American Institute of Mathematical Sciences

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May  2016, 15(3): 871-892. doi: 10.3934/cpaa.2016.15.871

Traveling waves for a diffusive SEIR epidemic model

Zhiting Xu 1,

1. 

School of Mathematical Sciences, South China Normal University, Guangzhou, Guangdong 510631

Received  August 2015 Revised  December 2015 Published  February 2016

In this paper, we propose a diffusive SEIR epidemic model with saturating incidence rate. We first study the well posedness of the model, and give the explicit formula of the basic reproduction number $\mathcal{R}_0$. And hence, we show that if $\mathcal{R}_0>1$, then there exists a positive constant $c^*>0$ such that for each $c>c^*$, the model admits a nontrivial traveling wave solution, and if $\mathcal{R}_0\leq1$ and $c\geq 0$ (or, $\mathcal{R}_0>1$ and $c\in[0,c^*)$), then the model has no nontrivial traveling wave solutions. Consequently, we confirm that the constant $c^*$ is indeed the minimal wave speed. The proof of the main results is mainly based on Schauder fixed theorem and Laplace transform.
Keywords: saturating incidence rate., minimal wave speed, SEIR epidemic model, Traveling wave solutions.
Mathematics Subject Classification: Primary: 35C07, 35K57; Secondary: 92D3.
Citation: Zhiting Xu. Traveling waves for a diffusive SEIR epidemic model. Communications on Pure & Applied Analysis, 2016, 15 (3) : 871-892. doi: 10.3934/cpaa.2016.15.871
References:
[1]

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. Simul., (1-3) (2015), 1370-1381. doi: 10.1016/j.cnsns.2014.07.005.  Google Scholar

[2]

F. Braner and C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology, Springer, New York, 2001. doi: 10.1007/978-1-4757-3516-1.  Google Scholar

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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

[4]

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

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

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H. W. Hethcote and van den Driessche, Some epidemiological models with nonlinear incidence, J. Math. Biol., 29 (1991), 271-287. doi: 10.1007/BF00160539.  Google Scholar

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

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J. Huang and X. Zou, Existence of traveling wavefronts of delayed reaction diffusion systems without monotonicity, Discrete Contin. Dyn. Sys., 9 (2003), 925-936. doi: 10.3934/dcds.2003.9.925.  Google Scholar

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W. O. Kermack and A. G. McKendrick, Contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. (Ser. A), 115 (1927), 700-721; part II, Proc. R. Soc. Lond. (Ser. A), 138 (1932), 55-83; part III, Proc. R. Soc. Lond. (Ser. A), 141 (1933), 94-112. Google Scholar

[11]

W.-T. Li, G. Lin and S. Ruan, Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems, Nonlinearity, 19 (2006), 1253-1257. doi: 10.1088/0951-7715/19/6/003.  Google Scholar

[12]

W.-T. Li, G. Lin, C. Ma and F.-Y. Yang, Travelling wave solutions of a nonlocal delayed SIR model with outbreak threshold, Discrete Contin. Dyn. Sys. (Ser.B), 19 (2014), 467-484. doi: 10.3934/dcdsb.2014.19.467.  Google Scholar

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X. Liang and X.-Q. Zhao, Asymptotic speed of spread and traveling waves for monotone semiflows with applications, Comm. Pure. Appl. Math., 60 (2007) 1-40. doi: 10.1002/cpa.20154.  Google Scholar

[14]

Y. Lv, R. Yuan and Y. Pei, The imact of predation on the coexistence and competitive exclusion of pathogens in prey, Math. Biosci., 251 (2014), 16-29. doi: 10.1016/j.mbs.2014.02.005.  Google Scholar

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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

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S. Ruan and W. Wang, Dynamical behavior of an epidemic models with a nonlinear incidence rate, J. Differential Equations, 188 (2003), 135-163. doi: 10.1016/S0022-0396(02)00089-X.  Google Scholar

[19]

H. Wang, Spreading speeds and traveling waves for non-cooperative reaction-diffusion stsyems, J. Nonlinear Sciences, 21 (2011), 747-783. doi: 10.1007/s00332-011-9099-9.  Google Scholar

[20]

