Dynamical behaviour of tuberculosis transmission

Abstract

Tuberculosis(TB) is a contagious disease in human caused by infection withВ Mycobacterium tuberculosis(Mtb). Most infections results a clinically asymptotic state termed as latent TB infection(LTBI) whereas a smaller portion ofВ infected individuals grow symptomatic active pulmonary TB. The main difference between TB and other infectious diseases is that, the disease progressionВ from primary infection(LTBI) to active pulmonary TB is signicantly time-consuming.
We proposed and study an SEIR type mathematical model for TB transmission incorporating roles of both exogenous re-infection and endogenous reactivation. Our model possesses two kinds of steady states: infection free andВ endemic. The epidemiological threshold key that is, basic reproduction numberВ R0 has been obtained by using next-generation matrix. We observe that theВ disease transmission rate and exogenous re-infection level plays a signicantВ role in order to determine the qualitative behaviour of our proposed model system. Our results demonstrate that when exogenous re-infection level crosses aВ critical value our system undergoes backward bifurcation and hence a stable endemic equilibrium exists in spite of the fact R0 < 1. Therefore, reducing R0 lessВ than unity is not sucient to eradicate TB completely. We further investigateВ that proposed model experience stable periodic solutions as increases throughВ a critical value. Various numerical simulations have been conducted coveringВ the breadth of feasible parameter space to support analytical establishments.