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May  2020, 19(5): 2853-2886. doi: 10.3934/cpaa.2020125

Traveling waves in a nonlocal dispersal epidemic model with spatio-temporal delay

1. 

Nonlinear Scientific Research Center, Faculty of Science, Jiangsu University, Zhenjiang, Jiangsu 212013, China

2. 

School of Mathematical Sciences, Nanjing Normal University, Nanjing, Jiangsu 210046, China

*Corresponding author

Received  June 2019 Revised  November 2019 Published  March 2020

Fund Project: This research was supported by National Natural Science Foundation of China (No. 11731014) and China Postdoctoral Science Foundation (No. 2018M642173)

In this paper, we investigate the existence and nonexistence of traveling wave solutions in a nonlocal dispersal epidemic model with spatio-temporal delay. It is shown that this model admits a nontrivial positive traveling wave solution when the basic reproduction number $ R_0>1 $ and the wave speed $ c\geq c^* $ ($ c^* $ is the critical speed) and this model has no traveling wave solutions when $ R_0\leq1 $ or $ c<c^* $. This indicates that $ c^* $ is the minimal wave speed.

Citation: Jingdong Wei, Jiangbo Zhou, Wenxia Chen, Zaili Zhen, Lixin Tian. Traveling waves in a nonlocal dispersal epidemic model with spatio-temporal delay. Communications on Pure & Applied Analysis, 2020, 19 (5) : 2853-2886. doi: 10.3934/cpaa.2020125
References:
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S. Ai and R. Albashaireh, Traveling waves in spatial SIRS models, J. Dyn. Differ. Equ., 26 (2014), 143-164.  doi: 10.1007/s10884-014-9348-3.  Google Scholar

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F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, Vol. 165, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/surv/165.  Google Scholar

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F. Brauer, P. van den Driessche and J. Wu, Mathematical Epidemiology, Lecture Notes in Mathematics, Vol. 1945, Springer, New York, 2008. doi: 10.1007/978-3-540-78911-6.  Google Scholar

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Y. ChenJ. Guo and F. Hamel, Traveling waves for a lattice dynamical system arising in a diffusive endemic model, Nonlinearity, 30 (2017), 2334-2359.  doi: 10.1088/1361-6544/aa6b0a.  Google Scholar

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H. Cheng and R. Yuan, Traveling wave solutions for a nonlocal dispersal Kermack-Mckendrick epidemic model with spatio-temporal delay, Sci. Sin. Math., 45 (2015), 765-788.  doi: 10.1007/s00028-016-0362-2.  Google Scholar

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W. DingW. Huang and S. Kansakar, Traveling wave solutions for a diffusive SIS epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1291-1304.  doi: 10.3934/dcdsb.2013.18.1291.  Google Scholar

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A. DucrotM. Langlais and P. Magal, Qualitative analysis and travelling wave solutions for the SI model with vertical transmission, Commun. Pure Appl. Anal., 11 (2012), 97-113.  doi: 10.3934/cpaa.2012.11.97.  Google Scholar

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A. Ducrot and P. Magal, Traveling wave solutions for an infection-age structured epidemic model with external supplies, Nonlinearity, 23 (2011), 2891-2911.  doi: 10.1088/0951-7715/24/10/012.  Google Scholar

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S. Fu, Traveling waves for a diffusive SIR model with delay, J. Math. Anal. Appl., 435 (2016), 20-37.  doi: 10.1016/j.jmaa.2015.09.069.  Google Scholar

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J. He and J. Tsai, Traveling waves in the Kermack-Mckendrick epidemic model with latent period, Z. Angew. Math. Phys., 70 (2019), 27. doi: 10.1007/s00033-018-1072-0.  Google Scholar

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H. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar

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Y. Hosono and B. Ilyas, Traveling waves for a simple diffusive SIR epidemic model, Math. Mod. Meth. Appl. S., 5 (1995), 935-966.  doi: 10.1142/S0218202595000504.  Google Scholar

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

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W. LiG. LinC. Ma and F. Yang, Traveling wave solutions of a nonlocal delayed SIR model without outbreak threshold, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 467-484.  doi: 10.3934/dcdsb.2014.19.467.  Google Scholar

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Y. LiW. Li and G. Lin, Traveling waves of a delayed diffusive SIR epidemic model, Commun. Pure Appl. Anal., 14 (2015), 1001-1022.  doi: 10.3934/cpaa.2015.14.1001.  Google Scholar