H. Wang and X.-S. Wang, Travelling waves phenomena in a Kermack-McKendrick SIR model,, \emph{J. Dyn. Diff. Equat}., (): 10884.   Google Scholar

[21]

W. Wang and X.-Q. Zhao, A nonlocal and time-delayed reaction-diffusion model of dengue transmission, SIAM J. Appl. Math., 71 (2011), 147-168. doi: 10.1137/090775890.  Google Scholar

[22]

X.-S. Wang, H. Wang and J. Wu, Travelling waves of diffusive predator-prey systems: disease outbreak propagation, Discrete Contin. Dyn. Sys., 32 (2012), 3303-3324. doi: 10.3934/dcds.2012.32.3303.  Google Scholar

[23]

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

[24]

Z.-C. Wang and J. Wu, Travelling waves in a bio-reactor model with stage-structure, J. Math. Anal. Appl., 385 (2012), 683-692. doi: 10.1016/j.jmaa.2011.06.084.  Google Scholar

[25]

Z.-C. Wang, J. Wu and R. Liu, Traveling waves of Avian influenza spread, Proc. Amer. Math. Soc., 149 (2012), 3931-3946. doi: 10.1090/S0002-9939-2012-11246-8.  Google Scholar

[26]

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

[27]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, New York, 1996. doi: 10.1007/978-1-4612-4050-1.  Google Scholar

[28]

D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 208 (2007), 419-492. doi: 10.1016/j.mbs.2006.09.025.  Google Scholar

[29]

R. Xu and Z. Ma, Global stablity of a SIR epidemic model with nonlinear incidence rate and time delay, Nonlinear Analysis: Real World Applications, 10 (2009), 3175-3189. doi: 10.1016/j.nonrwa.2008.10.013.  Google Scholar

[30]

Z. Xu, Traveling waves in a Kermack-Mckendrick epidemic model with diffusion and latent period, Nonlinear Analysis, 111 (2014), 66-81. doi: 10.1016/j.na.2014.08.012.  Google Scholar

[31]

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

[32]

L. Zhang, B. Li and J. Shang, Stability and travelling waves for a time-delayed population system with stage structure, Nonlinear Analysis: Real World Applications, 13 (2012), 1429-1440. doi: 10.1016/j.nonrwa.2011.11.007.  Google Scholar

[33]

T. Zhang and W. Wang, Existence of thaveling wave solutions for influenza model with treatment, J. Math. Anal. Appl., 419 (2014), 469-495. doi: 10.1016/j.jmaa.2014.04.068.  Google Scholar

show all references

References:
[1]

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. Simul., (1-3) (2015), 1370-1381. doi: 10.1016/j.cnsns.2014.07.005.  Google Scholar

[2]

F. Braner and C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology, Springer, New York, 2001. doi: 10.1007/978-1-4757-3516-1.  Google Scholar

[3]

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

[4]

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

[5]

J. Fang and X.-Q. Zhao, Monotone wavefronts for partially degenerate reaction-diffusion systems, J. Dyn. Diff. Equat., 21 (2009), 663-680. doi: 10.1007/s10884-009-9152-7.  Google Scholar

[6]

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

[7]

H. W. Hethcote and van den Driessche, Some epidemiological models with nonlinear incidence, J. Math. Biol., 29 (1991), 271-287. doi: 10.1007/BF00160539.  Google Scholar

[8]

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

[9]

J. Huang and X. Zou, Existence of traveling wavefronts of delayed reaction diffusion systems without monotonicity, Discrete Contin. Dyn. Sys., 9 (2003), 925-936. doi: 10.3934/dcds.2003.9.925.  Google Scholar

[10]

W. O. Kermack and A. G. McKendrick, Contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond. (Ser. A), 115 (1927), 700-721; part II, Proc. R. Soc. Lond. (Ser. A), 138 (1932), 55-83; part III, Proc. R. Soc. Lond. (Ser. A), 141 (1933), 94-112. Google Scholar

[11]

W.-T. Li, G. Lin and S. Ruan, Existence of travelling wave solutions in delayed reaction-diffusion systems with applications to diffusion-competition systems, Nonlinearity, 19 (2006), 1253-1257. doi: 10.1088/0951-7715/19/6/003.  Google Scholar

[12]

