# 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 and Continuous Dynamical Systems - B, 2016, 21 (1) : 291-311. doi: 10.3934/dcdsb.2016.21.291
##### 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. [2] G. Aronsson and R. B. Kellogg, On a differential equation arising from compartmental analysis, Math. Biosci., 38 (1978), 113-122. doi: 10.1016/0025-5564(78)90021-4. [3] N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53 (2006), 421-436. doi: 10.1007/s00285-006-0015-0. [4] R. M. Bolker and B. T. Grenfell, Space, persistence, and dynamics of measles epidemics, Phil. Trans. Roy. Soc. Lond. Ser. B., 348 (1995), 309-320. doi: 10.1098/rstb.1995.0070. [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), 135-154. doi: 10.1016/S0025-5564(98)10016-0. [6] C. Corduneanu, Almost Periodic Functions, Chelsea Publishing Company New York, N.Y., 1989. [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), 329-358. doi: 10.1007/s00285-012-0548-3. [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), 365-382. doi: 10.1007/BF00178324. [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), 5-14. doi: 10.1093/ije/11.1.5. [10] A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1974. [11] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, in: Applied Mathematical Sciences, Vol. 99, Springer, Berlin, Heidelberg, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [12] H. W. Hethcote, Qualitative analysis of communicable disease models, Math. Biosci., 28 (1976), 335-356. doi: 10.1016/0025-5564(76)90132-2. [13] Y. Hino, S. Murakami and T. Naiko, Functional Differential Equations with Infinite Delay, in: Lecture Notes in Mathematics, Vol. 1473, Springer, Berlin, Heidelberg, 1991. [14] P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275. doi: 10.1137/S0036141003439173. [15] X. Liu and X.-Q. Zhao, A periodic epidemic model with age structure in a patchy environment, SIAM J. Appl. Math., 71 (2011), 1896-1917. doi: 10.1137/100813610. [16] A. McKendrick, Applications of mathematics to medical problems, Proc. Edinb. Math. Soc., 44 (1925), 98-130. doi: 10.1017/S0013091500034428. [17] S. Novo and R. Obaya, Strictly ordered mininal subsets of a class of convex monotone skew-product semiflows, J. Differential Equations, 196 (2004), 249-288. doi: 10.1016/S0022-0396(03)00152-9. [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), 86-123. doi: 10.1016/j.jde.2004.01.002. [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), 1899-1925. doi: 10.1016/j.jde.2009.12.007. [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), iv+67 pp. doi: 10.1090/memo/0190. [21] G. Sell, Topological Dynamics and Ordinary Differential Equations, Van Nostrand Reinhold, London, 1971. [22] 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. [23] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, Amer. Math. Soc., Providence, 1995. [24] H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511530043. [25] J. W.-H. So, J. Wu and X. Zou, Structured population on two patches: Modeling dispersal and delay, J. Math. Biol., 43 (2001), 37-51. doi: 10.1007/s002850100081. [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), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [27] 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. [28] 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. [29] W. Wang and X.-Q. Zhao, An age-structured epidemic model in a patchy environment, SIAM J. Appl. Math., 65 (2005), 1597-1614. doi: 10.1137/S0036139903431245. [30] W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Diff. Equ., 20 (2008), 699-717. doi: 10.1007/s10884-008-9111-8. [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), 143-152. [32] 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. [33] 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. [34] X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.

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. [2] G. Aronsson and R. B. Kellogg, On a differential equation arising from compartmental analysis, Math. Biosci., 38 (1978), 113-122. doi: 10.1016/0025-5564(78)90021-4. [3] N. Bacaër and S. Guernaoui, The epidemic threshold of vector-borne diseases with seasonality, J. Math. Biol., 53 (2006), 421-436. doi: 10.1007/s00285-006-0015-0. [4] R. M. Bolker and B. T. Grenfell, Space, persistence, and dynamics of measles epidemics, Phil. Trans. Roy. Soc. Lond. Ser. B., 348 (1995), 309-320. doi: 10.1098/rstb.1995.0070. [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), 135-154. doi: 10.1016/S0025-5564(98)10016-0. [6] C. Corduneanu, Almost Periodic Functions, Chelsea Publishing Company New York, N.Y., 1989. [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), 329-358. doi: 10.1007/s00285-012-0548-3. [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), 365-382. doi: 10.1007/BF00178324. [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), 5-14. doi: 10.1093/ije/11.1.5. [10] A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1974. [11] J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, in: Applied Mathematical Sciences, Vol. 99, Springer, Berlin, Heidelberg, New York, 1993. doi: 10.1007/978-1-4612-4342-7. [12] H. W. Hethcote, Qualitative analysis of communicable disease models, Math. Biosci., 28 (1976), 335-356. doi: 10.1016/0025-5564(76)90132-2. [13] Y. Hino, S. Murakami and T. Naiko, Functional Differential Equations with Infinite Delay, in: Lecture Notes in Mathematics, Vol. 1473, Springer, Berlin, Heidelberg, 1991. [14] P. Magal and X.-Q. Zhao, Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal., 37 (2005), 251-275. doi: 10.1137/S0036141003439173. [15] X. Liu and X.-Q. Zhao, A periodic epidemic model with age structure in a patchy environment, SIAM J. Appl. Math., 71 (2011), 1896-1917. doi: 10.1137/100813610. [16] A. McKendrick, Applications of mathematics to medical problems, Proc. Edinb. Math. Soc., 44 (1925), 98-130. doi: 10.1017/S0013091500034428. [17] S. Novo and R. Obaya, Strictly ordered mininal subsets of a class of convex monotone skew-product semiflows, J. Differential Equations, 196 (2004), 249-288. doi: 10.1016/S0022-0396(03)00152-9. [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), 86-123. doi: 10.1016/j.jde.2004.01.002. [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), 1899-1925. doi: 10.1016/j.jde.2009.12.007. [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), iv+67 pp. doi: 10.1090/memo/0190. [21] G. Sell, Topological Dynamics and Ordinary Differential Equations, Van Nostrand Reinhold, London, 1971. [22] 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. [23] H. L. Smith, Monotone Dynamical Systems: An Introduction to the Theory of Competitive and Cooperative Systems, Mathematical Surveys and Monographs, Amer. Math. Soc., Providence, 1995. [24] H. L. Smith and P. Waltman, The Theory of the Chemostat, Cambridge University Press, Cambridge, 1995. doi: 10.1017/CBO9780511530043. [25] J. W.-H. So, J. Wu and X. Zou, Structured population on two patches: Modeling dispersal and delay, J. Math. Biol., 43 (2001), 37-51. doi: 10.1007/s002850100081. [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), 29-48. doi: 10.1016/S0025-5564(02)00108-6. [27] 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. [28] 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. [29] W. Wang and X.-Q. Zhao, An age-structured epidemic model in a patchy environment, SIAM J. Appl. Math., 65 (2005), 1597-1614. doi: 10.1137/S0036139903431245. [30] W. Wang and X.-Q. Zhao, Threshold dynamics for compartmental epidemic models in periodic environments, J. Dyn. Diff. Equ., 20 (2008), 699-717. doi: 10.1007/s10884-008-9111-8. [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), 143-152. [32] 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. [33] 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. [34] X.-Q. Zhao, Dynamical Systems in Population Biology, Springer-Verlag, New York, 2003. doi: 10.1007/978-0-387-21761-1.
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