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

Abba B. Gumel, Baojun Song

Research output: Contribution to journalArticle

21 Scopus citations

Abstract

The resurgence of multi-drug-resistant tuberculosis in some parts of Europe and North America calls for a mathematical study to assess the impact of the emergence and spread of such strain on the global effort to effectively control the burden of tuberculosis. This paper presents a deterministic compartmental model for the transmission dynamics of two strains of tuberculosis, a drug-sensitive (wild) one and a multi-drug-resistant strain. The model allows for the assessment of the treatment of people infected with the wild strain. The qualitative analysis of the model reveals the following. The model has a disease-free equilibrium, which is locally asymptotically stable if a certain threshold, known as the effective reproduction number, is less than unity. Further, the model undergoes a backward bifurcation, where the disease-free equilibrium coexists with a stable endemic equilibrium. One of the main novelties of this study is the numerical illustration of tri-stable equilibria, where the disease-free equilibrium coexists with two stable endemic equilibrium when the aforementioned threshold is less than unity, and a bi-stable setup, involving two stable endemic equilibria, when the effective reproduction number is greater than one. This, to our knowledge, is the first time such dynamical features have been observed in TB dynamics. Finally, it is shown that the backward bifurcation phenomenon in this model arises due to the exogenous re-infection property of tuberculosis.

Original languageEnglish
Pages (from-to)437-455
Number of pages19
JournalMathematical Biosciences and Engineering
Volume5
Issue number3
StatePublished - 1 Jul 2008

Keywords

  • Bifurcation analysis
  • Dynamical system
  • Epidemic model
  • Multi-drug-resistant Tb
  • Multi-stable steady state

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