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Y. LiW. Li and F. Yang, Traveling waves for a nonlocal dispersal SIR model with delay and external supplies, Appl. Math. Comput., 247 (2014), 723-740.  doi: 10.1016/j.amc.2014.09.072.  Google Scholar

[21]

Y. LiW. Li and G. Zhang, Stability and uniqueness of traveling waves of a non-local dispersal SIR epidemic model, Dyn. Partial Differ. Equ., 14 (2017), 87-123.  doi: 10.4310/DPDE.2017.v14.n2.a1.  Google Scholar

[22]

G. Lin, Minimal wave speed of competitive diffusive systems with time delays, Appl. Math. Lett., 76 (2018), 164-169.  doi: 10.1016/j.aml.2017.08.018.  Google Scholar

[23]

Z. Ma and R. Yuan, Traveling wave solutions of a nonlocal dispersal SIRS model with spatio-temporal delay, Int. J. Biomath., 10 (2017), 5. doi: 10.1142/S1793524517500711.  Google Scholar

[24]

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

[25]

N. Rawal and W. Shen, Criteria for the existence and lower bounds of principal eigenvalues of time periodic nonlocal dispersal operators and applications, J. Dyn. Differ. Equ., 24 (2012), 927-954.  doi: 10.1007/s10884-012-9276-z.  Google Scholar

[26]

W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differ. Equ., 249 (2010), 747-795.  doi: 10.1016/j.jde.2010.04.012.  Google Scholar

[27] N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford University Press, Oxford, 1997.   Google Scholar
[28]

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

[29]

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

[30]

H. Wang and X. Wang, Traveling waves phenomena in a Kemack-Mckendrick SIR model, J. Dyn. Differ. Equ., 28 (2016), 143-166.  doi: 10.1007/s10884-015-9506-2.  Google Scholar

[31]

Z. Wang and J. Wu, Travelling waves of a diffusive Kermack-Mckendrick epidemic model with non-local delayed transmission, Proc. R. Soc. A - Math. Phys. Eng. Sci., 466 (2010), 237-261.  doi: 10.1098/rspa.2009.0377.  Google Scholar

[32]

X. WangH. Wang and J. Wu, Traveling waves of diffusive predator-prey systems: disease outbreak propagation, Discret. Contin. Dyn. Syst., 32 (2012), 3303-3324.  doi: 10.3934/dcds.2012.32.3303.  Google Scholar

[33]

P. Weng and X. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, J. Differ. Equ., 229 (2006), 270-296.  doi: 10.1016/j.jde.2006.01.020.  Google Scholar

[34] D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, 1941.   Google Scholar
[35]

C. Wu, Existence of traveling waves with the critical speed for a discrete diffusive epidemic model, J. Differ. Equ., 262 (2017), 272-282.  doi: 10.1016/j.jde.2016.09.022.  Google Scholar

[36]

C. Wu and P. Weng, Asymptotic speed of propagation and traveling wavefronts for a SIR epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 15 (2012), 867-892.  doi: 10.3934/dcdsb.2011.15.867.  Google Scholar

[37]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dyn. Differ. Equ., 13 (2001), 651–687. [Erratum in J. Dyn. Differ. Equ. 20 (2008), 531–533]. doi: 10.1007/s10884-007-9090-1.  Google Scholar

[38]

F. Yang and W. Li, Traveling waves in a nonlocal dispersal SIR model with critical wave speed, J. Math. Anal. Appl., 458 (2018), 1131-1146.  doi: 10.1016/j.jmaa.2017.10.016.  Google Scholar

[39]

F. Yang, W. Li and J. Wang, Wave propagation for a class of non-local dispersal non-cooperative systems, Proc. R. Soc. Edinb. Sect. A Math., (2019), 1-33 doi: 10.1017/prm.2019.4.  Google Scholar

[40]

F. YangW. Li and Z. Wang, Traveling waves in a nonlocal dispersal SIR epidemic model, Nonlinear Anal. Real World Appl., 23 (2015), 129-147.  doi: 10.1016/j.nonrwa.2014.12.001.  Google Scholar

[41]

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

[42]

F. ZhanG. HuoQ. LiuG. Sun and Z. Jin, Existence of traveling waves in nonlinear SI epidemic models, J. Biol. Syst., 17 (2009), 643-657.  doi: 10.1142/S0218339009003101.  Google Scholar

[43]

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

[44]

