Research Article An Optimal Treatment Control of TB-HIV …downloads.hindawi.com/journals/ijmms/2016/8261208.pdf · 2019-07-30 · Research Article An Optimal Treatment Control of

Research ArticleAn Optimal Treatment Control of TB-HIV Coinfection

Fatmawati1 and Hengki Tasman2

1Departemen Matematika, Fakultas Sains dan Teknologi, Universitas Airlangga, Surabaya 60115, Indonesia2Departemen Matematika, Fakultas Matematika dan Ilmu Pengetahuan Alam, Universitas Indonesia, Depok 16424, Indonesia

Correspondence should be addressed to Fatmawati; [email protected]

Received 6 January 2016; Revised 5 March 2016; Accepted 7 March 2016

Academic Editor: Ram U. Verma

Copyright © 2016 Fatmawati and H. Tasman. This is an open access article distributed under the Creative Commons AttributionLicense, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properlycited.

An optimal control on the treatment of the transmission of tuberculosis-HIV coinfection model is proposed in this paper. We usetwo treatments, that is, anti-TB and antiretroviral, to control the spread of TB and HIV infections, respectively. We first present anuncontrolled TB-HIV coinfection model. The model exhibits four equilibria, namely, the disease-free, the HIV-free, the TB-free,and the coinfection equilibria. We further obtain two basic reproduction ratios corresponding to TB and HIV infections. Theseratios determine the existence and stability of the equilibria of the model. The optimal control theory is then derived analyticallyby applying the Pontryagin Maximum Principle. The optimality system is performed numerically to illustrate the effectiveness ofthe treatments.

1. Introduction

Tuberculosis (TB) is an infectious disease caused by bacteriaMycobacterium tuberculosis that most often attack the lungs.The bacteria are spread through air from one person toanother when the people with active TB cough, sneeze, speak,or sing. People nearby may breathe in these bacteria andbecome infected. According to the World Health Organiza-tion (WHO), one-third of the world’s population is infectedwith TB [1]. In 2013, 9 million people around the worldbecame sick with TB disease and around 1.5 million TB-related deaths worldwide were reported. TB is the mostcommon opportunistic disease that affects people infectedwith HIV [2].

HIV stands for human immunodeficiency virus that canlead to acquired immunodeficiency syndrome (AIDS). HIVcan be transmitted via the exchange of a variety of body fluidsfrom infected individuals, such as blood, breast milk, semen,and vaginal secretions. There is no cure for HIV infection.However, effective treatment with antiretroviral (ARV) drugscan control the virus so that people with HIV can enjoyhealthy and productive lives [3]. As reported in WHO factsheet (2013), at least one-third of the 34 million people livingwith HIV worldwide are infected with TB. HIV and TB form

a lethal combination, each speeding the other’s progress. TBis one of the leading causes of death among people living withHIV. Almost 25% of deaths among people with HIV are dueto TB [1]. Therefore, an effective strategy is needed to controlthe transmission of TB-HIV coinfection in the population.

Mathematical models provide an important tool inunderstanding the spread and control of TB-HIV coinfectiondiseases. The dynamics of the transmission of TB-HIVcoinfection model have been studied by many researchers[4–7]. Gakkhar and Chavda [4] formulated the dynamics ofTB-HIV coinfection model with the population divided intofour subclasses: the susceptible class, the TB infective class,the HIV infective class, and the TB-HIV coinfection class.They found the basic reproduction number for each of thediseases and checked the stability results for the equilibriumpoints. Naresh et al. [5] proposed a model to study the effectof tuberculosis on the transmission dynamics of HIV in alogistically growing human population. Roeger et al. [6] focuson the joint dynamics of HIV and TB in a pseudocompet-itive environment, at the population level. Sharomi et al.[7] discussed the synergistic interaction between HIV andMycobacterium tuberculosisusing a deterministicmodel, withmany of the essential biological and epidemiological featuresof the two infections.

Hindawi Publishing CorporationInternational Journal of Mathematics and Mathematical SciencesVolume 2016, Article ID 8261208, 11 pageshttp://dx.doi.org/10.1155/2016/8261208

2 International Journal of Mathematics and Mathematical Sciences

Λ

S It

𝛽tSIt

𝛿S

𝛽hSIh

u1𝛼1It

𝛿It

𝜙𝛽hItIh

𝜎1𝛽tIhItIh

Iht

𝛿Ih𝛿Iht

u1𝛼2Iht

𝜎2𝛽tAhIt

(1 − u2)𝛾1Ih (1 − u2)𝛾2Iht

AhAht

u1𝛼3Aht

𝛿Ah 𝜇1Ah 𝛿Aht 𝜇

2Aht

Figure 1: TB-HIV coinfection transmission diagram.

