years of age for the
prevention of anal cancer caused by HPV types 16, 18, 31, 33, 45, 52, and 58,
genital warts caused by HPV types 6 and 11, and precancerous or dysplastic
lesions caused by above HPV types.
The HPV types 6/11 are LR and the remainder (16, 18, 31, 33, 45, 52,
and 58) are HR. Thus, GARDASIL9 prevents against 90% of genital wart
cases and 90% of the cancer cases [43]. Although GARDASIL9 may prevent
partially against other HPV types, for modeling purposes we assume that it
does not happen. Therefore, it would be interesting to introduce changes in
order to monitor if a node is infected by HPV types included in GARDASIL9
or not, and then, to simulate accurately the prevention effect of the vaccine.
Following the study conducted in [9], if a woman is infected by HPV
LR, the probability to be only infected by 6/11 is 34.23%, 63.06% only
infected by others than 6/11 and 2.70% to be infected by both. Also,
if a woman is infected by HPV HR, the probability to be infected only
by 16/18/31/33/45/52/58 is 30.44%, 23.66% only infected by others than
16/18/31/33/45/52/58 and 45.90% by both. Due to the lack of information
about men, we will also use the above percentages for men.
Then, before starting the vaccination, we label men and women as in-
fected of HPV LR 6/11 or infected of HPV LR other than 6/11 or infected
of both, following the above percentages. Analogously, we label men and
women as infected of HPV HR 16/18/31/33/45/52/58 or infected of HPV
HR other than 16/18/31/33/45/52/58 or infected of both.
Once these assignments are done, we continue with the HPV transmission
dynamics including the vaccination, taking into account the new labels we
included. If a node has been vaccinated, it can be infected by the types of
HPV different to those that GARSDASIL9 prevents and, in this case, they
will never be infected of 6/11 nor 16/18/31/33/45/52/58. If a node has not
been vaccinated, can be infected by any HPV type.
The assumed effectiveness of the vaccine is 96.5%.
43
4.4 Introducing vaccination loss protection
In the previous section we assume that the protection of GARDASIL9 is for-
ever. In fact, until now, people vaccinated by GARDASIL (previous version
of GARDASIL9) do not have experienced any loss in the protection. But
this does not mean that it could happen in the future.
In fact, we want to simulate the worst possible scenario, that is, the
sudden drop to zero of the protection. To simulate this possibility, we will
introduce a new parameter that represents the time after the vaccination
where the protection is complete. Therefore, after this time, the vaccinated
individual will behave as a non-vaccinated individual.
4.5 Introducing variations in the vaccination
coverage
One of our goals is to simulate scenarios where variations in the vaccination
coverage have occurred and we want to study the effect in the global protec-
tion against HPV due to these variation of the coverage. To simulate these
scenarios, we will include into the model vectors of coverage, indicating the
vaccine coverage every month, and vaccinating the people following these
variable coverages.
44
Chapter 5
Model validation: The
Australian case
In this chapter we are going to check if we can obtain similar results to those
in [1,22] using the calibration parameters.
As we introduced in Chapter 1[1], there is a decrease on the number of
infected persons and the number of persons with GW is already reported for
Australia after two years of administering vaccinations to young girls. These
results were more impressive than predicted by continuous models.
To check the reliability of the model, we simulated the HPV vaccination
campaign carried out in Australia [1], and compared them with the actual
impact published [1]. In 2007, Australian health authorities started a vacci-
nation program for 12–13 year-old girls with a coverage of 73% (83% in the
first dose, 80% in the second dose and 73% in the third dose). In addition,
from 2007 to 2009, there was a catch-up vaccination program for women aged
13–26 with a decreasing coverage with age until 52% in women aged 20–26.
The results can be summarized as follows [1]:
Two years after the vaccine was introduced, the proportion of genital
warts diagnosed declined by a 59% in vaccine eligible young women
aged 12–26 years in 2007, and by 39% in men of the same age.
No significant decline was observed in women or men older than 26
years old, non-resident young women, or men who have sex with men
(MSM).
Two different scenarios were considered to be simulated:
Scenario 1: vaccination of 83% of the 14 year-old girls (or younger girls)
plus a catch-up with coverage 73% for 14–26 year-old women.
45
Scenario 2: vaccination of 73% of 14 year-old girls (or younger girls)
plus a catch-up with a vaccination coverage of 52% for 14–26 year-old
women.
These simulations represented the upper and lower bounds of the scenario
implemented in Australia.
5.1 How to measure the decline
We call Ithe number of infected women of LR HPV 6 and/or 11 just be-
fore the starting of the vaccination campaign; we call V= (v1, . . . , vN)
to the number of infected women of LR HPV 6 and/or 11 every month
from the starting of the vaccination program until the end of the simulation.
Then, the vector
100 ×1v1
I,...,1vN
I(5.1)
is a measure of the percentage of decline of the number of infected women
of LR HPV 6 and/or 11 after the beginning of the vaccination campaign. This
will also be applied to men and MSM.
In order to compare GW data given in [1] with our model, results referred
to infected women of LR HPV 6 and/or 11, we should take into account that,
whether a fixed proportion of HPV 6 and/or 11 infected individuals develops
warts, the percentage of decline in warts and in infected women of LR HPV
6 and/or 11 will be comparable.
Another important issue for the natural history of the disease is the per-
sistence of the infection [44]. Our model does not consider the persistence “a
priori”, but we derive the cases of genital warts from the number of cases of
infected individuals by taking this data into account.
5.2 Does our model return similar values to
those in [1]?
In this section, we are going to show figures about prevalence and decline
of the percentage of women, men and MSM infected of LR 6/11, the HPV
type responsible of 90% of genital warts. Taking into account that genital
warts, in average, use to appear 6 months after the infection, the figures
about prevalence or decline will be a good estimation of the prevalence and
the decline of genital warts.
46
(a) (b)
(c)
Figure 5.1: Percentage of women (a), men (b) and MSM (c) aged 14-26
infected of LR HPV 6 and/or 11 after the implementation of the vaccination
program. The cyan lines correspond to the average and 95% confidence
interval for Scenario 1 and the yellow lines to Scenario 2. We can see the
fast decrease for women and men in both scenarios from the very beginning.
However, there is not effect on MSM.
Figure 5.1 shows the percentage of women, men and MSM aged 14-26
infected of LR 6/11 after starting the vaccination program in both simulated
scenarios. We can see the fast decrease for women and men in both scenarios
from the very beginning. MSM remain constant.
In Figure 5.2, we have plotted the same data as in Figure 5.1 but from
another point of view: the average percentage of decline of women and men
infected of LR HPV 6 and/or 11. As the vaccination program progresses
over time, the percentage of decline obviously grows. After 2 years, the
model shows a decline of
47
(a) (b)
Figure 5.2: Percentage of decline of women (a) and men (b) aged 14-26
infected of LR HPV 6 and/or 11 (and consequently of genital warts) after
the implementation of the vaccination program. The cyan lines correspond
to the average and 95% confidence interval for Scenario 1 and the yellow lines
to Scenario 2. After 2 years, the model shows a decline of 72% for Scenario
1 and 54.8% for Scenario 2, in average, for women and 38.9% for Scenario 1
and 27.7% for Scenario 2, in average, for men.
Scenario 1: 72.0% with CI 95% [67.7%,76.5%] for women and 38.9%
with CI 95% [32.0%,45.5%] for men.
Scenario 2: 54.8% with CI 95% [48.5%,59.0%] for women and 27.7%
with CI 95% [21.3%,34.5%] for men.
Australian reported levels of decline (59% in women and 39% in men aged
14-26) will be reached by the model after
Scenario 1: 1.66 years with CI 95% [1.5,1.75] for women and 2.0 years
with CI 95% [1.75,2.16] for men,
Scenario 2: 2.1 years with CI 95% [2.0,2.33] for women and 2.42 years
with CI 95% [2.08,2.83] for men.
No significant impact on the rate of infection was observed in men aged
27-64, 2 years after the implementation of the vaccination program (Figure
5.3) and the same in women and MSM agreeing the observations reported
in [1]. It can be explained by the fact that, usually, individuals have sexual
intercourses with people more or less the same age.
Then, our model predicts figures close to the ones given in [1].
48
(a) (b)
(c)
Figure 5.3: Percentage of decline of women (a), men (b) and (c) MSM aged
27-64 infected of LR HPV 6 and/or 11 (and consequently of genital warts)
after the implementation of the vaccination program in both scenarios. The
cyan lines correspond to the average and 95% confidence interval for Scenario
1 and the yellow lines to Scenario 2. Notice that, in average, no significant
decline appears in the 5 years after the implementation of the vaccination
program.
49
5.3 Study of the herd immunity effect over
HPV LR infection
The herd immunity effect in both scenarios is shown in Figure 5.4 for women,
men and MSM. Notice that, in men and MSM, any decline is due to herd
immunity. The decline in the whole female population appears when the lines
representing their decline are over the vaccination lines (blue for Scenario 1
and green for Scenario 2) also shown in this figure. The herd immunity effect
starts after
for women
Scenario 1: 0.58 years with CI95% [0.0,22.1].
Scenario 2: 0.58 years with CI95% [0.0,22.1].
for men
Scenario 1: 0.0 years with CI95% [0.0,0.83].
Scenario 2: 0.0 years with CI95% [0.0,0.83].
The herd immunity effect starts very quickly for men. For MSM, there is
not a clear herd immunity effect because the decline is stable over the time.
For women, we can see that, practically, the CI95% decline lines are over
the vaccination lines in both scenarios. This means that, in the worst case,
only the vaccinated women will be protected and in the best case, almost all
women will be protected by vaccination or by herd immunity effect.
Notice that the herd immunity effect is very clear within the CI95% both
for women and men, but it does not appear in the MSM population. In the
best case scenario, the MSM subpopulation achieves a constant protection
level of 10% 15%. This could be attributed to the way in which the MSM
individuals are connected: with a very large number of LSPs among them
and some casual links with women with large LSPs.
