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Fitting the HIV Epidemic in Zambia: A Two-Sex Micro-
Simulation Model
Pauline M. Leclerc
1
, Alan P. Matthews
2
, Michel L. Garenne
1,3
*
1Institut Pasteur, Unite
´d’Epide
´miologie des Maladies Emergentes, Paris, France, 2School of Physics, University of Kwazulu-Natal, Durban, South Africa, 3Institut pour la
Recherche et le De
´veloppement, Paris, France
Abstract
Background:
In describing and understanding how the HIV epidemic spreads in African countries, previous studies have not
taken into account the detailed periods at risk. This study is based on a micro-simulation model (individual-based) of the
spread of the HIV epidemic in the population of Zambia, where women tend to marry early and where divorces are not
frequent. The main target of the model was to fit the HIV seroprevalence profiles by age and sex observed at the
Demographic and Health Survey conducted in 2001.
Methods and Findings:
A two-sex micro-simulation model of HIV transmission was developed. Particular attention was paid
to precise age-specific estimates of exposure to risk through the modelling of the formation and dissolution of relationships:
marriage (stable union), casual partnership, and commercial sex. HIV transmission was exclusively heterosexual for adults or
vertical (mother-to-child) for children. Three stages of HIV infection were taken into account. All parameters were derived
from empirical population-based data. Results show that basic parameters could not explain the dynamics of the HIV
epidemic in Zambia. In order to fit the age and sex patterns, several assumptions were made: differential susceptibility of
young women to HIV infection, differential susceptibility or larger number of encounters for male clients of commercial sex
workers, and higher transmission rate. The model allowed to quantify the role of each type of relationship in HIV
transmission, the proportion of infections occurring at each stage of disease progression, and the net reproduction rate of
the epidemic (R
0
= 1.95).
Conclusions:
The simulation model reproduced the dynamics of the HIV epidemic in Zambia, and fitted the age and sex
pattern of HIV seroprevalence in 2001. The same model could be used to measure the effect of changing behaviour in the
future.
Citation: Leclerc PM, Matthews AP, Garenne ML (2009) Fitting the HIV Epidemic in Zambia: A Two-Sex Micro-Simulation Model. PLoS ONE 4(5): e5439.
doi:10.1371/journal.pone.0005439
Editor: Nitika Pant Pai, McGill University Health Center, Montreal Chest Institute, Canada
Received October 22, 2008; Accepted March 27, 2009; Published May 5, 2009
Copyright: ß2009 Leclerc et al. This is an open-access article distributed under the terms of the Creative Commons Attribution License, which permits
unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited.
Funding: The French Ministry of Higher Education and Research for supporting the work of PL; the Franco-South African Program (Protea) for supporting the
project; SACEMA, South Africa, and the IAS HIV Trust for supporting the work of AM. The funders had no role in study design, data collection and analysis, decision
to publish, or preparation of the manuscript.
Competing Interests: The authors have declared that no competing interests exist.
* E-mail: mgarenne@pasteur.fr
Introduction
The dynamics of HIV epidemics in Africa remain poorly
understood, and virtually no mathematical model has been able to
reproduce them accurately. By the year 2000, after some 20 years
of transmission of the virus, some countries had high or very high
levels of HIV seroprevalence, while others remained with low or
very low levels [1]. For a long time the evidence showing the
differences between countries remained weak and based on biased
and erratic data on HIV seroprevalence among pregnant women.
With the development of HIV testing in the Demographic and
Health Surveys [2] (DHS) and other large-scale seroprevalence
surveys conducted on representative samples of adult populations,
major differences in seroprevalence emerged clearly, ranging for
instance from 0.7% (Senegal, 2005) to 25.9% (Swaziland, 2006).
Despite these large differences in levels, some features seem to
be common to the African epidemics: similar age profiles for
adults, and similar differences between men and women.
Typically, the HIV seroprevalence is very low before sexual
debut, which occurs around age 11 years on the average, rises
quickly with age, up to a peak in the 30’s, then declines less rapidly
with age, the last age available being usually 49 years for women
and 59 years for men in DHS surveys. For women the rise of
seroprevalence by age is sharper than for men, the peak is around
32 years (range among 21 countries: 27–36), the maximum
seroprevalence is about 25% higher than for men (range 0% to
72%), and the decline with age somewhat faster. For men, the rise
is slower; the peak is around age 37 (range 34–41), the maximum
lower or equal, and the right tail longer than for women. As a
result, the lifetime risk of infection is quite similar for both sexes,
and women tend to be infected earlier.
The gap between the age at peak infection of men and women is
similar whatever the level of seroprevalence, with an average of 6
years for the surveys available (range 3 to 9). These common
features of African epidemics are due to the same dominant mode
of HIV transmission for adults: unprotected heterosexual contact
[1,3]. This mode of infection implies a priori that equivalent
numbers of men and women will be infected in the long run,
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because of repeated exposure, the age gap being explained by the
age differences between sexual partners, within and outside
marriage, and by special features of sexual behaviour, in particular
commercial sex work.
One way to better understand the common features of the HIV
epidemics in Africa is to build a mathematical model able to
reproduce the patterns of infection, in particular the age and sex
patterns found in demographic surveys. We showed in an earlier
paper [4] that this was not possible by using compartmental
models, primarily because of the constraints imposed on changing
sexual partners. We propose here a more complex two-sex micro-
simulation model, based on detailed age and sex specific individual
behaviours. The main target of this model is to fit the detailed age
and sex profiles of HIV seroprevalence, and therefore the age gaps
between men and women. Our model differs from previous
models, which rarely use detailed information by age and sex, nor
realistic values of key parameters such as sexual debut, marriage,
divorce or commercial sex. Many other models published in the
literature are compartmental models and have different targets,
such as to account for an overall level of seroprevalence, to
account for the effect of various sexual networks, to evaluate the
effect of age difference between partners, or to model the
population impact of interventions, such as changing number of
partners, improving management of other STIs or mass
circumcision [5–14]. Some of the previous models are closer to
our approach. The work by Anderson and colleagues [9,10]
provided detailed age and sex patterns, but tended to ignore
marriage, and made very strong simplifications on sexual debut.
