A Novel Stochastic SVIR Model Capturing Transmission Variability Through Mean-Reverting Processes and Stationary Reproduction Thresholds

This study presents a stochastic SVIR epidemic model in which disease transmission rates fluctuate randomly over time, driven by independent, mean-reverting processes with multiplicative noise. These dynamics capture environmental variability and behavioral changes affecting disease spread. We derive analytical expressions for the conditional moments of the transmission rates and establish the existence of their stationary distributions under broad conditions. By averaging over these distributions, we define a stationary effective reproduction number that enables a probabilistic classification of outbreak scenarios. Specifically, we estimate the likelihood of disease persistence or extinction based on transmission uncertainty. Sensitivity analyses reveal that the shape and intensity of transmission variability play a decisive role in epidemic outcomes. Monte Carlo simulations validate our theoretical findings, showing strong agreement between empirical distributions and theoretical predictions. Our results underscore how randomness in disease transmission can fundamentally alter epidemic trajectories, offering a robust mathematical framework for risk assessment under uncertainty.