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Symbolic dynamics for the $N$-centre problem at negative energies
Traveling waves of diffusive predator-prey systems: Disease outbreak propagation
1. | Mprime Centre for Disease Modelling, York Institute for Health Research, Laboratory for Industrial and Applied Mathematics, Department of Mathematics and Statistics, York University, Toronto, M3J 1P3, Canada, Canada |
2. | Division of Mathematical and Natural Sciences, Arizona State University, Phoenix, AZ 85069-7100, United States |
References:
[1] |
G. Abramson and V. M. Kenkre, Spatiotemporal patterns in hantavirus infection,, Phys. Rev. E, 66 (2002). Google Scholar |
[2] |
G. Abramson, V. M. Kenkre, T. L. Yates and R. R. Parmenter, Traveling waves of infection in the hantavirus epidemics,, Bull. Math. Biol., 65 (2003), 519.
doi: 10.1016/S0092-8240(03)00013-2. |
[3] |
M. S. Abual-Rub, Vaccination in a model of an epidemic,, Int. J. Math. Math. Sci., 23 (2000), 425.
doi: 10.1155/S0161171200002696. |
[4] |
S. Ai and W. Huang, Traveling waves for a reaction-diffusion system in population dynamics and epidemiology,, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 663.
doi: 10.1017/S0308210500004054. |
[5] |
C. Atkinson and G. E. H. Reuter, Deterministic epidemic waves,, Math. Proc. Camb. Phil. Soc., 80 (1976), 315.
doi: 10.1017/S0305004100052944. |
[6] |
M. S. Bartlett, Measles periodicity and community size (with discussion),, J. Roy. Stat. Soc. A, 120 (1957), 48.
doi: 10.2307/2342553. |
[7] |
F. van den Bosch, J. A. J. Metz and O. Diekmann, The velocity of spatial population expansion,, J. Math. Biol., 28 (1990), 529.
doi: 10.1007/BF00164162. |
[8] |
F. Brauer and C. Castillo-Chávez, "Mathematical Models in Population Biology and Epidemiology,", Texts in Applied Mathematics, 40 (2001).
|
[9] |
K. Brown and J. Carr, Deterministic epidemic waves of critical velocity,, Math. Proc. Camb. Phil. Soc., 81 (1977), 431.
doi: 10.1017/S0305004100053494. |
[10] |
V. Capasso and L. Maddalena, A nonlinear diffusion system modelling the spread of oro-faecal diseases,, in, (1981). Google Scholar |
[11] |
T. Caraco, S. Glavanakov, G. Chen, J. E. Flaherty, T. K. Ohsumi and B. K. Szymanski, Stage-structured infection transmission and a spatial epidemic: A model for lyme disease,, Am. Nat., 160 (2002), 348.
doi: 10.1086/341518. |
[12] |
J. Carr and A. Chmaj, Uniqueness of traveling waves for nonlocal monostable equations,, Proc. Amer. Math. Soc., 132 (2004), 2433.
doi: 10.1090/S0002-9939-04-07432-5. |
[13] |
O. Diekmann, Thresholds and traveling waves for the geographical spread of an infection,, J. Math. Biol., 6 (1978), 109.
doi: 10.1007/BF02450783. |
[14] |
O. Diekmann and H. Kaper, On the bounded solutions of a nonliear convolution equation,, Nonlinear Analysis, 2 (1978), 721.
doi: 10.1016/0362-546X(78)90015-9. |
[15] |
S. Djebali, Traveling front solutions for a diffusive epidemic model with external sources,, Annales de la Faculté des Sciences de Toulouse Sér. 6, 10 (2001), 271.
|
[16] |
A. Ducrot, Travelling wave solutions for a scalar age-structured equation,, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 251.
doi: 10.3934/dcdsb.2007.7.251. |
[17] |
A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured model with diffusion,, Proc. R. Soc. Edin. Sect. A, 139 (2009), 459.
doi: 10.1017/S0308210507000455. |
[18] |
S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations,, J. Math. Biol., 17 (1983), 11.
doi: 10.1007/BF00276112. |
[19] |
S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations: A heteroclinic connection in $R^4$,, Trans. Amer. Math. Soc., 286 (1984), 557.
