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Published under licence by IOP Publishing Ltd GFTA 2019
Journal of Physics: Conference Series1562 (2020) 012017
IOP Publishing
doi:10.1088/1742-6596/1562/1/012017
1
Optimal Control of HIV/AIDS Epidemic Model with Two
Latent Stages, Vertical Transmission and Treatment
Nur Shoanah

, Sa'adatul Fitri, Trisilowati and Karunia Theda Kristanti
Mathematics Department, Faculty of Mathematics and Natural Sciences, University of Brawijaya,
Indonesia

Corresponding author:[email protected]
Abstract.In this research, we discussed about optimal control of HIV/AIDS epidemic model with two
latent stages, vertical transmission and treatment. In this model, the population is divided into ve sub-
populations, namely susceptible subpopulation, slow latent subpopulation, fast latent subpopulation, symp-
tomatic subpopulation and AIDS subpopulation. The latent stage is divided into slow latent and fast la-
tent stage depend on the condition of immune system which is di erent for each individual. Treatment
(ART/antiretroviral) is given to infected individu in symptomatic stage. The rate of treatment from symp-
tomatic stage to slow latent stage and to fast latent stage are set asu1(t)andu2(t)control variable, respec-
tively. Here, the objective of optimal control is to minimize the number of infected as well as the cost of
controls. The optimal control is obtained by applying Pontryagin's Principle. In the end, we show some nu-
merical simulations by using Forward-Backward Sweep Method. Numerical simulation result show that the
combination ofu1andu2control is the most e ective control to reduce the number of infected/symptomatic
subpopulation with minimum cost of controls.
Keywords: HIV/AIDS; two latent stages; vertical transmission; treatment; optimal control, Pontryagin's
Principle.`
1. Introduction
Acquired Immune Deciency Syndrome (AIDS) is the disease in human immune system that caused by
human immune deciency virus (HIV). HIV spread through both horizontal and vertical transmission.
Horizontal transmission occurs through direct or indirect contact with infected individu, such as sexual
intercourse, blood transfusion, using the HIV-contamined injection equipment and direct contact with
HIV-infected blood or uid. Vertical transmission is the process of spreading HIV/AIDS from a mother
who has positive HIV to her baby, that can be happened during pregnancy, childbirth, or breastfeeding
[1],[2].
AIDS has developed into a global epidemic in the world since rst identied as a disease in 1981.
There is no e ective medicine to cure HIV /AIDS. One of the prevention strategy is avoid the contact
with the virus. The spreading of HIV/AIDS can be represented in mathematical model. By analyzing the
appropriate mathematical model, the better understanding of the major factors that caused the pandemic
of HIV/AIDS can be obtained and be useful information to know the best prevention strategy. Many
researches have been developed this model. May and Anderson [3] introduce the HIV/AIDS model
for the rst time at 1986. Li et al. [1] discussed about global dynamics of an SEIR epidemic model
with vertical transmission. At 2013, Huo and Feng [4] constructed and analyzed an HIV/AIDS epidemic
model with di erent latent stages (slow latent and fast latent) and treatment. Mahato et al., [2] proposed

GFTA 2019
Journal of Physics: Conference Series1562 (2020) 012017
IOP Publishing
doi:10.1088/1742-6596/1562/1/012017
2
a mathematical model SEIA with vertical transmission of AIDS epidemic. Shoanah et al [5] developed
model in Huo and Feng [4] by adding vertical transmission.
The optimal control of HIV/AIDS epidemic model also have been developed. Generally, the aim of
optimal control on these cases is to minimize the infected subpopulation as well as the cost of control.
In 2014, Sule and Abdullah disscussed the treatment and education as control strategy. Numerical
simulation show that treatment and education for infected individu has positive impact for the control
of HIV/AIDS spreading. Silva and Torres (2017) studied the optimal control of HIV/AIDS epidemic
model through PrEP. Based on numerical simulation, PrEP signicantly reduce the spreading of HIV.
Marsudi (2019) studied optimal control of HIV model with changing behavior through an education
campaign, screening and treatment.
Here, we construct HIV/AIDS model with control and solve this optimal control problem. We
construct HIV/AIDS model with control from model in [5], that is the HIV/AIDS epidemic model with
two latent stages, vertical transmission and treatment. This model has some assumptions: some of the
suspectible individuals have other chronic diseases, the vertical transmission is happened in case infected
mother (in fast latent stage), the infant from infected mother always infected and enter the symptomatic
stage, the number of infected baby is less than the number of individu who died because AIDS disease.
After that, we solve optimal control problem by applying Minimum Pontryagin Principle.
2. Mathematical Model
HIV/AIDS epidemic model with two latent stages, vertical transmission and treatment as in [5] consists
of ve subpopulations, namely suspectible (S), slow laten (I1), fast latent (I2), symptomatic (J) and AIDS
(A). The model can be written as follows
dS
dt
=I2( 1I2S+ 2JS)S;
dI1
dt
=p 1I2S+q 2JS+1J("+)I1;
dI2
dt
=(1p) 1I2S+(1q) 2JS+"I1+2J(p1+)I2; (1)
dJ
dt
=(p1+)I2(1+2+p2+)J;
dA
dt
=p2J(+ )A:
The parameters that used in the models are described in Table 1.
Based on dynamical analysis in [5], model (1) has two eqilibrium point, that is disease-free
equilibrium and endemi equilibrium point. IfR0<1, the disease-free equilibrium point will be globally
asymptotically stable, while ifR0>1, the endemic equilibrium point globally asymptotically stable.
Therefore, whenR0>1, the outbreak of HIV/AIDS occured. We need to apply control in this condition.
We add two time-dependent controls in the model (1), that is treatmentu1(t)andu2(t)by changing
constant treatment rate1and2, respectively. Theu1(t)control treatment is the treatment for individu
in slow latent stageI1, while theu2(t)control treatment is the treatment for individu in fast latent stage

