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January  2016, 21(1): 271-289. doi: 10.3934/dcdsb.2016.21.271

An almost periodic epidemic model in a patchy environment

Bin-Guo Wang 1, , Wan-Tong Li 2, and  Lizhong Qiang 1,

1. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China, China
2. 

School of Mathematics and Statistics, Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, Gansu 730000

Received  August 2014 Revised  June 2015 Published  November 2015

The persistence and extinction of an almost periodic epidemic model in a patchy environment are studied. It is shown that the disease cannot invade the disease-free state if the exponential growth bound is less than zero and can invade if it is greater than zero. It is also shown that there exists an almost periodic solution which is globally attractive when each patch admits the same dispersal rate. Finally, numerical simulations illustrate the above results.
Keywords: exponential growth bound., epidemic model, Almost periodic, skew-product semiflow, threshold dynamics.
Mathematics Subject Classification: 35B15, 37B55, 92D3.
Citation: Bin-Guo Wang, Wan-Tong Li, Lizhong Qiang. An almost periodic epidemic model in a patchy environment. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 271-289. doi: 10.3934/dcdsb.2016.21.271
References:
[1]

S. Altizer, A. Dobson, P. Hosseini, P. Hudson, M. Pascual and P. Rohani, Seasonality and the dynamics of infectious diseases, Ecology Letters, 9 (2006), 467-484. doi: 10.1111/j.1461-0248.2005.00879.x.  Google Scholar

[2]

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[3]

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F. Cordova-Lepe, G. Robledo, M. Pinto and E. Gonzalez-Olivares, Modeling pulse infectious events irrupting into a controlled context: a SIS disease with almost periodic parameters, Appl. Math. Model., 36 (2012), 1323-1337. doi: 10.1016/j.apm.2011.07.085.  Google Scholar

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[6]

S. R. Dunbar, K. P. Rybakowski and K. Schmitt, Persistence in models of predator-prey populations with diffusion, J. Differential Equations, 65 (1986), 117-138. doi: 10.1016/0022-0396(86)90044-6.  Google Scholar

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A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1974.  Google Scholar

[8]

D. Gao and S. Ruan, Malaria models with spatial effects, Analyzing and Modeling Spatial and Temporal Dynamics of Infectious Disease, D. Chen, B. Moulin, and J. Wu,eds., John Wiley Sons., Hoboken, New Jersey, 2014, 111-138. Google Scholar

[9]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys and Monographs 25, Amer. Math. Soc., Providence, RI, 1988.  Google Scholar

[10]

H. W. Hethcote, Qualitative analysis of communicable disease models, Math. Biosci., 28 (1976), 335-356. doi: 10.1016/0025-5564(76)90132-2.  Google Scholar

[11]

M. W. Hirsch, H. L. Smith and X.-Q. Zhao, Chain transitivity, attractivity and strong repellors for semidynamical systems, J. Dyn. Diff. Equ., 13 (2001), 107-131. doi: 10.1023/A:1009044515567.  Google Scholar

[12]

X. Liu and P. Stechlinski, Transmission dynamics of a switched multi-city model with transport-related infections, Nonlinear Anal. Real Word Appl., 14 (2013), 264-279. doi: 10.1016/j.nonrwa.2012.06.003.  Google Scholar

[13]

Y. Nakata, On the global stability of a delayed epidemic model with transport-related infection, Nonlinear Anal. Real Word Appl., 12 (2011), 3028-3034. doi: 10.1016/j.nonrwa.2011.05.004.  Google Scholar

[14]

G. Sell, Topological Dynamics and Ordinary Differential Equations, Van Nostrand Reinhold, London, 1971.  Google Scholar

[15]

W. Shen and Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows, Memoirs of Amer. Math. Soc., 136 (1998), x+93 pp.. doi: 10.1090/memo/0647.  Google Scholar

[16]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative systems, Math. Surveys and Monographs, 41, Amer. Math. Soc., Providence, RI, 1995.  Google Scholar

[17]

H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511530043.  Google Scholar

[18]

B.-G. Wang and X.-Q. Zhao, Basic reproduction ratios for almost periodic compartmental epidemic models, J. Dyn. Diff. Equ., 25 (2013), 535-562. doi: 10.1007/s10884-013-9304-7.  Google Scholar

[19]

W. Wang and X.-Q. Zhao, An epidemic model in a patchy environment, Math. Biosci., 190 (2004), 97-112. doi: 10.1016/j.mbs.2002.11.001.  Google Scholar

[20]

W. Wang, Epidemic models with population dispersal, Mathematics for Life Sciences and Medicine, Biol. Med. Phys. Biomed. Eng., Springer, Berlin, 2007, 67-95.  Google Scholar

[21]

F. Zhang and X.-Q. Zhao, A periodic epidemic model in a patchy environment, J. Math. Anal. Appl., 325 (2007), 496-516. doi: 10.1016/j.jmaa.2006.01.085.  Google Scholar

[22]

X.-Q. Zhao, Global attractivity and stability in some monotone discrete dynamical systems, Bull. Austral. Math. Soc., 53 (1996), 305-324. doi: 10.1017/S0004972700017032.  Google Scholar

[23]

X.-Q. Zhao, Global attractivity in monotone and subhomogeneous almost periodic systems, J. Differential Equations, 187 (2003), 494-509. doi: 10.1016/S0022-0396(02)00054-2.  Google Scholar

[24]

X.-Q. Zhao, Persistence in almost periodic predator-prey reaction-diffusion systems, Fields Institute Communications, 36 (2003), 259-268.  Google Scholar

