# 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 & 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,, \emph{Commun. Nonlinear Sci. Numer. Simul}., (1-3) (2015), 1. doi: 10.1016/j.cnsns.2014.07.005. [2] F. Braner and C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology,, Springer, (2001). doi: 10.1007/978-1-4757-3516-1. [3] V. Capasso and G. Serio, A generalization of the Kermack-Mckendrick deterministic epidemic model,, \emph{Math. Biosci}., 42 (1978), 43. doi: 10.1016/0025-5564(78)90006-8. [4] J. Carr and A. Chmaj, Uniquence of the travling waves for nonlocal monostabe equations,, \emph{Proc. Amer. Math. Soc}., 132 (2004), 2433. doi: 10.1090/S0002-9939-04-07432-5. [5] J. Fang and X.-Q. Zhao, Monotone wavefronts for partially degenerate reaction-diffusion systems,, \emph{J. Dyn. Diff. Equat}., 21 (2009), 663. doi: 10.1007/s10884-009-9152-7. [6] H. W. Hethcote, The mathematics of infectious diseases,, \emph{SIAM Review}, 42 (2000), 599. doi: 10.1137/S0036144500371907. [7] H. W. Hethcote and van den Driessche, Some epidemiological models with nonlinear incidence,, \emph{J. Math. Biol.}, 29 (1991), 271. doi: 10.1007/BF00160539. [8] Y. Hosono and B. Ilyas, Traveling waves for a simple diffusive epidemic model,, \emph{Math. Models. Methods Appl. Sci}., 5 (1995), 935. doi: 10.1142/S0218202595000504. [9] J. Huang and X. Zou, Existence of traveling wavefronts of delayed reaction diffusion systems without monotonicity,, \emph{Discrete Contin. Dyn. Sys}., 9 (2003), 925. doi: 10.3934/dcds.2003.9.925. [10] W. O. Kermack and A. G. McKendrick, Contribution to the mathematical theory of epidemics,, \emph{Proc. R. Soc. Lond. (Ser. A)}, 115 (1927), 700. [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,, \emph{Nonlinearity}, 19 (2006), 1253. 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,, \emph{Discrete Contin. Dyn. Sys. (Ser.B)}, 19 (2014), 467. 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,, \emph{Comm. Pure. Appl. Math}., 60 (2007), 1. 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,, \emph{Math. Biosci}., 251 (2014), 16. doi: 10.1016/j.mbs.2014.02.005. [15] S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem,, \emph{J. Differential Equations}, 171 (2001), 294. doi: 10.1006/jdeq.2000.3846. [16] R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems,, \emph{Trans. Amer. Math. Soc}., 321 (1990), 1. doi: 10.2307/2001590. [17] J. D. Murray, Mathematical Biology, I and II,, third edn., (2002). [18] S. Ruan and W. Wang, Dynamical behavior of an epidemic models with a nonlinear incidence rate,, \emph{J. Differential Equations}, 188 (2003), 135. doi: 10.1016/S0022-0396(02)00089-X. [19] H. Wang, Spreading speeds and traveling waves for non-cooperative reaction-diffusion stsyems,, \emph{J. Nonlinear Sciences}, 21 (2011), 747. doi: 10.1007/s00332-011-9099-9. [20] H. Wang and X.-S. Wang, Travelling waves phenomena in a Kermack-McKendrick SIR model,, \emph{J. Dyn. Diff. Equat}., (): 10884. [21] W. Wang and X.-Q. Zhao, A nonlocal and time-delayed reaction-diffusion model of dengue transmission,, \emph{SIAM J. Appl. Math}., 71 (2011), 147. doi: 10.1137/090775890. [22] X.