Abstract
We consider the spread of a supercritical stochastic SIR (Susceptible, Infectious, Recovered) epidemic on a configuration model random graph. We mainly focus on the final stages of a large outbreak and provide limit results for the duration of the entire epidemic, while we allow for non-exponential distributions of the infectious period and for both finite and infinite variance of the asymptotic degree distribution in the graph. Our analysis relies on the analysis of some subcritical continuous time branching processes and on ideas from first passage percolation. As an application we investigate the effect of vaccination with an all-or-nothing vaccine on the duration of the epidemic. We show that if vaccination fails to prevent the epidemic, it often – but not always – increases the duration of the epidemic.
Highlights
Mathematical models have been widely used to study the spread of infectious diseases and to design control strategies for reducing the impact of those diseases [13]
In this manuscript we obtain asymptotic results on both the time of strong extinction T ∗(n) and the time until weak extinction T †(n) of an SIR epidemic on a configuration model random graph with n vertices. In these concluding remarks we only consider the time of strong extinction
We show that conditioned on a large outbreak and under some further mild conditions T ∗(n)/ log n converges in probability to (α )−1 + |α∗|−1, where α and α∗ are Malthusian parameters of branching processes that approximate respectively the early phase and the final phase of the epidemic
Summary
Mathematical models have been widely used to study the spread of infectious diseases and to design control strategies for reducing the impact of those diseases [13]. Volz [25] studied a deterministic model for the spread of an SIR epidemic through a network using a set of differential equations, keeping track of the probability that a vertex of given degree avoids infection as a function of time. Using a different mathematical approach Barbour and Reinert [7] study (among other things) a stochastic model for the spread of SIR epidemics on a configuration model with bounded degrees and minor conditions on the infectious period distribution. The duration of a supercritical SIR epidemic on a configuration model of that uniformly chosen vertex in an SI epidemic (i.e. an SIR epidemic with infinite infectious period) In this setting the question regarding the time until the last infection in the epidemic corresponds to the flooding time of the giant component of the random graph [1]. This vaccination strategy is asymptotically equivalent to vaccinating a uniform fraction of the population with a perfect vaccine, i.e. a vaccine which gives complete immunity
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