Numerical methods in the context of compartmental models in epidemiology

TL;DR

This paper explores numerical methods for analyzing compartmental epidemiological models, focusing on deterministic and stochastic approaches, including finite difference schemes, dynamic programming, and tau-leaping algorithms, to study disease extinction and model divergence.

Contribution

It introduces three numerical techniques—finite difference, dynamic programming, and tau-leaping—for analyzing epidemiological models and their large deviations behavior.

Findings

Finite difference scheme effectively solves the ODE model.

Dynamic programming computes disease extinction times.

Tau-leaping accelerates stochastic model simulations.

Abstract

We consider compartmental models in epidemiology. For the study of the divergence of the stochastic model from its corresponding deterministic limit (i.e., the solution of an ODE) for long time horizon, a large deviations principle suggests a thorough numerical analysis of the two models. The aim of this paper is to present three such motivated numerical works. We first compute the solution of the ODE model by means of a non-standard finite difference scheme. Next we solve a constraint optimization problem via discrete-time dynamic programming: this enables us to compute the leading term in the large deviations principle of the time of extinction of a given disease. Finally, we apply the {\tau}-leaping algorithm to the stochastic model in order to simulate its solution efficiently. We illustrate these numerical methods by applying them to two examples.