A Linear Algebraic Framework for SIR Epidemic Modelling with Applications to Infectious Disease Dynamics and Healthcare Planning

The Susceptible–Infected–Recovered (SIR) model is a fundamental framework in mathematical epidemiology used to analyse the spread and control of infectious disease within a population. Although classical SIR models provide valuable insights into epidemic dynamics, their nonlinear structure can make analytical investigation of local stability and epidemic thresholds challenging. In this study, a linear algebraic framework for the SIR model is developed by expressing the system in matrix form and analysing it through eigenvalues and eigenvectors. By linearizing the system around equilibrium points, the proposed approach enables systematic evaluation of local stability conditions and epidemic thresholds, providing insight into the early-stage growth or decay of an infection. In addition to its theoretical advantages, the framework has important medical and public health applications. The eigenvalue-based analysis facilitates assessment of epidemic growth rates, local stability characteristics, and identification of critical control thresholds. These insights can support healthcare planning, including hospital resource allocation, intervention strategies, and epidemic preparedness. Overall, the study demonstrates that linear algebraic techniques provide a useful complementary framework for analysing local epidemic dynamics and supporting epidemiological decision-making. A comparison between the nonlinear SIR model and its linearized approximation is also presented to highlight the applicability and limitations of the linear algebraic approach.