A predator-prey SIR type dynamics on large complete graphs with three phase transitions

Abstract

We are interested in a variation of the SIR (Susceptible/Infected/Recovered) dynamics on the complete graph, in which infected individuals may only spread to neighboring susceptible individuals at fixed rate $λ>0$ while recovered individuals may only spread to neighboring infected individuals at fixed rate $1$. This is also a variant of the so-called chase-escape process introduced by Kordzakhia and then Bordenave & Ganesan. Our work is the first study of this dynamics on complete graphs. Starting with one infected and one recovered individuals on the complete graph with $N+2$ vertices, and stopping the process when one type of individuals disappears, we study the asymptotic behavior of the probability that the infection spreads to the whole graph as $N\rightarrow\infty$ and show that for $λ\in (0,1)$ (resp. for $λ>1$), the infection dies out (resp. does not die out) with probability tending to one as $N\rightarrow\infty$, and that the probability that the infection dies out tends to $1/2$ for $λ=1$. We also establish limit theorems concerning the asymptotic state of the system in all regimes and show that two additional phase transitions occur in the subcritical phase $λ\in (0,1)$: at $λ=1/2$ the behavior of the expected number of remaining infected individuals changes, while at $λ=(\sqrt {5}-1)/2$ the behavior of the expected number of remaining recovered individuals changes. We also study the outbreak sizes of the infection, and show that the outbreak sizes are small if $λ\in(0,1/2]$, exhibit a power-law behavior for $1/2<λ<1$, and are pandemic for $λ\geq 1$. Our method relies on different couplings: we first couple the dynamics with two independent Yule processes by using an Athreya-Karlin embedding, and then we couple the Yule processes with Poisson processes thanks to Kendall's representation of Yule processes.