A mathematical model of the dynamics of lymphatic filariasis in Caraga Region, the Philippines

Abstract

Despite being one of the first countries to implement mass drug administration (MDA) for elimination of lymphatic filariasis (LF) in 2001 after a pilot study in 2000, the Philippines is yet to eliminate the disease as a public health problem with 6 out of the 46 endemic provinces still implementing MDA for LF as of 2018. In this work, we propose a mathematical model of the transmission dynamics of LF in the Philippines and a control strategy for its elimination using MDA. Sensitivity analysis using the Latin hypercube sampling and partial rank correlation coefficient methods suggests that the infected human population is most sensitive to the treatment parameters. Using the available LF data in Caraga Region from the Philippine Department of Health, we estimate the treatment rates r 1 and r 2 using the least-squares parameter estimation technique. Parameter bootstrapping showed small variability in the parameter estimates. Finally, we apply optimal control theory with the objective of minimizing the infected human population and the corresponding implementation cost of MDA, using the treatment coverage γ as the control parameter. Simulation results highlight the importance of maintaining a high MDA coverage per year to effectively minimize the infected population by the year 2030.

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royalsocietypublishing.org/journal/rsos
Research
Cite this article: Salonga PKN, Mendoza VMP,
Mendoza RG, Belizario Jr VY. 2021 A
mathematical model of the dynamics of
lymphatic filariasis in Caraga Region, the
Philippines. R. Soc. Open Sci. 8: 201965.
https://doi.org/10.1098/rsos.201965
Received: 31 October 2020
Accepted: 7 June 2021
Subject Areas:
health and disease and epidemiology/differential
equations/mathematical modelling
Keywords:
lymphatic filariasis, mathematical modelling,
sensitivity analysis, parameter estimation,
optimal control, Caraga Region, the Philippines
Author for correspondence:
Pamela Kim N. Salonga
e-mails: pksalonga@math.upd.edu.ph,
pnsalonga@up.edu.ph
A mathematical model of
the dynamics of lymphatic
filariasis in Caraga Region,
the Philippines
Pamela Kim N. Salonga
1,2
, Victoria May P. Mendoza
1,2
,
Renier G. Mendoza
1,2
and Vicente Y. Belizario Jr
3
1
Institute of Mathematics, and
2
Natural Sciences Research Institute, University of the
Philippines Diliman, Quezon City, Philippines
3
College of Public Health and Neglected Tropical Diseases Study Group, National Institutes of
Health, University of the Philippines Manila, Philippines
PKNS, 0000-0002-5522-1658; VMPM, 0000-0003-0953-9822;
RGM, 0000-0003-3507-0327
Despite being one of the first countries to implement mass drug
administration (MDA) for elimination of lymphatic filariasis (LF)
in 2001 after a pilot study in 2000, the Philippines is yet to
eliminate the disease as a public health problem with 6 out of
the 46 endemic provinces still implementing MDA for LF as of
2018. In this work, we propose a mathematical model of the
transmission dynamics of LF in the Philippines and a control
strategy for its elimination using MDA. Sensitivity analysis
using the Latin hypercube sampling and partial rank correlation
coefficient methods suggests that the infected human population
is most sensitive to the treatment parameters. Using the
available LF data in Caraga Region from the Philippine
Department of Health, we estimate the treatment rates r
1
and r
2
using the least-squares parameter estimation technique.
Parameter bootstrapping showed small variability in the
parameter estimates. Finally, we apply optimal control theory
with the objective of minimizing the infected human population
and the corresponding implementation cost of MDA, using the
treatment coverage γas the control parameter. Simulation results
highlight the importance of maintaining a high MDA coverage
per year to effectively minimize the infected population by the
year 2030.
1. Introduction
As a tropical country, the Philippines is endemic to several
neglected tropical diseases (NTDs) such as lymphatic filariasis
(LF), schistosomiasis, soil-transmitted helminths, food-borne
© 2021 The Authors. Published by the Royal Society under the terms of the Creative
Commons Attribution License http://creativecommons.org/licenses/by/4.0/, which permits
unrestricted use, provided the original author and source are credited.

trematodiases, rabies and leprosy [1]. These diseases are known to affect the most neglected fraction of
society—the people living in poverty in remote areas with little to no access to clean water, quality
education and proper sanitation [2].
Although no individual NTD demands a global priority in terms of burden of disease due to
disability and mortality, collectively, the disability-adjusted life years associated with NTDs is
tantamount to that of HIV/AIDS, tuberculosis or malaria [3]. At present, NTDs threaten more than
one billion people in tropical and subtropical countries worldwide [2].
LF is a parasitic disease caused by the filarial roundworms Wuchereria bancrofti,Brugia malayi and Brugia
timori which are transmitted to humans by mosquitoes of the genera Aedes,Anopheles,Culex and Mansonia.
These parasites are known to target the human lymphatic system causing different degrees of clinical signs
and symptoms. While majority of the infected population are asymptomatic, many develop acute clinical
disease that commonly manifests as episodic occurrence of painful inflammation of the lymph nodes and
lymphatic vessels. These episodes are usually accompanied by fever, malaise, chills and headache, and are a
sign of the presence of immature larvae called microfilariae (Mf ) in the lymphatics. Without proper care and
treatment, these periodic acute manifestations can develop into chronic disease characterized by the
abnormal enlargement or swelling of body parts, usually the limbs and genitalia, due to chronic lymphatic
obstruction, lymph fluid accumulation, tissue swelling and later, skin thickening. This explains why LF is
commonly known as elephantiasis [4]. Although not fatal, the acute and chronic manifestations of LF can
significantly diminish the quality of life of affected individuals. The debilitating and disabling symptoms
associated with LF reduce the economic productivity of an infected person and their caretakers which further
contributes to poverty. This, together with the social stigma caused by the disfiguring manifestations of the
disease, threaten the psychosocial health of affected individuals forcing them into isolation and depression [5].
In 2000, the World Health Organization (WHO) launched the Global Programme to Eliminate
Lymphatic Filariasis (GPELF) as a response to the World Health Assembly’s resolution to eliminate LF
as a public health problem. The GPELF endorses a two-point strategy in eliminating lymphatic filariasis:
interrupting transmission through mass drug administration (MDA) and controlling morbidity [6]. The
objective of MDA is to reduce microfilaremia prevalence in infected individuals to levels where infection
is deemed to be intransmissible by delivering a two-drug combination of antifilarial drugs (albendazole
(ALB) plus diethylcarbamazine citrate (DEC) or ivermectin) once a year to eligible people in all
established endemic areas for at least 5 years. If the Mf prevalence in an area reaches elimination levels,
MDA is stopped. The area then goes under surveillance to monitor infection levels for several years. If
low infection levels are consistently maintained for 5 years, then the area is declared free from LF [7].
The general impact of MDA in eliminating LF is heavily dependent on the treatment coverage or the
proportion of the total population in an endemic area that ingested the antifilarial drugs [8]. As per WHO
guidelines [9], a minimum target coverage of 65% of the total population must be achieved per round of
implementation for MDA to be effective. Reports from countries that successfully eliminated the disease
highlight the importance of high MDA coverage along with the commitment and dedication of the
government and health workers, and the active participation of the community in achieving
elimination of LF [10–14].
The WHO also acknowledges the impact of vectorcontrol strategies in interrupting disease transmission
by reducing host–vector contacts, and promotes integrated vector management against many vector-borne
diseases such as malaria, dengue and LF [15]. Several studies highlight the importance of the integration of
vector control with MDA in enhancing the control and elimination of lymphatic filariasis especially in
highly endemic areas [16–20]. Some studies have also shown that vector control alone (i.e. without
MDA) can result to a significant reduction in LF infection in the community as observed in Togo [21],
Solomon Islands [22,23], The Gambia [24], Kenya [25,26], Nigeria [27] and Papua New Guinea [28–30].
In the Philippines, LF was first described in 1907 by foreign workers, and had reportedly become
prevalent in several Philippine provinces towards the end of the twentieth century [31]. According to the
Philippine Department of Health (DOH), LF was endemic in 46 (out of 81) provinces in 12 (out of 17)
regions with an at risk population of 45 000 000 Filipinos [1]. As part of the DOH’s commitment to
eliminate the disease as a public health problem in the country, it has created the National Filariasis
Elimination Program (NFEP), previously called National Filariasis Control Program, which has adopted
GPELF’s elimination strategy. Since 2001, MDA using the combination drug DEC-plus-ALB is
implemented once every year for a minimum of 5 years in all established endemic areas in the country.
The DOH has set a target programme coverage of at least 85% annually to increase the chances of
achieving Mf prevalence and antigen rate of less than 1%, which is the set criterion for stopping LF
MDA in the country [1]. The NFEP also continues to strengthen the morbidity management and
disability prevention strategy to alleviate the suffering and disability of chronically infected individuals.
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2

