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Dynamic behaviors of a cholera model with nonlinear incidences, multiple transmission pathways, and imperfect vaccine

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Abstract

In this article, we propose a cholera model to study the effects of multiple transmission pathways, imperfect vaccine, nonlinear incidences, and differential infectivity of vibrios. The expression of the basic reproductive number \(\mathcal {R}_0\) is derived. There is only the disease-free equilibrium \(E_0\) if \(\mathcal {R}_0\le 1\), while, besides \(E_0\), there is also a unique endemic equilibrium \(E^*\) if \(\mathcal {R}_0>1\). When \(\mathcal {R}_0<1\), \(E_0\) is globally asymptotically stable by using the technique of linearization and the fluctuation lemma. When \(\mathcal {R}_0>1\), \(E^*\) is globally asymptotically stable by the Lyapunov direct method. These theoretical results are supported with numerical simulations for the case with Beddington-DeAngelis incidences. We further perform the sensitivity analyses of \(\mathcal {R}_0\) and the infection level at \(E^*\) to determine the significant parameters affecting disease outbreak and severity, respectively. Influences of the vaccination rate \(\phi \) and the waning rate of vaccine \(\eta \) on the dynamical behaviors of the model are also discussed.

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References

  1. WHO: Report of World Health Organization, 2022, accessed 20 September 2022. https://www.who.int/news-room/fact-sheets/detail/cholera

  2. Nelson, E.J., Harris, J.B., Morris, J.G., Calderwood, S.B., Camilli, A.: Cholera transmission: the host, pathogen and bacteriophage dynamic. Nat. Rev. Microbiol. 7(10), 693–702 (2009). https://doi.org/10.1038/nrmicro2204

    Article  Google Scholar 

  3. Capasso, V., Paveri-Fontana, S.L.: A mathematical model for the 1973 cholera epidemic in the european mediterranean region. Rev. Epidemol. Sante Publique 27(2), 121–132 (1979)

    Google Scholar 

  4. Codeo, C.T.: Endemic and epidemic dynamics of cholera: the role of the aquatic reservoir. BMC Infect. Dis. 1(1), 1–14 (2001). https://doi.org/10.1186/1471-2334-1-1

    Article  Google Scholar 

  5. Goh, K.T., Teo, S.H., Lam, S., Ling, M.K.: Person-to-person transmission of cholera in a psychiatric hospital. J. Infect. 20(3), 193–200 (1990). https://doi.org/10.1016/0163-4453(90)90994-J

    Article  Google Scholar 

  6. Mukandavire, Z., Liao, S., Wang, J., Gaff, H., Smith, D.L., Morris, J.G., Jr.: Estimating the reproductive numbers for the 2008–2009 cholera outbreaks in zimbabwe. Proc. Natl. Acad. Sci. 108(21), 8767–8772 (2011). https://doi.org/10.1073/pnas.1019712108

    Article  Google Scholar 

  7. Kumar Gupta, R., Kumar Rai, R., Kumar Tiwari, P., Kumar Misra, A., Martcheva, M.: A mathematical model for the impact of disinfectants on the control of bacterial diseases. J. Biol. Dyn. 17(1), 2206859 (2023). https://doi.org/10.1080/17513758.2023.2206859

    Article  MathSciNet  Google Scholar 

  8. Shuai, Z., van den Driessche, P.: Global dynamics of cholera models with differential infectivity. Math. Biosci. 234(2), 118–126 (2011). https://doi.org/10.1016/j.mbs.2011.09.003

    Article  MathSciNet  Google Scholar 

  9. Wang, J., Liao, S.: A generalized cholera model and epidemic-endemic analysis. J. Biol. Dyn. 6(2), 568–589 (2012). https://doi.org/10.1080/17513758.2012.658089

    Article  MathSciNet  Google Scholar 

  10. Wang, Y., Cao, J.: Global stability of general cholera models with nonlinear incidence and removal rates. J. Franklin Inst. 352(6), 2464–2485 (2015). https://doi.org/10.1016/j.jfranklin.2015.03.030

    Article  MathSciNet  Google Scholar 

  11. Song, C., Xu, R., Bai, N., Tian, X., Lin, J.: Global dynamics and optimal control of a cholera transmission model with vaccination strategy and multiple pathways. Math. Biosci. Eng. 17(4), 4210–4224 (2020). https://doi.org/10.3934/mbe.2020233

    Article  MathSciNet  Google Scholar 

  12. Bai, N., Song, C., Xu, R.: Mathematical analysis and application of a cholera transmission model with waning vaccine-induced immunity. Nonlinear Anal. Real World Appl. 58, 103232 (2021). https://doi.org/10.1016/j.nonrwa.2020.103232

