Stochastic Analysis of a Closed SIS Malaria Model

Abstract

A pseudo-equilibrium approximation model for the dynamics and transmission of malaria in human populations is studied. A stochastic version of this model is then formulated and analyzed based on the fact that the disease is endemic and therefore has a basic reproduction number greater than unity. Using a comprehensive theory on asymptotic approximation techniques in recurrent epidemics, approximations for the quasi-stationary distribution and the time to extinction are derived. We find that whenever the reproduction number is greater than unity, the time to extinction of the disease is exponentially distributed with positive exponent and therefore becomes very large within very large human population sizes. We then interpret the fact that it has been difficult to eradicate malaria with the exponentially large time to extinction of the number of infected individuals in the population of humans.

Keywords: Pseudo-equilibrium Approximation, Stationary Distribution, Quasi-stationary Distribution