Stability Analysis of Tuberculosis SITS Model

. Tuberculosis (TB) is an infectious disease caused by mycobacterium tuberculosis. The purpose of this study is to investigate the dynamics of TB spread by using mathematical model. We develop SITS model which expressed as system of differential equations. The system has two equilibrium points, namely disease-free equilibrium point and endemic equilibrium point. The stability condition of the equilibrium points is proved. We perform several numerical simulations to support our theoretical results.


I. INTRODUCTION
Tuberculosis (TB) is an infectious disease caused by Mycobacterium tuberculosis. TB is one of the deadliest diseases and transmitted through droplets (sputum splashes) [1].
TB is the most dangerous infectious disease in the world. The World Health Organization (WHO) reported that 1.5 million people died from TB (1.1 million is HIV negative and 0.4 million is HIV positive), with 89,000 men, 480,000 women and 140,000 children. It is estimated that 9.6 million people suffer from TB in 2014 with 12% of whom are HIV-positive [2]. WHO stated that 58% of new TB cases occurs in Southeast Asia and Western Pacific region in 2014. The number of TB cases in India, Indonesia and China are 23%, 10% and 10% of the total incidence worldwide, respectively. It is estimated that one million new cases per year occur in Indonesia (WHO, 2015). In West Sulawesi, the number of TB patients in 2007 is 5,100 clinical TB patients. The number of TB clinical patients in 2007 had increased compared to 2006. One intervention used by the West Sulawesi Provincial government in controlling TB is increasing the effectiveness of TB control through strengthening DOTS [3]. In recent years, a deterministic model is used study tuberculosis spread e.g. SEI model [4], SIR and SEIR models [5] and SITR model [6].
In this paper, we discuss SITS (Susceptible-Infected-Treatment-Susceptible) model. We divide the population into three subpopulations, i.e. susceptible individuals, infected individuals, and individuals who take medication. (Treatment). We assume that human who recover from TB can get reinfected.

Model Formulation
In this study, SITS model incorporating the effect of vaccination is formulated. We determine the equilibrium points of the system and stability condition of the equlibrium point . The assumptions made to formulate the model are:  Recruitment of susceptible individuals is constant.  Susceptible individual who interacts with infected individual can be infected.  Infected individual can die due to TB.  Human who revover from TB can get reinfected.  The population is homogen, which implies the death rate and transmission rate of all individuals are equal. The parameter used are presented in Table 1. Based on the assumptions that have been made, we obtain a diagram as follows  Figure 1, we obtain the following system of differential equations

Equilibrium Points
The system has two equilibrium points. The disease free equilibrium point and endemic equilibrium point are respectively

III. Model Analysis
We use next generation approach [7] to determine the basic reproduction number. The reproduction number of the model is Now, we present stability condition of the disease free equilibrium point.

Teorema 2. dfe
x is locally asymptotically stable If

R 
Proof. The Jacobian matrix of the model at dfe (3)
Hence, we get the following theorem. Teorema 3. ee x is locally asymptotically stable if

IV. Numerical Simulations
To verify and to support our theoretical results, we perform numerical simulation using the following parameter value The initial value used is (0) 10, (0) 1, (0) 0.
result is shown in Figure 2 and Figure 3. It is clear that the infected and treatment compartment tend to zero and positive equilibrium if 0 1 R  and 0 1 R  , respectively.

R  
Now, we present numerical simulation result with parameter values presented in Table 2   V. Conclussion In this paper, tuberculosis SITS model is discussed. We prove stability condition of all equilibrium points. Intervention which can increase treatment rate e.g. health education related TB can reduce TB prevalence.