In this paper, mathematical model with an optimalcontrol on the treatment of TB-HIV coinfection is proposed.The optimal control strategies have been applied to thestudies on epidemiological models such as HIV, TB, HepatitisC, Malaria, coinfection Malaria-Cholera, and HIV-Malariadiseases dynamics [8–16]. Very few studies have been appliedin the area of optimal control theory to TB-HIV coinfectionmodels. Recently, Agusto and Adekunle [17] have used opti-mal control strategies associated with treating symptomaticindividuals with TB using the two-strain TB-HIV/AIDStransmission model. The aim of this study is to analyze theeffect of two treatment scenarios, that is, anti-TB and ARV, tocontrol the spread of TB-HIV coinfection diseases.

The organization of this paper is as follows. In Section 2,we derive a model of tuberculosis-HIV coinfection transmis-sion with controls on anti-TB and ARV treatment.Themodelis analyzed in Section 3. In Section 4, we show the numericalsimulations to illustrate the effectiveness of the treatments.The conclusion of this paper could be seen in Section 5.

2. Model Formulation

We assume that human population is homogeneous andclosed. The total population, denoted by 𝑁, is classified intosix classes, namely, the susceptible class (𝑆), the infected withTB only and susceptible to HIV class (𝐼

𝑡), the infected with

HIV only and susceptible to TB class (𝐼ℎ), the infected with

TB and HIV both class (𝐼ℎ𝑡), the infected with AIDS only and

susceptible to TB class (𝐴ℎ), and the infected with TB and

AIDS both class (𝐴ℎ𝑡). We also assume that the susceptible

cannot get TB and HIV infection simultaneously at the sametime.

We consider the anti-TB treatment control 𝑢1and the

ARV control 𝑢2. The control functions 𝑢

1and 𝑢

2are defined

on interval [0, 𝑡𝑓], where 0 ≤ 𝑢

𝑖(𝑡) ≤ 1, 𝑡 ∈ [0, 𝑡

𝑓], 𝑖 = 1, 2,

and 𝑡𝑓denotes the end time of the controls.

We use the transmission diagram as in Figure 1 for deriv-ing our model.

The model is as follows:

𝑑𝑆

𝑑𝑡= Λ + 𝑢

1𝛼1𝐼𝑡− 𝛽𝑡𝑆𝐼𝑡− 𝛽ℎ𝑆𝐼ℎ− 𝛿𝑆,

𝑑𝐼𝑡

𝑑𝑡= 𝛽𝑡𝑆𝐼𝑡− 𝑢1𝛼1𝐼𝑡− 𝛿𝐼𝑡− 𝜙𝛽ℎ𝐼𝑡𝐼ℎ,

𝑑𝐼ℎ

𝑑𝑡= 𝛽ℎ𝑆𝐼ℎ+ 𝑢1𝛼2𝐼ℎ𝑡− 𝜎1𝛽𝑡𝐼ℎ𝐼𝑡− (1 − 𝑢

2) 𝛾1𝐼ℎ

− 𝛿𝐼ℎ,

𝑑𝐼ℎ𝑡

𝑑𝑡= 𝜎1𝛽𝑡𝐼ℎ𝐼𝑡− 𝑢1𝛼2𝐼ℎ𝑡− (1 − 𝑢

2) 𝛾2𝐼ℎ𝑡− 𝛿𝐼ℎ𝑡

+ 𝜙𝛽ℎ𝐼𝑡𝐼ℎ,

𝑑𝐴ℎ

𝑑𝑡= (1 − 𝑢

2) 𝛾1𝐼ℎ+ 𝑢1𝛼3𝐴ℎ𝑡− 𝜎2𝛽𝑡𝐴ℎ𝐼𝑡

− (𝛿 + 𝜇1) 𝐴ℎ,

𝑑𝐴ℎ𝑡

𝑑𝑡= (1 − 𝑢

2) 𝛾2𝐼ℎ𝑡+ 𝜎2𝛽𝑡𝐴ℎ𝐼𝑡− 𝑢1𝛼3𝐴ℎ𝑡

− (𝛿 + 𝜇2) 𝐴ℎ𝑡.

(1)

The region of biological interest of model (1) is

Ω = {(𝑆, 𝐼𝑡, 𝐼ℎ, 𝐼ℎ𝑡, 𝐴ℎ, 𝐴ℎ𝑡) ∈ R6

+: 0 ≤ 𝑁 ≤

Λ

𝛿} , (2)

and all of the parameters used in model (1) are nonnegative.The description of the parameters is given below.

Parameters of Model (1). Consider the following:

Λ: recruitment rate into the population.𝛿: natural death rate.𝛽𝑡: infection rate for TB.

𝛽ℎ: infection rate for HIV.

𝜎1: progression rate from HIV only to TB infection.

𝜎2: progression rate from AIDS only to TB infection.

𝜙: progression rate from TB only to HIV infection.𝛼1: recovery rate from TB.