[25]

X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

[26]

X.-Q. Zhao and Z. J. Jing, Global asymptotic behavior of some cooperative systems of functional differential equations, Canad. Appl. Math. Quart., 4 (1996), 421-444.  Google Scholar

show all references

References:
[1]

S. Altizer, A. Dobson, P. Hosseini, P. Hudson, M. Pascual and P. Rohani, Seasonality and the dynamics of infectious diseases, Ecology Letters, 9 (2006), 467-484. doi: 10.1111/j.1461-0248.2005.00879.x.  Google Scholar

[2]

J. Arino, Diseases in metapopulations. Modeling and dynamics of infectious diseases, 64-122, Ser. Contemp. Appl. Math. CAM, 11, Higher Ed. Press, Beijing, 2009.  Google Scholar

[3]

G. Butler and P. Waltman, Persistence in dynamical systems, J. Differential Equations, 63 (1986), 255-263. doi: 10.1016/0022-0396(86)90049-5.  Google Scholar

[4]

F. Cordova-Lepe, G. Robledo, M. Pinto and E. Gonzalez-Olivares, Modeling pulse infectious events irrupting into a controlled context: a SIS disease with almost periodic parameters, Appl. Math. Model., 36 (2012), 1323-1337. doi: 10.1016/j.apm.2011.07.085.  Google Scholar

[5]

C. Corduneanu, Almost Periodic Functions, Chelsea Publishing Company New York, N.Y., 1989. Google Scholar

[6]

S. R. Dunbar, K. P. Rybakowski and K. Schmitt, Persistence in models of predator-prey populations with diffusion, J. Differential Equations, 65 (1986), 117-138. doi: 10.1016/0022-0396(86)90044-6.  Google Scholar

[7]

A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1974.  Google Scholar

[8]

D. Gao and S. Ruan, Malaria models with spatial effects, Analyzing and Modeling Spatial and Temporal Dynamics of Infectious Disease, D. Chen, B. Moulin, and J. Wu,eds., John Wiley Sons., Hoboken, New Jersey, 2014, 111-138. Google Scholar

[9]

J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys and Monographs 25, Amer. Math. Soc., Providence, RI, 1988.  Google Scholar

[10]

H. W. Hethcote, Qualitative analysis of communicable disease models, Math. Biosci., 28 (1976), 335-356. doi: 10.1016/0025-5564(76)90132-2.  Google Scholar

[11]

M. W. Hirsch, H. L. Smith and X.-Q. Zhao, Chain transitivity, attractivity and strong repellors for semidynamical systems, J. Dyn. Diff. Equ., 13 (2001), 107-131. doi: 10.1023/A:1009044515567.  Google Scholar

[12]

X. Liu and P. Stechlinski, Transmission dynamics of a switched multi-city model with transport-related infections, Nonlinear Anal. Real Word Appl., 14 (2013), 264-279. doi: 10.1016/j.nonrwa.2012.06.003.  Google Scholar

[13]

Y. Nakata, On the global stability of a delayed epidemic model with transport-related infection, Nonlinear Anal. Real Word Appl., 12 (2011), 3028-3034. doi: 10.1016/j.nonrwa.2011.05.004.  Google Scholar

[14]

G. Sell, Topological Dynamics and Ordinary Differential Equations, Van Nostrand Reinhold, London, 1971.  Google Scholar

[15]

W. Shen and Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows, Memoirs of Amer. Math. Soc., 136 (1998), x+93 pp.. doi: 10.1090/memo/0647.  Google Scholar

[16]

H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative systems, Math. Surveys and Monographs, 41, Amer. Math. Soc., Providence, RI, 1995.  Google Scholar

[17]

H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511530043.  Google Scholar

[18]

B.-G. Wang and X.-Q. Zhao, Basic reproduction ratios for almost periodic compartmental epidemic models, J. Dyn. Diff. Equ., 25 (2013), 535-562. doi: 10.1007/s10884-013-9304-7.  Google Scholar

[19]

W. Wang and X.-Q. Zhao, An epidemic model in a patchy environment, Math. Biosci., 190 (2004), 97-112. doi: 10.1016/j.mbs.2002.11.001.  Google Scholar

[20]

W. Wang, Epidemic models with population dispersal, Mathematics for Life Sciences and Medicine, Biol. Med. Phys. Biomed. Eng., Springer, Berlin, 2007, 67-95.  Google Scholar

[21]

F. Zhang and X.-Q. Zhao, A periodic epidemic model in a patchy environment, J. Math. Anal. Appl., 325 (2007), 496-516. doi: 10.1016/j.jmaa.2006.01.085.  Google Scholar

[22]

X.-Q. Zhao, Global attractivity and stability in some monotone discrete dynamical systems, Bull. Austral. Math. Soc., 53 (1996), 305-324. doi: 10.1017/S0004972700017032.  Google Scholar

[23]

X.-Q. Zhao, Global attractivity in monotone and subhomogeneous almost periodic systems, J. Differential Equations, 187 (2003), 494-509. doi: 10.1016/S0022-0396(02)00054-2.  Google Scholar

[24]

X.-Q. Zhao, Persistence in almost periodic predator-prey reaction-diffusion systems, Fields Institute Communications, 36 (2003), 259-268.  Google Scholar

[25]

X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.  Google Scholar

[26]

X.-Q. Zhao and Z. J. Jing, Global asymptotic behavior of some cooperative systems of functional differential equations, Canad. Appl. Math. Quart., 4 (1996), 421-444.  Google Scholar

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