Some new results in the mathematical theory of phage-reproduction

Abstract

SummaryIn 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 timetthe bacterial burst has not taken place, its occurence betweentandt+ Δtis a random event with probabilityf(· |t)Δt+o(Δt), wherefis a non-negative and non-decreasing function of the numberX(t) of vegetative phages and ofZ(t), the number of mature bacteriophages at timet.More specifically it is assumed thatf=b(t)X(t) +c(t)Z(t) withb(t),c(t) ≦ 0. HereX(t) denotes the survivors in a linear birth and death process andZ(t) the number of deaths until timet.The joint distribution ofXTandZT, the respective numbers of vegetative and mature bacteriophages at the burst time is considered. The distribution ofZTis then fitted to some observed data of Delbrück [2].