Analysis and event-triggered control for a stochastic epidemic model with logistic growth

Tingting Cai; Yuqian Wang; Liang Wang; Zongying Tang; Jun Zhou; Tingting Cai; Yuqian Wang; Liang Wang; Zongying Tang; Jun Zhou

doi:10.3934/mbe.2023105

Mathematical Biosciences and Engineering

2023, Volume 20, Issue 2: 2243-2260. doi: 10.3934/mbe.2023105

Previous Article Next Article

Research article Special Issues

Analysis and event-triggered control for a stochastic epidemic model with logistic growth

1.
College of Eco-Environmental Engineering, Yunnan Forestry Technological College, Kunming 650224, China
2.
Innovation Group of Orchid Conservation and Utilization, Yunnan Forestry Technological College, Kunming 650224, China
3.
Yunnan Forestry Senior Technical School, Kunming 650200, China
4.
China Rongtong Agricultural Development Group Corporation Limited, Kunming 650031, China
5.
College of Mathematics and Physics, Southwest Forestry University, Kunming 650224, China

Academic Editor: Xiaodi Li
^† The authors contributed equally to this work.

Received: 13 September 2022 Revised: 28 October 2022 Accepted: 31 October 2022 Published: 17 November 2022

In this paper, a stochastic epidemic model with logistic growth is discussed. Based on stochastic differential equation theory, stochastic control method, etc., the properties of the solution of the model nearby the epidemic equilibrium of the original deterministic system are investigated, the sufficient conditions to ensure the stability of the disease-free equilibrium of the model are established, and two event-triggered controllers to drive the disease from endemic to extinction are constructed. The related results show that the disease becomes endemic when the transmission coefficient exceeds a certain threshold. Furthermore, when the disease is endemic, we can drive the disease from endemic to extinction by choosing suitable event-triggering gains and control gains. Finally, the effectiveness of the results is illustrated by a numerical example.
- stochastic epidemic model,
- Gaussian white noise,
- event-triggered control,
- almost sure exponential stability,
- mean square stability
Citation: Tingting Cai, Yuqian Wang, Liang Wang, Zongying Tang, Jun Zhou. Analysis and event-triggered control for a stochastic epidemic model with logistic growth[J]. Mathematical Biosciences and Engineering, 2023, 20(2): 2243-2260. doi: 10.3934/mbe.2023105

Related Papers:

Abstract

In this paper, a stochastic epidemic model with logistic growth is discussed. Based on stochastic differential equation theory, stochastic control method, etc., the properties of the solution of the model nearby the epidemic equilibrium of the original deterministic system are investigated, the sufficient conditions to ensure the stability of the disease-free equilibrium of the model are established, and two event-triggered controllers to drive the disease from endemic to extinction are constructed. The related results show that the disease becomes endemic when the transmission coefficient exceeds a certain threshold. Furthermore, when the disease is endemic, we can drive the disease from endemic to extinction by choosing suitable event-triggering gains and control gains. Finally, the effectiveness of the results is illustrated by a numerical example.

