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

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.


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


[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
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.