An epidemic in a dynamic population with importation of infectives

Article, Preprint, Other literature type English OPEN
Ball, Frank ; Britton, Tom ; Trapman, Pieter (2017)
  • Publisher: Institute of Mathematical Statistics
  • Journal: (issn: 1050-5164)
  • Related identifiers: doi: 10.1214/16-AAP1203
  • Subject: 60F05 | 92D30 | Branching process | 60J28 | 60K05 | SIR epidemic | Mathematics - Probability | 60J80 | Skorohod metric | regenerative process | weak convergence
    arxiv: Quantitative Biology::Populations and Evolution

Consider a large uniformly mixing dynamic population, which has constant birth rate and exponentially distributed lifetimes, with mean population size $n$. A Markovian SIR (susceptible $\to$ infective $\to$ recovered) infectious disease, having importation of infectives, taking place in this population is analysed. The main situation treated is where $n\to\infty$, keeping the basic reproduction number $R_{0}$ as well as the importation rate of infectives fixed, but assuming that the quotient of the average infectious period and the average lifetime tends to 0 faster than $1/\log n$. It is shown that, as $n\to\infty$, the behaviour of the 3-dimensional process describing the evolution of the fraction of the population that are susceptible, infective and recovered, is encapsulated in a 1-dimensional regenerative process $S=\{S(t);t\ge0\}$ describing the limiting fraction of the population that are susceptible. The process $S$ grows deterministically, except at one random time point per regenerative cycle, where it jumps down by a size that is completely determined by the waiting time since the start of the regenerative cycle. Properties of the process $S$, including the jump size and stationary distributions, are determined.
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