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Bifurcation analysis of a nonlinear pulse SIR model with media coverage

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Abstract

In this study, we propose a nonlinear pulse SIR model with media coverage to describe the effects of vaccination and isolation measures determined by the numbers of susceptible individuals. The dynamic behaviour of the model without impulse is discussed and the basic reproduction number defined. The existence and stability of disease-free periodic solutions(DFPS) are investigated when \(R_0<1\). Even if \(R_0>1\) the DFPS is still stable when \(S_H<1/R_0\), which indicates that a state-dependent pulse strategy is still effective in preventing the outbreak of infectious diseases by choosing suitable \(S_H\). Further, by defining the Poincaré map and using the bifurcation theorem, transcritical and pitchfork bifurcations near the DFPS with respect to some key parameters were investigated. We found that complex dynamic behaviour, with important biological significance, and the rich biological significance, can be exhibited with the introduction of impulse control into the model.

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References

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

    MathSciNet  MATH  Google Scholar 

  2. Xiao, Y., Tang, S.: Dynamics of infection with nonlinear incidence in a simple vaccination model. Nonlinear Anal. Real World Appl. 11(5), 4154–4163 (2010)

    MathSciNet  MATH  Google Scholar 

  3. Yuan, Z., Wang, L.: Global stability of epidemiological models with group mixing and nonlinear incidence rates. Nonlinear Anal. Real World Appl. 11(2), 995–1004 (2010)

    MathSciNet  MATH  Google Scholar 

  4. Guo, H., Li, M.Y., Shuai, Z.: Global stability of the endemic equilibrium of multigroup SIR epidemic models. Can. Appl. Math. Quart. 14(3), 259–284 (2006)

    MathSciNet  MATH  Google Scholar 

  5. Kermack, W.O., McKendrick, A.G.: A contribution to the mathematical theory of epidemics. Proc. R. Soc. London 115(772), 700–721 (1927)

    MATH  Google Scholar 

  6. Li, J., Yang, Y., Xiao, Y., Liu, S.: A class of Lyapunov functions and the global stability of some epidemic models with nonlinear incidence. J. Appl. Anal. Comput. 6(1), 38–46 (2016)

    MathSciNet  MATH  Google Scholar 

  7. Sun, R.: Global stability of the endemic equilibrium of multigroup SIR models with nonlinear incidence. Comput. Math. Appl. 60(8), 2286–2291 (2010)

    MathSciNet  MATH  Google Scholar 

  8. Wang, A., Xiao, Y., Zhu, H.: Dynamics of a filippov epidemic model with limited hospital beds. Math. Biosci. Eng. 15(3), 739 (2018)

    MathSciNet  MATH  Google Scholar 

  9. Xia, F., Yang, X., Cheke, R.A., Xiao, Y.: Quantifying competitive advantages of mutant strains in a population involving importation and mass vaccination rollout. Infect. Dis. Modell. 6, 988–996 (2021)

    Google Scholar 

  10. Li, T., Xiao, Y.: Complex dynamics of an epidemic model with saturated media coverage and recovery. Nonlinear Dyn. 107(3), 2995–3023 (2022)

    Google Scholar 

  11. Li, Q., Xiao, Y.: Analysis of a mathematical model with nonlinear susceptibles-guided interventions. Math. Biosci. Eng. 16(5), 5551–5583 (2019)

    MathSciNet  MATH  Google Scholar 

  12. Zhang, J., Wang, L., Wang, J.: SIR model-based prediction of infected population of coronavirus in hubei province. arXiv preprint arXiv:2003.06419 (2020)

  13. Zhang, Y., Song, P.: Dynamics of the piecewise smooth epidemic model with nonlinear incidence. Chaos Solitons Fractals 146, 110903 (2021)

    MathSciNet  Google Scholar 

  14. Li, K., Zhang, H., Zhu, G., Small, M., Fu, X.: Suboptimal control and targeted constant control for semi-random epidemic networks. IEEE Trans. Syst. Man Cyb. Syst. 51(4), 2602–2610 (2019)

