Abstract
In this paper, we propose a time-periodic and diffusive SIR epidemic model with constant infection period. By introducing the basic reproduction number \({\mathcal{R}_0}\) via a next generation operator for this model, we show that the disease goes extinction if \({\mathcal{R}_0 < 1}\) ; while the disease is uniformly persistent if \({\mathcal{R}_0 > 1}\).
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References
Bacaër N., Ait Dads E.: Genealogy with seasonality, the basic reproduction number, and the influenza pandemic. J. Math. Biol. 62, 741–762 (2011)
Bacaër N., Ait Dads E.: On the biological interpretation of a definition for the parameter R 0 in periodic population models. J. Math. Biol. 65, 601–621 (2012)
Bacaër N., Guernaoui S.: The epidemic threshold of vector-borne diseases with seasonality. J. Math. Biol. 53, 421–436 (2006)
Capasso V., Serio G.: A generalization of the Kermack-Mckendrick deterministic epidemic model. Math. Biosci. 42, 41–61 (1978)
Daners D., Koch Medina P.: Abstract Evolution Equations, Periodic Problems and Applications, Pitman Research Notes in Mathematics Series. Vol. 279, Longman, Harlow (1992)
Diekmann O., Heesterbeek J.A.P., Metz J.A.J.: On the definition and the computation of the basic reproduction ratio R 0 in models of infectious disease in heterogeneous populations. J. Math. Biol. 28, 365–382 (1990)
Drnovšek R.: Bounds for the spectral radius of positive operators. Comment. Math. Univ. Carolin. 41, 459–467 (2000)
Friedman A.: Partial Differential Equations of Parabolic Type. Prentice-Hall, Englewood Cliffs (1964)
Hess P.: Periodic-Parabolic Boundary Value Problems and Positivity. Longman Scientific and Technical, Harlow (1991)
Hirsch M.W., Smith H.L., Zhao X.-Q.: Chain transitivity, attractivity and strong repellors for semidynamical systems. J. Dyn. Differ. Equ. 13, 107–131 (2001)
Inaba H.: On a new perspective of the basic reproduction number in heterogeneous environments. J. Math. Biol. 65, 309–348 (2012)
Kato T.: Perturbation Theory for Linear Operators. Springer, Berlin (1976)
Lloyd A.: Realistic distributions of infectious periods in epidemic models: Changing patterns of persistence and dynamics. Theor. Popul. Biol. 60, 59–71 (2001)
Lou Y., Zhao X.-Q.: Threshold dynamics in a time-delayed periodic SIS epidemic model. Discret. Contin. Dyn. Syst. Ser. B 12, 169–186 (2009)
Magal P., Zhao X.-Q.: Global attractors and steady states for uniformly persistent dynamical systems. SIAM J. Math. Anal. 37, 251–275 (2005)
Martin R.H., Smith H.L.: Abstract functional differential equations and reaction-diffusion systems. Trans. Am. Math. Soc. 321, 1–44 (1990)
Metz J.A.J., Diekmann O.: The Dynamics of Physiologically Structured Populations. Springer, New York (1986)
Peng R., Zhao X.-Q.: A reaction-diffusion SIS epidemic model in a time-periodic environment. Nonlinearity 25, 1451–1471 (2012)
Rebelo C., Margheri A., Bacaër N.: Persistence in some periodic epidemic models with infection age or constant periods of infection. Discret. Contin. Dyn. Syst. B 19, 1451–1471 (2014)
Thieme H.R.: Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity. SIAM J. Appl. Math. 70, 188–211 (2009)
van den Driessche P., Watmough J.: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48 (2002)
Wang W., Zhao X.-Q.: An epidemic model with population dispersal and infection period. SIAM J. Appl. Math. 66, 1454–1472 (2006)
Wang W., Zhao X.-Q.: Threshold dynamics for compartmental epidemic models in periodic environments. J. Dynam. Differ. Equ. 20, 699–717 (2008)
Zhang L., Wang Z.-C., Zhao X.-Q.: Threshold dynamics of a time periodic reaction-diffusion epidemic model with latent period. J. Differ. Equ. 258, 3011–3036 (2015)
Zhang Y., Zhao X.-Q.: A reaction-diffusion Lyme disease model with seasonality. SIAM. J. Appl. Math. 73, 2077–2099 (2013)
Zhao X.-Q.: Dynamical Systems in Population Biology. Springer, New York (2003)
Zhao X.-Q.: Basic reproduction ratios for periodic compartmental models with time delay, J. Dynam. Diff. Eqns. (2015). doi:10.1007/s10884-015-9425-2
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Research was partially supported by NSF of China (11371179).
Research was partially supported by the China Scholarship Council and the Fundamental Research Funds for the Central Universities (lzujbky-2015-210).
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Zhang, L., Wang, ZC. A time-periodic reaction–diffusion epidemic model with infection period. Z. Angew. Math. Phys. 67, 117 (2016). https://doi.org/10.1007/s00033-016-0711-6
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DOI: https://doi.org/10.1007/s00033-016-0711-6