Splitting Feller Diffusion for Cell Division with Parasite Infection

Abstract

We describe the evolution of the quantity of parasites in a population of cells which divide in continuous-time. The quantity of parasites in a cell follows a Feller diffusion, which is splitted randomly between the two daughter cells when a division occurs. The cell division rate may depend on the quantity of parasites inside the cell and we are interested in the cases of constant or monotone division rate. We first determine the asymptotic behavior of the quantity of parasites in a cell line, which follows a Feller diffusion with multiplicative jumps. We then consider the evolution of the infection of the cell population and give criteria to determine whether the proportion of infected cells goes to zero (recovery) or if a positive proportion of cells becomes largely infected (proliferation of parasites inside the cells).A fragment with mass x splits with rate r(x) in two fragments whose mass are respectively equal to Θx and (1 − Θ)x. Let us notice similarities with [BD86, BD87] for splitting intervals of size x at rate r(x) = x α (α > 0), which amount to considering constant quantity of parasites within cells.We aim at determining how the infection evolves in the cell population. We stress that the results we give in this paper do not depend on the initial quantity of parasites x 0 > 0.First, in Section 3, we determine the asymptotic behavior of the quantity of parasites in a cell line (Y t : t ≥ 0). This amounts to following the infection in a cell until it divides and then choose at random one of the two daughter cells. This process is a Feller diffusion X with multiplicative jumps Θ occurring at rate r(.) and we prove the following extinction criterion.Proposition. (i) In the case where r(.) = r is constant,(ii) In the case where r is an increasing function, ⋆ If there exists x 0 such that g ≤ E(log(1/Θ))r(x 0 ), then a.s. there exists t > 0 such that Y t = 0.⋆ If g > E(log(1/Θ)) sup x∈R+ r(x), then P(lim t→∞ Y t = ∞) > 0.(iii) In the case where r is a decreasing function, ⋆ If g ≤ E(log(1/Θ)) inf x∈R+ r(x), then a.s. there exists t > 0 such that Y t = 0.

Branching Processes in a Lévy Random Environment

“…Since after this time both processes are equal to ∞, we obtain that Z and Z are indistinguishable. In other words, there is a unique strong solution to (5). The strong Markov property follows from the fact that we have a strong solution, the integrators are Lévy processes and the integrand functions doesn't depend on time.…”

“…It is not difficult to see that Z is a weak solution to (5). Finally, we consider two solutions to (5), Z and Z , and consider τ m = inf{t ≥ 0 :…”

“…where the process S = (S t , t ≥ 0) is a Brownian motion with drift which is independent of B. Bansaye and Tran [5] studied a cells dividing model which are infected by parasites. Informally, the quantity of parasites in a cell evolves as a Feller diffusion.…”

Uniform sampling in a structured branching population

“…This model is a continuous version of Kimmel's multilevel model for plasmids [34] which has already been studied in the case of a constant or monotone division rate by Bansaye and Tran in [7]. It models the proliferation of a parasite infection in a cell population.…”

The combined effects of Feller diffusion and transcriptional/translational bursting in simple gene networks

“…Diffusion processes on bounded domains with random jumps from the boundary have fine ergodic and spectral properties [3,36]. Yet, a different approach is given in [2], where the authors study extinction.…”