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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 & Continuous Dynamical Systems - B, 2016, 21 (10) : 3723-3742. doi: 10.3934/dcdsb.2016118
References:
[1]

Appl. Math. Comput., 263 (2015), 221-232. doi: 10.1016/j.amc.2015.04.048.  Google Scholar

[2]

Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 1370-1381. doi: 10.1016/j.cnsns.2014.07.005.  Google Scholar

[3]

Arch. Ration. Mech. Anal., 178 (2005), 57-80. doi: 10.1007/s00205-005-0367-4.  Google Scholar

[4]

Springer, New York, 2001. doi: 10.1007/978-1-4757-3516-1.  Google Scholar

[5]

Proc. Amer. Math. Soc., 132 (2004), 2433-2439. doi: 10.1090/S0002-9939-04-07432-5.  Google Scholar

[6]

J. Dyn. Diff. Equat., 21 (2009), 663-680. doi: 10.1007/s10884-009-9152-7.  Google Scholar

[7]

J. Math. Anal. Appl., 435 (2016), 20-37. doi: 10.1016/j.jmaa.2015.09.069.  Google Scholar

[8]

SIAM Review, 42 (2000), 599-653. doi: 10.1137/S0036144500371907.  Google Scholar

[9]

Math. Models. Methods Appl. Sci., 5 (1995), 935-966. doi: 10.1142/S0218202595000504.  Google Scholar

[10]

Nonlinearity, 26 (2013), 121-139. doi: 10.1088/0951-7715/26/1/121.  Google Scholar

[11]

Proc. R. Soc. Lond., B. 115 (1927), 700-721. Google Scholar

[12]

Discrete Contin. Dyn. Sys., Ser.B. 19 (2014), 467-484. doi: 10.3934/dcdsb.2014.19.467.  Google Scholar

[13]

Springer, New York, 2002.  Google Scholar

[14]

third edition, Springer, New York, 2001. doi: 10.1007/978-1-4613-0003-8.  Google Scholar

[15]

J. Differential Equations, 195 (2003), 430-470. doi: 10.1016/S0022-0396(03)00175-X.  Google Scholar

[16]

J. Nonlinear Sciences, 21 (2011), 747-783. doi: 10.1007/s00332-011-9099-9.  Google Scholar

[17]

J. Dyn. Diff. Equat., 28 (2016), 143-166. doi: 10.1007/s10884-015-9506-2.  Google Scholar

[18]

SIAM J. Appl. Math., 71 (2011), 147-168. doi: 10.1137/090775890.  Google Scholar

[19]

Discrete Contin. Dyn. Sys., 32 (2012), 3303-3324. doi: 10.3934/dcds.2012.32.3303.  Google Scholar

[20]

Proc. R. Soc., A, 466 (2010), 237-261. doi: 10.1098/rspa.2009.0377.  Google Scholar

[21]

J. Differential Equations, 229 (2006), 270-296. doi: 10.1016/j.jde.2006.01.020.  Google Scholar

[22]

Trans. Amer. Math. Soc., 368 (2016), 6033-6062. doi: 10.1090/tran/6526.  Google Scholar

[23]

J. Differential Equations, 258 (2015), 1058-1105. doi: 10.1016/j.jde.2014.10.009.  Google Scholar

[24]

Nonlinear Analysis, 111 (2014), 66-81. doi: 10.1016/j.na.2014.08.012.  Google Scholar

[25]

Commun. Pure Appl. Anal. 15 (2016), 871-892. doi: 10.3934/cpaa.2016.15.871.  Google Scholar

[26]

Appl. Math. Modelling. 40 (2016), 7265-7280. doi: 10.1016/j.apm.2016.03.021.  Google Scholar

[27]

Discrete Contin. Dyn. Sys., Ser.B. 18 (2013), 1969-1993. doi: 10.3934/dcdsb.2013.18.1969.  Google Scholar

[28]

Nonlinear Analysis: Real World Applications, 23 (2015), 129-147. doi: 10.1016/j.nonrwa.2014.12.001.  Google Scholar

[29]

J. Math. Anal. Appl., 419 (2014), 469-495. doi: 10.1016/j.jmaa.2014.04.068.  Google Scholar

show all references

References:
[1]

