Age structured population dynamics with applications in epidemiology

Abstract

Models describing the dynamics of biological phenomena that evolve in stages have been studied extensively using time-delay mathematical models. Recently, these models have evolved into age structured models with age dependent variables. Various mathematical tools have been employed to study these models and investigate the effects of age structure. This work explores two age structured models to address the issues of latent infection of cells by Human Immuno-deffciency Virus (HIV) and the effects of latent Tu- berculosis (TB) infection on the dynamics of HIV. We consider models discussed on the transmission dynamics of HIV by multiple cell types through two transmission routes within-host and on the co-epidemic of HIV and TB. Latency of infected cells provides a major challenge to the elimination of HIV within-host since the virus persists at low levels within the latent population. Furthermore, the spread of viral particles through each transmission route may facilitate the progression of the disease due to continued infection of cells by infected cells or free viral particles. Investigating the dynamics of a HIV and TB co-epidemic provides insights into the effects of latency and the long term behavior of the synergistic relationship between HIV and TB. In this work, we extend the integer order systems of differential equations studied in [Xia, 2017] and [Xiaoyan, 2013] to fractional order. The within-host dynamics are described by a system of Caputo fractional derivatives while the co-epidemic by a system of Caputo-Fabrizio fractional derivatives. The equilibrium points of each system are obtained and the reproduction numbers of the diseases are computed. It is shown that the reproduction number of HIV through each transmission route contribute to the reproduction number of HIV through each cell type. Furthermore, the local asymptotic stability of the disease-free equilibrium

is established.