5.4 Effect of the reduction of the vaccine ef-
fectiveness in the catch-up vaccination
In the reference [45], the authors perform a literature review where they
show that the effectiveness of the HPV vaccine on cervical abnormalities is
between 20% 54% instead of the assumed 96.5%. This means that the vac-
cine effectiveness experiments a reduction in those women who had previous
contact with the virus.
50
(a) (b)
(c)
Figure 5.4: Percentage of decline of women (a), men (b) and MSM (c) aged
14-64 for the vaccination program in Australia. The cyan lines correspond to
the average and 95% confidence interval for Scenario 1 and the yellow lines
to Scenario 2. In the figure (a), blue and green lines correspond to women
vaccination percentage for Scenario 1 and 2, respectively. Notice that the
herd immunity effect contributes to the decline in the number of infections
in men and the decline in the number of infections for unvaccinated women.
This latter can be seen when the decline lines are over the vaccination line.
However, any herd immunity effect can be seen in MSM.
51
To study the effect of this catch-up loss of the effectiveness, we propose
the simulation of the following two scenarios:
Scenario 1: vaccination of 83% of the 14 year-old girls (or younger girls)
plus a catch-up with coverage 73% for 14–26 year-old women with 54%
of catch-up effectiveness.
Scenario 2: vaccination of 73% of 14 year-old girls (or younger girls)
plus a catch-up with a vaccination coverage of 52% for 14–26 year-old
women with 20% of catch-up effectiveness.
The above scenarios consist of a variation of the scenarios already pro-
posed for Australia at the beginning of this chapter, but now, considering the
reduction of the effectiveness in the catch-up vaccination. The simulations
consider upper and lower scenarios.
In the following, we present the compared results with the scenarios with-
out loss of effectiveness in the catch-up vaccination. Then, after 2 years, the
model declines are
Ali et al. [1]: 59% for women and 39% for men.
Catch-up without loss of effectiveness:
Scenario 1: 72.0% with IC 95% [67.7%,76.5%] for women and
38.9% with IC 95% [32.0%,45.5%] for men.
Scenario 2: 54.8% with IC 95% [48.5%,59.0%] for women and
27.7% with IC 95% [21.3%,34.5%] for men.
Catch-up with loss of effectiveness:
Scenario 1: 45.68% with IC 95% [40%,51.9%] for women and
23.2% IC 95% [18%,32.5%] for men.
Scenario 2: 17.5% IC 95% [11%,25.4%] for women and 9% IC 95%
[4.7%,13%] for men.
The levels of decline reported in [1], that is 59% in women and 39% in
men aged 14-26, will be reached after
Ali et al. [1]: 2 years
Catch-up without loss of effectiveness:
Scenario 1: 1.66 years with CI 95% [1.5,1.75] for women and 2.0
years with CI 95% [1.75,2.16] for men,
52
Scenario 2: 2.1 years with CI 95% [2.0,2.33] for women and 2.42
years with CI 95% [2.08,2.83] for men.
Catch-up with loss of effectiveness:
Scenario 1: 2.75 years with CI 95% [2.25,4] for women and 2.9
years with CI 95% [2.4,4.6] for men,
Scenario 2: 8.41 years with CI 95% [6.7,9.3] for women and 10
years with CI 95% [6,11.2] for men.
The differences among the obtained results seem to point out that the
catch-up effectiveness for genital warts may be higher than the figures col-
lected in [45] for cervical abnormalities.
5.5 Discussion
The random network of sexually transmitted HPV including up to 100,000
nodes, was developed to fit the data of surveys concerning the number of
sexual partners throughout life. Standard continuous models are insuffi-
cient to accurately predict transmission because they do not account for the
individual to individual transmission of the infection, the role of hubs in
disseminating the virus through the rest of the population and neither the
vaccination campaigns targeting specific groups of individuals.
This network has successfully been applied to the stable state of infections
by LR and HR HPV genotypes in Spain [33]. In this study we mimicked the
results found in the HPV vaccination campaign in Australia [1], and showed
very reliable results.
Models based upon continuous differential equations predict a slower de-
crease in the number of infected individuals after implementing similar vacci-
nation campaigns [10]. Hence, the case of the HPV vaccination in Australia
provides one of the best real scenarios for testing new network models in
mathematical epidemiology. There is an on-going debate on the pertinence
of an approach based upon networks on epidemiology [46] and this work
contributes to show the necessity of such an approach in many cases, in
particular, in those corresponding to STD.
To validate the model, we used the Australian experience, with two dif-
ferent vaccination coverages: routinely vaccination campaign for 12 13
year-old girls with a coverage of 73% and 83% and a catch-up program in
the 14 26 age group with an average coverage of 52% and 73%. This pro-
gram revealed an important herd immunity effect [1], so that vaccination
decreased the incidence of genital warts (GW) even in the non-vaccinated
53
men because of the protection of infection conferred by the vaccine, and the
decreased transmission of the virus.
The model predicted a fast decline in the number of infections parallel
to the decline in the number of GW in Australia with very similar values.
However, this model was built with Spanish data on sexual behavior [25]
and prevalence of HPV infection [9], that might differ from the Australian
one, and may explain the minor differences found between the model and
the actual data published. Herd immunity in this model of STD is predicted
much sooner than in other highly transmitted aerial transported infectious
diseases as influenza or RSV, due to the structure of the network. This
supports the need to build appropriate LSP networks.
Also, we performed simulations taking into account the loss of effective-
ness in the catch-up vaccination following the figures collected in [45]. The
results of decline were compared with the previous simulations for Australia
and those presented in [1]. It seems that the catch-up effectiveness for genital
warts may be higher than the figures collected in [45] for cervical abnormal-
ities.
Other models have also predicted the protection of males by vaccinat-
ing girls and women, but only for men, as the model used by Bogaards et
al. [23]. This model uses Bayesian techniques to study the herd immunity
effect. However, in contrast with our model, it does not take into account
the dynamics of the HPV transmission, the importance of age-groups and
the different roles they play in the propagation of these viruses or the links
among the MSM subpopulation and the heterosexual network. In this sense,
a network model is required to study the impact of the vaccination strategies
in short, medium and long time scales.
Vaccination strategies should seek an optimal effectiveness and efficiency.
In this case, it can be seen the quick apparition of the herd immunity effect
on males and females only vaccinating women. However, the herd immunity
effect does not appear in MSM. This can be the consequence of the large
LSP numbers for MSM and their casual connections with women with large
LSP numbers in the heterosexual subnetwork.
The model considers a quiet close community, where there is not much
contact with other communities. This may not be the case in Spain which in
2016 received over 75 million tourists [47], representing almost the double of
the number of Spanish inhabitants, and where sexual contacts are frequent.
This may bias the results, as the herd immunity in Spain may not be so clear
as in countries with less tourism. Nevertheless we will study the effect of
tourism in Spain.
Another issue that we must take into account, is the modelling of the
population with a high number of contacts because these individuals are
54
hubs in the network whose vaccination may induce a faster decline of the
virus prevalence. Our approach is rather conservative in the assignment of
LSP for men and women with 10 or more links because we assume that all of
them have similar LSP. However, it is expected that individuals with extreme
values of LSP are favouring the transmission of HPV in such a way that a
targeted vaccination can show its benefit in a very short time.
55
Chapter 6
Simulations
In the previous chapters, we have built the network model and it has been
validated. Now, in this chapter, we are going to use the model to perform
several simulations, including different strategies in order to find out what
we can expect applying these plans of action and also, to understand the
mechanisms behind the transmission dynamics of the HPV. Some of the
simulations have been suggested in the previous chapter. These simulations
will focus in Spain, although they can be extrapolated to other countries.
Some of these simulations will be, in fact, a sensitivity analysis, with the
aim to simulate scenarios that may occur at the present time or in the future.
We should not forget that the data used to build and calibrate the model
have more than 10 years and the sexual behavior of humans seems to have
changed in the last ten years.
The simulations we are going to perform are:
the study of the decline of warts with the current vaccination campaign
in Spain: vaccination of girls with a coverage of 70%;
the study of the decline of oncogenic HPV with the current vaccination
campaign in Spain;
the study of what would happen if the effect of the vaccine disappears
suddenly after 20 years;
the study to determine if the tourism in Spain has a significant effect
on the HPV infection;
the study of the decline of oncogenic HPV if we vaccinate boys and
girls with a high coverage;
56
the study of how long the decline is recovered after a drop in the cov-
erage;
the study of how the decline of HPV is affected if the average number
of LSPs increases significantly;
the study of how the decline of HPV is affected if the number of MSMs
increases significantly.
6.1 Decline of HPV LR infections and cases
of genital warts in the long-term in Spain
Let us study the decline of HPV LR infections and, consequently, of cases of
genital warts in the long-term in Spain. We are going to use the procedure
explained in Section 5.1. The vaccination program started in Oct 2007.
The results can be seen in Figure 6.1. For women, we also include the per-
centage of vaccinated women over the time that will help us to visualize the
herd immunity effect, because this effect arises when the decline line is over
the vaccination line. To be precise, the herd immunity is present, in average,
from 2008.3 (0.6 years after the starting of the vaccination program) with
CI95% [2007.75,2037.83], that is, in the worst case, the herd immunity effect
in women will start in 2037.83, 30 years after the starting of the vaccination
program.
In case of men, the herd immunity appears from the very beginning 0.58
year with CI95% [0.0,2.17]. As men are not vaccinated, the herd immunity
can be visualized if the line is over zero.
The herd immunity in MSM does not exist. There is an almost con-
stant band between 25% and 20% gathering the best and worst decline
percentages with non appreciable changes.
