×

Super-critical and critical traveling waves in a two-component lattice dynamical model with discrete delay. (English) Zbl 1433.34101

Summary: A delayed lattice dynamical system arising in a disease model is proposed to model the transmission of communicable disease. The existence of nontrivial, positive, bounded super-critical and critical traveling waves for this system are established, respectively. Meanwhile, the non-existence of such traveling waves for the system is obtained. Moreover, the effects of diffusive rate of infective individuals and time-delay on critical speed are discussed, accordingly.

MSC:

34K31 Lattice functional-differential equations
34K30 Functional-differential equations in abstract spaces
35K40 Second-order parabolic systems
39A12 Discrete version of topics in analysis
39B72 Systems of functional equations and inequalities
34K60 Qualitative investigation and simulation of models involving functional-differential equations
46N60 Applications of functional analysis in biology and other sciences
92D30 Epidemiology
PDFBibTeX XMLCite
Full Text: DOI

References:

[1] Ai, S.; Albashaireh, R., Traveling waves in spatial SIRS models, J. Dyn. Differ. Equ., 26, 143-164 (2014) · Zbl 1293.35069
[2] Bai, Z.; Zhang, S., Traveling waves of a diffusive SIR epidemic model with a class of nonlinear incidence rates and distributed delay, Commun. Nonlinear Sci., 22, 1370-1381 (2015) · Zbl 1331.92142
[3] Beretta, E.; Takeuchi, Y., Global stability of an SIR epidemic model with time delays, J. Math. Biol., 33, 250-260 (1995) · Zbl 0811.92019
[4] Capasso, V.; Serio, G., A generalization of the Kermack-Mckendrick deterministic epidemic model, Math. Biosci., 42, 43-61 (1978) · Zbl 0398.92026
[5] Chen, Y.; Guo, J.; Hamel, F., Traveling waves for a lattice dynamical system arising in a diffusive endemic model, Nonlinearity, 30, 2334-2359 (2017) · Zbl 1373.34022
[6] Ducrot, A.; Magal, P., Travelling wave solutions for an infection-age structured epidemic model with external supplies, Nonlinearity, 24, 2891-2911 (2011) · Zbl 1227.92044
[7] Ducrot, A.; Magal, P.; Ruan, S., Travelling wave solutions in multigroup age-structure epidemic models, Arch. Ration. Mech. Anal., 195, 311-331 (2010) · Zbl 1181.92071
[8] Feng, Z.; Thieme, H. R., Endemic models with arbitrarily distributed periods of infection, i fundamental properties of the model, SIAM J. Appl. Math., 61, 803-833 (2000) · Zbl 0991.92028
[9] Feng, Z.; Thieme, H. R., Endemic models with arbitrarily distributed periods of infection, II fast disease dynamics and permanent recovery, SIAM J. Appl. Math., 61, 983-1012 (2000) · Zbl 1016.92035
[10] Fu, S., Traveling waves for a diffusive SIR model with delay, J. Math. Anal. Appl., 435, 20-37 (2016) · Zbl 1347.34101
[11] Fu, S.; Guo, J.; Wu, C., Traveling wave solutions for a discrete diffusive epidemic model, J. Nonlinear Convex. A., 17, 1739-1751 (2016) · Zbl 1353.39005
[12] Chen, X.; Guo, J., Uniqueness and existence of traveling waves for discrete quasilinear monostable dynamics, Math. Ann., 326, 123-146 (2003) · Zbl 1086.34011
[13] He, J.; Tsai, J., Traveling waves in the Kermack-Mckendrick epidemic model with latent period, Z. Angew. Math. Phys., 70, 27 (2019) · Zbl 1407.92126
[14] Hethcote, H. W., The mathematics of infectious disease, SIAM Rev., 42, 599-653 (2000) · Zbl 0993.92033
[15] Hosono, Y.; Ilyas, B., Traveling waves for a simple diffusive epidemic model, Math. Mod. Meth. Appl. S., 5, 935-966 (1995) · Zbl 0836.92023
[16] Kuang, Y., Delay Differential Equations with Applications in Population Dynamics (1993), Academic Press: Academic Press San Diego · Zbl 0777.34002
[17] Korobeinikov, A., Global properties of infectious disease models with nonlinear incidence, Bull. Math. Biol., 69, 1871-1886 (2007) · Zbl 1298.92101
[18] Kuniyas, T.; Oizumi, R., Existence result for an age-structures SIS epidemic model with spatial diffusion, Nonlinear Anal. Real, 23, 196-208 (2015) · Zbl 1319.35274
[19] Lam, K.; Wang, X.; Zhang, T., Traveling waves for a class of diffusive disease-transmission models with network structures, SIAM J. Math. Anal., 50, 5719-5748 (2018) · Zbl 1402.35149