𝛼2: recovery rate from TB of TB-HIV coinfection.

𝛼3: recovery rate from TB of TB-AIDS coinfection.

𝜇1: AIDS disease induced death rate.

𝜇2: TB-AIDS disease induced death rate.

𝛾1: progression rate fromHIV only to AIDS infection.

𝛾2: progression rate from TB-HIV coinfection to TB-

AIDS coinfection.

International Journal of Mathematics and Mathematical Sciences 3

Model (1) is well posed in the nonnegative region R6+

because the vector field on the boundary does not point tothe exterior. So, if it is given an initial condition in the region,then the solution is defined for all time 𝑡 ≥ 0 and remains inthe region.

We seek to minimize the number of TB-HIV/AIDScoinfections while keeping the costs of applying anti-TB andARV treatment controls as low as possible. We consider anoptimal control problem with the objective function given by

𝐽 (𝑢1, 𝑢2)

= ∫

𝑡𝑓

0

(𝐼𝑡+ 𝐼ℎ𝑡+ 𝐴ℎ+ 𝐴ℎ𝑡+𝑐1

2𝑢2

1+𝑐2

2𝑢2

2)𝑑𝑡,

(3)

where 𝑐1and 𝑐2are the weighting constants for anti-TB and

ARV treatment efforts, respectively.We take a quadratic formfor measuring the control cost [12, 13, 17]. The terms 𝑐

1𝑢2

1and

𝑐2𝑢2

2describe the cost associated with the anti-TB and ARV

treatment controls, respectively. Larger values of 𝑐1and 𝑐2will

imply more expensive implementation cost for anti-TB andARV treatment efforts.

Our goal is to find an optimal control pair 𝑢∗1and 𝑢∗2such

that

𝐽 (𝑢∗

1, 𝑢∗

2) = minΓ

𝐽 (𝑢1, 𝑢2) , (4)

where Γ = {(𝑢1, 𝑢2) | 0 ≤ 𝑢

𝑖≤ 1, 𝑖 = 1, 2}.

3. Model and Sensitivity Analysis

Consider model (1) without the control functions 𝑢1and 𝑢

2.

Let

𝑅𝑡=Λ𝛽𝑡

𝛿2

𝑅ℎ=

Λ𝛽ℎ

𝛿 (𝛾1+ 𝛿)

.

(5)

The parameters 𝑅𝑡and 𝑅

ℎare basic reproduction ratios for

TB infection and HIV infection, respectively. These ratiosdescribe the number of secondary cases of primary caseduring the infectious period due to the type of infection[18, 19].

By setting 𝑢1= 𝑢2= 0, model (1) has four equilibria (with

respect to the coordinates (𝑆, 𝐼𝑡, 𝐼ℎ, 𝐼ℎ𝑡, 𝐴ℎ, 𝐴ℎ𝑡)); these are as

follows:

(i) The disease-free equilibrium 𝐸0= (Λ/𝛿, 0, 0, 0, 0, 0).

This equilibrium always exists.

(ii) The TB-endemic equilibrium 𝐸𝑡= (𝛿/𝛽

𝑡, (𝛿/𝛽𝑡)(𝑅𝑡−

1), 0, 0, 0, 0). The equilibrium 𝐸𝑡exists if 𝑅

𝑡> 1.

(iii) The HIV-endemic equilibrium 𝐸ℎ= ((𝛾

1+ 𝛿)/𝛽

ℎ,

0, (𝛿/𝛽ℎ)(𝑅ℎ− 1), 0, (𝛾

1𝛿/𝛽ℎ(𝛿 + 𝜇

1))(𝑅ℎ− 1), 0). The

equilibrium 𝐸ℎexists if 𝑅

ℎ> 1.

Rt

Rh

1

10

E0

Et

Et Eht

E0

E0 E0

Et

Eh

Eh

E0 Eh

Figure 2: The diagram of equilibria with respect to 𝑅ℎand 𝑅

𝑡.

(iv) The TB-HIV-endemic equilibrium 𝐸ℎ𝑡= (𝑆∗, 𝐼∗

𝑡, 𝐼∗

ℎ,

𝐼∗

ℎ𝑡, 𝐴∗

ℎ, 𝐴∗

ℎ𝑡), where

𝑆∗=𝜎1𝛽𝑡𝐼∗

𝑡+ 𝛾1+ 𝛿

𝛽ℎ

,

𝐼∗

ℎ=𝛽2

𝑡𝜎1𝐼∗

𝑡+ 𝛿𝛽ℎ(𝑅𝑡/𝑅ℎ− 1)

𝛽2

ℎ

,

𝐼∗

ℎ𝑡=(𝜎1𝛽𝑡+ 𝜙𝛽ℎ)

(𝛾2+ 𝛿)

