Modeling Infectious Diseases: From SIR Models to Diffusion-Based Approaches and Numerical Solutions

As global living standards improve and medical technology advances, many infectious diseases have been effectively controlled. However, certain diseases, such as the recent COVID-19 pandemic, continue to pose significant threats to public health. This paper explores the evolution of infectious disease modeling, from early ordinary differential equation-based models like the SIR framework to more complex reaction-diffusion models that incorporate both temporal and spatial dynamics. The study highlights the importance of numerical methods, such as the Runge-Kutta method, implicit-explicit time-discretization techniques, and finite difference methods, in solving these models. By analyzing the development and application of these methods, this research underscores their critical role in predicting disease spread, informing public health strategies, and mitigating the impact of future pandemics.