W.-T. Li, G. Lin, C. Ma and F.-Y. Yang, Travelling wave solutions of a nonlocal delayed SIR model with outbreak threshold, Discrete Contin. Dyn. Sys. (Ser.B), 19 (2014), 467-484. doi: 10.3934/dcdsb.2014.19.467.  Google Scholar

[13]

X. Liang and X.-Q. Zhao, Asymptotic speed of spread and traveling waves for monotone semiflows with applications, Comm. Pure. Appl. Math., 60 (2007) 1-40. doi: 10.1002/cpa.20154.  Google Scholar

[14]

Y. Lv, R. Yuan and Y. Pei, The imact of predation on the coexistence and competitive exclusion of pathogens in prey, Math. Biosci., 251 (2014), 16-29. doi: 10.1016/j.mbs.2014.02.005.  Google Scholar

[15]

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

[16]

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

[17]

J. D. Murray, Mathematical Biology, I and II, third edn., Springer-Verlag, New York, 2002.  Google Scholar

[18]

S. Ruan and W. Wang, Dynamical behavior of an epidemic models with a nonlinear incidence rate, J. Differential Equations, 188 (2003), 135-163. doi: 10.1016/S0022-0396(02)00089-X.  Google Scholar

[19]

H. Wang, Spreading speeds and traveling waves for non-cooperative reaction-diffusion stsyems, J. Nonlinear Sciences, 21 (2011), 747-783. doi: 10.1007/s00332-011-9099-9.  Google Scholar

[20]

H. Wang and X.-S. Wang, Travelling waves phenomena in a Kermack-McKendrick SIR model,, \emph{J. Dyn. Diff. Equat}., (): 10884.   Google Scholar

[21]

W. Wang and X.-Q. Zhao, A nonlocal and time-delayed reaction-diffusion model of dengue transmission, SIAM J. Appl. Math., 71 (2011), 147-168. doi: 10.1137/090775890.  Google Scholar

[22]

X.-S. Wang, H. Wang and J. Wu, Travelling waves of diffusive predator-prey systems: disease outbreak propagation, Discrete Contin. Dyn. Sys., 32 (2012), 3303-3324. doi: 10.3934/dcds.2012.32.3303.  Google Scholar

[23]

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

[24]

Z.-C. Wang and J. Wu, Travelling waves in a bio-reactor model with stage-structure, J. Math. Anal. Appl., 385 (2012), 683-692. doi: 10.1016/j.jmaa.2011.06.084.  Google Scholar

[25]

Z.-C. Wang, J. Wu and R. Liu, Traveling waves of Avian influenza spread, Proc. Amer. Math. Soc., 149 (2012), 3931-3946. doi: 10.1090/S0002-9939-2012-11246-8.  Google Scholar

[26]

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

[27]

J. Wu, Theory and Applications of Partial Functional Differential Equations, Springer, New York, 1996. doi: 10.1007/978-1-4612-4050-1.  Google Scholar

[28]

D. Xiao and S. Ruan, Global analysis of an epidemic model with nonmonotone incidence rate, Math. Biosci., 208 (2007), 419-492. doi: 10.1016/j.mbs.2006.09.025.  Google Scholar

[29]

R. Xu and Z. Ma, Global stablity of a SIR epidemic model with nonlinear incidence rate and time delay, Nonlinear Analysis: Real World Applications, 10 (2009), 3175-3189. doi: 10.1016/j.nonrwa.2008.10.013.  Google Scholar

[30]

Z. Xu, Traveling waves in a Kermack-Mckendrick epidemic model with diffusion and latent period, Nonlinear Analysis, 111 (2014), 66-81. doi: 10.1016/j.na.2014.08.012.  Google Scholar

[31]

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

[32]

L. Zhang, B. Li and J. Shang, Stability and travelling waves for a time-delayed population system with stage structure, Nonlinear Analysis: Real World Applications, 13 (2012), 1429-1440. doi: 10.1016/j.nonrwa.2011.11.007.  Google Scholar

[33]

T. Zhang and W. Wang, Existence of thaveling wave solutions for influenza model with treatment, J. Math. Anal. Appl., 419 (2014), 469-495. doi: 10.1016/j.jmaa.2014.04.068.  Google Scholar

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