𝐼∗

ℎ𝐼∗

𝑡,

𝐴∗

ℎ=

𝛾1𝐼∗

ℎ

𝜎2𝛽𝑡𝐼∗

𝑡+ 𝛿 + 𝜇

1

,

𝐴∗

ℎ𝑡=𝛾2𝐼∗

ℎ𝑡+ 𝜎2𝛽𝑡𝐴∗

ℎ𝐼∗

𝑡

𝛿 + 𝜇2

,

(6)

and 𝐼∗𝑡satisfies the quadratic equation

𝐴0(𝐼∗

𝑡)2

+ 𝐴1𝐼∗

𝑡+ 𝐴2= 0, (7)

where

𝐴0= 𝛽2

𝑡𝜎1(𝛽𝑡𝜎1+ 𝛽ℎ𝜙) ,

𝐴1

=𝛿3𝑅𝑡(𝛿 + 𝛾

1) (𝛿𝜎1𝑅ℎ(𝜙 − 1) + 2𝛿𝜎

1𝑅𝑡+ 𝜙𝑅𝑡(𝛿 + 𝛾

1))

Λ2,

𝐴2= −

𝛿2(𝛿 + 𝛾

1)2

(𝑅ℎ− 𝑅𝑡+ 𝜙𝑅ℎ(𝑅ℎ− 1))

Λ.

(8)

The HIV-TB coinfection equilibrium 𝐸ℎ𝑡exists if 𝑅

ℎ, 𝑅𝑡> 1

and 𝜙𝑅2ℎ+ 𝑅ℎ(1 − 𝜙) > 𝑅

𝑡.

Summarizing the above results, we get diagram of exis-tence of equilibriawith respect to𝑅

ℎand𝑅

𝑡as in Figure 2.The


35

30

25

20

15

10

5

0.5 1.0 1.5 2.0 2.5 3.0

Ih

Rt

Rh > 1

Figure 3: The bifurcation diagram for 𝑅𝑡versus 𝐼

ℎ.

40

30

20

10

0.5 1.0 1.5 2.0 2.5 3.0

It

Rt

Rh > 1


𝑡.

numerical bifurcation diagrams for the basic reproductionnumbers𝑅

𝑡versus the infective classes 𝐼

ℎ, 𝐼𝑡, and 𝐼

ℎ𝑡are given

in Figures 3–5, respectively. In Figures 3–5, 𝑅ℎis fixed for a

value larger than one.The following theorems give the stability criteria of the

equilibriums.

Theorem 1. The disease-free equilibrium 𝐸0is locally asymp-

totically stable if 𝑅𝑡, 𝑅ℎ< 1 and unstable if 𝑅

𝑡, 𝑅ℎ> 1.

Proof. Linearizing model (1) near the equilibrium 𝐸0gives

eigenvalues −𝛿, −(𝛾2+ 𝛿), −(𝜇

1+ 𝛿), −(𝜇

2+ 𝛿), 𝛿(𝑅

𝑡− 1),

and (𝛾1+ 𝛿)(𝑅

ℎ− 1). It is clear that all of the eigenvalues are

negative if 𝑅𝑡, 𝑅ℎ< 1. So, if 𝑅

𝑡, 𝑅ℎ< 1, the equilibrium 𝐸

0is

locally asymptotically stable. Otherwise, it is unstable.

Theorem 2. Suppose that the TB-endemic equilibrium 𝐸𝑡

exists. It is locally asymptotically stable if 𝑅ℎ/𝑅𝑡< 1; otherwise

it is unstable.

Proof. Linearizing model (1) near the equilibrium 𝐸𝑡gives

eigenvalues −𝛿, −(𝛾2+𝛿), −(𝛿+𝜇

2), 𝛿(1−𝑅

𝑡), −𝜇1−𝛿[𝜎2(𝑅𝑡−

1)+1], and (𝑅ℎ/𝑅𝑡−1)(𝛿+𝛾

1)−𝛿𝜎1(𝑅𝑡−1). So, the equilibrium

2.0

1.5

1.0

0.5

0.5 1.0 1.5 2.0 2.5 3.0

Iht

Rt

Rh > 1


ℎ𝑡.

𝐸𝑡is locally asymptotically stable if 𝑅

𝑡/𝑅ℎ< 1; otherwise it is

unstable.

Theorem 3. Suppose that the HIV-endemic equilibrium 𝐸ℎ

exists. It is locally asymptotically stable if 𝑅𝑡/𝑅ℎ< 1; otherwise

it is unstable.

Proof. Linearizing model (1) near the equilibrium 𝐸ℎgives

eigenvalues −(𝜇1+ 𝛿), −(𝜇

2+ 𝛿), −(𝛾

2+ 𝛿), and 𝛿(𝑅

𝑡/𝑅ℎ−

1−𝜙(𝑅ℎ−1)) and the roots of quadratic equation 𝑥2+𝛿𝑅

ℎ𝑥+

𝛿(𝛿+𝛾1)(𝑅ℎ−1) = 0. So, if𝑅

𝑡/𝑅ℎ< 1, then the equilibrium𝐸

ℎ

is locally asymptotically stable; otherwise it is unstable.

