Traveling waves and spread rates for a West Nile virus model. (English) Zbl 1334.92414
Summary: A reaction-diffusion model for the spatial spread of West Nile virus is developed and analysed. Infection dynamics are based on a modified version of a model for cross infection between birds and mosquitoes [M. J. Wonham et al., “An epidemiological model for West-Nile virus: invasion analysis and control application”, Proc. R. Soc. Lond. B 271, 501–507 (2004)], and diffusion terms describe movement of birds and mosquitoes. Working with a simplified version of the model, the cooperative nature of cross-infection dynamics is utilized to prove the existence of traveling waves and to calculate the spatial spread rate of infection. Comparison theorem results are used to show that the spread rate of the simplified model may provide an upper bound for the spread rate of a more realistic and complex version of the model.
References:
| [1] | Hadeler, K., Lewis, M., 2002. Spatial dynamics of the diffusive logistic equation with a sedentary compartment. CAMQ 10, 473–499. · Zbl 1065.35141 |
| [2] | Komar, N., Langevin, S., Hinten, S., Nemeth, N., Edwards, E., Hettler, D., Davis, B., Bowen, R., Bunning, M., 2003. Experimental Infection of North American birds with the New York 1999 strain of West Nile virus. Emerg. Inform. Dis. 93, 311–322. · doi:10.3201/eid0903.020628 |
| [3] | Lewis, M., Li, B., Weinberger, H., 2002. Spreading speed and linear determinacy for two species competition models. J. Math. Biol. 45, 219–233. · Zbl 1032.92031 · doi:10.1007/s002850200144 |
| [4] | Li, B., Weinberger, H., Lewis, M., in press. Spreading speeds as slowest wave speeds for cooperative systems. Math. Biosci. · Zbl 1075.92043 |
| [5] | Lu, G., Sleeman, B.D., 1993. Maximum principles and comparison theorems for semilinear parabolic systems and their applications. Proc. R. Soc. Edinburgh A 123, 857–885. · Zbl 0791.35006 · doi:10.1017/S0308210500029541 |
| [6] | Lui, R., 1989a. Biological growth and spread modeled by system of recursions. I. Math. Theor. Math. Biosci. 93, 269–295. · Zbl 0706.92014 · doi:10.1016/0025-5564(89)90026-6 |
| [7] | Lui, R., 1989b. Biological growth and spread modeled by system of recursions. II. Biological theory. Math. Biosci. 93, 297–312. · Zbl 0706.92015 · doi:10.1016/0025-5564(89)90027-8 |
| [8] | Okubo, A., 1998. Diffusion-type models for avian range expansion. In: Henri Quellet, I. (Ed.), Acta XIX Congressus Internationalis Ornithologici. National Museum of Natural Sciences, University of Ottawa Press, pp. 1038–1049. |
| [9] | Smoller, J., 1983. Shock Waves and Reaction-Diffusion Systems. Springer-Verlag, New York. · Zbl 0508.35002 |
| [10] | van den Driessche, P., Watmough, J., 2002. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29–48. · Zbl 1015.92036 · doi:10.1016/S0025-5564(02)00108-6 |
| [11] | Volpert, A.I., Volpert, V.A., Volpert, V.A., 1994. Traveling Wave Solutions of Parabolic Systems, Vol. 140. American Mathematical Society, Providence, RI. · Zbl 0805.35143 |
| [12] | Weinberger, H., Lewis, M., Li, B., 2002. Analysis of linear determinacy for spread in cooperative models. J. Math. Biol. 45, 183–218. · Zbl 1023.92040 · doi:10.1007/s002850200145 |
| [13] | Wonham, M.J., de Camino-Beck, T., Lewis, M., 2004. An epidemiological model for West-Nile virus: Invasion analysis and control application. Proc. R. Soc. Lond. B 271, 501–507. · doi:10.1098/rspb.2003.2608 |
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.
