Duration of a minor epidemic

Abstract

Disease outbreaks in stochastic SIR epidemic models are characterized as either minor or major. When ${ℛ}_0<1$ , all epidemics are minor, whereas if ${ℛ}_0>1$ , they can be minor or major. In 1955, Whittle derived formulas for the probability of a minor or a major epidemic. A minor epidemic is distinguished from a major one in that a minor epidemic is generally of shorter duration and has substantially fewer cases than a major epidemic. In this investigation, analytical formulas are derived that approximate the probability density, the mean, and the higher-order moments for the duration of a minor epidemic. These analytical results are applicable to minor epidemics in stochastic SIR, SIS, and SIRS models with a single infected class. The probability density for minor epidemics in more complex epidemic models can be computed numerically applying multitype branching processes and the backward Kolmogorov differential equations. When ${ℛ}_0$ is close to one, minor epidemics are more common than major epidemics and their duration is significantly longer than when ${ℛ}_0\ll 1$ or ${ℛ}_0\gg 1$ .

Keywords: Birth-death process; Branching process; Epidemic model; Markov chain.