Mathematical Modeling of Epidemics Using Stochastic Differential Equations: A Review - Prabhat Kushwaha (Recent Trends in Mathematics, 2025)
The accurate modeling of infectious disease dynamics is crucial for predicting outbreaks and informing public health interventions. While deterministic models such as the SIR (Susceptible-Infected-Recovered) framework have traditionally been used to understand disease transmission, they often fail to account for the randomness inherent in real-world scenarios. Disease spread is influenced by numerous uncertain factors, including individual behavioral changes, environmental fluctuations, and imperfect data reporting. These uncertainties can significantly impact model predictions, particularly during the initial phase of an outbreak when the number of infected individuals is low and random events play a critical role. Stochastic Differential Equations (SDEs) have emerged as a powerful mathematical tool for incorporating randomness into epidemic models. By adding stochastic components to classical compartmental models, SDEs provide a more realistic and flexible representation of disease dynamics. These models allow for the simulation of random perturbations in transmission, recovery, and contact rates, enabling researchers to explore a wider range of possible epidemic outcomes. Additionally, SDE-based models are better suited to capturing extinction events, variability in peak infection times, and the probability of outbreak containment under uncertainty. This review presents a comprehensive overview of the application of SDEs in epidemic modeling. It discusses the mathematical foundation of SDEs, methods of converting deterministic models into their stochastic counterparts, and commonly used numerical approaches such as the Euler–Maruyama method for solving SDEs. We also highlight recent applications of these models in analyzing diseases like COVID-19, Ebola, and seasonal influenza. Furthermore, the review addresses current challenges in parameter estimation and model validation, and suggests future research directions including hybrid modeling, data-driven approaches, and the integration of machine learning with stochastic frameworks. Overall, SDE-based epidemic modeling is a rapidly evolving field with significant potential to enhance our understanding and response to infectious diseases.
Keywords: Disease Transmission Uncertainty; Epidemiological Forecasting; Randomness in Disease Dynamics; SIR Model with Noise; Stochastic Differential Equations (SDEs); Stochastic Epidemic Models