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

Ahmed A.F.  Osman; Aijaz  Magray; Snehal  Rathi; D.  Raju; Madhulika  Mishra; Aafaq A.  Rather; Mohammed  Ataelfadiel; Yasser Ayad  Soliman; Ali Salem  Bin Sama; Ala  Abdullah

doi:10.6000/1929-6029.2026.15.23

Authors

Ahmed A.F. Osman Applied College, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia
Aijaz Magray Department of Mathematics, JB Institute of Engineering and Technology, Hyderabad-500075, India
Snehal Rathi Department of Computer Engineering, Vishwakarma Institute of Technology, Pune, India
D. Raju Department of Mathematics, Jawaharlal Nehru Technical University, Hyderabad-500085, India
Madhulika Mishra Symbiosis Statistical Institute, Symbiosis International (Deemed University), Pune, India
Aafaq A. Rather Symbiosis Statistical Institute, Symbiosis International (Deemed University), Pune, India
Mohammed Ataelfadiel Applied College, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia
Yasser Ayad Soliman Applied College, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia
Ali Salem Bin Sama Department of Management Information Systems (MIS), College of Business Administration, King Faisal University, Al-Ahsa, Eastern Province, Kingdom of Saudi Arabia
Ala Abdullah Applied College, King Faisal University, P.O. Box 400, Al-Ahsa 31982, Saudi Arabia

DOI:

https://doi.org/10.6000/1929-6029.2026.15.23

Keywords:

SIR model, Infectious disease modelling, Linear algebra, Eigenvalue analysis, Epidemic dynamics, Healthcare planning, Stability analysis, Basic reproduction number

Abstract

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.

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