A modification of the general stochastic epidemic motivated by AIDS modelling

Abstract

This paper considers a model for the spread of an epidemic in a closed, homogeneously mixing population in which new infections occur at rate βxy/(x + y), where x and y are the numbers of susceptible and infectious individuals, respectively, and β is an infection parameter. This contrasts with the standard general epidemic in which new infections occur at rate βxy. Both the deterministic and stochastic versions of the modified epidemic are analysed. The deterministic model is completely soluble. The time-dependent solution of the stochastic model is derived and the total size distribution is considered. Threshold theorems, analogous to those of Whittle (1955) and Williams (1971) for the general stochastic epidemic, are proved for the stochastic model. Comparisons are made between the modified and general epidemics. The effect of introducing variability in susceptibility into the modified epidemic is studied.