One of the closest to our model is probably STD-SIM, a micro-
simulation model which is population-based, details sexual
behaviour, and allows for co-infection with other sexually
transmitted diseases [15]. The main differences between our
model and STD-SIM are the more detailed parameterisation of
the risk periods, in particular sexual debut, marriage, and divorces,
and the mode of partnership formation. On the other hand we
ignored the dynamics of co-infection with other STI’s, and our
aim was focused only on one country. Also very close is the recent
micro-simulation model developed at the University of Pau,
France, which attempts to fit the HIV situation in Cameroon [16].
The main difference with this new model is the emphasis on
commercial sex and the lack of precise reference to marriage.
However, the Kamla & Artzouni model is more sophisticated in
the transmission module, because it includes detailed dynamics of
the viral load during infection, and relates transmission to viral
load.
Our model is applied to the case of Zambia. There were several
reasons for choosing this country. The HIV/AIDS epidemic was
early, large, and well documented: the HIV prevalence increased
steadily since 1980, reaching nearly 15% in 2001 [17]. Zambia
was one of the first countries to conduct a detailed DHS in 2001
[17], which included age and sex profiles of HIV prevalence, as
well as most of the variables needed to build the model. Zambia
has also a wealth of detailed and reliable demographic and
epidemiologic data, which can be used for the modelling exercise.
Materials and Methods
Micro-simulation model
The model is a stochastic discrete-event process with a time-step
driven approach, typical of a Markovian process. The model is
structured as three main modules. The ‘‘demographic module’’
includes simple renewal processes: births, ageing and deaths; its
time step is the month. The ‘‘marital module’’ includes the
marriage market (entry and exit from marriage) and the other
types of relationships (within marriage, outside of marriage and
commercial sex); its time step is the week. Sexual behaviour is
modelled within each type of relationship, and its time step is also
the week. The ‘‘epidemiological module’’ includes HIV/AIDS
infections, either by heterosexual transmission among couples or
by vertical transmission from an HIV-positive mother to her
newborn child; its time-step is one week for adult transmission, and
associated with sexual behaviour, whereas the time-step is one
month for vertical transmission, and associated with births.
Disease progression after primo-infection is included in this
module, up to death. The simulations start from a baseline in
1980, when a seed of HIV is introduced, and run for 25 years, the
main target being to fit the situation in 2001. The model is written
in C++, and the code is available from the authors on request. The
essential equations of the model are presented in Appendix S1.
Demographic module
The demographic module determines the population structure,
by age and sex, with a one month time-step for evolution in time.
The baseline population is a stable population generated to fit the
Zambian population in 1980, with baseline values of age-specific
fertility and mortality rates derived from empirical data (DHS
surveys). The associated stable population was computed from
Lotka’s equations and fertility and mortality rates. Fertility and
mortality rates for the non-infected population are assumed to
remain constant over time. The corresponding total fertility rate
(TFR) was 6.12 children per woman, life expectancy was 47.8
years for males and 50.5 years for females (see Appendix S2,
Figure S1, S2). The corresponding intrinsic growth rate was 0.029,
which is close to the empirical growth rate estimated for Zambia
between the 1980 and 2000 censuses (0.028). More details are
available in Leclerc, 2009 [18].
Marital and sexual behaviour module
Three types of relationship were considered, during which HIV
heterosexual transmission could occur: marriages (stable unions),
casual partnerships (short-term relationships within or outside
marriage), and commercial sex. One of the main characteristics of
our model is the detailed process of the marital market. At each
point in time, each person has a strictly defined status, and during
their life course people enter and leave unions, casual partnerships,
or commercial sex. There are several possible statuses, with
transitions between them (Figure 1): ‘‘Virgin’’ (V) represents
people who never had sex; ‘‘Single’’ (S) represents never married
people who are sexually experienced; ‘‘Couple’’ (C) represents
never married people who are in a casual partnership; ‘‘Union’’
(U) represents people who are married (whether first marriage or
remarriage); ‘‘Widowed’’ (W) represents previously-married peo-
ple who are now widowed; ‘‘divorced’’ (D) represents previously-
married people who are now divorced; the two ‘‘in partnership’’
groups (P
W
and P
D
) represent people who are in a casual
partnership and who have been respectively widowed or divorced.
Some married men and married women may have casual
partnerships, up to three concomitant partners. Some unmarried
men and women may also have up to three partners at the same
time. Polygyny is also allowed, with up to three wives per husband.
All transitions are random, and controlled by a set of age and
sex specific parameters. Individuals enter adolescence as ‘‘Virgin’’
and experience sexual debut, either by first marriage or by couple
formation. Transition to first sexual experience follows a
parametric model, called the Picrate model [19], fitted with
DHS data. The Picrate model is a 3-parameter mathematical
function, based on recruitment rates which increase from 0 to a
maximum, following a cumulative Weibull function. This
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parameterization allows one to compute transition rates by week or
month. Transitions to first marriage are given by a Picrate model
[19], also fitted on DHS data. These functions are displayed in
Appendix S2, Figure S3. Casual partnerships end by break-up, at a
constant rate. Marriages end by divorce at a constant rate, or by
mortality of the partner. Remarriage also occurs at a constant rate,
and break as do first marriages. In addition to this main scheme, two
types of multiple partnerships were added: polygamy for men, and
concomitant marital and extra-marital relationships for both sexes.
