A FRACTIONAL SITR MODEL FOR DYNAMIC OF TUBERCULOSIS SPREAD

. This work presents a fractional SITR mathematical model that investigates the Tuberculosis (TB) spread in a human population. It was shown that disease-free and endemic equilibrium stability depended on the basic reproduction number. These results are in accordance with the epidemic theory. A numerical example is given to demonstrate the validity of the results. The results show that the infected subpopulation increases in the absence of special treatment


Tuberculosis (TB) is an infectious disease caused by Mycobacterium tuberculosis, and is
usually acquired through airborne infection from active TB cases [1,2]. According to the World Health Organization, one third of the world's population is infected with tuberculosis either latently or actively. Despite effective antimicrobial chemotherapy, tuberculosis infection remains a leading cause of death from an infectious disease [3].
with the initial conditions S(0) = S 0 , I(0) = I 0 , T (0) = T 0 , R(0) = R 0 and the involve various parameters in (1) are described in Table 1. The population total is N = S + I + T + R. Along with the development of the fractional-order differential equation, recently the issue on development of mathematical models in form of the fractional-order non-linear differential equation are widely discussed by many researchers, see [10,11,12,13,14,15,16]. In this paper, we modify the model (1) by replacing usual derivative into fractional-order derivative such that the model (1) can be written as a following new model: where ∆ (δ ) is the Caputo fractional derivative operator of oder δ with 0 < δ < 1. As a new SIT R model, we study the stability of the disease-free equilibrium and endemic equilibrium of the model (2). To the best of the author's knowledge, this issue has not been solved yet to date.
Therefore the results of this work constitute a novelties at once a new contribution in the field of fractional-order epidemic dynamic.
The paper is organized as follows: Section 2 considers some useful results about Caputo fractional derivative and stability of the fractional-order nonlinear system. The main result of this article is presented in the section 3. Section 4 concludes the paper.

SOME USEFUL RESULTS
In this section we recall several mathematical tools used in this study. The Caputo fractional where Γ(.) is the Euler Gamma function [17], and ∆ (k) x(.) is the usual kth derivative of function Let us consider the fractional-order nonlinear system involving Caputo derivative (4) can be written as where A ∈ R n×n . The point x * is said the equilibrium point of the system (4) if f(t, x * ) = 0.
[18] Let x = x * is an equilibrium of the the fractional-order system (4) with δ ∈ (0, 1). The equilibrium point x = x * is asymptotically stable if for all roots λ of the equation where J x * is the Jacobian matrix of system (4) at the equilibrium x * .

STABILITY ANALYSIS
The dynamical behavior of the model can be classified by the basic reproductive number [6]. By applying the next generation technique presented in [6], the basic reproduction number, denoted by R 0 , for the model (2) is .
The equilibrium points of the model (2) is evaluated by solving the following equations: The disease-free equilibrium, denoted by K 0 , of the fractional order TB model (2) is obtained by assuming I = 0, such that the disease-free equilibrium is The endemic equilibrium, denoted by K 1 , of the fractional order TB model (2) exists if R 0 > 1.
Thus the endemic equilibrium of the model (2) is K 1 = (S * , I * , T * , R * ), where S * = (µ + α)N β , We will analyze the stability of these two equilibrium points. First of all, the Jacobian matrix of the vector field corresponding to model (2) is The stability of the disease-free equilibrium K 0 is given in the following theorem.
We now consider the stability of the endemic equilibrium K 1 . The Jacobian matrix of (13) at The stability of the endemic equilibrium K 1 is given in the following theorem.
In order to show the validity of the results, let us consider the following numerical example. For the model (2)

CONFLICT OF INTERESTS
The author(s) declare that there is no conflict of interests.