# American Institute of Mathematical Sciences

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 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.

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##### 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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