Chalmers Conferences, 9th European Conference on Mathematical and Theoretical Biology

Backward bifurcations in a model coupling within-host and between-host dynamics
Zhilan Feng

Last modified: 2014-06-09




Analysis of a mathematical model that couples explicitly the within-host and between-host dynamics in an environmentally-driven infectious disease will be presented. The within-host sub-system is linked to a contaminated environment (E) via an inoculation rate g(E) for the hosts. The within-host parasite load V(E) can affect the environment contamination, which directly contributes to the infection rate of hosts for the between-host sub-system. When the two sub-systems are considered in isolation (i.e., g(E)=0), the dynamics are standard and simple. That is, either the infection-free equilibrium is stable or a unique positive equilibrium is stable depending on the relevant reproduction number being less or greater than 1. However, when the two sub-systems are explicitly coupled (i.e., g(E)>0), the full system exhibits more complex dynamics including backward bifurcations; that is, multiple positive equilibria can exist with one of which being stable even when the relevant reproduction number is less than 1. Analytic results are obtained which help determine the conditions for various bifurcations as well as global dynamics of the sub-systems. The analysis is carried out by separating the fast and slow variables based on the fact that the immunological and epidemiological processes occur on very different time scales, which provides the possibility to use tools from singular perturbation theory. Numerical simulations of the full system confirm the analytic results.