Preface

The application of mathematics to biology has led to tremendous advances in the understanding of plant and animal dynamics and growth. Paramount among these areas is the spread of diseases. The application described here motivated the adaption of diffusive wave analysis to the spread of haemorrhagic disease among rabbit populations in New Zealand. The disease was introduced as an attempt, initially illegal but subsequently legalised, to control the burgeoning rabbit population in highly productive farming areas. The conceptual basis adopted was that there is a threshold maximum value of the spatial density of healthy rabbits (“susceptibles”) below which the disease will not propagate. The dependence of the wave speed on the density (when it is above the threshold) can also be evaluated.

The procedure is generic and can be applied to a wide range of modelling scenarios. The nonlinear dynamics involves a range of parameters which are determined from data: the infectivity of the disease, the dispersion constant, and the infected death rate.

The purpose of this project was to see if the predicted threshold matches that seen in practice, with a view to assisting the understanding of the disease spread and the ability of it to deal with the huge problem of the endemic rabbit population many farming areas have.

The values obtained for the threshold density are close to those observed in practice and indicate that the model used is approximately correct. Of course many questions remain about the method of disease transmission. The effect of wind is taken into account. The methods employed are a healthy mix of analytical and numerical techniques, demonstrating again the interplay that underpins many successful solutions of nonlinear systems.