In the following we investigate the sensitivity of thebasic reproduction numbers 𝑅

𝑡and 𝑅

ℎto the parameters

in the model. The sensitivity analysis determines the modelrobustness to parameter values. Here, we could know theparameters that have a high impact on the reproductionnumbers (𝑅

𝑡and 𝑅

ℎ). Using the approach in [20], we derived

the analytical expression for sensitivity index of 𝑅𝑡and 𝑅

ℎto

each parameter.The normalized forward sensitivity index of a variable, ℎ,

that depends differentially on a parameter, 𝑙, is defined as

Υℎ

𝑙fl𝜕ℎ

𝜕𝑙

𝑙

ℎ. (9)

Now, using the parameter values in Table 1, we have thefollowing results in Table 2. The sensitivity index of 𝑅

𝑡with

respect to 𝛽𝑡is

Υ𝑅𝑡

𝛽𝑡fl𝜕𝑅𝑡

𝜕𝛽𝑡

𝛽𝑡

𝑅𝑡

= 1. (10)

The sensitivity indices of the basic reproduction numbers(𝑅𝑡and 𝑅

ℎ) to parameters (see Table 2), such as recruitment

rate of the population (Λ), natural death rate (𝛿), infectionrate for HIV (𝛽

ℎ), and progression rate from HIV only to

AIDS infection (𝛾1), can be derived in the same way as (10).

In the sensitivity indices of 𝑅𝑡, since Υ𝑅𝑡

𝛽𝑡= 1, increasing

(or decreasing) infection rate for TB, 𝛽𝑡, by 10%, increases

(or decreases) the reproduction number 𝑅𝑡by 10%. In the


same way, increasing (or decreasing) recruitment rate of thepopulation Λ by 10% increases (or decreases) 𝑅

𝑡by 10% and

in like manner, increasing (or decreasing) natural death rate𝛿 by 10% decreases (increases) 𝑅

𝑡by 20%.

Similarly, for the sensitivity indices of 𝑅ℎ, since Υ𝑅ℎ

𝛽ℎ=

1, increasing (or decreasing) infection rate for HIV, 𝛽ℎ, by

10%, increases (or decreases) the reproduction number 𝑅ℎ

by 10%. Thus, increasing (or decreasing) recruitment rateof the population Λ, by 10%, increases (or decreases) 𝑅

ℎby

10%. Also increasing (or decreasing) natural death rate 𝛿 by10% decreases (increases) 𝑅

ℎby 16,67%. In a similar manner,

increasing (or decreasing) progression rate fromHIV only toAIDS infection 𝛾

1by 10% decreases (increases) 𝑅

ℎby 3,33%.

4. Analysis of Optimal Control

Next, we analyze model (1) with its control functions 𝑢1and

𝑢2. Consider the objective function (3) for model (1). The

necessary conditions to determine the optimal controls 𝑢∗1

and 𝑢∗2such as condition (4) with constraint model (1) could

be obtained using the Pontryagin Maximum Principle [21].The principle converts (1)–(4) into minimizing Hamiltonianfunction𝐻 problem with respect to (𝑢

1, 𝑢2); that is,

𝐻(𝑆, 𝐼𝑡, 𝐼ℎ, 𝐼ℎ𝑡, 𝐴ℎ, 𝐴ℎ𝑡, 𝑢1, 𝑢2, 𝜆1, 𝜆2, . . . , 𝜆

6)

= 𝐼𝑡+ 𝐼ℎ𝑡+ 𝐴ℎ+ 𝐴ℎ𝑡+𝑐1

2𝑢2

1+𝑐2

2𝑢2

2+

6

∑

𝑖=1

𝜆𝑖𝑔𝑖,

(11)

where 𝑔𝑖denotes the right hand side of model (1) which is the

𝑖th state variable equation. The variables 𝜆𝑖, 𝑖 = 1, 2, . . . , 6,

are called adjoint variables satisfying the following costateequations:

