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 b iological systems distinguish themselves from those found in engineering by the way the control acts. We illustrate this by considering the specific e xample of the spread of rabies among foxes. We first give a brief description of a model proposed by Murray et al. (Mur ray, J. D., Stanley, E. A. & Brown, D. L., Proc. R. Sec. Lend., 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 distributio n of bait in a region around an observed outbreak. The extended model can be formulated as a nonlinear time-varying control sy stem described by partial differential equations. In contrast to most engin eering 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 u se of a fixed point theorem. The problem and the theory are illustrated by some numerical simulations.