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

We study 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. 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→∞ and show that for λ∈(0,1) (resp. for λ>1), the infection dies out (resp. does not die out) with probability tending to one as N→∞, and that the probability that the infection dies out tends to 1/2 for λ=1. We also establish limit theorems concerning the final state of the system in all regimes and show that two additional phase transitions occur in the subcritical phase λ∈(0,1): at λ=1/2 the behavior of the expected number of remaining infected individuals changes, while at λ=(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 (or self-limiting) if λ∈(0,1/2], exhibit a power-law behavior for 1/2<λ<1, and are pandemic for λ⩾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.