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June  2017, 22(4): 1719-1741. doi: 10.3934/dcdsb.2017082

Traveling waves for a nonlocal dispersal SIR model with general nonlinear incidence rate and spatio-temporal delay

School of Science and Technology, Zhejiang International Studies University, Hangzhou 310012, China

* Corresponding author: yangyyj@126.com

Received  December 2015 Revised  November 2016 Published  February 2017

Fund Project: Supported by the National Natural Science Fund of China (Nos. 11501518, 11501519 and 11626219)

In this paper, we study a nonlocal dispersal SIR model with general nonlinear incidence rate and spatio-temporal delay. By Schauder's fixed point theorem and Laplace transform, we show that the existence and nonexistence of traveling wave solutions are determined by the basic reproduction number and the minimal wave speed. Some examples are listed to illustrate the theoretical results. Our results generalize some known results.

Citation: Jinling Zhou, Yu Yang. Traveling waves for a nonlocal dispersal SIR model with general nonlinear incidence rate and spatio-temporal delay. Discrete & Continuous Dynamical Systems - B, 2017, 22 (4) : 1719-1741. doi: 10.3934/dcdsb.2017082
References:
[1]

M. E. Alexander and S. M. Moghadas, Periodicity in an epidemic model with a generalized non-linear incidence, Math. Biosci., 189 (2004), 75-96.  doi: 10.1016/j.mbs.2004.01.003.  Google Scholar

[2]

J. F. M. Al-Omari and S. A. Gourley, Monotone wave-fronts in a structured population model with dsitributed maturation delay, IMA J. Appl. Math., 70 (2005), 858-879.  doi: 10.1093/imamat/hxh073.  Google Scholar

[3]

Z. Bai and S. Wu, Traveling waves in a delayed SIR epidemic model with nonlinear incidence, Appl. Math. Comput., 263 (2015), 221-232.  doi: 10.1016/j.amc.2015.04.048.  Google Scholar

[4]

Z. Bai and S. Zhang, Traveling waves of a diffusive SIR epidemic model with a class of nonlinear incidence rates and distributed delay, Commun.Nonlinear Sci. Numer. Simulat., 22 (2015), 1370-1381.  doi: 10.1016/j.cnsns.2014.07.005.  Google Scholar

[5]

A. Boumenir and V. Nguyen, Perron theorem in the monotone iteration method for traveling waves in delayed reaction-diffusion equations, J. Differential Equations, 244 (2008), 1551-1570.  doi: 10.1016/j.jde.2008.01.004.  Google Scholar

[6]

N. F. Britton, Aggregation and the competitive exclusion principle, J. Theor. Biol., 136 (1989), 57-66.  doi: 10.1016/S0022-5193(89)80189-4.  Google Scholar

[7]

N. F. Britton, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688.  doi: 10.1137/0150099.  Google Scholar

[8]

V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43-61.  doi: 10.1016/0025-5564(78)90006-8.  Google Scholar

[9]

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

[10]

H. Cheng and R. Yuan, Traveling wave solutions for a nonlocal dispersal Kermack-McKendrick epidemic model with spatio-temporal delay (in Chinese), Sci. China Math., 45 (2015), 765-788.   Google Scholar

[11]

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

[12]

G. B. Ermentrout and J. B. McLeod, Existence and uniqueness of travelling waves for a neural network, Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 461-478.  doi: 10.1017/S030821050002583X.  Google Scholar

[13]

Z. Feng and H. R. Thieme, Endemic models with arbitrarily distributed periods of infection Ⅰ: fundamental properties of the model, SIAM J. Appl. Math., 61 (2000), 803-833.  doi: 10.1137/S0036139998347834.  Google Scholar

[14]

Z. Feng and H. R. Thieme, Endemic models with arbitrarily distributed periods of infection Ⅱ: fast disease dynamics and permanent recovery, SIAM J. Appl. Math., 61 (2000), 983-1012.  doi: 10.1137/S0036139998347846.  Google Scholar