    Google Scholar 

  15. Lv, W., He, H., Li, K., Jiang, N.: Control strategies of an SIVS network model with two vaccinations. J. Franklin Inst. 359(4), 1724–1746 (2022)

    MathSciNet  MATH  Google Scholar 

  16. Lu, X., Wang, S., Liu, S., Li, J.: An SEI infection model incorporating media impact. Math. Biosci. Eng. 14(5), 1317 (2017)

    MathSciNet  MATH  Google Scholar 

  17. Wang, A., Xiao, Y.: A filippov system describing media effects on the spread of infectious diseases. Nonlinear Anal. Hybrid Syst. 11, 84–97 (2014)

  18. Li, Y., Cui, J.: The effect of constant and pulse vaccination on SIS epidemic models incorporating media coverage. Commun. Nonlinear Sci. Numer. Simul. 14(5), 2353–2365 (2009)

    MathSciNet  MATH  Google Scholar 

  19. Cui, J., Sun, Y., Zhu, H.: The impact of media on the control of infectious diseases. J. Dyn. Differ. Equ. 20(1), 31–53 (2008)

    MathSciNet  MATH  Google Scholar 

  20. Sun, C., Yang, W., Arino, J., Khan, K.: Effect of media-induced social distancing on disease transmission in a two patch setting. Math. Biosci. 230(2), 87–95 (2011)

    MathSciNet  MATH  Google Scholar 

  21. Cui, J.A., Tao, X., Zhu, H.: An SIS infection model incorporating media coverage. Rocky Mount. J. Math. 38(5), 1323–1334 (2008)

    MathSciNet  MATH  Google Scholar 

  22. Tchuenche, J.M., Dube, N., Bhunu, C.P., Smith, R.J., Bauch, C.T.: The impact of media coverage on the transmission dynamics of human influenza. BMC Public Health 11(1), 1–14 (2011)

    Google Scholar 

  23. Tang, S., Xiao, Y., Yang, Y., Zhou, Y., Wu, J., Ma, Z.: Community-based measures for mitigating the 2009 H1N1 pandemic in China. PLoS ONE 5(6), e10911 (2010)

    Google Scholar 

  24. Cai, L.M., Li, X.Z.: Analysis of a SEIV epidemic model with a nonlinear incidence rate. Appl. Math. Modell. 33(7), 2919–2926 (2009)

    MathSciNet  MATH  Google Scholar 

  25. Zhou, W., Xiao, Y., Heffernan, J.M.: Optimal media reporting intensity on mitigating spread of an emerging infectious disease. PLOS ONE 14(3), e0213898 (2019)

    Google Scholar 

  26. Keeling, M., Woolhouse, M., May, R., Davies, G., Grenfell, B.T.: Modelling vaccination strategies against foot-and-mouth disease. Nature 421(6919), 136–142 (2003)

    Google Scholar 

  27. Scherer, A., McLean, A.: Mathematical models of vaccination. Br Med Bull 62(1), 187–199 (2002)

    Google Scholar 

  28. Smith, T., Killeen, G.F., Maire, N., Ross, A., Molineaux, L., Tediosi, F., Hutton, G., Utzinger, J., Dietz, K., Tanner, M.: Mathematical modeling of the impact of malaria vaccines on the clinical epidemiology and natural history of plasmodium falciparum malaria: overview. Am. J. Tropic. Med. Hyg. 75(2), 1–10 (2006)

    Google Scholar 

  29. Grassly, N.C., Fraser, C.: Mathematical models of infectious disease transmission. Nat. Rev. Microbiol. 6(6), 477–487 (2008)

    Google Scholar 

  30. Kribs-Zaleta, C.M., Velasco-Hernández, J.X.: A simple vaccination model with multiple endemic states. Math. Biosci. 164(2), 183–201 (2000)