[20] Li, W.; Lin, G.; Ma, C.; Yang, F., Traveling wave solutions of a nonlocal delayed SIR model without outbreak threshold, Discrete Cont. Dyn. B, 19, 467-484 (2014) · Zbl 1311.35051
[21] Li, Y.; Li, W.; Lin, G., Traveling waves of a delayed diffusive SIR epidemic model, Commun. Pur. Appl. Anal., 14, 1001-1022 (2015) · Zbl 1422.35106
[22] Li, Y.; Li, W.; Yang, Y., Stability of traveling waves of a diffusive susceptible-infective-removed (SIR) epidemic model, J. Math. Phys., 57, 041504 (2016) · Zbl 1339.35335
[23] Li, Y.; Li, W.; Zhang, G., Stability and uniqueness of traveling waves of a non-local dispersal SIR epidemic model, Dyn. Part. Differ. Equ., 14, 87-123 (2017) · Zbl 1380.37147
[24] Shu, H.; Wang, L.; Watmough, J., Global stability of a nonlinear viral infection model with infinitely distributed intracellular delays and CTL immune responses, SIAM J. Appl. Math., 73, 1280-1302 (2013) · Zbl 1272.92031
[25] Wang, X.; Wang, H.; Wu, J., Traveling waves of diffusive predator-prey system: disease outbreak propagation, Discrete Contin. Dyn. A, 32, 3303-3324 (2012) · Zbl 1241.92069
[26] Wang, X.; Wu, J.; Yang, Y., Richards model revisited: validation by and application to infection dynamics, J. Theor. Biol., 313, 12-19 (2012) · Zbl 1337.92219
[27] Wang, Z.; Wu, J., Traveling waves of a diffusive Kermack-Mckendrick epidemic model with nonlocal delayed transmission, P. Roy. Soc. A, 466, 237-261 (2010) · Zbl 1195.35291
[28] Wang, Z.; Wu, J.; Liu, R., Traveling waves of avian influenza spread, Proc. Am. Math. Soc., 140, 3931-3946 (2012) · Zbl 1275.35068
[29] Wang, Z.; Zhang, L.; Zhao, X., Time periodic traveling waves for a periodic and diffusive SIR epidemic model, J. Dyn. Differ. Equ., 30, 379-403 (2018) · Zbl 1384.35044
[30] Weng, P.; Zhao, X., Spreading speed and traveling waves for a multi-type SIS epidemic model, J. Differ. Equ., 229, 270-296 (2006) · Zbl 1126.35080
[31] Widder, D., The Laplace Transform (1941), NJ: Princeton University Press: NJ: Princeton University Press Princeton · JFM 67.0384.01
[32] Wu, C., Existence of traveling waves with the critical speed for a discrete diffusive epidemic model, J. Differ. Equ., 262, 272-282 (2017) · Zbl 1387.34094
[33] Wu, C.; Weng, P., Asymptotic speed of propagation and traveling wavefronts for a SIR epidemic model, Discrete Conti. Dyn.-B, 15, 867-892 (2012) · Zbl 1221.35097
[34] Wu, J., Theory and Applications of Partial Functional Differential Equations (1996), Springer: Springer New York · Zbl 0870.35116
[35] Wu, J.; Zou, X., Traveling wave front solutions in reaction-diffusion systems with delay, J. Dyn. Diff. Equ., 13, 651-687 (2001) · Zbl 0996.34053
[36] Xu, Z., Traveling waves in a Kermack-Mckendrick epidemic model with diffusion and latent period, Nonlinear Anal.-Theor., 111, 66-81 (2014) · Zbl 1338.92144
[37] Yang, F.; Li, Y.; Li, W.; Wang, Z., Traveling waves in a nonlocal dispersal Kermack-Mckendrick epidemic model, Discrete Cont. Dyn. B, 18, 1969-1993 (2013) · Zbl 1277.35107
[38] Zhao, L.; Wang, Z., Traveling wave fronts in a diffusive epidemic model with multiple parallel infectious stages, IMA J. Appl. Math., 81, 795-823 (2016) · Zbl 1406.35444
[39] Zhen, Z.; Wei, J.; Zhou, J.; Tian, L., Wave propagation in a nonlocal diffusion epidemic model with nonlocal delayed effects, Appl. Math. Comput., 339, 15-37 (2018) · Zbl 1428.35176
[40] Zhen, Z.; Wei, J.; Tian, L.; Zhou, J.; Chen, W., Wave propagation in a diffusive SIR epidemic model with spatiotemporal delay, Math. Method. Appl. Sci., 41, 7074-7098 (2018) · Zbl 1402.35152
[41] Zhou, J.; Song, L.; Wei, J.; Xu, H., Critical traveling waves in a diffusive disease model, J. Math. Anal. Appl., 476, 522-538 (2019) · Zbl 1416.35071
[42] Zhou, J.; Xu, J.; Wei, J.; Xu, H., Existence and non-existence of traveling wave solutions for a nonlocal dispersal SIR epidemic model with nonlinear incidence rate, Nonlinear Anal. Real, 41, 204-231 (2018) · Zbl 1383.35232
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.