Traveling wave solutions for a delayed SIRS infectious disease model with nonlocal diffusion and nonlinear incidence. (English) Zbl 1406.92622

Tian, Xiaohong; Xu, Rui

Abstr. Appl. Anal. 2014, Article ID 795320, 9 p. (2014).

Summary: A delayed SIRS infectious disease model with nonlocal diffusion and nonlinear incidence is investigated. By constructing a pair of upper-lower solutions and using Schauder’s fixed point theorem, we derive the existence of a traveling wave solution connecting the disease-free steady state and the endemic steady state.

MSC:

92D30	Epidemiology
35Q92	PDEs in connection with biology, chemistry and other natural sciences

Keywords:

SIRS infectious disease model; traveling wave; nonlocal diffusion

References:

[1]	Gan, Q.; Xu, R.; Yang, P., Travelling waves of a delayed SIRS epidemic model with spatial diffusion, Nonlinear Analysis, 12, 1, 52-68 (2011) · Zbl 1202.35046
[2]	Blyuss, K. B., On a model of spatial spread of epidemics with long-distance travel, Physics Letters A: General, Atomic and Solid State Physics, 345, 1-3, 129-136 (2005) · Zbl 1345.92134
[3]	Cui, M.; Ma, T.; Li, X., Spatial behavior of an epidemic model with migration, Nonlinear Dynamics, 64, 4, 331-338 (2011)
[4]	Lou, Y.; Zhao, X., A reaction-diffusion malaria model with incubation period in the vector population, Journal of Mathematical Biology, 62, 4, 543-568 (2011) · Zbl 1232.92057
[5]	Wang, Z.; Li, W.; Ruan, S., Travelling wave fronts in reaction-diffusion systems with spatio-temporal delays, Journal of Differential Equations, 222, 1, 185-232 (2006) · Zbl 1100.35050
[6]	Weng, P.; Zhao, X., Spreading speed and traveling waves for a multi-type SIS epidemic model, Journal of Differential Equations, 229, 1, 270-296 (2006) · Zbl 1126.35080
[7]	Chen, X., Existence, uniqueness, and asymptotic stability of traveling waves in nonlocal evolution equations, Advances in Differential Equations, 2, 1, 125-160 (1997) · Zbl 1023.35513
[8]	Coville, J.; Dávila, J.; Martínez, S., Nonlocal anisotropic dispersal with monostable nonlinearity, Journal of Differential Equations, 244, 12, 3080-3118 (2008) · Zbl 1148.45011
[9]	Fife, P., Some nonclassical trends in parabolic and parabolic-like evolutions, Trends in Nonlinear Analysis, 153-191 (2003), Berlin, Germany: Springer, Berlin, Germany · Zbl 1072.35005
[10]	Sun, Y.; Li, W.; Wang, Z., Entire solutions in nonlocal dispersal equations with bistable nonlinearity, Journal of Differential Equations, 251, 3, 551-581 (2011) · Zbl 1228.35017
[11]	Sun, Y.; Li, W.; Wang, Z., Traveling waves for a nonlocal anisotropic dispersal equation with monostable nonlinearity, Nonlinear Analysis A: Theory and Methods, 74, 3, 814-826 (2011) · Zbl 1211.35068
[12]	Zhang, G.; Wang, Y., Critical exponent for nonlocal diffusion equations with Dirichlet boundary condition, Mathematical and Computer Modelling, 54, 1-2, 203-209 (2011) · Zbl 1228.35039
[13]	Li, W. T.; Sun, Y. J.; Wang, Z. C., Entire solutions in the Fisher-KPP equation with nonlocal dispersal, Nonlinear Analysis, 11, 4, 2302-2313 (2010) · Zbl 1196.35015
[14]	Capasso, V.; Serio, G., A generalization of the Kermack-McKendrick deterministic epidemic model, Mathematical Biosciences, 42, 1-2, 43-61 (1978) · Zbl 0398.92026
[15]	Xu, R.; Ma, Z., Stability of a delayed SIRS epidemic model with a nonlinear incidence rate, Chaos, Solitons & Fractals, 41, 5, 2319-2325 (2009) · Zbl 1198.34098
[16]	Yu, X.; Wu, C.; Weng, P., Traveling waves for a SIRS model with nonlocal diffusion, International Journal of Biomathematics, 5, 5 (2012) · Zbl 1291.92106