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.

A nonlinear system of two delay differential equations is proposed to modelhematopoietic stem cell dynamics. Each equation describes the evolution of asub-population, either proliferating or nonproliferating. The nonlinearityaccounting for introduction of nonproliferating cells in the proliferating phaseis assumed to depend upon the total number of cells. Existence and stabilityof steady states are investigated. A Lyapunov functional is built to obtain theglobal asymptotic stability of the trivial steady state. The study ofeigenvalues of a second degree exponential polynomial characteristic equationallows to conclude to the existence of stability switches for the uniquepositive steady state. A numerical analysis of the role of each parameter on theappearance of stability switches completes this analysis.

Many models in biology and ecology can be described by reaction-diffusionequations wit time delay. One of important solutions for these type of equationsis the traveling wave solution that shows the phenomenon of wave propagation.The existence of traveling wave fronts has been proved for large class ofequations, in particular, the monotone systems, such as the cooperativesystems and some competition systems. However, the problem on the uniqueness oftraveling wave (for a fixed wave speed) remains unsolved. In this paper, we showthat, for a class of monotone diffusion systems with time delayed reaction term,the mono-stable traveling wave font is unique whenever it exists.

We propose and analyze a nonlinear mathematical model of hematopoiesis,describing the dynamics of stem cell population subject to impulsiveperturbations. This is a system of two age-structured partial differentialequations with impulses. By integrating these equations over theage, we obtain a system of two nonlinear impulsive differential equations withseveral discrete delays. This system describes the evolution of the totalhematopoietic stem cell populations with impulses. We first examine theasymptotic behavior of the model in the absence of impulsions.Secondly, we add the impulsive perturbations and we investigate the qualitativebehavior of the model including the global asymptotic stability of the trivialsolution and the existence of periodic solution in the case of periodicimpulsive perturbations. Finally, numerical simulations are carriedout to illustrate the behavior of the model. This study maybe helpful tounderstand the reactions observed in the hematopoietic system after differentforms of stress as direct destruction by some drugs or irradiation.

In this paper, with the assumptions that an infectious disease has a fixedlatent period in a population and the latent individuals of the population maydisperse, we reformulate an SIR model for the population living in two patches(cities, towns, or countries etc.), which is a generalization of the classicKermack-McKendrick SIR model. The model is given by a system of delaydifferential equations with a fixed delay accounting for the latency andnon-local terms caused by the mobility of the individuals during the latentperiod. We analytically show that the model preserves some properties that theclassic Kermack-McKendrick SIR model possesses: the disease always dies out,leaving a certain portion of the susceptible population untouched (calledfinal sizes). Although we can not determine the two final sizes, we are able toshow that the ratio of the final sizes in the two patches is totally determinedby the ratio of the dispersion rates of the susceptible individuals between thetwo patches. We also explore numerically the patterns by which the disease diesout, and find that the new model may have very rich patterns for the diseaseto die out. In particular, it allows multiple outbreaks of the disease before itgoes to extinction, strongly contrasting to the classic Kermack-McKendrick SIRmodel.

In this paper, we consider a system of nonlinear delay-differential equations(DDEs) which models the dynamics of the interaction between chronic myelogenousleukemia (CML), imatinib, and the anti-leukemia immune response. Because of thechaotic nature of the dynamics and the sparse nature of experimental data, welook for ways to use computation to analyze the model without employing directnumerical simulation. In particular, we develop several tools usingLyapunov-Krasovskii analysis that allow us to test the robustness of the modelwith respect to uncertainty in patient parameters. The methods developed in thispaper are applied to understanding which model parameters primarily affect thedynamics of the anti-leukemia immune response during imatinib treatment. Thegoal of this research is to aid the development of more efficient modelingapproaches and more effective treatment strategies in cancer therapy.

In this survey, we briefly review some of our recent studies on predator-preymodels with discrete delay. We first study the distribution of zeros of a seconddegree transcendental polynomial. Then we apply the general results on thedistribution of zeros of the second degree transcendental polynomial to variouspredator-prey models with discrete delay, including Kolmogorov-typepredator-prey models, generalized Gause-type predator-prey models withharvesting, etc. Bogdanov-Takens bifurcations in delayed predator-prey modelswith nonmonotone functional response and in delayed predator-prey model withpredator harvesting are also introduced.