# American Institute of Mathematical Sciences

doi: 10.3934/era.2020118

## Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment

 Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu 730050, China

* Corresponding author: Hai-Feng Huo

Received  August 2020 Revised  October 2020 Published  November 2020

Fund Project: This work is supported by the National Natural Science Foundation of China (11861044 and 11661050), and the HongLiu first-class disciplines Development Program of Lanzhou University of Technology

A reaction-diffusion SEIR model, including the self-protection for susceptible individuals, treatments for infectious individuals and constant recruitment, is introduced. The existence of traveling wave solution, which is determined by the basic reproduction number $R_0$ and wave speed $c,$ is firstly proved as $R_0>1$ and $c\geq c^*$ via the Schauder fixed point theorem, where $c^*$ is minimal wave speed. Asymptotic behavior of traveling wave solution at infinity is also proved by applying the Lyapunov functional. Furthermore, when $R_0\leq1$ or $R_0>1$ with $c\in(0,\ c^*),$ we derive the non-existence of traveling wave solution with utilizing two-sides Laplace transform. We take advantage of numerical simulations to indicate the existence of traveling wave, and show that self-protection and treatment can reduce the spread speed at last.

Citation: Hai-Feng Huo, Shi-Ke Hu, Hong Xiang. Traveling wave solution for a diffusion SEIR epidemic model with self-protection and treatment. Electronic Research Archive, doi: 10.3934/era.2020118
##### References:

show all references

##### References:
The numerical simulations of existence for traveling wave solution of system (2)
Cross section curve of traveling wave solution for system (2) as $t = 200.$
Show the effects of self-protection $\sigma$ and treatment $\theta$ on minimal spread speed $c^*,$ where $\sigma$ and $\theta$ are taken from $0.05$ to $1.$
 [1] Lateef Olakunle Jolaoso, Maggie Aphane. Bregman subgradient extragradient method with monotone self-adjustment stepsize for solving pseudo-monotone variational inequalities and fixed point problems. Journal of Industrial & Management Optimization, 2020  doi: 10.3934/jimo.2020178 [2] Christian Beck, Lukas Gonon, Martin Hutzenthaler, Arnulf Jentzen. On existence and uniqueness properties for solutions of stochastic fixed point equations. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020320 [3] Peng Luo. Comparison theorem for diagonally quadratic BSDEs. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020374 [4] Marc Homs-Dones. A generalization of the Babbage functional equation. Discrete & Continuous Dynamical Systems - A, 2021, 41 (2) : 899-919. doi: 10.3934/dcds.2020303 [5] Lars Grüne. Computing Lyapunov functions using deep neural networks. Journal of Computational Dynamics, 2020  doi: 10.3934/jcd.2021006 [6] Peter Giesl, Sigurdur Hafstein. System specific triangulations for the construction of CPA Lyapunov functions. Discrete & Continuous Dynamical Systems - B, 2020  doi: 10.3934/dcdsb.2020378 [7] Fabian Ziltener. Note on coisotropic Floer homology and leafwise fixed points. Electronic Research Archive, , () : -. doi: 10.3934/era.2021001 [8] Peter Giesl, Zachary Langhorne, Carlos Argáez, Sigurdur Hafstein. Computing complete Lyapunov functions for discrete-time dynamical systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (1) : 299-336. doi: 10.3934/dcdsb.2020331 [9] Yoshitsugu Kabeya. Eigenvalues of the Laplace-Beltrami operator under the homogeneous Neumann condition on a large zonal domain in the unit sphere. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3529-3559. doi: 10.3934/dcds.2020040 [10] Ugo Bessi. Another point of view on Kusuoka's measure. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020404 [11] Xin Zhang, Jie Xiong, Shuaiqi Zhang. Optimal reinsurance-investment and dividends problem with fixed transaction costs. Journal of Industrial & Management Optimization, 2021, 17 (2) : 981-999. doi: 10.3934/jimo.2020008 [12] Isabeau Birindelli, Françoise Demengel, Fabiana Leoni. Boundary asymptotics of the ergodic functions associated with fully nonlinear operators through a Liouville type theorem. Discrete & Continuous Dynamical Systems - A, 2020  doi: 10.3934/dcds.2020395 [13] Sumit Arora, Manil T. Mohan, Jaydev Dabas. Approximate controllability of a Sobolev type impulsive functional evolution system in Banach spaces. Mathematical Control & Related Fields, 2020  doi: 10.3934/mcrf.2020049 [14] Shuang Liu, Yuan Lou. A functional approach towards eigenvalue problems associated with incompressible flow. Discrete & Continuous Dynamical Systems - A, 2020, 40 (6) : 3715-3736. doi: 10.3934/dcds.2020028 [15] Alberto Bressan, Wen Shen. A posteriori error estimates for self-similar solutions to the Euler equations. Discrete & Continuous Dynamical Systems - A, 2021, 41 (1) : 113-130. doi: 10.3934/dcds.2020168 [16] Parikshit Upadhyaya, Elias Jarlebring, Emanuel H. Rubensson. A density matrix approach to the convergence of the self-consistent field iteration. Numerical Algebra, Control & Optimization, 2021, 11 (1) : 99-115. doi: 10.3934/naco.2020018 [17] Fanni M. Sélley. A self-consistent dynamical system with multiple absolutely continuous invariant measures. Journal of Computational Dynamics, 2021, 8 (1) : 9-32. doi: 10.3934/jcd.2021002 [18] Gang Luo, Qingzhi Yang. The point-wise convergence of shifted symmetric higher order power method. Journal of Industrial & Management Optimization, 2021, 17 (1) : 357-368. doi: 10.3934/jimo.2019115 [19] Meng Ding, Ting-Zhu Huang, Xi-Le Zhao, Michael K. Ng, Tian-Hui Ma. Tensor train rank minimization with nonlocal self-similarity for tensor completion. Inverse Problems & Imaging, , () : -. doi: 10.3934/ipi.2021001 [20] Sabira El Khalfaoui, Gábor P. Nagy. On the dimension of the subfield subcodes of 1-point Hermitian codes. Advances in Mathematics of Communications, 2021, 15 (2) : 219-226. doi: 10.3934/amc.2020054

Impact Factor: 0.263