    MATH  Google Scholar 

  31. Fraser, C., Donnelly, C.A., Cauchemez, S., Hanage, W.P., Van Kerkhove, M.D., Hollingsworth, T.D., Griffin, J., Baggaley, R.F., Jenkins, H.E., Lyons, E.J., et al.: Pandemic potential of a strain of influenza A (H1N1): early findings. Science 324(5934), 1557–1561 (2009)

    Google Scholar 

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

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

    Google Scholar 

  34. Yang, Y., Xiao, Y.: Threshold dynamics for compartmental epidemic models with impulses. Nonlinear Anal. Real World Appl. 13(1), 224–234 (2012)

    MathSciNet  MATH  Google Scholar 

  35. Yang, Y., Xiao, Y., Wu, J.: Pulse HIV vaccination: feasibility for virus eradication and optimal vaccination schedule. Bull. Math. Biol. 75(5), 725–751 (2013)

    MathSciNet  MATH  Google Scholar 

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

    MATH  Google Scholar 

  37. Meng, X., Chen, L.: The dynamics of a new SIR epidemic model concerning pulse vaccination strategy. Appl. Math. Comput. 197(2), 582–597 (2008)

    MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  39. Zhang, Q., Tang, B., Tang, S.: Vaccination threshold size and backward bifurcation of SIR model with state-dependent pulse control. J. Theor. Biol. 455, 75–85 (2018)

    MathSciNet  MATH  Google Scholar 

  40. Li, Q., Xiao, Y.: Dynamical behavior and bifurcation analysis of the SIR model with continuous treatment and state-dependent impulsive control. Int. J. Bifurc. Chaos 29(10), 1950131 (2019)

    MathSciNet  MATH  Google Scholar 

  41. Corless, R.M., Gonnet, G.H., Hare, D., Jeffrey, D.J., Knuth, D.E.: On the Lambert function. Adv. Comput. Math. 5(1), 329–359 (1996)

    MathSciNet  MATH  Google Scholar 

  42. Simeonov, P.S., Bainov, D.D.: Orbital stability of periodic solutions of autonomous systems with impulse effect. Int. J. Syst. Sci. 19(12), 2561–2585 (1988)

    MathSciNet  MATH  Google Scholar 

  43. Bainov, D., Simeonov, P.: Impulsive Differential Equations: Periodic Solutions and Applications. CRC Press (1993)

    MATH  Google Scholar 

  44. Grandmont, J.M.: Nonlinear difference equations, bifurcations and chaos: an introduction. Res. Econ. 62(10), 122–177 (2008)

    Google Scholar 

  45. Hale, J.K.: Ordinary Differential Equations. Wiley Press, New York (1972)

    Google Scholar 

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Acknowledgements

This work was supported by the National Natural Science Foundation of China under Grants (11961024,12271068 (Y. Tan), 11801047 (J. Yang)), and by Joint Training Base Construction Project for Graduate Students in Chongqing (JDLHPYJD2021016), and by the Program of Chongqing Municipal Education Commission (KJQN201900707 (Z. Liu)), and by the Natural Science Foundation of Chongqing under Grant (cstc2019jcyj-msxmX0755 (Z. Liu)), and by Group Building Scientific Innovation Project for universities in Chongqing (CXQT21021).

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Authors and Affiliations

  1. College of Mathematics and Statistics, Chongqing Jiaotong University, Chongqing, 400074, China

    Jin Yang, Likun Guan, Zhuo Chen, Yuanshun Tan & Zijian Liu

  2. Natural Resources Institute, University of Greenwich, Medway, Central Avenue, Chatham Maritime, Chatham-Kent, ME4 4TB, UK

    Robert A. Cheke

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  1. Jin Yang
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Correspondence to Jin Yang.

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Yang, J., Guan, L., Chen, Z. et al. Bifurcation analysis of a nonlinear pulse SIR model with media coverage. Nonlinear Dyn 111, 19543–19562 (2023). https://doi.org/10.1007/s11071-023-08869-x

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