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

Ruibo Ma, Tai Heng Liu, Baghdadi Othmane, Dong Yao

Abstract

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

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

Abstract

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, and . 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 and , 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.
Paper Structure (4 sections, 3 theorems, 25 equations, 2 figures, 2 tables)

This paper contains 4 sections, 3 theorems, 25 equations, 2 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Single-type SIR on Galton-Watson trees
  3. Two-type SIR on Galton-Watson trees
  4. Conclusion and future directions

Key Result

Theorem 1

Let the infection rate be $\alpha$ and the recovery rate be $\mu$ for a single-type SIR process on $GW (p)$. At time 0, only the root is infected. The probability of survival is greater than 0 if and only if $\alpha/\mu > 1/{(m-1)}$, where $m$ is the mean of $p$. Thus, if $\mu$ is fixed, then the cr

Figures (2)

  • Figure 1: The average number of $B$ infections in each generation. Five parameters are fixed: $\alpha _1 = 5$, $\alpha_2 = 0.75$, $\beta_1 = 8$, and $\mu _1 = \mu _2 = 1$. The value of $\beta_2$ varies from 1.0 to 1.8.
  • Figure 2: The proportion of simulations with some $B$ infections in each generation. Five parameters are fixed: $\alpha _1 = 5$, $\alpha_2 = 0.75$, $\beta_1 = 8$, and $\mu _1 = \mu _2 = 1$. The value of $\beta_2$ varies from 1.0 to 1.8.

Theorems & Definitions (6)

  • Theorem 1
  • Theorem 2
  • Remark 1
  • Lemma 1
  • proof
  • proof : Proof of Theorem \ref{['general']}