CRITICAL TRAVELING WAVES IN A DELAYED DIFFUSIVE EPIDEMIC SYSTEM

Jingdong Wei; Han Jiang; Zaili Zhen; Jiangbo Zhou

doi:10.11948/20250194

2026 Volume 16 Issue 2

Article Contents

Article navigation > Journal of Applied Analysis & Computation > 2026 > 16(2): 1016-1034

Next Article Previous Article

Jingdong Wei, Han Jiang, Zaili Zhen, Jiangbo Zhou. CRITICAL TRAVELING WAVES IN A DELAYED DIFFUSIVE EPIDEMIC SYSTEM[J]. Journal of Applied Analysis & Computation, 2026, 16(2): 1016-1034. doi: 10.11948/20250194

Citation:

Jingdong Wei, Han Jiang, Zaili Zhen, Jiangbo Zhou. CRITICAL TRAVELING WAVES IN A DELAYED DIFFUSIVE EPIDEMIC SYSTEM[J]. Journal of Applied Analysis & Computation, 2026, 16(2): 1016-1034. doi: 10.11948/20250194

CRITICAL TRAVELING WAVES IN A DELAYED DIFFUSIVE EPIDEMIC SYSTEM

1.
School of Mathematical Sciences, Jiangsu University, Zhenjiang, Jiangsu 212013, China

Author Bio: Email: weijingdong@ujs.edu.cn(J. Wei); Email: 18018321165@163.com(H. Jiang); Email: ujszjb@126.com(J. Zhou)
Corresponding author: Email: zhenzaili@ujs.edu.cn(Z. Zhen)

Abstract

Abstract

In this paper, we investigate the existence of critical traveling wave solutions in a diffusive epidemic system with delay. The existence of super-critical traveling wave solutions is well-known. By constructing suitable upper-lower solutions of wave system and applying Schauder’s fixed point theorem coupled with delicate analysis, we derive the existence of non-trivial positive bounded critical traveling wave solution for the first time. Moreover, if the transmission rate equals to the removed rate, then the nonexistence of traveling wave solutions with any positive wave velocity is obtained.
- Diffusive epidemic system /
- critical traveling wave /
- time delay
MSC: 35Q92, 35K57, 35C07, 92D30

FullText(HTML)