Partner selection is achieved through a complex algorithm,
designed for fitting both male and female distributions at the same
time. In brief, individuals choose partners from the opposite sex in
the group of people susceptible to the corresponding status
(marriage, casual partnership or commercial sex) depending on an
age preference matrix for each type of relation. These preferences
are initially given by a bivariate gamma distribution fitted on Zambia
DHS-2001 data, specific to the type of relationship, and then fine-
tuned by a marital market algorithm in order to balance supply and
demand of both sexes. The function is displayed in Appendix S2,
Figure S4, S5, S6, S7. Furthermore, the relationship formation rates
for previously-married are attenuated at later ages, since empirical
data show that the frequency of partnership formation declines with
age. For the same reason, the mean number of acts of sexual
intercourse was attenuated at older ages of the male partner.
The frequency of intercourse was set at one or two contacts per
week per relationship, in order to produce 100 contacts a year for a
continuous relationship. This number was independent of the
number of concomitant relationships. This assumption reflects the
fact that persons who are more sexually mobile (more partners) also
tend to have more sex acts per year. The overall number is consistent
with the values found in Zambia: 48 contacts per year while taking
into account the periods without relationship, and consistent with
values found elsewhere, as in the French population [20] and with
values used in other models [7,21]. At each time-step and for each
type of relationship, the number of sexual contacts is calculated and
applied to the ongoing relationship.
In addition to marriages and casual partnerships, the module
allows for commercial sex. Female commercial sex workers are
recruited between age 15 and 49, and retire at 50. Women enter
the CSW market randomly, selected from the unmarried female
15–49 age group in order to represent, at each point of time, 1%
of the unmarried female population aged 15–49 [22]. For males,
being a potential CSW client is determined at birth, and some
30% of males are assumed to become clients during their life. This
number was derived from an analysis of the 2001 DHS survey
[21]. Men who are in this group contact CSW’s randomly after
their first sexual encounter, and the frequency of contacts depends
on their marital status [22]. For more details on this module, see
Leclerc et al., 2008 [18].
All the transition rates, constant or age-specific, were calculated
beforehand from the 2001 Zambia DHS and are summarised in
Table 1.
Epidemiological module
Heterosexual transmission of HIV occurs in one of the three
types of exposure status (marriage, casual partnership or
commercial sex). In the case of sero-discordant couples (one
partner HIV-positive and the other partner HIV-negative), HIV
transmission occurs randomly at each sexual encounter with a
given probability. Contamination is therefore simulated by
computing the probability of infection given the number of sexual
contacts during the at-risk period, that is the duration of the
relationship. The basic male to female transmission per act is
allowed to change with the stage of the infection of the index case,
with age for women, and for the various simulations (see below).
After infection, a person moves through three HIV stages before
dying of AIDS: the ‘‘primary infection’’ which lasts 6 months on
average, the ‘‘latent period’’ which lasts several years on average,
and the final stage ‘‘clinical AIDS’’ which lasts two years on
average. Each transition to the next stage, including death, is
random, and follows a Weibull distribution defined by the
associated median waiting time. Since life expectancy with AIDS
Figure 1. Transition scheme of marital statuses (both sexes).
doi:10.1371/journal.pone.0005439.g001
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decreases with age, the ‘‘latent period’’ is considered to be
variable, from more than ten years for people infected at young
ages, to about 3 years for people infected after age 50 [23].
All these parameters were derived from the published literature.
The duration of primary infection and the stage-specific
transmission probabilities were derived from the Rakai study
[24]. The age-specific male to female transmission was derived
from the Masaka study [25].
Vertical transmission was treated separately. Infected mothers
could transmit HIV to children at a constant rate (30%). HIV-
infected women had reduced fertility, by a constant ratio (30%).
Survival of the HIV-positive new-born children was calculated
independently, and was fitted with a double Weibull distribution,
to match the two forms of the disease: rapid-evolution and slower-
evolution [26], following the recommendations made by UNAIDS
to model child mortality of infected children [27].
Assumptions about heterosexual transmission of HIV
As will be seen below, the basic parameters described above did
not permit the fitting of the empirical data. Therefore, in order to
fit the age patterns of prevalence observed in Zambia in 2001, we
developed various assumptions concerning the heterosexual
transmission of HIV, and in particular: differential susceptibility
of young women, and healthy carriage.
Differential susceptibility of young women is based on an
observation made in the Masaka study [25]. In sero-discordant
couples, the transmission from males to females was higher for
women below age 25 than for women above age 25. We built on
this observation to test the impact of differential susceptibility by
age with our model. The pattern of differential susceptibility by
age is presented below.
The second assumption, called ‘‘healthy carriage’’, was a
speculative hypothesis made earlier in order to reconcile the
incompatibilities between male and female age profiles of HIV
seroprevalence. This hypothesis, developed by MG [28], builds
upon the complex mode of infection of the HIV virus, from
epithelial cells to blood cells, both processes being highly
probabilistic. It assumes that, after exposure to an infected
woman, a man could host for a few days the HIV virus in his
epithelial cells without being fully infected, therefore remaining
seronegative. If such a man had intercourse with a second woman,
not infected with HIV, within a short period of time (about one
Table 1. Main parameters used in the model: values and sources.