Appl. Math. Comput., 263 (2015), 221-232. doi: 10.1016/j.amc.2015.04.048.  Google Scholar

[2]

Commun. Nonlinear Sci. Numer. Simul., 22 (2015), 1370-1381. doi: 10.1016/j.cnsns.2014.07.005.  Google Scholar

[3]

Arch. Ration. Mech. Anal., 178 (2005), 57-80. doi: 10.1007/s00205-005-0367-4.  Google Scholar

[4]

Springer, New York, 2001. doi: 10.1007/978-1-4757-3516-1.  Google Scholar

[5]

Proc. Amer. Math. Soc., 132 (2004), 2433-2439. doi: 10.1090/S0002-9939-04-07432-5.  Google Scholar

[6]

J. Dyn. Diff. Equat., 21 (2009), 663-680. doi: 10.1007/s10884-009-9152-7.  Google Scholar

[7]

J. Math. Anal. Appl., 435 (2016), 20-37. doi: 10.1016/j.jmaa.2015.09.069.  Google Scholar

[8]

SIAM Review, 42 (2000), 599-653. doi: 10.1137/S0036144500371907.  Google Scholar

[9]

Math. Models. Methods Appl. Sci., 5 (1995), 935-966. doi: 10.1142/S0218202595000504.  Google Scholar

[10]

Nonlinearity, 26 (2013), 121-139. doi: 10.1088/0951-7715/26/1/121.  Google Scholar

[11]

Proc. R. Soc. Lond., B. 115 (1927), 700-721. Google Scholar

[12]

Discrete Contin. Dyn. Sys., Ser.B. 19 (2014), 467-484. doi: 10.3934/dcdsb.2014.19.467.  Google Scholar

[13]

Springer, New York, 2002.  Google Scholar

[14]

third edition, Springer, New York, 2001. doi: 10.1007/978-1-4613-0003-8.  Google Scholar

[15]

J. Differential Equations, 195 (2003), 430-470. doi: 10.1016/S0022-0396(03)00175-X.  Google Scholar

[16]

J. Nonlinear Sciences, 21 (2011), 747-783. doi: 10.1007/s00332-011-9099-9.  Google Scholar

[17]

J. Dyn. Diff. Equat., 28 (2016), 143-166. doi: 10.1007/s10884-015-9506-2.  Google Scholar

[18]

SIAM J. Appl. Math., 71 (2011), 147-168. doi: 10.1137/090775890.  Google Scholar

[19]

Discrete Contin. Dyn. Sys., 32 (2012), 3303-3324. doi: 10.3934/dcds.2012.32.3303.  Google Scholar

[20]

Proc. R. Soc., A, 466 (2010), 237-261. doi: 10.1098/rspa.2009.0377.  Google Scholar

[21]

J. Differential Equations, 229 (2006), 270-296. doi: 10.1016/j.jde.2006.01.020.  Google Scholar

[22]

Trans. Amer. Math. Soc., 368 (2016), 6033-6062. doi: 10.1090/tran/6526.  Google Scholar

[23]

J. Differential Equations, 258 (2015), 1058-1105. doi: 10.1016/j.jde.2014.10.009.  Google Scholar

[24]

Nonlinear Analysis, 111 (2014), 66-81. doi: 10.1016/j.na.2014.08.012.  Google Scholar

[25]

Commun. Pure Appl. Anal. 15 (2016), 871-892. doi: 10.3934/cpaa.2016.15.871.  Google Scholar

[26]

Appl. Math. Modelling. 40 (2016), 7265-7280. doi: 10.1016/j.apm.2016.03.021.  Google Scholar

[27]

Discrete Contin. Dyn. Sys., Ser.B. 18 (2013), 1969-1993. doi: 10.3934/dcdsb.2013.18.1969.  Google Scholar

[28]

Nonlinear Analysis: Real World Applications, 23 (2015), 129-147. doi: 10.1016/j.nonrwa.2014.12.001.  Google Scholar

[29]

J. Math. Anal. Appl., 419 (2014), 469-495. doi: 10.1016/j.jmaa.2014.04.068.  Google Scholar

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