Local approximation of Markov chains in time and space

Abstract

In epidemic modelling, the emergence of a disease is characterized by the low numbers of infectious individuals. Environmental randomness impacts outcomes such as the outbreak or extinction of the disease in this case. This randomness can be accounted for by modelling the system as a continuous time Markov chain, $X\left(t\right)$ . The probability of extinction given some initial state is the probability of hitting a subset of the state space associated with extinction for the initial state. This hitting probability can be studied by passing to the discrete time Markov chain (DTMC), $X_n$ . An approach is presented to approximate a DTMC on a countably infinite state space by a DTMC on a finite state space for the purpose of solving general hitting problems. This approach is applied to approximate the probability of disease extinction in an epidemic model. It is also applied to evaluate a heterogeneous disease control strategy in a metapopulation.

Keywords: Epidemiology; Markov chain; population dynamics; public health.