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A reaction–diffusion epidemic model with incubation period in almost periodic environments

Published online by Cambridge University Press:  25 September 2020

LIZHONG QIANG
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, People’s Republic of China emails: [email protected]; [email protected]; [email protected]
BIN-GUO WANG
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, People’s Republic of China emails: [email protected]; [email protected]; [email protected]
ZHI-CHENG WANG
Affiliation:
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, People’s Republic of China emails: [email protected]; [email protected]; [email protected]

Abstract

In this paper, we propose and study an almost periodic reaction–diffusion epidemic model in which disease latency, spatial heterogeneity and general seasonal fluctuations are incorporated. The model is given by a spatially nonlocal reaction–diffusion system with a fixed time delay. We first characterise the upper Lyapunov exponent λ* for a class of almost periodic reaction–diffusion equations with a fixed time delay and provide a numerical method to compute it. On this basis, the global threshold dynamics of this model is established in terms of λ* It is shown that the disease-free almost periodic solution is globally attractive if λ* < 0, while the disease is persistent if λ* > 0. By virtue of numerical simulations, we investigate the effects of diffusion rate, incubation period and spatial heterogeneity on disease transmission.

Type
Papers
Copyright
© The Author(s), 2020. Published by Cambridge University Press

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Footnotes

Supported by NSF of China (11501269, 11731005, 11371179, 11801241) and the Fundamental Research Funds for the Central Universities (lzujbky-2017-ot09, lzujbky-2020-13).

