van den Bosch
J. A. J.
First author: Mathematical Methods and Models, Section of Mathematics, Wageningen Agricultural University, Dreijenlaan 4, 6703 HA Wageningen, the Netherlands; second author: Institute of Evolutionary and Ecological Sciences, Theoretical Biology Section, Kaiserstraat 63, PO Box 9516, 2300 RA Leiden, the Netherlands, and Adaptive Dynamics Network, International Institute for Applied Systems Analysis, A-2361 Laxenburg, Austria; and third author: Department of Phytopathology, Wageningen Agricultural University, PO Box 8025, 6700 EE Wageningen, the Netherlands
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Accepted for publication 22 January 1999.
An analytical model of a pandemic, initiated by a single focus and spreading over a continent, is developed using foci as the smallest units of disease and fields as the smallest units of host. A few generalizing assumptions lead to a parameter-sparse model that may answer general questions on pandemics in a qualitative manner. For pandemic spread of disease during one season, a ‘within-season velocity of pandemic spread,’ C, is expressed as a set of integral equations. Reduction of inoculum during the off-season is expressed by a ‘survival ratio’ of inoculum, ε. The effect of the off-season is a ‘push-back’ of the pandemic front over a distance Δh. It will be shown how Δh is related to C and ε. The mean pandemic spread over successive years is calculated as the ‘polyetic velocity of pandemic spread,’ V, which depends on C and the push-back distance. The concept of ‘pandemic effectiveness’ is parameterized. Relations between the two velocities of pandemic spread and several model parameters are studied. Somewhat unexpectedly, velocities of pandemic spread depend only in a very limited way on field density represented by the ‘cropping ratio’ ζ. This implies that our model and methods will also apply to situations with inhomogeneous field distributions. The effect of parameter values on rates of severity increase are analyzed. Finally, generalizations of the model are developed and their applications discussed.
© 1999 The American Phytopathological Society