A control theoretic approach to containing the spread of rabies

Many problems in medicine and biology involve some kind of spatial spread, and quite often the need to control it. A large proportion of medical and biological systems distinguish themselves from those found in engineering by the way the control acts. We illustrate this by considering the specific example of the spread of rabies among foxes. We first give a brief description of a model proposed by Murray et al. (Murray, J. D., Stanley, E. A. & Brown, D. L., Proc. R. Soc. Lond., B 229, 111-150 (1986)), which we extend to include the control mechanism. The problem is to prevent the spread of rabies by vaccinating foxes via the distribution of bait in a region around an observed outbreak. The extended model can be formulated as a nonlinear time-varying control system described by partial differential equations. In contrast to most engineering type control problems, the control does not continuously affect the system but only acts through the initial distributions. We briefly outline a general theory developed for dealing with such nonlinear systems by the use of a fixed point theorem. The problem and the theory are illustrated by some numerical simulations.