Traveling wave solutions in a delayed local and nonlocal diffusion model for a generalized SIR epidemic model.
Saved in:
| Title: | Traveling wave solutions in a delayed local and nonlocal diffusion model for a generalized SIR epidemic model. |
|---|---|
| Authors: | Darazirar R; Department of Mathematics, Faculty of Exact Sciences and Informatics, Hassiba Benbouali University, 02000, Chlef, Algeria. rassimrassim269@gmail.com.; Laboratory of Mathematics and Applications, Faculty of Exact Sciences and Informatics, Hassiba Benbouali University, Hay Essalam, 02000, Chlef, Algeria. rassimrassim269@gmail.com. |
| Source: | Theory in biosciences = Theorie in den Biowissenschaften [Theory Biosci] 2025 Nov; Vol. 144 (3-4), pp. 243-259. Date of Electronic Publication: 2025 Jul 17. |
| Publication Type: | Journal Article |
| Language: | English |
| Journal Info: | Publisher: Urban & Fischer Country of Publication: Germany NLM ID: 9708216 Publication Model: Print-Electronic Cited Medium: Internet ISSN: 1611-7530 (Electronic) Linking ISSN: 14317613 NLM ISO Abbreviation: Theory Biosci Subsets: MEDLINE |
| Imprint Name(s): | Publication: Jena : Urban & Fischer Original Publication: Jena : Gustav Fischer, c1997- |
| MeSH Terms: | Epidemics* , Communicable Diseases*/epidemiology , Communicable Diseases*/transmission , Models, Biological* , Epidemiological Models*, Humans ; Basic Reproduction Number ; Computer Simulation ; Diffusion ; Disease Susceptibility ; Models, Theoretical ; Time Factors |
| Abstract: | Competing Interests: Declarations. Conflict of interest: The authors declare no Conflict of interest. In this paper, we investigate traveling wave solutions for a delayed reaction-diffusion epidemic model incorporating both local and nonlocal diffusion mechanisms. The model describes the dynamics of susceptible, infected, and recovered populations, with infection spreading influenced by delayed interactions. The susceptible and recovered populations follow a nonlocal diffusion process, while the infected population undergoes local diffusion. We derive comprehensive results regarding the existence and nonexistence of traveling wave solutions. Specifically, we demonstrate that if the basic reproduction number INLINEMATH satisfies INLINEMATH , the system does not admit any traveling wave solutions. Conversely, when INLINEMATH we identify a critical wave speed INLINEMATH , such that for any INLINEMATH , the system admits a non-critical bounded traveling wave solution. For INLINEMATH , however, the model does not admit bounded, non-negative traveling wave solutions. Numerical simulations are performed to validate these theoretical results, highlighting the influence of both diffusion and delay mechanisms on wave propagation in the SIR model. (© 2025. The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.) |
| References: | Alberti G, Bellettini G (1998) A nonlocal anisotropic model for phase transitions Part I: the optimal profile problem. Math Ann 310:527–560. (PMID: 10.1007/s002080050159) Bagheri S, Akrami MH, Loghmani GB (2024) Traveling wave in an eco-epidemiological model with diffusion and convex incidence rate: dynamics and numerical simulation. Math Comput Simul 216:347–366. (PMID: 10.1016/j.matcom.2023.10.001) Berestycki H, Roques L, Perthame B (2012) Nonlocal reaction-diffusion equations and traveling waves. Nonlinear Anal Theory Methods Appl 74(3):744–760. Cai Y, Wang W (2016) Fish-hook bifurcation branch in a spatial heterogeneous epidemic model with cross-diffusion. Nonlinear Anal Real World Appl 30:99–125. (PMID: 10.1016/j.nonrwa.2015.12.002) Cheng H, Yuan R (2017) Traveling waves of a