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September  2019, 24(9): 4703-4720. doi: 10.3934/dcdsb.2018330

The diffusive model for Aedes aegypti mosquito on a periodically evolving domain

1. 

School of Mathematical Science, Yangzhou University, Yangzhou 225002, China

2. 

Key Laboratory of PCBHFASQ, Yangzhou University, Yangzhou 225002, China

* Corresponding author

Received  June 2018 Revised  August 2018 Published  September 2019 Early access  January 2019

Fund Project: The work is partially supported by the NNSF of China (Grant No. 11771381, 61877052).

This paper deals with a reaction-diffusion model on a periodically and isotropically evolving domain in order to explore the diffusive dynamics of Aedes aegypti mosquito, where we divide it into two sub-populations: the winged population and an aquatic form. The spatial-temporal risk index $ R_0(\rho) $ depending on the domain evolution rate $ \rho(t) $ as well as its analytical properties is investigated. The long-time behaviors of the periodic solutions under the condition $ R_0(\rho)>1 $ and $ R_0(\rho)\leq1 $ are explored, respectively. Moreover, we consider the specific case where $ \rho(t)\equiv1 $ to better understand the impact of the periodic evolution rate on the persistence and extinction of Aedes aegypti mosquito. Numerical simulations further verify our analytical results that the periodic domain evolution has a significant impact on the dispersal of Aedes aegypti mosquito.

Citation: Mengyun Zhang, Zhigui Lin. The diffusive model for Aedes aegypti mosquito on a periodically evolving domain. Discrete and Continuous Dynamical Systems - B, 2019, 24 (9) : 4703-4720. doi: 10.3934/dcdsb.2018330
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L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst. Ser. A, 21 (2008), 1-20. doi: 10.3934/dcds.2008.21.1.

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P. $\acute{A}$lvarez-Caudevilla, Y. H. Du and R. Peng, Qualitative analysis of a cooperative reaction-diffusion system in a spatiotemporally degenerate environment, SIAM J. Math. Anal., 46 (2014), 499-531. doi: 10.1137/13091628X.

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I. Ant$\acute{o}$n and J. L$\acute{o}$pez-G$\acute{o}$mez, The strong maximum principle for cooperative periodic-parabolic systems and the existence of principal eigenvalues, World Congress of Nonlinear Analysts '92, Vol. Ⅰ-Ⅳ (Tampa, FL, 1992), 323-334, de Gruyter, Berlin, 1996.

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N. Baca$\ddot{e}$r 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.

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P. A. Bliman, M. S. Aronna, F. C. Coelho and M. A. H. B. da Silva, Ensuring successful introduction of Wolbachia in natural populations of Aedes aegypti by means of feedback control, J. Math. Biol., 76 (2018), 1269-1300. doi: 10.1007/s00285-017-1174-x.

[8]

S. Cauchemez, M. Ledrans, C. Poletto, P.Quenel, H. De Valk, V. Colizza and P. Y. Bo$\ddot{e}$lle, Local and regional spread of chikungunya fever in the Americas, Euro surveill, Biometrika, 19 (2014), 20854.

[9]

E. J. Crampin, E. A. Gaffney and P. K. Maini, Mode-doubling and tripling in reaction-diffusion patterns on growing domains: A piecewise linear model, J. Math. Biol., 44 (2002), 107-128. doi: 10.1007/s002850100112.

[10]

R. H. Cui and Y. Lou, A spatial SIS model in advective heterogeneous environments, J. Differential Equations, 261 (2016), 3305-3343. doi: 10.1016/j.jde.2016.05.025.

[11]

W. W. Ding, R. Peng and L. Wei, The diffusive logistic model with a free boundary in a heterogeneous time-periodic environment, J. Differential Equations, 263 (2017), 2736-2779. doi: 10.1016/j.jde.2017.04.013.

[12]

Y. H. Du, Z. M. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment, J. Funct. Anal., 265 (2013), 2089-2142. doi: 10.1016/j.jfa.2013.07.016.

[13]

Y. H. Du and Z. G. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405. doi: 10.1137/090771089.

[14]

D. J. Gubler, Dengue and dengue hemorrhagic fever, Clin. Microb. Rev., 11 (1998), 480-496.

[15]

P. Hess, Periodic-parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics, Vol. 247. Longman Sci. Tech., Harlow, 1991.

[16]

D. H. Jiang and Z. C. Wang, The diffusive Logistic equation on periodically evolving domains, J. Math. Anal. Appl., 458 (2018), 93-111. doi: 10.1016/j.jmaa.2017.08.059.

[17]

Y. Kaneko and H. Matsuzawa, Spreading speed and sharp asymptotic profiles of solutions in free boundary problems for nonlinear advection-diffusion equations, J. Math. Anal. Appl., 428 (2015), 43-76. doi: 10.1016/j.jmaa.2015.02.051.