Vector control measures such as spraying and use of bednets are generally not a supported strategy for
most LF endemic areas in the country due to their underwhelming impact in reducing host–vector contacts.
This inefficiency can be attributed to the exophilic, exophagic and day-biting behaviours of Aedes poicilius
mosquitoes, which are the principal vectors of LF in the Philippines [32]. These mosquitovectors are known
to breed on the leaf axils of abaca and banana plants; hence, any form of vector control that aims to destroy
their breeding sites may result to the loss of livelihood of farmers and field workers in these areas [33].
However, in areas wherein Anopheles mosquitoes are the main vectors of LF, vector control measures can
enhance the impact of MDA in interrupting the transmission [7].
As of 2018, LF threatens around 893 million people in 49 countries worldwide and a global baseline
estimate of 36 million individuals are suffering from chronic disease manifestations [8]. This is a
remarkable progress compared to the 1.3 billion people at risk and 120 million people infected with
40 million disfigured and incapacitated in 2000 [34]. However, the magnitude of these numbers reflect
how far we are still from our goal of eliminating LF as a global public health problem. Because of
this, the WHO has reset its target for elimination of LF to the year 2030 from its initial target of 2020.
According to the new NTD roadmap [35], at least 58 out of 72 (81%) endemic countries need to be
validated for the elimination of LF as a public health problem by 2030. Unfortunately, the Philippines
is one of the countries that are yet to eliminate LF as a public health problem with approximately 5.3
million Filipinos in 6 endemic provinces still requiring MDA as of 2018, which is the DOH’s national
elimination target date for LF [1,36].
Three general simulation models of LF transmission and control (LYMFASIM [37], EPIFIL [38,39] and
TRANSFIL [40]) are currently being used to support policy making and designing elimination strategies for
LF. LYMFASIM is an individual-based modelling framework wherein different models for the transmission
and control of LF can be composed by choosing different assumptions and by varying parameter values
[37]. EPIFIL is a coupled partial differential equation and ordinary differential equation model describing
patterns of LF infection and associated diseases through the changes in the human and mosquito
populations over age and time [38]. TRANSFIL is described to be the individual-based stochastic
equivalent of EPIFIL [40]. All three models account for the human, mosquito and parasite dynamics in the
transmission and control of LF. Some studies on these simulation models are discussed in [41–46].
Several mathematical models of LF have also looked into the dynamics of the disease by considering
the interaction of the human and mosquito populations. In 2009, Supriatna and Soewono investigated the
long-term effects of targeted medical treatment in the disease dynamics in Jati Sampurna, West Java,
Indonesia using an SI–SI model [47]. Supriatna extended this model by considering an additional
human compartment consisting of latent individuals to account for the ‘delay’in the infection period
[48]. Numerical simulations using an SEI-SI model developed by Bhunu and Mushayabasa suggest
that treatment of both exposed and infected humans might lead to a more effective control of the
disease compared to the treatment of infected population alone [49]. This model is extended by
Bhunu to assess the potential of pre-exposure vaccination and chemoprophylaxis in the control of the
disease [50]. In a 2016 study, Iddi et al. assessed the impact of MDA, health education campaigns and
the vector control sterile insect technique on the transmission dynamics of LF using a deterministic
model [51]. In 2017, Mwamtobe et al. proposed an SEI-SI LF model with quarantine and treatment as
control strategies [52]. In these studies, the parameter values are either assumed or obtained from
existing malaria models.
Herein, we develop a mathematical model of LF to investigate how mass drug administration impacts
the disease dynamics in the Philippines. Compared to the existing models which mostly explored
scenarios wherein the treatment is given only to the infectious population, we consider a more
realistic representation of MDA wherein the antifilarial drugs are given to all eligible individuals in
the population, infected and uninfected alike. Hence, the treatment affects not only the dynamics of
the infectious human population but also those who are infected but not yet infectious. Since this
study is Philippine-specific, we also use Philippine filariasis data to estimate model parameters to
have more meaningful insights. The model considers no intervention on the vector population due to
the reasons mentioned previously.
Ultimately, we aim to use optimal control theory (OCT) to determine the best implementation strategy
for MDA until 2030 that will lead to a reduction of the infected population with a minimized MDA
implementation cost. There are various applications for OCT in many different fields. For instance, in
modelling cancer dynamics, OCT can be used to determine the optimal treatment regimen which will
minimize tumour density and drug side-effects over a defined period of time [53]. In epidemic
modelling, OCT can be used to determine optimal vaccination schedule which will lead to the
elimination of the epidemic in the population [54]. In modelling disease transmission and control, OCT
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can provide insights on the optimal intervention strategy with the least implementation cost [55]. Other
studies with applications of optimal control theory can be found in [56,57] (tuberculosis), [58] (dengue)
and [59] (HIV/AIDS). An in-depth discussion on optimal control theory and a number of insightful
applications are provided by Lenhart et al. [60].
To the authors’knowledge, this study provides the first mathematical model of LF in the Philippines.
Although the DOH begun MDA in 2001, we still see Mf prevalence rates of greater than 1% in some areas
in the Philippines. This motivates us to study the transmission dynamics of LF and suggest ways to
accelerate the elimination of the disease especially in the remaining endemic areas in the country. This
study aims to assist the DOH and similar programmes in the region in designing more effective and
cost-efficient implementation approaches for MDA fit for each endemic area, to achieve LF elimination
in the whole country in the near future.
The rest of the paper is organized as follows. In the Methods section, the proposed model for LF
transmission is described. Information on the epidemiological data and the parameter values and initial
conditions used are also presented in this section. In the Results and discussion section, stability analysis
of the steady state solutions of the model is presented. Results of the sensitivity analysis using the Latin
hypercube sampling and partial rank correlation coefficient method are also discussed. The obtained
results provide information on the critical model parameters with respect to the infected population,
which guide the parameter estimation using the available filariasis data from the Philippine DOH. In this
section, numerical simulations on the application of optimal control theory using the forward–backward
sweep method are also presented. We investigate the efforts in minimizing the number of infected
individuals and the corresponding implementation cost of MDA. We summarize the results of our work
and recommend possible extensions of this research in the last section.
2. Methods
2.1. Mathematical model of lymphatic filariasis
Filarial parasites that cause LF need two host species to complete their five-stage life cycle: a definitive host
(humans or animals) wherein the development from third-stage larva (L3) to adult worm and
the reproduction of microfilariae occurs, and an intermediate host (mosquito) wherein the development
from microfilaria to L3 occurs. The mosquito also acts as a vector of the parasite that physically carries and
transmits infective larvae from one human to another. Without one host or the other, disease transmission
will not be sustained in the population. Mosquitoes are infectious to humans if they harbour third-stage
larval parasites and humans are infectious to mosquitoes if they harbour microfilariae [61].
The LF transmission dynamics is summarized as follows. When an infected mosquito bites an
uninfected human, the infective L3 are introduced onto the skin. The infection in humans begins when
these parasites enter the human body through the mosquito bitewound and migrate to the lymphatics
where they develop to maturity within 6–12 months [62]. Adult worms reproduce and fecund female
worms release thousands of microfilariae. These microfilariae travel through the lymphatic channels
into the blood stream where they are taken up by a mosquito through a blood meal. Within 10–12 days,
the ingested microfilariae move from the mosquito’s gut to its thoracic cavity where they mature to
infective L3. At this point, the L3 larvae migrate to the mosquito’s proboscis, and the transmission cycle
continues when the infected mosquito bites an uninfected human [61].
We propose a deterministic model of LF transmission involving the interaction of the human and
mosquito populations. First, we assume that in an LF endemic area, the human population can be
categorized into three epidemiological classes based on each individual’s infection level: uninfected
U
h
(t), latent L
h
(t), and infectious I
h
(t). The latent stage accounts for the development of infective L3
larvae into fecund adult worms. Hence, individuals in the latent stage are considered infected but not
yet infectious. Meanwhile, all infected individuals who are able to transmit the infection are in the
infectious class. Thus, the total human population is represented by N
h
(t)=U
h
(t)+L
h
(t)+I
h
(t). On the
other hand, mosquitoes can only be either uninfected U
v
(t), or infected I
v
(t). Here, the latent stage is
considered negligible; thus, infected vectors are assumed to be infectious. We note that the model
considers only the female mosquito population since only adult female mosquitoes contribute to the
transmission. Hence, the total mosquito population at time tis defined as N
v
(t)=U
v
(t)+I
v
(t).
Moreover, the model considers one species of worm and one species of mosquito.
We assume that recruitment into both human and mosquito populations is only by birth. Moreover, since
there is no vertical transmission of the infection (i.e. infected pregnant mothers cannot pass the infection to
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4

