Skip to main content

Advertisement

SpringerLink
Log in

Global dynamic analysis of a nonlinear state-dependent feedback control SIR model with saturation incidence

  • Regular Article
  • Published:
The European Physical Journal Plus Aims and scope Submit manuscript

Abstract

This paper investigates an SIR model with nonlinear state-dependent feedback control. The saturation incidence rate is introduced into the classical SIR model, and the specific expression of the equilibrium point of the model without pulse control is derived, and the stability of the equilibrium point is analyzed. When the number of susceptible individuals reaches a threshold \(S_{h}\), comprehensive prevention and control strategies such as treatment, isolation, and vaccination will be implemented. The existence and global stability of the disease-free periodic solution (DFPS) of the model with state-dependent feedback control are discussed, and the properties of the Poincaré map are analyzed. Based on the bifurcation theory related to a one-parameter family of maps associated with the Poincaré map, we define the control reproduction number \(R_{c}\) based on different parameters and observe the effect of these parameters on bifurcation. Numerical simulations show that transcritical and pitchfork bifurcations, as well as backward bifurcations, may occur under certain conditions. Additionally, a sensitivity analysis of \(R_{c}\) is conducted, and the optimal control strategy for the SIR model without pulse control is discussed. These results can provide some suggestions for virus control and vaccine production, among other issues.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3
Fig. 4
Fig. 5
Fig. 6
Fig. 7
Fig. 8
Fig. 9

Similar content being viewed by others

Data availability

Data sharing not applicable to this article.

References

  1. V. Capasso, G. Serio, A generalization of the Kermack–McKendrick deterministic epidemic model. Math. Biosci. 42(1), 43–61 (1978)

    Article  MathSciNet  MATH  Google Scholar 

  2. B. Shulgin, L. Stone, Z. Agur, Pulse vaccination strategy in the SIR epidemic model. Bull. Math. Biol. 60(6), 1123–1148 (1998)

    Article  MATH  Google Scholar 

  3. S.J. Gao, L.S. Chen, J.J. Nieto, A. Torres, Analysis of a delayed epidemic model with pulse vaccination and saturation incidence. Vaccine 24(35–36), 6037–6045 (2006)

    Article  Google Scholar 

  4. S.J. Gao, L.S. Chen, Z.D. Teng, Impulsive vaccination of an SEIRS model with time delay and varying total population size. Bull. Math. Biol. 69(2), 731–745 (2007)

    Article  MATH  Google Scholar 

  5. A. D’Onofrio, Stability properties of pulse vaccination strategy in SEIR epidemic model. Math. Biosci. 179(1), 57–72 (2002)

    Article  MathSciNet  MATH  Google Scholar 

  6. F.C. Kong, J.J. Nieto, Impact of discontinuous treatments on the generalized epidemic model. Topol. Methods Nonlinear Anal. 56(1), 349–378 (2020)

    MathSciNet  MATH  Google Scholar 

  7. I. Area, F.J. Fernández, J.J. Nieto, F.A.F. Tojo, Concept and solution of digital twin based on a stieltjes differential equation. Math. Methods Appl. Sci. 45(12), 7451–7465 (2022)

    Article  ADS  MathSciNet  Google Scholar 

  8. R.J. Smith, E.J. Schwartz, Predicting the potential impact of a cytotoxic Tlymphocyte HIV vaccine: how often should you vaccinate and how strong should the vaccine be? Math. Biosci. 212(2), 180–187 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  9. J.C. Panetta, A mathematical model of periodically pulsed chemotherapy: tumor recurrence and metastasis in a competitive environment. Bull. Math. Biol. 58(3), 425–447 (1996)

    Article  MATH  Google Scholar 

  10. J.C. Panetta, A mathematical model of drug resistance: heterogeneous tumors. Math. Biosci. 147(1), 41–61 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  11. S. Bunimovich-Mendrazitsky, H. Byrne, L. Stone, Mathematical model of pulsed immunotherapy for superficial bladder cancer. Bull. Math. Biol. 70(7), 2055–2076 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  12. T.Y. Cheng, S.Y. Tang, R.A. Cheke, Threshold dynamics and bifurcation of a state-dependent feedback nonlinear control SIR model. J. Comput. Nonlinear Dyn. 14(7), 071001 (2019)

