American Institute of Mathematical Sciences

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November  2014, 19(9): 2993-3018. doi: 10.3934/dcdsb.2014.19.2993

A reaction-diffusion model of dengue transmission

Zhiting Xu 1, and  Yingying Zhao 2,

1. 

School of Mathematical Sciences, South China Normal University, Guangzhou, Guangdong 510631
2. 

School of Mathematical Sciences, South China Normal University, Guangzhou, 510631, China

Received  July 2013 Revised  April 2014 Published  September 2014

This paper is devoted to the mathematical analysis of a reaction-diffusion model of dengue transmission. In the case of a bounded spatial habitat, we obtain the local stability as well as the global stability of either disease-free or endemic steady state in terms of the basic reproduction number $\mathcal{R}_0$. In the case of an unbounded spatial habitat, we establish the existence of the traveling wave solutions connecting the two constant steady states when $\mathcal{R}_0>1$, and the nonexistence of the traveling wave solutions that connect the disease-free steady state itself when $\mathcal{R}_0<1$. Numerical simulations are performed to illustrate the main analytic results.
Keywords: global stability, local stability, Traveling wave solutions, reaction-diffusion model, dengue transmission..
Mathematics Subject Classification: Primary: 34D20, 35C07; Secondary: 35Q92, 92D3.
Citation: Zhiting Xu, Yingying Zhao. A reaction-diffusion model of dengue transmission. Discrete and Continuous Dynamical Systems - B, 2014, 19 (9) : 2993-3018. doi: 10.3934/dcdsb.2014.19.2993
References:
[1]

L. Cai, S. Guo, X. Li and M. Ghosh, Global dynamics of a dengue epidemic mathematical model, Chaos, Solitons and Fractals, 42 (2009), 2297-2304. doi: 10.1016/j.chaos.2009.03.130.

[2]

V. Capasso and G. Serio, A generalization of the Kermack-Mckendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43-61. doi: 10.1016/0025-5564(78)90006-8.

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, Centers for Diease Control and Prevention,, Available from: , (). 

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S. Chen and J. Shi, Global attractivity of equilibrium in Gierer-Meinhardt system with activator production satutation and gene expression time delays, Nonlinear Analysis: Real Wirld Applications, 14 (2013), 1871-1886. doi: 10.1016/j.nonrwa.2012.12.004.

[5]

Y. Du and S. H. Hsu, A diffusive predator-prey model: In heterogeneous envirenmen, J. Differential Equations, 203 (2004), 331-364. doi: 10.1016/j.jde.2004.05.010.

[6]

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.

[7]

J. Fang and J. Wu, Monotone traveling waves for delayed Lotka-Volterra competition systems, Discrete Contin. Dyn. Syst., 32 (2012), 3043-3058. doi: 10.3934/dcds.2012.32.3043.

[8]

Q. Gan, R. Xu and P. Yang, Travelling waves of a hepatitis B virus infection model with spatial diffusion and time delay, IMA J. Appl. Math., 75 (2010), 392-417. doi: 10.1093/imamat/hxq009.

[9]

S. M. Garba, A. B. Gumel and M. R. Abu Bakar, Backward bifurcations in dengue transmission dynamics, Math. Biosci., 215 (2008), 11-25. doi: 10.1016/j.mbs.2008.05.002.

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D. J. Gubler, Dengue/dengue haemorrhagic fever: History and current status, Novartis Foundation Symposium, 277 (2006), 3-16. doi: 10.1002/0470058005.ch2.

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D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, New York, 1981.

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C.-H. Hsu and T.-S. Yang, Existence, uniqueness, monotonicity and asymptotic behaviour of travelling waves for epidemic models, Nonlinearity, 26 (2013), 121-139. doi: 10.1088/0951-7715/26/1/121.

[13]

J. Huang and X. Zou, Existence of travelling wavefronts of delayed reaction diffusion systems without monotonicity, Discrete Contin. Dyn. Syst., 9 (2003), 925-936. doi: 10.3934/dcds.2003.9.925.

