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A spatially heterogeneous predator-prey model
An almost periodic Dengue transmission model with age structure and time-delayed input of vector in a patchy environment
School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China |
In this paper, we propose an almost periodic multi-patch SIR-SEI model with age structure and time-delayed input of vector. The existence of the almost periodic disease-free solution and the definition of the basic reproduction ratio $ R_{0} $ are given. It is shown that the disease is uniformly persistent if $ R_0>1 $, and it dies out if $ R_0<1 $ under the assumptions that there exists a small invasion and the same travel rate of susceptible, infective and recovered host population in different patches. Finally, we illustrate the above results by numerical simulations. In addition, a simple example shows that the basic reproduction ratio may be underestimated or overestimated if an almost periodic coefficient is approximated by a periodic one.
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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] |
J. Arino and P. van den Driessche,
A multicity epidemic model, Math. Popul. Stud., 10 (2003), 175-193.
doi: 10.1080/08898480306720. |
[3] |
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. |
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C. Corduneanu, Almost Periodic Functions, Chelsea Publishing Company New York, N.Y., 1989. Google Scholar |
[5] |
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 models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.
doi: 10.1007/BF00178324. |
[6] |
D. J. D. Earn, P. Rohani, B. M. Bolker and B. T. Grenfell,
A simple model for complex dynamical transitions in epidemics, Science, 287 (2000), 667-670.
doi: 10.1126/science.287.5453.667. |
[7] |
L. Esteva and C. Vargas,
Analysis of a dengue disease transmission model, Math. Biosci., 150 (1998), 131-151.
doi: 10.1016/S0025-5564(98)10003-2. |
[8] |
L. Esteva and C. Vargas,
A model for dengue disease with variable human population, J. Math. Biol., 38 (1999), 220-240.
doi: 10.1007/s002850050147. |
[9] |
A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1974. |
[10] |
S. Gakkhar and N. C. Chavda,
Impact of awareness on the spread of Dengue infection in human population, Appl. Math., 4 (2013), 142-147.
doi: 10.4236/am.2013.48A020. |
[11] |
D. Gubler, Dengue and Dengue hemorrhagic fever., Clinical Microbiology Reviews, 3 (1998), 480-496. Google Scholar |
[12] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys and Monographs 25, Amer. Math. Soc., Providence, RI, 1988. |
[13] |
J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Appl. Math. Sci., Vol. 99, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-4342-7. |
[14] |
Y. Hino, S. Murakami and T. Naiko, Functional Differential Equations with Infinite Delay, Lecture Notes in Mathematics, Vol. 1473, Springer-Verlag, Berlin, 1991.
doi: 10.1007/BFb0084432. |
[15] |
W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond., 115 (1927), 700-721. Google Scholar |
[16] |
S. Lee and C. Castillo-Chavez,
The role of residence times in two-patch dengue transmission dynamics and optimal strategies, J. Theoret. Biol., 374 (2015), 152-164.
doi: 10.1016/j.jtbi.2015.03.005. |
[17] |
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. |
[18] |
Y. Lou and X.-Q. Zhao,
Threshold dynamics in a time-delayed periodic SIS epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 169-186.
doi: 10.3934/dcdsb.2009.12.169. |
[19] |
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. |
[20] |
G. R. Phaijoo and D. B. Gurung,
Mathematical study of dengue disease transmission in multi-patch environment, Appl. Math., 7 (2016), 1521-1533.
doi: 10.4236/am.2016.714132. |
[21] |
G. R. Phaijoo and D. B. Gurung, Mathematical study of dengue disease with and without awareness in host population, Int. J. Adv. Eng. Res. Appl., 1 (2015), 239-245. Google Scholar |
[22] |
P. Pongsumpun, Mathematical model of dengue disease with the incubation period of virus, World Academy of Science, Engineering and Technology, 44 (2008), 328-332. Google Scholar |
[23] |
L. Qiang and B.-G. Wang,
An almost periodic malaria transmission model with time- delayed input of vector, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1525-1546.
doi: 10.3934/dcdsb.2017073. |
[24] |
L. Qiang, B.-G. Wang and X.-Q. Zhao, Basic reproduction ratios for almost periodic compartmental epidemic with time delay, J. Diff. Equ., 269 (), 4440–4476.
