# American Institute of Mathematical Sciences

January  2016, 21(1): 291-311. doi: 10.3934/dcdsb.2016.21.291

## An almost periodic epidemic model with age structure in a patchy environment

 1 School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China 2 School of Mathematics and Statistics, Key Laboratory of Applied Mathematics and Complex Systems, Lanzhou University, Lanzhou, Gansu 730000

Received  October 2014 Revised  August 2015 Published  November 2015

An almost periodic epidemic model with age structure in a patchy environment is considered. The existence of the almost periodic disease-free solution and the definition of the basic reproduction ratio $R_{0}$ are given. Based on those, it is shown that a disease dies out if the basic reproduction number $R_{0}$ is less than unity and persists in the population if it is greater than unity.
Citation: Bin-Guo Wang, Wan-Tong Li, Liang Zhang. An almost periodic epidemic model with age structure in a patchy environment. Discrete & Continuous Dynamical Systems - B, 2016, 21 (1) : 291-311. doi: 10.3934/dcdsb.2016.21.291
##### References:
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Verduyn Lunel, Introduction to Functional Differential Equations,, in: Applied Mathematical Sciences, (1993). doi: 10.1007/978-1-4612-4342-7. Google Scholar [12] H. W. Hethcote, Qualitative analysis of communicable disease models,, Math. Biosci., 28 (1976), 335. doi: 10.1016/0025-5564(76)90132-2. Google Scholar [13] Y. Hino, S. Murakami and T. Naiko, Functional Differential Equations with Infinite Delay,, in: Lecture Notes in Mathematics, (1473). Google Scholar [14] P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems,, SIAM J. Math. Anal., 37 (2005), 251. doi: 10.1137/S0036141003439173. Google Scholar [15] X. Liu and X.-Q. Zhao, A periodic epidemic model with age structure in a patchy environment,, SIAM J. Appl. Math., 71 (2011), 1896. doi: 10.1137/100813610. Google Scholar [16] A. McKendrick, Applications of mathematics to medical problems,, Proc. Edinb. Math. Soc., 44 (1925), 98. doi: 10.1017/S0013091500034428. Google Scholar [17] S. Novo and R. Obaya, Strictly ordered mininal subsets of a class of convex monotone skew-product semiflows,, J. Differential Equations, 196 (2004), 249. doi: 10.1016/S0022-0396(03)00152-9. Google Scholar [18] S. Novo, R. Obaya and A. M. Sanz, Attractor minimal sets for cooperative and strongly convex delay differential system,, J. Differential Equations, 208 (2005), 86. doi: 10.1016/j.jde.2004.01.002. Google Scholar [19] C. Núñez, R. Obaya and A. M. Sanz, Minimal sets in monotone and sublinear skew-product semiflows I: The general case,, J. Differential Equations, 248 (2010), 1899. doi: 10.1016/j.jde.2009.12.007. Google Scholar [20] R. J. Sacker and G. R. Sell, Lifting properties in skew-product flows with applications to differential equations,, in Memoirs of the American Mathematical Society, 11 (1977). doi: 10.1090/memo/0190. Google Scholar [21] G. Sell, Topological Dynamics and Ordinary Differential Equations,, Van Nostrand Reinhold, (1971). Google Scholar [22] W. Shen and Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows,, Memoirs of Amer. Math. Soc., 136 (1998). doi: 10.1090/memo/0647. Google Scholar [23] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs,, Amer. Math. Soc., (1995). Google Scholar [24] H. L. Smith and P. Waltman, The Theory of the Chemostat,, Cambridge University Press, (1995). doi: 10.1017/CBO9780511530043. Google Scholar [25] J. W.