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

In the standard SIR model on a graph, infected vertices infect their neighbors at rate and recover at rate .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.