Simple Approximations for Epidemics with Exponential and Fixed Infectious Periods

Abstract

Analytical approximations have generated many insights into the dynamics of epidemics, but there is only one well-known approximation which describes the dynamics of the whole epidemic. In addition, most of the well-known approximations for different aspects of the dynamics are for the classic susceptible–infected–recovered model, in which the infectious period is exponentially distributed. Whilst this assumption is useful, it is somewhat unrealistic. Equally reasonable assumptions are that the infectious period is finite and fixed or that there is a distribution of infectious periods centred round a nonzero mean. We investigate the effect of these different assumptions on the dynamics of the epidemic by deriving approximations to the whole epidemic curve. We show how the well-known sech-squared approximation for the infective population in ‘weak’ epidemics (where the basic reproduction rate $R_0\approx 1$) can be extended to the case of an arbitrary distribution of infectious periods having finite second moment, including as examples fixed and gamma-distributed infectious periods. Further, we show how to approximate the time course of a ‘strong’ epidemic, where $R_0\gg 1$, demonstrating the importance of estimating the infectious period distribution early in an epidemic.