Traveling waves for a diffusive SEIR epidemic model

Zhiting Xu

doi:10.3934/cpaa.2016.15.871

Article Contents

2016, Volume 15, Issue 3: 871-892. Doi: 10.3934/cpaa.2016.15.871

This issue Previous Article A class of generalized quasilinear Schrödinger equations Next Article Qualitative properties of solutions to an integral system associated with the Bessel potential

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: May 2016

Abstract Related Papers Cited by

Abstract

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:

Mathematics Subject Classification: Primary: 35C07, 35K57; Secondary: 92D30.

Citation:

Related Papers

Cited by

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.