-S. Wang, H. Wang and J. Wu, Travelling waves of diffusive predator-prey systems: disease outbreak propagation,, \emph{Discrete Contin. Dyn. Sys}., 32 (2012), 3303. 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,, \emph{Proc. R. Soc. Lond. (Ser. A)}, 466 (2010), 237. doi: 10.1098/rspa.2009.0377. [24] Z.-C. Wang and J. Wu, Travelling waves in a bio-reactor model with stage-structure,, \emph{J. Math. Anal. Appl}., 385 (2012), 683. doi: 10.1016/j.jmaa.2011.06.084. [25] Z.-C. Wang, J. Wu and R. Liu, Traveling waves of Avian influenza spread,, \emph{Proc. Amer. Math. Soc}., 149 (2012), 3931. 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,, \emph{J. Differential Equations}, 229 (2006), 270. doi: 10.1016/j.jde.2006.01.020. [27] J. Wu, Theory and Applications of Partial Functional Differential Equations,, Springer, (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,, \emph{Math. Biosci}., 208 (2007), 419. 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,, \emph{Nonlinear Analysis: Real World Applications}, 10 (2009), 3175. doi: 10.1016/j.nonrwa.2008.10.013. [30] Z. Xu, Traveling waves in a Kermack-Mckendrick epidemic model with diffusion and latent period,, \emph{Nonlinear Analysis}, 111 (2014), 66. 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,, \emph{Discrete Contin. Dyn. Sys. (Ser.B)}, 13 (2010), 195. 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,, \emph{Nonlinear Analysis: Real World Applications}, 13 (2012), 1429. doi: 10.1016/j.nonrwa.2011.11.007. [33] T. Zhang and W. Wang, Existence of thaveling wave solutions for influenza model with treatment,, \emph{J. Math. Anal. Appl}., 419 (2014), 469. 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,, \emph{Commun. Nonlinear Sci. Numer. Simul}., (1-3) (2015), 1. doi: 10.1016/j.cnsns.2014.07.005. [2] F. Braner and C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology,, Springer, (2001). doi: 10.1007/978-1-4757-3516-1. [3] V. Capasso and G. Serio, A generalization of the Kermack-Mckendrick deterministic epidemic model,, \emph{Math. Biosci}., 42 (1978), 43. doi: 10.1016/0025-5564(78)90006-8. [4] J. Carr and A. Chmaj, Uniquence of the travling waves for nonlocal monostabe equations,, \emph{Proc. Amer. Math. Soc}., 132 (2004), 2433. doi: 10.1090/S0002-9939-04-07432-5. [5] J. Fang and X.-Q. Zhao, Monotone wavefronts for partially degenerate reaction-diffusion systems,, \emph{J. Dyn. Diff. Equat}., 21 (2009), 663. doi: 10.1007/s10884-009-9152-7. [6] H. W. Hethcote, The mathematics of infectious diseases,, \emph{SIAM Review}, 42 (2000), 599. doi: 10.1137/S0036144500371907. [7] H. W. Hethcote and van den Driessche, Some epidemiological models with nonlinear incidence,, \emph{J. Math. Biol.}, 29 (1991), 271. doi: 10.1007/BF00160539. [8] Y. Hosono and B. Ilyas, Traveling waves for a simple diffusive epidemic model,, \emph{Math. Models. Methods Appl. Sci}., 5 (1995), 935. doi: 10.1142/S0218202595000504. [9] J. Huang and X. Zou, Existence of traveling wavefronts of delayed reaction diffusion systems without monotonicity,, \emph{Discrete Contin. Dyn. Sys}., 9 (2003), 925. doi: 10.3934/dcds.2003.9.925. [10] W. O. Kermack and A. G. McKendrick, Contribution to the mathematical theory of epidemics,, \emph{Proc. R. Soc. Lond. (Ser. A)}, 115 (1927), 700. [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,, \emph{Nonlinearity}, 19 (2006), 1253. 