|
[20] |
M. J. Faddy and I. H. Slorach, Bounds on the velocity of spread of infection for a spatially connected epidemic process,, J. Appl. Probab., 17 (1980), 839.
doi: 10.2307/3212977. |
[21] |
J. Fang and J. Wu, Monotone traveling waves for delayed Lotka-Volterra competition systems,, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 3043.
doi: 10.3934/dcds.2012.32.3303. |
[22] |
R. Fisher, The wave of advance of advantageous genes,, Ann. of Eugenics, 7 (1937), 355.
doi: 10.1111/j.1469-1809.1937.tb02153.x. |
[23] |
W. E. Fitzgibbon, M. E. Parrott and G. F. Webb, Diffusion epidemic models with incubation and crisscross dynamics,, Math. Biosci., 128 (1995), 131.
doi: 10.1016/0025-5564(94)00070-G. |
[24] |
Q. Gan, R. Xu and P. Yang, Travelling waves of a delayed SIRS epidemic model with spatial diffusion,, Nonlinear Anal. Real World Appl., 12 (2011), 52.
doi: 10.1016/j.nonrwa.2010.05.035. |
[25] |
B. T. Grenfell, O. N. Bjornstad and J. Kappey, Travelling waves and spatial hierarchies in measles epidemics,, Nature, 414 (2001), 716.
doi: 10.1038/414716a. |
[26] |
Y. Hosono and B. Ilyas, Existence of traveling waves with any positive speed for a diffusive epidemic model,, Nonlinear World, 1 (1994), 277.
|
[27] |
Y. Hosono and B. Ilyas, Traveling waves for a simple diffusive epidemic model,, Math. Models Methods Appl. Sci., 5 (1995), 935.
doi: 10.1142/S0218202595000504. |
[28] |
J.-H. Huang and X.-F. Zou, Travelling wave solutions in delayed reaction diffusion systems with partial monotonicity,, Acta Mathematicae Applicatae Sinica Engl. Ser., 22 (2006), 243.
doi: 10.1007/s10255-006-0300-0. |
[29] |
W. Huang, Traveling waves for a biological reaction-diffusion model,, J. Dynam. Differential Equations, 16 (2004), 745.
|
[30] |
A. Källén, Thresholds and travelling waves in an epidemic model for rabies,, Nonlinear Anal., 8 (1984), 851.
doi: 10.1016/0362-546X(84)90107-X. |
[31] |
A. Källén, P. Arcuri and J. D. Murray, A simple model for the spatial spread and control of rabies,, J. Theor. Biol., 116 (1985), 377.
doi: 10.1016/S0022-5193(85)80276-9. |
[32] |
W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics,, Proc. R. Soc. Lond. B, 115 (1927), 700. Google Scholar |
[33] |
D. G. Kendall, Discussion on Professor Bartlett's paper,, J. Roy. Stat. Soc. A, 120 (1957), 64. Google Scholar |
[34] |
D. G. Kendall, Mathematical models of the spread of infection,, in, (1965), 213. Google Scholar |
[35] |
C. R. Kennedy and R. Aris, Traveling waves in a simple population model involving growth and death,, Bull. Math. Biol., 42 (1980), 397.
|
[36] |
M. N. Kuperman and H. S. Wio, Front propagation in epidemiological models with spatial dependence,, Physica A, 272 (1999), 206.
doi: 10.1016/S0378-4371(99)00284-8. |
[37] |
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. |
[38] |
X. Liang, Y. Yi and X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems,, J. Differential Equations, 231 (2006), 57.
|
[39] |
X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications,, Comm. Pure Appl. Math., 60 (2007), 1.
|
[40] |
S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem,, J. Differential Equations, 171 (2001), 294.
|
[41] |
N. A. Maidana and H. M. Yang, Describing the geographic spread of dengue disease by traveling waves,, Math. Biosci., 215 (2008), 64.
doi: 10.1016/j.mbs.2008.05.008. |
[42] |
D. Mollison, Possible velocities for a simple epidemic,, Adv. Appl. Prob., 4 (1972), 233.
doi: 10.2307/1425997. |
[43] |
D. Mollison, Spatial contact models for ecological and epidemic spread,, J. Roy. Stat. Soc. B, 39 (1977), 283.
|
[44] |
B. Mukhopadhyay and R. Bhattacharyya, Existence of epidemic waves in a disease transmission model with two-habitat population,, Internat. J. Systems Sci., 38 (2007), 699.
