Abstract
A susceptible–infected–susceptible almost periodic reaction–diffusion epidemic model is studied by means of establishing the theories and properties of the basic reproduction ratio \({R_{0}}\). Particularly, the asymptotic behaviors of \({R_{0}}\) with respect to the diffusion rate \({D_{I}}\) of the infected individuals are obtained. Furthermore, the uniform persistence, extinction and global attractivity are presented in terms of \({R_{0}}\). Our results indicate that the interaction of spatial heterogeneity and temporal almost periodicity tends to enhance the persistence of the disease.
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Allen L.J.S., Bolker B.M., Lou Y., Nevai A.L.: Asymptotic profiles of the steady states for an SIS epidemic disease patch model. SIAM J. Appl. Math. 67, 1283–1309 (2007)
Allen L.J.S., Bolker B.M., Lou Y., Nevai A.L.: Asymptotic profiles of the steady states for an SIS epidemic reaction–diffusion model. Discrete Contin. Dyn. Syst. A 21, 1–20 (2008)
Altizer S., Dobson A., Hosseini P., Hudson P., Pascual M., Rohani P.: Seasonality and the dynamics of infectious diseases. Ecol. Lett. 9, 467–484 (2006)
Bacaër N., Guernaoui S.: The epidemic threshold of vector-borne diseases with seasonality. J. Math. Biol. 53, 421–436 (2006)
Caraco T., Glavanakov S., Chen G., Flaherty J.E., Ohsumi T.K., Szymanski B.K.: Stage structured infection transmission and a spatial epidemic: a model for Lyme disease. Am. Nat. 160, 348–359 (2002)
Corduneanu C.: Almost Periodic Functions. Chelsea Publishing Company, New York, NY (1989)
Daners, D., Koch Medina, P.: Abstract Evolution Equations, Periodic Problems and Applications (Pitman Research Notes in Mathematics Series vol. 279). Harlow, Longman Scientific & Technical (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 the models for infectious disease in heterogeneous populations. J. Math. Biol. 28, 365–382 (1990)
Fink, A.M.: Almost Periodic Differential Equations, Lecture Notes in Mathematics, Springer, Berlin (1974)
Gilbarg I.D., Trudinger N.S.: Elliptic Partial Differential Equations of Second Order, 2nd ed. Springer, Berlin (1983)
Hale, J.K.: Asymptotic Behavior of Dissipative Systems, Mathematical Surveys and Monographs, vol. 25. American Mathematical Society, Providence, RI (1988)
Henry, D.: Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics 840. Springer, Berlin (1981)
Hess, P.: Periodic-Parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics, Series 247. Longman Scientific and Technical (1991)
Huang W., Han M., Liu K.: Dynamics of an SIS reaction–diffusion epidemic model for disease transmission. Math. Biosci. Eng. 7, 51–66 (2010)
Hutson V., Mischaikow K., Polácik P.: The evolution of dispersal rates in heterogeneous time-periodic environment. J. Math. Biol. 43, 501–533 (2001)
Hutson V., Shen W., Vickers G.T.: Estimates for the principal spectrum point for certain time-dependent parabolic operators. Proc. Am. Math. Soc. 129, 1669–1679 (2000)
Lewis M., Renclawowicz J., van den Driessche P.: Traveling waves and spread rates for a West Nile virus model. Bull. Math. Biol. 68, 3–23 (2006)
Lieberman G.M.: Second Order Parabolic Differential Equations. World Scientific, River Edge, NJ (1996)
Lou Y., Zhao X.-Q.: A reaction–diffusion malaria model with incubation period in the vector population. J. Math. Biol. 62, 543–568 (2011)
Murray J.D., Stanley E.A., Brown D.L.: On the spatial spread of rabies among foxes. Proc. R. Soc. Lond. Ser. B 229, 111–150 (1986)
Ou C., Wu J.: Spatial spread of rabies revisited: Influence of age-dependent diffusion on nonlinear dynamics. SIAM J. Appl. Math. 67, 138–163 (2006)
Pang P.Y.H., Wang M.: Strategy and stationary pattern in a three-species predator-prey model. J. Diff. Equ. 200, 245–273 (2004)
Peng R.: Asymptotic profile of the positive steady state for an SIS epidemic reaction–diffusion model: I. J. Diff. Equ. 247, 1096–1119 (2009)
Peng R., Liu S.: Global stability of the steady states of an SIS epidemic reaction–diffusion model. Nonlinear Anal. TMA 71, 239–247 (2009)
Peng R., Zhao X.-Q.: A reaction–diffusion SIS epidemic model in a time-periodic environment. Nonlinearity 25, 1451–1471 (2012)
Protter M.H., Weinberger H.F.: Maximum Principles in Differential Equations. Springer, Berlin (1984)
Sacker R., Sell G.: A spectral theory for linear differential systems. J. Diff. Equ. 27, 320–358 (1978)
Sell G.: Topological Dynamics and Ordinary Differential Equations. Van Nostrand Reinhold, London (1971)
Shen W., Yi Y.: Almost automorphic and almost periodic dynamics in skew-product semiflows. Mem. Am. Math. Soc. 647, 136 (1998)
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 B.-G., Zhao X.-Q.: Basic reproduction ratios for almost periodic compartmental epidemic models. J. Dyn. Diff. Equ. 25, 535–562 (2013)
Wang W., Zhao X.-Q.: A nonlocal and time-delayed reaction–diffusion model of dengue transmission. SIAM J. Appl. Math. 71, 147–168 (2011)
Wang W., Zhao X.-Q.: Threshold dynamics for compartmental epidemic models in periodic environments. J. Dyn. Diff. Equ. 20, 699–717 (2008)
Wang Y., Zhao X.-Q.: Global convergence in monotone and uniformly stable recurrent skew-product semiflows. Infinite dimensional dynamical systems. Fields Inst. Commun. 64, 391–406 (2013)
Zhao X.-Q.: Persistence in almost periodic predator-prey reaction–diffusion systems. Fields Inst. Commun. 36, 259–268 (2003)
Zhao X.-Q.: Global attractivity in monotone and subhomogeneous almost periodic systems. J. Diff. Equ. 187, 494–509 (2003)
Zhao X.-Q.: Dynamical Systems in Population Biology. Springer, New York (2003)
Zhao X.-Q.: Global dynamics of a reaction and diffusion model for Lyme disease. J. Math. Biol. 65, 787–808 (2012)
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Wang, BG., Li, WT. & Wang, ZC. A reaction–diffusion SIS epidemic model in an almost periodic environment. Z. Angew. Math. Phys. 66, 3085–3108 (2015). https://doi.org/10.1007/s00033-015-0585-z
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DOI: https://doi.org/10.1007/s00033-015-0585-z