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December  2016, 21(10): 3723-3742. doi: 10.3934/dcdsb.2016118

Traveling waves in an SEIR epidemic model with the variable total population

Zhiting Xu 1,

1. 

School of Mathematical Sciences, South China Normal University, Guangzhou, Guangdong 510631

Received  December 2015 Revised  February 2016 Published  November 2016

In the present paper, we propose a simple diffusive SEIR epidemic model where the total population is variable. We first give the explicit formula of the basic reproduction number $\mathcal{R}_0$ for the model. And hence, we show that if $\mathcal{R}_0>1$, then there exists a constant $c^*>0$ such that for any $c>c^*$, the model admits a nontrivial traveling wave solution, and if $\mathcal{R}_0<1$ and $c>0$ (or, $\mathcal{R}_0>1$ and $c\in(0,c^*)$), then the model has no nontrivial traveling wave solution. Consequently, we obtain the full information about the existence and non-existence of traveling wave solutions of the model by determined by the constants $\mathcal{R}_0$ and $c^*$. The proof of the main results is mainly based on Schauder fixed point theorem and Laplace transform.
Keywords: Laplace transform., the basic reproduction number, SEIR epidemic model, Traveling wave solutions, Schauder fixed point theorem.
Mathematics Subject Classification: Primary: 35C07, 92D30; Secondary: 35K5.
Citation: Zhiting Xu. Traveling waves in an SEIR epidemic model with the variable total population. Discrete and Continuous Dynamical Systems - B, 2016, 21 (10) : 3723-3742. doi: 10.3934/dcdsb.2016118
References:
[1]

Z. Bai and S.-L. Wu, Traveling waves in a delayed SIR epidemic model with nonlinear incidence, Appl. Math. Comput., 263 (2015), 221-232. doi: 10.1016/j.amc.2015.04.048.

[2]

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., 22 (2015), 1370-1381. doi: 10.1016/j.cnsns.2014.07.005.

[3]

H. Berestycki, F. Hamel, A. Kiselev and L. Ryzhik, Quenching and propagation in KPP reaction-diffusion equations with a heat loss, Arch. Ration. Mech. Anal., 178 (2005), 57-80. doi: 10.1007/s00205-005-0367-4.

[4]

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.

[5]

J. Carr and A. Chmaj, Uniquence of the traveling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439. doi: 10.1090/S0002-9939-04-07432-5.

[6]

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.

[7]

S.-C. Fu, Traveling waves for a diffusive SIR model with delay, J. Math. Anal. Appl., 435 (2016), 20-37. doi: 10.1016/j.jmaa.2015.09.069.

[8]

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

[9]

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.

[10]

C.-H. Hsu and T.-S. Yang, Existence, uniqueness, monotonicity and asymptotic behaviour of travelling waves for epidemic models, Nonlinearity, 26 (2013), 121-139. doi: 10.1088/0951-7715/26/1/121.

[11]

W. O. Kermack and A. G. McKendrick, Contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond., B. 115 (1927), 700-721.

[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]

J. D. Murray, Mathematical Biology, I and II, third edition. Springer, New York, 2002.

[14]

L. Perko, Differential Equations and Dynamical Systems, third edition, Springer, New York, 2001. doi: 10.1007/978-1-4613-0003-8.

[15]

H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations, 195 (2003), 430-470. doi: 10.1016/S0022-0396(03)00175-X.

[16]

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.

[17]

H. Wang and X.-S. Wang, Travelling waves phenomena in a Kermack-McKendrick SIR model, J. Dyn. Diff. Equat., 28 (2016), 143-166. doi: 10.1007/s10884-015-9506-2.

[18]

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.

[19]

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.

[20]

Z.-C. Wang and J. Wu, Travelling waves of a diffusive Kermack-McKendrick epidemic model with non-local delayed transmission, Proc. R. Soc., A, 466 (2010), 237-261. doi: 10.1098/rspa.2009.0377.

[21]

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.

[22]

S.-L. Wu and C.-H. Hsu, Existence of entire solutions for delayed monostable epidemic models, Trans. Amer. Math. Soc., 368 (2016), 6033-6062. doi: 10.1090/tran/6526.

[23]

S.-L. Wu, C.-H. Hsu and Y. Xiao, Global attractivity, spreading speeds and traveling wave of delayed nonlocal reaction-diffusion systems, J. Differential Equations, 258 (2015), 1058-1105. doi: 10.1016/j.jde.2014.10.009.

[24]

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.

[25]

Z. Xu, Traveling waves for a diffusive SEIR epidemic model, Commun. Pure Appl. Anal. 15 (2016), 871-892. doi: 10.3934/cpaa.2016.15.871.

[26]

Z. Xu, C. Ai, Traveling waves in a diffusive influenza epidemic model with vaccination, Appl. Math. Modelling. 40 (2016), 7265-7280. doi: 10.1016/j.apm.2016.03.021.

