Stochastic SIR model with Poisson noises
DOI:
https://doi.org/10.17721/1812-5409.2025/1.2Keywords:
stochastic SIR model, environmental noise, centered and non-centered Poisson noises, global positive solution, disease-free equilibrium, endemic equilibrium, asymptotic behaviorAbstract
Due to the negative impact of infectious diseases on population growth, it is important to understand the dynamic behavior of such diseases. Mathematical deterministic SIR models are widely used to study the spread of infectious diseases. In real life, there is a lot of randomness and stochasticity such as environmental noise, so using stochastic models is more suitable. Suppose we want to take into account abrupt environmental perturbations, such as epidemics, fires, earthquakes, etc. in the considered models. In that case, we must introduce Poisson noises into the population models for describing such discontinuous systems. The existence and uniqueness of the global positive solution are proved for the system of stochastic differential equations describing a non-autonomous SIR model disturbed by white noise, centered and non-centered Poisson noises. In the deterministic case there is a threshold of the system for an epidemic to occur, so called the basic reproduction number. Depending on the value of the reproduction number there is the disease-free equilibrium, or there is an endemic equilibrium, which implies the disease always remains. In the case of the autonomous stochastic SIR model, we study the asymptotic behavior of the solution to the corresponding system of stochastic differential equations around these points of equilibrium.
Pages of the article in the issue: 8 - 16
Language of the article: English
References
Anderson, R.M., & May, R.M. (1979). Population biology of infectious diseases, part I. Nature, v.280, 361–367
Jiang, D., Yua, J., Ji, Ch., & Shi, N. (2011). Asymptotic behavior of global positive solution to a stochastic SIR model. Mathematical and Computer Modelling, 54, 221–232. https://doi.org/10.1016/j.mcm.2011.02.004
Zhang, X., & Wang, Ke. (2013). Stochastic SIR model with jumps. Applied Mathematics Letters, 26, 867–874. https://doi.org/10.1016/j.aml.2013.03.013
Borysenko, O.D., & Borysenko, O.V. (2022). A stochastic predator-prey model that depends on the population density of the predator. Bulletin of Taras Shevchenko National University of Kyiv, Series: Physics & Mathematics, 4, 11–17. https://doi.org/10.17721/1812-5409.2022/4.1
Gikhman, I.I. & Skorokhod, A.V. (1982). Stochastic Differential Equations and its Applications. Kyiv, Naukova Dumka [in Russian].
Downloads
Published
Issue
Section
License
Copyright (c) 2025 Oleksandr Borysenko, Olga Borysenko
This work is licensed under a Creative Commons Attribution 4.0 International License.
Authors who publish with this journal agree to the following terms:
- Authors retain copyright and grant the journal right of first publication with the work simultaneously licensed under a Creative Commons Attribution License that allows others to share the work with an acknowledgement of the work's authorship and initial publication in this journal.
- Authors are able to enter into separate, additional contractual arrangements for the non-exclusive distribution of the journal's published version of the work (e.g., post it to an institutional repository or publish it in a book), with an acknowledgement of its initial publication in this journal.
- Authors are permitted and encouraged to post their work online (e.g., in institutional repositories or on their website) prior to and during the submission process, as it can lead to productive exchanges, as well as earlier and greater citation of published work (See The Effect of Open Access).
