Global Dynamics of a TB Model with Classes Age Structure and Environmental Transmission

TLDR: In this paper, an age structured SVEIR epidemic model for TB is formulated and analyzed by considering three types of ages e.g., latent age, infection age and vaccination age.

Abstract: In this article, an age structured SVEIR epidemic model for TB is formulated and analyzed by considering three types of ages e.g., latent age, infection age and vaccination age. The presented model also incorporates the environmental transmission of TB. The dynamics of the disease is governed by a system of differential-integral equations. We assume that the vaccines for TB are partially effective. Some vaccinated individuals get permanent immunity to this disease, but some vaccinated individuals lose its protective power over a time and become susceptible again. The dynamical property of the model is established by using LaSalle’s invariance principle and constructing suitable Lyapunov functions. It has been shown that the dynamics of the model is governed by basic reproductive number \(\mathcal {R}(\xi )\). The disease-free equilibrium is globally stable if the basic reproductive number \(\mathcal {R}(\xi ) 1\). As the basic reproduction number plays an important role in determining the stability of the system, reducing this number below one through vaccination can lead to decrease in the transmission of this disease. Additionally, contaminated environment also contributes to the increase in \(\mathcal {R}(\xi )\), so we also need to keep the environment clean to decrease the basic reproduction number \(\mathcal {R}(\xi )\) below one. These types of control measures are easy to implement in our society and certainly this will improve the well-being of the society.