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

## TL;DR

This paper models how two successive epidemics spread through a network, considering that some people might be partially immune after the first outbreak.

## Contribution

The paper introduces a novel branching process method to analyze the second epidemic based on the structure of the first.

## Key findings

- A threshold parameter for the second epidemic is calculated using a branching process approximation.
- The probability of a large outbreak in the second epidemic is determined based on the first epidemic's structure.
- The model accurately approximates outcomes for finite populations, matching known cases from the literature.

## Abstract

We present a stochastic model for two successive SIR (Susceptible \documentclass[12pt]{minimal}
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\begin{document}$$\rightarrow $$\end{document}→ Infectious \documentclass[12pt]{minimal}
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\begin{document}$$\rightarrow $$\end{document}→ 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.