GFTA 2019
Journal of Physics: Conference Series1562 (2020) 012017
IOP Publishing
doi:10.1088/1742-6596/1562/1/012017
3
I2. Now we have the model (1) with controls,
dS
dt
=I2( 1I2S+ 2JS)S;
dI1
dt
=p 1I2S+q 2JS+u1J("+)I1;
dI2
dt
=(1p) 1I2S+(1q) 2JS+"I1+u2J(p1+)I2; (2)
dJ
dt
=(p1+)I2(u1+u2+p2+)J;
dA
dt
=p2J(+ )A:
withS(0)=S0;I1(0)=I10;I2(0)=I20;J(0)=J0;A(0)=A0as initial conditions.
3. Optimal Control
The aim for optimal control problem of model (2) is to minimize this cost function as objective function
that given by
F[u1;u2]=
Z
T
0
w
2
J+
w1
2
u
2
1
+
w2
2
u
2
2
dt (3)
with equation system (2) as the constraint. Herew;w1;w2are weight related to symptomatic
subpopulation, controlu1and controlu2, respectively.Tis nal time of control. The optimal control is
determined such that
F[u

1
;u

2
]=minfF[u1;u2]ju1;u22Ug
whereU=f0u1;u21g. Then we construct Hamiltonian function,
H=
w
2
J+
w1
2
u
2
1
+
w2
2
u
2
2
+S(I2( 1I2S+ 2JS)S)
+I1
(p 1I2S+q 2JS+u1J("+)I1)
+I2
((1p) 1I2S+(1q) 2JS+"I1+u2J(p1+)I2)
+J((p1+)I2(u1+u2+p2+)J)+A(p2J(+ )A)
whereS; I1
; I2
; J; Aare costate variables.
Based on Pontryagin's Principle, the optimal solution of Hamiltonian function can be obtained if it
satisfy these conditions
3.1. State Equations
By di erentiating the Hamiltonian function with respect to each costate variable, we get the state
equations:
dS
dt
=
@H
@S
=I2( 1I2S+ 2JS)S;
dI1
dt
=
@H
@I1
=p 1I2S+q 2JS+u1J("+)I1;
dI2
dt
=
@H
@I2
=(1p) 1I2S+(1q) 2JS+"I1+u2J(p1+)I2;
dJ
dt
=
@H
@J
=(p1+)I2(u1+u2+p2+)J;
dA
dt
=
@H
@A
=p2J(+ )A;

GFTA 2019
Journal of Physics: Conference Series1562 (2020) 012017
IOP Publishing
doi:10.1088/1742-6596/1562/1/012017
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withS(0)=S0;I1(0)=I10;I2(0)=I20;J(0)=J0;A(0)=A0as initial conditions.
3.2. Costate Equations
Costate equation is negative value of the Hamiltonian function di erentiated with respect to costate
variable:
dS
dt
=
@H
@S
=( 1I2+ 2J+)S(p 1I2+q 2J)I1
((1p) 1I2+(1q) 2J)I2
;
dI1
dt
=
@H
@I1
=(+)I1
I2
;
dI2
dt
=
@H
@I2
=( 1S+)Sp 1SI1
((1p) 1Sp1+)I2
p1J;
dJ
dt
=
@H
@J
=
w
2
+ 2SS(q 2S=u1)I1
((1q) 2S+u2)I2
(u1u2p2)Jp2A;
dA
dt
=
@H
@A
=(+ )A;
withS(T)=I1
(T)=I2
(T)=J(T)=A(T)=0 as transversal conditions.
3.3. Stationary Conditions
By di erentiating the Hamiltonian function with respect to each variable control, we get the stationary
conditions:
@H
@u1
=0
w1u1(t)+I1
J(t)J(t)J=0
u