    Article  MathSciNet  Google Scholar 

  13. Onuorah, M.O., Atiku, F.A., Juuko, H.: Mathematical model for prevention and control of cholera transmission in a variable population. Res. Math. (2022). https://doi.org/10.1080/27658449.2021.2018779

    Article  MathSciNet  Google Scholar 

  14. Tian, J.P., Jin, W.: Global stability for cholera epidemic models. Math. Biosci. 232(1), 31–41 (2011). https://doi.org/10.1016/j.mbs.2011.04.001

    Article  MathSciNet  Google Scholar 

  15. Wang, X., Chen, Y., Martcheva, M., Rong, L.: Asymptotic analysis of a vector-borne disease model with the age of infection. J. Biol. Dyn. 14(1), 332–367 (2020). https://doi.org/10.1080/17513758.2020.1745912

    Article  MathSciNet  Google Scholar 

  16. Li, B., Zhang, F., Wang, X.: A delayed diffusive hbv model with nonlinear incidence and ctl immune response. Math. Methods Appl. Sci. 45(17), 11930–11961 (2022). https://doi.org/10.1002/mma.8547

    Article  MathSciNet  Google Scholar 

  17. Wang, X., Zhang, Z., Jia, C.: A SEIARV model with asymptomatic infection and saturation rates. J. Xinyang Normal Univ. (Natl. Sci. Edn.) 36(01), 16–21 (2023). https://doi.org/10.3969/j.issn.1003-0972.2023.01.003

  18. Korobeinikov, A.: Global properties of infectious disease models with nonlinear incidence. Bull. Math. Biol. 69(6), 1871–1886 (2007). https://doi.org/10.1007/s11538-007-9196-y

    Article  MathSciNet  Google Scholar 

  19. Ge, Q., Wang, X., Rong, L.: A delayed reaction-diffusion viral infection model with nonlinear incidences and cell-to-cell transmission. Int. J. Biomath. 14(8), 305–342 (2021). https://doi.org/10.1142/S179352452150100X

    Article  MathSciNet  Google Scholar 

  20. Hartley, D.M., Morris, J.G., Jr., Smith, D.L.: Hyperinfectivity: A critical element in the ability of v. cholerae to cause epidemics? PLoS Med. 3(1), 63–69 (2006). https://doi.org/10.1371/journal.pmed.0030007

    Article  Google Scholar 

  21. Shuai, Z., Tien, J.H., van den Driessche, P.: Cholera models with hyperinfectivity and temporary immunity. Bull. Math. Biol. 74(10), 2423–2445 (2012). https://doi.org/10.1007/s11538-012-9759-4

    Article  MathSciNet  Google Scholar 

  22. WHO: Global Task Force on Cholera Control, Cholera Country Profile, 2011, accessed 20 September 2022. http://www.who.int/cholera/countries/Haiti Country Profile

  23. Sharma, S., Singh, F.: Bifurcation and stability analysis of a cholera model with vaccination and saturated treatment. Chaos, Solitons & Fractals 146(6), 110912 (2021). https://doi.org/10.1016/j.chaos.2021.110912

    Article  MathSciNet  Google Scholar 

  24. Tian, X., Xu, R., Lin, J.: Mathematical analysis of a cholera infection model with vaccination strategy. Appl. Math. Comput. 361, 517–535 (2019). https://doi.org/10.1016/j.amc.2019.05.055

    Article  MathSciNet  Google Scholar 

  25. Cui, J., Wu, Z., Zhou, X.: Mathematical analysis of a cholera model with vaccination. J. Appl. Math. 13(2), 1–16 (2014). https://doi.org/10.1155/2014/324767

    Article  MathSciNet  Google Scholar 

  26. Khajanchi, S., Mondal, J., Tiwari, P.K.: Optimal treatment strategies using dendritic cell vaccination for a tumor model with parameter identifiability. J. Biol. Syst. 31(02), 487–516 (2023). https://doi.org/10.1142/S0218339023500171

    Article  MathSciNet  Google Scholar 

  27. van den Driessche, P., Watmough, J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180(1), 29–48 (2002). https://doi.org/10.1016/S0025-5564(02)00108-6

    Article  MathSciNet  Google Scholar 

  28. Tineo, A.: Asymptotic behaviour of positive solutions of the nonautonomous lotka-volterra competition equations. Differ. Integr. Equ. 6(2), 449–457 (1993)