doi: 10.1080/00207720701596417. |
[45] |
J. D. Murray and W. L. Seward, On the spatial spread of rabies among foxes with immunity,, J. Theor. Biol., 156 (1992), 327.
doi: 10.1016/S0022-5193(05)80679-4. |
[46] |
J. D. Murray, E. A. Stanley and D. L. Brown, On the spatial spread of rabies among foxes,, Proc. R. Soc. Lond. B, 229 (1986), 111.
doi: 10.1098/rspb.1986.0078. |
[47] |
S. Pan, Traveling wave fronts in an epidemic model with nonlocal diffusion and time delay,, Int. Journal of Math. Analysis, 2 (2008), 1083.
|
[48] |
L. Rass and J. Radcliffe, "Spatial Deterministic Epidemics,", Math. Surveys Monogr., 102 (2003).
|
[49] |
E. Renshaw, Waveforms and velocities for models of spatial infection,, J. Appl. Probab., 18 (1981), 715.
doi: 10.2307/3213325. |
[50] |
S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models,, in, (2007), 97.
|
[51] |
S. Ruan and J. Wu, Modeling spatial spread of communicable diseases involving animal hosts,, in, (2009), 293. Google Scholar |
[52] |
S. Ruan and D. Xiao, Stability of steady states and existence of travelling waves in a vector-disease model,, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 991.
doi: 10.1017/S0308210500003590. |
[53] |
I. Sazonov and M. Kelbert, Travelling waves in a network of SIR epidemic nodes with an approximation of weak coupling,, Math. Med. Biol., 28 (2011), 165.
doi: 10.1093/imammb/dqq016. |
[54] |
I. Sazonov, M. Kelbert and M. B. Gravenor, The speed of epidemic waves in a one-dimensional lattice of SIR models,, Mathematical Modelling of Natural Phenomena, 3 (2008), 28.
doi: 10.1051/mmnp:2008069. |
[55] |
H. L. Smith and X.-Q. Zhao, Traveling waves in a bio-reactor model,, Nonlinear Anal. Real World Appl., 5 (2004), 895.
doi: 10.1016/j.nonrwa.2004.05.001. |
[56] |
L. T. Takahashi, N. A. Maidana, W. C. Ferreira, Jr., 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.
doi: 10.1016/j.bulm.2004.08.005. |
[57] |
H. Thieme, Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread,, J. Math. Biol., 8 (1979), 173.
doi: 10.1007/BF00279720. |
[58] |
H. 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.
|
[59] |
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.
doi: 10.1016/S0025-5564(02)00108-6. |
[60] |
H. Wang, On the existence of traveling waves for delayed reaction-diffusion equations,, J. Differential Equations, 247 (2009), 887.
|
[61] |
X.-S. Wang, J. Wu and Y. Yang, Richards model revisited: Validation by and application to infection dynamics,, submitted., (). Google Scholar |
[62] |
Z.-C. Wang and J. Wu, Traveling waves of a diffusive Kermack-McKendrick epidemic model with non-local delayed transmission,, Proc. R. Soc. Lond. Ser. A Math. Phys. Eng. Sci., 466 (2010), 237.
doi: 10.1098/rspa.2009.0377. |
[63] |
Z.-C. Wang and J. Wu, Traveling waves in a bio-reactor model with stage-structure,, J. Math. Anal. Appl., 385 (2012), 683.
doi: 10.1016/j.jmaa.2011.06.084. |
[64] |
P. Weng and X.-Q. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model,, J. Differential Equations, 229 (2006), 270.
|
[65] |
S.-L. Wu and S.-Y. Liu, Existence and uniqueness of traveling waves for non-monotone integral equations with application,, J. Math. Anal. Appl., 365 (2010), 729.
doi: 10.1016/j.jmaa.2009.11.028. |
[66] |
J. Wylie, H. Huang and R. M. Miura, Systems of coupled diffusion equations with degenerate nonlinear source terms: Linear stability and traveling waves,, Discrete Contin. Dyn. Syst., 23 (2009), 561.
