In this work, we concern with the numerical approach for delay differential equations with random coefficients. We first show that the exact solution of the problem considered admits good regularity in the random space, provided that the given data satisfy some reasonable assumptions. A stochastic collocation method is proposed to approximate the solution in the random space, and we use the Legendre spectral collocation method to solve the resulting deterministic delay differential equations. Convergence property of the proposed method is analyzed. It is shown that the numerical method yields the familiar exponential order of convergence in both the random space and the time space. Numerical examples are given to illustrate the theoretical results.

A mathematical model, coupling the dynamics of short-term stem-like cells and matureleukocytes in leukemia with that of the immune system, is investigated. The model isdescribed by a system of seven delay differential equations with seven delays. Threeequilibrium points E0, E1,E2 are highlighted. The stability and the existence of theHopf bifurcation for the equilibrium points are investigated. In the analysis of themodel, the rate of asymmetric division and the rate of symmetric division are veryimportant.

We are interested in optimizing the co-administration of two drugs for some acute myeloidleukemias (AML), and we are looking for in vitro protocols as a first step. This issue canbe formulated as an optimal control problem. The dynamics of leukemic cell populations inculture is given by age-structured partial differential equations, which can be reduced toa system of delay differential equations, and where the controls represent the action ofthe drugs. The objective function relies on eigenelements of the uncontrolled model and ongeneral relative entropy, with the idea to maximize the efficiency of the protocols. Theconstraints take into account the toxicity of the drugs. We present in this paper themodeling aspects, as well as theoretical and numerical results on the optimal controlproblem that we get.

A general framework for age-structured predator-prey systems is introduced. Individualsare distinguished into two classes, juveniles and adults, and several possibleinteractions are considered. The initial system of partial differential equations isreduced to a system of (neutral) delay differential equations with one or two delays.Thanks to this approach, physically correct models for predator-prey with delay areprovided. Previous models are considered and analysed in view of the above results. ARosenzweig-MacArthur model with delay is presented as an example.

This paper is concerned with the asymptotic expansion and numerical solution of systems of linear delay differential equations with highly oscillatory forcing terms. The computation of such problems using standard numerical methods is exceedingly slow and inefficient, indeed standard software is practically useless for this purpose. We propose an alternative, consisting of an asymptotic expansion of the solution, where each term can be derived either by recursion or by solving a non-oscillatory problem. This leads to methods which, counter-intuitively to those developed according to standard numerical reasoning, exhibit improved performance with growing frequency of oscillation.

We illustrate the appearance of oscillating solutions in delay differential equationsmodeling hematopoietic stem cell dynamics. We focus on autonomous oscillations, arising asconsequences of a destabilization of the system, for instance through a Hopf bifurcation.Models of hematopoietic stem cell dynamics are considered for their abilities to describeperiodic hematological diseases, such as chronic myelogenous leukemia and cyclicalneutropenia. After a review of delay models exhibiting oscillations, we focus on threeexamples, describing different delays: a discrete delay, a continuous distributed delay,and a state-dependent delay. In each case, we show how the system can have oscillatingsolutions, and we characterize these solutions in terms of periods and amplitudes.

Recent discovery of cancer stem cells in tumorigenic tissues has raised many questionsabout their nature, origin, function and their behavior in cell culture. Most of currentexperiments reporting a dynamics of cancer stem cell populations in culture show theeventual stability of the percentages of these cell populations in the whole population ofcancer cells, independently of the starting conditions. In this paper we propose amathematical model of cancer stem cell population behavior, based on specific features ofcancer stem cell divisions and including, as a mathematical formalization of cell-cellcommunications, an underlying field concept. We compare the qualitative behavior ofmathematical models of stem cells evolution, without and with an underlying signal. Inabsence of an underlying field, we propose a mathematical model described by a system ofordinary differential equations, while in presence of an underlying field it is describedby a system of delay differential equations, by admitting a delayed signal originated byexisting cells. Under realistic assumptions on the parameters, in both cases (ODE withoutunderlying field, and DDE with underlying field) we show in particular the stability ofpercentages, provided that the delay is sufficiently small. Further, for the DDE case (inpresence of an underlying field) we show the possible existence of, either damped orstanding, oscillations in the cell populations, in agreement with some existingmathematical literature. The outcomes of the analysis may offer to experimentalists a toolfor addressing the issue regarding the possible non-stem to stem cells transition, bydetermining conditions under which the stability of cancer stem cells population can beobtained only in the case in which such transition can occur. Further, the provideddescription of the variable corresponding to an underlying field may stimulate furtherexperiments for elucidating the nature of “instructive" signals for cell divisions,underlying a proper pattern of the biological system.