[14]

W. Huang, Traveling wave solutions for a class of predator-prey systems, J. Dyn. Differ. Equ., 24 (2012), 633-644. doi: 10.1007/s10884-012-9255-4.

[15]

T. W. Hwang and F. B. Wang, Dynamics of dengue fever trasmission model with crowding effect in human population and spatial variation, Discrete Contin. Dyn. Syst. (Ser. B), 18 (2013), 147-161. doi: 10.3934/dcdsb.2013.18.147.

[16]

X. Lin, P. Weng and C. Wu, Traveling wave solutions for a predator-prey system with sigmoidal response function, J. Dyn. Differ. Equ., 23 (2011), 903-921. doi: 10.1007/s10884-011-9220-7.

[17]

F. X. Jousset, Geographic A. aegypti strains and dengue-2 virus: Susceptibility, ability to transmit to vertebrate and transovarial transmission, Annales de Virologie, 132 (1981), 357-370. doi: 10.1016/S0769-2617(81)80006-8.

[18]

M. Y. L and H. Shu, Global dynamics of an in-host viral model with intracelluar delay, Bull. Math. Biol., 72 (2010), 1492-1505. doi: 10.1007/s11538-010-9503-x.

[19]

W.-T. Li, G. Lin and S. Ruan, Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems, Nonlinearity, 19 (2006), 1253-1257. doi: 10.1088/0951-7715/19/6/003.

[20]

S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential. Equations, 171 (2001), 294-314. doi: 10.1006/jdeq.2000.3846.

[21]

R. Martin and H. Smith, Absract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44. doi: 10.2307/2001590.

[22]

C. C. McCluskey, Complete global stablity for an SIR epidemic model with delay-distributed or discrete, Nonlinear Anaysis: Real World Applications, 11 (2010), 55-59. doi: 10.1016/j.nonrwa.2008.10.014.

[23]

, MedicineNet.com,, Available from: , (). 

[24]

J. D. Murray, Mathematical Biology: I. An Introduction, Springer, New York, 2002.

[25]

C. V. Pao, Global asymptotic stability of Lotka-Volterra competition systems with diffusion and time delays, Nonlinear Analalysis: Real World Application, 5 (2004), 91-104. doi: 10.1016/S1468-1218(03)00018-X.

[26]

L. Rosen and D. A. Shroyer, Transovarial transmission of dengue viruses by mosquitoes: A. Albopictus and A. Aegypti, Am. J. Trop. Med. Hyg., 32 (1983), 1108-1119.

[27]

J. J. Tewa, J. L. Dimi and S. Bowong, Lyapunov functions for a dengue disease transmission model, Chaos, Solitions and Fractals, 39 (2009), 936-941. doi: 10.1016/j.chaos.2007.01.069.

[28]

H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations, 195 (2003), 430-470. doi: 10.1016/S0022-0396(03)00175-X.

[29]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, in Translations of Mathematical Monographs, 140, Amer. Math. Soc., Providence, 1994.

[30]

W. Wang and X.-Q. Zhao, A nonlocal and time-delayed reaction-diffusion model of dengue transmission, SIAM J. Appl. Math., 71 (2011), 147-168. doi: 10.1137/090775890.

[31]

X.-S. Wang, H. Wang and J. Wu, Traveling waves of diffusive perdator-prey systems: disease putbreak propagatton, Discrete Contin. Dyn. Syst., 32 (2013), 3302-3324.

[32]

Z. C. Wang, W. T. Li and S. G. Ruan, Traveling wave fronts in reaction-diffusion systems with spatiotemporal delays, J. Differential Equations, 222 (2006), 185-232. doi: 10.1016/j.jde.2005.08.010.

[33]

P. Weng and Z. Xu, Wavefronts for a global reaction-diffusion systems with inifinite distributed delay, J. Math. Anal. Appl., 345 (2008), 522-534. doi: 10.1016/j.jmaa.2008.04.039.

