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

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

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.
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

[1]

Yu Yang, Jinling Zhou, Cheng-Hsiung Hsu. Critical traveling wave solutions for a vaccination model with general incidence. Discrete & Continuous Dynamical Systems - B, 2021  doi: 10.3934/dcdsb.2021087

[2]

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

[3]

Wei-Jian Bo, Guo Lin, Shigui Ruan. Traveling wave solutions for time periodic reaction-diffusion systems. Discrete & Continuous Dynamical Systems, 2018, 38 (9) : 4329-4351. doi: 10.3934/dcds.2018189

[4]

Kazeem Olalekan Aremu, Chinedu Izuchukwu, Grace Nnenanya Ogwo, Oluwatosin Temitope Mewomo. Multi-step iterative algorithm for minimization and fixed point problems in p-uniformly convex metric spaces. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2161-2180. doi: 10.3934/jimo.2020063

[5]

Chin-Chin Wu. Existence of traveling wavefront for discrete bistable competition model. Discrete & Continuous Dynamical Systems - B, 2011, 16 (3) : 973-984. doi: 10.3934/dcdsb.2011.16.973

[6]

Shangzhi Li, Shangjiang Guo. Permanence and extinction of a stochastic SIS epidemic model with three independent Brownian motions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2693-2719. doi: 10.3934/dcdsb.2020201

[7]

Christos Sourdis. A Liouville theorem for ancient solutions to a semilinear heat equation and its elliptic counterpart. Electronic Research Archive, , () : -. doi: 10.3934/era.2021016

[8]

Yan Zhang, Peibiao Zhao, Xinghu Teng, Lei Mao. Optimal reinsurance and investment strategies for an insurer and a reinsurer under Hestons SV model: HARA utility and Legendre transform. Journal of Industrial & Management Optimization, 2021, 17 (4) : 2139-2159. doi: 10.3934/jimo.2020062

[9]

Renhao Cui. Asymptotic profiles of the endemic equilibrium of a reaction-diffusion-advection SIS epidemic model with saturated incidence rate. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2997-3022. doi: 10.3934/dcdsb.2020217

[10]

Juntao Sun, Tsung-fang Wu. The number of nodal solutions for the Schrödinger–Poisson system under the effect of the weight function. Discrete & Continuous Dynamical Systems, 2021, 41 (8) : 3651-3682. doi: 10.3934/dcds.2021011

[11]

Alberto Ibort, Alberto López-Yela. Quantum tomography and the quantum Radon transform. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021021

[12]

Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems, 2021, 41 (6) : 2543-2557. doi: 10.3934/dcds.2020374

[13]

Thomas Alazard. A minicourse on the low Mach number limit. Discrete & Continuous Dynamical Systems - S, 2008, 1 (3) : 365-404. doi: 10.3934/dcdss.2008.1.365

[14]

José Raúl Quintero, Juan Carlos Muñoz Grajales. On the existence and computation of periodic travelling waves for a 2D water wave model. Communications on Pure & Applied Analysis, 2018, 17 (2) : 557-578. doi: 10.3934/cpaa.2018030

[15]

Naeem M. H. Alkoumi, Pedro J. Torres. Estimates on the number of limit cycles of a generalized Abel equation. Discrete & Continuous Dynamical Systems, 2011, 31 (1) : 25-34. doi: 10.3934/dcds.2011.31.25

[16]

Yuta Ishii, Kazuhiro Kurata. Existence of multi-peak solutions to the Schnakenberg model with heterogeneity on metric graphs. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021035

[17]

Haibo Cui, Haiyan Yin. Convergence rate of solutions toward stationary solutions to the isentropic micropolar fluid model in a half line. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2899-2920. doi: 10.3934/dcdsb.2020210

[18]

Lara Abi Rizk, Jean-Baptiste Burie, Arnaud Ducrot. Asymptotic speed of spread for a nonlocal evolutionary-epidemic system. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021064

[19]

Ugo Bessi. Another point of view on Kusuoka's measure. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3241-3271. doi: 10.3934/dcds.2020404

[20]

Umberto Biccari. Internal control for a non-local Schrödinger equation involving the fractional Laplace operator. Evolution Equations & Control Theory, 2021  doi: 10.3934/eect.2021014

2019 Impact Factor: 1.27

Metrics

  • PDF downloads (91)
  • HTML views (0)
  • Cited by (7)

Other articles
by authors

[Back to Top]