References

Altizer, S., Dobson, A., Hosseini, P., Hudson, P., Pascual, M. & Rohani, P. (2006) Seasonality and the dynamics of infectious diseases. Ecol. Lett. 180, 467484.CrossRefGoogle Scholar
Anderson, R. & May, R. (1979) Population biology of infectious diseases I. Nature 280, 361367.CrossRefGoogle Scholar
Anderson, R. & May, R. (1991) Infectious Diseases of Humans: Dynamics and Control, Oxford University Press, Oxford.Google Scholar
Bai, Zh., Peng, R. & Zhao, X.-Q. (2018) A reaction-diffusion malaria model with seasonality and incubation period. J. Math. Biol. 77, 201228.CrossRefGoogle Scholar
Bezandry, P. H. & Diagana, T. (2011) Almost Periodic Stochastic Processes, Springer, New York.CrossRefGoogle Scholar
Brauer, F. & Castillo-Chavez, C. (2012) Mathematical Models in Population Biology and Epidemiology, 2nd ed., Texts in Applied Mathematics, vol. 40, Springer, New York.CrossRefGoogle Scholar
Caraballo, T., Langa, J., Obaya, R. & Sanz, A. (2018) Global and cocycle attractors for non-autonomous reaction-diffusion equations. The case of null upper Lyapunov exponent. J. Differ. Equ. 265, 39143953.CrossRefGoogle Scholar
Cooke, K. L. & van den Driessche, P. (1996) Analysis of an SEIRS epidemic model with two delays. J. Math. Biol. 35, 240260.CrossRefGoogle ScholarPubMed
Córdova-Lepe, F., Robledo, G., Pinto, M. & González-Olivares, E. (2012) Modeling pulse infectious events irrupting into a controlled context: a SIS disease with almost periodic parameters. Appl. Math. Model. 36, 13231337.CrossRefGoogle Scholar
Diagana, T., Elaydi, S. & Yakubu, A.-Z. (2007) Population models in almost periodic environments. J. Differ. Equ. Appl. 13, 239260.CrossRefGoogle Scholar
Fink, A. M. (1969) Compact families of almost periodic functions and an application of the Schauder fixed-point theorem. SIAM J. Appl. Math. 17, 12581262.CrossRefGoogle Scholar
Fink, A. M. (1974) Almost Periodic Differential Equations, Lecture Notes in Mathematics, Springer-Verlag, Berlin.Google Scholar
Friedman, A. (1964) Partial Differential Equations of Parabolic Type, Prentice-Hall, Englewood Cliffs, NJ.Google Scholar
Guo, Z., Wang, F. & Zou, X. (2012) Threshold dynamics of an infective disease model with a fixed latent period and non-local infections. J. Math. Biol. 65, 13871410.CrossRefGoogle ScholarPubMed
Hale, J. K. (1988) Asymptotic Behavior of Dissipative Systems, American Mathematical Society, Providence RI.Google Scholar
Hess, P. (1991) Periodic-Parabolic Boundary Value Problems and Positivity, Longman Scientific and Technical, Harlow, UK. Hutson, V., Shen, W. & Vickers, G. T. (2001) Estimates for the principal spectrum point for certain time-dependent parabolic operators. Proc. Amer. Math. Soc. 129, 16691679.Google Scholar
Jin, Y. & Zhao, X.-Q. (2009) Spatial dynamics of a non-local periodic reaction-diffusion model with stage structure. SIAM J. Math. Anal. 40, 24962516.CrossRefGoogle Scholar
Kermack, W. O. & Mckendrich, A. G. (1927) A contribution to the mathematical theory of epidemic. Proc. R. Soc. 115, 700721.Google Scholar
Li, F. & Zhao, X.-Q. (2019) A periodic SEIRS epidemic model with a time-dependent latent period. J. Math. Biol. 78, 15531579.CrossRefGoogle ScholarPubMed
Li, J. & Zou, X. (2009) Generalization of the Kermack-McKendrick SIR model to a patchy environment for a disease with latency. Math. Model. Nat. Phenom. 4, 92118.CrossRefGoogle Scholar
Li, J. & Zou, X. (2010) Dynamics of an epidemic model with non-local infections for diseases with latency over a patchy environment. J. Math. Biol. 60, 645686.CrossRefGoogle Scholar
Liang, X., Zhang, L. & Zhao, X.-Q. (2019) Basic reproduction ratios for periodic abstract functional differential equations (with application to a spatial model for Lyme disease). J. Dyn. Diff. Equ. 31, 12471278.CrossRefGoogle Scholar
Lou, Y. & Zhao, X.-Q. (2011) A reaction-diffusion malaria model with incubation period in the vector population. J. Math. Biol. 62, 543568.CrossRefGoogle ScholarPubMed
Lunardi, A. (1995) Analytic Semigroups and Optimal Regularity in Parabolic Problems, Birkhäuser, Boston.Google Scholar
Magal, P. & Zhao, X.-Q. (2005) Global attractors and steady states for uniformly persistent dynamical systems. SIAM J. Math. Anal. 37, 251275.CrossRefGoogle Scholar
Martin, R. H. & Smith, H. L. (1990) Abstract functional differential equations and reaction-diffusion systems. Trans. Amer. Math. Soc. 321, 144.Google Scholar
Metz, J. A. J. & Diekmann, O. (1986) The Dynamics of Physiologically Structured Populations, Springer, New York.CrossRefGoogle Scholar
Novo, S., NúÑez, C. & Obaya, R. (2014) Skew-product semiflows for non-autonomous partial functional differential equations with delay. Discrete Contin. Dyn. Syst. Ser. A 34, 42914321.CrossRefGoogle Scholar
Novo, S., Obaya, R. & Sanz, A. M. (2013) Topological dynamics for monotone skew-product semiflows with applications. J. Dyn. Differ. Equ. 25, 12011231.CrossRefGoogle Scholar
Obaya, R. & Sanz, A. M. (2016) Uniform and strict persistence in monotone skew-product semiflows with applications to non-autonomous Nicholson systems. J. Differ. Equ. 261, 41354163.CrossRefGoogle Scholar
Sell, G. (1971) Topological Dynamics and Ordinary Differential Equations, Van Nostrand Reinhold, London.Google Scholar
Wang, B.-G., Li, W.-T. & Wang, Z.-C. (2015) A reaction-diffusion SIS epidemic model in an almost periodic environment. Z. Angew. Math. Phys. 66, 30853108.CrossRefGoogle Scholar
Wang, B.-G. & Zhao, X.-Q. (2013) Basic reproduction ratios for almost periodic compartmental epidemic models. J. Dyn. Differ. Equ. 25, 535562.CrossRefGoogle Scholar
Wu, J. (1996) Theory and Applications of Partial Functional Differential Equations, Springer, New York.CrossRefGoogle Scholar
Wu, R. & Zhao, X.-Q. (2019) A reaction-diffusion model of vector-borne disease with periodic delays. J. Nonlinear Sci. 29, 2964.CrossRefGoogle Scholar
Zhang, L. & Wang, Z.-C. (2016) A time-periodic reaction-diffusion epidemic model with infection period. Z. Angew. Math. Phys. 67, 117.CrossRefGoogle Scholar
Zhang, L., Wang, Z.-C. & Zhao, X.-Q. (2015) Threshold dynamics of a time periodic reaction-diffusion epidemic model with latent period. J. Differ. Equ. 258, 30113036.CrossRefGoogle Scholar
Zhao, X.-Q. (2017) Dynamical Systems in Population Biology, 2nd ed., Springer-Verlag, New York.CrossRefGoogle Scholar
Zhao, X.-Q. (2017) Basic reproduction ratios for periodic compartmental models with time delay. J. Dyn. Differ. Equ. 258, 6782.CrossRefGoogle Scholar