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Dynamical analysis of an age-structured multi-group SIVS epidemic model

1 Complex Systems Research Center, Shanxi University, Taiyuan, Shanxi 030006, P.R. China
2 Shanxi Key Laboratory of Mathematical Techniques and Big Data Analysis on Disease Control and Prevention, Shanxi University, Taiyuan, Shanxi 030006, P.R. China

Host heterogeneities such as space, gender, and age etc are intrinsic characters for investigating diseases mechanisms and transmission routes. First, we incorporate inter-group, intra-group and age structure to propose a multi-group SIVS epidemic model. Then we obtain the basic reproduction number of the system which is the spectral radius of the next generation operator by the renewal equation. Based on some assumptions for parameters, we obtain the existence and uniqueness of endemic equilibrium. By means of integral semigroup theory and Lyapunov methods, we show that the threshold dynamics of the system is completely determined by the basic
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Keywords multi-group epidemic model; the next generation operator; global attractivity

Citation: Junyuan Yang, Rui Xu, Xiaofeng Luo. Dynamical analysis of an age-structured multi-group SIVS epidemic model. Mathematical Biosciences and Engineering, 2019, 16(2): 636-666. doi: 10.3934/mbe.2019031

References

  • 1. R. M. Anderson and R. M. May, Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, 1992.
  • 2. G.P. Sahu and J. Dhar, Dynamics of an SEQIHRS epidemic model with media coverage, quarantine and isolation in a community with pre-existing immunity, J. Math. Anal. Appl., 421 (2015), 1651–1672.
  • 3. X. Peng, X. Xu, X. Fu and T. Zhou, Vaccination intervention on epidemic dynamics in networks, Phys. Rev. E, 87 (2013), 022813.
  • 4. W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. Ser. A, 115 (1927), 700–721.
  • 5. W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics: II, Proc. R. Soc. Lond. Ser. B, 138 (1932), 55–83.
  • 6. W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics: III, Proc. R. Soc. Lond. Ser. B, 141 (1933), 94–112.
  • 7. J. Li and Y. Yang, SIR-SVS epidemic models with continuous and impulsive vaccination strategies, J. Theor. Biol., 280 (2011), 108–116.
  • 8. C. M. Kribs-Zaleta and J. X. Velasco-Hern´andez, A simple vaccination model with multiple endemic states, Math. Biosci., 164 (2000), 183-201.
  • 9. L. Nie, J. Shen and C. Yang, Dynamic behavior analysis of SIVS epidemic models with statedependent pulse vaccination, Nonlinear Anal.-Hybrid Syst., 27 (2018), 258–270.
  • 10. D. Zhao, T. Zhang and S. Yuan, The threshold of a stochastic SIVS epidemic model with nonlinear saturated incidence, Physica A, 443 (2016), 372–379.
  • 11. Y. Xiao and S. Tang, Dynamics of infection with nonlinear incidence in a simple vaccination model, Nonlinear Anal: RWA, 11 (2010), 4154–4163.
  • 12. M. Martcheva, An Introduction to Mathematical Epidemiology, Springer, New York, 2015.
  • 13. J. S. Lavineabc, O. N. BjØrnstadab, B. F. de Blasiode and J. Storsaeter, Short-lived immunity against pertussis, age-specific routes of transmission, and the utility of a teenage booster vaccine, Vaccine, 30 (2012), 544–551.
  • 14. A. Lajmanovich and J. A. York, A deterministic model for gonorrhea in a nonhomogeneous population, Math. Biosci. 28 (1976), 221–236.
  • 15. T. Kuniya, J. Wang and H. Inaba, Multi-group SIR epidemic model with age structure, Discrete Contin. Dyn. Syst.-Ser. B, 21 (2016), 3515–3550.
  • 16. M. Eichner and K. Dietz, Transmission potential of smallpox: estimates based on detailed data from an outbreak, Am. J. Epidemiol., 158 (2003), 110–117.
  • 17. H. Nishiura and M. Eichner, Infectiousness of smallpox relative to disease age: estimates based on transmission network and incubation period, Epidemiol. Infect., 135 (2007), 1145–1150.
  • 18. F. Cooper, Non-Markovian network epidemics, MA Thesis, 2013. Available from: www.dtc.ox. ac.uk/people/13/cooperf/files/MA469ThesisFergusCooper.pdf
  • 19. J. Wang and H. Shu, Global analysis on a class of multi-group SEIR model with latency and relapse, Math. Biosci. Eng., 13 (2016), 209–225.
  • 20. T. Kuniya, Global stability of a multi-group SVIR epidemic model, Nonlinear Anal: RWA, 14 (2013), 1135–1143.
  • 21. H. Guo, M. Y. Li and Z. Shuai, A graph-theoretic approach to the method of global Lyapunov functions, Proc. Amer. Math. Soc., 136 (2008), 2793–2802.
  • 22. P. Magal, C. C. McCluskey and G. F. Webb, Lyapunov functional and global asymptotic stability for an infection-age model, Appl. Anal., 89 (2010), 1109–1140.
  • 23. P. Magal, Compact attractors for time-periodic age-structured population models, Electron. J. Differ. Eq., 2001 (2001), 1–35.
  • 24. O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365–382.
  • 25. H. Inaba, The Malthusian parameter and R0 for heterogeneous populations in periodic environments, Math. Biosci. Eng., 9 (2012), 313–346.
  • 26. M. A. Krasnoselskii, Positive Solutions of Operator Equations, 1st edition, Noordhoff, Groningen, 1964.
  • 27. M. G. Krein and M. A. Rutman, Linear operators leaving invariant a cone in a Banach space, Am. Math. Soc. Transl., 1950, 128.
  • 28. J. K. Hale, Asymptotic Behavior of Dissipative Systems, in: Math. Surv. Monogr., vol. 25, Am. Math. Soc., Providence, RI, 1988.
  • 29. H. L. Smith and H. R. Thieme, Dynamical Systems and Population Persistence, 1st edition, Amer. Math. Soc., Providence, 2011.
  • 30. J. A. Walker, Dynamical Systems and Evolution Equations, 1st edition, Plenum Press, New York and London, 1980.
  • 31. A. N. Kolmogorov and S. V. Fomin, Elements of the Theory of Functions and Functional Analysis (Dover Books on Mathematics), Dover Publications, INC, Mineola, New York, 1999.
  • 32. G. F. Webb, Theory of Nonlinear Age-Dependent Population Dynamics, Marcel Dekker, New York, 1985.
  • 33. Centers for Disease Control and Prevention, Genital HPV Infection - Fact Sheet, 2017. Avaialbe from: https://www.cdc.gov/std/hpv/stdfact-hpv.htm.
  • 34. H. R. Thieme, Mathematics in Population Biology. Priceton University Press, Princeton, 2003.
  • 35. W. M. Hirsch, H. Hanisch, J. P. Gabriel, Differential equation models of some parasitic infections: methods for the study of asymptotic behavior, Comm. Pure Appl. Math., 38 (1985), 733–753.
  • 36. J. Y. Yang and F. Xu, Global stability of two SIS epidemic mean-field models on complex networks: Lyapunov functional approach. J. Franklin Inst., 355 (2018), 6763–6779.
  • 37. Z. H. Liu and R. Yuan, Zero-Hopf bifurcation for an infection-age structured epidemic model with a nonlinear incidence rate, Sci. China Math., 60 (2017), 1371–1398.

 

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