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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 tubercu-
losis, 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 cer-
tain 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 nov-
elties 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.
Mathematics Subject Classification: 92D30, 34C37, 37G35.
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