Stability analysis of the transmission dynamics of tuberculosis models∗

This paper addresses the spread of tuberculosis through one-strain and two-strain models. We first present a basic model that incorporates fast and slow progression, effective chemoprophylaxis and therapeutic treatment. The system exhibits the traditional behavior. We prove that if the basic reproduction ratio R0 ≤ 1, then the disease free equilibrium is globally asymptotically stable on the nonnegative orthant and if R0 > 1 an endemic equilibrium exists and is globally asymptotically stable. Based on the first model, the second model deals with the problem 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. Our objective is to characterize the role of multidrug-resistant in the transmission of tuberculosis. The coexistence and stability of the associated equilibria are discussed.