The differences between the best and the worst cases is due to the LSP
network structure. We do not know exactly how these networks are and the
uncertainty that involves the building of these networks may lead to extreme
scenarios. This happens in general and in MSM in particular. Thus, we
should say that, in our built networks of 100,000 nodes, less than 2,000 are
labeled MSM. This figure is very low to state predictions for MSM population
with a reliable uncertainty and it may explain why the Figure 6.1 for MSM
has so extreme and noisy 95% confidence interval.
In the Table 6.1, we can see when given percentages of decline will be
reached for women and men, in average. Note that it is not necessary a long
57
(a) (b)
(c)
Figure 6.1: Decline of HPV LR 6/11 infections, and consequently, of genital
warts (GW) in 14-64 years old women (a), men (b) and MSM (c) over the
time, from Oct 2007 when the vaccination campaign started in Spain. In the
women decline, we include the percentage of vaccinated women. It helps to
visualize the herd immunity effect when the decline line is over the vaccina-
tion line. In case of men, the lines over the yellow abscissa line means effect
of the herd immunity. However, for the MSM, as in the Australian case,
there is not herd immunity effect.
58
time to achieve high percentages of decline for both, women and men. Recall
that MSM are included in men.
Decline Women Men
30% year 2017, CI95% [2015,2035] year 2021, CI95% [2018,2044]
40% year 2021, CI95% [2017,2039] year 2027, CI95% [2021,2049]
50% year 2024, CI95% [2020,2045] year 2044, CI95% [2027,2056]
60% year 2029, CI95% [2023,2047] year 2068, CI95% [2032,]
70% year 2035, CI95% [2027,2052] year , CI95% [,]
80% year 2053, CI95% [2030,2101] year , CI95% [,]
Table 6.1: In this table we show when given percentages of decline of HPV
LR 6/11 will be reached over the time with a 95% confidence interval. The
symbol ”” means that this percentage is not reached in the simulation
period. Note that, in average, high percentages of decline for women and
men are achieved very soon.
6.2 Decline of HPV oncogenic HR infections
in the long-term in Spain
Here, we are going to repeat the study done in the previous section corre-
sponding to HPV oncogenic HR 16/18/31/33/45/52/58, those GARDASIL9
prevents. In this case, the study may give an idea about the future decline
of the cases of HPV-related cancer, but taking into account that cancer use
to appear after around 20 years of persistent infection.
As in the previous section, we are going to use the procedure explained
in Section 5.1. The vaccination program started in Oct 2007.
The results can be seen in Figure 6.2. For women, the herd immunity is
present, in average, from 2010 (2 years after the starting of the vaccination
program). In the worst case, the herd immunity effect in women appears
clearly after 39 years, around 2047.
In case of men, in average, the herd immunity appears from the very
beginning (0 years, 2007.75). As men are not vaccinated, the herd immunity
can be visualized if the line is over zero. In the worst case, 2.5 years (2010.25)
are necessary to arise the threshold of the herd immunity. As in the previous
cases, herd immunity effect does not appear for MSM.
We should note that, for HPV oncoviruses, the evolution of the decline is
more uncertain than for HPV 6/11 after some years of vaccination campaign
(see the confidence intervals). Then, if we want to predict when there will be
59
(a) (b)
(c)
Figure 6.2: Decline of HPV oncogenic HR infections in 14-64 years old
women, men and MSM over the time, from Oct 2007, when the vaccination
campaign started in Spain. In the women decline, we include the percentage
of vaccinated women. It helps to visualize the herd immunity effect when
the decline line is over the vaccination line. In case of men, the lines over the
yellow abscissa line means effect of the herd immunity. For the MSM, there
is not herd immunity effect.
a reduction of cases of cervical cancer after 2030, it may have a delay that
could be of 15 years, in the worst case, although the average suggests very
promising reductions.
In the Table 6.2, we can see when given percentages of decline of HPV
oncogenic HR will be reached for women and men. As in the HPV LR case, it
is not necessary a long time to achieve high percentages of decline for women
and men in average. Furthermore, the decline percentages for oncogenic HPV
types are worse than HPV LR 6/11.
60
Decline Women Men
30% year 2022, CI95% [2019,2040] year 2036, CI95% [2024,2054]
40% year 2027, CI95% [2023,2044] year 2063, CI95% [2029,2065]
50% year 2031, CI95% [2025,2047] year , CI95% [,]
60% year 2039, CI95% [2028,2051] year , CI95% [,]
70% year 2050, CI95% [2032,2055] year , CI95% [,]
80% year 2076, CI95% [2058,] year , CI95% [,]
Table 6.2: In this table we show when given percentages of oncogenic HPV
decline in Spain with the current vaccination program will be reached over
the time with a 95% confidence interval. The symbol ”” means that this
percentage is not reached in the simulation period.
6.3 What would happen in Spain if, after 20
years of the vaccination, the effect of the
vaccine disappear completely?
In this section, we are going to simulate what would happen if the vaccine is
protecting completely the vaccinated woman during 20 years and suddenly,
in the month after these 20 year, she losses completely the protection and
she becomes vulnerable to the HPV LR 6/11 and oncogenic HPV. In Figure
6.3 we can see the evolution of the protection simulated for every vaccinated
woman.
20 30 40 50 60
Age
20
40
60
80
100
%protection
Figure 6.3: Evolution of the vaccine protection for every vaccinated woman.
After vaccination until 20 years, the vaccine protects completely (100%) but
the month after, the protection drops to 0%, the vaccine does not protect
anymore.
In Figure 6.4, we can see a comparative of the decline of HPV LR 6/11
between the scenario where the effect of the vaccine is permanent, shown
61
in Sections 6.1 and 6.2, and the scenario where the effect of the vaccine
disappear completely after 20 years. In the graphs for women and men we
can see how this latter case stabilizes in levels of decline around 20% lower, in
average, than the case where the effect of the vaccine is permanent. However,
a significant fraction of the population remains protected despite the loss of
the protection.
In regard to MSM, we can see that the lost of the vaccine protection does
not affect the decline. This is a fact that also supports the statement that
there is not herd immunity for MSM, even if the properties of the vaccine
change, because the MSM decline does not change.
Now, in Figure 6.5, we can see an analogous comparative for the decline
of oncogenic HPV between the mentioned scenarios: permanent protection
and the drop of the protection after 20 years. The graphs are very similar to
the ones in Figure 6.4 with the only change that the difference between the
levels of decline are higher. The results for MSM are almost identical, what
supports that there is not herd immunity effect on them.
Now, we are going to perform a sensitivity analysis, consisting of the
comparison between the decline of HPV infection if there is a drop in the
vaccine protection after 20 years and after 30 years. This comparison will
be done only for men and women, because there are not differences in MSM
due to the lack of herd immunity effect. The results can be seen in Figure
6.6.
Figure 6.6 leads us to conclude that the longer is the vaccine protection
the lower will be the drop in the decline.
To finish the present section, we are going to show Figure 6.7 where we
can see the percentage of women protected by the vaccine if the protection
is permanent and if there is a loss of protection after 20 and 30 years.
6.4 Simulation of the effect of the tourism on
the contagion of HPV in Spain
In 2017, 82 millions of tourists visited Spain [47]. These visits may have
influence on the spread of HPV and this is what we are going to study in the
present section.
Unfortunately, there are not available data related to the number of in-
fections due to sexual intercourses with tourists, therefore, we are going to
introduce a more or less believable scenario, skewing a bit against the vaccine.
This scenario will simulate, every month, that 1% of Spanish individuals aged
18 29 years old with 4 or more LSP have sexual intercourses with infected
62
(a) (b)
(c)
Figure 6.4: Comparative of the decline of HPV LR 6/11 in 14-64 years
old women (a), men (b) and MSM (c) over the time, between the scenario
where the effect of the vaccine is permanent (magenta lines, representing the
mean and the 95% confidence interval) and the scenario where the effect of
the vaccine disappear completely after 20 years (cyan lines, representing the
mean and the 95% confidence interval). As we can see, for women and men,
the latter stabilizes in levels of decline about 20% less, in average, than the
case where the vaccine protects permanently. For MSM there are not changes
between both scenarios.
63
(a) (b)
(c)
Figure 6.5: Comparative of the decline of oncogenic HPV infections in 14-
64 years old women (a), men (b) and MSM (c) over the time, between the
scenario where the effect of the vaccine is permanent (magenta lines, repre-
senting the mean and the 95% confidence interval) and the scenario where
the effect of the vaccine disappear completely after 20 years (cyan lines, rep-
resenting the mean and the 95% confidence interval). The graphs are very
similar to the ones in Figure 6.4 with the only change that the difference
between the levels of decline are higher. For MSM there are not changes
between both scenarios.
64
(a) (b)
(c) (d)
Figure 6.6: Comparative of the decline of HPV 6/11 infections in 14-64 years
old women (a) and men (b) over the time, between the scenario where the
drop of the vaccine protection occurs after 20 years of vaccination (cyan lines)
and after 30 years (yellow lines). The same for figures (c) and (d) with HPV
oncogenic infection. The difference of 10 years in the drop results in around
10% of difference in the decline.
65
Figure 6.7: Percentage of women protected by the vaccine in three scenar-
ios: permanent protection (cyan line), total loss of protection after 30 years
(magenta line) and total loss of protection after 20 years (yellow line).
tourists.
To do so, we need to introduce a new feature in the computational
model to simulate the effect of the tourism. First, recall that, following
[9], among the infected women, 76.03% of them are infected of HPV HR,
11.72% are infected of HPV LR and 12.24% are infected by both, HR and
LR. Also, 50.35% of the HPV HR are with oncogenic types, that is, HPV
16/18/31/33/45/52/58. Furthermore, 37.075% of the HPV LR are with 6/11.
Then, for all individuals with 4 or more LSP aged 18 29 rnd() denotes a
computational function that generates a random number in the interval [0,1].
If rnd() <0.01 (the individual may be infected by a tourist) and let
x=rnd()
If x < 0.7603 the node gets infected by HPV HR. Also, if rnd() <
0.5035 and the individual is not vaccinated, the infection is by an
oncogenic HPV HR.
