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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, 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.
Highlights
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
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
Motivated by the above discussion, this paper investigates the impact of drug resistance as a competition between multiples types of strains of Mycobacterium tuberculosis: those that are sensitive to anti-tuberculosis drugs and those that are resistant
Summary
Tuberculosis (TB) remains a major health problem, especially in Africa and especially in the sub-Saharan countries. Drug resistant tuberculosis (MDR-TB) is a form of the disease caused by a bacillus that is not responsive to isoniazid and rifampicin, the two most effective first-line anti-tuberculosis drugs. Severe resistance may develop; we are talking about highly drug-resistant tuberculosis, which is an even more serious form of MDR-TB due to bacilli that do not respond to the most effective second-line drugs, often leaving patients without any other therapeutic option. The dynamics of tuberculosis is complex due to the multiple interactions between the human host and Mycobacterium tuberculosis, the increasing HIV epidemic in the early 1990s leading to HIV/TB co-infection, the emergence of drug resistant TB, immigration to the US from developing countries, increased mass transportation, malnutrition, heavy alcohol drinking, smoking, co infection with diabetes mellitus and indoor air pollution from solid fuels.
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