# American Institute of Mathematical Sciences

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:
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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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##### 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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