doi: 10.3934/dcds.2009.23.561. |
[67] |
D. Xu and X.-Q. Zhao, Bistable waves in an epidemic model,, J. Dynamics and Differential Equations, 16 (2004), 679.
|
[68] |
D. Xu and X.-Q. Zhao, Asymptotic speed of spread and traveling waves for a nonlocal epidemic model,, Discrete Contin. Dyn. Syst. Ser. B, 5 (2005), 1043.
doi: 10.3934/dcdsb.2005.5.1043. |
[69] |
J. Yang, S. Liang and Y. Zhang, Travelling waves of a delayed SIR epidemic model with nonlinear incidence rate and spatial diffusion,, PLoS ONE, 6 (2011).
doi: 10.1371/journal.pone.0021128. |
[70] |
Y.-R. Yang, W.-T. Li and S.-L. Wu, Exponential stability of traveling fronts in a diffusion epidemic system with delay,, Nonlinear Anal. Real World Appl., 12 (2011), 1223.
doi: 10.1016/j.nonrwa.2010.09.017. |
[71] |
F. Zhang and X.-Q. Zhao, Spreading speed and travelling waves for a spatially discrete SIS epidemic model,, Nonlinearity, 21 (2008), 97.
doi: 10.1088/0951-7715/21/1/005. |
[72] |
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. |
show all references
References:
[1] |
G. Abramson and V. M. Kenkre, Spatiotemporal patterns in hantavirus infection,, Phys. Rev. E, 66 (2002). Google Scholar |
[2] |
G. Abramson, V. M. Kenkre, T. L. Yates and R. R. Parmenter, Traveling waves of infection in the hantavirus epidemics,, Bull. Math. Biol., 65 (2003), 519.
doi: 10.1016/S0092-8240(03)00013-2. |
[3] |
M. S. Abual-Rub, Vaccination in a model of an epidemic,, Int. J. Math. Math. Sci., 23 (2000), 425.
doi: 10.1155/S0161171200002696. |
[4] |
S. Ai and W. Huang, Traveling waves for a reaction-diffusion system in population dynamics and epidemiology,, Proc. Roy. Soc. Edinburgh Sect. A, 135 (2005), 663.
doi: 10.1017/S0308210500004054. |
[5] |
C. Atkinson and G. E. H. Reuter, Deterministic epidemic waves,, Math. Proc. Camb. Phil. Soc., 80 (1976), 315.
doi: 10.1017/S0305004100052944. |
[6] |
M. S. Bartlett, Measles periodicity and community size (with discussion),, J. Roy. Stat. Soc. A, 120 (1957), 48.
doi: 10.2307/2342553. |
[7] |
F. van den Bosch, J. A. J. Metz and O. Diekmann, The velocity of spatial population expansion,, J. Math. Biol., 28 (1990), 529.
doi: 10.1007/BF00164162. |
[8] |
F. Brauer and C. Castillo-Chávez, "Mathematical Models in Population Biology and Epidemiology,", Texts in Applied Mathematics, 40 (2001).
|
[9] |
K. Brown and J. Carr, Deterministic epidemic waves of critical velocity,, Math. Proc. Camb. Phil. Soc., 81 (1977), 431.
doi: 10.1017/S0305004100053494. |
[10] |
V. Capasso and L. Maddalena, A nonlinear diffusion system modelling the spread of oro-faecal diseases,, in, (1981). Google Scholar |
[11] |
T. Caraco, S. Glavanakov, G. Chen, J. E. Flaherty, T. K. Ohsumi and B. K. Szymanski, Stage-structured infection transmission and a spatial epidemic: A model for lyme disease,, Am. Nat., 160 (2002), 348.
doi: 10.1086/341518. |
[12] |
J. Carr and A. Chmaj, Uniqueness of traveling waves for nonlocal monostable equations,, Proc. Amer. Math. Soc., 132 (2004), 2433.
doi: 10.1090/S0002-9939-04-07432-5. |
[13] |
O. Diekmann, Thresholds and traveling waves for the geographical spread of an infection,, J. Math. Biol., 6 (1978), 109.
doi: 10.1007/BF02450783. |
[14] |
O. Diekmann and H. Kaper, On the bounded solutions of a nonliear convolution equation,, Nonlinear Analysis, 2 (1978), 721.