1
(t)=
(JI1
)J

(t)
w1
(4)
@H
@u2
=0
w2u2(t)+I2
J(t)J(t)J=0
u

2
(t)=
(JI2
)J

(t)
w2
(5)
Sinceu1(t)andu2(t)are dened in 0u1(t);u2(t)1, then based on (4) and (5), we get
u

1
=
8
>>>>>>>>
<
>>>>>>>>:
0 ;if
(JI1
)J

(t)
w1
0
(JI1
)J

(t)
w1
;if 0<
(JI1
)J

(t)
w1
<1
1 ;if
(JI1
)J

(t)
w1
1
u

2
=
8
>>>>>>>>
<
>>>>>>>>:
0 ;if
(JI2
)J

(t)
w2
0
(JI2
)J

(t)
w2
;if 0<
(JI2
)J

(t)
w2
<1
1 ;if
(JI2
)J

(t)
w2
1

GFTA 2019
Journal of Physics: Conference Series1562 (2020) 012017
IOP Publishing
doi:10.1088/1742-6596/1562/1/012017
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and optimal controlu1(t)andu2(t)can be simplied as
u

1
=min
n
max

0;
(JI1
)J

(t)
w1

;1
o
u

2
=min
n
max

0;
(JI2
)J

(t)
w2

;1
o
4. Numerical Simulations
In order to solve the optimal control problem above, we use Forward-Backward Sweep method. We use
forward-di erence of fourth order Runge Kutta to solve the state equations, while backward-di erence
of fourth order of Runge Kutta to solve costate equations. In all numerical simulations we use parameter
as in Table 1,S(0)=561:8;I1(0)=51:1;I2(0)=14:3;J(0)=11:2;A(0)=5 as initial conditions and
w=20;w1=20;w2=2 as weights related to symptomatic subpopulation, controlu1and controlu2,
respectively.
Table 1:Parameters of The Model
Parameter Description value
 Recruitment rate of the population 0.545
1 Transmission coecient ofI2 0.0001
2 Transmission coecient ofJ 0.006
p Fraction ofSbeing infected byI2and enteringI10.9
q Fraction ofSbeing infected byJand enteringI1 0.9
" Progression rateI1toI2 0.002
p1 Progression rateI2toJ 0.01
p2 Progression rateJtoA 0.03
1 Treatment rate fromJtoI1 u1(t)
2 Treatment rate fromJtoI2 u2(t)
Vertical transmission rate 0.005
 Naturally death rate 0.01
The disease-related death rate 0.01
We simulate 3 strategies that possible to be applied:
(i) u1(t)
(a)The number of subpopulationJ(b)The optimal control prole ofu1(t)
Figure 1:Numerical simulation result of strategy A

GFTA 2019
Journal of Physics: Conference Series1562 (2020) 012017
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doi:10.1088/1742-6596/1562/1/012017
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(ii) u2(t)
(a)The number of subpopulationJ(b)The optimal control prole ofu2(t)
Figure 2:Numerical simulation result of strategy B
(iii) u1(t)andu2(t)
(a)The number of subpopulationJ(b)The optimal control prole ofu1(t)andu2(t)
Figure 3:Numerical simulation result of strategy C
Figure 1, 2 and 3 show the numerical simulations result for implementation of strategy A, B, C,
respectively. Based on Figure 1a, 2a and 3a, we can see that all strategies reduce the number of
infected subpopulation signicantly. There is no signicant di erences in all strategies based on the
number of infected subpopulation result but regarding the cost, strategy C has the minimum cost. The
optimal control proles for each strategy is depicted in Figure 1b, 2b and 3b. The number of infected
subpopulationJin condition without and with control and its implementation cost for strategy A, B and
C can be shown in Table 2,
Table 2:The number of infected subpopulationJin condition without and with control
and its implementation cost for each strategy
StrategyJwithout controlJwith control Cost
A 15.1836 0.6114 887.8016
B 15.1836 0.6746 778.7879
C 15.1835 0.3823 609.6369
Based on the simulation result, we can conclude that by usingw=20;w1=20;w2=2 as the weights,
the combination of controlu1(t)andu2(t)is e ective to reduce the number of infected/symptomatic
subpopulation with minimum cost of controls. For the next research, it can be useful if we also analyze
the cost e ectiveness for each strategy by using some methods such as Incremental Cost E ectiveness
Ratio (ICER).

GFTA 2019
Journal of Physics: Conference Series1562 (2020) 012017
IOP Publishing
doi:10.1088/1742-6596/1562/1/012017
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5. Conclusion
In this paper, we construct HIV/AIDS epidemic model with control by set the rate of treatments in
HIV/AIDS model as control variables. By using Pontryagin Minimum Principle, we obtained the optimal
control of HIV/AIDS model that have been constructed. Numerical simulations show that by using
w=20;w1=20;w2=2 as the weights, the combination of controlu1(t)andu2(t)is e ective to reduce
the number of infected subpopulation with minimum cost of controls.
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