[34]

, World Health Organization, 2013,, Available from: , (). 

[35]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delays, J. Dyn. Differ. Equ., 13 (2001), 651-687. doi: 10.1023/A:1016690424892.

[36]

R. Xu and Z. Ma, An HBV model with diffusion and time delay, J. Theor. Biol., 257 (2009), 449-509. doi: 10.1016/j.jtbi.2009.01.001.

[37]

X.-Q. Zhao and W. Wang, Fisher waves in an epidemic model, Discrete Contin. Dyn. Syst. (Ser.B), 4 (2004), 1117-1128. doi: 10.3934/dcdsb.2004.4.1117.

show all references

References:
[1]

L. Cai, S. Guo, X. Li and M. Ghosh, Global dynamics of a dengue epidemic mathematical model, Chaos, Solitons and Fractals, 42 (2009), 2297-2304. doi: 10.1016/j.chaos.2009.03.130.

[2]

V. Capasso and G. Serio, A generalization of the Kermack-Mckendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43-61. doi: 10.1016/0025-5564(78)90006-8.

[3]

, Centers for Diease Control and Prevention,, Available from: , (). 

[4]

S. Chen and J. Shi, Global attractivity of equilibrium in Gierer-Meinhardt system with activator production satutation and gene expression time delays, Nonlinear Analysis: Real Wirld Applications, 14 (2013), 1871-1886. doi: 10.1016/j.nonrwa.2012.12.004.

[5]

Y. Du and S. H. Hsu, A diffusive predator-prey model: In heterogeneous envirenmen, J. Differential Equations, 203 (2004), 331-364. doi: 10.1016/j.jde.2004.05.010.

[6]

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.

[7]

J. Fang and J. Wu, Monotone traveling waves for delayed Lotka-Volterra competition systems, Discrete Contin. Dyn. Syst., 32 (2012), 3043-3058. doi: 10.3934/dcds.2012.32.3043.

[8]

Q. Gan, R. Xu and P. Yang, Travelling waves of a hepatitis B virus infection model with spatial diffusion and time delay, IMA J. Appl. Math., 75 (2010), 392-417. doi: 10.1093/imamat/hxq009.

[9]

S. M. Garba, A. B. Gumel and M. R. Abu Bakar, Backward bifurcations in dengue transmission dynamics, Math. Biosci., 215 (2008), 11-25. doi: 10.1016/j.mbs.2008.05.002.

[10]

D. J. Gubler, Dengue/dengue haemorrhagic fever: History and current status, Novartis Foundation Symposium, 277 (2006), 3-16. doi: 10.1002/0470058005.ch2.

[11]

D. Henry, Geometric Theory of Semilinear Parabolic Equations, Springer, New York, 1981.

[12]

C.-H. Hsu and T.-S. Yang, Existence, uniqueness, monotonicity and asymptotic behaviour of travelling waves for epidemic models, Nonlinearity, 26 (2013), 121-139. doi: 10.1088/0951-7715/26/1/121.

[13]

J. Huang and X. Zou, Existence of travelling wavefronts of delayed reaction diffusion systems without monotonicity, Discrete Contin. Dyn. Syst., 9 (2003), 925-936. doi: 10.3934/dcds.2003.9.925.

[14]

W. Huang, Traveling wave solutions for a class of predator-prey systems, J. Dyn. Differ. Equ., 24 (2012), 633-644. doi: 10.1007/s10884-012-9255-4.

[15]

T. W. Hwang and F. B. Wang, Dynamics of dengue fever trasmission model with crowding effect in human population and spatial variation, Discrete Contin. Dyn. Syst. (Ser. B), 18 (2013), 147-161. doi: 10.3934/dcdsb.2013.18.147.

[16]

X. Lin, P. Weng and C. Wu, Traveling wave solutions for a predator-prey system with sigmoidal response function, J. Dyn. Differ. Equ., 23 (2011), 903-921. doi: 10.1007/s10884-011-9220-7.

