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Mean-field backward stochastic Volterra integral equations
Traveling waves in a nonlocal dispersal Kermack-McKendrick epidemic model
1. | School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China, China |
2. | School of Mathematic and Statistics, Lanzhou University, Lanzhou, Gansu 730000 |
References:
[1] |
F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. Toledo-Melero, "Nonlocal Diffusion Problems,", Mathematical Surveys and Monographs, (2010).
|
[2] |
P. Bates, P. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions,, Arch. Rational Mech. Anal., 138 (1997), 105.
doi: 10.1007/s002050050037. |
[3] |
P. Bates and G. Zhao, Existence, uniquenss and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal,, J. Math. Anal. Appl., 332 (2007), 428.
doi: 10.1016/j.jmaa.2006.09.007. |
[4] |
J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations,, Proc. Amer. Math. Soc., 132 (2004), 2433.
doi: 10.1090/S0002-9939-04-07432-5. |
[5] |
X. Chen, Existence, uniqueness and asymptotic stability of travelling waves in non-local evolution equations,, Adv. Differential Equations, 2 (1997), 125.
|
[6] |
J. Coville, J. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity,, J. Differential Equations, 244 (2008), 3080.
doi: 10.1016/j.jde.2007.11.002. |
[7] |
J. Coville, J. Dávila and S. Martínez, Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity,, SIAM J. Math. Anal., 39 (2008), 1693.
doi: 10.1137/060676854. |
[8] |
J. Coville and L. Dupaigne, On a nonlocal reaction diffusion equation arising in population dynamics,, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 727.
doi: 10.1017/S0308210504000721. |
[9] |
P. De Mottoni, E. Orlandi and A. Tesei, Asymptotic behavior for a system describing epidemics with migration and spatial spread of infection,, Nonlinear Anal., 3 (1979), 663.
doi: 10.1016/0362-546X(79)90095-6. |
[10] |
A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured model with diffusion,, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 459.
doi: 10.1017/S0308210507000455. |
[11] |
A. Ducrot, P. Magal and S. Ruan, Travelling wave solutions in multigroup age-structure epidemic models,, Arch. Ration. Mech. Anal., 195 (2010), 311.
doi: 10.1007/s00205-008-0203-8. |
[12] |
P. Fife, An integrodifferential analog of semilinear parabolic PDEs,, in, 177 (1996), 137.
|
[13] |
P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions,, in, (2003), 153.
|
[14] |
R. Fisher, The wave of advance of advantageous genes,, Ann. of Eugenics, 7 (1937), 355.
doi: 10.1111/j.1469-1809.1937.tb02153.x. |
[15] |
J. García-Melián and J. D. Rossi, A logistic equation with refuge and nonlocal diffusion,, Commun. Pure Appl. Anal., 8 (2009), 2037.
doi: 10.3934/cpaa.2009.8.2037. |
[16] |
V. Hutson and M. Grinfeld, Non-local dispersal and bistability,, Euro. J. Appl. Math., 17 (2006), 221.
doi: 10.1017/S0956792506006462. |
[17] |
V. Hutson, S. Martinez, K. Mischailow and G. T. Vickers, The evolution of dispersal,, J. Math. Biol., 47 (2003), 483.
doi: 10.1007/s00285-003-0210-1. |
[18] |
V. Hutson, W. Shen and G. T. Vickers, Spectral theory for nonlocal dispersal with periodic or almost-periodic time dependence,, Rocky Mountain J. Math., 38 (2008), 1147.
doi: 10.1216/RMJ-2008-38-4-1147. |
[19] |
Y. Hosono and B. Ilyas, Travelling waves for a simple diffusive epidemic model,, Math. Model Meth. Appl. Sci., 5 (1995), 935.
doi: 10.1142/S0218202595000504. |
[20] |
C. Y. Kao, Y. Lou and W. Shen, Random dispersal vs non-local dispersal,, Discrete Contin. Dyn. Syst., 26 (2010), 551.
doi: 10.3934/dcds.2010.26.551. |
[21] |
C.-Y. Kao, Y. Lou and W. Shen, Evolution of mixed dispersal in periodic environments,, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2047.
doi: 10.3934/dcdsb.2012.17.2047. |
[22] |
W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics,, Proc. Roy. Soc. London Ser. B, 115 (1927), 700.
doi: 10.1098/rspa.1927.0118. |
[23] |
W.-T. Li, G. Lin and S. Ruan, Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems,, Nonlinerity, 19 (2006), 1253.
doi: 10.1088/0951-7715/19/6/003. |
[24] |
W.-T. Li, Y.-J. Sun and Z.-C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal,, Nonlinear Anal. Real Word Appl., 11 (2010), 2302.
