Mathematical epidemiological models with finite time extinction : the case of African swine fever virus in wildlife areas

Abstract

In Mathematical Epidemiology disease free states are commonly represented as equilibria of dynamical systems which model the respective epidemiological processes. However, in cases when the equilibrium is zero and is related to extinction (of the population), due to the uniqueness property of a complete dynamical system, solutions may converge to an equilibrium but never reach it. This may give rise to qualitatively unrealistic behaviour such as a population that is practically extinct but is able to grow. An example of a case when this problem may arise is when modelling the dynamics of African Swine Fever (ASF), a contagious disease a ecting both domestic and wild pigs, in the Mkuze Game Reserve. In the paper by Arnot et. al.[3] it was established that although an increase in burrow infestation rates was observed, the disease was not detected within the game reserve. This situation cannot be captured using a model with exponential decay. In the following research project, we study various ODE and PDE models with the property that solutions approaching the disease free equilibrium 0, will reach it within nite time and remain at 0 thereafter. These include basic population models and epidemiological models with age and state structure. We then construct a model for ASF in order to accurately illustrate the phenomenon observed at the game reserve.