May  2017, 37(5): 2483-2512. doi: 10.3934/dcds.2017107

Traveling waves and entire solutions for an epidemic model with asymmetric dispersal

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

Corresponding author

Received  June 2016 Revised  December 2016 Published  February 2017

This paper is concerned with traveling waves and entire solutions of one epidemic model with asymmetric dispersal kernel function arising from the spread of an epidemic by oral-faecal transmission. The asymmetry of the kernel function will have an influence on two aspects: (ⅰ) the minimal wave speed of traveling wave fronts may be nonpositive, but we give a new restrictive condition on the kernel function to guarantee it is positive; (ⅱ) the two traveling wave solutions with the same speed spreading from right and left of $x$-axis may be different in shape, which further makes that the entire solutions with five or four parameters may be asymmetric and the entire solutions with three parameters increasing in $x$ may be different from those decreasing in $x$ in shape. As for traveling wave solutions, we get the existence, asymptotic behavior and uniqueness of the two traveling wave solutions spreading from right and left of $x$-axis, respectively. We further construct three new entire solutions with five, four or three parameters. Two comparison principles also be established.

Citation: Wan-Tong Li, Wen-Bing Xu, Li Zhang. Traveling waves and entire solutions for an epidemic model with asymmetric dispersal. Discrete & Continuous Dynamical Systems - A, 2017, 37 (5) : 2483-2512. doi: 10.3934/dcds.2017107
References:
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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: 10.1088/0951-7715/26/10/2925.  Google Scholar

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Y. J. Sun, L. Zhang, W. T. Li and Z. C. Wang, Entire solutions in nonlocal monostable equations: asymmetric case (2015), submitted. Google Scholar

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Y. J. SunW. T. Li and Z. C. Wang, Traveling waves for a nonlocal anisotropic dispersal equation with monostable nonlinearity, Nonlinear Anal., 74 (2011), 814-826.  doi: 10.1016/j.na.2010.09.032.  Google Scholar

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A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, Vol. 140, Amer. Math. Soc. , Providence Rhode Island, 1994.  Google Scholar

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Z. C. WangW. T. Li and J. Wu, Entire solutions in delayed lattice differential equations with monostable nonlinearity, SIAM J. Math. Anal., 40 (2009), 2392-2420.  doi: 10.1137/080727312.  Google Scholar

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S. L. Wu and C. H. Hsu, Entire solutions with merging fronts to a bistable periodic lattice dynamical system, Discrete Contin. Dyn. Syst., 36 (2016), 2329-2346.  doi: 10.3934/dcds.2016.36.2329.  Google Scholar

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[32]

S. L. Wu and H. Wang, Front-like entire solutions for monostable reaction-diffusion systems, J. Dynam. Differential Equations, 25 (2013), 505-533.  doi: 10.1007/s10884-013-9293-6.  Google Scholar

[33]

D. Xu and X. Q. Zhao, Erratum to "Bistable waves in an epidemic model", J. Dynam. Differential Equations, 17 (2005), 219-247.  doi: 10.1007/s10884-005-6294-0.  Google Scholar

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H. Yagisita, Backward global solutions characterizing annihilation dynamics of travelling fronts, Publ. Res. Inst. Math. Sci., 39 (2003), 117-164.  doi: 10.2977/prims/1145476150.  Google Scholar

[35]

L. Zhang, W. T. Li, Z. C. Wang and Y. J. Sun, Entire solutions in nonlocal bistable equations: asymmetric case (2016), submitted. Google Scholar

[36]

L. ZhangW. T. Li and S. L. Wu, Multi-type entire solutions in a nonlocal dispersal epidemic model, J. Dynam. Differential Equations, 28 (2016), 189-224.  doi: 10.1007/s10884-014-9416-8.  Google Scholar

[37]

X. Q. Zhao and W. Wang, Fisher waves in an epidemic model, Discrete Contin. Dyn. Syst. B, 4 (2004), 1117-1128.  doi: 10.3934/dcdsb.2004.4.1117.  Google Scholar

show all references

References:
[1]

V. Capasso and S. L. Paveri-Fontana, A mathematical model for the 1973 cholera epidemic in the European Mediterranean region, Rev. d'Epidemiol. Santé Publique, 27 (1979), 32-121.   Google Scholar