nonlocal dispersal Kermack-Mckendrick epidemic model with delayed transmission. J Evol Equ 17:979–1002. (PMID: 10.1007/s00028-016-0362-2) Crooks E, Dancer E, Hilhorst D, Mimura M, Ninomiya H (2004) Spatial segregation limit of a competition-diffusion system with Dirichlet boundary conditions. Nonlinear Anal Real World Appl 5(4):645–665. (PMID: 10.1016/j.nonrwa.2004.01.004) Darazirar R, Yaseen RM, Mohsen AA et al (2025) Minimal wave speed and traveling wave in nonlocal dispersion SIS epidemic model with delay. Bound Value Probl 2025:67. (PMID: 10.1186/s13661-025-02055-1) Djilali S, Darazirar R, Alraddadi I (2025) Traveling wave solution for a delayed reaction-diffusion two-group SIR epidemic model with a generalized nonlinear incidence function. J Appl Math Comput. https://doi.org/10.1007/s12190-025-02474-4. (PMID: 10.1007/s12190-025-02474-4) Djilali S, Sarmad G, Tridane A (2025) Dynamics and asymptotic profiles of a local-nonlocal dispersal SIR epidemic model with spatial heterogeneity. Infect Dis Model 10(2):387–409. (PMID: 39816755) Dong F, Li W, Wang J (2019) Propagation dynamics in a three-species competition model with nonlocal anisotropic dispersal. Nonlinear Anal. Real World Appl 48:232–266. (PMID: 10.1016/j.nonrwa.2019.01.012) Fu S-C (2014) The existence of traveling wave fronts for a reaction-diffusion system modelling the acidic nitrate-ferroin reaction. Quart Appl Math 72:649–664. (PMID: 10.1090/S0033-569X-2014-01349-5) Guenad B, Darazirar R, Djilali S, Alraddadi I (2024) Traveling waves in a delayed reaction-diffusion SIR epidemic model with a generalized incidence function. Nonlinear Dyn 113(4):3673–3693. (PMID: 10.1007/s11071-024-10413-4) Huang G, Takeuchi Y, Ma W, Wei D (2010) Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate. Bull Math Biol 72:1192–1207. (PMID: 10.1007/s11538-009-9487-620091354) Huang G, Takeuchi Y, Ma W, Wei D (2010) Global stability for delay SIR and SEIR epidemic models with nonlinear incidence rate. Bull Math Biol 72:1192–1207. (PMID: 10.1007/s11538-009-9487-620091354) Hutson V, Grinfeld M (2006) Non-local dispersal and bistability. Eur J Appl Math 17:221–232. (PMID: 10.1017/S0956792506006462) Korobeinikov A, Maini PK (2005) Non-linear incidence and stability of infectious disease models. Math Med Biol J IMA 22(2):113–128. (PMID: 10.1093/imammb/dqi001) Lai X, Wu J (2021) Traveling wave fronts in nonlocal epidemic models with quarantine. Phys Rev E 104:034310. Li Y, Li WT, Yang FY (2014) Traveling waves for a nonlocal dispersal SIR model with delay and external supplies. Appl Math Comput 247:723–740. Martcheva M (2015) An introduction to mathematical epidemiology, Springer. Mei M, Pao C, Zhang B (2012) Traveling wavefronts for a nonlocal dispersal population model with spatio-temporal delay. J Math Anal Appl 395(1):220–234. Mimura M, Murray JD (1978) On a diffusive prey-predator model which exhibits patchiness. J Theor Biol 75:249–262. (PMID: 10.1016/0022-5193(78)90332-6745441) Murray JD (2002) Mathematical Biology, 3rd ed, Springer. Okubo A, Levin SA (2001) Diffusion and ecological problems: modern perspectives, Springer. Thieme HR (1993) Persistence under relaxed point-dissipativity (with applications to an endemic model). SIAM J Math Anal 24(2):407–435. (PMID: 10.1137/0524026) Thieme H, Zhao X (2003) Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models. J Differ Equ 195(2):430–470. (PMID: 10.1016/S0022-0396(03)00175-X) Tomás C, Fatini M, Pettersson M (2018) A stochastic SIRI epidemic model with relapse and media coverage. Discrete Contin Dyn Syst 23(8):3483–3501. (PMID: 10.3934/dcdsb.2018250) Wang W, Zhao X-Q (2007) Basic reproduction numbers