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.  Google Scholar

[2]

F. Braner and C. Castillo-Chávez, Mathematical Models in Population Biology and Epidemiology,, Springer, (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,, \emph{Math. Biosci}., 42 (1978), 43.  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,, \emph{Proc. Amer. Math. Soc}., 132 (2004), 2433.  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,, \emph{J. Dyn. Diff. Equat}., 21 (2009), 663.  doi: 10.1007/s10884-009-9152-7.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   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,, \emph{Nonlinearity}, 19 (2006), 1253.  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,, \emph{Discrete Contin. Dyn. Sys. (Ser.B)}, 19 (2014), 467.  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,, \emph{Comm. Pure. Appl. Math}., 60 (2007), 1.  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,, \emph{Math. Biosci}., 251 (2014), 16.  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,, \emph{J. Differential Equations}, 171 (2001), 294.  doi: 10.1006/jdeq.2000.3846.  Google Scholar

[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.  Google Scholar

[17]

J. D. Murray, Mathematical Biology, I and II,, third edn., (2002).   Google Scholar

[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.  Google Scholar

[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.  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,, \emph{SIAM J. Appl. Math}., 71 (2011), 147.  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,, \emph{Discrete Contin. Dyn. Sys}., 32 (2012), 3303.  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,, \emph{Proc. R. Soc. Lond. (Ser. A)}, 466 (2010), 237.  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,, \emph{J. Math. Anal. Appl}., 385 (2012), 683.  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,, \emph{Proc. Amer. Math. Soc}., 149 (2012), 3931.  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,, \emph{J. Differential Equations}, 229 (2006), 270.  doi: 10.1016/j.jde.2006.01.020.  Google Scholar

[27]

J. Wu, Theory and Applications of Partial Functional Differential Equations,, Springer, (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,, \emph{Math. Biosci}., 208 (2007), 419.  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,, \emph{Nonlinear Analysis: Real World Applications}, 10 (2009), 3175.  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,, \emph{Nonlinear Analysis}, 111 (2014), 66.  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,, \emph{Discrete Contin. Dyn. Sys. (Ser.B)}, 13 (2010), 195.  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,, \emph{Nonlinear Analysis: Real World Applications}, 13 (2012), 1429.  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,, \emph{J. Math. Anal. Appl}., 419 (2014), 469.  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,, \emph{Commun. Nonlinear Sci. Numer. Simul}., (1-3) (2015), 1.  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, (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,, \emph{Math. Biosci}., 42 (1978), 43.  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,, \emph{Proc. Amer. Math. Soc}., 132 (2004), 2433.  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,, \emph{J. Dyn. Diff. Equat}., 21 (2009), 663.  doi: 10.1007/s10884-009-9152-7.  Google Scholar

[6]

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

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   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,, \emph{Nonlinearity}, 19 (2006), 1253.  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,, \emph{Discrete Contin. Dyn. Sys. (Ser.B)}, 19 (2014), 467.  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,, \emph{Comm. Pure. Appl. Math}., 60 (2007), 1.  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,, \emph{Math. Biosci}., 251 (2014), 16.  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,, \emph{J. Differential Equations}, 171 (2001), 294.  doi: 10.1006/jdeq.2000.3846.  Google Scholar

[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.  Google Scholar

[17]

J. D. Murray, Mathematical Biology, I and II,, third edn., (2002).   Google Scholar

[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.  Google Scholar

[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.  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,, \emph{SIAM J. Appl. Math}., 71 (2011), 147.  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,, \emph{Discrete Contin. Dyn. Sys}., 32 (2012), 3303.  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,, \emph{Proc. R. Soc. Lond. (Ser. A)}, 466 (2010), 237.  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,, \emph{J. Math. Anal. Appl}., 385 (2012), 683.  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,, \emph{Proc. Amer. Math. Soc}., 149 (2012), 3931.  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,, \emph{J. Differential Equations}, 229 (2006), 270.  doi: 10.1016/j.jde.2006.01.020.  Google Scholar

[27]

