Extinction times in the subcritical stochastic SIS logistic epidemic

Graham Brightwell, Thomas House, Malwina Luczak

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    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 (Formula presented.) but more slowly than (Formula presented.). 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.

    Original languageEnglish
    Pages (from-to)1-39
    Number of pages39
    JournalJournal of Mathematical Biology
    Issue number2
    Early online date31 Jan 2018
    Publication statusPublished - Aug 2018


    • Birth-and-death chain
    • Near-critical epidemic
    • Stochastic SIS logistic epidemic
    • Time to extinction


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