[2]

J. Carr and A. Chmaj, Uniqueness of travelling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439.  doi: 10.1090/S0002-9939-04-07432-5.  Google Scholar

[3]

X. Chen and J. S. Guo, Existence and uniqueness of entire solutions for a reaction-diffusion equation, J. Differential Equations, 212 (2005), 62-84.  doi: 10.1016/j.jde.2004.10.028.  Google Scholar

[4]

X. ChenJ. S. Guo and H. Ninomiya, Entire solutions of reaction-diffusion equations with balanced bistable nonlinearities, Proc. Roy. Soc. Edinburgh Sect. A, 136 (2006), 1207-1237.  doi: 10.1017/S0308210500004959.  Google Scholar

[5]

J. CovilleJ. Dávila and S. Martínez, Nonlocal anisotropic dispersal with monostable nonlinearity, J. Differential Equations, 244 (2008), 3080-3118.  doi: 10.1016/j.jde.2007.11.002.  Google Scholar

[6]

J. Coville, Travelling fronts in asymmetric nonlocal reaction diffusion equations: the bistable and ignition cases Prépublication du CMM Hal-00696208. Google Scholar

[7]

J. Coville and L. Dupaigne, On a non-local equation arising in population dynamics., Proc. Roy. Soc. Edinburgh. Sect. A, 137 (2007), 727-755.  doi: 10.1017/S0308210504000721.  Google Scholar

[8] W. Ellison and F. Ellison, Prime Numbers, A Wiley-Interscience Publication, John Wiley & Sons, New York; Hermann, Paris, 1985.   Google Scholar
[9]

Y. FukaoY. Morita and H. Ninomiya, Some entire solutions of the Allen-Cahn equation, Taiwanese J. Math., 8 (2004), 15-32.   Google Scholar

[10]

J. S. Guo and Y. Morita, Entire solutions of reaction-diffusion equations and an application to discrete diffusive equations, Discrete Contin. Dyn. Syst., 12 (2005), 193-212.  doi: 10.3934/dcds.2005.12.193.  Google Scholar

[11]

J. S. Guo and C. H. Wu, Entire solutions for a two-component competition system in a lattice, Tohoku. Math. J., 62 (2010), 17-28.  doi: 10.2748/tmj/1270041024.  Google Scholar

[12]

F. Hamel and N. Nadirashvili, Entire solutions of the KPP equation, Comm. Pure Appl. Math., 52 (1999), 1255-1276.  doi: 10.1002/(SICI)1097-0312(199910)52:10<1255::AID-CPA4>3.0.CO;2-W.  Google Scholar

[13]

F. Hamel and N. Nadirashvili, Travelling fronts and entire solutions of the Fisher-KPP equation in $\mathbb{R}^{N}$, Arch. Ration. Mech. Anal., 157 (2001), 91-163.  doi: 10.1007/PL00004238.  Google Scholar

[14]

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: 10.1088/0951-7715/26/10/2925.  Google Scholar

[15]

W. T. LiY. J Sun and Z. C. Wang, Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Anal. Real World Appl., 11 (2010), 2302-2313.  doi: 10.1016/j.nonrwa.2009.07.005.  Google Scholar

[16]

W. T. LiZ. C. Wang and J. Wu, Entire solutions in monostable reaction-diffusion equations with delayed nonlinearity, J. Differential Equations, 245 (2008), 102-129.  doi: 10.1016/j.jde.2008.03.023.  Google Scholar

[17]

W. T. LiL. Zhang and G. B. Zhang, Invasion entire solutions in a competition system with nonlocal dispersal, Discrete Contin. Dyn. Syst., 35 (2015), 1531-1560.  doi: 10.3934/dcds.2015.35.1531.  Google Scholar

[18]

G. Lv, Asymptotic behavior of traveling fronts and entire solutions for a nonlocal monostable equation, Nonlinear Anal., 72 (2010), 3659-3668.  doi: 10.1016/j.na.2009.12.047.  Google Scholar

[19]

R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems, Trans. Amer. Math. Soc., 321 (1990), 1-44.  doi: 10.2307/2001590.  Google Scholar

[20]

Y. Morita and H. Ninomiya, Entire solutions with merging fronts to reaction-diffusion equations, J. Dynam. Differential Equations, 18 (2006), 841-861.  doi: 10.1007/s10884-006-9046-x.  Google Scholar

