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

A reaction-diffusion model of dengue transmission

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.
Citation: Zhiting Xu, Yingying Zhao. A reaction-diffusion model of dengue transmission. Discrete & 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, 42 (2009), 2297. doi: 10.1016/j.chaos.2009.03.130. Google Scholar

[2]

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

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

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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. doi: 10.1016/j.nonrwa.2012.12.004. Google Scholar

[5]

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

[6]

L. Esteva and C. Vargas, Analysis of a dengue disease transmission model,, Math. Biosci., 150 (1998), 131. doi: 10.1016/S0025-5564(98)10003-2. Google Scholar

[7]

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

[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. doi: 10.1093/imamat/hxq009. Google Scholar

[9]

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

[10]

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

[11]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Springer, (1981). Google Scholar

[12]

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

[13]

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

[14]

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

[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. doi: 10.3934/dcdsb.2013.18.147. Google Scholar

[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. doi: 10.1007/s10884-011-9220-7. Google Scholar

[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. doi: 10.1016/S0769-2617(81)80006-8. Google Scholar

[18]

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

[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. doi: 10.1088/0951-7715/19/6/003. Google Scholar

[20]

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

[21]

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

[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. doi: 10.1016/j.nonrwa.2008.10.014. Google Scholar

[23]

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

[24]

J. D. Murray, Mathematical Biology: I. An Introduction,, Springer, (2002). Google Scholar

[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. doi: 10.1016/S1468-1218(03)00018-X. Google Scholar

[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. Google Scholar

[27]

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

[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. doi: 10.1016/S0022-0396(03)00175-X. Google Scholar

[29]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems,, in Translations of Mathematical Monographs, (1994). Google Scholar

[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. doi: 10.1137/090775890. Google Scholar

[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. Google Scholar

[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. doi: 10.1016/j.jde.2005.08.010. Google Scholar

[33]

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

[34]

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

[35]

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

[36]

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

[37]

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

show all references

References:
[1]

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

[2]

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

[3]

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

[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. doi: 10.1016/j.nonrwa.2012.12.004. Google Scholar

[5]

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

[6]

L. Esteva and C. Vargas, Analysis of a dengue disease transmission model,, Math. Biosci., 150 (1998), 131. doi: 10.1016/S0025-5564(98)10003-2. Google Scholar

[7]

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

[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. doi: 10.1093/imamat/hxq009. Google Scholar

[9]

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

[10]

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

[11]

D. Henry, Geometric Theory of Semilinear Parabolic Equations,, Springer, (1981). Google Scholar

[12]

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

[13]

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

[14]

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

[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. doi: 10.3934/dcdsb.2013.18.147. Google Scholar

[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. doi: 10.1007/s10884-011-9220-7. Google Scholar

[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. doi: 10.1016/S0769-2617(81)80006-8. Google Scholar

[18]

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

[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. doi: 10.1088/0951-7715/19/6/003. Google Scholar

[20]

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

[21]

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

[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. doi: 10.1016/j.nonrwa.2008.10.014. Google Scholar

[23]

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

[24]

J. D. Murray, Mathematical Biology: I. An Introduction,, Springer, (2002). Google Scholar

[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. doi: 10.1016/S1468-1218(03)00018-X. Google Scholar

[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. Google Scholar

[27]

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

[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. doi: 10.1016/S0022-0396(03)00175-X. Google Scholar

[29]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems,, in Translations of Mathematical Monographs, (1994). Google Scholar

[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. doi: 10.1137/090775890. Google Scholar

[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. Google Scholar

[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. doi: 10.1016/j.jde.2005.08.010. Google Scholar

[33]

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

[34]

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

[35]

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

[36]

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

[37]

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

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