𝑑𝜆1

𝑑𝑡= (𝜆1− 𝜆2) 𝛽𝑡𝐼𝑡+ (𝜆1− 𝜆3) 𝛽ℎ𝐼ℎ+ 𝜆1𝛿,

𝑑𝜆2

𝑑𝑡= −1 + (𝜆

2− 𝜆1) 𝑢1𝛼1+ (𝜆1− 𝜆2) 𝛽𝑡𝑆 + 𝜆2𝛿

+ (𝜆3− 𝜆4) 𝜎1𝛽𝑡𝐼ℎ+ (𝜆2− 𝜆4) 𝜙𝛽ℎ𝐼ℎ

+ (𝜆5− 𝜆6) 𝜎2𝛽𝑡𝐴ℎ,

𝑑𝜆3

𝑑𝑡= (𝜆1− 𝜆3) 𝛽ℎ𝑆 + (𝜆

2− 𝜆4) 𝜙𝛽ℎ𝐼𝑡

+ (𝜆3− 𝜆4) 𝜎1𝛽𝑡𝐼𝑡+ (𝜆3− 𝜆5) (1 − 𝑢

2) 𝛾1

+ 𝜆3𝛿,

𝑑𝜆4

𝑑𝑡= −1 + (𝜆

4− 𝜆3) 𝛼2𝑢1+ (𝜆4− 𝜆6) (1 − 𝑢

2) 𝛾2

+ 𝜆4𝛿,

𝑑𝜆5

𝑑𝑡= −1 + (𝜆

5− 𝜆6) 𝜎2𝛽𝑡𝐼𝑡+ 𝜆5(𝛿 + 𝜇

1) ,

𝑑𝜆6

𝑑𝑡= −1 + (𝜆

6− 𝜆5) 𝑢1𝛼3+ 𝜆6(𝛿 + 𝜇

2) ,

(12)

where the transversality conditions 𝜆𝑖(𝑡𝑓) = 0, 𝑖 = 1, . . . , 6.

By applying Pontryagin’s Maximum Principle and theexistence result for the optimal control pairs, the steps toobtain the optimal controls 𝑢 = (𝑢

∗

1, 𝑢∗

2) are as follows

[22, 23]:

(1) Minimize the Hamilton function𝐻with respect to 𝑢;that is, 𝜕𝐻/𝜕𝑢 = 0, which is the stationary condition.We obtain

𝑢∗

1=

{{{{{

{{{{{

{

0 for 𝑢1≤ 0

(𝜆4− 𝜆3) 𝛼2𝐼ℎ𝑡+ (𝜆2− 𝜆1) 𝛼1𝐼𝑡+ (𝜆6− 𝜆5) 𝛼3𝐴ℎ𝑡

𝑐1

for 0 < 𝑢1< 1

1 for 𝑢1≥ 1,

𝑢∗

2=

{{{{{

{{{{{

{

0 for 𝑢2≤ 0

(𝜆6− 𝜆4) 𝛾2𝐼ℎ𝑡+ (𝜆5− 𝜆3) 𝛾1𝐼ℎ

𝑐2

for 0 < 𝑢2< 1

1 for 𝑢2≥ 1.

(13)

(2) Solve the state system �̇�(𝑡) = 𝜕𝐻/𝜕𝜆 which is model(1), where 𝑥 = (𝑆, 𝐼

𝑡, 𝐼ℎ, 𝐼ℎ𝑡, 𝐴ℎ, 𝐴ℎ𝑡), 𝜆 = (𝜆

1, 𝜆2,

. . . , 𝜆6) with initial condition 𝑥(0).

(3) Solve the costate system �̇�(𝑡) = −𝜕𝐻/𝜕𝑥 which issystem (12) with the end condition 𝜆

𝑖(𝑡𝑓) = 0, 𝑖 =

1, . . . , 6.

Hence, we obtain the following theorem.

Theorem 4. The optimal controls (𝑢∗1, 𝑢∗

2) that minimize the

objective function 𝐽(𝑢1, 𝑢2) on Γ are given by

𝑢∗

1= max{0,

min(1,(𝜆4− 𝜆3) 𝛼2𝐼ℎ𝑡+ (𝜆2− 𝜆1) 𝛼1𝐼𝑡+ (𝜆6− 𝜆5) 𝛼3𝐴ℎ𝑡

𝑐1

)} ,

𝑢∗

2= max{0,min(1,

(𝜆6− 𝜆4) 𝛾2𝐼ℎ𝑡+ (𝜆5− 𝜆3) 𝛾1𝐼ℎ

𝑐2

)} ,

(14)


Table 1: Parameter values.

Parameter Value Reference Parameter Value ReferenceΛ 50000/year [7] 𝛼

12/year Assumed

𝛿 0.02/year [7] 𝛼2

1.2/year Assumed𝛽𝑡

0.00031/year Assumed 𝛼3

1/year Assumed𝛽ℎ

0.00045/year Assumed 𝜇1

0.03/year Assumed𝜎1

1.02/year Assumed 𝜇2

0.06/year Assumed𝜎2

1.04/year Assumed 𝛾1

0.01/year Assumed𝜙 1.0002/year Assumed 𝛾

20.05/year Assumed

Table 2: Sensitivity indices to parameter for the TB-HIV model.