doi: 10.1016/0362-546X(78)90015-9. |
[15] |
S. Djebali, Traveling front solutions for a diffusive epidemic model with external sources,, Annales de la Faculté des Sciences de Toulouse Sér. 6, 10 (2001), 271.
|
[16] |
A. Ducrot, Travelling wave solutions for a scalar age-structured equation,, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 251.
doi: 10.3934/dcdsb.2007.7.251. |
[17] |
A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured model with diffusion,, Proc. R. Soc. Edin. Sect. A, 139 (2009), 459.
doi: 10.1017/S0308210507000455. |
[18] |
S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations,, J. Math. Biol., 17 (1983), 11.
doi: 10.1007/BF00276112. |
[19] |
S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations: A heteroclinic connection in $R^4$,, Trans. Amer. Math. Soc., 286 (1984), 557.
|
[20] |
M. J. Faddy and I. H. Slorach, Bounds on the velocity of spread of infection for a spatially connected epidemic process,, J. Appl. Probab., 17 (1980), 839.
doi: 10.2307/3212977. |
[21] |
J. Fang and J. Wu, Monotone traveling waves for delayed Lotka-Volterra competition systems,, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 3043.
doi: 10.3934/dcds.2012.32.3303. |
[22] |
R. Fisher, The wave of advance of advantageous genes,, Ann. of Eugenics, 7 (1937), 355.
doi: 10.1111/j.1469-1809.1937.tb02153.x. |
[23] |
W. E. Fitzgibbon, M. E. Parrott and G. F. Webb, Diffusion epidemic models with incubation and crisscross dynamics,, Math. Biosci., 128 (1995), 131.
doi: 10.1016/0025-5564(94)00070-G. |
[24] |
Q. Gan, R. Xu and P. Yang, Travelling waves of a delayed SIRS epidemic model with spatial diffusion,, Nonlinear Anal. Real World Appl., 12 (2011), 52.
doi: 10.1016/j.nonrwa.2010.05.035. |
[25] |
B. T. Grenfell, O. N. Bjornstad and J. Kappey, Travelling waves and spatial hierarchies in measles epidemics,, Nature, 414 (2001), 716.
doi: 10.1038/414716a. |
[26] |
Y. Hosono and B. Ilyas, Existence of traveling waves with any positive speed for a diffusive epidemic model,, Nonlinear World, 1 (1994), 277.
|
[27] |
Y. Hosono and B. Ilyas, Traveling waves for a simple diffusive epidemic model,, Math. Models Methods Appl. Sci., 5 (1995), 935.
doi: 10.1142/S0218202595000504. |
[28] |
J.-H. Huang and X.-F. Zou, Travelling wave solutions in delayed reaction diffusion systems with partial monotonicity,, Acta Mathematicae Applicatae Sinica Engl. Ser., 22 (2006), 243.
doi: 10.1007/s10255-006-0300-0. |
[29] |
W. Huang, Traveling waves for a biological reaction-diffusion model,, J. Dynam. Differential Equations, 16 (2004), 745.
|
[30] |
A. Källén, Thresholds and travelling waves in an epidemic model for rabies,, Nonlinear Anal., 8 (1984), 851.
doi: 10.1016/0362-546X(84)90107-X. |
[31] |
A. Källén, P. Arcuri and J. D. Murray, A simple model for the spatial spread and control of rabies,, J. Theor. Biol., 116 (1985), 377.
doi: 10.1016/S0022-5193(85)80276-9. |
[32] |
W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics,, Proc. R. Soc. Lond. B, 115 (1927), 700. Google Scholar |
[33] |
D. G. Kendall, Discussion on Professor Bartlett's paper,, J. Roy. Stat. Soc. A, 120 (1957), 64. Google Scholar |
[34] |
D. G. Kendall, Mathematical models of the spread of infection,, in, (1965), 213. Google Scholar |
[35] |
C. R. Kennedy and R. Aris, Traveling waves in a simple population model involving growth and death,, Bull. Math. Biol., 42 (1980), 397.