    MathSciNet  Google Scholar 

  29. Yang, J., Bi, S.: Stability and hopf bifurcation of a delayed virus infection model with latently infected cells and beddington-deangelis incidence. Int. J. Biomath. 13(5), 2050045 (2020). https://doi.org/10.1142/S179352452050045X

    Article  MathSciNet  Google Scholar 

  30. Zhao, S.: Analysis on stochastic dynamics of two-consumers-one-resource competing systems with beddington-deangelis functional response. Int. J. Biomath. 14(2), 2050058 (2021). https://doi.org/10.1142/S1793524520500588

    Article  MathSciNet  Google Scholar 

  31. Modnak, C.: A model of cholera transmission with hyperinfectivity and its optimal vaccination control. Int. J. Biomath. 10(6), 1750084 (2017). https://doi.org/10.1142/S179352451750084X

    Article  MathSciNet  Google Scholar 

  32. Sanchez, M.A., Blower, S.M.: Uncertainty and sensitivity analysis of the basic reproductive rate: tuberculosis as an example. Am. J. Epidemiol. 145(12), 1127–1137 (1997). https://doi.org/10.1093/oxfordjournals.aje.a009076

    Article  Google Scholar 

  33. Wang, Y., Zhou, Y., Brauer, F., Heffernan, J.M.: Viral dynamics model with ctl immune response incorporating antiretroviral therapy. J. Math. Biol. 67(4), 901–934 (2013). https://doi.org/10.1007/s00285-012-0580-3

    Article  MathSciNet  Google Scholar 

  34. Wang, S., Xu, F., Rong, L.: Bistability analysis of an hiv model with immune response. J. Biol. Syst. 25(4), 677–695 (2017). https://doi.org/10.1142/S021833901740006X

    Article  MathSciNet  Google Scholar 

  35. Wang, X., Chen, Y., Song, X.: Global dynamics of a cholera model with age structures and multiple transmission modes. Int. J. Biomath. 12(5), 1950051 (2019). https://doi.org/10.1142/S1793524519500517

    Article  MathSciNet  Google Scholar 

  36. Berge, T., Bowong, S., Lubuma, J.M.-S.: Global stability of a two-patch cholera model with fast and slow transmissions. Math. Comput. Simul. 133, 142–164 (2017). https://doi.org/10.1016/j.matcom.2015.10.013

  37. Posny, D., Modnak, C., Wang, J.: A multigroup model for cholera dynamics and control. Int. J. Biomath. 9(1), 1–27 (2016). https://doi.org/10.1142/S1793524516500017

    Article  MathSciNet  Google Scholar 

  38. Robertson, S.L., Eisenberg, M.C., Tien, J.H.: Heterogeneity in multiple transmission pathways: modelling the spread of cholera and other waterborne disease in networks with a common water source. J. Biol. Dyn. 7(1), 254–275 (2013). https://doi.org/10.1080/17513758.2013.853844

    Article  MathSciNet  Google Scholar 

  39. Wang, F.-B., Wang, X.: A general multipatch cholera model in periodic environments. Discrete Contin. Dyn. Syst. 27(3), 1647–1670 (2022). https://doi.org/10.3934/dcdsb.2021105

    Article  MathSciNet  Google Scholar 

  40. Eisenberg, M.C., Shuai, Z., Tien, J.H., van den Driessche, P.: A cholera model in a patchy environment with water and human movement. Math. Biosci. 246(1), 105–112 (2013). https://doi.org/10.1016/j.mbs.2013.08.003

    Article  MathSciNet  Google Scholar 

  41. Brauer, F., Shuai, Z., van den Driessche, P.: Dynamics of an age-of-infection cholera model. Math. Biosci. Eng. 10(6), 1335–1349 (2013). https://doi.org/10.3934/mbe.2016.13.227

    Article  MathSciNet  Google Scholar 

  42. Lin, J., Xu, R., Tian, X.: Global dynamics of an age-structured cholera model with both human-to-human and environment-to-human transmissions and saturation incidence. Appl. Math. Model. 63, 688–708 (2018). https://doi.org/10.1016/j.apm.2018.07.013

    Article  MathSciNet  Google Scholar 

  43. Cai, L.-M., Modnak, C., Wang, J.: An age-structured model for cholera control with vaccination. Appl. Math. Comput. 299, 127–140 (2017). https://doi.org/10.1016/j.amc.2016.11.013

    Article  MathSciNet  Google Scholar 

  44. Kokomo, E., Emvudu, Y.: Mathematical analysis and numerical simulation of an age-structured model of cholera with vaccination and demographic movements. Nonlinear Anal. Real World Appl. 45, 142–156 (2019). https://doi.org/10.1016/j.cnsns.2018.06.023

