Stability of infection-free almost periodic solution in impulsive SEIR-model
DOI:
https://doi.org/10.17721/1812-5409.2026/1.17Keywords:
SEIR-model of differential equations, impulsive perturbation, infection-free solution, periodic solution, almost periodic solutionAbstract
Mathematical models of the infection spread dynamics have remained a relevant area of research for the past few decades. Moreover, these models have also proven to be suitable for modeling DDoS attacks in computer networks. The qualitative behaviour of solutions in such models is usually arranged as follows: there is some threshold value at which the global asymptotic stability of the solution with a zero component, which is responsible for the number of infected individuals, changes to the stability of the non-zero equilibrium position. From a meaningful point of view, this means that the self-exhaustion and disappearance of the infection changes to a constant circulation of the infection in the population.
However, the presence of an impulsive perturbation can make the infection-free solution asymptotically stable regardless of the threshold value, and such impulsive perturbations allow a natural interpretation within the framework of the epidemiological model as the process of vaccination of a certain part of individuals.
Such problems have already been considered for the SIR- and SEIR-models in the case of a simple structure of impulsive points, due to which the impulsive problem was actually reduced to a problem without impulsive perturbation. However, a more complex structure of the set and magnitudes of impulsive effects requires the application of the theory of impulsive differential equations, which is done in this work.
Namely, the article considers the SEIR-system of differential equations with impulsive perturbations that have an almost periodic character. The existence and properties of an infection-free impulsive almost periodic solution are investigated, and its asymptotic stability is proven.
Pages of the article in the issue: 130 - 135
Language of the article: English
References
Hethcote, H. W. (1989). Three Basic Epidemiological Models. In S. A. Levin, T. G. Hallam, L. J. Gross (Eds.), Applied Mathematical Ecology. Biomathematics, vol 18. Springer, Berlin, Heidelberg. https://doi.org/10.1007/978-3-642-61317-3_5
Liu, W. H., Hethcote, H. W., & Levin, S. A. (1987). Dynamical Behavior of Epidemiological Models with Nonlinear Incidence Rates. Journal of Mathematical Biology, 25, 359–380. https://doi.org/10.1007/bf00277162
Balarezo, J. F., Wang, S., Chavez, K. G., Al-Hourani, A. & Kandeepan, S. (2022). A Survey on DOS/DDOS Attacks Mathematical Modelling for Traditional, SDN and Virtual Networks. Engineering Science and Technology, an International Journal, 31, Article ID: 101065. https://doi.org/10.1016/j.jestch.2021.09.011
Li, M. Y., & Muldowney J. S. (1995). Global stability for the SEIR model in epidemiology. Math. Biosci, 125, 155–168. https://doi.org/10.1016/0025-5564(95)92756-5
Li, M. Y., & Muldowney, J. S. (1996). A Geometric Approach to Global-Stability Problems. SIAM Journal on Mathematical Analysis, 27, 1070–1083. http://dx.doi.org/10.1137/S0036141094266449
Samoilenko, A. M., & Perestyuk, N. A. (1995). Impulsive Differential Equations. World Scientific, Singapore. https://doi.org/10.1142/2892
Shulgin, B., Stone L., & Agur, Z. (1998). Pulse Vaccination Strategy in the SIR Epidemic Model. Bulletin of Mathematical Biology, 60(6), 1123–1148. https://doi.org/10.1016/S0895-7177(02)00177-2
D'Onofrio, A. (2002). Stability properties of pulse vaccination strategy in SEIR epidemic model. Mathematical Biosciences, 179(1), 57–72. https://doi.org/10.1016/S0025-5564(02)00095-0
Feketa, P., Klinshov, V., & Lücken, L. (2021). A survey on the modelling of hybrid behaviors: How to account for impulsive jumps properly. Communications in Nonlinear Science and Numerical Simulation, 103, 105955. https://doi.org/10.1016/j.cnsns.2021.105955
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Copyright (c) 2026 Oleksiy Kapustyan, Anna Sukretna, Dmytro Kapustian, Myrzagali Ospanov
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