|
[36] |
M. N. Kuperman and H. S. Wio, Front propagation in epidemiological models with spatial dependence,, Physica A, 272 (1999), 206.
doi: 10.1016/S0378-4371(99)00284-8. |
[37] |
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. |
[38] |
X. Liang, Y. Yi and X.-Q. Zhao, Spreading speeds and traveling waves for periodic evolution systems,, J. Differential Equations, 231 (2006), 57.
|
[39] |
X. Liang and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for monotone semiflows with applications,, Comm. Pure Appl. Math., 60 (2007), 1.
|
[40] |
S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem,, J. Differential Equations, 171 (2001), 294.
|
[41] |
N. A. Maidana and H. M. Yang, Describing the geographic spread of dengue disease by traveling waves,, Math. Biosci., 215 (2008), 64.
doi: 10.1016/j.mbs.2008.05.008. |
[42] |
D. Mollison, Possible velocities for a simple epidemic,, Adv. Appl. Prob., 4 (1972), 233.
doi: 10.2307/1425997. |
[43] |
D. Mollison, Spatial contact models for ecological and epidemic spread,, J. Roy. Stat. Soc. B, 39 (1977), 283.
|
[44] |
B. Mukhopadhyay and R. Bhattacharyya, Existence of epidemic waves in a disease transmission model with two-habitat population,, Internat. J. Systems Sci., 38 (2007), 699.
doi: 10.1080/00207720701596417. |
[45] |
J. D. Murray and W. L. Seward, On the spatial spread of rabies among foxes with immunity,, J. Theor. Biol., 156 (1992), 327.
doi: 10.1016/S0022-5193(05)80679-4. |
[46] |
J. D. Murray, E. A. Stanley and D. L. Brown, On the spatial spread of rabies among foxes,, Proc. R. Soc. Lond. B, 229 (1986), 111.
doi: 10.1098/rspb.1986.0078. |
[47] |
S. Pan, Traveling wave fronts in an epidemic model with nonlocal diffusion and time delay,, Int. Journal of Math. Analysis, 2 (2008), 1083.
|
[48] |
L. Rass and J. Radcliffe, "Spatial Deterministic Epidemics,", Math. Surveys Monogr., 102 (2003).
|
[49] |
E. Renshaw, Waveforms and velocities for models of spatial infection,, J. Appl. Probab., 18 (1981), 715.
doi: 10.2307/3213325. |
[50] |
S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models,, in, (2007), 97.
|
[51] |
S. Ruan and J. Wu, Modeling spatial spread of communicable diseases involving animal hosts,, in, (2009), 293. Google Scholar |
[52] |
S. Ruan and D. Xiao, Stability of steady states and existence of travelling waves in a vector-disease model,, Proc. Roy. Soc. Edinburgh Sect. A, 134 (2004), 991.
doi: 10.1017/S0308210500003590. |
[53] |
I. Sazonov and M. Kelbert, Travelling waves in a network of SIR epidemic nodes with an approximation of weak coupling,, Math. Med. Biol., 28 (2011), 165.
doi: 10.1093/imammb/dqq016. |
[54] |
I. Sazonov, M. Kelbert and M. B. Gravenor, The speed of epidemic waves in a one-dimensional lattice of SIR models,, Mathematical Modelling of Natural Phenomena, 3 (2008), 28.
doi: 10.1051/mmnp:2008069. |
[55] |
H. L. Smith and X.-Q. Zhao, Traveling waves in a bio-reactor model,, Nonlinear Anal. Real World Appl., 5 (2004), 895.
doi: 10.1016/j.nonrwa.2004.05.001. |
[56] |
L. T. Takahashi, N. A. Maidana, W. C. Ferreira, Jr., 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.
doi: 10.1016/j.bulm.2004.08.005. |
[57] |
H. Thieme, Density-dependent regulation of spatially distributed populations and their asymptotic speed of spread,, J. Math. Biol., 8 (1979), 173.
doi: 10.1007/BF00279720. |
[58] |
H. 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.
|
[59] |
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.
doi: 10.1016/S0025-5564(02)00108-6. |
[60] |
H. Wang, On the existence of traveling waves for delayed reaction-diffusion equations,, J. Differential Equations, 247 (2009), 887.
|
[61] |
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