J. Wu, Theory and Applications of Partial Functional Differential Equations,, Springer, (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,, \emph{Math. Biosci}., 208 (2007), 419.  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,, \emph{Nonlinear Analysis: Real World Applications}, 10 (2009), 3175.  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,, \emph{Nonlinear Analysis}, 111 (2014), 66.  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,, \emph{Discrete Contin. Dyn. Sys. (Ser.B)}, 13 (2010), 195.  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,, \emph{Nonlinear Analysis: Real World Applications}, 13 (2012), 1429.  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,, \emph{J. Math. Anal. Appl}., 419 (2014), 469.  doi: 10.1016/j.jmaa.2014.04.068.  Google Scholar

[1]

Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, , () : -. doi: 10.3934/era.2020118

[2]

Omid Nikan, Seyedeh Mahboubeh Molavi-Arabshai, Hossein Jafari. Numerical simulation of the nonlinear fractional regularized long-wave model arising in ion acoustic plasma waves. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020466

[3]

Jerry L. Bona, Angel Durán, Dimitrios Mitsotakis. Solitary-wave solutions of Benjamin-Ono and other systems for internal waves. I. approximations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 87-111. doi: 10.3934/dcds.2020215

[4]

Adel M. Al-Mahdi, Mohammad M. Al-Gharabli, Salim A. Messaoudi. New general decay result for a system of viscoelastic wave equations with past history. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2020273

[5]

Siyang Cai, Yongmei Cai, Xuerong Mao. A stochastic differential equation SIS epidemic model with regime switching. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020317

[6]

Ahmad Z. Fino, Wenhui Chen. A global existence result for two-dimensional semilinear strongly damped wave equation with mixed nonlinearity in an exterior domain. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5387-5411. doi: 10.3934/cpaa.2020243

[7]

Vivina Barutello, Gian Marco Canneori, Susanna Terracini. Minimal collision arcs asymptotic to central configurations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 61-86. doi: 10.3934/dcds.2020218

[8]

Zhenzhen Wang, Tianshou Zhou. Asymptotic behaviors and stochastic traveling waves in stochastic Fisher-KPP equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020323

[9]

Wei Feng, Michael Freeze, Xin Lu. On competition models under allee effect: Asymptotic behavior and traveling waves. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5609-5626. doi: 10.3934/cpaa.2020256

[10]

Dan Zhu, Rosemary A. Renaut, Hongwei Li, Tianyou Liu. Fast non-convex low-rank matrix decomposition for separation of potential field data using minimal memory. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2020076

[11]

Laurence Cherfils, Stefania Gatti, Alain Miranville, Rémy Guillevin. Analysis of a model for tumor growth and lactate exchanges in a glioma. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020457

[12]

Laurent Di Menza, Virginie Joanne-Fabre. An age group model for the study of a population of trees. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020464

[13]

Qingfang Wang, Hua Yang. Solutions of nonlocal problem with critical exponent. Communications on Pure & Applied Analysis, 2020, 19 (12) : 5591-5608. doi: 10.3934/cpaa.2020253

[14]

Weiwei Liu, Jinliang Wang, Yuming Chen. Threshold dynamics of a delayed nonlocal reaction-diffusion cholera model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020316

[15]

Yining Cao, Chuck Jia, Roger Temam, Joseph Tribbia. Mathematical analysis of a cloud resolving model including the ice microphysics. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 131-167. doi: 10.3934/dcds.2020219

[16]

Zhouchao Wei, Wei Zhang, Irene Moroz, Nikolay V. Kuznetsov. Codimension one and two bifurcations in Cattaneo-Christov heat flux model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020344

[17]

Shuang Chen, Jinqiao Duan, Ji Li. Effective reduction of a three-dimensional circadian oscillator model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020349

[18]

Barbora Benešová, Miroslav Frost, Lukáš Kadeřávek, Tomáš Roubíček, Petr Sedlák. An experimentally-fitted thermodynamical constitutive model for polycrystalline shape memory alloys. Discrete & Continuous Dynamical Systems - S, 2020  doi: 10.3934/dcdss.2020459

[19]

Cuicui Li, Lin Zhou, Zhidong Teng, Buyu Wen. The threshold dynamics of a discrete-time echinococcosis transmission model. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020339

[20]

Peter Poláčik, Pavol Quittner. Entire and ancient solutions of a supercritical semilinear heat equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 413-438. doi: 10.3934/dcds.2020136

2019 Impact Factor: 1.105

Metrics

  • PDF downloads (67)
  • HTML views (0)
  • Cited by (12)

Other articles
by authors

[Back to Top]