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1
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Co możemy opisać układem dynamicznym?

100%
Foryś U.
Annales Universitatis Paedagogicae Cracoviensis | Studia ad Didacticam Mathematicae Pertinentia
|
2014
|
tom 6
73-85
PL
In this paper we present several examples of simple dynamical systemsdescribing various real processes. We start from well know Fibonaccisequence, through Lotka-Volterra model of prey-predator system, love affairdynamics, ending with modelling of tumour growth.
2
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Influence of diffusion on interactions between malignant gliomas and immune system

100%
Foryś U.
Applicationes Mathematicae
|
2010
|
tom 37
|
nr 1
53-67
EN
We analyse the influence of diffusion and space distribution of cells in a simple model of interactions between an activated immune system and malignant gliomas, among which the most aggressive one is GBM Glioblastoma Multiforme. It turns out that diffusion cannot affect stability of spatially homogeneous steady states. This suggests that there are two possible outcomes-the solution is either attracted by the positive steady state or by the semitrivial one. The semitrivial steady state describes the healthy state, while the positive one reflects the chronic disease and typically the level of tumour cells in this state is very high, exceeding the threshold of lethal outcome. Results of numerical simulation show that the initial tumour cells distribution has an essential impact on the dynamics of the system. If the positive steady state exists, then we observe bistability and the initial distribution decides to which steady state the solution tends.
3
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Mathematical models in epidemiology and immunology

100%
Foryś U.
Mathematica Applicanda
|
2000
|
tom 28
|
nr 42/01
PL
Artykuł nie zawiera streszczenia
EN
The article contains no abstract
4
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Professor Wiesław Szlenk (1935-1995)

100%
Foryś U.
Applicationes Mathematicae
|
2000
|
tom 27
|
nr 1
1-20
5
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National Conference Applications of Mathematics to Biology and Medicine

100%
Foryś U.
Mathematica Applicanda
|
2018
|
tom 46
|
nr 1
EN
It is unbelievable that this year we will meet at the National Conference Applications of Mathematics to Biology and Medicine for the 24th time. The initiator, chairman of the Scientific Committee and tireless participant of all previous conferences is Prof. Mariusz Ziółko who in the 1970s participated in the Schools of Mathematical Modeling in Biology organized by Prof. Adam Łomnicki, an outstanding biologist and ecologist from the Jagiellonian University. In 1994, M. Ziółko suggested to Prof. Łomnicki reactivation of these schools, thanks to which many young people got a chance to become a recognizable researcher. Unfortunately, he met with refusal. However, Prof. Łomnicki suggested that Prof. Ziółko continue the work, but according to his own concept. In 1995, the first conference took place - we met in an elegant centre in Zakopane, and I owe my presence to this conference to my late professor, Wiesław Szlenk, who also participated schools organized by Prof. Łomnicki. Therefore he appeared on the list of people to whom M. Ziółko sent out invitations. While the supervisor sent me to the conference... That's how it was - I participated in all (except one) conferences, actively co-organizing many of them.This year we have another experiment - instead of the conference materials, we give up the readers a special issue of Mathematica Applicanda. I hope that thanks to this the articles will reach a wider group of potential readers, and the conference will gain new participants in the future.
6
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Jan Poleszczuk – the Winner of the Polish Mathematical Society’s Prize