References

[1]	J. S. Weitz, J. Dushoff, Modeling post-death transmission of Ebola: Challenges for inference and opportunities for control, Sci. Rep., 5 (2015), 1–7. https://doi.org/10.1038/srep08751 doi: 10.1038/srep08751
[2]	R. E. Morrison, Jr. A. Cunha, Embedded model discrepancy: A case study of Zika modeling, Chaos, 30 (2020), 051103. https://doi.org/10.1063/5.0005204 doi: 10.1063/5.0005204
[3]	K. A. Hattaf, New generalized definition of fractional derivative with non-singular kernel, Computation, 8 (2020), 49. https://doi.org/10.3390/computation8020049 doi: 10.3390/computation8020049
[4]	K. A. Hattaf, On the stability and numerical scheme of fractional differential equations with application to biology, Computation, 10 (2022), 97. https://doi.org/10.3390/computation10060097 doi: 10.3390/computation10060097
[5]	M. Lipsitch, T. Cohen, B. Cooper, J. M. Robins, S. Ma, L. James, et al., Transmission dynamics and control of severe acute respiratory syndrome, Science, 300 (2003), 1966–1970. https://doi.org/10.1126/science.1086616 doi: 10.1126/science.1086616
[6]	C. Nowzari, V. M. Preciado, G. J. Pappas, Analysis and control of epidemics: A survey of spreading processes on complex networks, IEEE Control Syst. Mag., 36 (2016), 26–46. https://doi.org/10.1109/MCS.2015.2495000 doi: 10.1109/MCS.2015.2495000
[7]	X. Li, S. Song, J. Wu, Exponential stability of nonlinear systems with delayed impulses and applications, IEEE Trans. Autom. Control, 64 (2019), 4024–4034. https://doi.org/10.1109/TAC.2019.2905271 doi: 10.1109/TAC.2019.2905271
[8]	I. Ahn, S. Heo, S. Ji, K. H. Kim, T. Kim, E. J. Lee, et al., Investigation of nonlinear epidemiological models for analyzing and controlling the MERS outbreak in Korea, J. Theor. Biol., 437 (2018), 17–28. https://doi.org/10.1016/j.jtbi.2017.10.004 doi: 10.1016/j.jtbi.2017.10.004
[9]	R. M. May, Stability and Complexity in Model Ecosystems, Princeton: Princeton University Press, 2001. https://doi.org/10.1515/9780691206912
[10]	D. Vasseur, P. Yodzis, The color of environmental noise, Ecology, 85 (2004), 1146–1152. https://doi.org/10.1890/02-3122 doi: 10.1890/02-3122
[11]	Q. Yang, X. Mao, Stochastic dynamics of SIRS epidemic models with random perturbation, Math. Biosci. Eng., 11 (2014), 1003–1025. https://doi.org/10.3934/mbe.2014.11.1003 doi: 10.3934/mbe.2014.11.1003
[12]	X. Xiao, S. Joshi, J. Cecil, Critical assessment of shape retrieval tools (SRTs), Int. J. Adv. Manuf. Technol., 116 (2021), 3431–3446. https://doi.org/10.1007/s00170-021-07681-4 doi: 10.1007/s00170-021-07681-4
[13]	Q. Liu, Q. Chen, Analysis of the deterministic and stochastic SIRS epidemic models with nonlinear incidence, Phys. A, 428 (2015), 140–153. https://doi.org/10.1016/j.physa.2015.01.075 doi: 10.1016/j.physa.2015.01.075
[14]	X. Xiao, C. Waddell, C. Hamilton, H. Xiao, Quality prediction and control in wire arc additive manufacturing via novel machine learning framework, Micromachines, 13 (2022), 137. https://doi.org/10.3390/mi13010137 doi: 10.3390/mi13010137
[15]	W. Zhao, J. Li, T. Zhang, X. Meng, T. Zhang, Persistence and ergodicity of plant disease model with markov conversion and impulsive toxicant input, Commun. Nonlinear Sci. Numer. Simul., 48 (2017), 70–84. https://doi.org/10.1016/j.cnsns.2016.12.020 doi: 10.1016/j.cnsns.2016.12.020
[16]	Q. Liu, D. Jiang, T. Hayat, A. Alsaedi, B. Ahmad, A stochastic SIRS epidemic model with logistic growth and general nonlinear incidence rate, Phys. A, 551 (2020), 124152. https://doi.org/10.1016/j.physa.2020.124152 doi: 10.1016/j.physa.2020.124152
[17]	X. Meng, X. Wang, Stochastic predator-prey system subject to Lévy jumps, Discrete Dyn. Nat. Soc., (2016), 5749892. https://doi.org/10.1155/2016/5749892 doi: 10.1155/2016/5749892
[18]	A. J. Gray, D. Greenhalgh, L. Hu, X. Mao, J. Pan, A stochastic differential equation SIS epidemic model, Soc. Ind. Appl. Math., 71 (2011), 876–902. https://doi.org/10.1137/10081856X doi: 10.1137/10081856X
