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.
