Threshold dynamics of an age-structured vaccinated epidemic model with both direct and indirect routes of infections

Abstract

Age-structured epidemic models play a crucial role in the study of epidemiological modeling. Motivated by this, in this article, we propose an epidemic model with age since infection and vaccination age of individuals, a coupled system of PDE and integro-differential equations (IDE). Here, two contagious routes, (a) direct human-to-human contact and (b) indirect environmentally contaminated surfaces or objects are considered. First, we establish the well-posedness of the model, followed by the basic reproduction number (R0) and the role of the threshold value of R0 in the asymptotic profile of the solution semi-flow is established. We observe the global stability of the disease-free steady state for R0<1, the uniform persistence of the disease and the existence of the endemic steady state for R0>1. This endemic steady state is also globally asymptotically stable for R0>1. We have further analyzed the influence of vaccination age and age since infection in the threshold parameter R0. Our analysis shows that the threshold parameter R0 does not depend explicitly on vaccination age, but it strictly decreases with the natural depletion rate of the contaminated environment. Finally, the model is discretized using the finite difference method to illustrate our theoretical results numerically.