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Dynamic analysis of the recurrent epidemic model

1 Department of Mathematics, Shaanxi University of Science and Technology, Xi’an, 710021, P.R. China
2 School of Science, Nanjing University of Posts and Telecommunications, Nanjing, 210023, P.R. China
3 Department of Applied Mathematics, The University of Western Ontario, London, N6A 5B7, Canada

Special Issues: Non-smooth biological dynamical systems and applications

In this work, an SIRS model with age structure is proposed for recurrent infectious disease by incorporating temporary immunity and delay. We formulate the model as an abstract non-densely defined Cauchy problem and derive the conditions for the global stability of disease free equilibrium, the local stability of endemic equilibrium, and the existence of Hopf bifurcation. Both non-periodic and periodic behaviors are possible when the disease persists in population, where time delay plays an important role. Numerical examples are provided to illustrate our theoretical results.
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Keywords age-structured model; recurrent epidemic; C0-semigroup; asymptotical stability; Hopf bifurcation

Citation: Hui Cao, Dongxue Yan, Ao Li. Dynamic analysis of the recurrent epidemic model. Mathematical Biosciences and Engineering, 2019, 16(5): 5972-5990. doi: 10.3934/mbe.2019299

References

  • 1. L. Simonsen, The global impact of influenza on morbidity and mortality, Vaccine, 17 (1999), 3–10.
  • 2. S.W. Huang, Y.W. Hsu, D.J. Smith, et.al., Reemergence of Enterovirus 71 in 2008 in Taiwan: Dynamics of Genetic and Antigenic Evolution from 1998 to 2008, J. Clin. Microbiol., 47 (2009), 3653–3662.
  • 3. S. Chiba, R. Kogasaka, M. Akihara, et al., Recurrent attack of rotavirus gastroenteritis after adenovirus-induced diarrhoea, Arch. Dis. Child., 54 (1979), 398–400.
  • 4. A. Johansen, A simple model of recurrent epidemics, J. Theor. Biol., 178 (1996), 45–51.
  • 5. B. F. Finkenstadt, A stochastic model for extinction and recurrence of epidemics: estimation and inference for measles, Biostatistics, 3(2002), 493–510.
  • 6. A. L. Lloyd, Estimating variability in models for recurrent epidemics: assessing the use of moment closure techniques, Theor. Popul. Biol., 65 (2004), 49–65.
  • 7. L. Stone, R. Olinky and A. Huppert, Seasonal dynamics of recurrent epidemics, Nature, 446 (2007), 533–536.
  • 8. R. Olinky, A. Huppert and L. Stone, Seasonal dynamics and thresholds governing recurrent epidemics, J. Math. Biol., 56(2008), 827–839.
  • 9. M. Begon, S. Telfer, M. J. Smith, et al., Seasonal host dynamics drive the timing of recurrent epidemics in a wildlife population, Proc. R. Soc. B, 276 (2009), 1063–1610.
  • 10. J. Verdasca, M. M. Telo da Gama, A. Numes, et al., Recurrent epidemics in small world networks, J. Theor. Biol., 233 (2005), 553–561.
  • 11. M. Zheng, C. Wang, J. Zhou, et al., Non-periodic outbreaks of recurrent epidemics and its network modelling, Sci. Rep., 5 (2015), 16010.
  • 12. H. Liu, M. Zheng, D. Wu, et al., Hysteresis loop of nonperiodic outbreaks of recurrent epidemics, Phys. Rev., 94 (2016), 062318.
  • 13. K. J. Engel and R. Nagel, One-Parameter Semigroups for Linear Evolution Equations, Springer, New York, 2000.
  • 14. A. Pazy, Semigroups of Linear Operators and Applications to Partial Differential Equations, Springer, New York, 1983.
  • 15. M. Martcheva and H. R. Thieme, Progression age enhanced backward bifurcation in an epidemic model with super-infection, J. Math. Biol., 46 (2003), 385–424.
  • 16. H. R. Thieme, Convergence results and a Poincaré-Bendixson trichotomy for asymptotically autonomous differential equations, J. Math. Biol., 30 (1992), 755–763.
  • 17. J. K. Hale, Theory of Function Differential Equations, Springer, Heidelberg, 1977.

 

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