[15]

P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153-191.  Google Scholar

[16]

S. A. Gourley, Travelling front solutions of a nonlocal Fisher equation, J. Math. Biol., 41 (2000), 272-284.  doi: 10.1007/s002850000047.  Google Scholar

[17]

S. A. Gourley and S. Ruan, Convergence and traveling fronts in functional differential equations with nonlocal terms: a competition model, SIAM J. Math. Anal., 35 (2003), 806-822.  doi: 10.1137/S003614100139991.  Google Scholar

[18]

S. Hsu and X. Zhao, Spreading speeds and traveling waves for non-monotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789.  doi: 10.1137/070703016.  Google Scholar

[19]

Z. HuP. BiW. Ma and S. Ruan, Bifurcations of an SIRS epidemic model with nonlinear incidence rate, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 93-112.  doi: 10.3934/dcdsb.2011.15.93.  Google Scholar

[20]

G. HuangY. TakeuchiW. Ma and D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 1192-1207.  doi: 10.1007/s11538-009-9487-6.  Google Scholar

[21]

G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence, J. Math. Biol., 63 (2011), 125-139.  doi: 10.1007/s00285-010-0368-2.  Google Scholar

[22]

V. HutsonS. MartinezK. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.  doi: 10.1007/s00285-003-0210-1.  Google Scholar

[23]

A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886.  doi: 10.1007/s11538-007-9196-y.  Google Scholar

[24]

W. LiG. 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.  Google Scholar

[25]

W. Li and F. Yang, Traveling waves for a nonlocal dispersal SIR model with standard incidence, J. Integral Equations Appl., 26 (2014), 243-273.  doi: 10.1216/JIE-2014-26-2-243.  Google Scholar

[26]

D. Liang and J. Wu, Travelling waves and numerical approximations in a reaction advection diffusion equation with nonlocal delayed effects, J. Nonlinear Sci., 13 (2003), 289-310.  doi: 10.1007/s00332-003-0524-6.  Google Scholar

[27]

W. LiuS. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204.  doi: 10.1007/BF00276956.  Google Scholar

[28]

S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations, 171 (2001), 294-314.  doi: 10.1006/jdeq.2000.3846.  Google Scholar

[29]

S. Pan, Traveling wave fronts of delayed non-local diffusion systems without quasimonotonicity, J. Math. Anal. Appl., 346 (2008), 415-424.  doi: 10.1016/j.jmaa.2008.05.057.  Google Scholar

[30]

K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615.  doi: 10.2307/2000859.  Google Scholar

[31]

Y. TangD. HuangS. Ruan and W. Zhang, Coexistence of limit cycles and homoclinic loops in a SIRS model with a nonlinear incidence rate, SIAM J. Appl. Math., 69 (2008), 621-639.  doi: 10.1137/070700966.  Google Scholar

[32]

H. R. Thieme and X. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations, 195 (2003), 430-470.  doi: 10.1016/S0022-0396(03)00175-X.  Google Scholar

[33]

Z. WangW. Li and S. Ruan, Travelling wave fronts in reaction-diffusion systems with spatio-temporal delays, J. Differential Equations, 222 (2006), 185-232.  doi: 10.1016/j.jde.2005.08.010.  Google Scholar

[34]

J. WangW. Li and F. Yang, Traveling waves in a nonlocal dispersal SIR model with nonlocal delayed transmission, Commun. Nonlinear Sci. Numer. Simulat., 27 (2015), 136-152.  doi: 10.1016/j.cnsns.2015.03.005.  Google Scholar

[35]

Z. 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), 237-261.  doi: 10.1098/rspa.2009.0377.  Google Scholar

[36]

S. WuC. H. Hsu and Y. Xiao, Global attractivity, spreading speeds and traveling waves of delayed nonlocal reaction-diffusion systems, J. Differential Equations, 258 (2015), 1058-1105.  doi: 10.1016/j.jde.2014.10.009.  Google Scholar

