Abstract
It has been commonly accepted that spatial diffusion and environmental heterogeneity are important factors that should be considered in the spread of infectious diseases. In order to understand the impact of spatial heterogeneity of the environment and movement of individuals on the persistence and extinction of a disease, Allen et al. [9] proposed a frequency-dependent SIS (susceptible-infected-susceptible) reaction–diffusion model for a population in a continuous spatial habitat. They assumed that both rates of the transmission and recovery of the disease depend on spatial variables. Another feature of this SIS model is that the total population number is constant. The habitat is characterized as low-risk (or high-risk) if the spatial average of the transmission rate of the disease is less than (or greater than) the spatial average of its recovery rate. The individual site is also characterized as low-risk (or high-risk) if the local transmission rate of the disease is less than (or greater than) its local recovery rate, which corresponds to the case where the local reproduction number is less than (or greater than) one.
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References
N.D. Alikakos, An application of the invariance principle to reaction–diffusion equations. J. Differ. Equ. 33, 201–225 (1979)
L.J.S. Allen, B.M. Bolker, Y. Lou, A.L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model. Discrete Cont. Dyn. Syst. 21, 1–20 (2008)
K.J. Brown, P.C. Dunne, R.A. Gardner, A semilinear parabolic system arising in the theory of superconductivity. J. Differ. Equ. 40, 232–252 (1981)
R. Cui, Y. Lou, A spatial SIS model in advective heterogeneous environments. J. Differ. Equ. 261, 3305–3343 (2016)
L.C. Evans, Partial Differential Equations. Graduate Studies in Mathematics, vol. 19 (American Mathematical Society, Providence, RI, 1998)
A. Friedman, Partial Differential Equations of Parabolic Type (Printice-Hall, Englewood Cliffs, N.J., 1964)
D. Henry, Geometric Theory of Semilinear Parabolic Equations. Lecture Notes in Mathematics, vol. 840 (Springer, Berlin, 1981)
P. Hess, Periodic-Parabolic Boundary Value Problems and Positivity. Pitman Research Notes in Mathematics Series, vol. 247 (Longman Scientific and Technical, Harlow, 1991)
W. Huang, M. Han, K. Liu, Dynamics of an SIS reaction-diffusion epidemic model for disease transmission. Math. Biosci. Eng. 7, 51–66 (2010)
V. Hutson, K. Mischaikow, P. Polác̆ik, The evolution of dispersal rates in a heterogeneous time-periodic environment. J. Math. Biol. 43, 501–533 (2001)
T. Kato, Perturbation Theory for Linear Operators (Springer, Berlin/Heidelberg, 1976)
O.A. Ladyzenskaya, V.A. Solonnikov, N.N. Ural’ceva, Linear and Quasilinear Equations of Parabolic Type (American Mathematical Society, Providence, RI, 1967)
D. Le, Dissipativity and global attractors for a class of quasilinear parabolic systems. Commun. Partial Differ. Equ. 22, 413–433 (1997)
H. Li, R. Peng, F.-B. Wang, Varying total population enhances disease persistence: qualitative analysis on a diffusive SIS epidemic model. J. Differ. Equ. 262, 885–913 (2017)
G.M. Lieberman, Second Order Parabolic Differential Equations (World Scientific Publishing Co., Inc., River Edge, NJ, 1996)
G. Nadin, The principal eigenvalue of a space-time periodic parabolic operator. Ann. Mat. Pura Appl. 188, 269–295 (2009)
W.-M. Ni, The Mathematics of Diffusion (Society for Industrial and Applied Mathematics (SIAM), Philadelphia, PA, 2011)
P.Y.H. Pang, M. Wang, Strategy and stationary pattern in a three-species predator-prey model. J. Differ. Equ. 200, 245–273 (2004)
R. Peng, Asymptotic profiles of the positive steady state for an SIS epidemic reaction-diffusion model. Part I. J. Differ. Equ. 247, 1096–1119 (2009)
R. Peng, S. Liu, Global stability of the steady states of an SIS epidemic reaction-diffusion model. Nonlinear Anal. TMA 71, 239–247 (2009)
R. Peng, F. Yi, Asymptotic profile of the positive steady state for an SIS epidemic reaction-diffusion model: effects of epidemic risk and population movement. Phys. D 259, 8–25 (2013)
R. Peng, X.-Q. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment. Nonlinearity 25, 1451–1471 (2012)
M.H. Protter, H.F. Weinberger, Maximum Principles in Differential Equations (Springer, New York, 1984)
H.R. Thieme, Spectral bound and reproduction number for infinite-dimensional population structure and time heterogeneity. SIAM J. Appl. Math. 70, 188–211 (2009)
W. Wang, X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments. J. Dyn. Differ. Equ. 20, 699–717 (2008)
W. Wang, X.-Q. Zhao, A nonlocal and time-delayed reaction-diffusion model of Dengue transmission. SIAM J. Appl. Math. 71, 147–168 (2011)
Z.-C. Wang, L. Zhang, X.-Q. Zhao, Time periodic traveling waves for a periodic and diffusive SIR epidemic model. J. Dyn. Differ. Equ. (2016). doi:10.1007/s10884-016-9546-2
Y. Wu, X. Zou, Asymptotic profiles of steady states for a diffusive SIS epidemic model with mass action infection mechanism. J. Differ. Equ. 261, 4424–4447 (2016)
X.-Q. Zhao, Uniform persistence and periodic coexistence states in infinite-dimensional periodic semiflows with applications. Can. Appl. Math. Q. 3, 473–495 (1995)
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Zhao, XQ. (2017). A Periodic Reaction–Diffusion SIS Model. In: Dynamical Systems in Population Biology. CMS Books in Mathematics. Springer, Cham. https://doi.org/10.1007/978-3-319-56433-3_13
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DOI: https://doi.org/10.1007/978-3-319-56433-3_13
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