Parameters Value Source
Demographic parameters
Males Females
Total fertility rate (age specific fertility rates) 6.12 DHS Zambia 2001 [17]
Sex ratio at birth 1.00 DHS Zambia 1992, 1996, 2001 [17,66,67]
Life expectancy (age specific death rates) 47.8 years 50.5 years Model life table fitted on DHS data
Marital parameters
Median age at first sex 17 years 16 years DHS Zambia 2001 [17]
Median age at first marriage 22 years 17 years DHS Zambia 2001 [17]
Couple formation rate 1.10 DHS Zambia 2001 [17]
Partnership formation rate 1.35 DHS Zambia 2001 [17]
Break-up rate 2.00 DHS Zambia 2001 [17]
Divorce rate 0.015 DHS Zambia 2001 [17]
Remarriage rate 0.254
Proportion of men with two wives 0.15 DHS Zambia 2001 [17]
Proportion of men with three wives 0.03 DHS Zambia 2001 [17]
Extramarital relation rate 0.19 DHS Zambia 2001 [17]
Sexual parameters
Mean number of intercourses by year 100 [7,20,21]
Proportion of CSW’s (clients for men) 0.30 0.01 DHS Zambia 2001 [17,22]
Mean number of visits to CSW per year (unmarried) 4.33* DHS Zambia 2001 [17,22]
Mean number of visits to CSW per year (married) 3.03* DHS Zambia 2001 [17,22]
HIV/AIDS parameters
MTCT transmission 0.30 Dabis et al. [68]
Fertility reduction 0.30 Hunter et al. [69]
Baseline transmission probability per act 0.0007* Wawer et al. [24]
Factor HIV stage 1 11.71 Wawer et al. [24]
Factor HIV stage 2 (ref) 1.0
Factor HIV stage 3 5.0 Wawer et al. [24]
Mean duration of stage 1 0.5 year Mindel et al. [70]
Mean duration of stage 3 2 years Mindel et al. [70]
*
Theses parameters are allowed to change during simulations.
doi:10.1371/journal.pone.0005439.t001
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week), he could theoretically transmit the virus to the second
woman. This could occur typically in case of concomitant
relationships, especially in the case of commercial sex.
Simulation process
Before starting the simulations, the demographic and marital
modules were run several years before the introduction of HIV/
AIDS in the population. Then, in 1980, the assumed date of the
first HIV cases in Zambia, the virus is introduced in the
population, by infecting 1% of the 15–49 age group. The
dynamics of the epidemic are then simulated year by year, until
2001. The population characteristics and the infections are
monitored over time and stored after each year, so that all details
can be retrieved at any point in time.
Several parameters were allowed to change in order to fit the
prevalence in 2001. (1) probability of transmission per sex-act; (2)
differential susceptibility of women; (3) number of visits to
commercial sex workers for unmarried and married men or
higher transmission for clients of sex workers; (4) healthy carriage.
The simulations explored the realistic combinations of these
parameters in their ability to fit the observed patterns in 2001.
Above all, we used the flexibility of the HIV transmission
probability to fit the level of prevalence in the population, the
other parameters being used for fitting the age and sex profiles.
For precise fitting of the overall level of seroprevalence, we used
Brent’s method and the Golden Section search procedure.
Because results of our simulations depended on a large number
of parameters, we had to set limits on the values to be taken by
varying parameters, called ‘‘realistic values’’. The heterosexual
probability per sex-act in stage 2 was allowed to vary between
0.0007 (value from the Rakai study) and 0.0050; the annual
number of visits to commercial sex workers was allowed to vary
between 3 and 12; healthy carriage was assumed to vary from 0 to
1, that is the probability to become HIV healthy carrier during
one week by contact with an HIV-positive female partner.
Different scenarios for differential susceptibility were tested,
allowing the female risk of acquiring HIV to be multiplied by a
factor between 1 and 5 depending on the age-group.
The empirical age and sex seroprevalence patterns derived from
the 2001 DHS survey were affected by random fluctuations. The
profiles were therefore fitted with a polynomial on the logit of the
seroprevalence: female HIV prevalence peaked at age 31, with
25.7% of infected women, and male prevalence peaked at age 36
with 20.6% of infected men, that is a ratio of female maximal
prevalence to male maximal prevalence of 1.25, all values quite
typical of African HIV epidemics. These were the main criteria
that we used to test the reliability of our simulations, in addition to
plotting the corresponding figures.
Finally, most of the events occurring during simulations being
stochastic, we used a large sample size (675,000 persons at
baseline, or about 12% of the total population) in order to reduce
variability in the results of simulations. This variability remained
small, especially with this large population size and given the high
prevalence levels seen in Zambia.
Results
Table 1 gives the baseline parameters used in our simulations.
With these values, and without adding any of the alternative
hypotheses, no epidemic could be generated: the HIV prevalence
decreased soon after the introduction of the virus in the
population, leading to extinction.
Changing the transmission probability allowed an epidemic, but
the final age and sex profiles were far from those expected. With a
transmission probability equal to 0.004379, overall prevalence rates
at age 15–49 were close to what was expected : 16.5% and 12.8% for
female and male groups respectively (as compared with 16.6% and
12.0% respectively in the DHS) but peaks were reached at ages 28
for female and 32 for males, much earlier than expected (31 and 36
years respectively). Moreover, assuming the same value of
transmission for males and females, the peak prevalence ratio (F/
M) was 1.13, lower than the value in the DHS (1.25).
Simulation H0
After exploring a wide range of parameters, we found a
combination giving a good fit to the age and sex patterns of
prevalence in 2001. This simulation is labelled ‘‘H0’’ in this paper,
and defined as follows:
– a baseline (stage 2) transmission probability from woman to
man of 0.002479;
– a differential susceptibility for young women equal to 1 for
women aged more than 40 years, 1.5 times higher for woman
aged between 30 and 40 years, and 2.5 times higher for woman
aged less than 30 years;
– a four-times higher annual number of visits to commercial sex
workers for eligible married men (12 visits), or a four-times
higher transmission rate in case of contact with a CSW.
The age patterns of prevalence in 2001, simulated and
observed, are displayed in Figure 2. The fit is of good quality,
even though it was difficult to obtain: the peak prevalence (25.8%)
for women occurs at age 31, and the peak prevalence for men
(20.5%) occurs at age 35. The peak prevalence ratio (F/M) equals
1.26, and the overall prevalence rates are 17.5% for women aged
15–49 and 11.5% for men aged 15–49 (as compared with 16.6%
and 12.0% respectively in the DHS). More important, the age
patterns obtained by the simulations were close to those found in
the DHS survey. The hypotheses underlying this simulation
remained within the range of acceptable values. The transmission
probability was 2.5 times that found in discordant couples in
Uganda, a value usually considered too low because of a selection
bias (couples who have a lower transmission rate are more likely to
be discordant). The pattern of differential susceptibility was close
to that found in Masaka. However, the annual number of visits to
CSW’s may seem unrealistic, since it is four times higher than that
found in surveys, and therefore assumes a large understatement,
but the same results could be obtained with four-times higher
transmission rates, which is consistent with the likely presence of
co-infection with STI’s.
