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Global stability of an age-structured cholera model

1. Department of Applied Mathematics, Nanjing University of Science and Technology, Nanjing 210094
2. Department of Mathematics, Xinyang Normal University, Xinyang 464000

In this paper, an age-structured epidemicmodel is formulated to describe the transmission dynamics ofcholera. The PDE model incorporates direct and indirect transmissionpathways, infection-age-dependent infectivity and variable periodsof infectiousness. Under some suitable assumptions, the PDE modelcan be reduced to the multi-stage models investigated in theliterature. By using the method of Lyapunov function, we establishedthe dynamical properties of the PDE model, and the results show thatthe global dynamics of the model is completely determined by thebasic reproduction number $\mathcal R_0$: if $\mathcal R_0 < 1$the cholera dies out, and if $\mathcal R_0>1$ the disease will persist at the endemicequilibrium. Then the global results obtained for multi-stage modelsare extended to the general continuous age model.
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Keywords Age of infection; global stability; Lyapunov function.; cholera

Citation: Jianxin Yang, Zhipeng Qiu, Xue-Zhi Li. Global stability of an age-structured cholera model. Mathematical Biosciences and Engineering, 2014, 11(3): 641-665. doi: 10.3934/mbe.2014.11.641

References

  • 1. Lancet, 377 (2011), 1248-1255.
  • 2. J. Math. Biol., 28 (1998), 365-382.
  • 3. Wiley, New York, 2000.
  • 4. J. Diff. Equs., 218 (2005), 292-324.
  • 5. Math. Biosci., 76 (1985), 69-86.
  • 6. Amer. Natur., 111 (1977), 135-143.
  • 7. Mathematical Surveys and Monographs 25, American Mathematical Society, Providence, RI, 1988.
  • 8. SIAM J. Math. Anal., 20 (1989), 388-395.
  • 9. Cambridge University Press, Cambridge, 1988.
  • 10. SIAM J. Appl. Math., 72 (2012), 25-38.
  • 11. Applied Mathematics Monographs 7, comitato nazionale per le scienze matematiche, Consiglio Nazionale delle Ricerche (C. N. R), Giardini, Pisa, 1995.
  • 12. Math. Popul. Studi., 17 (1988), 47-77.
  • 13. Proc. Roy. Soc. Lond. B, 268 (2001), 985-993.
  • 14. Theor. Popul. Biol., 60 (2001), 59-71.
  • 15. Electron. J. Diff. Equs., 65 (2001), 1-35.
  • 16. Communications on Pure and Applied Analysis, 3 (2004), 695-727.
  • 17. SIAM J. Math. Anal., 37 (2005), 251-275.
  • 18. Diff. Integr. Equs., 19 (2006), 573-600.
  • 19. Appl. Anal., 89 (2010), 1109-1140.
  • 20. SIAM J. Appl. Math., 73 (2013), 1058-1095.
  • 21. J. Math. Biol., 39 (1999), 471-492.
  • 22. Math. Biosci., 234 (2011), 118-126.
  • 23. Bull. Math. Biol., 74 (2010), 2423-2445.
  • 24. Diff. Integr. Equs., 3 (1990), 1035-1066.
  • 25. SIAM J. Appl. Math., 53 (1993), 1447-1479.
  • 26. 2010. Available from: http://www.math.wm.edu/~jptian/preprints/pr-7-ode-cholera.pdf.
  • 27. Math. Biosci., 232 (2011), 31-41.
  • 28. Bull. Math. Biol., 72 (2010), 1506-1533.
  • 29. Math. Biosci., 180 (2002), 29-48.
  • 30. Marcel Dekker, New York, 1985.

 

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