A stochastic epidemic model for the impact of contact heterogeneity on extinction probability, final size, and herd immunity

We propose a stochastic epidemic framework that incorporates contact heterogeneity into transmission dynamics through a negative binomial distribution, capturing the overdispersion associated with superspreading phenomena. Analytical results show that, although the basic reproduction number is independent of the dispersion parameter, heterogeneity significantly alters epidemic outcomes. In particular, stronger heterogeneity increases the probability of early extinction via a stochastic bottleneck effect, while simultaneously reducing both the final epidemic size and the herd immunity threshold. To characterize these effects, we formulate a discrete-time Markov chain (DTMC) model and derive its deterministic mean-field limit, establishing their asymptotic consistency as the population size tends to infinity. Stochastic simulations further reveal a dynamically sensitive regime under low dispersion and low mean contact, as well as a trade-off between epidemic peak and duration, consistent with empirical contact patterns reported in COVID-19 studies. From a complex systems perspective, these findings can be understood as a consequence of the interplay between contact topology and transmission dynamics: highly connected individuals dominate early spreading but are preferentially removed from the susceptible pool, leading to a self-limiting effect at the population level. This mechanism highlights the importance of incorporating contact heterogeneity when modeling epidemic processes on complex networks.