Global dynamics of discrete mathematical models of tuberculosis

Abstract

In this paper, we develop discrete models of Tuberculosis (TB). This includes SEI endogenous and exogenous models without treatment. These models are then extended to a SEIT model with treatment. We develop two types of net reproduction numbers, one is the traditional ${\mathcal{R}}_0$ which is based on the disease-free equilibrium, and a new net reproduction number ${\mathcal{R}}_0\left({\mathcal{E}}^{\ast }\right)$ based on the endemic equilibrium. It is shown that the disease-free equilibrium is globally asymptotically stable if ${\mathcal{R}}_0\le 1$ and unstable if ${\mathcal{R}}_0>1$. Moreover, the endemic equilibrium is locally asymptotically stable if ${\mathcal{R}}_0\left({\mathcal{E}}^{\ast }\right)<1<{\mathcal{R}}_0$.

KEYWORDS:

• Tuberculosis
• treatment
• disease-free
• endemic
• endogenous
• exogenous
• local stability
• global stability
• SEI models
• SEIT models
• disease control
• treatment
• epidemiology
• public health
• TB elimination

MATHEMATICS SUBJECT CLASSIFICATIONS:

• 92-10
• 39A30

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

The author(s) reported there is no funding associated with the work featured in this article.