their offspring), all new members of both populations enter their respective uninfected classes at per capita rates
b
h
and b
v
. Since there is no disease-induced death, both humans and mosquitoes leave their respective
population through natural death at respective per capita rates δ
h
and δ
v
.
The model also assumes homogeneous mixing of the human and mosquito populations. That is, each
uninfected individual in the population has an equal probability of being bitten by an infected mosquito
and each uninfected mosquito has the same probability of biting an infected human. Now, if βis the
average number of bites per mosquito per unit time, then there are β(N
v
(t)/N
h
(t)) mosquito bites per
human per unit time. However, only a fraction of these bites, which come from infected mosquitoes, are
potentially infective to humans. Further, only a proportion of the potentially infective bites actually result
to a successful transmission. Thus, we define the force of infection from mosquitoes to humans, λ
vh
(t), as
the product of the average mosquito bites per human per time β(N
v
(t)/N
h
(t)), the probability that the
mosquito is infected I
v
(t)/N
v
(t), and the probability of successful transmission of infection θ
vh
,i.e.
l
vh(t)¼
bu
vh
Iv(t)
Nh(t):
Without treatment, the force of infection from humans to mosquitoes, defined as
l
wo
hv(t)¼
bu
hv
Ih(t)
Nh(t),(2:1)
is just the product of the average number of bites per mosquito per unit time β, the probability that the
human is infectious I
h
(t)/N
h
(t), and the probability of successful transmission of infection θ
hv
.Itis
known that the antifilarial drugs given during MDA can instantaneously kill the microfilariae in humans
and can halt the reproduction of adult worms, thus reducing the probability of transmission for a
significant amount of time [63]. So, if we define pas the proportion of reduction in transmission due to
treatment, then the transmission probability becomes θ
hv
(1 −p). Hence, with treatment, the force of
infection is given by
l
w
hv(t)¼
bu
hv (1 p)Ih(t)
Nh(t):(2:2)
Now, suppose only a proportion γof the human population is given treatment. Then, the total force of
infection from humans to mosquitoes is computed using equations (2.1) and (2.2) as follows:
l
hv(t)¼
l
wo
hv(t)(1 
g
)þ
l
w
hv(t)
g
¼
bu
hv
Ih(t)
Nh(t)(1 p
g
):
Thus, upon sufficient bites to infectious humans, uninfected mosquitoes move to the infected class at the rate
λ
hv
(t). Similarly, after adequate infective mosquito bites, uninfected humans move to the latent class at the
rate λ
vh
(t). Note that the force of infection from mosquitoes to humans remains the same with or without
treatment since the antifilarial drugs ingested by humans do not directly affect the mosquitoes. Owing to
the growth and development of parasites in the human body, individuals in the latent class will
eventually become infectious at the rate α. Since there is no permanent nor temporary immunity, the
latent and infectious humans given treatment move back to the uninfected class at the rates r
1
and r
2
,
respectively. We note that r
1
≥r
2
, since the level of infection (i.e. density and developmental stage of
parasites in the body) of the latent individuals is assumed to be lower than those who are infectious.
We assume that the parameters b
h
,b
v
,δ
h
,δ
v
,θ
vh
,θ
hv
,β,αare strictly positive real numbers. We also
assume that the antifilarial drugs are effective; thus, r
1
,r
2
are strictly positive, and pis within (0, 1]. Since
the treatment coverage γis a controllable parameter, we assume that it can take any value within [0, 1],
where γ= 0 implies that no one in the population is given treatment while γ= 1 implies that the whole
population is given treatment. We also assume that all individuals given treatment actually ingest the
antifilarial drugs.
Figure 1 provides a graphical interpretation of our model. Based on our model descriptions and
assumptions, the proposed model is governed by the following system of differential equations:
dUh
dt¼bhNhþ(r1Lhþr2Ih)
g
(
l
vh þ
d
h)Uh,
dLh
dt¼
l
vhUh(
a
þr1
g
þ
d
h)Lh,
dIh
dt¼
a
Lh(r2
g
þ
d
h)Ih,
dUv
dt¼bvNv(
l
hv þ
d
v)Uv
and dIv
dt¼
l
hvUv
d
vIv,
9
>
>
>
>
>
>
>
=
>
>
>
>
>
>
>
;
(2:3)
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5

where
l
vh ¼
bu
vh
Iv
Nh
,
l
hv ¼
bu
hv(1 p
g
)Ih
Nh
:
2.2. Epidemiological data
The available filariasis data from the Philippine Department of Health’s Field Health Service Information
System (FHSIS) Annual Reports [64] were compiled for each Philippine province and region covering the
years 2009–2018. The 2009–2018 data report annual prevalence rates, computed as the percentage of
microfilariae-positive individuals over the total cases examined in the area. According to the DOH
FHSIS reports, 5 (out of 5) provinces and 3 (out of 6) cities in the administrative region of Caraga
have been endemic for LF [64]. As of 2018, one province (Surigao del Norte) and one city (Surigao
City) in Caraga Region are still implementing MDA for LF while the other four provinces have
already been declared LF-free (Agusan del Sur and Dinagat Islands in 2010, Surigao del Sur in 2013,
and Agusan del Norte in 2015). The 2018 DOH FHSIS report also suggests that there are still
confirmed cases of LF in the three provinces and 1 city previously declared LF-free [64]. Hence, the
Caraga filariasis data are used in our simulations. From the recorded total human population and
annual prevalence rates, the infectious human population per year was estimated as the proportion of
the total population that are microfilariae positive (i.e. I
h
= total population × prevalence rate). Table 1
gives a summary of our dataset.
2.3. Parameter values and initial conditions
The human natural death rate δ
h
, measured in weeks, is computed as the inverse of life expectancy, which
is calculated by subtracting the median age of the Caraga population in 2010 [65] from the recorded life
expectancy at birth in Caraga in 2010 [66]. An approximate for the obtained value is δ
h
= 0.00042 ( per
week). Using the obtained value and the Caraga population data for the years 2009–2018 in table 1,
we estimate the human birth rate b
h
from the exact solution of dN
h
/dt=(b
h
−δ
h
)N
h
which is
Nh(t)¼Nh(0) e(bh
d
h)t. Solving this numerically using the built-in Matlab function lsqcurvefit,we
find that the birth rate is approximately b
h
= 0.0006 ( per week).
Meanwhile, since LF and dengue are transmitted by mosquitoes of the same genus (Aedes), the
parameter values for the mosquito death rate δ
v
and mosquito biting rate βare obtained from a
dengue model presented by de los Reyes V & Escaner IV [67]. Here, we assume that b
v
=δ
v
.
The progression from L
h
to I
h
(1/α) is usually between 6 and 12 months [62], but we fix α=0.0288
which is about eight months, following a study by Jambulingam et al. [44]. In [39], Norman et al.seta
value of 1.13 × 10
−4
for the proportion of L3 filarial parasites entering a host which develop into adult
worms. In our model, this value is assigned to the transmission probability from vector to human, θ
vh
.In
the same paper, the proportion of mosquitoes which pick up infection when biting an infected host was
assigned a value of 0.37, which we set to be the transmission probability from human to vector, θ
hv
.
The treatment coverage γ= 0.619 is computed as the average of the recorded MDA programme coverages
in Caraga from 2010 to 2018 obtained from the DOH FHSIS data [64]. The treatment rates r
1
and r
2
in table 1
are defined as the inverse of the average duration of the treatment of humans in compartments L
h
and I
h
,
UhLhIh
Uv
Iv
dvIvdvUv
dhIh
dhLh
dhUh
bhNh
bvNv
lvhUh
lhvUv
aLh
r2gIh
r1gLh
Figure 1. Diagram of the LF transmission model. Solid lines represent the transition of humans and mosquitoes between different
states of infection. Dashed lines represent the transfer of parasites from human to mosquito and vice versa through a mosquito bite.
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6

respectively, i.e. the time it takes foran infected person to be treated and to move back to the uninfected class.
These parameters are estimated using the available filariasis data in Caraga.
The parameter prepresents the proportion of reduction in transmission due to treatment. From what
has been inferred from the literature, this parameter depends on many factors such as the percentage of
microfilariae and adult worms killed due to treatment, the reduction in Mf production, and the duration
of the sustained reduction in the density of these parasites in the human body following treatment. Here,
a value of 0.6 is assigned to the parameter p. We show in the next section that the infected human
population L
h
+I
h
is not sensitive to the parameter p(figure 2).
In our numerical simulations, we use the 2009 population data from Caraga as initial conditions.
Since only the total human population N
h
(0) and the infectious population I
h
(0) are available, it is
assumed that the remaining human population, N
h
(0) −I
h
(0), can be apportioned between the
uninfected U
h
(0) and latent L
h
(0) populations using a partitioning parameter m∈[0, 1] in the relation
Uh(0) ¼(Nh(0) Ih(0)) m
and Lh(0) ¼(Nh(0) Ih(0)) (1 m),(2:4)
Table 1. The DOH FHSIS filariasis data in Caraga Region from 2009 to 2018. The data under total population, total cases
examined, cases found positive and prevalence rate are obtained from the DOH FHSIS Annual Reports [64] while the data under
infectious population are computed by multiplying total population by prevalence rate.
year
total
population
total cases
examined
cases found
positive
prevalence
rate (%)
infectious
population
2009 2 501 400 5 036 54 1.07 26 822
2010 2 429 224 8 275 174 2.10 51 080
2011 2 611 700 3 984 25 0.63 16 389
2012 2 507 410 18 406 189 1.03 25 747
2013 2 544 172 18 406 189 1.03 26 125
2014 2 581 399 11 537 40 0.35 8950
2015 2 619 098 39 831 25 0.06 1644
2016 2 657 380 35 033 145 0.41 10 999
2017 2 828 583 49 744 73 0.15 4151
2018 2 694 944 42 943 56 0.13 3514
PRCC values
year 1
year 2
year 3
year 4
year 5
year 6
year 7
year 8
year 9
year 10
parameter
ba g mdummypr
2
r
1
–1.0
–0.5
0.5
1.0
0
Figure 2. PRCC values depicting the sensitivities of the model output L
h
+I
h
with respect to the model parameters at 10 different
time points.
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7

where the obtained values are rounded off to the nearest whole number to be biologically consistent
with the human population. This parameter mis also estimated along with r
1
and r
2
using the
available filariasis data from Caraga. Meanwhile, the initial values for the mosquito population are
assumed to be U
v
(0) = 1 000 000 and I
v
(0) = 100 000. Table 2 lists the parameter values of the LF
transmission model.
3. Results and discussion
3.1. Mathematical analysis of the model
Since all model parameters and state variables are assumed to be non-negative, the total populations are
also non-negative, i.e. N
h
(t)=U
h
(t)+L
h
(t)+I
h
(t)≥0 and N
v
(t)=U
v
(t)+I
v
(t)≥0. Hence, our region of
biological interest is
C
¼(Uh,Lh,Ih,Uv,Iv)[R5
0

