This study investigates the dynamics of measles epidemics using deterministic and stochastic SIR models. In the deterministic model, we analyze infection dynamics through the basic reproductive number \({\mathcal{R}_0} \), assessing equilibrium stability for \(\mathcal{R}_0 < 1 \) and \(\mathcal{R}_0 > 1 \). In the stochastic model, we establish the existence of global solutions and define the stochastic reproductive number \({\mathcal{R}^s_0}\), examining disease dynamics under varying environmental fluctuations. While the deterministic model suggests disease persistence, stochastic factors introduce the possibility of extinction events, highlighting the impact of randomness on disease transmission. Additionally, we present conditions for infection extinction and discuss the stochastic stability of solutions. Numerical simulations illustrate the theoretical findings.
