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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), 011912, 5 pp. |
[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), 519-534.
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), 425-429.
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), 663-675.
doi: 10.1017/S0308210500004054. |
[5] |
C. Atkinson and G. E. H. Reuter, Deterministic epidemic waves, Math. Proc. Camb. Phil. Soc., 80 (1976), 315-330.
doi: 10.1017/S0305004100052944. |
[6] |
M. S. Bartlett, Measles periodicity and community size (with discussion), J. Roy. Stat. Soc. A, 120 (1957), 48-70.
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), 529-565.
doi: 10.1007/BF00164162. |
[8] |
F. Brauer and C. Castillo-Chávez, "Mathematical Models in Population Biology and Epidemiology," Texts in Applied Mathematics, 40, Springer-Verlag, New York, 2001. |
[9] |
K. Brown and J. Carr, Deterministic epidemic waves of critical velocity, Math. Proc. Camb. Phil. Soc., 81 (1977), 431-433.
doi: 10.1017/S0305004100053494. |
[10] |
V. Capasso and L. Maddalena, A nonlinear diffusion system modelling the spread of oro-faecal diseases, in "Nonlinear Phenomena in Mathematical Sciences" (ed. V. Lakshmikantham), Academic Press, New York, 1981. |
[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), 348-359.
doi: 10.1086/341518. |
[12] |
J. Carr and A. Chmaj, Uniqueness of traveling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439.
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), 109-130.
doi: 10.1007/BF02450783. |
[14] |
O. Diekmann and H. Kaper, On the bounded solutions of a nonliear convolution equation, Nonlinear Analysis, 2 (1978), 721-737.
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), 271-292. |
[16] |
A. Ducrot, Travelling wave solutions for a scalar age-structured equation, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 251-273.
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), 459-482.
doi: 10.1017/S0308210507000455. |
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S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11-32.
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S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations: A heteroclinic connection in $R^4$, Trans. Amer. Math. Soc., 286 (1984), 557-594. |
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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), 839-845.
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J. Fang and J. Wu, Monotone traveling waves for delayed Lotka-Volterra competition systems, Discrete Contin. Dyn. Syst. Ser. A, 32 (2012), 3043-3058.
doi: 10.3934/dcds.2012.32.3303. |
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R. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369.
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W. E. Fitzgibbon, M. E. Parrott and G. F. Webb, Diffusion epidemic models with incubation and crisscross dynamics, Math. Biosci., 128 (1995), 131-155.
doi: 10.1016/0025-5564(94)00070-G. |
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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), 52-68.
doi: 10.1016/j.nonrwa.2010.05.035. |
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B. T. Grenfell, O. N. Bjornstad and J. Kappey, Travelling waves and spatial hierarchies in measles epidemics, Nature, 414 (2001), 716-723.
doi: 10.1038/414716a. |
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Y. Hosono and B. Ilyas, Existence of traveling waves with any positive speed for a diffusive epidemic model, Nonlinear World, 1 (1994), 277-290. |
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Y. Hosono and B. Ilyas, Traveling waves for a simple diffusive epidemic model, Math. Models Methods Appl. Sci., 5 (1995), 935-966.
doi: 10.1142/S0218202595000504. |
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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), 243-256.
doi: 10.1007/s10255-006-0300-0. |
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W. Huang, Traveling waves for a biological reaction-diffusion model, J. Dynam. Differential Equations, 16 (2004), 745-765. |
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A. Källén, Thresholds and travelling waves in an epidemic model for rabies, Nonlinear Anal., 8 (1984), 851-856.
doi: 10.1016/0362-546X(84)90107-X. |
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A. Källén, P. Arcuri and J. D. Murray, A simple model for the spatial spread and control of rabies, J. Theor. Biol., 116 (1985), 377-393.
doi: 10.1016/S0022-5193(85)80276-9. |
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W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. B, 115 (1927), 700-721. |
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D. G. Kendall, Discussion on Professor Bartlett's paper, J. Roy. Stat. Soc. A, 120 (1957), 64-67. |
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D. G. Kendall, Mathematical models of the spread of infection, in "Mathematics and Computer Science in Biology and Medicine," Medical Research Council, London, (1965), 213-225. |
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C. R. Kennedy and R. Aris, Traveling waves in a simple population model involving growth and death, Bull. Math. Biol., 42 (1980), 397-429. |
[36] |
M. N. Kuperman and H. S. Wio, Front propagation in epidemiological models with spatial dependence, Physica A, 272 (1999), 206-222.