doi: 10.1016/j.jde..03.027. |
[25] |
G. R. Sell, Topological Dynamics and Ordinary Differential Equations, Van Nostrand Reinhold Co., London, 1971. |
[26] |
W. Shen and Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows, Mem. Amer. Math. Soc., 136 (1998), 93pp.
doi: 10.1090/memo/0647. |
[27] |
H. L. Smith, Monotone Dynamics Systems: An Introductionto the Theory of Competitive and Cooperative Systems, Amer. Math. Soc., Providence, RI. 1995. |
[28] |
E. Soewono and A. K. Supriatna,
A two-dimensional model for the transmission of dengue fever disease, Bull. Malays. Math. Sci. Soc., 24 (2001), 49-57.
|
[29] |
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. |
[30] |
B.-G. Wang, W.-T. Li and L. Qiang,
An almost periodic epidemic model in a patchy environment, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 271-289.
doi: 10.3934/dcdsb.2016.21.271. |
[31] |
B.-G. Wang, W.-T. Li and L. Zhang,
An almost periodic epidemic model with age structure in a patchy environment, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 291-311.
doi: 10.3934/dcdsb.2016.21.291. |
[32] |
W. Wang and G. Mulone,
Threshold of disease transmission in a patch environment, J. Math. Anal. Appl., 285 (2003), 321-335.
doi: 10.1016/S0022-247X(03)00428-1. |
[33] |
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. |
[34] |
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. |
[35] |
D. M. Watts, D. S. Burke, B. A. Harrison, R. E. Whitmire and A. Nisalak,
Effect of temperature on the vector efficiency of Aedes aegypti for dengue 2 virus, Am. J. Trop. Hyg., 36 (1987), 143-152.
doi: 10.4269/ajtmh.1987.36.143. |
[36] |
World Health Organization (2012), Global Strategy for Dengue Prevention and Control 2012–, World Health Organization, Geneva. Google Scholar |
[37] |
F. Zhang and X.-Q. Zhao,
A periodic epidemic model in a patchy environment, J. Math. Appl., 325 (2007), 496-516.
doi: 10.1016/j.jmaa.2006.01.085. |
[38] |
X.-Q. Zhao, Dynamical Systems in Population Biology, Springer, Cham, 2017.
doi: 10.1007/978-3-319-56433-3. |
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] |
J. Arino and P. van den Driessche,
A multicity epidemic model, Math. Popul. Stud., 10 (2003), 175-193.
doi: 10.1080/08898480306720. |
[3] |
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. |
[4] |
C. Corduneanu, Almost Periodic Functions, Chelsea Publishing Company New York, N.Y., 1989. Google Scholar |
[5] |
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 models for infectious diseases in heterogeneous populations, J. Math. Biol., 28 (1990), 365-382.
doi: 10.1007/BF00178324. |
[6] |
D. J. D. Earn, P. Rohani, B. M. Bolker and B. T. Grenfell,
A simple model for complex dynamical transitions in epidemics, Science, 287 (2000), 667-670.
doi: 10.1126/science.287.5453.667. |
[7] |
L. Esteva and C. Vargas,
Analysis of a dengue disease transmission model, Math. Biosci., 150 (1998), 131-151.
doi: 10.1016/S0025-5564(98)10003-2. |
[8] |
L. Esteva and C. Vargas,
A model for dengue disease with variable human population, J. Math. Biol., 38 (1999), 220-240.
doi: 10.1007/s002850050147. |
[9] |
A. M. Fink, Almost Periodic Differential Equations, Lecture Notes in Mathematics, Springer-Verlag, Berlin, 1974. |
[10] |
S. Gakkhar and N. C. Chavda,
Impact of awareness on the spread of Dengue infection in human population, Appl. Math., 4 (2013), 142-147.
doi: 10.4236/am.2013.48A020. |
[11] |
D. Gubler, Dengue and Dengue hemorrhagic fever., Clinical Microbiology Reviews, 3 (1998), 480-496. Google Scholar |
[12] |
J. K. Hale, Asymptotic Behavior of Dissipative Systems, Math. Surveys and Monographs 25, Amer. Math. Soc., Providence, RI, 1988. |
[13] |
J. K. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Appl. Math. Sci., Vol. 99, Springer-Verlag, New York, 1993.
doi: 10.1007/978-1-4612-4342-7. |
[14] |
Y. Hino, S. Murakami and T. Naiko, Functional Differential Equations with Infinite Delay, Lecture Notes in Mathematics, Vol. 1473, Springer-Verlag, Berlin, 1991.