If 0.7603 x < 0.7603 + 0.1172 the node gets infected by HPV
LR. Also, if rnd() <0.3707 and the individual is not vaccinated,
the infection is by HPV 6/11.
If 0.7603 + 0.1172 xthe node gets infected by HPV LR and by
HPV HR (co-infection). Thus, if rnd() <0.5035 and the individ-
ual is not vaccinated, the infection is by an oncogenic HPV HR
and rnd() <0.3707 the infection is also by HPV 6/11.
Including the above features in the computational model and performing
66
a simulation, in the Figure 6.8 we can see the influence of the tourism in the
HPV infections.
Figure 6.8 shows that there are not remarkable differences between both
scenarios including the very beginning, where there are very few girls vacci-
nated. Therefore, the tourism does not seem to be a factor that influences
the decline of HPV 6/11. Here we do not show the same graphs for oncogenic
HPV because of their similarity with the ones in Figure 6.8.
6.5 Simulation of different vaccination strate-
gies in girls and boys with the aim to
eradicate the cancer diseases associated
HPV oncogenic HR
In this Chapter, we are going to simulate the scenarios where we vaccinate
GARDASIL9 to boys and girls with a coverage of 60%, 75% and 90%. Then,
we will see when a decline of 65%, 75%, 85% and 95% is reached in all the
cases for HPV oncogenic HR. Our results will be compared with the ones
in [48], where the authors consider that the elimination of the HPV HR
included in GARDASIL9 (the oncogenic ones) will be reached after 80 years
vaccinating boys and girls with a coverage of 80%. The results can be seen
in Table 6.3.
Now, in Figure 6.9, we show some graphs about the decline of HPV onco-
genic in 14-64 years old women, men and MSM with coverage for women
and men of 75% and 90%. The difference between the percentage of vacci-
nated individuals (yellow lines) and the decline of the HPV (cyan lines) is a
measure of the herd immunity effect on each population. As we can see, the
herd immunity effect on women and men are similar. The small differences
are because men include MSM even though the percentage of MSM is low
respect to the total of men. Nevertheless, in MSM, the herd immunity is low,
because their high number of LSP that MSM have.
As we can see in the Figure 6.9, the highest values of decline are reached
around 2060 in all the graphs, that is, after a complete generation has been
vaccinated. Therefore, a high coverage for women and men during a whole
generation has to be considered if we want to eradicate the cancer produced
by HPV oncogenic.
Nevertheless, Figure 6.9 shows that the variation of the decline in the
long-run between the vaccination of 75% to 90% is small if we think that we
need an increase of 15% in coverage. From year 2060 to 2100, the differences
67
(a) (b)
(c) (d)
(e) (f)
Figure 6.8: Comparative of the decline of HPV LR 6/11 in 14-64 years old
women (a), men (c) and MSM (e) over the time and a zoom of the graphs for
the first five years for women (b), men (d) and MSM (f), between the non-
tourism scenario (yellow lines) and the tourism scenario where every month
1% of Spanish individuals aged 18 29 years old with 4 or more LSP have
sexual intercourses with infected tourists (cyan lines). Magenta lines show
the percentage of vaccinate women and men (nobody in men nor MSM). No
remarkable changes appear. 68
Vaccination of boys and girls with a coverage of 60%
Decline Women Men MSM
65% year 2040, CI95% [2029,2051] year 2047, CI95% [2031,2053] year , CI95% [2053,]
75% year 2050, CI95% [2032,2055] year 2058, CI95% [2038,2070] year , CI95% [,]
85% year 2084, CI95% [2069,] year , CI95% [,] year , CI95% [,]
95% year , CI95% [,] year , CI95% [,] year , CI95% [,]
Vaccination of boys and girls with a coverage of 75%
Decline Women Men MSM
65% year 2034, CI95% [2028,2047] year 2039, CI95% [2028,2048] year 2045, CI95% [2042,2049]
75% year 2042, CI95% [2030,2050] year 2046, CI95% [2032,2052] year 2053, CI95% [2049,2059]
85% year 2050, CI95% [2033,2054] year 2054, CI95% [2042,2056] year , CI95% [,]
95% year , CI95% [2060,] year , CI95% [,] year , CI95% [,]
Vaccination of boys and girls with a coverage of 90%
Decline Women Men MSM
65% year 2032, CI95% [2027,2046] year 2035, CI95% [2027,2046] year 2039, CI95% [2036,2042]
75% year 2038, CI95% [2029,2050] year 2041, CI95% [2030,2049] year 2044, CI95% [2041,2047]
85% year 2046, CI95% [2032,2052] year 2048, CI95% [2033,2052] year 2050, CI95% [2047,2052]
95% year 2054, CI95% [2042,2056] year 2055, CI95% [2048,2057] year , CI95% [2056,]
Table 6.3: Here, we show when a decline of HPV oncogenic HR of 65%, 75%, 85% and 95% is reached when
vaccinating boys and girls with a coverage of 60%, 75% and 90%. MSM are included in men, and also considered
separately.
69
Decline. Women. Coverage 75% Decline. Women. Coverage 90%
Decline. Men. Coverage 75% Decline. Men. Coverage 90%
Decline. MSM. Coverage 75% Decline. MSM. Coverage 90%
Figure 6.9: Decline of HPV oncogenic in 14-64 years old women, men and
MSM with coverage for women and men of 75% and 90% over the time. The
horizontal dashed lines point out the decline percentages of 65%, 75%, 85%
and 95%. The difference between the cyan and the yellow lines is a measure
of the herd immunity effect on each population. In this case, due to the high
coverage, we can see herd immunity effect in MSM only in the long-run.
70
in the decline between coverage 90% and 75% are: in men vary from 6.3%
to 10.1%; in women vary from 3.3% to 6.4%; and for MSM vary from 10.2%
to 20%. Only in the MSM coverage increase is fully reflected in the similar
order of magnitude, around 15%. In other words, looking at Table 6.3,
with the increase in the coverage from 75% to 90%, in average, the levels of
decline 65%, 75% and 85% are reached 2 4 years before for women and
46 years before for men. For MSM, the levels of decline 65% and 75%
are reached 69 years before, in average. This fact may suggest that, reaching
coverages around 70%75% is enough to have a high protection in the whole
population without a significant increasing of the vaccination cost to achieve
small increases in the decline in the long-run.
Also, we would like to point out that the herd immunity effect, measured
by the difference between the decline lines (cyan) and the vaccination lines
(yellow), is similar in men and women, and there is a small herd immunity
in MSM in the long-run, due to the high vaccine coverage.
In recent papers as [48,49,50], the authors find possible the elimina-
tion of the oncogenic HPV included in GARDASIL9 in around a generation.
However, our model with its accurate replication of the herd immunity effect,
suggests that we have to be cautious. MSM have small herd immunity effect
and, therefore, if we are not able to vaccinate almost all the MSM with higher
coverages on them, the unprotected MSM may preserve the circulation of the
virus due to their high number of sexual partners.
6.6 Quantifying the delay to recover normal
levels of decline due to a drop in the vac-
cination coverage
In Denmark, the vaccination program started in 2009 with a coverage of 80%.
After 5 years, in 2 months the coverage dropped until 10%. It remained 2
years in 10% and then, in 5 years increased until 70% and remained constant.
In Figure 6.10 we show the coverage evolution.
Drops in the coverage have happened in some countries and it is a con-
cerning issue because the natural consequence is the delay in achieving high
rates of protection in order to reduce the pathologies associated to HPV.
Thus, here, we are going to perform simulations to study the delay pro-
duced due to a drop in the coverage under different circumstances. A first
approach has published in [51], where the authors consider an age-structured
model of differential equations [52] to simulate the following vaccination pro-
grams
71
2020 2040 2060 2080
0.2
0.4
0.6
0.8
Figure 6.10: Evolution of the HPV vaccine coverage in Denmark from the
program starting in 2009.
1. Case 1 or base case: vaccination of 80.5% girls aged 11-12 and 53.25%
of the girls aged 13-18.
2. Case 2: base case + vaccination of 80.50% of women aged 22-26 during
the year 2015.
3. Case 3: Case 2 + vaccination of 80.50% of the boys aged 11-12.
4. Case 4: Case 3 + vaccination of 80.50% of the boys aged 13-26 during
the year 2015.
The above strategies are consistent with the Swedish policies, where they
started vaccinating girls in 2007. The authors assume a vaccination effec-
tiveness of 95% in the vaccination of 11-12 years old boys and girls and 92%
for older ones.
The results obtained in [51] say that the vaccination programs are less
resilient when only girls are vaccinated, that is, the effect of a drop in the
coverage is greater if only girls are vaccinated because the decline in HPV in-
fection is recovered faster if boys and girls are vaccinated. Also, they say that
if vaccination coverage is high, 25 30 years after vaccination commences,
the effectiveness converges. The most salient impact of male vaccination is
the mitigation of loss of vaccine effectiveness in the face of an unexpected
reduction in coverage.
Our goal here is to see if using our model and scenarios adapted to the
Spanish situation, we are able to obtain results according to those in [51]. In
our case, we propose the following scenarios
1. Case 1 or base case: vaccination of 14 years old girls with a coverage
of 70% starting in Oct 2007 (current situation in Spain).
72
2. Case 2: base case + vaccination of 14 years old boys with a coverage
of 70% starting in January 2020.
3. Case 3: base case + in January 2025, in 2 months the coverage drops
until 10%, it remains 2 years in 10% and then, in 5 years increases until
70% and remains constant (Danish scenario).
4. Case 4: Case 2 + Danish scenario starting in January 2025 droping the
coverage in boys and girls (Danish scenario for both boys and girls).
Here, we assume an effectiveness of 96.5% and the use of GARDASIL9.
We do not have scenarios with catchup vaccination, because in [45], as we
mentioned previously, the authors perform a literature review where they
show that the effectiveness of the HPV vaccine on cervical abnormalities is
between 20% 54% instead of the assumed 96.5% or 92% assumed in [51].