doi: 10.1016/j.nonrwa.2009.07.005. |
[25] |
G. Lin, W.-T. Li and M.-J. Ma, Traveling wave solutions in delayed reaction diffusion systems with applications to multi-species models,, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 393.
doi: 10.3934/dcdsb.2010.13.393. |
[26] |
J. D. Murray, "Mathematical Biology, II, Spatial Models and Biomedical Applications,", Third edition. Interdisciplinary Applied Mathematics, 18 (2003).
|
[27] |
S. Ma, Travelling wavefronts for delayed reaction-diffusion systems via a fixed point theorem,, J. Differential Equations, 171 (2001), 294.
doi: 10.1006/jdeq.2000.3846. |
[28] |
J. Medlock and M. Kot, Spreading disease: Integro-differential equations old and new,, Math. Biosci., 184 (2003), 201.
doi: 10.1016/S0025-5564(03)00041-5. |
[29] |
S. Pan, Traveling wave fronts of delayed non-local diffusion systems without quasimonotonicity,, J. Math. Anal. Appl., 346 (2008), 415.
doi: 10.1016/j.jmaa.2008.05.057. |
[30] |
S. Pan, W.-T. Li and G. Lin, Travelling wave fronts in nonlocal reaction-diffusion systems and applications,, Z. Angew. Math. Phys., 60 (2009), 377.
doi: 10.1007/s00033-007-7005-y. |
[31] |
S. Pan, W.-T. Li and G. Lin, Existence and stability of traveling wavefronts in a nonlocal diffusion equation with delay,, Nonlinear Anal., 72 (2010), 3150.
doi: 10.1016/j.na.2009.12.008. |
[32] |
K. Schumacher, Travelling-front solutions for integro-differential equations.I.,, J. Reine Angew. Math., 316 (1980), 54.
doi: 10.1515/crll.1980.316.54. |
[33] |
W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats,, J. Differential Equations, 15 (2010), 747.
doi: 10.1016/j.jde.2010.04.012. |
[34] |
Y.-J. Sun, W.-T. Li and Z.-C. Wang, Entire solutions in nonlocal dispersal equations with bistable nonlinearity,, J. Differential Equations, 251 (2011), 551.
doi: 10.1016/j.jde.2011.04.020. |
[35] |
Y.-J. Sun, W.-T. Li and Z.-C. Wang, Traveling waves for a nonlocal anisotropic dispersal equation with monostable nonlinearity,, Nonlinear Anal., 74 (2011), 814.
doi: 10.1016/j.na.2010.09.032. |
[36] |
X. Wang, Metastability and stability of patterns in a convolution model for phase transitions,, J. Differential Equations, 183 (2002), 434.
doi: 10.1006/jdeq.2001.4129. |
[37] |
Z.-C. Wang, W.-T. Li and S. Ruan, Traveling fronts in monostable equations with nonlocal delayed effects,, J. Dynam. Diff. Eqns., 20 (2008), 573.
doi: 10.1007/s10884-008-9103-8. |
[38] |
Z.-C. Wang, W.-T. Li and S. Ruan, Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity,, Trans. Amer. Math. Soc., 361 (2009), 2047.
doi: 10.1090/S0002-9947-08-04694-1. |
[39] |
Z.-C. Wang and J. Wu, Travelling waves of a diffusive Kermack-McKendrick epidemic model with non-local delayed transmission,, Proc. R. Soc. Lond. Ser. A, 466 (2010), 237.
doi: 10.1098/rspa.2009.0377. |
[40] |
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. |
[41] |
D. V. Widder, "The Laplace Transform,", Princeton University Press, (1941).
|
[42] |
H. Yagisita, Existence and nonexistence of traveling waves for a nonlocal monostable equation,, Publ. Res. Inst. Math. Sci., 45 (2009), 925.
doi: 10.2977/prims/1260476648. |
[43] |
H. Yagisita, Existence of traveling waves for a nonlocal monostable equation: an abstract approach,, Publ. Res. Inst. Math. Sci., 45 (2009), 955.
doi: 10.2977/prims/1260476649. |
[44] |
L. Zhang, Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neural networks,, J. Differential Equations, 197 (2004), 162.
doi: 10.1016/S0022-0396(03)00170-0. |
[45] |
G.-B. Zhang, W.-T. Li and Z.-C. Wang, Spreading speeds and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity,, J. Differential Equations, 252 (2012), 5096.
doi: 10.1016/j.jde.2012.01.014. |
show all references
References:
[1] |
F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. Toledo-Melero, "Nonlocal Diffusion Problems,", Mathematical Surveys and Monographs, (2010).