[18]

C. X. LeiZ. G. Lin and Q. Y. Zhang, The spreading front of invasive species in favorable habitat or unfavorable habitat, J. Differential Equations, 257 (2014), 145-166.  doi: 10.1016/j.jde.2014.03.015.

[19]

C. Li, Y. M. Lu, J. N. Liu and X. X. Wu, Climate change and dengue fever transmission in China: Evidences and challenges, S. Total Environment, 622-623 (2018), 493-501.

[20]

H. C. Li, R. Peng and F. B. Wang, Varying total population enhances disease persistence: Qualitative analysis on a diffusive SIS epidemic model, J. Differential Equations, 262 (2017), 885-913. doi: 10.1016/j.jde.2016.09.044.

[21]

X. Liang, L. Zhang and X. Q. Zhao, Basic reproduction ratios for periodic abstract functional differential equations (with application to a spatial model for Lyme disease), J. Dyn. Diff. Equat., 261 (2016), 340-372. doi: 10.1016/j.jde.2016.03.014.

[22]

Z. G. Lin and H. P. Zhu, Spatial spreading model and dynamics of West Nile virus in birds and mosquitoes with free boundary, J. Math. Biol., 75 (2017), 1381-1409. doi: 10.1007/s00285-017-1124-7.

[23]

A. Madzvamuse, Time-stepping schemes for moving grid finite elements applied to reaction-diffusion systems on fixed and growing domains, J. Computational Physics, 214 (2006), 239-263.  doi: 10.1016/j.jcp.2005.09.012.

[24]

A. Madzvamuse, E. A. Gaffney and P. K. Maini, Stability analysis of non-autonomous reaction-diffusion systems: the effects of growing domains, J. Math. Biol., 61 (2010), 133-164. doi: 10.1007/s00285-009-0293-4.

[25]

A. Madzvamuse and P. K. Maini, Velocity-induced numerical solutions of reaction-diffusion systems on continuously growing domains, J. Computational Physics, 225 (2007), 100-119. doi: 10.1016/j.jcp.2006.11.022.

[26]

M. Mimura, Y. Yamada and S. Yotsutani, A free boundary problem in ecology, Japan J. Appl. Math., 2 (1985), 151-186. doi: 10.1007/BF03167042.

[27]

M. Mimura, Y. Yamada and S. Yotsutani, Free boundary problems for some reaction-diffusion equations, Hiroshima Math. J., 17 (1987), 241-280.

[28]

C. V. Pao, Stability and attractivity of periodic solutions of parabolic systems with time delays, J. Math. Anal. Appl., 304 (2005), 423-450. doi: 10.1016/j.jmaa.2004.09.014.

[29]

R. Peng and X. Q. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471. doi: 10.1088/0951-7715/25/5/1451.

[30]

L. T. Takahashi, N. A. Maidana, W. Jr. Castro Ferreira, P. Pulino and H. M. Yang, Mathematical models for the Aedes aegypti dispersal dynamics: travelling waves by wing and wind, Bull. Math. Biol., 67 (2005), 509-528. doi: 10.1016/j.bulm.2004.08.005.

[31]

Q. L. Tang and Z. G. Lin, The asymptotic analysis of an insect dispersal model on a growing domain, J. Math. Anal. Appl., 378 (2011), 649-656. doi: 10.1016/j.jmaa.2011.01.057.

[32]

Q. L. Tang, L. Zhang and Z. G. Lin, Asymptotic profile of species migrating on a growing habitat, Acta Appl. Math., 116 (2011), 227-235. doi: 10.1007/s10440-011-9639-1.

[33]

C. R. Tian and S. G. Ruan, A free boundary problem for Aedes aegypti mosquito invasion, Appl. Math. Model., 46 (2017), 203-217. doi: 10.1016/j.apm.2017.01.050.

[34]

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.

[35]

W. D. Wang and X. Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673. doi: 10.1137/120872942.

[36]

D. Weetman and B. Kamgang, etc, Aedes mosquitoes and aedes-borne arboviruses in africa: Current and future threats, Int. J. Environ. Res. Public Health, 15 (2018), 220.

[37]

X. Q. Zhao, Dynamical Systems in Population Biology. Second Edition, CMS Books in Mathematics/Ouvrages de Math$\acute{e}$matiques de la SMC. Springer, Cham, 2017. doi: 10.1007/978-3-319-56433-3.

show all references

References:
[1] D. Acheson, Elementary Fluid Dynamics, Oxford University Press, New York, 1990. 
[2]

L. J. S. Allen, B. M. Bolker, Y. Lou and A. L. Nevai, Asymptotic profiles of the steady states for an SIS epidemic reaction-diffusion model, Discrete Contin. Dyn. Syst. Ser. A, 21 (2008), 1-20. doi: 10.3934/dcds.2008.21.1.