The model allowed us to disentangle the modalities of the
transmission, in particular the type of relationship, the age at
infection and the stage of the disease at time of infection. For men,
a large proportion of infections resulted from contacts with female
sex workers (47.2%), followed by contacts during casual partner-
ships (30.3%), and contacts within marriage (22.5%). For married
men, 58.9% of infections resulted from contacts with commercial
sex workers, whereas this proportion was only 28.6% for
unmarried men. For women, and because they marry early, a
majority of infections occurred within marriage (62.5%), followed
by casual partnerships (34.9%), commercial sex accounting for a
tiny proportion (2.6%), because CSW’s account for only 1% of the
population. It should be noted however that after 21 years some
90% of CSW’s were infected. Moreover, 66.4% of women infected
within short-term relationships were infected before their first
marriage, which accounts for 22.2% of infections. For men, 75.1%
of male infections within short-term relationships occurred before
their first marriage, and account for 20.7% of infections.
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The difference in age at infection stems from the age
differences of partners by type of relationship involved. For
infections occurring within marriage, the mean age at infection
for men was 36.7 years (IQR = 27 to 45), and for women 27.9
years (IQR = 20 to 33). In contrast, for infections occurring
during casual partnerships, ages were younger and the age
difference was smaller: 28.1 years for men (IQR = 20 to 33), and
26.6 years for women (IQR = 17 to 33). For commercial sex, the
mean age at infection was 33.3 (IQR = 24 to 41) for men, and
26.4 for women. (IQR = 18 to 35). It is therefore primarily the
difference in age at marriage that explains the overall age
difference at infection.
With respect to the stage of infection, a majority of male
infections occurred with a partner in stage 2 (Table 2). Indeed, a
majority of infections occurred with a CSW, which explains the
large number of infections in stage 2, since CSW’s are infected in
large numbers and at an early age and therefore are in stage 2 for
a large part of their professional lives. In contrast, women get
infected mainly by partners in stage 1, because of the high risk
associated with this stage during stable relationships. For short-
term relationships, female infections occur more often with
partners in stage 2, because of the longer duration of this stage.
Changing pattern of transmission over time. The
proportion of male infections from a CSW varies with the
duration since the onset of the epidemic. Before 1985, male
infections due to commercial sex account for 73.1% of the total,
whereas after 2000 they account for only 36.4%. As many clients
of commercial sex workers are married, female infections
Figure 2. Age patterns of HIV prevalence in 2001 in Zambia, observed (from DHS data) and simulated (from simulation H0) (women
in red and men in blue).
doi:10.1371/journal.pone.0005439.g002
Table 2. Proportion of infections occurring at each stage of the disease in the partner, by type of relationship, and mean age at
infection, by sex and type of relationship (H0).
Women Men
Marriages Casual partnerships CSW Total Marriages Casual partnerships CSW Total
Proportions by stage
Stage 1 36.9% 10.8% 1.2% 48.9% 3.1% 9.0% 16.1% 28.2%
Stage 2 19.0% 18.1% 0.7% 37.8% 15.9% 16.7% 21.0% 53.6%
Stage 3 6.6% 6.0% 0.7% 13.3% 3.5% 4.6% 10.1% 18.2%
Total 62.5% 34.9% 2.6% 100% 22.5% 30.3% 47.2% 100%
Mean age at infection by stage
Stage 1 29.1 23.0 28.7 27.7 35.2 22.8 33.9 30.5
Stage 2 23.0 23.8 25.0 23.4 34.3 27.4 33.1 31.7
Stage 3 35.1 41.2 23.9 37.2 48.8 40.8 32.9 38.0
Total 27.9 26.6 26.4 27.4 36.7 28.1 33.3 32.5
doi:10.1371/journal.pone.0005439.t002
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occurring within marriage also decrease, from 70.0% of female
infections before 1985 to 60.3% of female infections after 2000.
Net reproduction rate (R
0
). The net reproduction rate (R
0
)
of the epidemic was calculated by computing the secondary attack
rate by year since infection and multiplying by the survivorship of
the index cases by year. Results give estimates of R
0
equal to 1.95
(2.28 for female to male infections, and 1.61 for male to female
infections), which was close to the empirical R
0
, defined as the
ratios of simulated new infections from 1985 and 2005 to the
infections that occurred from 1980 to 1984 (27,650 men and
17,607 women infected from 12,311 women and 11,585 men, or
R
0
= 1.89). This value matches quite well what is known of the
dynamics of the HIV infection in Zambia over the period, that is
an increase from a low value (about 1%) in 1980 to a high 15%
HIV seroprevalence in 2001.
Other simulations: impact of changing parameters
In this part, we investigate the effect of changing critical
parameters around H0: heterosexual transmission probability,
differential susceptibility, number of visits to CSW’s, and by
introducing healthy carriage. Table 3 summarises the various
assumptions made with their main results.
Removing differential susceptibility (H1). Removing the
differential susceptibility of young women induces lower
prevalence for both sexes, and higher mean age at infection,
especially for women (mean = 29.8 versus 27.4 in the previous
simulation). More female infections occur within marriage than
previously (71.0% versus 62.5%), and fewer infections occur
during short-term relationships (25.1% versus 34.9%). For men,
more infections occur while visiting a CSW (59.3% versus 47.2%).
