Stochastic modelling of viral infection spread via a Partial Integro-Differential Equation

In the present article we propose a Partial Integro-Differential Equation (PIDE) model to approximate a stochastic SIS compartmental model for viral infection spread. First, an appropriate set of reactions is considered, and the corresponding Chemical Master Equation (CME) that describes the evolution of the reaction network as a stochastic process is posed. In this way, the inherent stochastic behaviour of the infection spread is incorporated in the modelling approach. More precisely, by considering that infection is propagated in bursts we obtain the PIDE model as the continuous counterpart to approximate the CME. In this way, the model takes into account that one infectious individual can be in contact with more than one susceptible person at a given time. Moreover, an appropriate semi-Lagrangian numerical method is proposed to efficiently solve the PIDE model. Numerical results and computational times for CME and PIDE models are compared and discussed. We also include a comparison of the main statistics of the PIDE model with the deterministic ODE model. Moreover, we obtain an analytical expression for the stationary solution of the proposed PIDE model, which also allows us to study the disease persistence. The methodology presented in this work is also applied to a real scenario as the COVID-19 pandemic.