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), 1253-1273.
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), 57-77. |
[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; Erratum: 61 (2008), 137-138, MR2361306. |
[40] |
S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations, 171 (2001), 294-314. |
[41] |
N. A. Maidana and H. M. Yang, Describing the geographic spread of dengue disease by traveling waves, Math. Biosci., 215 (2008), 64-77.
doi: 10.1016/j.mbs.2008.05.008. |
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D. Mollison, Possible velocities for a simple epidemic, Adv. Appl. Prob., 4 (1972), 233-257.
doi: 10.2307/1425997. |
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D. Mollison, Spatial contact models for ecological and epidemic spread, J. Roy. Stat. Soc. B, 39 (1977), 283-326. |
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B. Mukhopadhyay and R. Bhattacharyya, Existence of epidemic waves in a disease transmission model with two-habitat population, Internat. J. Systems Sci., 38 (2007), 699-707.
doi: 10.1080/00207720701596417. |
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J. D. Murray and W. L. Seward, On the spatial spread of rabies among foxes with immunity, J. Theor. Biol., 156 (1992), 327-348.
doi: 10.1016/S0022-5193(05)80679-4. |
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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), 111-150.
doi: 10.1098/rspb.1986.0078. |
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S. Pan, Traveling wave fronts in an epidemic model with nonlocal diffusion and time delay, Int. Journal of Math. Analysis, 2 (2008), 1083-1088. |
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L. Rass and J. Radcliffe, "Spatial Deterministic Epidemics," Math. Surveys Monogr., 102, Amer. Math. Soc., Providence, RI, 2003. |
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E. Renshaw, Waveforms and velocities for models of spatial infection, J. Appl. Probab., 18 (1981), 715-720.
doi: 10.2307/3213325. |
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S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, in "Mathematics for Life Science and Medicine" (eds. Y. Iwasa, K. Sato and Y. Takeuchi), Biol. Med. Phys. Biomed. Eng., Springer, Berlin, (2007), 97-122. |
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S. Ruan and J. Wu, Modeling spatial spread of communicable diseases involving animal hosts, in "Spatial Ecology," Chapman & Hall/CRC, Boca Raton, FL, (2009), 293-316. |
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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), 991-1011.
doi: 10.1017/S0308210500003590. |
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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), 165-183.
doi: 10.1093/imammb/dqq016. |
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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), 28-47.
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show all references
References:
[1] |
G. Abramson and V. M. Kenkre, Spatiotemporal patterns in hantavirus infection, Phys. Rev. E, 66 (2002), 011912, 5 pp. |
[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), 519-534.
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), 425-429.
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), 663-675.
doi: 10.1017/S0308210500004054. |
[5] |
C. Atkinson and G. E. H. Reuter, Deterministic epidemic waves, Math. Proc. Camb. Phil. Soc., 80 (1976), 315-330.
doi: 10.1017/S0305004100052944. |
[6] |
M. S. Bartlett, Measles periodicity and community size (with discussion), J. Roy. Stat. Soc. A, 120 (1957), 48-70.
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), 529-565.
doi: 10.1007/BF00164162. |
[8] |
F. Brauer and C. Castillo-Chávez, "Mathematical Models in Population Biology and Epidemiology," Texts in Applied Mathematics, 40, Springer-Verlag, New York, 2001. |
[9] |
K. Brown and J. Carr, Deterministic epidemic waves of critical velocity, Math. Proc. Camb. Phil. Soc., 81 (1977), 431-433.
doi: 10.1017/S0305004100053494. |
[10] |
V. Capasso and L. Maddalena, A nonlinear diffusion system modelling the spread of oro-faecal diseases, in "Nonlinear Phenomena in Mathematical Sciences" (ed. V. Lakshmikantham), Academic Press, New York, 1981. |
[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), 348-359.
doi: 10.1086/341518. |
[12] |
J. Carr and A. Chmaj, Uniqueness of traveling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439.
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), 109-130.
doi: 10.1007/BF02450783. |
[14] |
O. Diekmann and H. Kaper, On the bounded solutions of a nonliear convolution equation, Nonlinear Analysis, 2 (1978), 721-737.
