A mathematical analysis of the two-strain tuberculosis model dynamics with exogenous re-infection

Abstract

The rise of drug resistance has become a major obstacle in treating tuberculosis (TB), significantly contributing to the increasing disease burden. Therefore, it is essential to study the transmission dynamics of the disease, considering the factors that contribute to the strain’s impact on the disease burden, using an epidemiological model. We present a deterministic mathematical model that explores the dynamics of TB with two strains: drug-susceptible and drug-resistant, taking into account exogenous re-infection. We thoroughly analyze to gain insights into the behavior of the model. The qualitative analysis of the model reveals an interesting phenomenon known as “backward bifurcation,” where both stable disease-free and stable endemic equilibria coexist when the associated reproduction number is less than one. In the absence of exogenous re-infection, the model shows the existence of unique positive endemic equilibria. Numerical simulations were conducted, yielding noteworthy results. Increasing the treatment rate for individuals infected with the drug-susceptible strain reduces the number of new cases of drug-susceptible TB while increasing the detection of drug-resistant TB cases. The simulations demonstrate that drug-susceptible and drug-resistant TB strains can coexist when their reproduction numbers exceed one without competitive exclusion occurring. In summary, this study sheds light on the challenges posed by drug resistance in TB treatment and highlights the importance of understanding the disease dynamics through mathematical modeling to develop effective strategies for its control.

Keywords

• Deterministic mathematical model
• Tuberculosis
• Reproduction number
• Equilibria
• Stability
• Bifurcation