References

[1]	M. Cao, J. Zhao, J. Wang and R. Zhang, Dynamical analysis of a reaction-diffusion vector-borne disease model incorporating age-space structure and multiple transmission routes, Commun. Nonlinear Sci., 2023,127, 107550. doi: 10.1016/j.cnsns.2023.107550 CrossRef Google Scholar
[2]	R. Darazirar, Traveling wave solutions in a delayed local and nonlocal diffusion model for a generalized SIR epidemic model, Theor. Biosci., 2025. https://doi.org/10.1007/s12064-025-00443-w. doi: 10.1007/s12064-025-00443-w CrossRef Google Scholar
[3]	D. Denu, S. Ngoma and R. B. Salako, Dynamics of solutions of a diffusive time-delayed HIV/AIDS epidemic model: Traveling wave solutions and spreading speeds, J. Differ. Equations, 2023,344,846–890. doi: 10.1016/j.jde.2022.11.009 CrossRef Google Scholar
[4]	O. Diekmann, Thresholds and travelling waves for the geographical spread of infection, J. Math. Biol., 1978, 69,109–130. Google Scholar
[5]	O. Diekmann, Run for your life. A note on the asymptotic speed of propagation of an epidemic, J. Differ. Equations, 1979, 33, 58–73. doi: 10.1016/0022-0396(79)90080-9 CrossRef Google Scholar
[6]	S. Djilali, R. Darazirar and I. Alraddadi, Traveling wave solution for a delayed reaction-diffusion two-group SIR epidemic model with a generalized nonlinear incidence function, J. Appl. Math. Comput., 2025. https://doi.org/10.1007/s12190-025-02474-4. doi: 10.1007/s12190-025-02474-4 CrossRef Google Scholar
[7]	A. Ducrot and P. Magal, Travelling wave solutions for an infection-age structured model with diffusion, Proc. R.Soc. Edin. Ser. A Math., 2009,139,459–482. doi: 10.1017/S0308210507000455 CrossRef Google Scholar
[8]	A. Ducrot, P. Magal and S. Ruan, Travelling wave solutions in multigroup age-structured epidemic models, Arch. Rational Mech. Anal., 2012,195,311–331. Google Scholar
[9]	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
[10]	B. Guenad, R. Darazirar, S. Djilali and I. Alraddadi, Traveling waves in a delayed reaction-diffusion SIR epidemic model with a generalized incidence function, Nonlinear Dyn., 2025,113, 3673–3693. doi: 10.1007/s11071-024-10413-4 CrossRef Google Scholar
[11]	Y. Hosono and B. Ilyas, Travelling waves for a simple diffusive epidemic model, Math. Model Meth. Appl. Sci., 1995, 5,935–966. doi: 10.1142/S0218202595000504 CrossRef Google Scholar
[12]	J. Li and X. Zou, Modeling spatial spread of infectious diseases with a fixed latend period in a spatially continuous domain, Bull. Math. Biol., 2009, 71, 2048–2079. doi: 10.1007/s11538-009-9457-z CrossRef Google Scholar
[13]	W. Li, G. Lin, C. Ma and F. Yang, Traveling wave solutions of a nonlocal delay SIR model without outbreak threshold, Discrete and Continuous Dynamical Systems - B, 2014, 19,467–484. doi: 10.3934/dcdsb.2014.19.467 CrossRef Google Scholar
[14]	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
[15]	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
[16]	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
[17]	K. Wang, H. Zhao, H. Wang and R. Zhang, Traveling wave of a reaction-diffusion vector-borne disease model with nonlocal effects and distributed delay, J. Dyn. Differ. Equ., 2023, 35, 3149–3185. doi: 10.1007/s10884-021-10062-w CrossRef Google Scholar
[18]	X. Wang, H. Wang and J. Wu, Traveling waves of diffusive predator-prey systems: Disease outbreak propagation, Discrete Cont. Dyn., 2012, 32, 3303–3324. doi: 10.3934/dcds.2012.32.3303 CrossRef Google Scholar
[19]	Z. Wang and J. Wu, Travelling waves of a diffusive Kermack-McKendrick epidemic model with nonlocal delayed transmission, Proc. R. Soc. Lond. Ser. A Math., 2010,466,237–261. Google Scholar
[20]	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
[21]	J. Wei, J. Li, Z. Zhen, J. Zhou and M. Dong, Unidirectional wave propagation in a nonlocal dispersal endemic model with nonlinear incidence, J. Appl. Anal. Comput., 2025, 15,391–421. Google Scholar
[22]	J. Wei, J. Li and J. Zhou, Boundedness of traveling waves in a discrete diffusion model with delay, Qual. Theor. Dyn. Syst., 2024, 23, 48. doi: 10.1007/s12346-023-00903-y CrossRef Google Scholar
[23]	J. Wei, Z. Zhen, J. Zhou and L. Tian, Traveling waves for a discrete diffusion epidemic model with delay, Taiwan. J. Math., 2021, 25,831–866. Google Scholar
[24]	J. Wei, J. Zhou, W. Chen, Z. Zhen and L. Tian, Traveling waves in a nonlocal dispersal epidemic model with spatio-temporal delay, Commun. Pure Appl. Anal., 2020, 19, 2853–2886. doi: 10.3934/cpaa.2020125 CrossRef Google Scholar
[25]	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
[26]	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
[27]	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, 32, 2150003. doi: 10.1142/S0129167X21500038 CrossRef Google Scholar
[28]	W. Wu and W. Zhang, Traveling wave solutions in a nonlocal dispersal SIR epidemic model with nonlocal time-delay and general nonlinear incidences, Int. J. Biomath., 2024, 17, 2350045. doi: 10.1142/S1793524523500456 CrossRef Google Scholar
[29]	L. Zhang, J. Wang and R. Zhang, Mathematical analysis for an age-space structured HIV model with latency, Math. Comput. Simul., 2024,220,595–617. doi: 10.1016/j.matcom.2024.02.017 CrossRef Google Scholar
[30]	R. Zhang and X. Ren, Lyapunov functions for some epidemic model with high risk and vaccinated class, Appl. Math. Lett., 2025,163, 109437. doi: 10.1016/j.aml.2024.109437 CrossRef Google Scholar
[31]	Z. Zhen, J. Wei, L. Tian, J. Zhou and W. Chen, Wave propagation in a diffusive SIR epidemic model with spatiotemporal delay, Math. Meth. Appl. Sci., 2018, 41, 7074–7098. doi: 10.1002/mma.5216 CrossRef Google Scholar
[32]	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
[33]	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
[34]	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
[35]	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