Marek Bodnar

The nonnegativity of solutions of delay differential equations

Abstract General conditions guaranteeing the nonnegativity of solutions of delay differential equations are proposed. Some examples when the nonnegativity is not preserved in time are given. The nonnegativity of solutions of the logistic equation with time delay is considered. Behaviour like in discrete time case is observed.

Time delays in proliferation process for solid avascular tumour

In the paper, a simple model of solid avascular tumour growth is studied. The model is based on the reaction-diffusion dynamics and mass conservation law and is considered with delay in cell proliferation process. In the case considered in this paper, the model reduces to one ordinary differential equation with time delay which describes the evolution of tumour radius. Basic properties of the model are studied in the paper. Steady-state analysis is presented with respect to the magnitude of delay. Existence of periodic solutions is proved for some parameter values. Global stability is proved for other ones. Results are illustrated by computer simulations.

THREE TYPES OF SIMPLE DDE'S DESCRIBING TUMOR GROWTH

In this paper, we compare three types of dynamical systems used to describe tumor growth. These systems are defined as solutions to three delay differential equations: the logistic, the Gompertz and the Greenspan types. We present analysis of these systems and compare with experimental data for Ehrlich Ascites tumor in mice.

ANGIOGENESIS MODEL WITH CARRYING CAPACITY DEPENDING ON VESSEL DENSITY

In this paper we consider a modification of the model of angiogenesis process proposed by Agur et al., in Discrete & Cont. Dyn. Sys. B4(1): 29–38, 2004. The number of steady states and their stability depending on the model parameters are studied. The hysteresis effect and cusp catastrophe are found for some parameters. The effect of hysteresis when the number of positive steady states changes from one to three is studied. The time delays are introduced. Numerical simulations, which show that oscillatory behaviour is possible, are performed.

Time delays in regulatory apoptosis for solid avascular tumour

In the paper, a model of multicellular spheroids dynamics is studied. It can be also interpreted as avascular tumour growth. The model is based on the diffusion of nutrient and mass conservation for two processes cell proliferation and apoptosis. Two types of apoptosis is taken into account-underlying and regulatory apoptosis. It is assumed that the process of regulatory apoptosis is delayed compared to proliferation and underlying apoptosis. Nonnegativity of solutions, steady state stability, and existence of periodic solutions are studied in the paper. Computer simulations of solutions are presented. Some conclusions for special values of the model parameters are stated.

New approach to modeling of antiangiogenic treatment on the basis of Hahnfeldt et al. model

In the paper we propose a new methodology in modeling of antiangiogenic treatment on the basis of well recognized model formulated by Hahnfeldt et al. in 1999. On the basis of the Hahnfeldt et al. model, with the usage of the optimal control theory, some protocols of antiangiogenic treatment were proposed. However, in our opinion the formulation of that model is valid only for the antivascular treatment, that is treatment that is focused on destroying endothelial cells. Therefore, we propose a modification of the original model which is valid in the case of the antiangiogenic treatment, that is treatment which is focused on blocking angiogenic signaling. We analyze basic mathematical properties of the proposed model and present some numerical simulations.

Stochastic Models of Gene Expression with Delayed Degradation

In many biochemical reactions occurring in living cells, number of various molecules might be low which results in significant stochastic fluctuations. In addition, most reactions are not instantaneous, there exist natural time delays in the evolution of cell states. It is a challenge to develop a systematic and rigorous treatment of stochastic dynamics with time delays and to investigate combined effects of stochasticity and delays in concrete models.We propose a new methodology to deal with time delays in biological systems and apply it to simple models of gene expression with delayed degradation. We show that time delay of protein degradation does not cause oscillations as it was recently argued. It follows from our rigorous analysis that one should look for different mechanisms responsible for oscillations observed in biological experiments.We develop a systematic analytical treatment of stochastic models of time delays. Specifically we take into account that some reactions, for example degradation, are consuming, that is: once molecules start to degrade they cannot be part in other degradation processes.We introduce an auxiliary stochastic process and calculate analytically the variance and the autocorrelation function of the number of protein molecules in stationary states in basic models of delayed protein degradation.

Delay can stabilize: Love affairs dynamics

Abstract We discuss two models of interpersonal interactions with delay. The first model is linear, and allows the presentation of a rigorous mathematical analysis of stability, while the second is nonlinear and a typical local stability analysis is thus performed. The linear model is a direct extension of the classic Strogatz model. On the other hand, as interpersonal relations are nonlinear dynamical processes, the nonlinear model should better reflect real interactions. Both models involve immediate reaction on partner’s state and a correction of the reaction after some time. The models we discuss belong to the class of two-variable systems with one delay for which appropriate delay stabilizes an unstable steady state. We formulate a theorem and prove that stabilization takes place in our case. We conclude that considerable (meaning large enough, but not too large) values of time delay involved in the model can stabilize love affairs dynamics.

Gompertz model with delays and treatment: Mathematical analysis

In this paper we study the delayed Gompertz model, as a typical model of tumor growth, with a term describing external interference that can reflect a treatment, e.g. chemotherapy. We mainly consider two types of delayed models, the one with the delay introduced in the per capita growth rate (we call it the single delayed model) and the other with the delay introduced in the net growth rate (the double delayed model). We focus on stability and possible stability switches with increasing delay for the positive steady state. Moreover, we study a Hopf bifurcation, including stability of arising periodic solutions for a constant treatment. The analytical results are extended by numerical simulations for a pharmacokinetic treatment function.

A simple model of carcinogenic mutations with time delay and diffusion

In the paper we consider a system of delay differential equations (DDEs) of Lotka-Volterra type with diffusion reflecting mutations from normal to malignant cells. The model essentially follows the idea of Ahangar and Lin (2003) where mutations in three different environmental conditions, namely favorable, competitive and unfavorable, were considered. We focus on the unfavorable conditions that can result from a given treatment, e.g. chemotherapy. Included delay stands for the interactions between benign and other cells. We compare the dynamics of ODEs system, the system with delay and the system with delay and diffusion. We mainly focus on the dynamics when a positive steady state exists. The system which is globally stable in the case without the delay and diffusion is destabilized by increasing delay, and therefore the underlying kinetic dynamics becomes oscillatory due to a Hopf bifurcation for appropriate values of the delay. This suggests the occurrence of spatially non-homogeneous periodic solutions for the system with the delay and diffusion.