:
The model system in equation (2.3) is mathematically and epidemiologically well posed in
C
.
Theorem 3.1. System (2.3)has two steady-state solutions:
(i)the disease-free equilibrium E
0
=(b
h
N
h
/δ
h
,0,0,b
v
N
v
/δ
v
,0)and
(ii)the endemic equilibrium E¼(U
h,L
h,I
h,U
v,I
v), where
U
h¼bhNh(
a
þr1
g
þ
d
h)(r2
g
þ
d
h)
d
h[(
a
þr1
g
þ
d
h)(r2
g
þ
d
h)þ
l

vh(
a
þr2
g
þ
d
h)] ,
L
h¼
l

vhbhNh(r2
g
þ
d
h)
d
h[(
a
þr1
g
þ
d
h)(r2
g
þ
d
h)þ
l

vh(
a
þr2
g
þ
d
h)] ,
I
h¼
l

vh
a
bhNh
d
h[(
a
þr1
g
þ
d
h)(r2
g
þ
d
h)þ
l

vh(
a
þr2
g
þ
d
h)] ,
U
v¼bvNv[
l

vh
d
h(
a
þr2
g
þ
d
h)þ
d
h(
a
þr1
g
þ
d
h)(r2
g
þ
d
h)]
l

vh[
abu
hv(1 p
g
)bhþ
d
h
d
v(
a
þr2
g
þ
d
h)] þ
d
v
d
h(
a
þr1
g
þ
d
h)(r2
g
þ
d
h)
and I
v¼
l

vh
abu
hv(1 p
g
)bhbvNv
l

vh
d
v[
abu
hv(1 p
g
)bhþ
d
h
d
v(
a
þr2
g
þ
d
h)] þ
d
2
v
d
h(
a
þr1
g
þ
d
h)(r2
g
þ
d
h),
Table 2. Parameters of the LF transmission model.
parameter description value per week references
b
h
human birth rate 0.0006 data-fitted
a
δ
h
human natural death rate 0.00042 calculated [65,66]
b
v
mosquito birth rate 0.1 assumed
δ
v
mosquito natural death rate 0.1 [67]
βmosquito biting rate 1 [67]
θ
vh
probability of transmission from mosquito to human 0.000113 [39]
θ
hv
probability of transmission from human to mosquito 0.37 [39]
αprogression rate from L
h
to I
h
0.0288 [44]
r
1
treatment rate of L
h
0.430848 data-fitted
b
r
2
treatment rate of I
h
0.010038 data-fitted
b
γtreatment coverage 0.619 [64]
pproportion of reduction in transmission due to treatment 0.6 assumed
mpartitioning parameter for the initial population 0.853284 data-fitted
b
a
From the Caraga population data.
b
From the Caraga filariasis data.
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8

where
l

vh ¼
ab
2
u
vh
u
hv(1 p
g
)bhbvNv
d
2
v
d
hNh(
a
þr1
g
þ
d
h)(r2
g
þ
d
h)
d
vNh[
abu
hv(1 p
g
)bhþ
d
v
d
h(
a
þr2
g
þ
d
h)] :
Since E
0
is a steady-state solution wherein the infected populations are zero, E
0
is referred to as the
disease-free equilibrium (DFE) solution. The steady-state solution Eis known as the endemic equilibrium
solution since the infection is constantly maintained in the population. The derivations of E
0
and E
are discussed in detail in appendix A.
The basic reproduction number, R0, is a threshold parameter that is used to assess whether or
not a disease will invade a population. In theory, if R0,1, then each infected person produces
less than one new infected individual in their entire period of infectiousness, which implies
that the infection will not be sustained in the population. On the contrary, if R0.1, then
each infected individual infects more than one person implying that the disease will eventually
invade the population. Here, the basic reproduction number is computed using the method of
next-generation matrix, a method introduced by Diekmann et al. [68]. The value of R0for
the model system (2.3) is presented in the next theorem. For the proof, we refer readers to
appendices B and C.
Theorem 3.2. The basic reproduction number for the model system (2.3)is
R0¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ab
2
u
vh
u
hv(1 p
g
)bhbvNv
d
2
v
d
hNh(
a
þr1
g
þ
d
h)(r2
g
þ
d
h)
s:
Moreover, the DFE E
0
is locally asymptotically stable when R0,1and unstable when R0.1. The endemic
equilibrium Eis locally asymptotically stable when R0.1.
3.2. Sensitivity analysis and parameter estimation
Sensitivity analysis is a tool used to identify and rank critical input parameters in a model with respect to
their impact on the reference model output. An input parameter is said to be influential if small
variations of its value result to significant changes in the model output. Based on the result of
sensitivity analysis, one can have an idea which input parameters need to be assigned accurate values
(i.e. the most influential parameters) and which ones can be roughly estimated (i.e. the less influential
parameters). In this study, we used one of the most efficient and reliable global sensitivity analysis
techniques—the partial rank correlation coefficient (PRCC) method combined with the Latin
hypercube sampling (LHS) technique for the sampling of parameters [69,70]. Numerical simulations
for sensitivity analysis using the LHS/PRCC method were carried out using modified versions of the
PRCC Matlab codes presented by Marino et al. [69].
The sign and magnitude of the PRCC values, which lie between −1 and +1, characterize the
qualitative relationship between the model input and the model output. A positive PRCC implies a
positive correlation between the input and the output; that is, an increase in the model input will
result in an increase in the model output. On the other hand, a negative PRCC implies a negative
correlation wherein an increase in the model input causes a decrease in the model output, and vice
versa. The absolute value of a PRCC measures the importance of the model input to the relative
model output. The greater the magnitude of a PRCC value, the greater the impact of the input to the
output. The p-value of the PRCC indicates the statistical significance of the value. In most cases, a
parameter is said to be sensitive or influential to the model output if the magnitude of the PRCC is
greater than or equal to 0.5 and the corresponding p-value is less than 0.05 [69]. We used these
criteria to determine the influential parameters in our model.
We tested the sensitivity of the model parameters β,α,γ,p,r
1
,r
2
,mand a dummy parameter, with
respect to the infected human population L
h
+I
h
. A dummy parameter is included to test the robustness
of the model as in [69]. That is, since it does not appear in the model equations and does not affect the
model in any other way, the dummy parameter should be assigned a sensitivity index of zero in the
simulations. The sample size is set to N= 10 000 and the time points of interest are every year for a
duration of 10 years or every after 52 weeks for T= 520 weeks.
We fix the values of b
h
and δ
h
based on the calculations using the available data. We also fix the
values of b
v
,δ
v
, and the transmission probabilities θ
vh
and θ
hv
, following the results in [39,67].
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9

Since there is limited information on the values for α,γand p, a uniform distribution is assigned to their
corresponding parameter ranges which we set to +50% of their respective baseline values given in
table 2. On the other hand, since βhas already been estimated in [67], we used a normal
distribution with the nominal value in table 2 as the mean and a standard deviation of 0.1. The range
of values for the treatment rates r
1
and r
2
is from 0.0019 to 0.5, or from two weeks to 10 years, while
the range of values for the partitioning parameter mis from 0 to 1. Both ranges are assigned a
uniform distribution.
The result of our simulations is shown in figure 2, where each coloured bar graph corresponds to a
particular time point. Based on our criteria for determining critical parameters, the results suggest that
the parameters γ,r
1
and r
2
are the most influential parameters throughout the running time of 10
years or 520 weeks. It can be observed that γ,r
1
and r
2
have a negative correlation to the model
output L
h
+I
h
. This implies that an increase in the treatment coverage and faster treatment rates will
most likely result in a decrease in the total infected human population. We also observed that all three
parameters have high magnitude of PRCC values for the first five years, after which the magnitude of
the PRCC values of γand r
1
decreased below 0.5. The PRCC values of r
2
decreased below 0.5 in the
last two years.
In our parameter estimation, we opted not to include the treatment coverage γsince there are
already recorded values for this in the DOH FHSIS filariasis data [64]. This left us with the
parameters r
1
and r
2
, which we estimate together with the partitioning parameter m, using the
available population data from Caraga Region [64] covering the years 2009–2018.
From the DOH FHSIS filariasis data in Caraga Region in table 1, we have 10 data points for I
h
(2009–2018) with the 2009 data as the initial condition. The parameters r
1
,r
2
and mare estimated by
minimizing the mean of the squared difference between the available data and the model output at
the corresponding time point using the Matlab function lsqcurvefit.
To establish unbiased choice of initial conditions for the parameter estimation, the LHS method is
used to generate 1000 sets of initial guesses for the estimates. For each set of initial guesses,
parameters are estimated using lsqcurvefit. The ‘best’estimate is then computed as the average of
the 1000 estimates from 1000 distinct initial guesses. The simulation results, along with the lower
bound and upper bound set for each of the three parameters for the Matlab simulations are given in
table 3.
Using the estimated value for the partitioning parameter m= 0.853284 and the 2009 filariasis
data from Caraga in table 1, we can now solve for the initial populations for U
h
and L
h
in equation
(2.4). Meanwhile, the estimated value for the treatment rate r
1
is 0.430848 which means that the
treatment period 1/r
1
is about 16 days, and the estimated value for the treatment rate r
2
is 0.010038
which implies that 1/r
2
is about 2 years. Recall that the progression from the latent stage to the
infectious stage lasts for about 6–12 months [62]. Hence, the obtained value for r
1
highlights the
effectiveness of MDA in preventing new infections. The obtained value for r
2
explains that although
transmission of L3 larvae from mosquitoes to humans is inefficient, the treatment of infectious
individuals takes a long time because of the 5-year reproductive life span of adult filarial worms [61].
This highlights the importance of maintaining a high treatment coverage since majority of the infected
population are asymptomatic. Figure 3 depicts the available filariasis data plotted with the
corresponding model fit from parameter estimation.
Parameter bootstrapping is also used to investigate the reliability of our estimated values.
Bootstrapping is a statistical technique used to quantify uncertainty and construct confidence intervals
in parameter estimates [71]. Following the algorithm presented by Chowell [71], 1000 samples of
synthetic datasets are generated from the best-fit model by assuming a Poisson error structure.
Parameters are then re-estimated from each synthetic dataset using lsqcurvefit to obtain a new set
Table 3. Results of parameter estimation using lsqcurvefitwith the bounds used. The estimate is computed as the
average of the estimates from 1000 distinct initial guesses generated by the LHS method.
parameter estimate lower bound upper bound
r
1
0.430848 0.0019 0.5
r
2
0.010038 0.0019 0.5
m0.853284 0 1
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10

