The extinction time of a subcritical branching process related to the SIR epidemic on a random graph (Q2794736)

The author studies a continuous-time Markov multitype branching process arising from the SIR (susceptible-infective-recovered) model on a random graph with given degrees. The model is defined as follows. Any individual in a population carries some number \(k\in \{1,2,\dots\}\) of spores. The number \(k\) is called the type of the individual. Each spore, at a given rate \(\beta>0\), is released, and gives rise to a new individual whose number of spores \(J\) is chosen according to some probability distribution \((p_j)_{j=1,2,\dots}\) with expectation \(\mu>0\) and finite second moment. Furthermore, each individual is removed from the population at rate \(\rho>0\). The author concentrates on the subcritical case when \(\lambda := \rho + \beta(1-\mu)>0\). For the probability \(q_k(t)\) that the population survives at time \(t>0\), given that it started from a single individual carrying \(k\) spores, he proves the asymptotic tail estimate NEWLINE\[NEWLINE q_k(t)= c k e^{-\lambda t} (1 + O(ke^{-at})),\quad t\to +\infty, NEWLINE\]NEWLINE for all \(k\in\mathbb N\), where \(a<\min (\lambda,\beta)\) and \(c\in (0,1]\) is a certain constant. Using this result, he proves a Gumbel (double exponential) limit law for the extinction time of a large population.