A note on the cooperative two-type SIR processes on Galton-Watson trees

TL;DR

This paper analyzes a two-type SIR epidemic process on Galton-Watson trees, showing that the critical threshold for disease survival remains unchanged despite the interaction between two infectious diseases.

Contribution

It introduces a two-type SIR model with cooperative interactions and proves that the critical threshold matches that of the single-type model on Galton-Watson trees.

Findings

Critical value for survival is unchanged by disease interaction.

Two-type model behaves similarly to single-type in terms of phase transition.

Initial infection with both types at the root suffices for analysis.

Abstract

In the standard SIR model on a graph, infected vertices infect their neighbors at rate $\alpha$ and recover at rate $\mu$. We consider a two-type SIR process where each individual in the graph can be infected with two types of diseases, $A$ and $B$. Moreover, the two diseases interact in a cooperative way so that an individual that has been infected with one type of disease can acquire the other at a higher rate. We prove that if the underlying graph is a Galton-Watson tree and initially the root is infected with both $A$ and $B$, while all others are susceptible, then the two-type SIR model has the same critical value for the survival probability as the classic single-type model.