Skip to main content
SpringerLink
Log in

Threshold dynamics of a stochastic infectious disease model with vaccination age under saturated media coverage

  • Original Research
  • Published:
Journal of Applied Mathematics and Computing Aims and scope Submit manuscript

Abstract

Vaccination and social media play pivotal roles in affecting disease transmission. Research has shown that disease transmission can be subject to random events. Consequently, we develop a stochastic infectious disease dynamical model that incorporates saturated media coverage and vaccination age. The Itô’s formula and the Lyapunov function method are applied to study the extinction behavior of the disease and the existence of a unique ergodic stationary distribution. The findings suggest that the media effect is delayed and cannot eliminate the disease completely. To directly control disease transmission, a combination of high-intensity noise disturbance and low vaccine wane rate is required. Furthermore, the shorter the disease incubation period, the more difficult it is to control.

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

Similar content being viewed by others

References

  1. Yang, J., Tang, S.Y., Cheke, R.A.: Impacts of varying strengths of intervention measures on secondary outbreaks of covid-19 in two different regions. Nonlinear Dyn. 104, 863–882 (2021)

    Article  PubMed  PubMed Central  Google Scholar 

  2. Kermack, W.O., McKendrick, A.G.: A contribution to the mathematical theory of epidemics. Proc. R. Soc. Lond. Ser. A Containing Pap. Math. Phys. Char. 115(772), 700–721 (1927)

    ADS  Google Scholar 

  3. Sun, C.J., Hsieh, Y.H.: Global analysis of an seir model with varying population size and vaccination. Appl. Math. Model. 34(10), 2685–2697 (2010)

    Article  MathSciNet  Google Scholar 

  4. Dubey, B., Patra, A., Srivastava, P., et al.: Modeling and analysis of an seir model with different types of nonlinear treatment rates. J. Biol. Syst. 21(03), 1350023 (2013)

    Article  MathSciNet  Google Scholar 

  5. Li, L., Zhang, J., Liu, C., et al.: Analysis of transmission dynamics for zika virus on networks. Appl. Math. Comput. 347, 566–577 (2019)

    MathSciNet  Google Scholar 

  6. Basnarkov, L.: Seair epidemic spreading model of covid-19. Chaos Solitons Fractals 142, 110394 (2021)

    Article  MathSciNet  PubMed  Google Scholar 

  7. Yu, Y., Tan, Y.S., Tang, S.Y.: Stability analysis of the covid-19 model with age structure under media effect. Comput. Appl. Math. 42(5), 204 (2023)

    Article  MathSciNet  Google Scholar 

  8. Qin, W.J., Tang, S.Y.: The selection pressures induced non-smooth infectious disease model and bifurcation analysis. Chaos Solitons Fractals 69, 160–171 (2014)

    Article  MathSciNet  PubMed  PubMed Central  ADS  Google Scholar 

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

    Article  PubMed  PubMed Central  Google Scholar 

  10. May, R.M., Macdonald, N.: Stability and complexity in model ecosystems. IEEE Trans. Syst. Man Cybern. 8(10), 779–779 (1978)

    Article  Google Scholar 

  11. Songer, J.R.: Influence of relative humidity on the survival of some airborne viruses. Appl. Microbiol. 15(1), 35–42 (1967)

    Article  CAS  PubMed  PubMed Central  Google Scholar 

  12. Glencross, D.A., Ho, T.R., Camina, N., et al.: Air pollution and its effects on the immune system. Free Radical Biol. Med. 151, 56–68 (2020)

    Article  CAS  Google Scholar 

  13. Wu, J.H., Robinson, S., Tsemg, J.S., et al.: Digital and physical factors influencing an individual’s preventive behavior during the covid-19 pandemic in taiwan: A perspective based on the s-o-r model. Comput. Hum. Behav. 139, 107525 (2023)

    Article  Google Scholar 

  14. Zhang, X.B., Wang, X.D., Huo, H.F.: Extinction and stationary distribution of a stochastic sirs epidemic model with standard incidence rate and partial immunity. Physica A 531, 121548 (2019)

    Article  MathSciNet  Google Scholar 

  15. Din, A., Li, Y.J., Yusuf, A.: Delayed hepatitis b epidemic model with stochastic analysis. Chaos Solitons Fractals 146, 110839 (2021)

    Article  MathSciNet  Google Scholar 

  16. Liu, Q., Jiang, D.Q., Hayat, T., Alsaedi, A., et al.: Dynamics of a multigroup siqs epidemic model under regime switching. Stoch. Anal. Appl. 38(5), 769–796 (2020)

    Article  MathSciNet  Google Scholar 

  17. Britton, T., Lindenstrand, D.: Epidemic modelling: aspects where stochasticity matters. Math. Biosci. 222(2), 109–116 (2009)

    Article  MathSciNet  PubMed  Google Scholar 

  18. Wang, N., Qi, L.X., Bessane, M., et al.: Global hopf bifurcation of a two-delay epidemic model with media coverage and asymptomatic infection. J. Differ. Equ. 369, 1–40 (2023)

    Article  MathSciNet  ADS  Google Scholar 

  19. Moghadas, S.M., Shoukat, A., Fitzpatrick, M.C., et al.: Projecting hospital utilization during the covid-19 outbreaks in the united states. Proc. Natl. Acad. Sci. 117(16), 9122–9126 (2020)

    Article  CAS  PubMed  PubMed Central  ADS  Google Scholar 

  20. Bärnighausen, T., Bloom, D.E., Cafiero-Fonseca, E.T., et al.: Valuing vaccination. Proc. Natl. Acad. Sci. 111(34), 12313–12319 (2014)