L. Zhao and Z. Wang, Traveling wave fronts in a diffusive epidemic model with multiple parallel infectious stages, IMA J. Appl. Math., 81 (2016), 795-823.  doi: 10.1093/imamat/hxw033.  Google Scholar

[45]

L. ZhaoZ. Wang and S. Ruan, Traveling wave solutions in a two-group SIR epidemic model with constant recruitment, J. Math. Biol., 1 (2018), 1-45.  doi: 10.1007/s00285-018-1227-9.  Google Scholar

[46]

Z. ZhenJ. WeiJ. Zhou and L. Tian, Wave propagation in a nonlocal diffusion epidemic model with nonlocal delayed effects, Appl. Math. Comput., 339 (2018), 15-37.  doi: 10.1016/j.amc.2018.07.007.  Google Scholar

[47]

Z. ZhenJ. WeiL. TianJ. Zhou and W. Chen, Wave propagation in a diffusive epidemic model with spatiotemporal delay, Math. Meth. Appl. Sci., 41 (2018), 7074-7098.  doi: 10.1002/mma.5216.  Google Scholar

[48]

J. ZhouL. SongJ. Wei and H. Xu, Critical traveling waves in a diffusive disease model, J. Math. Anal. Appl., 476 (2019), 522-538.  doi: 10.1016/j.jmaa.2019.03.066.  Google Scholar

[49]

J. ZhouJ. XuJ. Wei and H. Xu, Existence and non-existence of traveling wave solutions for a nonlocal dispersal SIR epidemic model with nonlinear incidence rate, Nonlinear Anal. Real World Appl., 41 (2018), 204-231.  doi: 10.1016/j.nonrwa.2017.10.016.  Google Scholar

[50]

J. Zhou and Y. Yang, Traveling waves for a nonlocal dispersal SIR model with general nonlinear incidence rate and spatio-temporal delay, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1719-1741.  doi: 10.3934/dcdsb.2017082.  Google Scholar

show all references

References:
[1]

S. Ai and R. Albashaireh, Traveling waves in spatial SIRS models, J. Dyn. Differ. Equ., 26 (2014), 143-164.  doi: 10.1007/s10884-014-9348-3.  Google Scholar

[2]

F. Andreu-Vaillo, J. M. Mazón, J. D. Rossi and J. Toledo-Melero, Nonlocal Diffusion Problems, Mathematical Surveys and Monographs, Vol. 165, American Mathematical Society, Providence, RI, 2010. doi: 10.1090/surv/165.  Google Scholar

[3]

F. Brauer and C. Castillo-Chvez, Mathematical Models in Population Biology and Epidemiology, 2$^nd$ edition, Springer, Berlin, 2012. doi: 10.1007/978-1-4614-1686-9.  Google Scholar

[4]

F. Brauer, P. van den Driessche and J. Wu, Mathematical Epidemiology, Lecture Notes in Mathematics, Vol. 1945, Springer, New York, 2008. doi: 10.1007/978-3-540-78911-6.  Google Scholar

[5]

Y. ChenJ. Guo and F. Hamel, Traveling waves for a lattice dynamical system arising in a diffusive endemic model, Nonlinearity, 30 (2017), 2334-2359.  doi: 10.1088/1361-6544/aa6b0a.  Google Scholar

[6]

H. Cheng and R. Yuan, Traveling wave solutions for a nonlocal dispersal Kermack-Mckendrick epidemic model with spatio-temporal delay, Sci. Sin. Math., 45 (2015), 765-788.  doi: 10.1007/s00028-016-0362-2.  Google Scholar

[7]

W. DingW. Huang and S. Kansakar, Traveling wave solutions for a diffusive SIS epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 18 (2013), 1291-1304.  doi: 10.3934/dcdsb.2013.18.1291.  Google Scholar

[8]

A. DucrotM. Langlais and P. Magal, Qualitative analysis and travelling wave solutions for the SI model with vertical transmission, Commun. Pure Appl. Anal., 11 (2012), 97-113.  doi: 10.3934/cpaa.2012.11.97.  Google Scholar

[9]

A. Ducrot and P. Magal, Traveling wave solutions for an infection-age structured epidemic model with external supplies, Nonlinearity, 23 (2011), 2891-2911.  doi: 10.1088/0951-7715/24/10/012.  Google Scholar

[10]

P. Fife, Some Nonclassic Trends in Parabolic and Parabolic-like Evolutions, Trends in Nonlinear Analysis, Springer, Berlin, 2003. doi: 10.1007/978-3-662-05281-5_3.  Google Scholar

[11]