[3]

P. $\acute{A}$lvarez-Caudevilla, Y. H. Du and R. Peng, Qualitative analysis of a cooperative reaction-diffusion system in a spatiotemporally degenerate environment, SIAM J. Math. Anal., 46 (2014), 499-531. doi: 10.1137/13091628X.

[4]

I. Ant$\acute{o}$n and J. L$\acute{o}$pez-G$\acute{o}$mez, The strong maximum principle for cooperative periodic-parabolic systems and the existence of principal eigenvalues, World Congress of Nonlinear Analysts '92, Vol. Ⅰ-Ⅳ (Tampa, FL, 1992), 323-334, de Gruyter, Berlin, 1996.

[5]

N. Baca$\ddot{e}$r 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.

[6] M. J. Baines, Moving Finite Element, Monographs on Numerical Analysis, Clarendon Press, Oxford, 1994. 
[7]

P. A. Bliman, M. S. Aronna, F. C. Coelho and M. A. H. B. da Silva, Ensuring successful introduction of Wolbachia in natural populations of Aedes aegypti by means of feedback control, J. Math. Biol., 76 (2018), 1269-1300. doi: 10.1007/s00285-017-1174-x.

[8]

S. Cauchemez, M. Ledrans, C. Poletto, P.Quenel, H. De Valk, V. Colizza and P. Y. Bo$\ddot{e}$lle, Local and regional spread of chikungunya fever in the Americas, Euro surveill, Biometrika, 19 (2014), 20854.

[9]

E. J. Crampin, E. A. Gaffney and P. K. Maini, Mode-doubling and tripling in reaction-diffusion patterns on growing domains: A piecewise linear model, J. Math. Biol., 44 (2002), 107-128. doi: 10.1007/s002850100112.

[10]

R. H. Cui and Y. Lou, A spatial SIS model in advective heterogeneous environments, J. Differential Equations, 261 (2016), 3305-3343. doi: 10.1016/j.jde.2016.05.025.

[11]

W. W. Ding, R. Peng and L. Wei, The diffusive logistic model with a free boundary in a heterogeneous time-periodic environment, J. Differential Equations, 263 (2017), 2736-2779. doi: 10.1016/j.jde.2017.04.013.

[12]

Y. H. Du, Z. M. Guo and R. Peng, A diffusive logistic model with a free boundary in time-periodic environment, J. Funct. Anal., 265 (2013), 2089-2142. doi: 10.1016/j.jfa.2013.07.016.

[13]

Y. H. Du and Z. G. Lin, Spreading-vanishing dichotomy in the diffusive logistic model with a free boundary, SIAM J. Math. Anal., 42 (2010), 377-405. doi: 10.1137/090771089.

[14]

D. J. Gubler, Dengue and dengue hemorrhagic fever, Clin. Microb. Rev., 11 (1998), 480-496.

[15]

P. Hess, Periodic-parabolic Boundary Value Problems and Positivity, Pitman Research Notes in Mathematics, Vol. 247. Longman Sci. Tech., Harlow, 1991.

[16]

D. H. Jiang and Z. C. Wang, The diffusive Logistic equation on periodically evolving domains, J. Math. Anal. Appl., 458 (2018), 93-111. doi: 10.1016/j.jmaa.2017.08.059.

[17]

Y. Kaneko and H. Matsuzawa, Spreading speed and sharp asymptotic profiles of solutions in free boundary problems for nonlinear advection-diffusion equations, J. Math. Anal. Appl., 428 (2015), 43-76. doi: 10.1016/j.jmaa.2015.02.051.

[18]

C. X. LeiZ. G. Lin and Q. Y. Zhang, The spreading front of invasive species in favorable habitat or unfavorable habitat, J. Differential Equations, 257 (2014), 145-166.  doi: 10.1016/j.jde.2014.03.015.

[19]

C. Li, Y. M. Lu, J. N. Liu and X. X. Wu, Climate change and dengue fever transmission in China: Evidences and challenges, S. Total Environment, 622-623 (2018), 493-501.

[20]

H. C. Li, R. Peng and F. B. Wang, Varying total population enhances disease persistence: Qualitative analysis on a diffusive SIS epidemic model, J. Differential Equations, 262 (2017), 885-913. doi: 10.1016/j.jde.2016.09.044.

[21]

X. Liang, L. Zhang and X. Q. Zhao, Basic reproduction ratios for periodic abstract functional differential equations (with application to a spatial model for Lyme disease), J. Dyn. Diff. Equat., 261 (2016), 340-372. doi: 10.1016/j.jde.2016.03.014.