of parameter estimates. Numerical simulations for parameter bootstrapping were carried out using
modified versions of the sample Matlab codes presented by Chowell [71]. Figure 4 shows the
parameter bootstrapping results with the means, standard deviations and 95% confidence intervals for
each parameter. We observe low standard deviations and narrow ranges of the 95% confidence
intervals for all the parameters. Moreover, the estimates obtained for m,r
1
and r
2
(table 3) all lie
within their corresponding confidence intervals.
3.3. Optimal control strategy
In implementing MDA programmes, the treatment coverage can vary through time depending on
the efforts of the government. In the LF transmission model, this translates to the constant
parameter γbecoming a function of time u(t). Since the main purpose of MDA is to limit and reduce
the infected humans by interrupting the transmission of infection, the proposed optimal control
problem is to minimize the number of infected (both latent and infectious) hosts as well as
2009
2010
2011
2012
2013
2014
2015
2016
2017
2018
0
1
2
3
4
5
6
incidence cases
model
data
×104
Figure 3. The available data points for I
h
and the corresponding fit from parameter estimation.
0
20
0.850
mean: 0.8533, s.d: 0.0010
95% CI: (0.8513, 0.8554)
mean: 0.4244, s.d: 0.0034
95% CI: (0.4177, 0.4310)
mean: 0.0100, s.d: 0.000006
95% CI: (0.0100, 0.0101)
0.855
m
0.42 0.43
r1
0.010040.01002 0.01006
r2
40
60
frequency
80
100
0
20
40
60
frequency
80
100
0
20
40
60
frequency
80
100
Figure 4. Results of parameter bootstrapping with the mean, standard deviation (s.d.) and 95% confidence interval (CI).
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11

the corresponding cost of implementing MDA. Here, minimizing the infected mosquito population is
not included in the objective since there is no control directly affecting the vector population.
From our simulations, we aim to gain insights about the optimal MDA coverage over a duration of T
years, T=(t
f
−t
0
) > 0, such that the objective is satisfied. Hence, our objective functional to be
minimized is
J(u)¼ðtf
t0
Lh(t)þIh(t)þc
2u2(t)
hi
dt,
where t
0
and t
f
are taken as 2018 and 2030, respectively. Here, cis a weighting parameter associated
with the implementation cost of MDA. We note that the parameter cdoes not represent per se
the actual monetary cost of implementing MDA. Instead, it is a constant parameter that balances
the size and importance of each term in the integrand. That is, if cis too high, more importance
will be given in reducing the cost of implementation of MDA compared to the infected
population. On the other hand, if cis low, the minimization problem will put equal importance
in minimizing the infectious population and the cost of implementation. It is assumed that the
control is a quadratic function to represent nonlinear implementation costs. Our goal is to find
optimal usatisfying
J(u)¼min
uJ(u)
subject to
dUh
dt¼bhNhþ(r1Lhþr2Ih)u(t)
bu
vh
Iv
Nh
Uh
d
hUh,
dLh
dt¼
bu
vh
Iv
Nh
Uh(
a
þr1u(t)þ
d
h)Lh,
dIh
dt¼
a
Lh(r2u(t)þ
d
h)Ih,
dUv
dt¼bvNv
bu
hv(1 pu(t)) Ih
Nh
Uv
d
vUv
and dIv
dt¼
bu
hv(1 pu(t)) Ih
Nh
Uv
d
vIv,
with the initial conditions U
h
(0) = U
h,0
,L
h
(0) = L
h,0
,I
h
(0) = I
h,0
,U
v
(0) = U
v,0
,I
v
(0) = I
v,0
, and such that u
min
≤
u(t)≤u
max
.
Pontryagin’s maximum principle [72] transforms the optimal control problem into a problem that
minimizes a Hamiltonian Hpointwise with respect to the control u(t). Using x(t)=[U
h
,L
h
,I
h
,U
v
,I
v
](t)
and λ(t)=[λ
1
,…,λ
5
](t), the Hamiltonian His formed as
H(t,x(t), u(t),
l
(t)) ¼LhþIhþc
2u2
þ
l
1bhNhþ(r1Lhþr2Ih)u(t)
bu
vh
Iv
Nh
Uh
d
hUh

þ
l
2
bu
vh
Iv
Nh
Uh(
a
þr1u(t)þ
d
h)Lh

þ
l
3[
a
Lh(r2u(t)þ
d
h)Ih]
þ
l
4bvNv
bu
hv(1 pu(t)) Ih
Nh
Uv
d
vUv

þ
l
5
bu
hv(1 pu(t)) Ih
Nh
Uv
d
vIv

:
Applying Pontryagin’s maximum principle, we obtain the following result. The proof of this theorem can
be found in appendix D.
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12

Theorem 3.3. There exists optimal control u(t)minimizing the objective functional J(u(t)) over
V
¼{umin u
umax;u[L2ð2018;2030Þ}. Given this optimal solution, there exist adjoint variables λ
1
(t), …,λ
5
(t), which satisfy
d
l
1
dt¼(
l
1
l
2)
bu
vh
Iv
Nh
þ
l
1
d
h,
d
l
2
dt¼1þ(
l
2
l
1)r1uþ(
l
2
l
3)
a
þ
l
2
d
h,
d
l
3
dt¼1þ(
l
3
l
1)r2uþ(
l
4
l
5)
bu
hv(1 pu)Uv
Nh
þ
l
3
d
h,
d
l
4
dt¼(
l
4
l
5)
bu
hv(1 pu)Ih
Nh
þ
l
4
d
v
and d
l
5
dt¼(
l
1
l
2)
bu
vh
Uh
Nh
þ
l
5
d
v,
with transversality conditions λ
i
(t
f
)=0, for i =1,…,5.Furthermore, the optimality equation is as follows:
u(t)¼min umax,umin ,1
cr1Lh(
l
2
l
1)þr2Ih(
l
3
l
1)þ
bu
hvpIhUv
Nh
(
l
5
l
4)

:
Since our compiled population data in table 1 suggest that there are still infectious individuals in
Caraga in 2018, we look at the disease dynamics after 2018 to obtain insights on the optimal strategy
for implementing MDA to eliminate LF in the population by 2030. Hence, the choice of t
0
= 2018 and
t
f
= 2030 in the objective functional.
Numerical simulations were carried out in Matlab using the forward–backward sweep method [60].
We set u
min
= 0 to account for the possibility that MDA is stopped after 2018, and u
max
= 0.95 as we
recognize that a 100% treatment coverage is quite difficult to achieve in reality. We also considered
different values for the weighting parameter cto investigate how the variations in the MDA
implementation cost affect the optimal control solution. The obtained optimal control solutions for c=
10
i
,i= 0, 2, 4, 6, were compared with the no control solution (u(t)¼0, 8t) and the constant control
solution (u(t)¼0:619, 8t). Figures 5 and 6 show the obtained control profile and the corresponding
effect in the dynamics of the infectious population I
h
(t).
Observe from figure 6 that if MDA is stopped after 2018, the decrease in the infectious population
over a span of 12 years is only about 15%. On the other hand, if the current control is maintained for
another 12 years the infectious population can decrease further by 98%.
One apparent conclusion from the optimal control solutions in figure 5 is that the MDA coverage is
inversely proportional to the implementation cost. That is, the lower the cost, the higher the MDA
coverage. Consequently, a higher MDA coverage leads to a lower number of infected individuals at the
end time as illustrated in figure 6. Here, we compare the optimal control results for different values of c
to the no MDA strategy. As recorded in table 4, with cvalues equal to 10
0
,10
2
or 10
4
, the I
h
population
can be reduced by at least 91% at the end of the control period. When c=10
6
, we observe that the
optimal control problem gives more emphasis on minimizing the MDA implementation cost compared
2018
2020
2022
2024
2026
2028
2030
y
ear
0
0.2
0.4
0.6
0.8
1.0
u(t)
current control
no MDA
optimal control with c = 1
optimal control with c = 10
2
optimal control with c = 10
4
optimal control with c = 10
6
Figure 5. Control profile.
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13