In order to study the drop of the coverage, we are going to compare the
delay produced in the decline of infections between, on the one hand, the
Base Case and the Case 3 (only girls vaccination), and on the other hand,
the Case 1 and the Case 4 (vaccination of boys and girls).
We show the obtained results in Figures 6.11 and 6.12. Figure 6.11 depicts
the comparison between the Base Case and the Case 3 and Figure 6.12 the
comparison between the Case 2 and the Case 4. In both figures, on the left
column we can see the evolution over the time of the decline of the oncogenic
HPV in men, women and MSM (only in Figure 6.12), from year 2025 (when
the drop starts). On the right column, we visualize the average number of
years the simulation with the drop in the coverage needs to achieve the same
level of decline as the simulation without drop, from year 2025 until 2055.
The difference of years to achieve the same decline is calculated as follows: for
each time instant, we find the average decline level in the case where there is
not drop. Then, we seek in which time instant the same average decline level
is reached in the case there is drop and we calculate the difference between
both time instants and divided by 12.
In Figure 6.11, MSM are not considered because there is not herd immu-
nity effect nor decline on them when only girls are vaccinated.
If we look at the left columns, the CI95% begin to converge from 2060 for
men and women, having a significant intersection. In MSM the parallelism
is the same all the time.
Furthermore, when the decline starts to saturate, a short increase in the
decline takes much longer time. Now, looking at the right columns:
Men: in men, we can see that when boys and girls are vaccinated,
the resilience is much more. We can see more clearly the differences
73
(a) (b)
(c) (d)
Figure 6.11: Comparative of Base Case and Case 3. On the left column, the
average decline of oncogenic HPV in men (a) and women (c), in both cases.
On the right column, the time (in years) the Case 3 needs to achieve the
same levels of decline as the Base Case for oncogenic HPV in men (b) and
women (d).
74
(a) (b)
(c) (d)
(e) (f)
Figure 6.12: Comparative of Case 2 and Case 4. On the left column, the
decline of oncogenic HPV in men (a), MSM (c) and women (e), in both
cases. On the right column, the average time (in years) the Case 4 needs to
achieve the same levels of decline as the Case 2 for oncogenic HPV in men
(b), MSM (d) and women (f).
75
(a) (b)
Figure 6.13: Comparative of the average time in years of the delay in the
case where only girls are vaccinated and boys and girls are vaccinated. On
the left (a), the average delay of oncogenic HPV in men. On the right (b),
the same as (a) but for women. The vaccination programs are less resilient
when only girls are vaccinated.
in Figure 6.13(a). In the maxima values, the difference is from 5% to
12%.
MSM: it is interesting to notice that in 6.12(d), the delay is practically
constant over the time and equal to 4 years. It may be because the
small herd immunity effect on MSM.
Women: here, the difference is lower because in all the cases women are
vaccinated. Nevertheless, there is an extra herd immunity effect when
boys and girls are vaccinated that reduces in some years the delay.
In Figure 6.13 we plot in the same graphic, on the one hand, 6.11(b) and
6.12(b) to compare the average delay in men, and on the other hand, 6.11(f)
and 6.12(f) to compare the average delay in women. Here, the differences
can be visualised more clearly.
Thus, we conclude that the vaccination programs are more resilient when
boys and girls are vaccinated and, with the simulated programs, it is neces-
sary 3035 years after the coverage has been recovered to see the convergence
in the decline.
76
6.7 How is the decline of HPV affected if the
average number of LSPs increases signif-
icantly?
The data used to build the network model and to perform the calibration have
more than 10 years. Also, the sexual behavior of the people has changed in
the last years and therefore, it would be interesting to perform some simula-
tions adapting the model parameters to the new sexual behavior and also, we
perform a sensitivity analysis of the model. Here, we propose an increasing
in the number of lifetime sexual partners (LSP) of the nodes.
The simulation is going to be a little bit tricky.
1. The assignment of heterosexual LSPs follows the data in Tables 2.1
and 2.2. We are going to maintain the percentages but changing the
number of LSPs. Thus, the proportion of people with only 1 LSP, in
the simulation will have 2; people with 2 LSPs will have 4; people with
34 LSPs will have 68; people with 5 9 LSPs will have 9 13 and
the people with 10 or more LSPs will have 14 or more.
2. The parameter determining the average number of LSPs in men will be
increased in 8. Then, the values of the average number of LSPs in men
corresponding to the 30 selected simulations during the calibration (3.1)
will increase in 8, in order to guarantee that there are enough sexual
partners available to create all the couples.
3. In [30], we have the number of LSPs in MSM: 18 LSPs for the age group
14-19; 25 LSPs for the age group 20-24; 33 LSPs for the age group 25-
29; 45 LSPs for the age group 30-39; 59 LSPs for the age group 40-49;
50 LSPs for the age group 50-59; 56 LSPs for the age group 60-85+.
There number of LSPs will be increased by 4.
Grosso modo, in this simulation, we are increasing the LSPs in more than
100%. We assume the Spanish scenario where only girls are vaccinated with
a 70% coverage. In Figure 6.14 we can see graphically the obtained results.
The simulated increase in the number of LSPs provide small changes in
the decline of the HPV infections in the long-run. The differences are between
4.3% 6.4% for oncogenic HPV in men,
0% 2.6% for HPV 6/11 in men,
3.3% 4.5% for oncogenic HPV in women,
77
Decline oncogenic HPV men Decline HPV 6/11 men
Decline oncogenic HPV MSM Decline HPV 6/11 MSM
Decline oncogenic HPV women Decline HPV 6/11 women
Figure 6.14: Comparative of the decline of HPV in case the global average
number of LSP increases. In the current vaccination scenario (vaccinating
only girls with a coverage of 70%, there are only significant changes in declines
for oncogenic HPV in men.
78
2.0% 3.7% for HPV 6/11 in women.
Despite the noticeable increase in the number of LSPs, there is only a
small effect on the decline of the HPV infections. Therefore, although the
sexual behavior has changed in the last years and the number of sexual
partners has increased, the herd immunity effect remains stable and it is not
expected perceptible changes in the decline of the infection.
However, it may change if the increase in LSPs is much higher. In the
Figure 2 of paper [12], the authors study the average number of contacts for
the transmission dynamics of the Respiratory Syncytial Virus (RSV). They
found that the average number of contacts should take values greater than
25 to explain the dynamics of the disease.
The network defined in [12] has the same structure as the MSM LSP
network, with average number of contacts (LSPs) of 39 [30]. In infectious
diseases as smallpox, varicella, influenza, RSV, etc., where there are not
restrictions in the possible contagious contacts, the classical and network
epidemiological models say that the herd immunity effect only appears when
the vaccination coverage achieves high percentages. This fact explain what
we have seen along this dissertation: MSM do not have herd immunity effect
when only girls are vaccinated and some herd immunity is achieved if boys,
and therefore MSM, are also vaccinated with high rates of coverage.
On the other hand, with the obvious caveat that the structure of the
heterosexual LSP network is not the same as the one in the random network
in [12], a strong increase in the number of average LSPs to values close to
the average number of LSPs of MSM, 39, or greater, it is possible that the
herd immunity effect provided by the vaccine may be reduced significantly
or disappears in the heterosexual network. As a consequence, HPV would
become a disease with a transmission dynamics similar to smallpox, varicella
or influenza.
6.8 How does the decline of HPV changes if
the percentage of MSMs increases signif-
icantly?
As we mentioned before, the data used in this work is a little bit old and
the sexual behavior of the people has experimented changes in the last years.
Thus, it would be interesting to perform some simulations adapting properly
the model parameters. Here, we propose an increasing in the percentage of
men who have sex with men (MSM). This simulation can be also considered
79
as a sensitivity analysis to support the robustness of the present study with
respect to the variation of the percentage of MSM.
In our proposal we have assumed that the percentage of MSM in the
population is 3.88%. This value is provided by [25]. However, this survey
was published in 2003. Then, here, we are going to increase the percentage of
MSM from 3.88% until 10%. Only girls are vaccinated with a 70% coverage
(Spanish current program).
As we can see in Figure 6.15, the decline hardly changes in heterosexual
men, MSM and women. This means that the increase of MSM only affects
themselves, because they do not have herd immunity effect.
Furthermore, Figure 6.15 shows the decline in heterosexual men. Note
that the levels of decline for HPV 6/11 achieves more than 70% in the long-
run vaccinating girls with a coverage of 70%, meanwhile the girls achieves a
decline higher than 80%. In the case of oncogenic HPV, the decline is not
so high and it is more uncertain, achieving around 50% in heterosexual men
and around 80% in women. With this comment we may have an idea about
the protection provided by GARDASIL and GARDASIL9 in the women and
heterosexual men network, in the long-run.
80
Decline oncogenic HPV hetero men Decline HPV 6/11 hetero men
Decline oncogenic HPV MSM Decline HPV 6/11 MSM
Decline oncogenic HPV women Decline HPV 6/11 women
Figure 6.15: Comparative of the decline of HPV in case the number of MSM
increases from 3.88% until 10%. In the current vaccination scenario (vacci-
nating only girls with a coverage of 70%, there are not significant declines in
heterosexual men, MSM and women. The graphs corresponding to hetero-
sexual men show a higher decline as those considering both heterosexual and
homosexual men.
81
Chapter 7
Conclusions and limitations
In this dissertation, we present a computational model to describe the trans-
mission dynamics of HPV. The model is based on networks of lifetime sexual
partners (LSP) determining the paths where HPV spreads. We also encoun-
tered a set of limitations inherent to this kind of models.
Here, we intend to help in the current discussion about two main points:
why the vaccination campaigns are more effective than expected and whether
the boys should be also vaccinated.
As a result of the work done with the above goals in mind, in the following,
we point out the main general conclusions and limitations of this dissertation.
