INVESTIGATION OF DETERMINISTIC AND STOCHASTIC SIR EPIDEMIOLOGICAL MODEL WITH NONLINEAR INCIDENCE RATE

Year 2025, Volume: 26 Issue: 3, 317 - 331, 25.09.2025

Ali Serdar Nazlıpınar , Kübra Erol

https://doi.org/10.18038/estubtda.1716669

Abstract

In this study, an expanded SIR-type model is provided that takes behavioral and environmental factors into account when analyzing the dynamics of transmission. Equilibrium points and their local stability are explored in a deterministic framework, and the fundamental reproduction number is also calculated. The model is then reconstructed using a discrete-time Markov chain (DTMC) technique to represent the random character of illness propagation in real-world settings. The evolution of the epidemic can be analyzed probabilistically using transition probabilities thanks to this stochastic framework. Numerical simulations are used to verify the outcomes of the deterministic and stochastic versions, and a comparison of their predictive tendencies is made. The results have demonstrated the need to include stochasticity in epidemiological models, particularly when taking variability and uncertainty in transmission dynamics into consideration. This dual viewpoint gives useful insights for public health policies as well as a fuller knowledge of how diseases spread.

Keywords

SIR Model , Stochastic Modeling , Markov chains

References

• [1] Van den Driessche P, Watmough J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Mathematical Biosciences, 2002;180(1-2):29-48. doi:10.1016/S0025-5564(02)00108-6.
• [2] Kermack WO, McKendrick AG. A contribution to the mathematical theory of epidemics. Proceedings of the Royal Society of London Series A, 1927;115(772):700-721.
• [3] Diekmann O, Heesterbeek JAP, Metz JAJ. On the definition and the computation of the basic reproduction ratio R₀ in models for infectious diseases in heterogeneous populations. Journal of Mathematical Biology. 1990;28(4):365-382. doi:10.1007/BF00178324.
• [4] Hethcote HW. The Mathematics of Infectious Diseases. SIAM Review, 2000;42(4):599-653. doi:10.1137/S0036144500371907.
• [5] Brauer F, Castillo-Chavez C, Feng Z. Mathematical Models in Epidemiology. New York: Springer; 2019. doi:10.1007/978-1-4939-9828-9.
• [6] Anderson RM, May RM. Infectious Diseases of Humans: Dynamics and Control. Oxford: Oxford University Press; 1991. ISBN:019854040X.
• [7] Chowell G, Hyman JM, Hernán MA. A discrete-time epidemic model with stochastic force of infection and varying population size: application to the 2009 H1N1 pandemic in the United States. Mathematical Biosciences, 2007;211(2):131–145. doi:10.1016/j.mbs.2007.01.006
• [8] Näsell I. Stochastic models of some endemic infections. Mathematical Biosciences, 2002;179(1):1–19.
• [9] Britton T. Stochastic epidemic models: a survey. Mathematical Biosciences. 2010;225(1):24–35.
• [10] Malik K, Althobaiti S. Impact of the infected population and nonlinear incidence rate on the dynamics of the SIR model. Advances in Continuous and Discrete Models, 2025;2025(1):e.g. Article 1. doi:10.1186/s13662-025-03897-w
• [11] Mouaouine A, Boukhouima A, Hattaf K, Yousfi N. A fractional order SIR epidemic model with nonlinear incidence rate. Advances in Difference Equations. 2018;2018:160. doi:10.1186/s13662-018-1613-z
• [12] Murray JD. Mathematical Biology I: An Introduction. New York: Springer Verlag; 2002.
• [13] Allen LJS. A primer on stochastic epidemic models: formulation, numerical simulation, and analysis. Infectious Disease Modelling, 2017;2(2):128–142.
• [14] Martcheva M. An Introduction to Mathematical Epidemiology. New York: Springer; 2015. doi:10.1007/978-1-4899-7612-3
• [15] Diekmann O, Heesterbeek JAP. Mathematical Epidemiology of Infectious Diseases. Chichester, UK: John Wiley & Sons; 2000. ISBN:0-471-95752-3
• [16] Kiss IZ, Miller JC, Simon PL. Mathematics of Epidemics on Networks. Cham, Switzerland: Springer; 2017.
• [17] Karlin S, Taylor HM. A First Course in Stochastic Processes. 2nd ed. New York: Academic Press; 1975.
• [18] Durrett R. Probability: Theory and Examples. 4th ed. Philadelphia: Duxbury Press; 2010.
• [19] DeJesus EX, Kaufman C. Routh–Hurwitz criterion in the examination of eigenvalues of a system of nonlinear ordinary differential equations. Physical Review A, 1987;35(12):5288–5290. doi:10.1103/PhysRevA.35.5288