to minimizing the infected population. As a result, we obtain a low MDA coverage which led to a higher
number of infected individuals at the end time. In this case, the optimal solution corresponds to only
18% reduction in the infectious population compared to the no MDA strategy. The optimal control
solutions suggest that the infected population can be further minimized if MDA coverage is scaled up.
Even though we do not have enough information on the actual cost of MDA implementation, the data
on MDA coverage in the Philippines since 2001 suggest that scaling up MDA programmes is possible
especially since only 6 out of 46 endemic provinces are implementing MDA as of 2018.
4. Conclusion
LF remains a public health problem in the Philippines with millions of Filipinos still at risk ofbeing infected as
of 2018. In this work, we developed a mathematical model of LF transmission involving the interaction of
human and mosquito populations. Using this model, we investigated how the implementation of the
annual MDA for LF affects the disease dynamics in the LF-endemic region of Caraga. Sensitivity analysis
using the LHS/PRCC method showed that the infected human population is most sensitive to the
treatment coverage (i.e. how much of the population receives treatment) and the treatment rates (i.e. how
effective the antifilarial drugs are in reducing the parasite density in infected humans). This highlights the
importance of strategic MDA implementation, and efficient data gathering and data reporting to obtain
more accurate simulation results and produce more realistic and relevant insights. The treatment
parameters were estimated using the available filariasis data in Caraga obtained from the Philippine
DOH’s FHSIS. The estimated values emphasize the importance of MDA in preventing new infections.
Parameter bootstrapping showed small variability in all the parameter estimates, indicated by the low
standard deviations and narrow confidence intervals.
The application of optimal control theory highlighted the importance of maintaining a high treatment
coverage per year to effectively minimize the infected population by the year 2030. To achieve this, it is
2018
2020
2022
2024
2026
2028
2030
year
infectious humans, I
h
(t)
current control
no MDA
optimal control with c = 1
optimal control with c = 10
2
optimal control with c = 10
4
optimal control with c = 10
6
104
103
102
10
Figure 6. The corresponding effect of the controls in the dynamics of the infectious population I
h
. The y-axis is in log scale to better
illustrate the difference in the dynamics towards the end time for different values of c.
Table 4. Per cent reduction in the infectious population I
h
when the optimal control using implementation cost cis applied. Per
cent reduction is the relative difference in percentage between the number of infectious population in 2030 when optimal
control is applied and without MDA.
implementation cost cper cent reduction in I
h
(%)
10
0
99.76
10
2
99.47
10
4
91.55
10
6
18.24
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14

important that there is an effective and systematic implementation of MDA from the national level down
to the implementation units. Activities such as the distribution of antifilarial drugs, collection of data,
and diagnostic examinations must be enhanced. To address the problems with MDA acceptance [73],
educational activities must be conducted before the distribution of antifilarial drugs to inform the
people and help them understand how participating in MDA will benefit them and their
communities. Field workers and volunteers must also be well informed about the MDA programme
and the disease. Most importantly, since LF is closely related to poverty, eliminating the disease
requires the active commitment of the government to improve the lives of its poorest and most
vulnerable citizens. This does not only include the distribution of treatment against LF and other
NTDs but also the provision of better living conditions by giving people better opportunities for
employment, proper education, clean water and other basic human needs.
One limitation of our study is the uncertainty in the values of our model inputs, both initial
conditions and model parameters. Since the model is applied specifically to Caraga, the parameter
values must be accurate to well represent the transmission dynamics of LF in the region. However,
majority of the existing studies on LF in the Philippines are outdated and may no longer give accurate
information on the disease dynamics at present. We urge the Philippine DOH to have a systematic,
correct, and timely data gathering that may be relevant to the study of not only LF but also other
diseases. We also encourage local researchers to look into LF in the country. One possible topic to
explore is the dynamics of Aedes poicilius mosquitoes, or other known mosquito species that transmit
LF in the Philippines, and review values of relevant parameters. Both clinical and field studies on the
efficacy of the antifilarial drugs, or a comprehensive study on the general effects of MDA in reducing
transmission in a population under varying circumstances, could also be helpful for this study.
A possible extension of our model is the inclusion of the parasite dynamics, such as parasite
aggregation, to better capture the transmission processes of LF. Future works could also focus on
calibrating the model to incorporate heterogeneity in susceptibility and infectivity of individuals in the
population. We also recommend cost analysis of MDA in the remaining endemic areas, such as the
work of Amarillo et al. in Sorsogon, Philippines in 2009 [74].
Data accessibility. The filariasis data of the total population, total cases examined, cases found positive, and prevalence
rate for Caraga Region, the Philippines, presented in table 1 are obtained from the Philippine Department of
Health’s Field Health Service Information System Annual Reports [64].
Authors’contributions. P.K.N.S. participated in the conceptualization, methodology, and formal analysis of the study,
curated the data, carried out the simulations and computations, drafted the manuscript, acquired funding, and
participated in the critical review of the manuscript; V.M.P.M. and R.G.M. participated in the conceptualization,
methodology, and formal analysis of the study, drafted the manuscript, acquired funding, supervised the study,
and participated in the critical review of the manuscript; V.Y.B.J. participated in the conceptualization and formal
analysis of the study, supervised the study, and participated in the critical review of the manuscript. All authors
participated in the final review and editing of the manuscript, and gave final approval for publication and agree to
be held accountable for the work performed therein.
Competing interests. The authors declare no conflict of interest.
Funding. This work was funded by the Natural Sciences Research Institute, University of the Philippines Diliman under
research grant nos. MAT-20-1-02 and MAT-21-1-04. The funders had no role in the design of the study; in the collection,
analyses or interpretation of data; in the writing of the manuscript; or in the decision to publish the results.
Appendix A. Computation of the steady-state solutions of the LF model
To find steady-state solutions of the system
dUh
dt¼bhNhþ(r1Lhþr2Ih)
g
(
l
vh þ
d
h)Uh,
dLh
dt¼
l
vhUh(
a
þr1
g
þ
d
h)Lh,
dIh
dt¼
a
Lh(r2
g
þ
d
h)Ih,
dUv
dt¼bvNv(
l
hv þ
d
v)Uv
and dIv
dt¼
l
hvUv
d
vIv,
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15

where
l
vh ¼
bu
vh
Iv
Nh
,
l
hv ¼
bu
hv(1 p
g
)Ih
Nh
,
we look for the corresponding solution to
dUh
dt¼dLh
dt¼dIh
dt¼dUv
dt¼dIv
dt¼0:
We have the following:
dUh
dt¼0)Uh¼bhNhþ(r1Lhþr2Ih)
g
l
vh þ
d
h
,(A1)
dLh
dt¼0)Lh¼
l
vhUh
a
þr1
g
þ
d
h
, (A2)
dIh
dt¼0)Ih¼
a
Lh
r2
g
þ
d
h
,(A3)
dUv
dt¼0)Uv¼bvNv
l
hv þ
d
v
(A 4)
and dIv
dt¼0)Iv¼
l
hvUv
d
v
:(A5)
We first solve the state variables in terms of λ
vh
, then express λ
vh
in terms of the model parameters.
Using equation (A 4) and equation (A 5), we get
Iv¼bvNv
l
hv
d
v(
l
hv þ
d
v):(A 6)
Moreover, if we plug-in equation (A 2) to equation (A 3), we get
Ih¼
al
vhUh
(r2
g
þ
d
h)(
a
þr1
g
þ
d
h):(A 7)
Plugging-in equation (A 7) and equation (A2) to equation (A 1) and doing a few algebraic manipulations,
we have
Uh¼bhNh(r2
g
þ
d
h)(
a
þr1
g
þ
d
h)
d
h[(r2
g
þ
d
h)(
a
þr1
g
þ
d
h)þ
l
vh(
a
þr2
g
þ
d
h)] :(A 8)
Using equation (A 8) in equation (A 2), we get
Lh¼
l
vhbhNh(r2
g
þ
d
h)
d
h[(r2
g
þ
d
h)(
a
þr1
g
þ
d
h)þ
l
vh(
a
þr2
g
þ
d
h)] :(A 9)
Similarly, using equations (A 9) and (A 2), we obtain
Ih¼
al
vhbhNh
d
h[(r2
g
þ
d
h)(
a
þr1
g
þ
d
h)þ
l
vh(
a
þr2
g
þ
d
h)] :(A 10)
Recall that
l
vh ¼
bu
vh Iv
Nh. Using equation (A 6), we have
l
vh ¼
bu
vhbvNv
d
vNh

l
hv
l
hv þ
d
v
:(A 11)
In the same manner, we find an expression for λ
hv
using equation (A 10):
l
hv ¼
l
vh
abu
hv(1 p
g
)bh
l
vh
d
h(
a
þr2
g
þ
d
h)þ
d
h(r2
g
þ
d
h)(
a
þr1
g
þ
d
h):
Then we have
l
hv þ
d
v¼
l
vh[
abu
hv(1 p
g
)bhþ
d
v
d
h(
a
þr2
g
þ
d
h)] þ
d
v
d
h(r2
g
þ
d
h)(
a
þr1
g
þ
d
h)
l
vh
d
h(
a
þr2
g
þ
d
h)þ
d
h(r2
g
þ
d
h)(
a
þr1
g
þ
d
h)(A 12)
and
l
hv
l
hv þ
d
v
¼
l
vh
abu
hv(1 p
g
)bh
l
vh[
abu
hv(1 p
g
)bhþ
d
v
d
h(
a
þr2
g
þ
d
h)] þ
d
v
d
h(r2
g
þ
d
h)(
a
þr1
g
þ
d
h):(A 13)
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16