100%
Foryś U.
Mathematica Applicanda
|
2012
|
tom 40
|
nr 1
PL
Jan Poleszczuk urodził się 19 października 1986 roku w Warszawie w rodzinie humanistów. Rodzice i starszybrat są socjologami. Może trochę z przekory Janek postanowił studiować matematykę, choć z drugiej stronynie można wykluczyć, że pewien wpływ na to miała fascynacja jego ojca narzędziami statystycznymi. Wybie-rając proseminarium licencjackie, postanowił połączyć „tradycję z nowoczesnością”, zapisał się na „Biomatematykę i teorię gier” i pierwszym jego pomysłem było zastosowanie modeli Lesliego — standardowo stosowanych w socjologii — do opisu dynamiki nowotworów. Jednak w miarę zagłębiania się w tematykę zorientował się, że ten sposób opisu nie jest najlepszy, gdyż dotyczy zmian w kolejnych generacjach, natomiast wzrost nowotworu jest procesemciągłym, więc lepsze w tym przypadku są równania różniczkowe. Zorientował się także, że badania nad nowotworami stale postępują, ciągle odkrywane są kolejne fragmenty „łamigłówki”. Zainteresował się nowymi wynikami dotyczącymi procesu angiogenezy (tworzenia naczyń krwionośnych) nowotworowej, które wykazały, że powstające naczynia mają niewłaściwą strukturę, co przyczynia się np. do niejednorodnego rozprowadzania aplikowanych leków. Dopingowany przez promotor Urszulę Foryś – pod kierunkiem której napisał później pracę magisterską i pracuje obecnie nad doktoratem – zaproponował własny model, opisany w pracy licencjackiej, a następnie zaprezentowany na XIV Krajowej Konferencji Zastosowań Matematyki w Biologii i Medycynie (2008) w Lesznie, gdzie jego referat pt. „Model rozwoju nowotworu uwzględniający patologię angiogenezy” wzbudził ogromne zainteresowanie. Trzeba podkreślić, że w konferencji brali udział nie tylko uczestnicy krajowi, ale także profesorowie Marek Kimmel i Urszula Ledzewicz z USA oraz dr (obecnie prof.) Anna Marciniak-Czochra z Heidelbergu. Wystąpienie J. Poleszczuka (przygotowanei wytrenowane wcześniej co do minuty!) zrobiło takie wrażenie, że wszyscy sądzili, że prezentuje wstępne tezy pracy doktorskiej. Wyniki dotyczące tego etapu badań zostały opublikowane w [1] i [2].
EN
Jan Poleszczuk was born on 19th of October 1986 in Warsaw in a family of humanists. His parents and older brother are sociologists. Maybe a little out of contrariness he decided to study mathematics. However, it is possible that he was influenced by his father's fascination with statistical tools. Choosing a bachelor's seminar he decided to combine ''tradition with modernity'', registered for ''Biomathematics and Game Theory'' and his first idea was to use Lesli matrices, which are typically used in sociology, to model tumours dynamics. However, going into details he figured out that this type of description is not proper, because Lesli models consider subsequent generations, while the tumour growth is a continuous process. Therefore, the approach of differential equations should be better. He became interested in new results concerning the process of tumour angiogenesis (i.e. formation of new blood vessels from existing ones), which showed that the vessels have the wrong structure, contributing for example to non-uniform distribution of drugs application. Encourageing by his supervisor, Urszula Foryś (who is still his supervisor of PhD), he proposed his own model, which was presented at XIV National Conference of Applications in Biology and Medicine (2008) in Leszno. His report ''A model of tumour development taking into account angiogenesis pathology'' aroused great interest among the foreign researchers present in the conference, among them  prof. Marek Kimmel, prof. Urszula Ledzewicz (both from USA) and Dr. (now prof.) Anna Marciniak-Czochra from Heidelberg. Presentation by J. Poleszczuk (he prepared and trained to make it perfect in time!) made such an impression that everyone thought that he presented a preliminary PhD thesis. Results concerning this stage of research has been published.
7
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XXV National Conference on Mathematics Applied to Biology and Medicine

100%
Foryś U.
Mathematica Applicanda
|
2019
|
tom 47
|
nr 2
EN
A round anniversary is a reflection-friendly moment ... Like last year, I~would like to start with ``I can not believe that...'' This time, however, it will be more personally. Therefore -- I can not believe that it has been 25 years since I walked through the empty (yes, it is unbelievable as well!) streets of Zakopane to the resort where the First National Conference Applications of Mathematics to Biology and Medicine took place. At the end of my journey with quite a heavy backpack (clearly, you need to take the official, everyday and tourist outfit) I met only a hedgehog, which at that time was rather rare. Just at the entrance, participants were welcomed by Przemek Sypka, who participated in the organization of the first conferences, and at the same time created the logo that we use to this day. Another person I met there was Prof. Mariusz Ziółko -- initiator, chairman of the Scientific Committee and the only person who participated in all our conferences. My humble person ranks second in this ranking, as so far I missed only the conference in Mądralin, while the third one is Prof. Antoni Leon Dawidowicz, who was absent twice.
8
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Models of interactions between heterotrophic and autotrophic organisms