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,, \emph{Discrete Contin. Dyn. Sys. (Ser.B)}, 19 (2014), 467. 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,, \emph{Comm. Pure. Appl. Math}., 60 (2007), 1. 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,, \emph{Math. Biosci}., 251 (2014), 16. doi: 10.1016/j.mbs.2014.02.005. [15] S. Ma, Traveling wavefronts for delayed reaction-diffusion systems via a fixed point theorem,, \emph{J. Differential Equations}, 171 (2001), 294. doi: 10.1006/jdeq.2000.3846. [16] R. H. Martin and H. L. Smith, Abstract functional differential equations and reaction-diffusion systems,, \emph{Trans. Amer. Math. Soc}., 321 (1990), 1. doi: 10.2307/2001590. [17] J. D. Murray, Mathematical Biology, I and II,, third edn., (2002). [18] S. Ruan and W. Wang, Dynamical behavior of an epidemic models with a nonlinear incidence rate,, \emph{J. Differential Equations}, 188 (2003), 135. doi: 10.1016/S0022-0396(02)00089-X. [19] H. Wang, Spreading speeds and traveling waves for non-cooperative reaction-diffusion stsyems,, \emph{J. Nonlinear Sciences}, 21 (2011), 747. doi: 10.1007/s00332-011-9099-9. [20] H. Wang and X.-S. Wang, Travelling waves phenomena in a Kermack-McKendrick SIR model,, \emph{J. Dyn. Diff. Equat}., (): 10884. [21] W. Wang and X.-Q. Zhao, A nonlocal and time-delayed reaction-diffusion model of dengue transmission,, \emph{SIAM J. Appl. Math}., 71 (2011), 147. doi: 10.1137/090775890. [22] X.-S. Wang, H. Wang and J. Wu, Travelling waves of diffusive predator-prey systems: disease outbreak propagation,, \emph{Discrete Contin. Dyn. Sys}., 32 (2012), 3303. 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,, \emph{Proc. R. Soc. Lond. (Ser. A)}, 466 (2010), 237. doi: 10.1098/rspa.2009.0377. [24] Z.-C. Wang and J. Wu, Travelling waves in a bio-reactor model with stage-structure,, \emph{J. Math. Anal. Appl}., 385 (2012), 683. doi: 10.1016/j.jmaa.2011.06.084. [25] Z.-C. Wang, J. Wu and R. Liu, Traveling waves of Avian influenza spread,, \emph{Proc. Amer. Math. Soc}., 149 (2012), 3931. 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,, \emph{J. Differential Equations}, 229 (2006), 270. doi: 10.1016/j.jde.2006.01.020. [27] J. Wu, Theory and Applications of Partial Functional Differential Equations,, Springer, (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,, \emph{Math. Biosci}., 208 (2007), 419. 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,, \emph{Nonlinear Analysis: Real World Applications}, 10 (2009), 3175. doi: 10.1016/j.nonrwa.2008.10.013. [30] Z. Xu, Traveling waves in a Kermack-Mckendrick epidemic model with diffusion and latent period,, \emph{Nonlinear Analysis}, 111 (2014), 66. 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,, \emph{Discrete Contin. Dyn. Sys. (Ser.B)}, 13 (2010), 195. 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,, \emph{Nonlinear Analysis: Real World Applications}, 13 (2012), 1429. doi: 10.1016/j.nonrwa.2011.11.007. [33] T. Zhang and W. Wang, Existence of thaveling wave solutions for influenza model with treatment,, \emph{J. Math. Anal. Appl}., 419 (2014), 469. doi: 10.1016/j.jmaa.2014.04.068.