[22]

Z. G. Lin and H. P. Zhu, Spatial spreading model and dynamics of West Nile virus in birds and mosquitoes with free boundary, J. Math. Biol., 75 (2017), 1381-1409. doi: 10.1007/s00285-017-1124-7.

[23]

A. Madzvamuse, Time-stepping schemes for moving grid finite elements applied to reaction-diffusion systems on fixed and growing domains, J. Computational Physics, 214 (2006), 239-263.  doi: 10.1016/j.jcp.2005.09.012.

[24]

A. Madzvamuse, E. A. Gaffney and P. K. Maini, Stability analysis of non-autonomous reaction-diffusion systems: the effects of growing domains, J. Math. Biol., 61 (2010), 133-164. doi: 10.1007/s00285-009-0293-4.

[25]

A. Madzvamuse and P. K. Maini, Velocity-induced numerical solutions of reaction-diffusion systems on continuously growing domains, J. Computational Physics, 225 (2007), 100-119. doi: 10.1016/j.jcp.2006.11.022.

[26]

M. Mimura, Y. Yamada and S. Yotsutani, A free boundary problem in ecology, Japan J. Appl. Math., 2 (1985), 151-186. doi: 10.1007/BF03167042.

[27]

M. Mimura, Y. Yamada and S. Yotsutani, Free boundary problems for some reaction-diffusion equations, Hiroshima Math. J., 17 (1987), 241-280.

[28]

C. V. Pao, Stability and attractivity of periodic solutions of parabolic systems with time delays, J. Math. Anal. Appl., 304 (2005), 423-450. doi: 10.1016/j.jmaa.2004.09.014.

[29]

R. Peng and X. Q. Zhao, A reaction-diffusion SIS epidemic model in a time-periodic environment, Nonlinearity, 25 (2012), 1451-1471. doi: 10.1088/0951-7715/25/5/1451.

[30]

L. T. Takahashi, N. A. Maidana, W. Jr. Castro Ferreira, P. Pulino and H. M. Yang, Mathematical models for the Aedes aegypti dispersal dynamics: travelling waves by wing and wind, Bull. Math. Biol., 67 (2005), 509-528. doi: 10.1016/j.bulm.2004.08.005.

[31]

Q. L. Tang and Z. G. Lin, The asymptotic analysis of an insect dispersal model on a growing domain, J. Math. Anal. Appl., 378 (2011), 649-656. doi: 10.1016/j.jmaa.2011.01.057.

[32]

Q. L. Tang, L. Zhang and Z. G. Lin, Asymptotic profile of species migrating on a growing habitat, Acta Appl. Math., 116 (2011), 227-235. doi: 10.1007/s10440-011-9639-1.

[33]

C. R. Tian and S. G. Ruan, A free boundary problem for Aedes aegypti mosquito invasion, Appl. Math. Model., 46 (2017), 203-217. doi: 10.1016/j.apm.2017.01.050.

[34]

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.

[35]

W. D. Wang and X. Q. Zhao, Basic reproduction numbers for reaction-diffusion epidemic models, SIAM J. Appl. Dyn. Syst., 11 (2012), 1652-1673. doi: 10.1137/120872942.

[36]

D. Weetman and B. Kamgang, etc, Aedes mosquitoes and aedes-borne arboviruses in africa: Current and future threats, Int. J. Environ. Res. Public Health, 15 (2018), 220.

[37]

X. Q. Zhao, Dynamical Systems in Population Biology. Second Edition, CMS Books in Mathematics/Ouvrages de Math$\acute{e}$matiques de la SMC. Springer, Cham, 2017. doi: 10.1007/978-3-319-56433-3.

Figure 1.  $ \rho(t) = 1 $. For fixed domain, we have $ R_0(1.1)<1 $. Graphs $ (a)-(c) $ show that $ M $ tends to 0
Figure 2.  $ \rho(t) = e^{0.1(1-\cos(4t))} $. For big evolution rate $ \rho(t) $ with $ \overline{\rho^{-2}}<1 $, we have $ R_0(\rho)>1 $. Graph $ (a) $ shows that $ M $ stabilizes to a positive periodic steady state. Graphs $ (b) $ and $ (c) $, which are the cross-sectional view and contour map respectively, present the periodic evolution of the domain
Figure 3.  $ \rho(t) = 1 $. For fixed domain, we have $ R_0(1.1)>1 $. Graphs $ (a)-(c) $ show that $ M $ tends to a positive steady state
Figure 4.  $ \rho(t) = e^{0.1(\cos(4t)-1)} $. For small evolution rate $ \rho(t) $ with $ \overline{\rho^{-2}}>1 $, we couldn't figure out the value of $ R_0(\rho) $. But we can see from graphs $ (a)-(c) $ that mosquitoes become extinct eventually
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