POSITIVE TRAVELING WAVES IN A DIFFUSIVE EPIDEMIC SYSTEM WITH DISTRIBUTED DELAY AND CONSTANT EXTERNAL SUPPLIES

Zaili Zhen; Jingdong Wei; Jiangbo Zhou; Lixin Tian

doi:10.11948/20210010

2021 Volume 11 Issue 6

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2021 > 11(6): 2840-2865

Next Article Previous Article

Zaili Zhen, Jingdong Wei, Jiangbo Zhou, Lixin Tian. POSITIVE TRAVELING WAVES IN A DIFFUSIVE EPIDEMIC SYSTEM WITH DISTRIBUTED DELAY AND CONSTANT EXTERNAL SUPPLIES[J]. Journal of Applied Analysis & Computation, 2021, 11(6): 2840-2865. doi: 10.11948/20210010

Citation:

Zaili Zhen, Jingdong Wei, Jiangbo Zhou, Lixin Tian. POSITIVE TRAVELING WAVES IN A DIFFUSIVE EPIDEMIC SYSTEM WITH DISTRIBUTED DELAY AND CONSTANT EXTERNAL SUPPLIES[J]. Journal of Applied Analysis & Computation, 2021, 11(6): 2840-2865. doi: 10.11948/20210010

POSITIVE TRAVELING WAVES IN A DIFFUSIVE EPIDEMIC SYSTEM WITH DISTRIBUTED DELAY AND CONSTANT EXTERNAL SUPPLIES

1.
School of Mathematical Sciences, Jiangsu University, Zhenjiang, Jiangsu 212013, China
2.
Jiangsu Key Laboratory for Numerical Simulation of Large Scale Complex Systems, Nanjing Normal University, Nanjing, Jiangsu 210023, China

Corresponding author: Email: weijingdong@ujs.edu.cn (J. Wei)
Fund Project: We would like to thank Prof. Maoan Han and one anonymous referee for their approval of the present article. This research is supported by grants from National Natural Science Foundation of China (Nos. 12001241 & 11731014), Basic Research Program of Jiangsu Province (No. BK20200885), Jiangsu Key Lab for NSLSCS (No. 202006) and Young Science and Technology Talents Promotion Project for Zhenjiang Science and Technology Association

Abstract

Abstract

This article is concerned with the existence and non-existence of positive traveling wave solutions for a diffusive epidemic system with distributed delay and constant external supplies. By Schauder's fixed point theorem and Lyapunov functional technique, we establish the existence of super-critical positive traveling wave solutions of the system. Meanwhile, applying a limiting method and the theory of asymptotic spreading speeds coupled with comparison principle, we obtain the existence of critical traveling wave solutions for the system. Moreover, utilizing contradictory argument, we deduce the non-existence of traveling wave solutions for the system. These results improve the existing ones in the literature.
- Epidemic system /
- traveling wave /
- distributed delay /
- constant external supplies
MSC: 35K57, 37C65, 92D30

FullText(HTML)