S. Fu, Traveling waves for a diffusive SIR model with delay, J. Math. Anal. Appl., 435 (2016), 20-37.  doi: 10.1016/j.jmaa.2015.09.069.  Google Scholar

[12]

J. He and J. Tsai, Traveling waves in the Kermack-Mckendrick epidemic model with latent period, Z. Angew. Math. Phys., 70 (2019), 27. doi: 10.1007/s00033-018-1072-0.  Google Scholar

[13]

H. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42 (2000), 599-653.  doi: 10.1137/S0036144500371907.  Google Scholar

[14]

Y. Hosono and B. Ilyas, Traveling waves for a simple diffusive SIR epidemic model, Math. Mod. Meth. Appl. S., 5 (1995), 935-966.  doi: 10.1142/S0218202595000504.  Google Scholar

[15]

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

[16]

C. Y. Kao, Y. Lou and W. Shen, Random dispersal vs non-local dispersal, Discrete Contin. Dyn. Syst., 26 (2010), 551-596. doi: 10.3934/dcds.2010.26.551.  Google Scholar

[17]

C. Y. KaoY. Lou and W. Shen, Evolution of mixed dispersal in periodic environments, Discrete Contin. Dyn. Syst. Ser. B, 17 (2012), 2047-2072.  doi: 10.3934/dcdsb.2012.17.2047.  Google Scholar

[18]

W. LiG. LinC. Ma and F. Yang, Traveling wave solutions of a nonlocal delayed SIR model without outbreak threshold, Discrete Contin. Dyn. Syst. Ser. B, 19 (2014), 467-484.  doi: 10.3934/dcdsb.2014.19.467.  Google Scholar

[19]

Y. LiW. Li and G. Lin, Traveling waves of a delayed diffusive SIR epidemic model, Commun. Pure Appl. Anal., 14 (2015), 1001-1022.  doi: 10.3934/cpaa.2015.14.1001.  Google Scholar

[20]

Y. LiW. Li and F. Yang, Traveling waves for a nonlocal dispersal SIR model with delay and external supplies, Appl. Math. Comput., 247 (2014), 723-740.  doi: 10.1016/j.amc.2014.09.072.  Google Scholar

[21]

Y. LiW. Li and G. Zhang, Stability and uniqueness of traveling waves of a non-local dispersal SIR epidemic model, Dyn. Partial Differ. Equ., 14 (2017), 87-123.  doi: 10.4310/DPDE.2017.v14.n2.a1.  Google Scholar

[22]

G. Lin, Minimal wave speed of competitive diffusive systems with time delays, Appl. Math. Lett., 76 (2018), 164-169.  doi: 10.1016/j.aml.2017.08.018.  Google Scholar

[23]

Z. Ma and R. Yuan, Traveling wave solutions of a nonlocal dispersal SIRS model with spatio-temporal delay, Int. J. Biomath., 10 (2017), 5. doi: 10.1142/S1793524517500711.  Google Scholar

[24]

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

[25]

N. Rawal and W. Shen, Criteria for the existence and lower bounds of principal eigenvalues of time periodic nonlocal dispersal operators and applications, J. Dyn. Differ. Equ., 24 (2012), 927-954.  doi: 10.1007/s10884-012-9276-z.  Google Scholar

[26]

W. Shen and A. Zhang, Spreading speeds for monostable equations with nonlocal dispersal in space periodic habitats, J. Differ. Equ., 249 (2010), 747-795.  doi: 10.1016/j.jde.2010.04.012.  Google Scholar

[27] N. Shigesada and K. Kawasaki, Biological Invasions: Theory and Practice, Oxford University Press, Oxford, 1997.   Google Scholar
[28]

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

[29]

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

[30]

H. Wang and X. Wang, Traveling waves phenomena in a Kemack-Mckendrick SIR model, J. Dyn. Differ. Equ., 28 (2016), 143-166.  doi: 10.1007/s10884-015-9506-2.  Google Scholar

[31]

Z. Wang and J. Wu, Travelling waves of a diffusive Kermack-Mckendrick epidemic model with non-local delayed transmission, Proc. R. Soc. A - Math. Phys. Eng. Sci., 466 (2010), 237-261.  doi: 10.1098/rspa.2009.0377.  Google Scholar

[32]

X. WangH. Wang and J. Wu, Traveling waves of diffusive predator-prey systems: disease outbreak propagation, Discret. Contin. Dyn. Syst., 32 (2012), 3303-3324.  doi: 10.3934/dcds.2012.32.3303.  Google Scholar

