Some new results in the mathematical theory of phage-reproduction

Abstract

Summary In the theory of phage reproduction, the mathematical models considered thus far (see Gani [5]) assume that the bacterial burst occurs a fixed time after infection, after a fixed number of generations of phage multiplication, or when the number of mature bacteriophages has reached a fixed threshold. In the present paper, a more realistic assumption is considered: given that until time t the bacterial burst has not taken place, its occurence between tand t + Δt is a random event with probability f(· | t)Δt + o(Δt), where f is a non-negative and non-decreasing function of the number X(t) of vegetative phages and of Z(t), the number of mature bacteriophages at time t. More specifically it is assumed that f = b(t)X(t) + c(t)Z(t) with b(t), c(t) ≦ 0. Here X(t) denotes the survivors in a linear birth and death process and Z(t) the number of deaths until time t. The joint distribution of XT and ZT , the respective numbers of vegetative and mature bacteriophages at the burst time is considered. The distribution of ZT is then fitted to some observed data of Delbrück [2].