A Host-Parasite Model for a Two-Type Cell Population

Abstract

We consider a host-parasite model for a population of cells that can be of two types, A or B, and exhibits unilateral reproduction: while a B-cell always splits into two cells of the same type, the two daughter cells of an A-cell can be of any type. The random mechanism that describes how parasites within a cell multiply and are then shared into the daughter cells is allowed to depend on the hosting mother cell as well as its daughter cells. Focusing on the subpopulation of A-cells and its parasites, our model differs from the single-type model recently studied by Bansaye [5] in that the sharing mechanism may be biased towards one of the two types. Our main results are concerned with the nonextinctive case and provide information on the behavior, as n → ∞, of the number A-parasites in generation n and the relative proportion of A-and B-cells in this generation which host a given number of parasites. As in [5], proofs will make use of a so-called random cell line which, when conditioned to be of type A, behaves like a branching process in random environment.

Parasite infection in a cell population with deaths and reinfections

“…This concludes step (b) in the case g(x) + E[I]λ(x) = gx for x ≥ x. Now, as p and r are non decreasing, if we choose the same Poisson Point processes N , Q, N 1 and N 2 , and Brownian motion B we may couple two solutions Y (1) and Y (2) of the SDE (1.5) with respective growth rate functions for the parasites g 1 and g 2 satisfying…”

“…□ Proof of Proposition 2.1. Let us begin with point (1). Recall that from Lemma 3.2 E[Y t ] ≤ y for every t ≥ 0.…”

“…To end the proof of point (2) we use again the fact that we may couple a solution Y (1) of (1.5) with g 1 (x) + E[I]λ 1 (x) = gx with a solution Y (2) with g 2 (x) + E[I]λ 2 (x) ≥ gx with the same initial condition in such a way that…”

“…We refer to [22] for references on biological studies and on previous mathematical works on the class of processes under study. In particular, references [1,2,4,5,17,24,26] focused on branching within branching to study host-parasites dynamics. Similarly as in [6,11,14,15,20,22], the proof strategy consists in introducing a spinal process, and investigating the long time behaviour of this process that corresponds to the trait of a uniformly sampled individual in the population.…”

“…(3.28)We may thus obtain the following inequality,inf x>1 P x (τ − (b) ≤ l(b, δ, η)) ≥ e −8b −δ(1−a)which ends the proof of (3.27).We may now prove Proposition 2.3. Let us begin with point(1). By the Many-to-One formula (1.4), and the definition of τ − (b) we havee −(b−d)t E x   u∈Vt 1 {X u t ≤b}   = P x (Y t ≤ b) ≤ P x (τ − (b) ≤ t) → 0, (x → ∞),…”