[21]

Y. Morita and K. Tachibana, An entire solution to the Lotka-Volterra competition-diffusion equations, SIAM J. Math. Anal., 40 (2009), 2217-2240.  doi: 10.1137/080723715.  Google Scholar

[22] J. D. Murray, Mathematical Biology, Springer, Berlin-Heidelberg-New York, 1993.   Google Scholar
[23]

S. PanW. T. Li and G. Lin, Travelling wave fronts in nonlocal delayed reaction-diffusion systems and applications, Z. Angew. Math. Phys., 60 (2009), 377-392.  doi: 10.1007/s00033-007-7005-y.  Google Scholar

[24]

Y. J. Sun, L. Zhang, W. T. Li and Z. C. Wang, Entire solutions in nonlocal monostable equations: asymmetric case (2015), submitted. Google Scholar

[25]

Y. J. SunW. T. Li and Z. C. Wang, Traveling waves for a nonlocal anisotropic dispersal equation with monostable nonlinearity, Nonlinear Anal., 74 (2011), 814-826.  doi: 10.1016/j.na.2010.09.032.  Google Scholar

[26]

A. I. Volpert, V. A. Volpert and V. A. Volpert, Traveling Wave Solutions of Parabolic Systems, Translations of Mathematical Monographs, Vol. 140, Amer. Math. Soc. , Providence Rhode Island, 1994.  Google Scholar

[27]

M. Wang and G. Lv, Entire solutions of a diffusive and competitive Lotka-Volterra type system with nonlocal delay, Nonlinearity, 23 (2010), 1609-1630.  doi: 10.1088/0951-7715/23/7/005.  Google Scholar

[28]

Z. C. WangW. T. Li and S. Ruan, Entire solutions in bistable reaction-diffusion equations with nonlocal delayed nonlinearity, Trans. Amer. Math. Soc., 361 (2009), 2047-2084.  doi: 10.1090/S0002-9947-08-04694-1.  Google Scholar

[29]

Z. C. WangW. T. Li and J. Wu, Entire solutions in delayed lattice differential equations with monostable nonlinearity, SIAM J. Math. Anal., 40 (2009), 2392-2420.  doi: 10.1137/080727312.  Google Scholar

[30]

S. L. Wu and C. H. Hsu, Entire solutions with merging fronts to a bistable periodic lattice dynamical system, Discrete Contin. Dyn. Syst., 36 (2016), 2329-2346.  doi: 10.3934/dcds.2016.36.2329.  Google Scholar

[31]

S. L. Wu and C. H. Hsu, Existence of entire solutions for delayed monostable epidemic models, Trans. Amer. Math. Soc., 368 (2016), 6033-6062.  doi: 10.1090/tran/6526.  Google Scholar

[32]

S. L. Wu and H. Wang, Front-like entire solutions for monostable reaction-diffusion systems, J. Dynam. Differential Equations, 25 (2013), 505-533.  doi: 10.1007/s10884-013-9293-6.  Google Scholar

[33]

D. Xu and X. Q. Zhao, Erratum to "Bistable waves in an epidemic model", J. Dynam. Differential Equations, 17 (2005), 219-247.  doi: 10.1007/s10884-005-6294-0.  Google Scholar

[34]

H. Yagisita, Backward global solutions characterizing annihilation dynamics of travelling fronts, Publ. Res. Inst. Math. Sci., 39 (2003), 117-164.  doi: 10.2977/prims/1145476150.  Google Scholar

[35]

L. Zhang, W. T. Li, Z. C. Wang and Y. J. Sun, Entire solutions in nonlocal bistable equations: asymmetric case (2016), submitted. Google Scholar

[36]

L. ZhangW. T. Li and S. L. Wu, Multi-type entire solutions in a nonlocal dispersal epidemic model, J. Dynam. Differential Equations, 28 (2016), 189-224.  doi: 10.1007/s10884-014-9416-8.  Google Scholar

[37]

X. Q. Zhao and W. Wang, Fisher waves in an epidemic model, Discrete Contin. Dyn. Syst. B, 4 (2004), 1117-1128.  doi: 10.3934/dcdsb.2004.4.1117.  Google Scholar

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