[37]

D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 208 (2007), 419-429.  doi: 10.1016/j.mbs.2006.09.025.  Google Scholar

[38]

Z. Xu and D. Xiao, Minimal wave speed and uniqueness of traveling waves for a nonlocal diffusion population model with spatio-temporal delays, Differ. Integral Equ., 27 (2014), 1073-1106.   Google Scholar

[39]

H. Yagisita, Existence and nonexistence of traveling waves for a nonlocal monostable equation, Publ. Res. Inst. Math. Sci., 45 (2009), 925-953.  doi: 10.2977/prims/1260476648.  Google Scholar

[40]

F. YangY. LiW. Li and Z. Wang, Traveling waves in a nonlocal dispersal Kermack-McKendrick epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1969-1993.  doi: 10.3934/dcdsb.2013.18.1969.  Google Scholar

[41]

Y. Yang and D. Xiao, Influence of latent period and nonlinear incidence rate on the dynamics of SIRS epidemiological models, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 195-211.  doi: 10.3934/dcdsb.2010.13.195.  Google Scholar

[42]

Z. Yu and R. Yuan, Traveling wave fronts in reaction-diffusion systems with spatio-temporal delay and applications, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 709-728.  doi: 10.3934/dcdsb.2010.13.709.  Google Scholar

[43]

G. ZhangW. Li and Z. Wang, Spreading speeds and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity, J. Differential Equations, 252 (2012), 5096-5124.  doi: 10.1016/j.jde.2012.01.014.  Google Scholar

[44]

X. Zou and J. Wu, Existence of traveling wave fronts in delayed reaction-diffusion systems via the monotone iteration method, Proc. Amer. Math. Soc., 125 (1997), 2589-2598.  doi: 10.1090/S0002-9939-97-04080-X.  Google Scholar

show all references

References:
[1]

M. E. Alexander and S. M. Moghadas, Periodicity in an epidemic model with a generalized non-linear incidence, Math. Biosci., 189 (2004), 75-96.  doi: 10.1016/j.mbs.2004.01.003.  Google Scholar

[2]

J. F. M. Al-Omari and S. A. Gourley, Monotone wave-fronts in a structured population model with dsitributed maturation delay, IMA J. Appl. Math., 70 (2005), 858-879.  doi: 10.1093/imamat/hxh073.  Google Scholar

[3]

Z. Bai and S. Wu, Traveling waves in a delayed SIR epidemic model with nonlinear incidence, Appl. Math. Comput., 263 (2015), 221-232.  doi: 10.1016/j.amc.2015.04.048.  Google Scholar

[4]

Z. Bai and S. Zhang, Traveling waves of a diffusive SIR epidemic model with a class of nonlinear incidence rates and distributed delay, Commun.Nonlinear Sci. Numer. Simulat., 22 (2015), 1370-1381.  doi: 10.1016/j.cnsns.2014.07.005.  Google Scholar

[5]

A. Boumenir and V. Nguyen, Perron theorem in the monotone iteration method for traveling waves in delayed reaction-diffusion equations, J. Differential Equations, 244 (2008), 1551-1570.  doi: 10.1016/j.jde.2008.01.004.  Google Scholar

[6]

N. F. Britton, Aggregation and the competitive exclusion principle, J. Theor. Biol., 136 (1989), 57-66.  doi: 10.1016/S0022-5193(89)80189-4.  Google Scholar

[7]

N. F. Britton, Spatial structures and periodic travelling waves in an integro-differential reaction-diffusion population model, SIAM J. Appl. Math., 50 (1990), 1663-1688.  doi: 10.1137/0150099.  Google Scholar

[8]

V. Capasso and G. Serio, A generalization of the Kermack-McKendrick deterministic epidemic model, Math. Biosci., 42 (1978), 43-61.  doi: 10.1016/0025-5564(78)90006-8.  Google Scholar

