American Institute of Mathematical Sciences

Advanced
doi: 10.3934/dcdsb.2020220

An almost periodic Dengue transmission model with age structure and time-delayed input of vector in a patchy environment

Jing Feng and  Bin-Guo Wang ,

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

* Corresponding author: Bin-Guo Wang

Received  March 2019 Revised  May 2020 Published  July 2020

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.

Keywords: Almost periodic equations, basic reproduction ratio, disease-free solution.
Mathematics Subject Classification: 34D08, 34C27, 34D45.
Citation: Jing Feng, Bin-Guo Wang. An almost periodic Dengue transmission model with age structure and time-delayed input of vector in a patchy environment. Discrete & Continuous Dynamical Systems - B, doi: 10.3934/dcdsb.2020220
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 and P. van den Driessche, A multicity epidemic model, Math. Popul. Stud., 10 (2003), 175-193.  doi: 10.1080/08898480306720.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[27]

H. L. Smith, Monotone Dynamics Systems: An Introductionto the Theory of Competitive and Cooperative Systems, Amer. Math. Soc., Providence, RI. 1995.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

X.-Q. Zhao, Dynamical Systems in Population Biology, Springer, Cham, 2017. doi: 10.1007/978-3-319-56433-3.  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 and P. van den Driessche, A multicity epidemic model, Math. Popul. Stud., 10 (2003), 175-193.  doi: 10.1080/08898480306720.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[9]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[25]

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

[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.  Google Scholar

[27]

H. L. Smith, Monotone Dynamics Systems: An Introductionto the Theory of Competitive and Cooperative Systems, Amer. Math. Soc., Providence, RI. 1995.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[38]

X.-Q. Zhao, Dynamical Systems in Population Biology, Springer, Cham, 2017. doi: 10.1007/978-3-319-56433-3.  Google Scholar

Figure 1.  The number of equilibrum
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Figure 2.  The graph of $ \|u(t)\| $ and $ \ln\|u(t)\| $ of (4.3) when $ k = 1 $
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Figure 3.  The long-term behavior host and vector populations at patch 1 and 2 when $ R_0<1 $
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Figure 4.  The graph of $ \|u(t)\| $ and $ \ln\|u(t)\| $ of (4.3) when $ k = 4 $
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Figure 5.  The long-term behavior host and vector populations at patch 1 and 2 when $ R_0>1 $
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Figure 6.  Relationship between $ k $ and $ R_{0} $
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Figure 7.  The graph of $ \|u(t)\| $ and $ \ln\|u(t)\| $ of (4.3) when $ m_{21}^I = 0 $
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Figure 8.  The graph of $ \|u(t)\| $ and $ \ln\|u(t)\| $ of (4.3) when $ m_{21}^I = 1 $
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Figure 9.  Relationship between $ m_{21}^I $ and $ R_{0} $
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Figure 10.  Relationship between $ n $ and $ R_{0} $
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