Mathematical Model Analysis of Disease Spread Using Differential Equations

Abstract

This study aims to analyze the mathematical model of infectious disease spread using systems of differential equations. Mathematical epidemiology has become an important scientific approach in understanding disease transmission dynamics, predicting epidemic behavior, and evaluating public health intervention strategies. The study employs a mathematical analytical method using the Susceptible–Infected–Recovered (SIR) model to describe interactions among susceptible, infected, and recovered populations over time. The research process includes model formulation, equilibrium point analysis, basic reproduction number analysis, stability analysis, and numerical simulation of disease transmission dynamics. The mathematical model is represented through systems of ordinary differential equations involving transmission and recovery parameters to explain the progression of infectious diseases within a population. The results show that the transmission rate and recovery rate significantly influence epidemic behavior and disease spread intensity. The analysis of the basic reproduction number (R₀)indicates that if R₀>1, the disease spreads rapidly and may lead to an epidemic outbreak, whereas if R₀<1, disease transmission gradually declines and eventually disappears. Stability analysis demonstrates that the disease-free equilibrium is stable when the reproduction number remains below one and unstable when the reproduction number exceeds one. Numerical simulations further illustrate epidemic phases consisting of initial growth, infection peak, and decline phases. The findings confirm that differential equations provide effective mathematical tools for understanding infectious disease dynamics and supporting evidence-based public health decision-making. However, the study also recognizes that simplified epidemiological models possess limitations because real-world disease transmission is influenced by demographic, environmental, behavioral, and social factors. Therefore, further research is recommended to develop more complex and realistic epidemiological models integrating additional variables and computational approaches.