A hybrid approach for the simulation of S-I-R epidemics

Kurzfassung

Stochastic reaction networks have been widely used to study biochemical reaction systems. The temporal evolution of such systems is governed by the Chemical Master Equation (CME) whose solution is a time-dependent probability distribution on all possible system states. However, with a large number of particles and reactions, the system state rapidly becomes intractable, rendering the CME practically unsolvable in most realistic scenarios. To study the system’s behavior despite this intractability, stochastic simulation algorithms can be used to generate sample trajectories of the underlying stochastic process. Yet, since a multitude of simulations is required to obtain a reliable approximation of the full probability distribution, this can quickly become a computationally expensive task. An alternative is the Method of Moments, which involves the derivation of moment equations describing the dynamics of the mean and all central moments of the CME through a system of ordinary differential equations. This method is particularly useful for systems exhibiting large stochasticity. However, as lower-order moments generally depend on higher-order moments, moment closure methods are necessary. Such closure methods approximate higher-order moments in terms of lower ones, inevitably introducing a trade-off between accuracy and computational efficiency. Mathematical epidemiological models have long been used to study transmission dynamics of infectious diseases and support public health decision-making. These models range from deterministic mean-field models assuming a homogeneously mixed population, to metapopulation models accounting for spatial heterogeneity, and detailed agent-based models. In this work, we apply stochastic reaction networks in an epidemiological context. Specifically, we present a spatially-resolved stochastic susceptible-infected-recovered epidemiological model with different regions, where the movement of individuals between regions influences the local and global infection dynamics. We derive the corresponding system of moment equations and investigate the performance of zero-cumulant closure under varying conditions. To balance computational cost and the accurate representation of stochastic effects, we propose a hybrid modeling approach that combines stochastic simulation with moment-equation-based approximation. We analyze different switching criteria used to transition between the stochastic and deterministic components of the hybrid model, comparing their errors in means and variances in relation to the computational effort. Furthermore, we demonstrate that the hybrid approach is able to accurately capture the system’s stochastic dynamics in regimes where the naive moment closure approach fails.