A periodic reaction-diffusion system modelling man-environment-man epidemics. (English) Zbl 1365.92114
Summary: The purpose of this work is to study the spatial dynamics of a periodic reaction-diffusion epidemic model arising from the spread of oral-faecal transmitted diseases. We first show that the disease spreading speed is coincident with the minimal wave speed for monotone periodic travelling waves. Then we obtain a threshold result on the global attractivity of either zero or the positive periodic solution in a bounded spatial domain.
References:
| [1] | Altizer, S., Dobson, A., Hosseini, P., Hudson, P., Pascual, M. and Rohani, P., Seasonality and the dynamics of infectious diseases, Ecol. Lett.9 (2006) 467-484. |
| [2] | Capasso, V., Mathematical Structures of Epidemic Systems, , Vol. 97 (Springer-Verlag, Heidelberg, 1993). · Zbl 0798.92024 |
| [3] | Capasso, V. and Kunisch, K., A reaction-diffusion system arising in modelling man-environment diseases, Quart. Appl. Math.46 (1988) 431-450. · Zbl 0704.35069 |
| [4] | Capasso, V. and Maddalena, L., Convergence to equilibrium states for a reaction-diffusion system modelling the spatial spread of a class of bacterial and viral diseases, J. Math. Biol.13 (1981) 173-184. · Zbl 0468.92016 |
| [5] | Capasso, V. and Wilson, R. E., Analysis of reaction-diffusion system modeling man-environment-man epidemics, SIAM. J. Appl. Math.57 (1997) 327-346. · Zbl 0872.35053 |
| [6] | Deimling, K., Nonlinear Functional Analysis (Springer, 1985). · Zbl 0559.47040 |
| [7] | Grassly, N. C. and Fraser, C., Seasonal infectious disease epidemiology, Proc. Roy. Soc. B273 (2006) 2541-2550. |
| [8] | Hsu, S.-B., Wang, F.-B. and Zhao, X.-Q., Dynamics of a periodically pulsed bio-reactor model with a hydraulic storage zone, J. Dynam. Differ. Equations23 (2011) 817-842. · Zbl 1242.35208 |
| [9] | Hsu, S.-B., Wang, F.-B. and Zhao, X.-Q., Global dynamics of zooplankton and harmful algae in flowing habitats, J. Differ. Equations255 (2013) 265-297. · Zbl 1286.35129 |
| [10] | Hsu, C.-H. and Yang, T.-S., Existence, uniqueness, monotonicity and asymptotic behavior of traveling waves for epidemic models, Nonlinearity26 (2013) 121-139. · Zbl 1264.35255 |
| [11] | Jiang, J., Liang, X. and Zhao, X.-Q., Saddle-point behavior for monotone semiflows and reaction-diffusion models, J. Differ. Equations203 (2004) 313-330. · Zbl 1063.35083 |
| [12] | Liang, X., Yi, Y. and Zhao, X.-Q., Spreading speeds and traveling waves for periodic evolution system, J. Differ. Equations231 (2006) 57-77. · Zbl 1105.37017 |
| [13] | Liang, X. and Zhao, X.-Q, Asymptotic speeds of spread and traveling waves for monotone semiflow with applications, Commun. Pure Appl. Math.60 (2007) 1-40. · Zbl 1106.76008 |
| [14] | Lou, Y. and Zhao, X.-Q., The periodic Ross-Macdonald model with diffusion and advection, Appl. Anal.89 (2010) 1067-1089. · Zbl 1201.35070 |
| [15] | Magal, P. and Zhao, X.-Q., Global attractors and steady states for uniformly persistent dynamical systems, SIAM J. Math. Anal.37 (2005) 251-275. · Zbl 1128.37016 |
| [16] | Nussbaum, R. D., Eigenvectors of nonlinear positive operator and the linear Krein-Rutman theorem, in Fixed Point Theory, eds. Fadell, E. and Fournier, G., , Vol. 886 (Springer, New York, 1981), pp. 309-331. |
| [17] | Thieme, H. R. and Zhao, X.-Q., Asymptotic speeds of spread and traveling waves for integral equations and delayed reaction-diffusion models, J. Differ. Equations195 (2003) 430-470. · Zbl 1045.45009 |
| [18] | Volpert, A. I., Volpert, V. A. and Volpert, V. A., Traveling Wave Solutions of Parabolic Systems, (Amer. Math. Soc., Providence, RI, 1994). · Zbl 0835.35048 |
| [19] | Wang, F., A periodic reaction-diffusion model with a quiescent stage, Discrete Contin. Dynam. Syst. Ser. B17 (2012) 283-295. · Zbl 1235.35165 |
| [20] | Weinberger, H. F., Long-time behavior of a class of biological models, SIAM J. Math. Anal.13 (1982) 353-396. · Zbl 0529.92010 |
| [21] | Weinberger, H. F., Lewis, M. A. and Li, B., Analysis of linear determinacy for spread in cooperative models, J. Math. Biol.45 (2002) 183-218. · Zbl 1023.92040 |
| [22] | Wu, S.-L., Hsu, C.-H. and Xiao, Y., Global attractivity, spreading speeds and traveling waves of delayed nonlocal reaction-diffusion systems, J. Differ. Equations258 (2015) 1058-1105. · Zbl 1316.35155 |
| [23] | Yu, X. and Zhao, X.-Q., A periodic reaction-advection-diffusion model for a stream population, J. Differ. Equations258 (2015) 3037-3062. · Zbl 1335.35132 |
| [24] | Zhang, F. and Zhao, X.-Q., A periodic epidemic model in a patchy environment, J. Math. Anal. Appl.325 (2007) 496-516. · Zbl 1101.92046 |
| [25] | Zhao, X.-Q., Dynamical Systems in Population Biology (Springer-Verlag, New York, 2003). · Zbl 1023.37047 |
| [26] | Zhao, X.-Q. and Xiao, D., The asymptotic speed of spread and traveling waves for a vector disease model, J. Dynam. Differ. Equations18 (2006) 1001-1019. · Zbl 1114.45001 |
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