Existence of multiple-stable equilibria for a multi-drug-resistant model of mycobacterium tuberculosis

Abstract

The resurgence of multi-drug-resistant tuberculosis in some partsof Europe and North America calls for a mathematical study to assess theimpact of the emergence and spread of such strain on the global effort to effectively control the burden of tuberculosis. This paper presents a deterministiccompartmental model for the transmission dynamics of two strains of tubercu-losis, a drug-sensitive (wild) one and a multi-drug-resistant strain. The modelallows for the assessment of the treatment of people infected with the wildstrain. The qualitative analysis of the model reveals the following. The modelhas a disease-free equilibrium, which is locally asymptotically stable if a cer-tain threshold, known as the effective reproduction number, is less than unity.Further, the model undergoes a backward bifurcation, where the disease-freeequilibrium coexists with a stable endemic equilibrium. One of the main nov-elties of this study is the numerical illustration of tri-stable equilibria, wherethe disease-free equilibrium coexists with two stable endemic equilibrium whenthe aforementioned threshold is less than unity, and a bi-stable setup, involving two stable endemic equilibria, when the effective reproduction number isgreater than one. This, to our knowledge, is the first time such dynamicalfeatures have been observed in TB dynamics. Finally, it is shown that thebackward bifurcation phenomenon in this model arises due to the exogenousre-infection property of tuberculosis.

Keywords

• key words and phrases. multi-drug-resistant tb
• dynamical system
• bifurcation analysis
• epidemic model
• multi-stable steady state.