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 and 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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

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

[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.

[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.

[20]

H. Wang and X.-S. Wang, Travelling waves phenomena in a Kermack-McKendrick SIR model, J. Dyn. Diff. Equat., DOI 10.1007/s10884-015-9506-2.

[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.

[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.

[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.

[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.

[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.

[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.

[27]

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

[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.

[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.

[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.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[6]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[17]

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

[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.

[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.

[20]

H. Wang and X.-S. Wang, Travelling waves phenomena in a Kermack-McKendrick SIR model, J. Dyn. Diff. Equat., DOI 10.1007/s10884-015-9506-2.

[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.

[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.

[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.

[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.

[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.

[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.

[27]

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

[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.

[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.

[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.

[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.

[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.

[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.

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