References

[1]	Z. Bai and S. Wu, Traveling waves in a delayed SIR epidemic model with nonlinear incidence, Appl. Math. Comput., 2015, 263, 221-232. Google Scholar
[2]	E. Beretta, T. Hara, W. Ma and Y. Takeuchi, Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonlinear Anal., 2001, 47, 4107-4115. doi: 10.1016/S0362-546X(01)00528-4 CrossRef Google Scholar
[3]	E. Beretta and Y. Takeuchi, Global stability of an SIR epidemic model with time delays, J. Math. Biol., 1995, 33, 250-260. Google Scholar
[4]	E. Beretta and Y. Takeuchi, Convergence results in SIR epidemic models with varying population size, Nonlinear Anal., 1997, 28, 1909-1921. doi: 10.1016/S0362-546X(96)00035-1 CrossRef Google Scholar
[5]	F. Brauer and C. Castillo-Chvez, Mathematical Models in Population Biology and Epidemiology, 2nd edn. Springer, Berlin, 2012. Google Scholar
[6]	K. Brown and J. Carr, Deterministic epidemic waves of critical velocity, Math. Proc. Cambridge. Philos. Soc., 1977, 81, 431-433. doi: 10.1017/S0305004100053494 CrossRef Google Scholar
[7]	A. Chekroun, Existence and asymptotics of traveling wave fronts for a coupled nonlocal diffusion and difference system with delay, Electron. J. Qual. Theo., 2019, 85, 1-23. Google Scholar
[8]	Y. Chen, J. Guo and F. Hamel, Traveling waves for a lattice dynamical system arising in a diffusive endemic model, Nonlinearity, 2017, 30, 2334-2359. doi: 10.1088/1361-6544/aa6b0a CrossRef Google Scholar
[9]	A. Ducrot and P. Magal, Travelling wave solutions for an infection age structured epidemic model with external supplies, Nonlinearity, 2011, 24, 2891-2911. doi: 10.1088/0951-7715/24/10/012 CrossRef Google Scholar
[10]	J. Epstein, Nonlinear Dynamics, Mathematical Biology, and Social Science: Wise Use of Alternative Therapies, Westview Press, Boulder, 1997. Google Scholar
[11]	Y. Enatsu, Y. Nakata and Y. Muroya, Lyapunov functional techniques for the global stability analysis of a delayed SIRS epidemic model, Nonlinear Anal. Real., 2012, 13, 2120-2133. doi: 10.1016/j.nonrwa.2012.01.007 CrossRef Google Scholar
[12]	S. Fu, Traveling waves for a diffusive SIR model with delay, J. Math. Anal. Appl., 2016, 435, 20-37. doi: 10.1016/j.jmaa.2015.09.069 CrossRef Google Scholar
[13]	J. Guo, A. Poh and M. Shimojo, The spreading speed of an SIR epidemic model with nonlocal dispersal, Asymptotic Anal., 2020, 120, 163-174. doi: 10.3233/ASY-191584 CrossRef Google Scholar
[14]	W. Hirsch, H. Hamisch and J. P. Gabriel, Differential equation models of some parasitic infections: Methods for the study of asymptotic behavior, Commun. Pure Appl. Math., 1985, 38, 733-753. doi: 10.1002/cpa.3160380607 CrossRef Google Scholar
[15]	Y. Hosono and B. Ilyas, Traveling waves for a simple diffusive epidemic model, Math. Mod. Meth. Appl. S., 1995, 5, 935-966. doi: 10.1142/S0218202595000504 CrossRef Google Scholar
[16]	W. O. Kermack and A. G. McKendrick, Contribution to the mathematical theory of epidemics, P. Roy. Soc. Lond. A., 1927, 115, 700-721. doi: 10.1098/rspa.1927.0118 CrossRef Google Scholar
[17]	W. O. Kermack and A. G. McKendrick, Contribution to the mathematical theory of epidemics, Part Ⅱ. P. Roy. Soc. Lond. A., 1932, 138, 55-83. doi: 10.1098/rspa.1932.0171 CrossRef Google Scholar
[18]	W. O. Kermack and A. G. McKendrick, Contribution to the mathematical theory of epidemics, Part Ⅲ. P. Roy. Soc. Lond. A., 1933, 141, 94-112. doi: 10.1098/rspa.1933.0106 CrossRef Google Scholar
[19]	Y. Li, W. Li and G. Lin, Traveling waves of a delayed diffusive SIR epidemic model, Commun. Pur. Appl. Anal., 2015, 14, 1001-1022. doi: 10.3934/cpaa.2015.14.1001 CrossRef Google Scholar
[20]	G. Lin, Invasion traveling wave solutions of a predator-prey system, Nonlinear Anal., 2014, 96, 47-58. doi: 10.1016/j.na.2013.10.024 CrossRef Google Scholar
[21]	G. Lin and S. Ruan, Traveling wave solutions for delayed reaction-diffusion systems and applications to Lotka-Volterrra competition-diffusion models with distributed delays, J. Dyn. Differ. Equ., 2014, 26, 583-605. doi: 10.1007/s10884-014-9355-4 CrossRef Google Scholar
