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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 |
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. |
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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