  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 $\lambda>0$ 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 $\lambda\in (0,1)$ (resp. for
$\lambda>1$), 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 $\lambda=1$. 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
$\lambda\in (0,1)$: at $\lambda=1/2$ the behavior of the expected number of
remaining infected individuals changes, while at $\lambda=(\sqrt {5}-1)/2$ 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 $\lambda \in(0,1/2]$, exhibit a power-law behavior for
$1/2<\lambda<1$, and are pandemic for $\lambda\geq 1$. 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.
