Dynamic analysis of SIR epidemic model based on feedback control

Hui Zhu; Yunping Liu; Juan Dong; Lihong Wu; Hui Zhu; Yunping Liu; Juan Dong; Lihong Wu

doi:10.3934/era.2025337

Electronic Research Archive

2025, Volume 33, Issue 12: 7619-7636. doi: 10.3934/era.2025337

Previous Article Next Article

Theory article

Dynamic analysis of SIR epidemic model based on feedback control

1.
School of Engineering Science, Shandong Xiehe University, Jinan 250107, China
2.
School of Automation, Nanjing University of Information Science and Technology, Nanjing 210044, China
3.
School of Electronics and Information Engineering, Nanjing University of Information Science and Technology, Nanjing 210044, China
4.
Department of Mechanical and Electrical Engineering, Heilongjiang Institute of Construction Technology, 150025, China
5.
School of Mechanical Engineering, Nantong Institute of Technology, Nantong 226002, China

Received: 14 October 2025 Revised: 17 November 2025 Accepted: 25 November 2025 Published: 19 December 2025

In this paper, we study the SIR model and give a strategy to control the epidemic. Based on discrete-time observation, the linear control term is added and the upper bound of time delay is given in order to make the epidemic disappear. Both the deterministic and stochastic cases are considered. The novelty of this paper lies in the fact that it only controls for the infected population. Numerical examples verify the theory results.
- feedback control,
- SIR,
- extinction,
- stochastic models,
- noise
Citation: Hui Zhu, Yunping Liu, Juan Dong, Lihong Wu. Dynamic analysis of SIR epidemic model based on feedback control[J]. Electronic Research Archive, 2025, 33(12): 7619-7636. doi: 10.3934/era.2025337

Related Papers:

Abstract

In this paper, we study the SIR model and give a strategy to control the epidemic. Based on discrete-time observation, the linear control term is added and the upper bound of time delay is given in order to make the epidemic disappear. Both the deterministic and stochastic cases are considered. The novelty of this paper lies in the fact that it only controls for the infected population. Numerical examples verify the theory results.