Under the Public Health point of view:
1. Our model reproduces the singular situation occurred in Australia
where a special vaccination strategy was carried out.
2. Also, the model explains why the vaccination campaigns are more ef-
fective than expected and describes the herd immunity effect more ac-
curately.
3. The model shows that, only vaccinating girls, there is not herd immu-
nity effect on MSM.
4. The resilience is much more when vaccinating boys and girls.
5. An increasing in the number of LSP does not have a significant effect
on the decline of HPV infections.
6. An increasing in MSM has effect only on MSM.
Therefore, if we vaccinate women, they are protected and heterosexual
men are also protected by herd immunity. MSM are not protected at all.
82
If we want to eradicate the diseases associated to oncogenic HPV, we
should vaccinate boys and girls with high coverage, making sure that MSM
are vaccinated with the highest coverage possible to avoid the HPV keeps
circulating among unprotected MSM subnetwork.
We expect to give tools to those responsible for public health to be able
to design appropriate HPV NIPs, or expand to new cohorts combining effec-
tiveness and economy.
Under the technical point of view:
1. We have designed an epidemiological model to study the transmission
dynamics of HPV using LSP networks.
2. We use known real data to build big LSP networks (in networks, size
may matters, see Figure 5 in [38]).
3. We have designed a complex and innovative system to calibrate big
network models.
4. We have designed algorithms for model calibration taking into account
the uncertainty in the network model building and in the data.
5. We have provided some advances in the computational treatment of
the uncertainty quantification in computation demanding models.
Below we also remark the limitations of the model:
1. Data collected from CLEOPATRA used in calibration, is only for women,
so biased information must be considered.
2. Sexual habits data are collected from 2003, biased information may
also found here, although a sensitivity analysis has been performed.
3. The lack of specific data about sexual behavior of the population do
not allow us to check the reliability of the network structure. However,
the computational network model built reproduces accurately real sit-
uations.
4. In the model we have not considered that males have lower (or no)
immune protection against HPV compared to females [53]. In fact, we
have not considered any immune protection against HPV after infec-
tion.
5. The underlying demographic model assumes constant population using
real Spanish demographic data. However, even though the population
83
is constant, its dynamics makes that the age groups do not have con-
stant population. This irregularity may be the cause of small cyclicity
of the model output every complete generation of 50 years and the
lower levels of decline under the coverage in the period 2040 years.
84
Bibliography
[1] H. Ali, B. Donovan, H. Wand, T. R. Read, D. G. Regan, A. E. Grulich,
C. K. Fairley, and R. J. Guy, “Genital warts in young australians five
years into national human papillomavirus vaccination programme: na-
tional surveillance data,” BMJ, vol. 346, p. f2032, 2013.
[2] M. A. Stanley, “Epithelial cell responses to infection with human
papillomavirus,” Clinical Microbiology Reviews, vol. 25, no. 2, pp. 215–
222, 2012. [Online]. Available: https://cmr.asm.org/content/25/2/215
[3] J. Olsen and T. R. Jørgensen, “Revisiting the cost-effectiveness of uni-
versal HPV-vaccination in Denmark accounting for all potentially vac-
cine preventable HPV-related diseases in males and females,” Cost Ef-
fectiveness and Resource Allocation, vol. 13, no. 1, p. 1, 2015.
[4] WikimediaCommons, “Harald zur Hausen,” 2010, [On-
line; accessed 6 September 2018]. [Online]. Avail-
able: https://commons.wikimedia.org/wiki/File:Harald zur Hausen 03.
jpg#/media/File:Harald zur Hausen 03.jpg
[5] G. M. Clifford, R. K. Rana, S. Franceschi, J. S. Smith, G. Gough,
and J. M. Pimenta, “Human papillomavirus genotype distribution
in low-grade cervical lesions: Comparison by geographic region
and with cervical cancer,” Cancer Epidemiology and Prevention
Biomarkers, vol. 14, no. 5, pp. 1157–1164, 2005. [Online]. Available:
http://cebp.aacrjournals.org/content/14/5/1157
[6] C. J. Lacey, C. M. Lowndes, and K. V. Shah, “Burden and management
of non-cancerous HPV-related conditions: HPV-6/11 disease,” Vaccine,
vol. 24, pp. S35–S41, 2006.
[7] J. L. Marx, “Human papilloma virus and cervical cancer,” The Medical
journal of Australia, vol. 144 3, p. 164, 1986.
85
[8] J. S. C. Roberts, T. ONeill, and C. Lacey, “Vaccine for genital warts,”
Vaccines for Human Papillomavirus Infections and Anogenital disease,
Medical Intellegence Unit, vol. 14, 1999.
[9] X. Castellsagu´e, T. Iftner, E. Roura, J. A. Vidart, S. K. Kjaer,
F. X. Bosch, N. Mu˜noz, S. Palacios, M. S. M. Rodriguez,
L. Serradell, L. Torcel-Pagnon, and J. C. and, “Prevalence and
genotype distribution of human papillomavirus infection of the
cervix in Spain: The CLEOPATRE study,” Journal of Medical
Virology, vol. 84, no. 6, pp. 947–956, apr 2012. [Online]. Available:
https://doi.org/10.1002%2Fjmv.23282
[10] E. H. Elbasha, E. J. Dasbach, and R. P. Insinga, “Model for assessing
human papillomavirus vaccination strategies,” Emerging Infectious
Diseases, vol. 13, no. 1, pp. 28–41, jan 2007. [Online]. Available:
https://doi.org/10.3201%2Feid1301.060438
[11] E. H. Elbasha and A. P. Galvani, “Vaccination against multiple HPV
types,” Mathematical Biosciences, vol. 197, no. 1, pp. 88–117, 2005.
[12] L. Acedo, J.-A. Mora˜no, R.-J. Villanueva, J. Villanueva-Oller,
and J. D´ıez-Domingo, “Using random networks to study the
dynamics of respiratory syncytial virus (RSV) in the Spanish
region of Valencia,” Mathematical and Computer Modelling, vol. 54,
no. 7-8, pp. 1650–1654, oct 2011. [Online]. Available: https:
//doi.org/10.1016%2Fj.mcm.2010.11.068
[13] L. Acedo and J.-A. Mora˜no, “Brain oscillations in a random neural
network,” Mathematical and Computer Modelling, vol. 57, no. 7-8, pp.
1768–1772, 2013.
[14] S. N. Dorogovtsev and J. F. Mendes, Evolution of networks: From bio-
logical nets to the Internet and WWW. OUP Oxford, 2013.
[15] D. J. Watts and S. H. Strogatz, “Collective dynamics of ”small-world”
networks,” Nature, vol. 393, no. 6684, p. 440, 1998.
[16] N. A. Christakis and J. H. Fowler, “The spread of obesity in a large
social network over 32 years,” New England Journal of Medicine, vol.
357, no. 4, pp. 370–379, 2007.
[17] F. Liljeros, C. R. Edling, L. A. N. Amaral, H. E. Stanley, and Y. ˚
Aberg,
“The web of human sexual contacts,” Nature, vol. 411, no. 6840, p. 907,
2001.
86
[18] P. S. Bearman, J. Moody, and K. Stovel, “Chains of affection: The struc-
ture of adolescent romantic and sexual networks,” American Journal of
Sociology, vol. 110, no. 1, pp. 44–91, 2004.
[19] A. N. Burchell, H. Richardson, S. M. Mahmud, H. Trottier, P. P. Tellier,
J. Hanley, F. Coutl´ee, and E. L. Franco, “Modeling the sexual trans-
missibility of human papillomavirus infection using stochastic computer
simulation and empirical data from a cohort study of young women in
Montreal, Canada,” American Journal of Epidemiology, vol. 163, no. 6,
pp. 534–543, 2006.
[20] S. Helleringer and H.-P. Kohler, “Sexual network structure and the
spread of HIV in Africa: evidence from Likoma island, Malawi,” AIDS,
vol. 21, no. 17, pp. 2323–2332, 2007.
[21] B. V. Schmid and M. Kretzschmar, “Determinants of sexual network
structure and their impact on cumulative network measures,” PLoS
Computational Biology, vol. 8, no. 4, p. e1002470, 2012.
[22] C. K. Fairley, J. S. Hocking, L. C. Gurrin, M. Y. Chen, B. Donovan,
and C. Bradshaw, “Rapid decline in presentations for genital warts after
the implementation of a national quadrivalent human papillomavirus
vaccination program for young women,” Sexually transmitted infections,
vol. 85, pp. 499–502, 2009.
[23] J. A. Bogaards, J. Wallinga, R. H. Brakenhoff, C. J. Meijer, and
J. Berkhof, “Direct benefit of vaccinating boys along with girls against
oncogenic human papillomavirus: bayesian evidence synthesis,” BMJ,
vol. 350, p. h2016, 2015.
[24] IVE, “Portal estadistico de la Generalitat Valenciana (statistical
portal of the government of the Community of Valencia). Valencian
Institute of Statistics,” 2013, [Online; accessed 6 March 2017]. [Online].
Available: http://www.ive.es
[25] INE, “Encuesta de Salud y H´abitos Sexuales. (Health and
Sexual Habits Survey). Instituto Nacional de Estad´ıstica,” 2003,
[Online; accessed 6 March 2017]. [Online]. Available: http:
//www.ine.es/dyngs/INEbase/es/operacion.htm?c=Estadistica C&
cid=1254736176785&menu=resultados&idp=1254735573175g
[26] A. Chandra, C. E. Copen, and W. D. Mosher, “Sexual behavior, sexual
attraction, and sexual identity in the united states: Data from the 2006–
87
2010 national survey of family growth,” in International Handbook on
the Demography of Sexuality. Springer, 2013, pp. 45–66.
[27] W. D. Mosher, A. Chandra, J. Jones et al., “Sexual behavior and se-
lected health measures: men and women 15-44 years of age, united
states, 2002,” US Department of Health and Human Services, Centers
for Disease Control and Prevention, National Center for Health Statis-
tics Atlanta, GA, 2005.