Parameter Sensitivityindex (𝑅

𝑡) Parameter Sensitivity

index (𝑅ℎ)

Λ 1 Λ 1𝛽𝑡 1 𝛽

ℎ 1𝛿 −2 𝛿 −1.667

𝛾1 −0.333

where 𝜆𝑖, 𝑖 = 1, . . . , 6, is the solution of the costate equations

(12) with the transversality conditions 𝜆𝑖(𝑡𝑓) = 0, 𝑖 = 1, . . . , 6.

Substituting the optimal controls (𝑢∗

1, 𝑢∗

2) which are

obtained from the state system (1) and the costate system(12), we obtain the optimal system. The solutions of the opti-mality system will be solved numerically for some parameterchoices. Most of the parameter values are assumed withinrealistic ranges for a typical scenario due to lack of data.

5. Numerical Simulation

In this section, we investigate the numerical simulations ofmodel (1) with and without optimal control. The optimalcontrol strategy is obtained by the iterative method of Runge-Kuttamethod of order 4 [24].We start to solve the state equa-tions by the forward Runge-Kutta method of order 4. Thenwe use the backward Runge-Kutta method of order 4 to solvethe costate equations with the terminal conditions. Then, thecontrols are updated by using a convex combination of theprevious controls and the value from the characterizations of𝑢∗

1and 𝑢∗

2. This process is repeated and iteration is stopped

if the values of unknowns at the previous iteration are veryclose to the ones at the present iteration.

We consider three scenarios. In the first scenario, weconsider only the anti-TB treatment control. In the secondscenario, we consider only the ARV treatment control. Inthe last one, we use the optimal anti-TB and ARV treatmentcontrols. Parameters used in these simulations are givenin Table 1. In these simulations, we use initial condition(𝑆(0), 𝐼

𝑡(0), 𝐼ℎ(0), 𝐼ℎ𝑡(0), 𝐴

ℎ(0), 𝐴

ℎ𝑡(0)) = (500, 50, 10, 5, 5, 5)

and weighting constants 𝑐1= 80, 𝑐

2= 100.

5.1. First Scenario. In this scenario, we set the ARV control𝑢2to zero and activate only the anti-TB treatment control

𝑢1. The profile of the optimal treatment control 𝑢∗

1for

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Con

trol p

rofil

e

2 4 6 8 100Time (years)

u1

Figure 6: The profile of the optimal anti-TB control 𝑢∗1.

this scenario could be seen in Figure 6. To eliminate TB-HIV/AIDS coinfection in 10 years, the anti-TB treatmentshould be given intensively almost 7.5 years before decreasingto the lower bound in the end of 10th year.

The dynamics of the infected populations of this scenarioare given in Figures 7 and 8. We observe in Figure 7 that thiscontrol strategy results in a significant decrease in the numberof TB infected (𝐼

𝑡) and TB-HIV coinfection (𝐼

ℎ𝑡) populations

compared with the case without control. Specifically, usingthe control strategy, the TB-HIV coinfection population startto decrease from the third year. Also in the right of Figure 8,this control strategy results in a significant decrease in thenumber of TB-AIDS coinfections (𝐴

ℎ𝑡) as against an increase

in the uncontrolled case. On the contrary, the result in theleft of Figure 8 shows that the number of AIDS infected(𝐴ℎ) populations with and without the control does not

differ significantly because there is no intervention againstAIDS infection. Hence, the anti-TB treatment control givesa significant effect in controlling infected TB and also TB-HIV/AIDS coinfection.

5.2. Second Scenario. In the second scenario, we set the anti-TB treatment control 𝑢

1to zero and activate only the ARV

treatment control 𝑢2. The control profile of ARV treatment is

shown in Figure 9. We see that, to eliminate TB-HIV/AIDS


2 4 6 8 100Time (years)

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000TB

infe

cted

Without treatmentOptimal anti-TB treatment

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

TB-H

IV in

fect

ed

2 4 6 8 100Time (years)


Figure 7: The dynamics of 𝐼𝑡and 𝐼ℎ𝑡using control 𝑢∗

1.

2 4 6 8 100Time (years)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

AID

S in

fect

ed

×104



2 4 6 8 100Time (years)

0

500

1000

1500

2000

2500

3000

3500

4000

TB-A

IDS

infe

cted

Figure 8: The dynamics of 𝐴ℎand 𝐴

ℎ𝑡using control 𝑢∗

1.

coinfection in 10 years, the ARV treatment should be givenintensively during 10 years.

The dynamics of the TB-HIV/AIDS coinfection of thisscenario are given in Figures 10 and 11. We observe inFigure 10 that there is no significant difference in the numberof TB infected populations with and without the ARV controltreatment only. This may be due to the absence of thetreatment against TB infection. It was also observed thatthe number of TB-HIV coinfection populations increaseswith this control strategy compared to the number without

control. The positive impact of this strategy is shown inFigure 11, where the number of the AIDS infected and theTB-AIDS coinfection populations decreases significantly atthe end of the intervention period.

