Duration of a minor epidemic

Disease outbreaks in stochastic SIR epidemic models are characterized as either minor or major. When $$ , all epidemics are minor, whereas if $$ , 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 $$ is close to one, minor epidemics are more common than major epidemics and their duration is significantly longer than when $$ or $$ .