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), 271-292. |
[16] |
A. Ducrot, Travelling wave solutions for a scalar age-structured equation, Discrete Contin. Dyn. Syst. Ser. B, 7 (2007), 251-273.
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), 459-482.
doi: 10.1017/S0308210507000455. |
[18] |
S. R. Dunbar, Traveling wave solutions of diffusive Lotka-Volterra equations, J. Math. Biol., 17 (1983), 11-32.
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), 557-594. |
[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), 839-845.
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), 3043-3058.
doi: 10.3934/dcds.2012.32.3303. |
[22] |
R. Fisher, The wave of advance of advantageous genes, Ann. of Eugenics, 7 (1937), 355-369.
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), 131-155.
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), 52-68.
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), 716-723.
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), 277-290. |
[27] |
Y. Hosono and B. Ilyas, Traveling waves for a simple diffusive epidemic model, Math. Models Methods Appl. Sci., 5 (1995), 935-966.
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), 243-256.
doi: 10.1007/s10255-006-0300-0. |
[29] |
W. Huang, Traveling waves for a biological reaction-diffusion model, J. Dynam. Differential Equations, 16 (2004), 745-765. |
[30] |
A. Källén, Thresholds and travelling waves in an epidemic model for rabies, Nonlinear Anal., 8 (1984), 851-856.
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), 377-393.
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), 700-721. |
[33] |
D. G. Kendall, Discussion on Professor Bartlett's paper, J. Roy. Stat. Soc. A, 120 (1957), 64-67. |
[34] |
D. G. Kendall, Mathematical models of the spread of infection, in "Mathematics and Computer Science in Biology and Medicine," Medical Research Council, London, (1965), 213-225. |
[35] |
C. R. Kennedy and R. Aris, Traveling waves in a simple population model involving growth and death, Bull. Math. Biol., 42 (1980), 397-429. |
[36] |
M. N. Kuperman and H. S. Wio, Front propagation in epidemiological models with spatial dependence, Physica A, 272 (1999), 206-222.
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), 1253-1273.
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), 57-77. |
[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; Erratum: 61 (2008), 137-138, MR2361306. |
[40] |
S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations, 171 (2001), 294-314. |
[41] |
N. A. Maidana and H. M. Yang, Describing the geographic spread of dengue disease by traveling waves, Math. Biosci., 215 (2008), 64-77.
doi: 10.1016/j.mbs.2008.05.008. |
[42] |
D. Mollison, Possible velocities for a simple epidemic, Adv. Appl. Prob., 4 (1972), 233-257.
doi: 10.2307/1425997. |
[43] |
D. Mollison, Spatial contact models for ecological and epidemic spread, J. Roy. Stat. Soc. B, 39 (1977), 283-326. |
[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), 699-707.
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), 327-348.
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), 111-150.
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), 1083-1088. |
[48] |
L. Rass and J. Radcliffe, "Spatial Deterministic Epidemics," Math. Surveys Monogr., 102, Amer. Math. Soc., Providence, RI, 2003. |
[49] |
E. Renshaw, Waveforms and velocities for models of spatial infection, J. Appl. Probab., 18 (1981), 715-720.
doi: 10.2307/3213325. |
[50] |
S. Ruan, Spatial-temporal dynamics in nonlocal epidemiological models, in "Mathematics for Life Science and Medicine" (eds. Y. Iwasa, K. Sato and Y. Takeuchi), Biol. Med. Phys. Biomed. Eng., Springer, Berlin, (2007), 97-122. |
[51] |
S. Ruan and J. Wu, Modeling spatial spread of communicable diseases involving animal hosts, in "Spatial Ecology," Chapman & Hall/CRC, Boca Raton, FL, (2009), 293-316. |
[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), 991-1011.
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), 165-183.
doi: 10.1093/imammb/dqq016. |
[54] |
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