[17]

F. X. Jousset, Geographic A. aegypti strains and dengue-2 virus: Susceptibility, ability to transmit to vertebrate and transovarial transmission, Annales de Virologie, 132 (1981), 357-370. doi: 10.1016/S0769-2617(81)80006-8.

[18]

M. Y. L and H. Shu, Global dynamics of an in-host viral model with intracelluar delay, Bull. Math. Biol., 72 (2010), 1492-1505. doi: 10.1007/s11538-010-9503-x.

[19]

W.-T. Li, G. Lin and S. Ruan, Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems, Nonlinearity, 19 (2006), 1253-1257. doi: 10.1088/0951-7715/19/6/003.

[20]

S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential. Equations, 171 (2001), 294-314. doi: 10.1006/jdeq.2000.3846.

[21]

R. Martin and H. Smith, Absract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44. doi: 10.2307/2001590.

[22]

C. C. McCluskey, Complete global stablity for an SIR epidemic model with delay-distributed or discrete, Nonlinear Anaysis: Real World Applications, 11 (2010), 55-59. doi: 10.1016/j.nonrwa.2008.10.014.

[23]

, MedicineNet.com,, Available from: , (). 

[24]

J. D. Murray, Mathematical Biology: I. An Introduction, Springer, New York, 2002.

[25]

C. V. Pao, Global asymptotic stability of Lotka-Volterra competition systems with diffusion and time delays, Nonlinear Analalysis: Real World Application, 5 (2004), 91-104. doi: 10.1016/S1468-1218(03)00018-X.

[26]

L. Rosen and D. A. Shroyer, Transovarial transmission of dengue viruses by mosquitoes: A. Albopictus and A. Aegypti, Am. J. Trop. Med. Hyg., 32 (1983), 1108-1119.

[27]

J. J. Tewa, J. L. Dimi and S. Bowong, Lyapunov functions for a dengue disease transmission model, Chaos, Solitions and Fractals, 39 (2009), 936-941. doi: 10.1016/j.chaos.2007.01.069.

[28]

H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations, 195 (2003), 430-470. doi: 10.1016/S0022-0396(03)00175-X.

[29]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, in Translations of Mathematical Monographs, 140, Amer. Math. Soc., Providence, 1994.

[30]

W. Wang and X.-Q. Zhao, A nonlocal and time-delayed reaction-diffusion model of dengue transmission, SIAM J. Appl. Math., 71 (2011), 147-168. doi: 10.1137/090775890.

[31]

X.-S. Wang, H. Wang and J. Wu, Traveling waves of diffusive perdator-prey systems: disease putbreak propagatton, Discrete Contin. Dyn. Syst., 32 (2013), 3302-3324.

[32]

Z. C. Wang, W. T. Li and S. G. Ruan, Traveling wave fronts in reaction-diffusion systems with spatiotemporal delays, J. Differential Equations, 222 (2006), 185-232. doi: 10.1016/j.jde.2005.08.010.

[33]

P. Weng and Z. Xu, Wavefronts for a global reaction-diffusion systems with inifinite distributed delay, J. Math. Anal. Appl., 345 (2008), 522-534. doi: 10.1016/j.jmaa.2008.04.039.

[34]

, World Health Organization, 2013,, Available from: , (). 

[35]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delays, J. Dyn. Differ. Equ., 13 (2001), 651-687. doi: 10.1023/A:1016690424892.

[36]

R. Xu and Z. Ma, An HBV model with diffusion and time delay, J. Theor. Biol., 257 (2009), 449-509. doi: 10.1016/j.jtbi.2009.01.001.

[37]

X.-Q. Zhao and W. Wang, Fisher waves in an epidemic model, Discrete Contin. Dyn. Syst. (Ser.B), 4 (2004), 1117-1128. doi: 10.3934/dcdsb.2004.4.1117.

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