Urszula Foryś

BIOLOGICAL DELAY SYSTEMS AND THE MIKHAILOV CRITERION OF STABILITY

This paper deals with the stability analysis of biological delay systems. The Mikhailov criterion of stability is presented (and proved in the Appendix) for the case of discrete delay and distributed delay (i.e., delay in integral form). This criterion is used to check stability regions for some well-known equations, especially for the delay logistic equation and other equations with one discrete delay which appear in many applications. Some illustrations of the behavior of Mikhailov hodograph are shown.

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.

Marchuk's Model of Immune System Dynamics with Application to Tumour Growth

Marchuks model of a general immune reaction is presented in the paper. The results of investigation of the model are summarized. The qualitative behaviour of solutions to the model and its simplification is described. Many illustrations of recovery process, oscillations or lethal outcomes of a disease are shown. The model with time-dependent immune reactivity is also considered. Periodic dynamics caused by different reasons are compared.

Cellular Immunotherapy for High Grade Gliomas: Mathematical Analysis Deriving Efficacious Infusion Rates Based on Patient Requirements

To date, no effective cure has been found for high grade malignant glioma (HGG), current median survival for HGG patients under treatment ranging from 18 months (grade IV) to 5 years (grade III). Recently, T cell therapy for HGG has been suggested as a promising avenue for treating such resistant tumors, but clinical outcome has not been conclusive. For studying this new therapy option, a mathematical model for tumor–T cell interactions was developed, where tumor immune response was modeled by six coupled ordinary differential equations describing tumor cells, T cells, and their respective secreted cytokines and immune mediating receptors. Here we mathematically analyze the model in an untreated case and under T cell immunotherapy. For both settings we classify steady states, determine stability properties, and explore the global behavior of the model. Analysis suggests that in untreated patients, the system always converges to a steady state of a large tumor mass. An increase in the patients pro-inflamm...

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.

ANTI-TUMOR IMMUNITY AND TUMOR ANTI-IMMUNITY IN A MATHEMATICAL MODEL OF TUMOR IMMUNOTHERAPY

A two-dimensional system of ordinary differential equations is used to characterize the basic types of phase portraits of the immune system — tumor interactions model, and to study the impact of anti-immune activity by tumor on the outcome of immunotherapy. The focus is on specific (acquired) immunity and different forms of immunotherapy as active therapy with in vivo stimulation of the immunity and passive one with infusion of ex vivo produced specific immunity. The analysis is performed for two families of stimulation function, which describes the dynamics of the stimulation of the immune system by tumor antigens: (1) antigen dependent and (2) antigen per one immunity unit dependent functions, with Michaelis-Menten and sigmoid functions in each family. We show that there are no limit cycles in the system and that anti-immune activity by tumor changes all equilibrium points from global to local ones. In the latter case, the immune system has no control over the growth of large tumors. Furthermore, if the immunity is weak, the immune system cannot eradicate even small tumors. The weak immunity and stimulation strength result in unrestricted tumor growth. The patterns of asymptotic behavior of the system do not depend on the type of the stimulation function, but do depend on its parameters. Our results reflect the basic clinical and experimental knowledge about immunotherapy and its effectiveness and yield new suggestions for an efficient immunotherapy.

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.

The nature of Hopf bifurcation for the Gompertz model with delays

Abstract In this paper, we study the influence of time delays on the dynamics of the classical Gompertz model. We consider the models with one discrete delay introduced in two different ways and the model with two delays which generalise those with one delay. We study asymptotic behaviour and bifurcations with respect to the ratio of delays τ = τ 1 / τ 2 . Our results show that in such model with two delays there is only one stability switch and for a threshold value of bifurcation parameter, Hopf bifurcation (HB) occurs. However, the type of HB, and therefore its stability (i.e. stability of periodic orbits arising due to it), strongly depends on the magnitude of τ . The function describing stability of HB is periodic with respect to τ . Within one period of length 4 five changes of HB stability are observed. We also introduce the second model with two delays which has a better biological interpretation than the first one. In that model several stability switches can occur, depending on the model parameters. We illustrate analytical results on the example of tumour growth model with parameters estimated on the basis of experimental data.

Dynamical Models of Dyadic Interactions with Delay

We investigate a general class of linear models of dyadic interactions with a constant discrete time delay. We prove that the changes in stability of the stationary states occur for various intervals of the parameters that determine the strength and nature of emotional interactions between the partners. The dynamics of interactions depend on both reactivity of partners to their own emotional states as well as to the partners states. The results suggest that reactivity to the partners states has greater impact on the dynamics of the relationship than the reactivity to ones own states. Moreover, the results underscore the importance of deliberation in maintaining the stability of the relationship. Moreover, we have found that multiple stability switches are only possible when one of the partners reacts with delay to their own emotional states. We also propose a generalization to triadic interactions.