Concerning ages at infection, the main difference is observed
within casual partnerships. Female mean age at infection within
casual partnerships is now 30.7 years (versus 26.6 under H0),
because they occur more often after the breaking of the first
marriage. In order to fit the levels of prevalence after removing the
differential susceptibility for women (H19), a higher transmission
probability by sex-act of 0.003194 is needed (29% higher than
H0). With such a transmission probability, female maximal
prevalence equals 25.8%, close to what we expected, but male
maximal prevalence now equals 21.9% with the result that the
ratio of female to male maximal prevalence becomes 1.18
(Table 3), which is lower than what is found in the DHS.
Removing differential susceptibility has therefore major
shortcomings for the quality of the fit, because too many
infections occurred within the male group.
Changing the number of visits to CSW for married men
(H2). In this simulation, the mean number of visits to CSW for
married men is changed back to its original value (3.03 visits a
year). This implies that fewer male infections occur through client-
CSW relationships (34.5% versus 47.2% previously) and, as a
result, that fewer female infections occur within marriage (54.4%
versus 62.5%). Then, because sexual activity for the high risk
group of married men clients of CSWs is reduced, fewer infections
occur at older ages for men as well as for women. As a result, the
mean ages at infection are younger than previously (31.5 years for
men, 26.8 years for women), primarily because late infections no
longer occur (Table 3). Peaks of prevalence are reached 2 or 3
years before those obtained under H0 (28 years for women and 33
years for men). The main difference in terms of stage of the disease
is observed for married women: the proportion occurring in stage
1 falls to 42.2%, whereas the proportion of infections in stage 2
increases to 43.9%. In order to fit the correct levels of prevalence
(H29), the transmission probability by sex-act should be 0.003279
(32% higher than H0). Under this new assumption, HIV
prevalence peaks at 27 for women and 32 for men, and the age
patterns no longer fit the DHS data.
Assumption of healthy carriage (H3). The assumption of
healthy carriage was added this way: all men are susceptible to be
a healthy carrier and, for each contact with an HIV-positive
partner, they have a probability of 30% to carry the virus during
one week without getting infected. After that, if they have contact
with other HIV-negative women, during this same week, they
could transmit the virus to them with the same transmission
probabilities as if they were really infected. The choice of 30% is
the result of several simulations and will be explained below.
Adding the healthy carriage hypothesis to H0 implies higher
prevalence rates for both sexes with a maximum prevalence equal
to 41.9% for female and 33.8% for male (Table 3). The overall
prevalence for men and women aged 15–49 were respectively
18.7% and 28.7% in 2001. As women become more susceptible
because of healthy carriage, age at maximum prevalence is
younger than under H0 (29 years old versus 31). This is the result
of a lower mean age at infection for females (26.0 years). As
previously, a majority of female infections occur within marriage
Table 3. Assumptions made to test the impact of key parameters and their results on key indicators after simulations.
Hypothesis for simulations Key indicators from simulations
(Changes from H0) Age at peak
Prevalence at peak
(%) Maximal prevalence
ratio F:M
15–49 years-old
prevalence (%)
Women Men Women Men Women Men
DHS Survey values 31 36 25.7 20.6 1.25 16.6 12.0
H0 Realistic simulation 31 35 25.8 20.5 1.26 17.5 11.5
H1 No differential susceptibility 34 35 18.8 15.6 1.21 11.6 8.4
H19H1+Pt = 0.003194 32 34 25.8 21.9 1.18 16.7 12.1
H2 Mean number of visits to CSW for married men = 3.03 28 33 13.8 10.4 1.33 9.4 6.3
H29H2+Pt = 0.003279 27 32 26.1 21.9 1.19 17.9 12.6
H3 Healthy carriage 29 34 41.9 33.8 1.24 28.7 18.7
H39H4+Pt = 0.002459 with no differential susceptibility 33 35 25.9 21.3 1.22 16.5 11.4
Note: Hn9are made to fit the overall seroprevalence with other parameters equal to Hn.
Pt = probability of HIV transmission per sex-act.
doi:10.1371/journal.pone.0005439.t003
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(63.1%), then within short-term relationships (32.4%) and a small
proportion within the commercial sex market (1.6%). Overall,
28.1% of female infections are due to healthy carriage, especially
within marriage (35.4%), then within short-term relationships
(16.1%). Few female infections attributable to healthy carriage
occur within commercial sex (5.6%). For men, the main difference
is that they are now as infected in stage 1 (41.9%) as in stage 2 of
the disease (45.9%). The proportion of male infections occurring
within marriage is twice that under H0 (43.4% versus 22.5%
under H0). In summary, men visit commercial sex workers and
become healthy carriers; returning home they infect their wives,
but get infected soon after that because of the very infective stage 1
of their wives. In order to fit the correct levels of prevalence for
both sexes (H39), we had to remove differential susceptibility.
Indeed, it was impossible to fit levels of prevalence for both sexes
by combining differential susceptibility and healthy carriage.
Under this assumption, we need a transmission probability by sex-
act of 0.002459, which is similar to the one used under H0.
Prevalence rates peak at ages 33 and 35, and the age patterns no
longer fit the DHS data.
Discussion
To our knowledge, no other model has tried to fit the HIV
seroprevalence age patterns for both sexes simultaneously, while
taking into account the detailed periods at risk and fitting precisely
entry into sexual life, entry into first marriage, and marriage
dissolutions as well as re-marriage. To give a simple example, most
models assume that all adults enter their sexual life (or first
marriage) at exactly age 15, whereas in our model men and
women may enter sexual life at any age between 10 and 30, as they
do in real life. This is obviously very important to enable proper
fitting of the age and sex pattern of infection.
Our simulation exercise aimed at being as realistic as possible,
and used as much as possible empirical and detailed age-specific
values of the main parameters controlling couple formation and
transmission of the virus. Above all, it shows the very heavy
constraints for fitting properly the observed data on seroprevalence
by age and sex. Changing one parameter has an impact on the
whole transmission process, and when it affects directly one sex, it
also affects as a consequence the other sex, changing therefore the
dynamics of the epidemic and the age and sex profiles. Our
reference simulation (H0) was obtained after more than one
hundred simulation trials, all the others leading to inconsistent
patterns. Even if H0 could be criticised, it has the main advantage
of reproducing the pattern observed in the Zambian population
and therefore providing a plausible scenario.