|
[2] |
P. Bates, P. Fife, X. Ren and X. Wang, Traveling waves in a convolution model for phase transitions,, Arch. Rational Mech. Anal., 138 (1997), 105.
doi: 10.1007/s002050050037. |
[3] |
P. Bates and G. Zhao, Existence, uniquenss and stability of the stationary solution to a nonlocal evolution equation arising in population dispersal,, J. Math. Anal. Appl., 332 (2007), 428.
doi: 10.1016/j.jmaa.2006.09.007. |
[4] |
J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations,, Proc. Amer. Math. Soc., 132 (2004), 2433.
doi: 10.1090/S0002-9939-04-07432-5. |
[5] |
X. Chen, Existence, uniqueness and asymptotic stability of travelling waves in non-local evolution equations,, Adv. Differential Equations, 2 (1997), 125.
|
[6] |
J. Coville, J. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity,, J. Differential Equations, 244 (2008), 3080.
doi: 10.1016/j.jde.2007.11.002. |
[7] |
J. Coville, J. Dávila and S. Martínez, Existence and uniqueness of solutions to a nonlocal equation with monostable nonlinearity,, SIAM J. Math. Anal., 39 (2008), 1693.
doi: 10.1137/060676854. |
[8] |
J. Coville and L. Dupaigne, On a nonlocal reaction diffusion equation arising in population dynamics,, Proc. Roy. Soc. Edinburgh Sect. A, 137 (2007), 727.
doi: 10.1017/S0308210504000721. |
[9] |
P. De Mottoni, E. Orlandi and A. Tesei, Asymptotic behavior for a system describing epidemics with migration and spatial spread of infection,, Nonlinear Anal., 3 (1979), 663.
doi: 10.1016/0362-546X(79)90095-6. |
[10] |
A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured model with diffusion,, Proc. Roy. Soc. Edinburgh Sect. A, 139 (2009), 459.
doi: 10.1017/S0308210507000455. |
[11] |
A. Ducrot, P. Magal and S. Ruan, Travelling wave solutions in multigroup age-structure epidemic models,, Arch. Ration. Mech. Anal., 195 (2010), 311.
doi: 10.1007/s00205-008-0203-8. |
[12] |
P. Fife, An integrodifferential analog of semilinear parabolic PDEs,, in, 177 (1996), 137.
|
[13] |
P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions,, in, (2003), 153.
|
[14] |
R. Fisher, The wave of advance of advantageous genes,, Ann. of Eugenics, 7 (1937), 355.
doi: 10.1111/j.1469-1809.1937.tb02153.x. |
[15] |
J. García-Melián and J. D. Rossi, A logistic equation with refuge and nonlocal diffusion,, Commun. Pure Appl. Anal., 8 (2009), 2037.
doi: 10.3934/cpaa.2009.8.2037. |
[16] |
V. Hutson and M. Grinfeld, Non-local dispersal and bistability,, Euro. J. Appl. Math., 17 (2006), 221.
doi: 10.1017/S0956792506006462. |
[17] |
V. Hutson, S. Martinez, K. Mischailow and G. T. Vickers, The evolution of dispersal,, J. Math. Biol., 47 (2003), 483.
doi: 10.1007/s00285-003-0210-1. |
[18] |
V. Hutson, W. Shen and G. T. Vickers, Spectral theory for nonlocal dispersal with periodic or almost-periodic time dependence,, Rocky Mountain J. Math., 38 (2008), 1147.
doi: 10.1216/RMJ-2008-38-4-1147. |
[19] |
Y. Hosono and B. Ilyas, Travelling waves for a simple diffusive epidemic model,, Math. Model Meth. Appl. Sci., 5 (1995), 935.
doi: 10.1142/S0218202595000504. |
[20] |
C. Y. Kao, Y. Lou and W. Shen, Random dispersal vs non-local dispersal,, Discrete Contin. Dyn. Syst., 26 (2010), 551.
doi: 10.3934/dcds.2010.26.551. |
[21] |
C.-Y. Kao, Y. Lou and W. Shen, Evolution of mixed dispersal in periodic environments,, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2047.
doi: 10.3934/dcdsb.2012.17.2047. |
[22] |
W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics,, Proc. Roy. Soc. London Ser. B, 115 (1927), 700.
doi: 10.1098/rspa.1927.0118. |
[23] |
W.-T. Li, G. Lin and S. Ruan, Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems,, Nonlinerity, 19 (2006), 1253.
doi: 10.1088/0951-7715/19/6/003. |
[24] |
W.-T. Li, Y.-J. Sun and Z.-C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal,, Nonlinear Anal. Real Word Appl., 11 (2010), 2302.
