Mathematical Analysis of a General Two-Patch Model of Tuberculosis Disease with Lost Sight Individuals

Abstract

A two-patch model, ${SE}_{i1},\dots ,E_{in}{I}_i{L}_i$, $i=\mathrm{1,2}$, is used to analyze the spread of tuberculosis, with an arbitrary number $n$ of latently infected compartments in each patch. A fraction of infectious individuals that begun their treatment will not return to the hospital for the examination of sputum. This fact usually occurs in sub-Saharan Africa, due to many reasons. The model incorporates migrations from one patch to another. The existence and uniqueness of the associated equilibria are discussed. A Lyapunov function is used to show that when the basic reproduction ratio is less than one, the disease-free equilibrium is globally and asymptotically stable. When it is greater than one, there exists at least one endemic equilibrium. The local stability of endemic equilibria can be illustrated using numerical simulations. Numerical simulation results are provided to illustrate the theoretical results and analyze the influence of lost sight individuals.