[28] D. Gentner and A. B. Markman, “Structure mapping in analogy and
similarity.” American Psychologist, vol. 52, no. 1, p. 45, 1997.
[29] P. Miret, “La similitud entre los componentes de las parejas j´ovenes en
Espa˜na en la primera d´ecada del siglo XXI ¿Cada vez m´as iguales?
(the similarity among the components of young couples in spain in
the first decade of the xxist century. increasingly equal?),” Revista de
Estudios de Juventud, no. 90, pp. 225–255, 2010. [Online]. Available:
http://www.injuve.es/sites/default/files/RJ90-16.pdf
[30] “Estudio de conducta sexual entre homosexuales (Study of sex-
ual behavior among homosexuals),” Durex, Technical Report,
2002. [Online]. Available: http://www.sidastudi.org/es/registro/
2c9391e41fb402cc011fb442355a4176
[31] T. H. Cormen, C. E. Leiserson, R. L. Rivest, and C. Stein, Introduction
to algorithms. MIT press, 2009.
[32] T. A. Feo and M. G. Resende, “Greedy randomized adaptive search
procedures,” Journal of Global Optimization, vol. 6, no. 2, pp. 109–133,
1995.
[33] L. Acedo, C. Burgos, J.-I. Hidalgo, V. S´anchez-Alonso, R.-J. Villanueva,
and J. Villanueva-Oller, “Calibrating a large network model describing
the transmission dynamics of the human papillomavirus using a
particle swarm optimization algorithm in a distributed computing
environment,” The International Journal of High Performance
Computing Applications, p. 109434201769786, apr 2017. [Online].
Available: https://doi.org/10.1177/1094342017697862
[34] J. D´ıez-Domingo, V. S´anchez-Alonso, R.-J. Villanueva, L. Acedo,
J.-A. Mora˜no, and J. Villanueva-Oller, “Random network models
to predict the long-term impact of HPV vaccination on genital
warts,” Viruses, vol. 9, no. 10, p. 300, oct 2017. [Online]. Available:
https://doi.org/10.3390/v9100300
88
[35] A. R. Giuliano, J.-H. Lee, W. Fulp, L. L. Villa, E. Lazcano,
M. R. Papenfuss, M. Abrahamsen, J. Salmeron, G. M. Anic, D. E.
Rollison, and D. Smith, “Incidence and clearance of genital human
papillomavirus infection in men (HIM): a cohort study,” The Lancet,
vol. 377, no. 9769, pp. 932–940, mar 2011. [Online]. Available:
https://doi.org/10.1016/s0140-6736(10)62342-2
[36] A. G. Nyitray, M. Chang, L. L. Villa, R. J. C. da Silva, M. L.
Baggio, M. Abrahamsen, M. Papenfuss, M. Quiterio, J. Salmer´on,
E. Lazcano-Ponce, and A. R. Giuliano, “The natural history of
genital human papillomavirus among HIV-negative men having sex
with men and men having sex with women,” Journal of Infectious
Diseases, vol. 212, no. 2, pp. 202–212, feb 2015. [Online]. Available:
https://doi.org/10.1093/infdis/jiv061
[37] N. Khemka and C. Jacob, “Exploratory toolkit for evolutionary and
swarm-based optimization,” The Mathematica Journal, vol. 11, no. 3,
pp. 376–391, feb 2010. [Online]. Available: https://doi.org/10.3888%
2Ftmj.11.3-5
[38] J. Villanueva-Oller, L. Acedo, J. A. Mora˜no, and A. S´anchez-S´anchez,
“Epidemic random network simulations in a distributed computing
environment,” Abstract and Applied Analysis, vol. 2013, pp. 1–10, 2013.
[Online]. Available: https://doi.org/10.1155%2F2013%2F462801
[39] Python3, “Pyhton Software Foundation,” https://www.python.org,
2019.
[40] Wolfram Research, “Wolfram Mathematica: Modern Technical Com-
puting,” https://www.wolfram.com/mathematica/, 2019.
[41] R. Storn and K. Price, “Differential evolution–a simple and efficient
heuristic for global optimization over continuous spaces,” Journal of
Global Optimization, vol. 11, no. 4, pp. 341–359, 1997.
[42] Merck & Co., “Official Site for GARDASIL9,” https://www.gardasil9.
com, 2019.
[43] S. Hartwig, J.-J. Baldauf, G. Dominiak-Felden, F. Simondon,
L. Alemany, S. de Sanjos´e, and X. Castellsagu´e, “Estimation
of the epidemiological burden of HPV-related anogenital cancers,
precancerous lesions, and genital warts in women and men in europe:
Potential additional benefit of a nine-valent second generation HPV
89
vaccine compared to first generation HPV vaccines,” Papillomavirus
Research, vol. 1, pp. 90–100, dec 2015. [Online]. Available:
https://doi.org/10.1016/j.pvr.2015.06.003
[44] N. G. Campos, E. A. Burger, S. Sy, M. Sharma, M. Schiffman, A. C.
Rodriguez, A. Hildesheim, R. Herrero, and J. J. Kim, “An updated nat-
ural history model of cervical cancer: derivation of model parameters,”
American Journal of Epidemiology, vol. 180, no. 5, pp. 545–555, 2014.
[45] J. Skufca, M. Baay, G. P. Yen, S. Kothari, R. Drury, and C. Velicer,
“Effectiveness and impact of the quadrivalent human papillomavirus
vaccine on cervical abnormalities: A systematic literature review
update.” IPCV 2018, Sydney, Australia, October 2-6 2018. [Online].
Available: https://bit.ly/2Nf5FzA
[46] S. Eubank, S. Kumar, M. Marathe, A. Srinivasan, and N. Wang,
“Structure of social contact networks and their impact on epidemics,”
in AMS-DIMACS Special Volume on Epidemiology, 01 2006,
vol. 70. [Online]. Available: http://staff.vbi.vt.edu/seubank/papers/
AMS-DIMACS.pdf
[47] (Jan 31st, 2017) Estad´ıstica de movimientos tur´ısticos en fronteras 2017
(statistics of the turistic movements in the Spanish borders 2017. press
release from the Instituto Nacional de Estad´ıstica). [Online]. Available:
http://www.ine.es/daco/daco42/frontur/frontur1217.pdf
[48] M. Brisson, ´
E. B´enard, M. Drolet, J. A. Bogaards, I. Baussano,
S. V¨ansk¨a, M. Jit, M.-C. Boily, M. A. Smith, J. Berkhof, K. Canfell,
H. W. Chesson, E. A. Burger, Y. H. Choi, B. F. D. Blasio, S. J. D.
Vlas, G. Guzzetta, J. A. C. Hontelez, J. Horn, M. R. Jepsen, J. J.
Kim, F. Lazzarato, S. M. Matthijsse, R. Mikolajczyk, A. Pavelyev,
M. Pillsbury, L. A. Shafer, S. P. Tully, H. C. Turner, C. Usher, and
C. Walsh, “Population-level impact, herd immunity, and elimination
after human papillomavirus vaccination: a systematic review and
meta-analysis of predictions from transmission-dynamic models,” The
Lancet Public Health, vol. 1, no. 1, pp. e8–e17, nov 2016. [Online].
Available: https://doi.org/10.1016/s2468-2667(16)30001-9
[49] K. T. Simms, J. Steinberg, M. Caruana, M. A. Smith, J.-B.
Lew, I. Soerjomataram, P. E. Castle, F. Bray, and K. Canfell,
“Impact of scaled up human papillomavirus vaccination and cervical
screening and the potential for global elimination of cervical
cancer in 181 countries, 202099: a modelling study,” The Lancet
90
Oncology, vol. 20, no. 3, pp. 394 – 407, 2019. [Online]. Available:
http://www.sciencedirect.com/science/article/pii/S1470204518308362
[50] M. Brisson and M. Drolet, “Global elimination of cervical cancer as a
public health problem,” The Lancet Oncology, vol. 20, no. 3, pp. 319 –
321, 2019. [Online]. Available: http://www.sciencedirect.com/science/
article/pii/S1470204519300725
[51] K. M. Elfstr¨om, F. Lazzarato, S. Franceschi, J. Dillner, and I. Baussano,
“Human papillomavirus vaccination of boys and extended catch-up
vaccination: Effects on the resilience of programs,” Journal of Infectious
Diseases, vol. 213, no. 2, pp. 199–205, jul 2015. [Online]. Available:
https://doi.org/10.1093/infdis/jiv368
[52] I. Baussano, K. M. Elfstr¨om, F. Lazzarato, A. Gillio-Tos, L. D. Marco,
F. Carozzi, A. D. Mistro, J. Dillner, S. Franceschi, and G. Ronco,
“Type-specific human papillomavirus biological features: Validated
model-based estimates,” PLoS ONE, vol. 8, no. 11, p. e81171, nov 2013.
[Online]. Available: https://doi.org/10.1371/journal.pone.0081171
[53] D. C. Beachler, L. A. Pinto, T. J. Kemp, A. G. Nyitray, A. Hildesheim,
R. Viscidi, J. Schussler, A. R. Kreimer, and A. R. Giuliano, “An
examination of HPV16 natural immunity in men who have sex with
men (MSM) in the HPV in men (HIM) study,” Cancer Epidemiology
Biomarkers & Prevention, vol. 27, no. 4, pp. 496–502, Feb. 2018.
[Online]. Available: https://doi.org/10.1158/1055-9965.epi-17-0853
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Article
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The Human papillomaviruses (HPV) vaccine induces a herd immunity effect in genital warts when a large number of the population is vaccinated. This aspect should be taken into account when devising new vaccine strategies, like vaccination at older ages or male vaccination. Therefore, it is important to develop mathematical models with good predictive capacities. We devised a sexual contact network that was calibrated to simulate the Spanish epidemiology of different HPV genotypes. Through this model, we simulated the scenario that occurred in Australia in 2007, where 12–13 year-old girls were vaccinated with a three-dose schedule of a vaccine containing genotypes 6 and 11, which protect against genital warts, and also a catch-up program in women up to 26 years of age. Vaccine coverage were 73 % in girls with three doses and with coverage rates decreasing with age until 52 % for 20–26 year-olds. A fast 59 % reduction in the genital warts diagnoses occurred in the model in the first years after the start of the program, similar to what was described in the literature.
