American Institute of Mathematical Sciences

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September  2013, 18(7): 1969-1993. doi: 10.3934/dcdsb.2013.18.1969

Traveling waves in a nonlocal dispersal Kermack-McKendrick epidemic model

Fei-Ying Yang 1, , Yan Li 1, , Wan-Tong Li 2, and  Zhi-Cheng Wang 1,

1. 

School of Mathematics and Statistics, Lanzhou University, Lanzhou, Gansu 730000, China, China
2. 

School of Mathematic and Statistics, Lanzhou University, Lanzhou, Gansu 730000

Received  March 2012 Revised  March 2013 Published  May 2013

In this paper, we consider a Kermack-McKendrick epidemic model with nonlocal dispersal. We find that the existence and nonexistence of traveling wave solutions are determined by the reproduction number. To prove the existence of nontrivial traveling wave solutions, we construct an invariant cone in a bounded domain with initial functions being defined on, and apply Schauder's fixed point theorem as well as limiting argument. Here, the compactness of the support set of dispersal kernel is needed when passing to an unbounded domain in the proof. Moreover, the nonexistence of traveling wave solutions is obtained by Laplace transform if the speed is less than the critical velocity.
Keywords: Kermack-McKendrick, traveling waves, nonlocal dispersal, Schauder's fixed point theorem, Laplace transform..
Mathematics Subject Classification: 35K57, 37C65, 92D3.
Citation: Fei-Ying Yang, Yan Li, Wan-Tong Li, Zhi-Cheng Wang. Traveling waves in a nonlocal dispersal Kermack-McKendrick epidemic model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (7) : 1969-1993. doi: 10.3934/dcdsb.2013.18.1969
References:
[1]

F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. Toledo-Melero, "Nonlocal Diffusion Problems,", Mathematical Surveys and Monographs, (2010).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

X. Chen, Existence, uniqueness and asymptotic stability of travelling waves in non-local evolution equations,, Adv. Differential Equations, 2 (1997), 125.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

P. Fife, An integrodifferential analog of semilinear parabolic PDEs,, in, 177 (1996), 137.   Google Scholar

[13]

P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions,, in, (2003), 153.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

V. Hutson and M. Grinfeld, Non-local dispersal and bistability,, Euro. J. Appl. Math., 17 (2006), 221.  doi: 10.1017/S0956792506006462.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

J. D. Murray, "Mathematical Biology, II, Spatial Models and Biomedical Applications,", Third edition. Interdisciplinary Applied Mathematics, 18 (2003).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

D. V. Widder, "The Laplace Transform,", Princeton University Press, (1941).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[5]

X. Chen, Existence, uniqueness and asymptotic stability of travelling waves in non-local evolution equations,, Adv. Differential Equations, 2 (1997), 125.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[12]

P. Fife, An integrodifferential analog of semilinear parabolic PDEs,, in, 177 (1996), 137.   Google Scholar

[13]

P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions,, in, (2003), 153.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

V. Hutson and M. Grinfeld, Non-local dispersal and bistability,, Euro. J. Appl. Math., 17 (2006), 221.  doi: 10.1017/S0956792506006462.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

J. D. Murray, "Mathematical Biology, II, Spatial Models and Biomedical Applications,", Third edition. Interdisciplinary Applied Mathematics, 18 (2003).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[41]

D. V. Widder, "The Laplace Transform,", Princeton University Press, (1941).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

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