5.3. Third Scenario. In this scenario, we consider the anti-TBand ARV treatment controls simultaneously. The profile ofthe optimal anti-TB treatment control 𝑢∗

1and ARV control

𝑢∗

2of this scenario is in Figure 12. To eliminate TB-HIV/AIDS

coinfection in 10 years, the anti-TB treatment should be given


0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Con

trol p

rofil

e

2 4 6 8 100Time (years)

u2

Figure 9: The profile of the optimal ARV control 𝑢∗2.

2 4 6 8 100Time (years)

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000

TB in

fect

ed

Without treatmentOptimal ARV treatment

2 4 6 8 100Time (years)

0

2000

4000

6000

8000

10000

12000

14000

16000

18000

TB-H

IV in

fect

ed


Figure 10: The dynamics of 𝐼𝑡and 𝐼ℎ𝑡using control 𝑢∗

2.

intensively almost 8 years before dropping gradually untilreaching the lower bound in the end of 10th year, and theARVtreatment is also given similar to anti-TB treatment, except atthe beginning of the treatment.

Using the optimal controls in Figure 12, the dynamicsof the TB-HIV/AIDS coinfection populations are given inFigures 13 and 14, respectively. For this strategy, we observedin Figure 13 that the control strategies resulted in a decreasein the number of TB infected and TB-HIV coinfectionpopulations compared to the number without control. Asimilar decrease is observed in Figure 14 for AIDS infectedand TB-AIDS coinfection populations in the control strategy,while an increased number for the uncontrolled case resulted.

Our numerical results show that the combination of anti-TB treatment and ARV treatment has the highest impact todiminish the size of TB-HIV/AIDS coinfection. When usingonly one control, the anti-TB treatment is more effectivethan ARV treatment to reduce the number of TB-HIV/AIDScoinfection populations.

6. Conclusion

In this paper, we have studied a deterministic model for thetransmission of TB-HIV coinfection that includes use of anti-TB and ARV treatment as optimal control strategies. Themodel without controls exhibits four equilibria, namely, the


2 4 6 8 100Time (years)

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

AID

S in

fect

ed


0

500

1000

1500

2000

2500

3000

3500

4000

TB-A

IDS

infe

cted

2 4 6 8 100Time (years)


×104


ℎ𝑡using control 𝑢∗

2.

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Con

trol p

rofil

es

2 4 6 8 100Time (years)

u1

u2

Figure 12: The profile of the optimal controls 𝑢∗1and 𝑢∗

2.

disease-free equilibrium, the HIV-free equilibrium, the TB-free equilibrium, and the endemic equilibrium. We furtherobtain two thresholds, 𝑅

𝑡and 𝑅

ℎ, which are basic reproduc-

tion ratios for TB and HIV infections, respectively. Theseratios determine the existence and stability of the equilibria ofthe model. The existence of the equilibria with respect to thethresholds 𝑅

𝑡and 𝑅

ℎis summarized in Figure 2. If both the

thresholds are less than unity then the diseases-free equilib-rium is locally asymptotically stable. But if 𝑅

𝑡is greater than

unity with the condition 𝑅𝑡> 𝑅ℎand 𝑅

ℎis greater than unity

with the condition 𝑅ℎ> 𝑅𝑡, then the HIV-free and TB-free

equilibriums are locally asymptotically stable, respectively.Finally, the optimal control theory for TB-HIV coinfectionmodel is derived analytically by applying the PontryaginMaximum Principle.The numerical simulations were carriedout to perform the optimal anti-TB and ARV treatment con-trols. From our analysis and numerical results, we concludethat the combination of anti-TB and ARV treatments is themost effective to reduce the TB-HIV coinfection. However, ifwe have to use only one control, then the anti-TB treatment


0

2000

4000

6000

8000

10000

12000

14000

16000

18000

TB-H

IV in

fect

ed

2 4 6 8 100Time (years)

Without treatmentOptimal treatment

2 4 6 8 100Time (years)

0

500

1000

1500

2000

2500

3000

3500

4000

4500

5000TB

infe

cted


Figure 13: The dynamics of 𝐼𝑡and 𝐼ℎ𝑡using controls 𝑢∗

1and 𝑢∗

2.

0

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8

2

AID

S in

fect

ed

2 4 6 8 100Time (years)

×104


0

500

1000

1500

2000

2500

3000

3500

4000

TB-A

IDS

infe

cted

2 4 6 8 100Time (years)



ℎ𝑡using controls 𝑢∗

1and 𝑢∗

2.

is better than ARV treatment to eliminate the number of TB-HIV/AIDS coinfection populations.

Competing Interests

The authors declare that there are no competing interestsregarding the publication of this paper.

Acknowledgments

Parts of this research are funded by the Indonesian Direc-torate General for Higher Education (DIKTI) through DIPA

Universitas Airlangga/BOPTN 2014 according to SK Rektorno. 965/UN3/2014.

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