Modelling the spread of two successive SIR epidemics on a configuration model network

TL;DR

This paper develops a stochastic model to analyze the spread of two successive SIR epidemics on a network, accounting for partial immunity and using branching processes to estimate outbreak probabilities and sizes.

Contribution

It introduces a novel combined percolation and branching process framework to model successive epidemics with immunity effects on a configuration model network.

Findings

Threshold parameter for second epidemic identified

Probability of large second outbreak calculated

Fraction of population infected in second outbreak estimated

Abstract

We present a stochastic model for two successive SIR (Susceptible, Infectious, Recovered) epidemics in the same network structured population. Individuals infected during the first epidemic might have (partial) immunity for the second one. The first epidemic is analysed through a bond percolation model, while the second epidemic is approximated by a three-type branching process in which the types of individuals depend on their position in the percolation clusters used for the first epidemic. This branching process approximation enables us to calculate, in the large population limit and conditional upon a large outbreak in the first epidemic, a threshold parameter and the probability of a large outbreak for the second epidemic. A second branching process approximation enables us to calculate the fraction of the population that are infected by such a second large outbreak. We illustrate our results through some specific cases which have appeared previously in the literature and show that our asymptotic results give good approximations for finite populations.