[9]

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

[10]

H. Cheng and R. Yuan, Traveling wave solutions for a nonlocal dispersal Kermack-McKendrick epidemic model with spatio-temporal delay (in Chinese), Sci. China Math., 45 (2015), 765-788.   Google Scholar

[11]

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

[12]

G. B. Ermentrout and J. B. McLeod, Existence and uniqueness of travelling waves for a neural network, Proc. Roy. Soc. Edinburgh Sect. A, 123 (1993), 461-478.  doi: 10.1017/S030821050002583X.  Google Scholar

[13]

Z. Feng and H. R. Thieme, Endemic models with arbitrarily distributed periods of infection Ⅰ: fundamental properties of the model, SIAM J. Appl. Math., 61 (2000), 803-833.  doi: 10.1137/S0036139998347834.  Google Scholar

[14]

Z. Feng and H. R. Thieme, Endemic models with arbitrarily distributed periods of infection Ⅱ: fast disease dynamics and permanent recovery, SIAM J. Appl. Math., 61 (2000), 983-1012.  doi: 10.1137/S0036139998347846.  Google Scholar

[15]

P. Fife, Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in Nonlinear Analysis, Springer, Berlin, (2003), 153-191.  Google Scholar

[16]

S. A. Gourley, Travelling front solutions of a nonlocal Fisher equation, J. Math. Biol., 41 (2000), 272-284.  doi: 10.1007/s002850000047.  Google Scholar

[17]

S. A. Gourley and S. Ruan, Convergence and traveling fronts in functional differential equations with nonlocal terms: a competition model, SIAM J. Math. Anal., 35 (2003), 806-822.  doi: 10.1137/S003614100139991.  Google Scholar

[18]

S. Hsu and X. Zhao, Spreading speeds and traveling waves for non-monotone integrodifference equations, SIAM J. Math. Anal., 40 (2008), 776-789.  doi: 10.1137/070703016.  Google Scholar

[19]

Z. HuP. BiW. Ma and S. Ruan, Bifurcations of an SIRS epidemic model with nonlinear incidence rate, Discrete Contin. Dyn. Syst. Ser. B, 15 (2011), 93-112.  doi: 10.3934/dcdsb.2011.15.93.  Google Scholar

[20]

G. HuangY. TakeuchiW. Ma and D. Wei, Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate, Bull. Math. Biol., 72 (2010), 1192-1207.  doi: 10.1007/s11538-009-9487-6.  Google Scholar

[21]

G. Huang and Y. Takeuchi, Global analysis on delay epidemiological dynamic models with nonlinear incidence, J. Math. Biol., 63 (2011), 125-139.  doi: 10.1007/s00285-010-0368-2.  Google Scholar

[22]

V. HutsonS. MartinezK. Mischaikow and G. T. Vickers, The evolution of dispersal, J. Math. Biol., 47 (2003), 483-517.  doi: 10.1007/s00285-003-0210-1.  Google Scholar

[23]

A. Korobeinikov, Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69 (2007), 1871-1886.  doi: 10.1007/s11538-007-9196-y.  Google Scholar

[24]

W. LiG. 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.  Google Scholar

[25]

W. Li and F. Yang, Traveling waves for a nonlocal dispersal SIR model with standard incidence, J. Integral Equations Appl., 26 (2014), 243-273.  doi: 10.1216/JIE-2014-26-2-243.  Google Scholar

[26]

D. Liang and J. Wu, Travelling waves and numerical approximations in a reaction advection diffusion equation with nonlocal delayed effects, J. Nonlinear Sci., 13 (2003), 289-310.  doi: 10.1007/s00332-003-0524-6.  Google Scholar

[27]

W. LiuS. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204.  doi: 10.1007/BF00276956.  Google Scholar

[28]

S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem, J. Differential Equations, 171 (2001), 294-314.  doi: 10.1006/jdeq.2000.3846.  Google Scholar

