Large deviations in non-Markovian stochastic epidemics

Abstract

We develop a framework for non-Markovian, well-mixed SIR and SIS models beyond mean field, utilizing the continuous-time random walk formalism. Using a gamma distribution for the infection and recovery inter-event times as a test case, we derive asymptotical late-time master equations with effective memory kernels and obtain analytical predictions for the final outbreak size distribution in the SIR model, and quasistationary distribution and disease lifetime in the SIS model. We show that varying the width of the inter-event time distribution can greatly alter the outbreak size distribution or the disease lifetime. We also show that rescaled Markovian models may fail to capture fluctuations in the non-Markovian case. Overall, our analysis, confirmed against numerical simulations, paves the way for studying large deviations in structured populations on degree-heterogeneous networks