-H. So, J. Wu and X. Zou, Structured population on two patches: Modeling dispersal and delay,, J. Math. Biol., 43 (2001), 37. doi: 10.1007/s002850100081. Google Scholar [26] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math. Biosci., 180 (2002), 29. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar [27] B.-G. Wang and X.-Q. Zhao, Basic reproduction ratios for almost periodic compartmental epidemic models,, J. Dyn. Diff. Equ., 25 (2013), 535. doi: 10.1007/s10884-013-9304-7. Google Scholar [28] W. Wang and X.-Q. Zhao, An epidemic model in a patchy environment,, Math. Biosci., 190 (2004), 97. doi: 10.1016/j.mbs.2002.11.001. Google Scholar [29] W. Wang and X.-Q. Zhao, An age-structured epidemic model in a patchy environment,, SIAM J. Appl. Math., 65 (2005), 1597. doi: 10.1137/S0036139903431245. Google Scholar [30] W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments,, J. Dyn. Diff. Equ., 20 (2008), 699. doi: 10.1007/s10884-008-9111-8. Google Scholar [31] D. Watts, D. Burke, B. Harrison, R. Whitmire and A. Nisalak, Effect of temperature on the vector efficiency of Aedes aegypti for dengue 2 virus,, Am. J. Trop. Med. Hyg., 36 (1987), 143. Google Scholar [32] F. Zhang and X.-Q. Zhao, A periodic epidemic model in a patchy environment,, J. Math. Anal. Appl., 325 (2007), 496. doi: 10.1016/j.jmaa.2006.01.085. Google Scholar [33] X.-Q. Zhao, Global attractivity in monotone and subhomogeneous almost periodic systems,, J. Differential Equations, 187 (2003), 494. doi: 10.1016/S0022-0396(02)00054-2. Google Scholar [34] X.-Q. Zhao, Dynamical Systems in Population Biology,, Springer-Verlag, (2003). doi: 10.1007/978-0-387-21761-1. 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. doi: 10.1111/j.1461-0248.2005.00879.x. Google Scholar [2] G. Aronsson and R. B. Kellogg, On a differential equation arising from compartmental analysis,, Math. Biosci., 38 (1978), 113. doi: 10.1016/0025-5564(78)90021-4. Google Scholar [3] N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality,, J. Math. Biol., 53 (2006), 421. doi: 10.1007/s00285-006-0015-0. Google Scholar [4] R. M. Bolker and B. T. Grenfell, Space, persistence, and dynamics of measles epidemics,, Phil. Trans. Roy. Soc. Lond. Ser. B., 348 (1995), 309. doi: 10.1098/rstb.1995.0070. Google Scholar [5] C. Castillo-Chavez and Z. Feng, Global stability of an age-structure model for TB and its applications to optimal vaccination strategies,, Math. Biosci., 151 (1998), 135. doi: 10.1016/S0025-5564(98)10016-0. Google Scholar [6] C. Corduneanu, Almost Periodic Functions,, Chelsea Publishing Company New York, (1989). Google Scholar [7] R. Cressman and V. K$\hatr$ivan, Two-patch population models with adaptive dispersal: The effects of varying dispersal speeds,, J. Math. Biol., 67 (2013), 329. doi: 10.1007/s00285-012-0548-3. Google Scholar [8] O. Diekmann, J. A. P. Heesterbeek and J. A. J. Metz, 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 (1990), 365. doi: 10.1007/BF00178324. Google Scholar [9] P. E. M. Fine and J. Clarkson, Measles in England and Wales 1: An analysis of factors underlying seasonal patterns,, Int. J. Epidemiol., 11 (1982), 5. doi: 10.1093/ije/11.1.5. Google Scholar [10] A. M. Fink, Almost Periodic Differential Equations,, Lecture Notes in Mathematics, (1974). Google Scholar [11] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations,, in: Applied Mathematical Sciences, (1993). doi: 10.1007/978-1-4612-4342-7. Google Scholar [12] H. W. Hethcote, Qualitative analysis of communicable disease models,, Math. Biosci., 28 (1976), 335. doi: 10.1016/0025-5564(76)90132-2. Google Scholar [13] Y. Hino, S. Murakami and T. Naiko, Functional Differential Equations with Infinite Delay,, in: Lecture Notes in Mathematics, (1473). Google Scholar [14] P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems,, SIAM J. Math. Anal., 37 (2005), 251. doi: 10.1137/S0036141003439173. Google Scholar [15] X. Liu and X.