The probability of epidemic fade-out is non-monotonic in transmission rate for the Markovian SIR model with demography.

Epidemic fade-out refers to infection elimination in the trough between the first and second waves of an outbreak. The number of infectious individuals drops to a relatively low level between these waves of infection, and if elimination does not occur at this stage, then the disease is likely to become endemic. For this reason, it appears to be an ideal target for control efforts. Despite this obvious public health importance, the probability of epidemic fade-out is not well understood. Here we present new algorithms for approximating the probability of epidemic fade-out for the Markovian SIR model with demography. These algorithms are more accurate than previously published formulae, and one of them scales well to large population sizes. This method allows us to investigate the probability of epidemic fade-out as a function of the effective transmission rate, recovery rate, population turnover rate, and population size. We identify an interesting feature: the probability of epidemic fade-out is very often greatest when the basic reproduction number, R0, is approximately 2 (restricting consideration to cases where a major outbreak is possible, i.e., R0>1). The public health implication is that there may be instances where a non-lethal infection should be allowed to spread, or antiviral usage should be moderated, to maximise the chance of the infection being eliminated before it becomes endemic.