Comparing the Behaviour of Deterministic and Stochastic Model of SIS Epidemic

TL;DR

This paper compares deterministic and stochastic SIS epidemic models, analyzing their behaviors around the basic reproduction number $R_0$, and highlights differences in epidemic predictions and outcomes.

Contribution

It provides a detailed comparison of deterministic and stochastic SIS models, especially regarding their behavior when $R_0$ exceeds 1.

Findings

Deterministic models predict endemic equilibrium for $R_0 > 1$.

Stochastic models allow for the possibility of epidemic die-out even when $R_0 > 1$.

Large populations lead stochastic models to resemble deterministic predictions.

Abstract

Studies about epidemic modelling have been conducted since before 19th century. Both deterministic and stochastiic model were used to capture the dynamic of infection in the population. The purpose of this project is to investigate the behaviour of the models when we set the basic reproduction number, $R_0$. This quantity is defined as the expected number of contacts made by a typical infective to susceptibles in the population. According to the epidemic threshold theory, when $R_0 \leq 1$, minor epidemic occurs with probability one in both approaches, but when $R_0 > 1$, the deterministic and stochastic models have different interpretation. In the deterministic approach, major epidemic occurs with probability one when $R_0 > 1$ and predicts that the disease will settle down to an endemic equilibrium. Stochastic models, on the other hand, identify that the minor epidemic can possibly occur. If it does, then the epidemic will die out quickly. Moreover, if we let the population size be large and the major epidemic occurs, then it will take off and then reach the endemic level and move randomly around the deterministic's equilibrium.