    Article  Google Scholar 

  13. S.Y. Tang, R.A. Cheke, State-dependent impulsive models of integrated pest management (IPM) strategies and their dynamic consequences. J. Math. Biol. 50(3), 257–292 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  14. S.Y. Tang, Y.N. Xiao, L.S. Chen, R.A. Cheke, Integrated pest management models and their dynamical behaviour. Bull. Math. Biol. 67(1), 115–135 (2005)

    Article  MathSciNet  MATH  Google Scholar 

  15. S.Y. Tang, B. Tang, A.L. Wang, Y.N. Xiao, Holling-II predator-prey impulsive semi-dynamic model with complex Poincaré map. Nonlinear Dyn. 81(3), 1575–1596 (2015)

    Article  MATH  Google Scholar 

  16. S.Y. Tang, W.H. Pang, On the continuity of the function describing the times of meeting impulsive set and its application. Math. Biosci. Eng. 14(5–6), 1399–1406 (2017)

    Article  MathSciNet  MATH  Google Scholar 

  17. J.H. Liang, S.Y. Tang, J.J. Nieto, R.A. Cheke, Analytical methods for detecting pesticide switches with evolution of pesticide resistance. Math. Biosci. 245(2), 249–257 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. T.Y. Wang, Microbial insecticide model and homoclinic bifurcation of impulsive control system. Int. J. Biomath. 14(6), 2150043 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  19. S.Y. Tang, Y.N. Xiao, R.A. Cheke, Multiple attractors of host-parasitoid models with integrated pest management strategies: Eradication, persistence and outbreak. Theor. Popul. Biol. 73(2), 181–197 (2008)

    Article  MATH  Google Scholar 

  20. S.Y. Tang, J.H. Liang, Y.S. Tan, R.A. Cheke, Threshold conditions for integrated pest management models with pesticides that have residual effects. J. Math. Biol. 66(1–2), 1–35 (2013)

    MathSciNet  MATH  Google Scholar 

  21. Y.Q. Liu, X.Y. Li, Dynamics of a discrete predator-prey model with Holling-II functional response. Int. J. Biomath. 14(8), 2150068 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  22. J. Xu, M.Z. Huang, X.Y. Song, Dynamical analysis of a two-species competitive system with state feedback impulsive control. Int. J. Biomath. 13(05), 2050007 (2020)

    Article  MathSciNet  MATH  Google Scholar 

  23. S.Y. Tang, R.A. Cheke, Models for integrated pest control and their biological implications. Math. Biosci. 215(1), 115–125 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  24. Y. Tian, Y. Gao, K. Sun, Qualitative analysis of exponential power rate fishery model and complex dynamics guided by a discontinuous weighted fishing strategy. Commun. Nonlinear Sci. 118, 107011 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  25. Y. Tian, C. Li, J. Liu, Complex dynamics and optimal harvesting strategy of competitive harvesting models with interval-valued imprecise parameters. Chaos Soliton Fractal 167, 113084 (2023)

    Article  MathSciNet  Google Scholar 

  26. Y. Tian, Y. Gao, K. Sun, A fishery predator-prey model with anti-predator behavior and complex dynamics induced by weighted fishing strategies. Math. Biosci. Eng. 20(2), 1558–1579 (2023)

    Article  MathSciNet  Google Scholar 

  27. H. Li, Y. Tian, Dynamic behavior analysis of a feedback control predator-prey model with exponential fear effect and hassell-varley functional response. J. Frankl. Inst. 360(4), 3479–3498 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  28. Y. Tian, Y. Gao, K. Sun, Global dynamics analysis of instantaneous harvest fishery model guided by weighted escapement strategy. Chaos Soliton Fractal 164, 112597 (2022)

    Article  MathSciNet  Google Scholar 

  29. N.M. Ferguson, D.A. Cummings, S. Cauchemez, C. Fraser, S. Riley, A. Meeyai, S. Iamsirithaworn, D.S. Burke, Strategies for containing an emerging influenza pandemic in Southeast Asia. Nature 437(7056), 209–214 (2005)