References

[1]	E. Shagdar, B. Batgerel, Extending nonstandard finite difference scheme for SIR epidemic model, in International Conference on Optimization, Simulation and Control, Cham: Springer International Publishing, (2023), 187–200.
[2]	X. Wei, Global analysis of a network-based SIR epidemic model with a saturated treatment function, Int. J. Biomath., 17 (2024), 2350112. https://doi.org/10.1142/S1793524523501127 doi: 10.1142/S1793524523501127
[3]	P. Kaklamanos, A. Pugliese, M. Sensi, S. Sottile, A geometric analysis of the SIRS model with secondary infections, SIAM J. Appl. Math., 84 (2024), 661–686. https://doi.org/10.1137/23M1565632 doi: 10.1137/23M1565632
[4]	H. Bao, M. Wang, S. Yao, Qualitative properties of solutions to nonlocal infectious SIR epidemic models, Math. Methods Appl. Sci., 48 (2025), 13260–13275. https://doi.org/10.1002/mma.11101 doi: 10.1002/mma.11101
[5]	S. Dey, T. Kar, Dynamics of a nonlocal diffusive SIR epidemic model with nonlocal infection, Commun. Nonlinear Sci. Numer. Simul., 152 (2026), 109161. https://doi.org/10.1016/j.cnsns.2025.109161 doi: 10.1016/j.cnsns.2025.109161
[6]	H. Wu, B. Song, L. Zhang, H. Li, Z. Teng, Turing-Hopf bifurcation and inhomogeneous pattern for a reaction-diffusion SIR epidemic model with chemotaxis and delay, Math. Comput. Simul., 240 (2026), 1000–1022. https://doi.org/10.1016/j.matcom.2025.08.010 doi: 10.1016/j.matcom.2025.08.010
[7]	L. Zhu, X. Huang, Y. Liu, Z. Zhang, Spatiotemporal dynamics analysis and optimal control method for an SI reaction-diffusion propagation model, J. Math. Anal. Appl., 493 (2021), 124539. https://doi.org/10.1016/j.jmaa.2020.124539 doi: 10.1016/j.jmaa.2020.124539
[8]	L. Zhu, W. Zheng, S. Shen, Dynamical analysis of a SI epidemic-like propagation model with non-smooth control, Chaos Solitons Fractals, 169 (2023), 113273. https://doi.org/10.1016/j.chaos.2023.113273 doi: 10.1016/j.chaos.2023.113273
[9]	M. Lakhal, T. El Guendouz, R. Taki, M. El Fatini, The threshold of a stochastic SIRS epidemic model with a general incidence, Bull. Malays. Math. Sci. Soc., 47 (2024), 100. https://doi.org/10.1007/s40840-024-01696-2 doi: 10.1007/s40840-024-01696-2
[10]	D. Kuang, Q. Yin, J. Li, The threshold of a stochastic SIRS epidemic model with general incidence rate under regime-switching, J. Franklin Inst., 360 (2023), 13624–13647. https://doi.org/10.1016/j.jfranklin.2022.04.027 doi: 10.1016/j.jfranklin.2022.04.027
[11]	A. Singh, M. Mehra, S. Gulyani, A modified variable-order fractional SIR model to predict the spread of COVID-19 in India, Math. Methods Appl. Sci., 46 (2023), 8208–8222. https://doi.org/10.1002/mma.7655 doi: 10.1002/mma.7655
[12]	B. Li, L. Zhu, Turing instability analysis of a reaction-diffusion system for rumor propagation in continuous space and complex networks, Inf. Process. Manage., 61 (2024), 103621. https://doi.org/10.1016/j.ipm.2023.103621 doi: 10.1016/j.ipm.2023.103621
[13]	H. Sha, L. Zhu, Dynamic analysis of pattern and optimal control research of rumor propagation model on different networks, Inf. Process. Manage., 62 (2025), 104016. https://doi.org/10.1016/j.ipm.2024.104016 doi: 10.1016/j.ipm.2024.104016
[14]	T. Easlick, W. Sun, A unified stochastic SIR model driven by L$\acute{e}$vy noise with time-dependency, Adv. Contin. Discrete Models, (2024), 22. https://doi.org/10.1186/s13662-024-03818-3
[15]	Q. Liu, Q. Chen, Dynamics of a stochastic SIR epidemic model with saturated incidence, Appl. Math. Comput., 282 (2016), 155–166. https://doi.org/10.1016/j.amc.2016.02.022 doi: 10.1016/j.amc.2016.02.022
[16]	C. Ferraz, J. Lima, Critical behavior of the stochastic SIR model on random bond-diluted lattices, J. Stat. Phys., 191 (2024), 77. https://doi.org/10.1007/s10955-024-03295-8 doi: 10.1007/s10955-024-03295-8
[17]	T. Tuong, D. Nguyen, N. Nguyen, Stochastic multi-group epidemic SVIR models: degenerate case, Commun. Nonlinear Sci. Numer. Simul., 128 (2024), 107588. https://doi.org/10.1016/j.cnsns.2023.107588 doi: 10.1016/j.cnsns.2023.107588
[18]	P. D$\ddot{o}$nges, T. G$\ddot{o}$tz, N. Kruchinina, T. Kr$\ddot{u}$ger, K. Niedzielewski, V. Priesemann, et al., SIR model for households, SIAM J. Appl. Math., 84 (2024), 1460–1481. https://doi.org/10.1137/23M1556861 doi: 10.1137/23M1556861
[19]	Z. Han, Y. Wang, Z. Jin, Final and peak epidemic sizes of immuno-epidemiological SIR models, Discrete Contin. Dyn. Syst. Ser. B, 29 (2024), 4432–4462. https://doi.org/10.3934/dcdsb.2024049 doi: 10.3934/dcdsb.2024049
[20]	J. Wang, X. Yan, X. Chang, Dynamics analysis and optimal control of SIR-w infectious disease model under the influence of multidimensional information, J. Math. Anal. Appl., 553 (2026), 129855. https://doi.org/10.1016/j.jmaa.2025.129855 doi: 10.1016/j.jmaa.2025.129855
[21]	J. Broekaert, D. La Torre, F. Hafiz, Competing control scenarios in probabilistic SIR epidemics on social-contact networks, Ann. Oper. Res., 336 (2024), 2037–2060. https://doi.org/10.1007/s10479-022-05031-5 doi: 10.1007/s10479-022-05031-5
[22]	L. Zhu, Y. Ding, S. Shen, Green behavior propagation analysis based on statistical theory and intelligent algorithm in data-driven environment, Math. Biosci., 379 (2025), 109340. https://doi.org/10.1016/j.mbs.2024.109340 doi: 10.1016/j.mbs.2024.109340
[23]	L. Zhu, T. Zheng, Pattern dynamics analysis and application of West Nile virus spatiotemporal models based on higher-order network topology, Bull. Math. Biol., 87 (2025), 121. https://doi.org/10.1007/s11538-025-01501-6 doi: 10.1007/s11538-025-01501-6
[24]	A. Azouani, E. Titi, Feedback control of nonlinear dissipative systems by finite determining parameters-a reaction-diffusion paradigm, Evol. Equations Control Theory, 3 (2014), 579–594. https://doi.org/10.3934/eect.2014.3.579 doi: 10.3934/eect.2014.3.579
[25]	A. Azouani, E. Olson, E. Titi, Continuous data assimilation using general interpolant observables, J. Nonlinear Sci., 24 (2014), 277–304. https://doi.org/10.1007/s00332-013-9189-y doi: 10.1007/s00332-013-9189-y
[26]	G. Lv, Y. Shan, Continuous data assimilation and feedback control of fractional reaction-diffusion equations, Evol. Equations Control Theory, 13 (2024), 1151–1161. https://doi.org/10.3934/eect.2024020 doi: 10.3934/eect.2024020
[27]	G. Lv, W. Liu, Y. Wang, G. Zou, Data assimilation algorithm for evolution equations based on discrete-time observation, J. Math. Phys., 66 (2025), 042711. https://doi.org/10.1063/5.0238918 doi: 10.1063/5.0238918
[28]	X. Mao, Almost sure exponential stabilization by discrete-time stochastic feedback control, IEEE Trans. Autom. Control, 61 (2016), 1619–1624. https://doi.org/10.1109/TAC.2015.2471696 doi: 10.1109/TAC.2015.2471696
[29]	C. Fei, W. Fei, X. Mao, D. Xia, L. Yan, Stabilization of highly nonlinear hybrid systems by feedback control based on discrete-time state observations, IEEE Trans Autom. Control, 65 (2020), 2899–2912. https://doi.org/10.1109/TAC.2019.2933604 doi: 10.1109/TAC.2019.2933604
[30]	H. Zhu, Y. Liu, J. Dong, L. Wu, Analysis of rumor model based on discrete observation, sumbitted to Mathematical Methods in the Applied Sciences.
[31]	F. Jia, G. Lv, Dynamic analysis of a stochastic rumor propagation model, Physica A, 490 (2018), 613–623. https://doi.org/10.1016/j.physa.2017.08.125 doi: 10.1016/j.physa.2017.08.125
[32]	X. Mao, Stochastic Differential Equations and Applications, Horwood Publishing, Chichester, 1997.

Reader Comments

Your name:*

Email:*
© 2025 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)