[33]

P. Weng and X. Zhao, Spreading speed and traveling waves for a multi-type SIS epidemic model, J. Differ. Equ., 229 (2006), 270-296.  doi: 10.1016/j.jde.2006.01.020.  Google Scholar

[34] D. V. Widder, The Laplace Transform, Princeton University Press, Princeton, 1941.   Google Scholar
[35]

C. Wu, Existence of traveling waves with the critical speed for a discrete diffusive epidemic model, J. Differ. Equ., 262 (2017), 272-282.  doi: 10.1016/j.jde.2016.09.022.  Google Scholar

[36]

C. Wu and P. Weng, Asymptotic speed of propagation and traveling wavefronts for a SIR epidemic model, Discrete Contin. Dyn. Syst. Ser. B, 15 (2012), 867-892.  doi: 10.3934/dcdsb.2011.15.867.  Google Scholar

[37]

J. Wu and X. Zou, Traveling wave fronts of reaction-diffusion systems with delay, J. Dyn. Differ. Equ., 13 (2001), 651–687. [Erratum in J. Dyn. Differ. Equ. 20 (2008), 531–533]. doi: 10.1007/s10884-007-9090-1.  Google Scholar

[38]

F. Yang and W. Li, Traveling waves in a nonlocal dispersal SIR model with critical wave speed, J. Math. Anal. Appl., 458 (2018), 1131-1146.  doi: 10.1016/j.jmaa.2017.10.016.  Google Scholar

[39]

F. Yang, W. Li and J. Wang, Wave propagation for a class of non-local dispersal non-cooperative systems, Proc. R. Soc. Edinb. Sect. A Math., (2019), 1-33 doi: 10.1017/prm.2019.4.  Google Scholar

[40]

F. YangW. Li and Z. Wang, Traveling waves in a nonlocal dispersal SIR epidemic model, Nonlinear Anal. Real World Appl., 23 (2015), 129-147.  doi: 10.1016/j.nonrwa.2014.12.001.  Google Scholar

[41]

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

[42]

F. ZhanG. HuoQ. LiuG. Sun and Z. Jin, Existence of traveling waves in nonlinear SI epidemic models, J. Biol. Syst., 17 (2009), 643-657.  doi: 10.1142/S0218339009003101.  Google Scholar

[43]

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

[44]

L. Zhao and Z. Wang, Traveling wave fronts in a diffusive epidemic model with multiple parallel infectious stages, IMA J. Appl. Math., 81 (2016), 795-823.  doi: 10.1093/imamat/hxw033.  Google Scholar

[45]

L. ZhaoZ. Wang and S. Ruan, Traveling wave solutions in a two-group SIR epidemic model with constant recruitment, J. Math. Biol., 1 (2018), 1-45.  doi: 10.1007/s00285-018-1227-9.  Google Scholar

[46]

Z. ZhenJ. WeiJ. Zhou and L. Tian, Wave propagation in a nonlocal diffusion epidemic model with nonlocal delayed effects, Appl. Math. Comput., 339 (2018), 15-37.  doi: 10.1016/j.amc.2018.07.007.  Google Scholar

[47]

Z. ZhenJ. WeiL. TianJ. Zhou and W. Chen, Wave propagation in a diffusive epidemic model with spatiotemporal delay, Math. Meth. Appl. Sci., 41 (2018), 7074-7098.  doi: 10.1002/mma.5216.  Google Scholar

[48]

J. ZhouL. SongJ. Wei and H. Xu, Critical traveling waves in a diffusive disease model, J. Math. Anal. Appl., 476 (2019), 522-538.  doi: 10.1016/j.jmaa.2019.03.066.  Google Scholar

[49]

J. ZhouJ. XuJ. Wei and H. Xu, Existence and non-existence of traveling wave solutions for a nonlocal dispersal SIR epidemic model with nonlinear incidence rate, Nonlinear Anal. Real World Appl., 41 (2018), 204-231.  doi: 10.1016/j.nonrwa.2017.10.016.  Google Scholar

[50]

J. Zhou and Y. Yang, Traveling waves for a nonlocal dispersal SIR model with general nonlinear incidence rate and spatio-temporal delay, Discrete Contin. Dyn. Syst. Ser. B, 22 (2017), 1719-1741.  doi: 10.3934/dcdsb.2017082.  Google Scholar

Figure 1.  $ I_+^*(z) $ and $ I_-^*(z) $
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