    Article  MathSciNet  Google Scholar 

  45. Liu, W., Wang, J., Zhang, R.: Dynamics of an infection age-space structured cholera model with neumann boundary condition. Eur. J. Appl. Math. 33(3), 393–422 (2022). https://doi.org/10.1017/S095679252100005X

    Article  MathSciNet  Google Scholar 

  46. Yang, J., Modnak, C.: Dynamical analysis and optimal control simulation for an age-structured cholera transmission model. J. Franklin Inst. 356(15), 8438–8467 (2019). https://doi.org/10.1016/j.jfranklin.2019.08.016

    Article  MathSciNet  Google Scholar 

  47. Yang, J., Qiu, Z., Li, X.: Global stability of an age-structured cholera model. Math. Biosci. Eng. 11(3), 641–665 (2014). https://doi.org/10.3934/mbe.2014.11.641

    Article  MathSciNet  Google Scholar 

  48. Cai, L., Fan, G., Yang, C., Wang, J.: Modeling and analyzing cholera transmission dynamics with vaccination age. J. Franklin Inst. 357(12), 8008–8034 (2020). https://doi.org/10.1016/j.jfranklin.2020.05.030

    Article  MathSciNet  Google Scholar 

  49. Wang, X., Posny, D., Wang, J.: A reaction-convection-diffusion model for cholera spatial dynamics. Discrete Contin. Dyn. Syst. 21(8), 2785–2809 (2016). https://doi.org/10.3934/dcdsb.2016073

    Article  MathSciNet  Google Scholar 

  50. Capone, F., De Cataldis, V., De Luca, R.: Influence of diffusion on the stability of equilibria in a reaction-diffusion system modeling cholera dynamic. J. Math. Biol. 71(5), 1107–1131 (2015). https://doi.org/10.1007/s00285-014-0849-9

    Article  MathSciNet  Google Scholar 

  51. Misra, A., Gupta, A.: A reaction-diffusion model for the control of cholera epidemic. J. Biol. Syst. 24(4), 431–456 (2016). https://doi.org/10.1142/S0218339016500224

    Article  MathSciNet  Google Scholar 

  52. Shu, H., Ma, Z., Wang, X.-S.: Threshold dynamics of a nonlocal and delayed cholera model in a spatially heterogeneous environment. J. Math. Biol. 83(4), 1–33 (2021). https://doi.org/10.1007/s00285-021-01672-5

    Article  MathSciNet  Google Scholar 

  53. Wang, X., Zhao, X.-Q., Wang, J.: A cholera epidemic model in a spatiotemporally heterogeneous environment. J. Math. Anal. Appl. 468(2), 893–912 (2018). https://doi.org/10.1016/j.jmaa.2018.08.039

    Article  MathSciNet  Google Scholar 

  54. Tiwari, P.K., Rai, R.K., Khajanchi, S., Gupta, R.K., Misra, A.K.: Dynamics of coronavirus pandemic: effects of community awareness and global information campaigns. Eur. Phys. J. Plus 136(994), 11930–11961 (2021). https://doi.org/10.1140/epjp/s13360-021-01997-6

    Article  Google Scholar 

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Acknowledgements

This work is supported partially by the National Natural Science Foundation of China (12171413), the Natural Science Foundation of Henan Province (222300420016), the Program for Science and Technology Innovation Teams in Henan (21IRTSTHN014), and the Natural Science Foundation of Hunan Province (2019JJ40027).

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Authors and Affiliations

  1. School of Mathematics and Statistics, Xinyang Normal University, Xinyang, 464000, People’s Republic of China

    Hongyan Zhao & Xia Wang

  2. School of Mathematics, Hunan University, Changsha, 410082, People’s Republic of China

    Shaofen Zou

  3. Department of Mathematics, Wilfrid Laurier University, Waterloo, ON, N2L 3C5, Canada

    Yuming Chen

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  1. Hongyan Zhao
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HZ: Writing-original draft, Software. SZ: Writing-Reviewing and Editing. XW: Methodology, Supervision, Writing-Reviewing and Editing. YC: Supervision, Writing-Reviewing and Editing.

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Correspondence to Xia Wang.

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Zhao, H., Zou, S., Wang, X. et al. Dynamic behaviors of a cholera model with nonlinear incidences, multiple transmission pathways, and imperfect vaccine. J. Appl. Math. Comput. 70, 917–946 (2024). https://doi.org/10.1007/s12190-024-01994-9

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