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Background: Modelling studies have been widely used to inform human papillomavirus (HPV) vaccination policy decisions; however, many models exist and it is not known whether they produce consistent predictions of population-level effectiveness and herd effects. We did a systematic review and meta-analysis of model predictions of the long-term population-level effectiveness of vaccination against HPV 16, 18, 6, and 11 infection in women and men, to examine the variability in predicted herd effects, incremental benefit of vaccinating boys, and potential for HPV-vaccine-type elimination. Methods: We searched MEDLINE and Embase for transmission-dynamic modelling studies published between Jan 1, 2009, and April 28, 2015, that predicted the population-level impact of vaccination on HPV 6, 11, 16, and 18 infections in high-income countries. We contacted authors to determine whether they were willing to produce new predictions for standardised scenarios. Strategies investigated were girls-only vaccination and girls and boys vaccination at age 12 years. Base-case vaccine characteristics were 100% efficacy and lifetime protection. We did sensitivity analyses by varying vaccination coverage, vaccine efficacy, and duration of protection. For all scenarios we pooled model predictions of relative reductions in HPV prevalence (RRprev) over time after vaccination and summarised results using the median and 10th and 90th percentiles (80% uncertainty intervals [UI]). Findings: 16 of 19 eligible models from ten high-income countries provided predictions. Under base-case assumptions, 40% vaccination coverage and girls-only vaccination, the RRprev of HPV 16 among women and men was 0·53 (80% UI 0·46–0·68) and 0·36 (0·28–0·61), respectively, after 70 years. With 80% girls-only vaccination coverage, the RRprev of HPV 16 among women and men was 0·93 (0·90–1·00) and 0·83 (0·75–1·00), respectively. Vaccinating boys in addition to girls increased the RRprev of HPV 16 among women and men by 0·18 (0·13–0·32) and 0·35 (0·27–0·39) for 40% coverage, and 0·07 (0·00–0·10) and 0·16 (0·01–0·25) for 80% coverage, respectively. The RRprev were greater for HPV 6, 11, and 18 than for HPV 16 for all scenarios investigated. Finally at 80% coverage, most models predicted that girls and boys vaccination would eliminate HPV 6, 11, 16, and 18, with a median RRprev of 1·00 for women and men for all four HPV types. Variability in pooled findings was low, but increased with lower vaccination coverage and shorter vaccine protection (from lifetime to 20 years). Interpretation: Although HPV models differ in structure, data used for calibration, and settings, our population-level predictions were generally concordant and suggest that strong herd effects are expected from vaccinating girls only, even with coverage as low as 20%. Elimination of HPV 16, 18, 6, and 11 is possible if 80% coverage in girls and boys is reached and if high vaccine efficacy is maintained over time. Funding: Canadian Institutes of Health Research.
Article
Background: Cervical screening and human papillomavirus (HPV) vaccination have been implemented in most high-income countries; however, coverage is low in low-income and middle-income countries (LMICs). In 2018, the Director-General of WHO announced a call to action for the elimination of cervical cancer as a public health problem. WHO has called for global action to scale-up vaccination, screening, and treatment of precancer, early detection and prompt treatment of early invasive cancers, and palliative care. An elimination threshold in terms of cervical cancer incidence has not yet been defined, but an absolute rate of cervical cancer incidence could be chosen for such a threshold. In this study, we aimed to quantify the potential cumulative effect of scaled up global vaccination and screening coverage on the number of cervical cancer cases averted over the 50 years from 2020 to 2069, and to predict outcomes beyond 2070 to identify the earliest years by which cervical cancer rates could drop below two absolute levels that could be considered as possible elimination thresholds-the rare cancer threshold (six new cases per 100 000 women per year, which has been observed in only a few countries), and a lower threshold of four new cases per 100 000 women per year. Methods: In this statistical trends analysis and modelling study, we did a statistical analysis of existing trends in cervical cancer worldwide using high-quality cancer registry data included in the Cancer Incidence in Five Continents series published by the International Agency for Research on Cancer. We then used a comprehensive and extensively validated simulation platform, Policy1-Cervix, to do a dynamic multicohort modelled analysis of the impact of potential scale-up scenarios for cervical cancer prevention, in order to predict the future incidence rates and burden of cervical cancer. Data are presented globally, by Human Development Index (HDI) category, and at the individual country level. Findings: In the absence of further intervention, there would be 44·4 million cervical cancer cases diagnosed globally over the period 2020-69, with almost two-thirds of cases occurring in low-HDI or medium-HDI countries. Rapid vaccination scale-up to 80-100% coverage globally by 2020 with a broad-spectrum HPV vaccine could avert 6·7-7·7 million cases in this period, but more than half of these cases will be averted after 2060. Implementation of HPV-based screening twice per lifetime at age 35 years and 45 years in all LMICs with 70% coverage globally will bring forward the effects of prevention and avert a total of 12·5-13·4 million cases in the next 50 years. Rapid scale-up of combined high-coverage screening and vaccination from 2020 onwards would result in average annual cervical cancer incidence declining to less than six new cases per 100 000 individuals by 2045-49 for very-high-HDI countries, 2055-59 for high-HDI countries, 2065-69 for medium-HDI countries, and 2085-89 for low-HDI countries, and to less than four cases per 100 000 by 2055-59 for very-high-HDI countries, 2065-69 for high-HDI countries, 2070-79 for medium-HDI countries, and 2090-2100 or beyond for low-HDI countries. However, rates of less than four new cases per 100 000 would not be achieved in all individual low-HDI countries by the end of the century. If delivery of vaccination and screening is more gradually scaled up over the period 2020-50 (eg, 20-45% vaccination coverage and 25-70% once-per-lifetime screening coverage by 2030, increasing to 40-90% vaccination coverage and 90% once-per-lifetime screening coverage by 2050, when considered as average coverage rates across HDI categories), end of the century incidence rates will be reduced by a lesser amount. In this scenario, average cervical cancer incidence rates will decline to 0·8 cases per 100 000 for very-high-HDI countries, 1·3 per 100 000 for high-HDI countries, 4·4 per 100 000 for medium-HDI countries, and 14 per 100 000 for low-HDI countries, by the end of the century. Interpretation: More than 44 million women will be diagnosed with cervical cancer in the next 50 years if primary and secondary prevention programmes are not implemented in LMICs. If high coverage vaccination can be implemented quickly, a substantial effect on the burden of disease will be seen after three to four decades, but nearer-term impact will require delivery of cervical screening to older cohorts who will not benefit from HPV vaccination. Widespread coverage of both HPV vaccination and cervical screening from 2020 onwards has the potential to avert up to 12·5-13·4 million cervical cancer cases by 2069, and could achieve average cervical cancer incidence of around four per 100 000 women per year or less, for all country HDI categories, by the end of the century. A draft global strategy to accelerate cervical cancer elimination, with goals and targets for the period 2020-30, will be considered at the World Health Assembly in 2020. The findings presented here have helped inform initial discussions of elimination targets, and ongoing comparative modelling with other groups is supporting the development of the final goals and targets for cervical cancer elimination. Funding: National Health and Medical Research Council (NHMRC) Australia, part-funded via the NHMRC Centre of Excellence for Cervical Cancer Control (C4; APP1135172).
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Background: Evidence suggests that natural antibodies developed after HPV16 infection may protect some women but not men against subsequent HPV16 re-acquisition. Less is known whether antibodies developed following HPV16 infection are protective among men-who-have-sex-with-men (MSM). Methods: 475 MSM from the Human Papillomavirus in Men (HIM) study were tested for serum antibodies to HPV16 L1 using enzyme-linked immunosorbent assays, and for anal and genital HPV16 DNA using PCR consensus primer system (PGMY 09/11). Adjusted Cox regression was used to evaluate whether baseline HPV16 seropositivity impacts subsequent genital or anal HPV16 DNA. Results: The risk of subsequent genital HPV16 (aHR=1.05, 95%CI= 0.66-1.68) and anal HPV16 infections among MSM (aHR=2.34, 95%CI=0.92-5.98) was similar or non-significantly higher in HPV16 seropositive than HPV16 seronegative MSM. The risk of genital HPV16 was also similar between HPV16 seronegative and HPV16 seropositive MSM in the highest tertile of HPV16 antibody levels and when restricting to those with new sex partners during follow-up (p-values>0.20). Among the 118 MSM who were HPV16 seropositive, 90% remained HPV16 seropositive up to 4 years later. When tested together, MSM with the highest antibody titers (top tertile) had similar levels to females (mean=130.3 vs. 134.5 EU/mL, p-value=0.84). Discussion: Despite years of HPV16 seropositivity persistence and antibody titers comparable to females, this study suggested no evidence of HPV16 natural antibodies protecting against subsequent genital or anal HPV16 infection in MSM. Impact: This could help partially explain the high incidence of genital and anal HPV16 infection and related anal cancer seen in middle aged and older MSM.
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Working in large networks applied to epidemiological-type models has led us to design a simple but effective computed distributed environment to perform a large amount of model simulations in a reasonable time in order to study the behavior of these models and to calibrate them. Finding the model parameters that best fit the available data in the designed distributed computing environment becomes a challenge and it is necessary to implement reliable algorithms for model calibration. In this article, we have adapted the random particle swarm optimization algorithm to our distributed computing environment to be applied to the calibration of a papillomavirus transmission dynamics model on a lifetime sexual partners network. And we have obtained a good fitting saving time and calculations compared with the exhaustive searching strategy we have been using so far.