Analysis of a Deterministic SIR Model with Numerical Simulations for Infectious Disease Dynamics

In this paper, we revisit the classical deterministic SIR (Susceptible–Infectious–Recovered) framework as a baseline model for the transmission of directly communicable infectious diseases. Building on the foundational work of Kermack and McKendrick and subsequent developments in mathematical epidemiology, we formulate an SIR system with vital dynamics and provide a self-contained qualitative analysis. We first establish the positivity and boundedness of solutions and identify a biologically meaningful invariant region. The disease-free and endemic equilibria are derived explicitly, and the basic reproduction number R ₀ is computed via the next-generation matrix approach. We then investigate the local stability of the equilibria in terms of R ₀ , showing that the disease-free equilibrium is locally asymptotically stable when R ₀ <1 and unstable when R ₀ >1, while an endemic equilibrium exists and becomes locally asymptotically stable for R ₀ >1. For a specific parameter regime, we complement the local analysis with global stability results based on Lyapunov-type arguments. On the numerical side, we discretize the model using a classical fourth-order Runge–Kutta scheme and perform systematic simulations to illustrate the threshold behavior associated with R ₀ and the influence of key parameters such as the transmission and recovery rates. The numerical experiments highlight how changes in contact patterns or treatment efficacy affect the epidemic peak, the timing of the outbreak, and the long-term burden of infection. The results confirm the consistency between the qualitative theory and time-dependent simulations and underline the role of simple deterministic models as a first step toward more elaborate data-driven formulations for real-world epidemics.