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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 dierent 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 eective 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 eective 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 dierent 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
7
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 eective to reduce
the number of infected subpopulation with minimum cost of controls.
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