# American Institute of Mathematical Sciences

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 and Continuous Dynamical Systems, 2017, 37 (5) : 2483-2512. doi: 10.3934/dcds.2017107
##### References:
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Pan, W. 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. [24] Y. J. Sun, L. Zhang, W. T. Li and Z. C. Wang, Entire solutions in nonlocal monostable equations: asymmetric case (2015), submitted. [25] 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), 814-826.  doi: 10.1016/j.na.2010.09.032. [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. [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. [28] 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), 2047-2084.  doi: 10.1090/S0002-9947-08-04694-1. [29] Z. C. Wang, W. 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. [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. [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. [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. [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. [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. [35] L. Zhang, W. T. Li, Z. C. Wang and Y. J. Sun, Entire solutions in nonlocal bistable equations: asymmetric case (2016), submitted. [36] L. Zhang, W. 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. [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.

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##### 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. [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. [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. [4] X. Chen, J. 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. [5] J. Coville, J. 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. [6] J. Coville, Travelling fronts in asymmetric nonlocal reaction diffusion equations: the bistable and ignition cases Prépublication du CMM Hal-00696208. [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. [8] W. Ellison and F. Ellison, Prime Numbers, A Wiley-Interscience Publication, John Wiley & Sons, New York; Hermann, Paris, 1985. [9] Y. Fukao, Y. Morita and H. Ninomiya, Some entire solutions of the Allen-Cahn equation, Taiwanese J. Math., 8 (2004), 15-32. [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. [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. [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. [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. [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. [15] W. T. Li, Y. 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. [16] W. T. Li, Z. 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. [17] W. T. Li, L. 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. [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. [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. [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. [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. [22] J. D. Murray, Mathematical Biology, Springer, Berlin-Heidelberg-New York, 1993. [23] S. Pan, W. 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. [24] Y. J. Sun, L. Zhang, W. T. Li and Z. C. Wang, Entire solutions in nonlocal monostable equations: asymmetric case (2015), submitted. [25] 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), 814-826.  doi: 10.1016/j.na.2010.09.032. [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. [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. [28] 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), 2047-2084.  doi: 10.1090/S0002-9947-08-04694-1. [29] Z. C. Wang, W. 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. [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. [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. [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. [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. [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. [35] L. Zhang, W. T. Li, Z. C. Wang and Y. J. Sun, Entire solutions in nonlocal bistable equations: asymmetric case (2016), submitted. [36] L. Zhang, W. 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. [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.
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