A Mathematical Approach to the Problem of Achieving Selective Biological Effects

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Abstract
A biological system S is composed of k component systems S1,S2,...-Sk.-Embedded in these systems is a system S0 which is to be destroyed without destroying S. To accomplish this there are available various treatments [omega]=l,2,...n. If an amount x[omega] is used of treatment [omega], the response to the ith system, i=0,l,...k, is given by [SIGMA][omega] gi[omega] x . where gi[omega] is the sensitivity of the ith system to treatment [omega]. The systems have critical response levels a0,a1,...ak such that the ith system is destroyed if and only if [SIGMA][omega] gi[omega] x[omega] > ai. There are certain essential subsets V1V2,..-Vm of the set S1, S2, ...Sk, which correspond to altnative modes of functioning, and S can survive if all the systems in any one of the sets V1 survive. Then the existence of effective multiple treatment can be determined by solving the M. problem: maximize [SIGMA][omega]g0[omega] x[omega] over non-negative numbers (xj,X2...xn) satisfying [SIGMA][omega] gi[omega] x[omega] <- ai for each i corresponding to the members of an essential set. In this linear case, and even in certain more general cases, these problems can be solved by existing numerical techniques, e.g., linear programming. The approach is potentially applicable to a wide variety of practical problems in biological control. Various aspects of the model are discussed and illustrated.

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