# Estimating of the final size relation for SIR discrete epidemic model under intervention

### Abstract

We investigate the dynamic behaviors of a discrete-time $SIR$ epidemic model under intervention. Specifically, we examine the effects of reducing contacts among individuals at a particular time $m$ by a factor of $q > 1$, such that $\mathcal{R}_0/q < 1$. Our analysis reveals that in this case, the final size of the epidemic can be approximated by the formula $R_m\mathcal{R}_0(1-1/q)/(1-\mathcal{R}_0/q)$, where $R_m$ denotes the cumulative number of cases at date $m$ and $\mathcal{R}_0$ is the basic reproductive number. We also explore the situation where contacts are entirely eliminated at an early stage of the epidemic, i.e., when $q \to \infty$. Our results indicate that in such cases, the final size of the epidemic can be estimated by multiplying the cumulative number of cases $R_m$ at that date by the basic reproductive number $\mathcal{R}_0$. Our study provides valuable insights into the effectiveness of intervention strategies in controlling the spread of infectious diseases. We anticipate that our findings will be of interest to researchers, policymakers, and public health officials.