Mathematical modeling of the epidemic propagation with limited time spent in compartments taking and vaccination

  • S. Serovajsky Al-Farabi Kazakh National University
  • O. Turar Al-Farabi Kazakh National University
  • T. Imankulov Al-Farabi Kazakh National University


The paper proposes discrete and continuous mathematical models of epidemic development. A division of the population into nine compartments is suggested: susceptible, exposed, vaccinated, contact vaccinated, undetected patients, isolated patients, hospitalized patients, recovered and deceased. At the same time, the time spent in exposed and infected compartments is considered limited. According to the assumptions made in the models, a susceptible person can encounter the patient and go into the exposed compartment, and be vaccinated, and then also encounter the infection and go into the contact vaccinated compartment. Exposed people may become ill to any degree of severity or not, returning to the susceptible group. A contact vaccinated either does not become ill or becomes undetected or isolated patient. Every patient can recover. An undiagnosed patient may develop symptoms of the disease, because of which he moves into the isolated compartment. An isolated patient may be hospitalized, and a hospitalized patient may die. In the discrete model, discrete quantitative data for each day of the epidemic are considered, in the continuous one, these indicators are considered continuous functions. The article provides a qualitative and quantitative analysis of the proposed models. The influence of all parameters on the process under study is investigated.


[1] Ross, R. The Prevention of Malaria. – London: John Murray, 1911.
[2] Kermack, W.O. and McKendrick, A.G. A Contribution to the Mathematical Theory of Epidemics. – Proc. Roy. Soc. Lond. A, 1927, 115, 700–721.
[3] Keeling, M.J. and Rohani, P. Modeling Infectious Diseases in Humans and Animals Illustrated Edition. – Princeton: Princeton University Press, 2007.
[4] Sameni R. Mathematical Modeling of Epidemic Diseases; A Case Study of the COVID-19 Coronavirus. – arXiv:2003.11371. 2020.
[5] Krivorotko O.I., Kabanikhin S.I., Ziatkov N. Iu. et al. Shishlenin M.A., Matematicheskoe modelirovanie i prognozirovanie COVID-19 v Moskve i Novosibirskoi oblasti. (Mathematical modeling and forecasting COVID-19 in Moscow and Novosibirsk regions). – 2020
[6] Almeida, R., Cruz, A., Martins, N., and Monteiro N. An epidemiological MSEIR Model Described by the Caputo Fractional Derivative. – Int. J. of Dynamics and Control, 2019, 7, 776–784.
[7] Mwalili, S., Kimathi, M., Ojiambo, V. et al. SEIR model for COVID-19 dynamics incorporating the environment and social distancing. – BMC Res Notes 13, 352,2020.
[8] Unlu, E., Leger, H. Motornyi, O. et al Epidemic Analysis of COVID-19 Outbreak and Counter-Measures in France. – 2020. medRxiv. 2020.04.27.20079962. DOI: 10.1101/2020.04.27.20079962.
[9] Krivorotko O.I., Kabanikhin S.I. Matematicheskie modeli rasprostranenia COVID-19. (Mathematical models of COVID-19 development) – Mathematical town in Academcity, Novosibirsk, 2021. – arXiv:2112.05315v1 [q-bio.PE] 10 Dec 2021.
[10] Brauer, F., Feng, Z. and Castillo-Chavez, C. Discrete epidemic models // Math. Biosci. Eng. – 2010, 7. – P. 1–15.
[11] Turar, O., Serovajsky, S., Azimov, A. and Mustafin M Mathematical modeling of the epidemic propagation with a limited time spent in compartments / Proceedings of the 13th International ISAAC Congress, Birkhauser, Springer Int. Publ., Ghent, 2022 (to appear).
[12] Serovajsky, S. Turar, O. Mathematical Model of the Epidemic Propagation with Limited Time Spent in Exposed and Infected Compartments // Journal of Mathematics, Mechanics and Computer Science. – 2021. – No. 4 (112). – P. 162–169.
[13] Serovajsky, S. Mathematical modelling. – Chapman and Hall/CRC, London, 2021.
[14] Vynnycky, E. and White, R.G., eds. An Introduction to Infectious Disease Modelling. – Oxford: Oxford University Press, 2010.
[15] D’Onofrio, A. Stability properties of pulse vaccination strategy in SEIR epidemic model // Math. Biosci. – 2002, vol. 179, P. 57–72. [CrossRef]
[16] Gao, S., Teng, Z., Nieto, J. and Torres, A. Analysis of an SIR Epidemic Model with Pulse Vaccination and Distributed Time Delay. – Journal of Biomedicine and Biotechnology. 2007, 64870. doi:10.1155/2007/64870. PMC 2217597. PMID 18322563.
[17] De La Sen, M., Agarwal, R.P., Ibeas, A. and Alonso-Quesada, S. On a Generalized Time-Varying SEIR Epidemic Model with Mixed Point and Distributed Time-Varying Delays and Combined Regular and Impulsive Vaccination // Controls. Adv. Differ. Equ. 2010, 2010, 281612.
[18] Etxeberria-Etxaniz, M., Alonso-Quesada S. and De la Sen, M. On an SEIR Epidemic Model with Vaccination of Newborns and Periodic Impulsive Vaccination with Eventual On-Line Adapted Vaccination Strategies to the Varying Levels of the Susceptible Subpopulation // Appl. Sci. 2020, 10, 8296; doi:10.3390/app10228296. – 24 p.
[19] Schlickeiser, R. and Kroger, M. Analytical Modeling of the Temporal Evolution of Epidemics Outbreaks Accounting for Vaccinations // Physics. – 2021, 3: 386. doi:10.3390/physics3020028. S2CID 233589998.
[20] Li-Ming Cai, Zhaoqing Li, and Xinyu Song. Global Analysis of an Epidemic Model with Vaccination // J. Appl. Math. Comput. – 2018. – 57(1). – P. 605–628.
[21] Ghostine, R., Gharamti M., Hassrouny, S. and Hoteit, I. An Extended SEIR Model with Vaccination for Forecasting the COVID-19 Pandemic in Saudi Arabia Using an Ensemble Kalman Filter // Mathematics 2021, 9, 636. – 16 p.
[22] Parolinia, N., Luca Dede’a L., Ardenghia G., and Quarteroni A. Modelling the COVID-19 Epidemic and the Vaccination Campaign in Italy by the SUIHTER Model / arXiv:2112.11722v1 [q-bio.PE] 22 Dec 2021.
How to Cite
SEROVAJSKY, S.; TURAR, O.; IMANKULOV, T.. Mathematical modeling of the epidemic propagation with limited time spent in compartments taking and vaccination. Journal of Mathematics, Mechanics and Computer Science, [S.l.], v. 116, n. 4, dec. 2022. ISSN 2617-4871. Available at: <>. Date accessed: 28 jan. 2023. doi:
Keywords mathematical model, epidemic, vaccination