Using equation (A 12) and equation (A 4), we get
Uv¼bvNv[
l
vh
d
h(
a
þr2
g
þ
d
h)þ
d
h(r2
g
þ
d
h)(
a
þr1
g
þ
d
h)]
l
vh[
abu
hv(1 p
g
)bhþ
d
v
d
h(
a
þr2
g
þ
d
h)] þ
d
v
d
h(r2
g
þ
d
h)(
a
þr1
g
þ
d
h):(A 14)
Furthermore, plugging-in equation (A 13) to equation (A 6), we get
Iv¼
l
vh
abu
hv(1 p
g
)bhbvNv
l
vh
d
v[
abu
hv(1 p
g
)bhþ
d
v
d
h(
a
þr2
g
þ
d
h)] þ
d
2
v
d
h(r2
g
þ
d
h)(
a
þr1
g
þ
d
h):(A 15)
Plugging-in equation (A 12) to equation (A 11) and after doing a few algebraic computations, we obtain
l
vh(B
l
vh A)¼0,
where
A¼
ab
2
u
vh
u
hv(1 p
g
)bhbvNv
d
2
v
d
hNh(
a
þr1
g
þ
d
h)(r2
g
þ
d
h)
and
B¼
d
vNh[
abu
hv(1 p
g
)bhþ
d
v
d
h(
a
þr2
g
þ
d
h)]:
Here, we have two possible cases: either λ
vh
=0 or Bλ
vh
−A=0.
Case 1. Suppose λ
vh
= 0. Then it follows from equations (A 9), (A 10), and (A 8), respectively, that L
h
=
0, I
h
= 0 and U
h
=b
h
N
h
/δ
h
. It also follows from equations (A 15) and (A 14) that I
v
= 0 and Uv¼bvNv
d
v:
Hence, we have the solution
E0¼bhNh
d
h
,0,0, bvNv
d
v
,0

:
Case 2. Suppose Bλ
vh
−A= 0 and λ
vh
≠0. Then
l
vh ¼
ab
2
u
vh
u
hv(1 p
g
)bhbvNv
d
2
v
d
hNh(
a
þr1
g
þ
d
h)(r2
g
þ
d
h)
d
vNh[
abu
hv(1 p
g
)bhþ
d
v
d
h(
a
þr2
g
þ
d
h)] :
We denote the above as
l

vh. We have the solution
E¼(U
h,L
h,I
h,U
v,I
v),
where
U
h¼bhNh(
a
þr1
g
þ
d
h)(r2
g
þ
d
h)
d
h[(
a
þr1
g
þ
d
h)(r2
g
þ
d
h)þ
l

vh(
a
þr2
g
þ
d
h)] ,
L
h¼
l

vhbhNh(r2
g
þ
d
h)
d
h[(
a
þr1
g
þ
d
h)(r2
g
þ
d
h)þ
l

vh(
a
þr2
g
þ
d
h)] ,
I
h¼
l

vh
a
bhNh
d
h[(
a
þr1
g
þ
d
h)(r2
g
þ
d
h)þ
l

vh(
a
þr2
g
þ
d
h)] ,
U
v¼bvNv[
l

vh
d
h(
a
þr2
g
þ
d
h)þ
d
h(
a
þr1
g
þ
d
h)(r2
g
þ
d
h)]
l

vh[
abu
hv(1 p
g
)bhþ
d
h
d
v(
a
þr2
g
þ
d
h)] þ
d
v
d
h(
a
þr1
g
þ
d
h)(r2
g
þ
d
h)
and I
v¼
l

vh
abu
hv(1 p
g
)bhbvNv
l

vh
d
v[
abu
hv(1 p
g
)bhþ
d
h
d
v(
a
þr2
g
þ
d
h)] þ
d
2
v
d
h(
a
þr1
g
þ
d
h)(r2
g
þ
d
h):
Note that Eis biologically feasible if
l

vh .0. Our assumptions on the positivity of parameter values
imply that Bis always positive. It can also be shown that A> 0, i.e.
ab
2
u
vh
u
hv(1 p
g
)bhbvNv
d
2
v
d
hNh(
a
þr1
g
þ
d
h)(r2
g
þ
d
h)
.1:
Hence, U
h,L
h,I
h,U
v,I
vare all positive.
Appendix B. Computation of the basic reproduction number
Following van Driessche & Watmough [75], we first rearrange our system such that the first nentries
contain the infected populations. We have _
X¼(_
Lh,_
Ih,_
Iv,_
Uh,_
Uv)Twith solution X=(L
h
,I
h
,I
v
,U
h
,U
v
)
T
.
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17

Since there are three compartments for the infected population in our model, we have n=3. Now, _
Xcan
be expressed as
_
X¼F(X)V(X),
where Fi(X) is the rate of appearance of new infections in compartment i, and Vi(X)¼V
i(X)Vþ
i(X)
is the rate of transfer of individuals into compartment iby all other means (V
i(X)) and out of
compartment i(Vþ
i(X)), for i=1,…, 5. In our model, we have
F¼
bu
vh Iv
NhUh
0
bu
hv(1 p
g
)Ih
NhUv
0
0
2
6
6
6
6
4
3
7
7
7
7
5
and V¼
(
a
þr1
g
þ
d
h)Lh

a
Lhþ(r2
g
þ
d
h)Ih
d
vIv
bhNh(r1Lhþr2Ih)
g
þ
bu
vh Iv
NhUhþ
d
hUh
bvNvþ
bu
hv(1 p
g
)Ih
NhUvþ
d
vUv
2
6
6
6
6
6
4
3
7
7
7
7
7
5
:
From here, we obtain the matrices
F¼
@
Fi
@
Xj
(X0)

and V¼
@
Vi
@
Xj
(X0)

,
where X
0
is a disease-free equilibrium and i,j∈{1, 2, 3}. Diekmann et al. [68] defined R0as the spectral
radius, or the dominant eigenvalue, of the matrix FV
−1
. Following this, we have
F(X)¼
00
bu
vh Uh
Nh
00 0
0
bu
hv(1 p
g
)Uv
Nh0
2
43
5,
and evaluated at the DFE,
F(E0)¼
00
bu
vh bh
d
h
00 0
0
bu
hv(1 p
g
)bvNv
d
vNh0
2
43
5:
We also obtain
V(E0)¼
a
þr1
g
þ
d
h00

a
r2
g
þ
d
h0
00
d
v
2
43
5
and
V1(E0)¼
1
a
þr1
g
þ
d
h00
a
(
a
þr1
g
þ
d
h)(r2
g
þ
d
h)1
r2
g
þ
d
h0
00
1
d
v
2
6
6
43
7
7
5
:
Now, solving for the product of F(E
0
) and V
−1
(E
0
), we have
F(E0)V1(E0)¼
00
bu
vhbh
d
h
d
v
000
abu
hv(1p
g
)bvNv
d
vNh(
a
þr1
g
þ
d
h)(r2
g
þ
d
h)
bu
hv(1p
g
)bvNv
d
vNh(r2
g
þ
d
h)0
2
43
5:
Taking the largest eigenvalue of the above, we have
R0¼ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
ab
2
u
vh
u
hv(1 p
g
)bhbvNv
d
2
v
d
hNh(
a
þr1
g
þ
d
h)(r2
g
þ
d
h)
s:
Appendix C. Stability analysis of the steady-state solutions
To study the stability of our equilibrium solutions, we first linearize the system (2.3) near the steady states
and calculate the corresponding Jacobian.
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At the DFE E
0
=(b
h
N
h
/δ
h
,0,0,b
v
N
v
/δ
v
, 0), we have
JE0¼

d
hr1
g
r2
g
0
bu
vhbh
d
h
0(
a
þr1
g
þ
d
h)0 0
bu
vhbh
d
h
0
a
(r2
g
þ
d
h)0 0
00
bu
hv(1p
g
)bvNv
d
vNh
d
v0
00
bu
hv(1p
g
)bvNv
d
vNh0
d
v
2
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
5
:(C 1)
The corresponding characteristic equation is
0¼j
l
IJE0j
¼(
l
þ
d
h)(
l
þ
d
v)(
l
þ
a
þr1
g
þ
d
h)(
l
þr2
g
þ
d
h)(
l
þ
d
v)
ab
2
u
vh
u
hv(1 p
g
)bhbvNv
d
v
d
hNh

:
To prove that E
0
is locally asymptomatically stable, we need to show that the eigenvalues of (C1), i.e. the
roots of the characteristic equation above, are all negative. By assumption, δ
h
,δ
v
are strictly positive, thus
we only need to show that the roots of the cubic polynomial
(
l
þ
a
þr1
g
þ
d
h)(
l
þr2
g
þ
d
h)(
l
þ
d
v)
ab
2
u
vh
u
hv(1 p
g
)bhbvNv
d
v
d
hNh
(C 2)
are negative. We have
(
l
þ
a
þr1
g
þ
d
h)(
l
þr2
g
þ
d
h)(
l
þ
d
v)
ab
2
u
vh
u
hv(1 p
g
)bhbvNv
d
v
d
hNh
¼
l
3þ(
a
þr1
g
þ
d
hþr2
g
þ
d
hþ
d
v)
l
2þ[(
a
þr1
g
þ
d
h)(r2
g
þ
d
h)
þ
d
v(
a
þr1
g
þ
d
hþr2
g
þ
d
h)]
l
þ
d
v(
a
þr1
g
þ
d
h)(r2
g
þ
d
h)
ab
2
u
vh
u
hv(1 p
g
)bhbvNv
d
v
d
hNh
¼:a0
l
3þa1
l
2þa2
l
þa3:
By the Routh–Hurwitz criterion for stability [76], the roots of equation (C 2) are negative if the coefficients
of the cubic polynomial above are all positive and a
1
a
2
>a
0
a
3
. It follows from our assumption on
parameter values that
a0¼1.0, a1¼
a
þr1
g
þ
d
hþr2
g
þ
d
hþ
d
v.0 and
a2¼(
a
þr1
g
þ
d
h)(r2
g
þ
d
h)þ
d
v(
a
þr1
g
þ
d
hþr2
g
þ
d
h).0:
Note that a
3
> 0 if and only if R0<1. Indeed,
d
v(
a
þr1
g
þ
d
h)(r2
g
þ
d
h)bhbvNv
ab
2
u
vh
u
hv(1 p
g
)
d
v
d
hNh
.0
1bhbvNv
ab
2
u
vh
u
hv(1 p
g
)
d
2
v
d
hNh(
a
þr1
g
þ
d
h)(r2
g
þ
d
h)
.0
1R2
0.0
R0,1:
Now, to show that a
1
a
2
>a
0
a
3
, let
A1¼
a
þr1
g
þ
d
h,B1¼r2
g
þ
d
h,C1¼
d
v,D1¼bhbvNv
ab
2
u
vh
u
hv(1 p
g
)
d
v
d
hNh
:
Then, we can write
a1¼A1þB1þC1,a2¼A1B1þC1(A1þB1), a3¼A1B1C1D1:
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19