Among the main constraints found in the simulation was the age
at peak infection for males. It was almost impossible to reach
values greater than 35 or 36 years for men while keeping the main
parameters within a range of realistic values. This point definitely
deserves further research, but this observation seems to match
observations in empirical data throughout Africa.
A nice feature of this model is that it disentangles the
transmission process. The way the disease is transmitted appears
complex, because it involves differently the various types of union
formation, and the various stages of transmission. The role of each
factor evolves over time, and is sensitive to changing any of the
parameters. This is probably why we received conflicting evidence
from field surveys conducted over the past 20 years in many
African countries. For example, some authors found a correlation
between the number of CSWs and HIV prevalence levels across
African countries [29], whereas the 4-city study concluded that sex
work could not explain the differential spread of HIV among the
four cities [30], even though authors acknowledged that it could
have played a major role at the onset of the epidemic.
Despite its nice explanatory power, our model has a number of
limitations, firstly the values of its parameters. The heterosexual
transmission probability is one of the parameters most open to
criticism. The Rakai study gives a baseline value of 0.0007 per act
[24] (prevalent cases group, stage 2), and an overall transmission
parameter of 0.0011 [31] (prevalent and incident cases together),
which is about half the values selected for H0. However, our value
does not seem too unrealistic, and compares with that selected by
other authors [32–36].
Differential susceptibility of men and women remains a matter
of controversy. Some studies found that women are twice as
susceptible as men [25,32], whereas other studies found no
significant difference of transmission between the two sexes
[31,37,38]. Note that in some studies the differences between
male to female and female to male transmission is hampered by
male circumcision. It is striking to note that in Europe as in
Uganda, where male circumcision is rare, transmission is the same
either way. This is why we chose the same value of transmission
for males and females for older ages, since circumcision is rare in
Zambia. Differential susceptibility data were derived from a study
in Uganda. Fortunately, the situation in Zambia is quite similar to
the situation in Southern Uganda, with little circumcision, same
religion (Christian), and roughly the same level of economic
development.
Our assumption of a differential susceptibility by age among
women was necessary to fit the observed patterns. It has however
an impact on the overall susceptibility of women. Assuming that
women have intercourse between age 16 years (median age at first
sex) and age 50 years (end of reproductive life), with an average
frequency of 80 sex acts per year, and assuming a differential
susceptibility as assumed in H0, a simple calculation gives an
increased risk for all women of 1.74, which matches other
observations in the literature, and the assumptions made in other
modelling projects [39,40].
Differential susceptibility induces more infection at young ages
for females and results in a ratio of maximal prevalence (female/
male) close to that observed in DHS surveys (1.25) [17]. The
assumption of differential susceptibility is supported by studies
which found that age might be an important co-factor of HIV
infection for women [41], which might be a biological effect.
Indeed, the vaginal epithelium of adolescent and young women is
thinner than at older ages. In animal models, age was found as a
factor of thickness and integrity of the vaginal epithelium [42].
Removing differential susceptibility and adjusting transmission
probability leads to a ratio of 1.18. So, discrepancies in gender
prevalence are in part explained by the sexual network, but not
sufficiently to explain all the differences. To reach such prevalence
differences between the two sexes, women have to be more
susceptible than men.
Our mean number of partners was derived from DHS data,
after a detailed analysis by age and sex. It is somewhat lower than
the number used in other models, but we feel that it is realistic for
the Zambian situation.
The mean number of sex acts by year for a steady relationship
was set to 100 which is a little higher but remained consistent with
the values used in other models [7,21]. We also included a decline
of this mean with age, corresponding to a lower sexual activity for
older age-groups [43]. Considering a lower number of annual sex
acts would simply imply higher transmission per act in our model,
but will not change very much the age and sex patterns.
In our simulations, a large proportion of female infections occur
within marriage, because of extra-marital relationships of men,
Fitting Zambia HIV Epidemic
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including commercial sex. This conclusion is supported by
epidemiological [44–47] and serological [48] studies. Lurie et al.
[44] found that in 71.4% of discordant couples the male was the
infected partner and that he was infected mostly outside his regular
relationships. Another study concluded that men were four times
more likely than women to introduce HIV infection in
concordant-negative couples [45]. Moreover, Glynn et al. [49]
estimated that at least 25% of the infections in recently married
men were acquired from outside the marriage, by extra-marital
partnerships. This is mainly explained by gender differences in
sexual behaviour, as men more frequently engage in extra-marital
relationships, including commercial sex.
Age at first marriage has been shown to be an important factor
of HIV prevalence at country level, also demonstrated with its
correlate, the prevalence of premarital fertility [50,51]. Late
marriage for women implies long periods of premarital sexual
activity during which the rate of partner change can be high,
which facilitates the spread of the virus. The case of Zambia is
interesting and peculiar because this country has a high level of
HIV prevalence despite low median age at first marriage. As a
result, most infections occur after first marriage, a situation
different from other countries in Southern Africa, where infections
mainly occur in the premarital period.
Because the stage of primo-infection is very infectious, women
with an infected husband tend to become infected soon after their
spouse. In simulation H0, 59.3% of female infections occurring
through marriage occur during this 6-month period of primo-
infection.
Commercial sex was shown to play a major role in the spread of
HIV in the first years of the epidemic in Africa [52–55]. In
generalized epidemics, however, this role seems less important.