for reaction-diffusion epidemic models. SIAM J Appl Dyn Syst 5:988–1006. Wang ZC, Li WT, Ruan S (2008) Traveling fronts in monostable equations with nonlocal delayed effects. J Dyn Differ Equ 20:573–607. (PMID: 10.1007/s10884-008-9103-8) Wang K, Zhao H, Wang H, Zhang R (2023) Traveling wave of a reaction-diffusion vector-borne disease model with nonlocal effects and distributed delay. J Dyn Diff Equat 35(4):3149–3185. (PMID: 10.1007/s10884-021-10062-w) Wei J, Fan D (2017) Nonlocal diffusion in SIR epidemic models: wave propagation and its implications. J Differ Equ 263(8):4094–4115. Wu W, Teng Z (2023) Periodic traveling waves for a diffusive SIR epidemic model with general nonlinear incidence and external supplies. Commun Nonlinear Sci Numer Simul 116:106848. (PMID: 10.1016/j.cnsns.2022.106848) Wu W, Teng Z (2023) Traveling waves of a discrete diffusive waterborne pathogen model with general incidence. Commun Nonlinear Sci Numer Simul 126:107431. (PMID: 10.1016/j.cnsns.2023.107431) Xu D, Zhao X (2005) Asymptotic speed of spread and traveling waves for a nonlocal epidemic model. Discret Contin Dyn Syst -Ser B 5:1043–1056. Yang FY, Li Y, Li WT, Wang ZC (2013) Traveling waves in a nonlocal dispersal KermackMcKendrick epidemic model. Discret Contin Dyn Syst -Ser B 18:1969–1993. Zhou J, Xu J, Wei J, Xu H (2018) Existence and non-existence of traveling wave solutions for a nonlocal dispersal SIR epidemic model with nonlinear incidence rate. Nonlinear Anal Real World Appl 41:204–231. (PMID: 10.1016/j.nonrwa.2017.10.016) Zhou J, Xu J, Wei J, Xu H (2018) Existence and non-existence of traveling wave solutions for a nonlocal dispersal SIR epidemic model with nonlinear incidence rate. Nonlinear Anal Real World Appl 41:204–231. (PMID: 10.1016/j.nonrwa.2017.10.016) |
| Contributed Indexing: | Keywords: Basic reproduction number.; Local and nonlocal diffusion; Minimal wave speed (MWS); Upper-lower solutions |
| Entry Date(s): | Date Created: 20250717 Date Completed: 20251030 Latest Revision: 20251030 |
| Update Code: | 20251030 |
| DOI: | 10.1007/s12064-025-00443-w |
| PMID: | 40676411 |
| Database: | MEDLINE |
Full Text Finder 
Nájsť tento článok vo Web of Science | Abstract: | Competing Interests: Declarations. Conflict of interest: The authors declare no Conflict of interest.<br />In this paper, we investigate traveling wave solutions for a delayed reaction-diffusion epidemic model incorporating both local and nonlocal diffusion mechanisms. The model describes the dynamics of susceptible, infected, and recovered populations, with infection spreading influenced by delayed interactions. The susceptible and recovered populations follow a nonlocal diffusion process, while the infected population undergoes local diffusion. We derive comprehensive results regarding the existence and nonexistence of traveling wave solutions. Specifically, we demonstrate that if the basic reproduction number INLINEMATH satisfies INLINEMATH , the system does not admit any traveling wave solutions. Conversely, when INLINEMATH we identify a critical wave speed INLINEMATH , such that for any INLINEMATH , the system admits a non-critical bounded traveling wave solution. For INLINEMATH , however, the model does not admit bounded, non-negative traveling wave solutions. Numerical simulations are performed to validate these theoretical results, highlighting the influence of both diffusion and delay mechanisms on wave propagation in the SIR model.<br /> (© 2025. The Author(s), under exclusive licence to Springer-Verlag GmbH Germany, part of Springer Nature.) |
|---|---|
| ISSN: | 1611-7530 |
| DOI: | 10.1007/s12064-025-00443-w |