[27]

F.-Y. Yang, Y. Li, W.-T. Li and Z.-C. Wang, Traveling waves in a nonlocal dispersal Kermack-Mckendrick epidmic model, Discrete Contin. Dyn. Sys., Ser.B. 18 (2013), 1969-1993. doi: 10.3934/dcdsb.2013.18.1969.

[28]

F.-Y. Yang, Y. Li, W.-T. Li and Z.-C. Wang, Traveling waves in a nonlocal dispersal SIR epidmic model, Nonlinear Analysis: Real World Applications, 23 (2015), 129-147. doi: 10.1016/j.nonrwa.2014.12.001.

[29]

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.-L. Wu, Traveling waves in a delayed SIR epidemic model with nonlinear incidence, Appl. Math. Comput., 263 (2015), 221-232. doi: 10.1016/j.amc.2015.04.048.

[2]

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., 22 (2015), 1370-1381. doi: 10.1016/j.cnsns.2014.07.005.

[3]

H. Berestycki, F. Hamel, A. Kiselev and L. Ryzhik, Quenching and propagation in KPP reaction-diffusion equations with a heat loss, Arch. Ration. Mech. Anal., 178 (2005), 57-80. doi: 10.1007/s00205-005-0367-4.

[4]

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.

[5]

J. Carr and A. Chmaj, Uniquence of the traveling waves for nonlocal monostable equations, Proc. Amer. Math. Soc., 132 (2004), 2433-2439. doi: 10.1090/S0002-9939-04-07432-5.

[6]

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.

[7]

S.-C. Fu, Traveling waves for a diffusive SIR model with delay, J. Math. Anal. Appl., 435 (2016), 20-37. doi: 10.1016/j.jmaa.2015.09.069.

[8]

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

[9]

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.

[10]

C.-H. Hsu and T.-S. Yang, Existence, uniqueness, monotonicity and asymptotic behaviour of travelling waves for epidemic models, Nonlinearity, 26 (2013), 121-139. doi: 10.1088/0951-7715/26/1/121.

[11]

W. O. Kermack and A. G. McKendrick, Contribution to the mathematical theory of epidemics, Proc. R. Soc. Lond., B. 115 (1927), 700-721.

[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]

J. D. Murray, Mathematical Biology, I and II, third edition. Springer, New York, 2002.

[14]

L. Perko, Differential Equations and Dynamical Systems, third edition, Springer, New York, 2001. doi: 10.1007/978-1-4613-0003-8.

[15]

H. R. Thieme and X.-Q. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differential Equations, 195 (2003), 430-470. doi: 10.1016/S0022-0396(03)00175-X.

[16]

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.

[17]

H. Wang and X.-S. Wang, Travelling waves phenomena in a Kermack-McKendrick SIR model, J. Dyn. Diff. Equat., 28 (2016), 143-166. doi: 10.1007/s10884-015-9506-2.

[18]

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.

[19]

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.

[20]

Z.-C. Wang and J. Wu, Travelling waves of a diffusive Kermack-McKendrick epidemic model with non-local delayed transmission, Proc. R. Soc., A, 466 (2010), 237-261. doi: 10.1098/rspa.2009.0377.

[21]

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.

[22]

S.-L. Wu and C.-H. Hsu, Existence of entire solutions for delayed monostable epidemic models, Trans. Amer. Math. Soc., 368 (2016), 6033-6062. doi: 10.1090/tran/6526.

[23]

S.-L. Wu, C.-H. Hsu and Y. Xiao, Global attractivity, spreading speeds and traveling wave of delayed nonlocal reaction-diffusion systems, J. Differential Equations, 258 (2015), 1058-1105. doi: 10.1016/j.jde.2014.10.009.

[24]

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.

[25]

Z. Xu, Traveling waves for a diffusive SEIR epidemic model, Commun. Pure Appl. Anal. 15 (2016), 871-892. doi: 10.3934/cpaa.2016.15.871.

[26]

Z. Xu, C. Ai, Traveling waves in a diffusive influenza epidemic model with vaccination, Appl. Math. Modelling. 40 (2016), 7265-7280. doi: 10.1016/j.apm.2016.03.021.

[27]

F.-Y. Yang, Y. Li, W.-T. Li and Z.-C. Wang, Traveling waves in a nonlocal dispersal Kermack-Mckendrick epidmic model, Discrete Contin. Dyn. Sys., Ser.B. 18 (2013), 1969-1993. doi: 10.3934/dcdsb.2013.18.1969.

[28]

F.-Y. Yang, Y. Li, W.-T. Li and Z.-C. Wang, Traveling waves in a nonlocal dispersal SIR epidmic model, Nonlinear Analysis: Real World Applications, 23 (2015), 129-147. doi: 10.1016/j.nonrwa.2014.12.001.

[29]

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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