Brightwell, Graham 
ORCID: 0000-0001-5955-3628, House, Thomas and Luczak, Malwina J.
(2018)
Extinction times in the subcritical stochastic SIS logistic epidemic.
Journal of Mathematical Biology, 77 (2).
pp. 455-493.
ISSN 0303-6812
Abstract
Many real epidemics of an infectious disease are not straightforwardly super- or sub-critical, and the understanding of epidemic models that exhibit such complexity has been identified as a priority for theoretical work. We provide insights into the near-critical regime by considering the stochastic SIS logistic epidemic, a well-known birth-and-death chain used to model the spread of an epidemic within a population of a given size N. We study the behaviour of the process as the population size N tends to infinity. Our results cover the entire subcritical regime, including the “barely subcritical” regime, where the recovery rate exceeds the infection rate by an amount that tends to 0 as N→∞ but more slowly than N−1/2 . We derive precise asymptotics for the distribution of the extinction time and the total number of cases throughout the subcritical regime, give a detailed description of the course of the epidemic, and compare to numerical results for a range of parameter values. We hypothesise that features of the course of the epidemic will be seen in a wide class of other epidemic models, and we use real data to provide some tentative and preliminary support for this theory
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