doi: 10.1007/BFb0084432. |
[15] |
W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond., 115 (1927), 700-721. Google Scholar |
[16] |
S. Lee and C. Castillo-Chavez,
The role of residence times in two-patch dengue transmission dynamics and optimal strategies, J. Theoret. Biol., 374 (2015), 152-164.
doi: 10.1016/j.jtbi.2015.03.005. |
[17] |
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. |
[18] |
Y. Lou and X.-Q. Zhao,
Threshold dynamics in a time-delayed periodic SIS epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 12 (2009), 169-186.
doi: 10.3934/dcdsb.2009.12.169. |
[19] |
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. |
[20] |
G. R. Phaijoo and D. B. Gurung,
Mathematical study of dengue disease transmission in multi-patch environment, Appl. Math., 7 (2016), 1521-1533.
doi: 10.4236/am.2016.714132. |
[21] |
G. R. Phaijoo and D. B. Gurung, Mathematical study of dengue disease with and without awareness in host population, Int. J. Adv. Eng. Res. Appl., 1 (2015), 239-245. Google Scholar |
[22] |
P. Pongsumpun, Mathematical model of dengue disease with the incubation period of virus, World Academy of Science, Engineering and Technology, 44 (2008), 328-332. Google Scholar |
[23] |
L. Qiang and B.-G. Wang,
An almost periodic malaria transmission model with time- delayed input of vector, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1525-1546.
doi: 10.3934/dcdsb.2017073. |
[24] |
L. Qiang, B.-G. Wang and X.-Q. Zhao, Basic reproduction ratios for almost periodic compartmental epidemic with time delay, J. Diff. Equ., 269 (), 4440–4476.
doi: 10.1016/j.jde..03.027. |
[25] |
G. R. Sell, Topological Dynamics and Ordinary Differential Equations, Van Nostrand Reinhold Co., London, 1971. |
[26] |
W. Shen and Y. Yi, Almost automorphic and almost periodic dynamics in skew-product semiflows, Mem. Amer. Math. Soc., 136 (1998), 93pp.
doi: 10.1090/memo/0647. |
[27] |
H. L. Smith, Monotone Dynamics Systems: An Introductionto the Theory of Competitive and Cooperative Systems, Amer. Math. Soc., Providence, RI. 1995. |
[28] |
E. Soewono and A. K. Supriatna,
A two-dimensional model for the transmission of dengue fever disease, Bull. Malays. Math. Sci. Soc., 24 (2001), 49-57.
|
[29] |
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. |
[30] |
B.-G. Wang, W.-T. Li and L. Qiang,
An almost periodic epidemic model in a patchy environment, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 271-289.
doi: 10.3934/dcdsb.2016.21.271. |
[31] |
B.-G. Wang, W.-T. Li and L. Zhang,
An almost periodic epidemic model with age structure in a patchy environment, Discrete Contin. Dyn. Syst. Ser. B, 21 (2016), 291-311.
doi: 10.3934/dcdsb.2016.21.291. |
[32] |
W. Wang and G. Mulone,
Threshold of disease transmission in a patch environment, J. Math. Anal. Appl., 285 (2003), 321-335.
doi: 10.1016/S0022-247X(03)00428-1. |
[33] |
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. |
[34] |
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. |
[35] |
D. M. Watts, D. S. Burke, B. A. Harrison, R. E. Whitmire and A. Nisalak,
Effect of temperature on the vector efficiency of Aedes aegypti for dengue 2 virus, Am. J. Trop. Hyg., 36 (1987), 143-152.
doi: 10.4269/ajtmh.1987.36.143. |
[36] |
World Health Organization (2012), Global Strategy for Dengue Prevention and Control 2012–, World Health Organization, Geneva. Google Scholar |
[37] |
F. Zhang and X.-Q. Zhao,
A periodic epidemic model in a patchy environment, J. Math. Appl., 325 (2007), 496-516.
doi: 10.1016/j.jmaa.2006.01.085. |
[38] |
X.-Q. Zhao, Dynamical Systems in Population Biology, Springer, Cham, 2017.
doi: 10.1007/978-3-319-56433-3. |



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