[19]	Q. Liu, D. Jiang, N. Shi, T. Hayat, A. Alsaedi, Periodic solution for a stochastic nonautonomous SIR epidemic model with logistic growth, Phys. A, 462 (2016), 816–826. https://doi.org/10.1016/j.physa.2016.06.052 doi: 10.1016/j.physa.2016.06.052
[20]	S. Rajasekar, M. Pitchaimani, Q. Zhu, Progressive dynamics of a stochastic epidemic model with logistic growth and saturated treatment, Phys. A, (2019), 122649. https://doi.org/10.1016/j.physa.2019.122649 doi: 10.1016/j.physa.2019.122649
[21]	A. R. Khalili, A. Heydari, M. R. Zarrabi, Analysis and control of SEIR epidemic model via sliding mode control, Adv. Model. Optim., 18 (2016), 153–162.
[22]	M. Sharifi, H. Moradi, Nonlinear robust adaptive sliding mode control of influenza epidemic in the presence of uncertainty, J. Process Control, 56 (2017), 48–57. https://doi.org/10.1016/j.jprocont.2017.05.010 doi: 10.1016/j.jprocont.2017.05.010
[23]	G. Rohith, K. B. Devika, Dynamics and control of COVID-19 pandemic with nonlinear incidence rates, Nonlinear Dyn., 101 (2020), 2013–2026. https://doi.org/10.1007/s11071-020-05774-5 doi: 10.1007/s11071-020-05774-5
[24]	L. J. Chen, J. T. Sun, Global stability of an SI epidemic model with feedback controls, Appl. Math. Lett., 28 (2014), 53–55. https://doi.org/10.1016/j.aml.2013.09.009 doi: 10.1016/j.aml.2013.09.009
[25]	X. Xiao, B. Roh, C. Hamilton, Porosity management and control in powder bed fusion process through process-quality interactions, CIRP J. Manuf. Sci. Technol., 38 (2022), 120–128. https://doi.org/10.1016/j.cirpj.2022.04.005 doi: 10.1016/j.cirpj.2022.04.005
[26]	S. Gao, D. Zhong, Z. Yan, Analysis of novel stochastic switched SILI epidemic models with continuous and impulsive control, Phys. A, 495 (2018), 162–171. https://doi.org/10.1016/j.physa.2017.12.050 doi: 10.1016/j.physa.2017.12.050
[27]	X. Li, D. Peng, J. Cao, Lyapunov stability for impulsive systems via event-triggered impulsive control, IEEE Trans. Autom. Control, 65 (2020), 4908–4913. https://doi.org/10.1109/TAC.2020.2964558 doi: 10.1109/TAC.2020.2964558
[28]	M. Ogura, V. M. Preciado, Optimal containment of epidemics in temporal and adaptive networks, in Temporal Network Epidemiology, Springer, (2017), 241–266. https://doi.org/10.1007/978-981-10-5287-3 https://doi.org/10.1007/978-981-10-5287-3_11
[29]	C. Nowzari, V. M. Preciado, G. J. Pappas, Optimal resource allocation for control of networked epidemic models, IEEE Trans. Control Network Syst., 4 (2017), 159–169. https://doi.org/10.1109/TCNS.2015.2482221 doi: 10.1109/TCNS.2015.2482221
[30]	P. Tabuada, Event-triggered real-time scheduling of stabilizing control tasks, IEEE Trans. Autom. Control, 52 (2007), 1680–1685. https://doi.org/10.1109/TAC.2007.904277 doi: 10.1109/TAC.2007.904277
[31]	W. Heemels, K. H. Johansson, P. Tabuada, An introduction to event-triggered and self-triggered control, in Proceedings of the 51st IEEE Conference on Decision and Control, (2012), 3270–3285. https://doi.org/10.1109/CDC.2012.6425820
[32]	X. Li, X. Yang, J. Cao, Event-triggered impulsive control for nonlinear delay systems, Automatica, 117 (2020), 108981. https://doi.org/10.1016/j.automatica.2020.108981 doi: 10.1016/j.automatica.2020.108981
[33]	K. Hashimoto, Y. Onoue, M. Ogura, T. Ushio, Event-triggered control for mitigating SIS spreading processes, Ann. Rev. Control, 52 (2021), 479–494. https://doi.org/10.1016/j.arcontrol.2021.08.001 doi: 10.1016/j.arcontrol.2021.08.001
[34]	E. Allen, Modeling with Ito Stochastic Differential Equations, Springer-Verlag, Dordrecht, The Netherlands, 2007.
[35]	R. Z. Hasminskij, Stochastic Stability of Differential Equations, Sijthoof and Noordhoof, Alphen aan den Rijn, The Netherlands, 1980.
[36]	K. Hattaf, M. Mahrouf, J. Adnani, N. Yousfi, Qualitative analysis of a stochastic epidemic model with specific functional response and temporary immunity, Phys. A, 490 (2018), 591–600. https://doi.org/10.1016/j.physa.2017.08.043 doi: 10.1016/j.physa.2017.08.043
[37]	X. Mao, C. Yuan, Stochastic Differential Equations with Markovian Switching, Imperial College Press: London, 2006.

Reader Comments

Your name:*

Email:*
© 2023 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)