-Q. Zhao, A periodic epidemic model with age structure in a patchy environment,, SIAM J. Appl. Math., 71 (2011), 1896. doi: 10.1137/100813610. Google Scholar [16] A. McKendrick, Applications of mathematics to medical problems,, Proc. Edinb. Math. Soc., 44 (1925), 98. doi: 10.1017/S0013091500034428. Google Scholar [17] S. Novo and R. Obaya, Strictly ordered mininal subsets of a class of convex monotone skew-product semiflows,, J. Differential Equations, 196 (2004), 249. doi: 10.1016/S0022-0396(03)00152-9. Google Scholar [18] S. Novo, R. Obaya and A. M. Sanz, Attractor minimal sets for cooperative and strongly convex delay differential system,, J. Differential Equations, 208 (2005), 86. doi: 10.1016/j.jde.2004.01.002. Google Scholar [19] C. Núñez, R. Obaya and A. M. Sanz, Minimal sets in monotone and sublinear skew-product semiflows I: The general case,, J. Differential Equations, 248 (2010), 1899. doi: 10.1016/j.jde.2009.12.007. Google Scholar [20] R. J. Sacker and G. R. Sell, Lifting properties in skew-product flows with applications to differential equations,, in Memoirs of the American Mathematical Society, 11 (1977). doi: 10.1090/memo/0190. Google Scholar [21] G. Sell, Topological Dynamics and Ordinary Differential Equations,, Van Nostrand Reinhold, (1971). Google Scholar [22] W. Shen and Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows,, Memoirs of Amer. Math. Soc., 136 (1998). doi: 10.1090/memo/0647. Google Scholar [23] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs,, Amer. Math. Soc., (1995). Google Scholar [24] H. L. Smith and P. Waltman, The Theory of the Chemostat,, Cambridge University Press, (1995). doi: 10.1017/CBO9780511530043. Google Scholar [25] J. W.-H. So, J. Wu and X. Zou, Structured population on two patches: Modeling dispersal and delay,, J. Math. Biol., 43 (2001), 37. doi: 10.1007/s002850100081. Google Scholar [26] P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission,, Math. Biosci., 180 (2002), 29. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar [27] B.-G. Wang and X.-Q. Zhao, Basic reproduction ratios for almost periodic compartmental epidemic models,, J. Dyn. Diff. Equ., 25 (2013), 535. doi: 10.1007/s10884-013-9304-7. Google Scholar [28] W. Wang and X.-Q. Zhao, An epidemic model in a patchy environment,, Math. Biosci., 190 (2004), 97. doi: 10.1016/j.mbs.2002.11.001. Google Scholar [29] W. Wang and X.-Q. Zhao, An age-structured epidemic model in a patchy environment,, SIAM J. Appl. Math., 65 (2005), 1597. doi: 10.1137/S0036139903431245. Google Scholar [30] W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments,, J. Dyn. Diff. Equ., 20 (2008), 699. doi: 10.1007/s10884-008-9111-8. Google Scholar [31] D. Watts, D. Burke, B. Harrison, R. Whitmire and A. Nisalak, Effect of temperature on the vector efficiency of Aedes aegypti for dengue 2 virus,, Am. J. Trop. Med. Hyg., 36 (1987), 143. Google Scholar [32] F. Zhang and X.-Q. Zhao, A periodic epidemic model in a patchy environment,, J. Math. Anal. Appl., 325 (2007), 496. doi: 10.1016/j.jmaa.2006.01.085. Google Scholar [33] X.-Q. Zhao, Global attractivity in monotone and subhomogeneous almost periodic systems,, J. Differential Equations, 187 (2003), 494. doi: 10.1016/S0022-0396(02)00054-2. Google Scholar [34] X.-Q. Zhao, Dynamical Systems in Population Biology,, Springer-Verlag, (2003). doi: 10.1007/978-0-387-21761-1. Google Scholar
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