doi: 10.1016/j.nonrwa.2009.07.005. |
[25] |
G. Lin, W.-T. Li and M.-J. Ma, Traveling wave solutions in delayed reaction diffusion systems with applications to multi-species models,, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 393.
doi: 10.3934/dcdsb.2010.13.393. |
[26] |
J. D. Murray, "Mathematical Biology, II, Spatial Models and Biomedical Applications,", Third edition. Interdisciplinary Applied Mathematics, 18 (2003).
|
[27] |
S. Ma, Travelling wavefronts for delayed reaction-diffusion systems via a fixed point theorem,, J. Differential Equations, 171 (2001), 294.
doi: 10.1006/jdeq.2000.3846. |
[28] |
J. Medlock and M. Kot, Spreading disease: Integro-differential equations old and new,, Math. Biosci., 184 (2003), 201.
doi: 10.1016/S0025-5564(03)00041-5. |
[29] |
S. Pan, Traveling wave fronts of delayed non-local diffusion systems without quasimonotonicity,, J. Math. Anal. Appl., 346 (2008), 415.
doi: 10.1016/j.jmaa.2008.05.057. |
[30] |
S. Pan, W.-T. Li and G. Lin, Travelling wave fronts in nonlocal reaction-diffusion systems and applications,, Z. Angew. Math. Phys., 60 (2009), 377.
doi: 10.1007/s00033-007-7005-y. |
[31] |
S. Pan, W.-T. Li and G. Lin, Existence and stability of traveling wavefronts in a nonlocal diffusion equation with delay,, Nonlinear Anal., 72 (2010), 3150.
doi: 10.1016/j.na.2009.12.008. |
[32] |
K. Schumacher, Travelling-front solutions for integro-differential equations.I.,, J. Reine Angew. Math., 316 (1980), 54.
doi: 10.1515/crll.1980.316.54. |
[33] |
W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats,, J. Differential Equations, 15 (2010), 747.
doi: 10.1016/j.jde.2010.04.012. |
[34] |
Y.-J. Sun, W.-T. Li and Z.-C. Wang, Entire solutions in nonlocal dispersal equations with bistable nonlinearity,, J. Differential Equations, 251 (2011), 551.
doi: 10.1016/j.jde.2011.04.020. |
[35] |
Y.-J. Sun, W.-T. Li and Z.-C. Wang, Traveling waves for a nonlocal anisotropic dispersal equation with monostable nonlinearity,, Nonlinear Anal., 74 (2011), 814.
doi: 10.1016/j.na.2010.09.032. |
[36] |
X. Wang, Metastability and stability of patterns in a convolution model for phase transitions,, J. Differential Equations, 183 (2002), 434.
doi: 10.1006/jdeq.2001.4129. |
[37] |
Z.-C. Wang, W.-T. Li and S. Ruan, Traveling fronts in monostable equations with nonlocal delayed effects,, J. Dynam. Diff. Eqns., 20 (2008), 573.
doi: 10.1007/s10884-008-9103-8. |
[38] |
Z.-C. Wang, W.-T. Li and S. Ruan, Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity,, Trans. Amer. Math. Soc., 361 (2009), 2047.
doi: 10.1090/S0002-9947-08-04694-1. |
[39] |
Z.-C. Wang and J. Wu, Travelling waves of a diffusive Kermack-McKendrick epidemic model with non-local delayed transmission,, Proc. R. Soc. Lond. Ser. A, 466 (2010), 237.
doi: 10.1098/rspa.2009.0377. |
[40] |
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. |
[41] |
D. V. Widder, "The Laplace Transform,", Princeton University Press, (1941).
|
[42] |
H. Yagisita, Existence and nonexistence of traveling waves for a nonlocal monostable equation,, Publ. Res. Inst. Math. Sci., 45 (2009), 925.
doi: 10.2977/prims/1260476648. |
[43] |
H. Yagisita, Existence of traveling waves for a nonlocal monostable equation: an abstract approach,, Publ. Res. Inst. Math. Sci., 45 (2009), 955.
doi: 10.2977/prims/1260476649. |
[44] |
L. Zhang, Existence, uniqueness and exponential stability of traveling wave solutions of some integral differential equations arising from neural networks,, J. Differential Equations, 197 (2004), 162.
doi: 10.1016/S0022-0396(03)00170-0. |
[45] |
G.-B. Zhang, W.-T. Li and Z.-C. Wang, Spreading speeds and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity,, J. Differential Equations, 252 (2012), 5096.
doi: 10.1016/j.jde.2012.01.014. |
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