[29]

S. Pan, Traveling wave fronts of delayed non-local diffusion systems without quasimonotonicity, J. Math. Anal. Appl., 346 (2008), 415-424.  doi: 10.1016/j.jmaa.2008.05.057.  Google Scholar

[30]

K. W. Schaaf, Asymptotic behavior and traveling wave solutions for parabolic functional differential equations, Trans. Amer. Math. Soc., 302 (1987), 587-615.  doi: 10.2307/2000859.  Google Scholar

[31]

Y. TangD. HuangS. Ruan and W. Zhang, Coexistence of limit cycles and homoclinic loops in a SIRS model with a nonlinear incidence rate, SIAM J. Appl. Math., 69 (2008), 621-639.  doi: 10.1137/070700966.  Google Scholar

[32]

H. R. Thieme and X. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations, 195 (2003), 430-470.  doi: 10.1016/S0022-0396(03)00175-X.  Google Scholar

[33]

Z. WangW. Li and S. Ruan, Travelling wave fronts in reaction-diffusion systems with spatio-temporal delays, J. Differential Equations, 222 (2006), 185-232.  doi: 10.1016/j.jde.2005.08.010.  Google Scholar

[34]

J. WangW. Li and F. Yang, Traveling waves in a nonlocal dispersal SIR model with nonlocal delayed transmission, Commun. Nonlinear Sci. Numer. Simulat., 27 (2015), 136-152.  doi: 10.1016/j.cnsns.2015.03.005.  Google Scholar

[35]

Z. 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), 237-261.  doi: 10.1098/rspa.2009.0377.  Google Scholar

[36]

S. WuC. H. Hsu and Y. Xiao, Global attractivity, spreading speeds and traveling waves of delayed nonlocal reaction-diffusion systems, J. Differential Equations, 258 (2015), 1058-1105.  doi: 10.1016/j.jde.2014.10.009.  Google Scholar

[37]

D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 208 (2007), 419-429.  doi: 10.1016/j.mbs.2006.09.025.  Google Scholar

[38]

Z. Xu and D. Xiao, Minimal wave speed and uniqueness of traveling waves for a nonlocal diffusion population model with spatio-temporal delays, Differ. Integral Equ., 27 (2014), 1073-1106.   Google Scholar

[39]

H. Yagisita, Existence and nonexistence of traveling waves for a nonlocal monostable equation, Publ. Res. Inst. Math. Sci., 45 (2009), 925-953.  doi: 10.2977/prims/1260476648.  Google Scholar

[40]

F. YangY. LiW. Li and Z. Wang, Traveling waves in a nonlocal dispersal Kermack-McKendrick epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1969-1993.  doi: 10.3934/dcdsb.2013.18.1969.  Google Scholar

[41]

Y. Yang and D. Xiao, Influence of latent period and nonlinear incidence rate on the dynamics of SIRS epidemiological models, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 195-211.  doi: 10.3934/dcdsb.2010.13.195.  Google Scholar

[42]

Z. Yu and R. Yuan, Traveling wave fronts in reaction-diffusion systems with spatio-temporal delay and applications, Discrete Contin. Dyn. Syst. Ser. B, 13 (2010), 709-728.  doi: 10.3934/dcdsb.2010.13.709.  Google Scholar

[43]

G. ZhangW. Li and Z. Wang, Spreading speeds and traveling waves for nonlocal dispersal equations with degenerate monostable nonlinearity, J. Differential Equations, 252 (2012), 5096-5124.  doi: 10.1016/j.jde.2012.01.014.  Google Scholar

[44]

X. Zou and J. Wu, Existence of traveling wave fronts in delayed reaction-diffusion systems via the monotone iteration method, Proc. Amer. Math. Soc., 125 (1997), 2589-2598.  doi: 10.1090/S0002-9939-97-04080-X.  Google Scholar

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