The 4-city study conducted in the late 1990’s concluded that
commercial sex could not explain differential prevalence within
the sites [30]. We showed from five case studies in the early 2000s
that the role of female sex workers seems limited, and accounts for
only 1.3 to 9.4% of infections in the general population, although
Zambia was not included in this sample [56]. To a certain extent,
results of our simulation H0 reflect this fact. At the onset of the
epidemic, the proportion of male infections occurring during
contact with CSW’s is high (73.1% before 1985), then it decreases
to reach 36.4% after year 2000. Moreover, using our first
estimation of commercial sex, and after adjustment of the
transmission probability (H29), some 39.8% of male infections
are due to commercial sex before 1985, whereas this proportion
falls to 21.8% after year 2000. As a matter of fact, the proportion
of female infections within marriage decreases from 54.8% before
1985 to 50.2% after 2000. It still remains high compared to our
estimate, but reflects, to a certain extent, the fact that the epidemic
in Zambia is now generalised to the whole population and not
restricted to some high-risk groups.
Our model had several other limits, beyond the values of its
parameters. Firstly, heterogeneity in sexual behaviour is repre-
sented only by behaviour associated with marital status, and by
random effects. There is evidence of more complex heterogeneity,
and in particular by more complex ‘‘assortative mixing’’, that is a
preference from both sexes to form sexual partnerships with
persons with similar behaviours (either with low or with high
number of partners). This is only partly taken into account in our
model. Also, we did not take into account preferences for CSW’s,
that is the fact that some men tend to have relations preferentially
with the same person for a long period of time. We also assumed
independence between many parameters, such as divorce rates
and sexual behaviour, which may differ from real life situations.
Some of the parameters were taken as constant, when in reality
they are dependent on some other factors. We took a weekly step
for the epidemiological module, which implies some constraints in
terms of sexual contacts and concomitant partnerships.
We also ignored deliberately other factors of HIV transmission,
such as herpes or other STIs, which would require much more
complex modelling, and has already been treated elsewhere [8,57].
The average effect of STI’s can be considered to be somehow
included in the average transmission rate, and in the excess risk for
intercourse with a CSW.
We also ignored the spatial dimension of disease spread, in
particular the role of migration, and the differential behaviour in
urban and rural areas. These other factors need a separate
treatment, and indeed other types of models. The role of the
mines, as they attract young unmarried adults and favour
commercial sex, is also of concern for Zambia.
We also ignored other routes of transmission, in particular
iatrogenic transmission (blood transfusion, unsafe injections or
medical practices), sometimes considered to be important [58]. We
acknowledge that they may exist, but we thought that they were
unlikely to explain the age and sex patterns of HIV infections
which was our main target [59].
Our study is primarily heuristic, and aims at explaining a
common feature of African epidemics: the age and sex patterns of
seroprevalence in the adult population. It may also have some
policy implications. By better understanding the likely routes of
transmission, one might better target prevention policies. Of
course, our findings are country and period specific, and cannot be
easily extrapolated to other situations, unless more simulations are
run with different parameters. However, they reveal at least two
major target groups: sex workers and their clients, and newly
married women. If the first group has been the target of many
interventions [60–63], the second group has largely been ignored
as a potentially high risk group [64]. Prevention programs among
this group are needed, and this conclusion is supported by
epidemiological studies [45].
Our model can also be used for measuring the effect of changing
behaviour. After fitting the 2001 situation, one could extrapolate
the trends to the next 5 or 10 years. This exercise shows that in
2006, one expects a rising epidemic, with 19.4% of infected
women aged 15–49, and 12.4% of men in the same age group;
corresponding figures for 2011 are: 20.7% for women and 13.1%
for men, with maximal prevalence of 30% for females and 24% for
males. On the contrary, seroprevalence seems to have been
levelling off and even going down in Zambia over the past years,
according to sentinel sites [65]. This tends to indicate that
prevention efforts have been successful, and that the course of the
epidemic has been curbed by changing behaviour, whether by
reducing the number of partners or by using condoms.
Much remains to be explored to better understand the dynamics
of HIV epidemics in Africa, and their wide diversity. In particular,
may such micro-simulation models help explain the differences in
prevalence levels and in age and sex patterns seen over the
continent? This remains to be seen by applying the model to other
situations.
Supporting Information
Appendix S1
Found at: doi:10.1371/journal.pone.0005439.s001 (0.12 MB
DOC)
Appendix S2
Found at: doi:10.1371/journal.pone.0005439.s002 (0.32 MB
DOC)
Fitting Zambia HIV Epidemic
PLoS ONE | www.plosone.org 9 May 2009 | Volume 4 | Issue 5 | e5439
Figure S1 Age pattern of fertility, Zambia.
Found at: doi:10.1371/journal.pone.0005439.s003 (0.11 MB TIF)
Figure S2 Age pattern of mortality, without HIV/AIDS,
Zambia.
Found at: doi:10.1371/journal.pone.0005439.s004 (0.12 MB TIF)
Figure S3 Proportions ever married, and ever had intercourse,
after from fitting with the Picrate model.
Found at: doi:10.1371/journal.pone.0005439.s005 (0.14 MB TIF)
Figure S4 Bivariate gamma distribution of age of husband and
wife, first marriage.
Found at: doi:10.1371/journal.pone.0005439.s006 (0.21 MB TIF)
Figure S5 Bivariate gamma distribution of age of husband and
wife, remarriage.
Found at: doi:10.1371/journal.pone.0005439.s007 (0.22 MB TIF)
Figure S6 Bivariate gamma distribution of age of partners,
premarital relationship.
Found at: doi:10.1371/journal.pone.0005439.s008 (0.20 MB TIF)
Figure S7 Bivariate gamma distribution of age of partners,
extra- or post-marital relationship.
Found at: doi:10.1371/journal.pone.0005439.s009 (0.22 MB TIF)
Author Contributions
Conceived and designed the experiments: PML AM MLG. Analyzed the
data: PML AM MLG. Wrote the paper: PML AM MLG. Initiated the
study: MLG. Contributed to the data analysis and to the writing: MLG.
Contributed to the design: PML, AM. Carried out all the simulations:
PML. Wrote the first draft: PML. Wrote the computer code for the
program: AM. Contributed to the writing: AM.
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