    Article  ADS  Google Scholar 

  30. ...C. Fraser, C.A. Donnelly, S. Cauchemez, W.P. Hanage, M.D.V. Kerkhove, T.D. Hollingsworth, J. Griffin, R.F. Baggaley, H.E. Jenkins, E.J. Lyons, T. Jombart, W.R. Hinsley, N.C. Grassly, F. Balloux, A.C. Ghani, N.M. Ferguson, A. Rambaut, O.G. Pybus, H. Lopez-Gatell, C.M. Alpuche-Aranda, I.B. Chapela, E.P. Zavala, D.M.E. Guevara, F. Checchi, E. Garcia, S. Hugonnet, C. Roth, Pandemic potential of a strain of influenza A (H1N1): early findings. Science 324(5934), 1557–1561 (2009)

    Article  ADS  Google Scholar 

  31. S.Y. Tang, Y.N. Xiao, Y. Lin, R.A. Cheke, J.H. Wu, Campus quarantine (Fengxiao) for curbing emergent infectious diseases: lessons from mitigating A/H1N1 in Xi’an, China. J. Theor. Biol. 295, 47–58 (2012)

    Article  ADS  MathSciNet  MATH  Google Scholar 

  32. Y.N. Xiao, S.Y. Tang, J.H. Wu, Media impact switching surface during an infectious disease outbreak. Sci. Rep. 5, 7838 (2015)

    Article  ADS  Google Scholar 

  33. W.H. Fleming, R.W. Rishel, Deterministic and stochastic optimal control. Adv. Math. 1, 222 (1976)

    MathSciNet  Google Scholar 

  34. D.L. Lukes, Differential Equations: Classical to Controlled, Mathematics in Science and Engineering (Academic Press, New York, 1982), p.162

    Google Scholar 

  35. L.S. Pontryagin, V.G. Boltyanskii, R.V. Gamkrelidze, E.F. Mishchenko, The mathematical theory of optimal processes. Bell Syst. Tech. J. 27, 623–656 (1986)

    Google Scholar 

  36. D.M. Xiao, S.G. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate. Math. Biosci. 208(2), 419–429 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  37. W.M. Liu, S.A. Levin, Y. Lwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models. J. Math. Biol. 23(2), 187–204 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  38. W.M. Liu, H.W. Hethcote, S.A. Levin, Dynamical behavior of epidemiological models with nonlinear incidence rates. J. Math. Biol. 25(4), 359–380 (1987)

    Article  MathSciNet  MATH  Google Scholar 

  39. C.X. Liu, R.H. Cui, Qualitative analysis on an SIRS reaction-diffusion epidemic model with saturation infection mechanism. Nonlinear Anal. Real 62, 103364 (2021)

    Article  MathSciNet  MATH  Google Scholar 

  40. C.X. Liu, R.H. Cui, Analysis on a diffusive SIRS epidemic model with logistic source and saturated incidence rate. Discret. Contin. Dyn. B 28(5), 2960–2980 (2023)

    Article  MathSciNet  MATH  Google Scholar 

  41. E. Avila-Vales, G.E. Garcia-Almeida, A.G.C. Perez, Qualitative analysis of a diffusive SIR epidemic model with saturated incidence rate in a heterogeneous environment. J. Math. Anal. Appl. 503(1), 125295 (2021)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

This work is supported by the National Natural Science Foundation of China (no. 12171193, 12071407).

Author information

Authors and Affiliations

  1. School of Mathematics and Information Science, Zhengzhou University of Light Industry, Zhengzhou, 450002, China

    Yongfeng Li & Song Huang

  2. School of Mathematics and Statistics, Huanghuai University, Zhumadian, 463000, China

    Xinyu Song

Authors
  1. Yongfeng Li
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Song Huang
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Xinyu Song
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Yongfeng Li.

Ethics declarations

Conflict of interest

The authors declare no competing interest.

Rights and permissions

Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Li, Y., Huang, S. & Song, X. Global dynamic analysis of a nonlinear state-dependent feedback control SIR model with saturation incidence. Eur. Phys. J. Plus 138, 636 (2023). https://doi.org/10.1140/epjp/s13360-023-04277-7

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1140/epjp/s13360-023-04277-7

Search

Navigation