Thus,
a1a2a0a3¼(A1þB1þC1)[A1B1þC1(A1þB1)] (A1B1C1D1)
¼(A1þB1)[A1B1þC1(A1þB1þC1)] þD1
.0
since A
1
,B
1
,C
1
> 0 and D
1
≥0. Thus, the DFE is locally asymptotically stable when R0,1. On the other
hand, when R0.1, the coefficient a
3
< 0. By Descartes’rule of signs [76], this implies that there will be
one positive eigenvalue which proves the instability of E
0
. Therefore, the disease-free steady state E
0
is
locally asymptotically stable when R0,1 and unstable when R0.1.
To study the stability of E, we look at the eigenvalues of the Jacobian matrix evaluated at E, that is,
JE¼
(
l

vh þ
d
h)r1
g
r2
g
0
bu
vh
U
h
Nh
l

vh (
a
þr1
g
þ
d
h)0 0
bu
vh
U
h
Nh
0
a
(r2
g
þ
d
h)0 0
00
bu
hv(1 p
g
)U
v
Nh(
l

hv þ
d
v)0
00
bu
hv(1 p
g
)U
v
Nh
l

hv 
d
v
2
6
6
6
6
6
6
6
6
6
6
4
3
7
7
7
7
7
7
7
7
7
7
5
:
The corresponding characteristic equation is
0¼j
l
IJEj
¼
l
þ
l

vh þ
d
hr1
g
r2
g
0
bu
vh
U
h
Nh

l

vh
l
þ
a
þr1
g
þ
d
h00
bu
vh
U
h
Nh
0
al
þr2
g
þ
d
h00
00
bu
hv(1 p
g
)U
v
Nh
l
þ
l

hv þ
d
v0
00
bu
hv(1 p
g
)U
v
Nh
l

hv
l
þ
d
v


¼(
l
þ
d
h)(
l
þ
d
v)((
l
þ
l

hv þ
d
v)[(
l
þ
a
þr1
g
þ
d
h)(
l
þr2
g
þ
d
h)þ
l

vh(
l
þ
a
þr2
g
þ
d
h)]

ab
2
u
vh
u
hv(1 p
g
)U
hU
v
N2
h):
Since δ
h
,δ
v
are strictly positive for all time t,t≥0, we only need to show that the roots of the cubic
polynomial
(
l
þ
l

hv þ
d
v)h(
l
þ
a
þr1
g
þ
d
h)(
l
þr2
g
þ
d
h)þ
l

vh(
l
þ
a
þr2
g
þ
d
h)i
ab
2
u
vh
u
hv(1 p
g
)U
hU
v
N2
h
(C 3)
are negative. Substituting the values for U
hand U
v, we obtain
(
l
þ
l

hv þ
d
v)h(
l
þ
a
þr1
g
þ
d
h)(
l
þr2
g
þ
d
h)þ
l

vh(
l
þ
a
þr2
g
þ
d
h)i
ab
2
u
vh
u
hv(1 p
g
)U
hU
v
N2
h
¼
l
3þ(
l

hv þ
d
vþ
l

vh þ
a
þr1
g
þ
d
hþr2
g
þ
d
h)
l
2þh(
a
þr1
g
þ
d
h)(r2
g
þ
d
h)þ
l

vh(
a
þr2
g
þ
d
h)
þ(
l

hv þ
d
v)(
l

vh þ
a
þr1
g
þ
d
hþr2
g
þ
d
h)i
l
þ(
l

hv þ
d
v)h(
a
þr1
g
þ
d
h)(r2
g
þ
d
h)þ
l

vh(
a
þr2
g
þ
d
h)i
bhbvNv
ab
2
u
vh
u
hv(1 p
g
)(
a
þr1
g
þ
d
h)(r2
g
þ
d
h)
d
hNh(
l

hv þ
d
v)[
l

vh(
a
þr2
g
þ
d
h)þ(
a
þr1
g
þ
d
h)(r2
g
þ
d
h)]
¼:b0
l
3þb1
l
2þb2
l
þb3:
royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 8: 201965
20

By the Routh–Hurwitz criterion for stability, the roots of (C 3) are negative if the coefficients of the cubic
polynomial above are positive and b
1
b
2
>b
0
b
3
. It follows from our assumption on the parameter values that
b0¼1.0,
b1¼(
l

hv þ
d
vþ
l

vh þ
a
þr1
g
þ
d
hþr2
g
þ
d
h).0, and
b2¼(
a
þr1
g
þ
d
h)(r2
g
þ
d
h)þ
l

vh(
a
þr2
g
þ
d
h)þ(
l

hv þ
d
v)(
l

vh þ
a
þr1
g
þ
d
hþr2
g
þ
d
h).0:
Note that b
3
> 0 if and only if
(
l

hv þ
d
v)h(
a
þr1
g
þ
d
h)(r2
g
þ
d
h)þ
l

vh(
a
þr2
g
þ
m
h)i

ab
2
u
vh
u
hv(1 p
g
)(
a
þr1
g
þ
d
h)(r2
g
þ
d
h)bhbvNv
d
hNh(
l

hv þ
d
v)[
l

vh(
a
þr2
g
þ
d
h)þ(
a
þr1
g
þ
d
h)(r2
g
þ
d
h)] .0:
Following a rigorous computation, the above inequality is equivalent to R2
0.R0which means R0.1
since R0.0. Now, to show that b
1
b
2
>b
0
b
3
, let
A2¼
a
þr1
g
þ
d
h,B2¼r2
g
þ
d
h,
C2¼
a
þr2
g
þ
d
h,D2¼
l

hv þ
d
v,
and E2¼bhbvNv
ab
2
u
vh
u
hv(1 p
g
)(
a
þr1
g
þ
d
h)(r2
g
þ
d
h)
d
hNh(
l

hv þ
d
v)[
l

vh(
a
þr2
g
þ
d
h)þ(
a
þr1
g
þ
d
h)(r2
g
þ
d
h)] :
Then, we can write
b1¼
l

vh þA2þB2þD2,b2¼A2B2þ
l

vhC2þ
l

vh þA2þB2D2and
b3¼D2A2B2þ
l

vhC2E2:
We have
b1b2b0b3¼(
l

vh þA2þB2)[A2B2þ
l

vhC2þ
l

vh þA2þB2D2]
þD2A2B2þ
l

vhC2þ
l

vh þA2þB2D2
2D2A2B2þ
l

vhC2þE2
.0
since
l

vh .0, A
2
,B
2
,C
2
,D
2
> 0 and E
2
≥0. Hence, the endemic equilibrium Eis locally asymptotically
stable when R0.1.
Appendix D. Proof of the existence of the optimal control u(t)
The existence of the optimal control u(t) such that J(u(t)) ¼min
V
u(t) with state system
dUh
dt¼bhNhþ(r1Lhþr2Ih)u(t)
bu
vh
Iv
Nh
Uh
d
hUh,
dLh
dt¼
bu
vh
Iv
Nh
Uh(
a
þr1u(t)þ
d
h)Lh,
dIh
dt¼
a
Lh(r2u(t)þ
d
h)Ih,
dUv
dt¼bvNv
bu
hv(1 pu(t)) Ih
Nh
Uv
d
vUv
and dIv
dt¼
bu
hv(1 pu(t)) Ih
Nh
Uv
d
vIv,
is given by the convexity of the objective functional integrand. The adjoint equations and transversality
conditions are obtained using Pontryagin’s maximum principle [72]. In particular, differentiation of the
royalsocietypublishing.org/journal/rsos R. Soc. Open Sci. 8: 201965
21

Hamiltonian Hwith respect to the state variables gives the following system:
d
l
1
dt¼
@
H
@
Uh
,d
l
2
dt¼
@
H
@
Lh
,d
l
3
dt¼
@
H
@
Ih
,d
l
4
dt¼
@
H
@
Uv
,d
l
5
dt¼
@
H
@
Iv
with λ
i
(t
f
) = 0, for i=1,…,5.
The optimal control u(t) is derived using the following optimality condition:
@
H
@
u¼cu þr1Lh(
l
1
l
2)þr2Ih(
l
1
l
3)þ
bu
hvpIhUv
Nh
(
l
4
l
5)¼0
at u=u(t) on the set
V
. Hence, we have
u(t)¼1
cr1Lh(
l
2
l
1)þr2Ih(
l
3
l
1)þ
bu
hvpIhUv
Nh
(
l
5
l
4)

:
Taking into account the bounds on the control, we obtain the following characterization of u(t):
u(t)¼min umax,umin ,1
cr1Lh(
l
2
l
1)þr2Ih(
l
3
l
1)þ
bu
hvpIhUv
Nh
(
l
5
l
4)

:
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