 [1] Wei Ding, Wenzhang Huang, Siroj Kansakar. Traveling wave solutions for a diffusive sis epidemic model. Discrete & Continuous Dynamical Systems - B, 2013, 18 (5) : 1291-1304. doi: 10.3934/dcdsb.2013.18.1291 [2] Zhaosheng Feng, Goong Chen. Traveling wave solutions in parametric forms for a diffusion model with a nonlinear rate of growth. Discrete & Continuous Dynamical Systems - A, 2009, 24 (3) : 763-780. doi: 10.3934/dcds.2009.24.763 [3] Jiamin Cao, Peixuan Weng. Single spreading speed and traveling wave solutions of a diffusive pioneer-climax model without cooperative property. Communications on Pure & Applied Analysis, 2017, 16 (4) : 1405-1426. doi: 10.3934/cpaa.2017067 [4] Zhixing Hu, Ping Bi, Wanbiao Ma, Shigui Ruan. Bifurcations of an SIRS epidemic model with nonlinear incidence rate. Discrete & Continuous Dynamical Systems - B, 2011, 15 (1) : 93-112. doi: 10.3934/dcdsb.2011.15.93 [5] Zhiting Xu. Traveling waves in an SEIR epidemic model with the variable total population. Discrete & Continuous Dynamical Systems - B, 2016, 21 (10) : 3723-3742. doi: 10.3934/dcdsb.2016118 [6] Chufen Wu, Peixuan Weng. Asymptotic speed of propagation and traveling wavefronts for a SIR epidemic model. Discrete & Continuous Dynamical Systems - B, 2011, 15 (3) : 867-892. doi: 10.3934/dcdsb.2011.15.867 [7] Dashun Xu, Xiao-Qiang Zhao. Asymptotic speed of spread and traveling waves for a nonlocal epidemic model. Discrete & Continuous Dynamical Systems - B, 2005, 5 (4) : 1043-1056. doi: 10.3934/dcdsb.2005.5.1043 [8] Junhao Wen, Peixuan Weng. Traveling wave solutions in a diffusive producer-scrounger model. Discrete & Continuous Dynamical Systems - B, 2017, 22 (2) : 627-645. doi: 10.3934/dcdsb.2017030 [9] Jiang Liu, Xiaohui Shang, Zengji Du. Traveling wave solutions of a reaction-diffusion predator-prey model. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1063-1078. doi: 10.3934/dcdss.2017057 [10] Liang Zhang, Bingtuan Li. Traveling wave solutions in an integro-differential competition model. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 417-428. doi: 10.3934/dcdsb.2012.17.417 [11] Bang-Sheng Han, Zhi-Cheng Wang. Traveling wave solutions in a nonlocal reaction-diffusion population model. Communications on Pure & Applied Analysis, 2016, 15 (3) : 1057-1076. doi: 10.3934/cpaa.2016.15.1057 [12] Wan-Tong Li, Guo Lin, Cong Ma, Fei-Ying Yang. Traveling wave solutions of a nonlocal delayed SIR model without outbreak threshold. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 467-484. doi: 10.3934/dcdsb.2014.19.467 [13] F. Berezovskaya, Erika Camacho, Stephen Wirkus, Georgy Karev. "Traveling wave'' solutions of Fitzhugh model with cross-diffusion. Mathematical Biosciences & Engineering, 2008, 5 (2) : 239-260. doi: 10.3934/mbe.2008.5.239 [14] Guo Lin, Wan-Tong Li. Traveling wave solutions of a competitive recursion. Discrete & Continuous Dynamical Systems - B, 2012, 17 (1) : 173-189. doi: 10.3934/dcdsb.2012.17.173 [15] Linghai Zhang. Wave speed analysis of traveling wave fronts in delayed synaptically coupled neuronal networks. Discrete & Continuous Dynamical Systems - A, 2014, 34 (5) : 2405-2450. doi: 10.3934/dcds.2014.34.2405 [16] Xichao Duan, Sanling Yuan, Kaifa Wang. Dynamics of a diffusive age-structured HBV model with saturating incidence. Mathematical Biosciences & Engineering, 2016, 13 (5) : 935-968. doi: 10.3934/mbe.2016024 [17] Shouying Huang, Jifa Jiang. Global stability of a network-based SIS epidemic model with a general nonlinear incidence rate. Mathematical Biosciences & Engineering, 2016, 13 (4) : 723-739. doi: 10.3934/mbe.2016016 [18] Jian Yang, Bendong Lou. Traveling wave solutions of competitive models with free boundaries. Discrete & Continuous Dynamical Systems - B, 2014, 19 (3) : 817-826. doi: 10.3934/dcdsb.2014.19.817 [19] Bingtuan Li. Some remarks on traveling wave solutions in competition models. Discrete & Continuous Dynamical Systems - B, 2009, 12 (2) : 389-399. doi: 10.3934/dcdsb.2009.12.389 [20] 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

2017 Impact Factor: 0.884