Tuberculosis (TB) is a lung infection caused by the bacteria Mycobacterium tuberculosis. The disease usually occurs at the top of the lungs. TB is easily transmitted from the patient itself to others through saliva splashes, sneezing, coughing and direct intermediate contact with the patient [1]. TB attacks more than 75% of the productive age population 20-30% of lost family income per year due to tuberculosis [2]. Also, an active TB patient will infect 10-15 people around an infected human per year, and without adequate treatment, 50-60% of TB patients will die [3]. In addition to vaccinations, treatment for people with TB disease can also be done through chemotherapy. However, treatment in TB sufferers also has some problems, among others, the Paradox effect. Paradoxical Reaction (PR) at the time of chemotherapy for tuberculosis infants is defined as the side effects or deterioration of TB-infected persons either clinically or radiologically for some time after initiating chemotherapy efforts for TB treatment [4].

This paper presents a mathematical model of TB transmission considering the paradoxical reaction intervention. Paradoxical reaction among children and adult population considered in the model to capture the effect of this intervention in a long-term implementation. The model described below.

EQUATIONS (1) (PLEASE SEE THE ABSTRACT FILE)

where S_i is susceptible human, E_i is exposed TB human, I_i is the infectious human and R_i is the recovered human. Please note i=a,c denote c for children during a for an adult. First, we investigate the existence and local stability of possible equilibrium points, i.e., disease-free equilibrium (\Omega_1^*) and the endemic equilibrium (\Omega_2^*) point.  The disease-free equilibrium (\Omega_1^*) point given by, EQUATIONS (2) (PLEASE SEE THE ABSTRACT FILE), while the endemic equilibrium (\Omega_2^*) point is undefine explicitely. Then, the form of basic reproduction number for the model is constructed using the next-generation matrix [5] approach and given by, EQUATIONS (3) (PLEASE SEE THE ABSTRACT FILE).

Last, numerical simulation for a various scenario for the autonomous system related to the sensitivity analysis result was conducted to give a better explanation of the model results.

FIGURE (1) (PLEASE SEE THE ABSTRACT FILE)

Figure 1. Dynamic of total susceptible and infected population, with R0 = 0.2995474664 < 1 (green) and R0 = 1.482661730 >1 (yellow).

From the analytic and numerical investigation, we find that the paradoxical reaction plays an important role to determine the TB will coexist or disappear.