A Markovian SuASR Epidemic Framework With Stochastic External Factors

Epidemic waves are often shaped by uncertain external drivers, such as environmental variability or behavioral and policy changes, that can randomly amplify or dampen transmission. Capturing this randomness requires models whose dynamics evolve under time-varying, stochastic influences. In this paper, we introduce a stochastic susceptible-asymptomatic-symptomatic-removed (SuASR) framework that embeds random external factors directly into the epidemic mechanism through an extended continuous-time Markov chain (CTMC). The proposed framework is flexible enough to represent epidemic dynamics both with reinfection and with infection-induced immunity, depending on the epidemiological setting under study. Unlike standard Markovian epidemic models, our formulation explicitly tracks two concurrent infectious sources (asymptomatic and symptomatic), yielding coupled infection pathways and substantially richer transient behavior. A key contribution is analytical tractability, where we derive computable expressions for the joint distribution of asymptomatic and symptomatic infection-event counts and develop tools tailored to extreme-outbreak risk, including the distribution of the maximum infectious burden. These results enable probabilistic quantification of epidemic severity beyond mean trajectories, providing direct estimates of the likelihood of rare but consequential surges. We validate the model through an extensive sensitivity analysis and an application to the H1N1 influenza epidemic, illustrating how stochastic external influences reshape outbreak profiles, total infections from both sources, and the probability of extreme epidemic scenarios.