Mathematical Modeling of Drug Resistance in Tuberculosis Transmission and Optimal Control Treatment

Tuberculosis is caused by Mycobacterium tuberculosis. The disease is still one of the major killer of humans. We derive and analyzed mathematical modeling of drug resistance in tuberculosis transmission. We use control on treatment to reduce of the number of infected population. For model without control optimal, we have the basic reproduction number for the sensitive and resistant tuberculosis infection in the first area and tuberculosis infection in the second area. This number determine the existence and stability of equilibria. Then, the Pontryagin Maximum Principle is applied to derive necessary conditions for the optimal control of the tuberculosis disease. Finally, numerical simulations are performed to describe the analytical results.


Introduction
Tuberculosis (TB) is a deadly infectious disease caused by Mycobacterium tuberculosis (MTB). There is about one-third of the world's population has latent TB, which means people have been infected by MTB but are not (yet) ill with disease and cannot transmit the disease. TB is spread from person to person through the air. When people with pulmonary TB cough, sneeze or spit, they propel the TB germs into the air. Someone needs to breathe just a few of these germs to be infected. In 2011, 8.7 million people fell ill with TB and 1.4 million died from TB. Standard anti-TB drugs have been used for decades, and resistance to the medicines is growing. Disease strains that are resistant to a single anti-TB drug have been documented in every country surveyed [11]. Hence, a good understanding of the effectiveness of the treatment and control strategies in different regions of the world still needed Study of the spread of TB disease have been conducted by several researchers [1,4,5,10]. Other forms of mathematical models can be used to control the spread of disease is to formulate the application of optimal control to prevent and control TB disease with minimum costs [6]. In [8], the authors have developed a model of the spread of tuberculosis by vaccination and treatment in patients with TB. While in [10], the authors have extended a model of the spread of TB disease by observing the migration of healthy subpopulations in the two regions without the factor of resistance to anti-TB drugs. Therefore, in this paper will be constructed a mathematical model that describes the dynamics of the spread of TB by a factor of resistance to anti-TB drugs and also the migration of healthy sub-population in the two regions. In addition, we further carried out qualitative optimal control analysis to reduce the number of TB patients with the resistance factor. We use Pontryagin Maximum Principle to find the necessary conditions for the optimal control of the tuberculosis disease.

Model Formulation
In this paper, we extended the model of tuberculosis that has been developed in [10]. Model of tuberculosis SIR by a factor of migration in human populations are vulnerable and transmission does not occur during the migration process. This model consists of two major subpopulations. Each subpopulation is divided into three classes based on the epidemiology status that is susceptible (Si), infected (Ii),

Model Analysis
First, we analyze the model (1) without control function u, that is, without treatment. Let defined the parameter is called the disease-free equilibrium in the first

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area, but endemic in the second area, with is called the disease-free equilibrium in second area but infective sensitive endemic in the first area, with The equilibrium 5 E exists if and only if it satisfies the following conditions

The equilibrium
exists if and only if it satisfies the following conditions A criterion for the stability of the disease-free equilibrium is given in the following theorem.

Theorem 1.
Disease-free equilibrium is locally asymptotically stable if and only if 1 0  R .

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Proof. Local stability of the disease-free equilibrium 1 E is determined by the eigenvalues of the Jacobian matrix of the model (1) at 1 E . We get the eigenvalues are , .  Next, we will be reviewed the stability of the five endemic equilibriums i E , for i{2,3,…,6}. From the calculation, the eigenvalues of the Jacobian matrix at the point i E , for i{2,3,…,6} , is difficult to determine analytically. Hence, the stability of the endemic equilibriums will be performed numerically. We use three initial values for simulations. It aims to find out where the convergence of each solution given initial value. Numerically, the endemic equilibrium 2 E is tend to locally asymptotically stable if 1 02  R , as given in Figure 2.  Based on Figure 3 and Figure 4, it is shown that each of the three different initial values, all the graphs tend to converge to the endemic equilibriums. Thus, the stability of endemic equilibriums 3 E -6 E can be expressed in the following conjectures.

Conjecture
The endemic equilibrium 6 E is locally

Analysis of Optimal Control
The application of optimal control in this study is to minimize the number of individuals infected with TB through treatment with minimal cost. The optimal control strategy can be achieved by minimizing the following objective function: where c is weighting constant for attempt treatment. The greater the value c will imply more expensive implementation costs for treatment. We seek an optimal control u* such that where Consider again the objective function (2) to the model (1). Necessary conditions to determine the optimal control * u so that satisfy the conditions (3) with the constraint (1) will be solved by the Pontriyagin Maximum Principle [9]. This principle is to convert equation (1) -(3) to minimize the problem to the Hamiltonian function:    [3,9], we obtain the following theorem. s s r r s s Solving u* for subject the constrains, the optimal control can be derived. 

Numerical Simulation
In this section, we present the numerically the optimal solution to the optimal control treatment on tuberculosis transmission with drug resistance by the fourth order Runge-Kutta [7]. The state system is solved forward in time with initial conditions x(0) = (4100, 7,5,4,4110,8,4), while the co-state system is solved backward in time. For numerical simulation, we use the following parameters: The dynamic of the sensitive and resistance infected population in the first region are given in Figure 5. We see in Figure 5 that there is a significant difference in the number of sensitive infected s I and drug resistant individuals r I between the case with control and case without control. We also observe in the left of Figure 6 that the number sensitive infected 2 I decreases with control compared to the situation where there is no control. The profile of the optimal control * u could be seen in the right of Figure 6. From this Figure, we see that to reduce the number of individuals infected with TB in the first and the second regions in 10 years, the treatment should be hold intensively almost 8 years and then reduced to near zero at the end of the 10-th year.

Conclusion
In this paper we have constructed a mathematical model of drug resistant in the tuberculosis disease transmission that include treatment measure as optimal control. For the model without control, we obtained three basic reproduction ratios corresponding to the sensitive and resistant TB infection in the first region and the sensitive TB infection in the second region. These ratios determine the existence and stability of the equilibrium of the model. Using Pontryagin Maximum Principle, we derived and analyzed the condition for optimal control which minimize the both sensitive and resistant infective in the first region and the sensitive only in the second region. The numerical simulations with and without control show that the control strategy has a positive impact in reducing the spread of the disease.