MATHEMATICAL MODELLING OF INFECTIOUS DISEASE SPREAD USING DIFFERENTIAL EQUATIONS: ANALYSIS AND APPLICATIONS OF THE SIR AND SEIR MODELS

Abstract

Mathematical epidemiology provides a quantitative framework for studying how infectious diseases move through populations. Differential equations are particularly valuable because infection, recovery, loss of susceptibility, and intervention are processes that evolve continuously in time. This paper develops a focused study of two foundational compartmental models: the Susceptible–Infected–Recovered (SIR) model and the Susceptible Exposed–Infected–Recovered (SEIR) model. The work explains how epidemiological assumptions are translated into ordinary differential equations, how initial information is incorporated, and how model properties can be interpreted through equilibrium analysis, reproduction numbers, stability, sensitivity analysis, and numerical solution. The SIR model represents susceptible, infectious, and recovered groups, while the SEIR model adds an exposed compartment to represent a latent period between infection and infectiousness. The paper derives governing equations, conservation relations, threshold conditions, and intervention formulations. Numerical approaches including Euler and fourth-order Runge–Kutta methods are discussed, together with positivity, boundedness, and numerical stability. Three diagrammatic outputs and two analytical tables clarify the model structure, epidemic trajectories, and intervention scenarios. The principal conclusion is that SIR and SEIR differential equations are powerful explanatory and decision-support tools, but their predictions depend strongly on assumptions, parameter estimation, data quality, population mixing, and the temporal stability of transmission conditions. Mathematical results should therefore be interpreted as scenario-dependent estimates rather than exact forecasts.