[22]	L. Rizk, J. Burie and A. Ducrot, Travelling wave solutions for a non-local evolutionary-epidemic system, J. Differ. Equations, 2019, 267, 1467-1509. doi: 10.1016/j.jde.2019.02.012 CrossRef Google Scholar
[23]	H. Smith and X. Zhao, Global asymptotic stability of traveling waves in delayed reaction-diffusion equations, SIAM J. Math. Anal., 2000, 31, 514-534. doi: 10.1137/S0036141098346785 CrossRef Google Scholar
[24]	Y. Takeuchi, W. Ma and E. Beretta, Global asymptotic properties of a delay SIR epidemic model with finite incubation times, Nonlinear Anal., 2000, 42, 931-947. doi: 10.1016/S0362-546X(99)00138-8 CrossRef Google Scholar
[25]	H. Thieme and X. Zhao, Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differ. Equations, 2003, 195, 430-470. doi: 10.1016/S0022-0396(03)00175-X CrossRef Google Scholar
[26]	X. Wang, H. Wang and J. Wu, Traveling waves of diffusive predator-prey systems: disease outbreak propagation, Discrete Cont. Dyn-A., 2012, 32, 3303-3324. doi: 10.3934/dcds.2012.32.3303 CrossRef Google Scholar
[27]	J. Wei, Asymptotic boundary and nonexistence of traveling waves in a discrete diffusive epidemic model, J. Differ. Equ. Appl., 2020, 26, 163-170. doi: 10.1080/10236198.2019.1709181 CrossRef Google Scholar
[28]	J. Wei, Z. Zhen, J. Zhou and L. Tian, Traveling waves for a discrete diffusion epidemic model with delay, Taiwan. J. Math., Doi: 10.11650/tjm/201209. CrossRef Google Scholar
[29]	J. Wei, J. Zhou, Z. Zhen and L. Tian, Time periodic traveling waves in a three-component non-autonomous and reaction-diffusion epidemic model, Int. J. Math., 2021, 31, 2150003. Google Scholar
[30]	J. Wei, J. Zhou, W. Chen, Z. Zhen and L. Tian, Traveling waves in a nonlocal dispersal epidemic model with spatio-temporal delay, Commun. Pur. Appl. Anal., 2020, 19, 2853-2886. doi: 10.3934/cpaa.2020125 CrossRef Google Scholar
[31]	J. Wei, J. Zhou, Z. Zhen and L. Tian, Super-critical and critical traveling waves in a two-component lattice dynamical model with discrete delay, Appl. Math. Comput., 2019, 363, 124621. Google Scholar
[32]	J. Wei, J. Zhou, Z. Zhen and L. Tian, Super-critical and critical traveling waves in a three-component delayed disease system with mixed diffusion, J. Comput. Appl. Math., 2020, 367, 112451. doi: 10.1016/j.cam.2019.112451 CrossRef Google Scholar
[33]	J. Wu, Theory and Applicatios of Partial Functional Differential Equations, New York, Springer, 1996. Google Scholar
[34]	Z. Xu, Traveling waves in a Kermack-Mckendrick epidemic model with diffusion and latent period, Nonlinear Anal. Theor., 2014, 111, 66-81. doi: 10.1016/j.na.2014.08.012 CrossRef Google Scholar
[35]	Q. Ye, Z. Li, M. Wang and Y. Wu, Introduction to Reaction-Diffusion Equations, Science Press, Beijing, 2011. Google Scholar
[36]	Z. Zhen, J. Wei, J. Zhou and L. Tian, Wave propagation in a nonlocal diffusion epidemic model with nonlocal delayed effects, Appl. Math. Comput., 2018, 339, 15-37. Google Scholar
[37]	Z. Zhen, J. Wei, L. Tian, J. Zhou and W. Chen, Wave propagation in a diffusive SIR epidemic model with spatiotemporal delay, Math. Method. Appl. Sci., 2018, 41, 7074-7098. doi: 10.1002/mma.5216 CrossRef Google Scholar
[38]	J. Zhou, J. Xu, J. Wei and H. Xu, Existence and non-existence of traveling wave solutions for a nonlocal dispersal SIR epidemic model with nonlinear incidence rate, Nonlinear Anal. Real., 2018, 41, 204-231. doi: 10.1016/j.nonrwa.2017.10.016 CrossRef Google Scholar
[39]	J. Zhou, L. Song and J. Wei, Mixed types of waves in a discrete diffusive epidemic model with nonlinear incidence and time delay, J. Differ. Equations, 2020, 268, 4491-4524. doi: 10.1016/j.jde.2019.10.034 CrossRef Google Scholar
[40]	J. Zhou, L. Song, J. Wei and H. Xu, Critical traveling waves in a diffusive disease model, J. Math. Anal. Appl., 2019, 476, 522-538. doi: 10.1016/j.jmaa.2019.03.066 CrossRef Google Scholar
[41]	K. Zhou, M. Han and Q. Wang, Traveling wave solutions for delayed diffusive SIR epidemic model with nonlinear incidence rate and external supplies, Math. Method. Appl. Sci., 2017, 40, 2772-2783. doi: 10.1002/mma.4197 CrossRef Google Scholar