# American Institute of Mathematical Sciences

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

## Traveling waves for a diffusive SEIR epidemic model

 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.
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.

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##### 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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