Unveiling the mechanisms through which the somitogenesis regulatory network exertsspatiotemporal control of the somitic patterning has required a combination ofexperimental and mathematical modeling strategies. Significant progress has been made forthe zebrafish clockwork. However, due to its complexity, the clockwork of the amniotesegmentation regulatory network has not been fully elucidated. Here, we address thequestion of how oscillations are arrested in the amniote segmentation clock. We do this byconstructing a minimal model of the regulatory network, which privileges architecturalinformation over molecular details. With a suitable choice of parameters, our model isable to reproduce the oscillatory behavior of the Wnt, Notch and FGF signaling pathways inpresomitic mesoderm (PSM) cells. By introducing positional information via a single Wnt3agradient, we show that oscillations are arrested following an infinite-period bifurcation.Notably: the oscillations increase their amplitude as cells approach the anterior PSM andremain in an upregulated state when arrested; the transition from the oscillatory regimeto the upregulated state exhibits hysteresis; and opposing Fgf8 and RA gradients along thePSM naturally arise in our simulations. We hypothesize that the interaction between alimit cycle (originated by the Notch delayed-negative feedback loop) and a bistable switch(originated by the Wnt-Notch positive cross-regulation) is responsible for the observedsegmentation patterning. Our results agree with previously unexplained experimentalobservations and suggest a simple plausible mechanism for spatiotemporal control ofsomitogenesis in amniotes.

We discuss how distributed delays arise in biological models and review theliterature on such models. We indicate why it is important to keep thedistributions in a model as general as possible. We then demonstrate, throughthe analysis of a particular example, what kind of information can be gainedwith only minimal information about the exact distribution of delays.In particular we show that a distribution independent stability region maybe obtained in a similar way that delay independent results are obtained forsystems with discrete delays. Further, we show how approximations to theboundary of the stability region of an equilibrium point may be obtained withknowledge of one, two or three moments of the distribution. We compare theapproximations with the true boundary for the case of uniform and gammadistributions and show that the approximations improve as more moments are used.

This paper is concerned with the stabilisation of linear time-delay systems by tuning a finite number of parameters. Such problems typically arise in thedesign of fixed-order controllers. As time-delay systems exhibit an infinite amount of characteristic roots, a full assignment of the spectrum is impossible. However, if the system is stabilisable for the given parameter set, stability can in principle always be achieved through minimising the real part of the rightmostcharacteristic root, or spectral abscissa, in function of the parameters to be tuned. In general, the spectral abscissa is a nonsmooth and nonconvex function, precludingthe use of standard optimisation methods. Instead, we use a recently developed bundle gradient optimisation algorithm which has already been successfully applied to fixed-order controller design problems for systems of ordinary differential equations. In dealing with systems of time-delay type, we extend the use of this algorithm toinfinite-dimensional systems. This is realised by combining the optimisation method with advanced numerical algorithms toefficiently and accurately compute the rightmost characteristic roots of such time-delay systems. Furthermore, the optimisation procedure is adapted, enabling it to perform a local stabilisation of a nonlinear time-delay system along a branch of steady state solutions. We illustrate the use of the algorithm by presenting results for some numerical examples.