    Article  PubMed  PubMed Central  ADS  Google Scholar 

  21. Moorlag, S., Arts, R., Van Crevel, R., et al.: Non-specific effects of bcg vaccine on viral infections. Clin. Microbiol. Infect. 25(12), 1473–1478 (2019)

    Article  CAS  PubMed  Google Scholar 

  22. He, W.Q., Guo, G.N., Li, C.X.: The impact of hepatitis b vaccination in the united states, 1999–2018. Hepatology 75(6), 1566–1578 (2022)

    Article  PubMed  Google Scholar 

  23. Zhang, Q.Y., Zhang, R.T., Wu, W., et al.: Impact of social media news on covid-19 vaccine hesitancy and vaccination behavior. Telematics Inform. 80, 101983 (2023)

    Article  Google Scholar 

  24. Hu, L., Wang, S.F., Zheng, T.T., et al.: The effects of migration and limited medical resources of the transmission of sars-cov-2 model with two patches. Bull. Math. Biol. 84(5), 1–25 (2022)

    Article  MathSciNet  ADS  Google Scholar 

  25. Ahmed, N., Elsonbaty, A., Raza, A., et al.: Numerical simulation and stability analysis of a novel reaction-diffusion covid-19 model. Nonlinear Dyn. 106(2), 1293–1310 (2021)

    Article  PubMed  PubMed Central  Google Scholar 

  26. Webb, G.F.: Theory of Nonlinear Age-dependent Population Dynamics. CRC Press (1985)

  27. Ma, S.S.: A class of stochastic sis model studies considering the impact of media coverage and vertical contagion. J. Univ. Shanghai Sci. Technol. 42(6), 533–542 (2020)

    Google Scholar 

  28. Mao, X.R.: Stochastic Differential Equations and Applications. Elsevier (2007)

  29. Bahar, A., Mao, X.R., et al.: Stochastic delay lotka-volterra model. J. Math. Anal. Appl. 292(2), 364–380 (2004)

    Article  MathSciNet  Google Scholar 

  30. Jiang, D.Q., Shi, N.Z., Li, X.Y.: Global stability and stochastic permanence of a non-autonomous logistic equation with random perturbation. J. Math. Anal. Appl. 340(1), 588–597 (2008)

    Article  MathSciNet  Google Scholar 

  31. Bellet, L.R.: Ergodic properties of markov processes 25 (2006)

  32. Meyn, S.P., Tweedie, R.L.: Stability of markovian processes iii: Foster-lyapunov criteria for continuous-time processes. Adv. Appl. Probab. 25(3), 518–548 (1993)

    Article  MathSciNet  Google Scholar 

  33. Hajri, Y., Allali, A., Amine, S.: A delayed deterministic and stochastic siricv model: Hopf bifurcation and stochastic analysis. Math. Comput. Simul. 215, 98–121 (2024)

    Article  Google Scholar 

  34. Athreya, A., Kolba, T., Mattingly, J.: Propagating lyapunov functions to prove noise-induced stabilization. Electron. J. Probab. 17, 1–38 (2012)

    Article  MathSciNet  Google Scholar 

  35. Aniţa, S., Arnăutu, V., Capasso, V., et al: An Introduction to Optimal Control Problems in Life Sciences and Economics: from Mathematical Models to Numerical Simulation with MATLAB®. Springer (2011)

  36. Higham, D.J.: An algorithmic introduction to numerical simulation of stochastic differential equations. SIAM Rev. 43(3), 525–546 (2001)

    Article  MathSciNet  ADS  Google Scholar 

  37. Al Basir, F., Ray, S., Venturino, E.: Role of media coverage and delay in controlling infectious diseases: A mathematical model. Appl. Math. Comput. 337, 372–385 (2018)

    MathSciNet  Google Scholar 

  38. Song, P.F., Xiao, Y.N.: Analysis of an epidemic system with two response delays in media impact function. Bull. Math. Biol. 81, 1582–1612 (2019)

    Article  MathSciNet  PubMed  Google Scholar 

  39. Lu, R.X., Wei, F.Y.: Persistence and extinction for an age-structured stochastic svir epidemic model with generalized nonlinear incidence rate. Physica A 513, 572–587 (2019)

    Article  MathSciNet  ADS  Google Scholar 

  40. Shangguan, D.C., Liu, Z.J., Wang, L.W., et al.: A stochastic epidemic model with infectivity in incubation period and homestead-isolation on the susceptible. J. Appl. Math. Comput. 67(1–2), 785–805 (2021)

    Article  MathSciNet  PubMed  PubMed Central  Google Scholar 

Download references

Acknowledgements

This work was supported by the National Natural Science Foundation of China (NOs. 12271068, 11961024), the Joint Training Base Construction Project for Graduate Students in Chongqing (JDLHPYJD2021016), Group Building Scientific Innovation Project for Universities in Chongqing [Grant Number 2023S0134].

Author information

Authors and Affiliations

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

    Yue Yu, Yuanshun Tan & Yu Mu

Authors
  1. Yue Yu
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yuanshun Tan
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Yu Mu
    View author publications

    You can also search for this author in PubMed Google Scholar

Contributions

YY: Methodology, Software, Writing—original draft. YT: Conceptualization, Writing—original draft. YM: Writing—review & editing.

Corresponding author

Correspondence to Yuanshun Tan.

Ethics declarations

Competing interest

Authors declare that they have no conflict of interest.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

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

Yu, Y., Tan, Y. & Mu, Y. Threshold dynamics of a stochastic infectious disease model with vaccination age under saturated media coverage. J. Appl. Math. Comput. 70, 657–688 (2024). https://doi.org/10.1007/s12190-023-01983-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s12190-023-01983-4

Keywords

Mathematics Subject Classification

Search

Navigation