Mathematical analysis of a drug resistance in a tuberculosis transmission model

Abstract

In this paper, we investigate the effect of the emergence of TB drug-resistant within a human population. We first propose a drug resistance in a tuberculosis transmission model with two strains of Mycobacterium tuberculosis: those that are sensitive to anti-tuberculosis drugs and those that are resistant. After, we present the theoretical results of the model. More precisely, we compute the disease-free equilibrium and derive the basic reproduction number R0 that determines the outcome of the disease. We show that there exists a threshold parameter ξ such that the disease-free equilibrium is globally asymptotically stable in a feasible region whenever R0 ≤ ξ < 1, while when ξ < R0 < 1, the model exhibits the phenomenon of backward bifurcation and if R0 > 1, the disease-free equilibrium is unstable and there exists an unique endemic equilibrium which is stable. Conditions for the coexistence of sensitive and resistant strains are derived. We also show that the model undergoes the Hopfbifurcation with respect to the transmission rates. A dynamically consistent non standard finite difference scheme is developed to illustrate and validate theoretical result. The motivation comes to the fact the classical Runge-Kutta scheme cannot preserve the positivity of solutions of the model.