On positive solutions and the Omega limit set for a class of delay differential equations

This paper studies the positive solutions of a class of delay differential equations with two delays. These equations originate from the modeling of hematopoietic cell populations. We give a sufficient condition on the initial function for $t\leq 0$ such that the solution is positive for all time $t>0$. The condition is "optimal". We also discuss the long time behavior of these positive solutions through a dynamical system on the space of continuous functions. We give a characteristic description of the $\omega$ limit set of this dynamical system, which can provide informations about the long time behavior of positive solutions of the delay differential equation.Comment: 15 pages, 2 figure

Run-and-tumble bacteria slowly approaching the diffusive regime

The run-and-tumble (RT) dynamics followed by bacterial swimmers gives rise first to a ballistic motion due to their persistence, and later, through consecutive tumbles, to a diffusive process. Here we investigate how long it takes for a dilute swimmer suspension to reach the diffusive regime as well as what is the amplitude of the deviations from the diffusive dynamics. A linear time dependence of the mean-squared displacement (MSD) is insufficient to characterize diffusion and thus we also focus on the excess kurtosis of the displacement distribution. Four swimming strategies are considered: (i) the conventional RT model with complete reorientation after tumbling, (ii) the case of partial reorientation, characterized by a distribution of tumbling angles, (iii) a run-and-reverse model with rotational diffusion, and (iv) a RT particle where the tumbling rate depends on the stochastic concentration of an internal protein. By analyzing the associated kinetic equations for the probability density function and simulating the models, we find that for models (ii), (iii), and (iv) the relaxation to diffusion can take much longer than the mean time between tumble events, evidencing the existence of large tails in the particle displacements. Moreover, the excess kurtosis can assume large positive values. In model (ii) it is possible for some distributions of tumbling angles that the MSD reaches a linear time dependence but, still, the dynamics remains non-Gaussian for long times. This is also the case in model (iii) for small rotational diffusivity. For all models, the long-time diffusion coefficients are also obtained. The theoretical approach, which relies on eigenvalue and angular Fourier expansions of the van Hove function, is in excellent agreement with the simulations.Comment: 12 pages, 4 captioned figures. Accepted for publication in Physical Review

An analysis of a stochastic model for bacteriophage systems

International audienceIn this article, we analyze a system modeling bacteriophage treatments for infections in a noisy context. In the small noise regime, we show that after a reasonable amount of time the system is close to a sane equilibrium (which is a relevant biologic information) with high probability. Mathematically speaking, our study hinges on concentration techniques for delayed stochastic differential equations

A Delay Reaction-Diffusion Model of the Spread of Bacteriophage Infection

This paper is a continuation of recent attempts to understand, via mathematical modeling, the dynamics of marine bacteriophage infections. Previous authors have proposed systems of ordinary differential delay equations with delay dependent coefficients. In this paper we continue these studies in two respects. First, we show that the dynamics is sensitive to the phage mortality function, and in particular to the parameter we use to measure the density dependent phage mortality rate. Second, we incorporate spatial effects by deriving, in one spatial dimension, a delay reactiondiffusion model in which the delay term is rigorously derived by solving a von Foerster equation. Using this model, we formally compute the speed at which the viral infection spreads through the domain and investigate how this speed depends on the system parameters. Numerical simulations suggest that the minimum speed according to linear theory is the asymptotic speed of propagation.</p

A Delay Reaction-Diffusion Model of the Spread of Bacteriophage Infection

This paper is a continuation of recent attempts to understand, via mathematical modeling, the dynamics of marine bacteriophage infections. Previous authors have proposed systems of ordinary differential delay equations with delay dependent coefficients. In this paper we continue these studies in two respects. First, we show that the dynamics is sensitive to the phage mortality function, and in particular to the parameter we use to measure the density dependent phage mortality rate. Second, we incorporate spatial effects by deriving, in one spatial dimension, a delay reactiondiffusion model in which the delay term is rigorously derived by solving a von Foerster equation. Using this model, we formally compute the speed at which the viral infection spreads through the domain and investigate how this speed depends on the system parameters. Numerical simulations suggest that the minimum speed according to linear theory is the asymptotic speed of propagation.</p

A delay reaction-diffusion model of the spread of bacteriophage infection

Abstract. This paper is a continuation of recent attempts to understand, via mathematical modeling, the dynamics of marine bacteriophage infections. Previous authors have proposed systems of ordinary differential delay equations with delay dependent coefficients. In this paper we continue these studies in two respects. First, we show that the dynamics is sensitive to the phage mortality function, and in particular to the parameter we use to measure the density dependent phage mortality rate. Second, we incorporate spatial effects by deriving, in one spatial dimension, a delay reactiondiffusion model in which the delay term is rigorously derived by solving a von Foerster equation. Using this model, we formally compute the speed at which the viral infection spreads through the domain and investigate how this speed depends on the system parameters. Numerical simulations suggest that the minimum speed according to linear theory is the asymptotic speed of propagation