64%
Foryś U., Szymańska Z.
Applicationes Mathematicae
|
2009
|
tom 36
|
nr 3
279-294
EN
We present two simple models describing relations between heterotrophic and autotrophic organisms in the land and water environments. The models are based on the Dawidowicz & Zalasiński models but we assume the boundedness of the oxygen resources. We perform a basic mathematical analysis of the models. The results of the analysis are complemented by numerical illustrations.
9
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Angiogenesis process with vessel impairment for Gompertzian and logistic type of tumour growth

64%
Poleszczuk J., Foryś U.
Applicationes Mathematicae
|
2009
|
tom 36
|
nr 3
313-331
EN
We propose two models of vessel impairment in the process of tumour angiogenesis and we consider three types of treatment: standard chemotherapy, antiangiogenic treatment and a combined treatment. The models are based on the idea of Hahnfeldt et al. that the carrying capacity for any solid tumour depends on its vessel density. In the models proposed the carrying capacity also depends on the process of vessel impairment. In the first model a logistic type equation is used to describe the neoplastic cell dynamics, while in the second one we use the Gompertz type of growth. Simulations of solutions show that a vascular dormant state of the tumour can be reached in two different ways. In addition in each case efficiency of treatments is different.
10
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Stability switches for some class of delayed population models

64%
Skonieczna J., Foryś U.
Applicationes Mathematicae
|
2011
|
tom 38
|
nr 1
51-66
EN
We study stability switches for some class of delay differential equations with one discrete delay. We describe and use a simple method of checking the change of stability which originally comes from the paper of Cook and Driessche (1986). We explain this method on the examples of three types of prey-predator models with delay and compare the dynamics of these models under increasing delay.
11
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Influence of time delays on the Hahnfeldt et al. angiogenesis model dynamics

64%
Bodnar M., Foryś U.
Applicationes Mathematicae
|
2009
|
tom 36
|
nr 3
251-262
EN
We study the influence of time delays on the dynamics of the general Hahnfeldt et al. model of an angiogenesis process. We analyse the dynamics of the system for different values of the parameter α which reflects the strength of stimulation of the vessel formation process. Time delays are introduced in three subprocesses: tumour growth, stimulation and inhibition of vessel formation (represented by endothelial cell dynamics). We focus on possible destabilisation of the positive steady state due to the delay. Results are illustrated by numerical simulations performed for parameter values estimated by Hahnfeldt et al. for tumour volume data of Lewis lung carcinoma implanted in mice.
12
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Logistic equations in tumour growth modelling

64%
Foryś U., Marciniak-Czochra A.
International Journal of Applied Mathematics and Computer Science
|
2003
|
tom 13
|
nr 3
317-325
EN
The aim of this paper is to present some approaches to tumour growth modelling using the logistic equation. As the first approach the well-known ordinary differential equation is used to model the EAT in mice. For the same kind of tumour, a logistic equation with time delay is also used. As the second approach, a logistic equation with diffusion is proposed. In this case a delay argument in the reaction term is also considered. Some mathematical properties of the presented models are studied in the paper. The results are illustrated using computer simulations.
13
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Time delays in proliferation and apoptosis for solid avascular tumour

64%
Foryś U., Kolev M.
Banach Center Publications
|
2003
|
tom 63
|
nr 1
187-196
EN
The role of time delays in solid avascular tumour growth is considered. The model is formulated in terms of a reaction-diffusion equation and mass conservation law. Two main processes are taken into account-proliferation and apoptosis. We introduce time delay first in underlying apoptosis only and then in both processes. In the absence of necrosis the model reduces to one ordinary differential equation with one discrete delay which describes the changes of tumour radius. Basic properties of the model depending on the magnitude of delay are studied. Nonnegativity of solutions is investigated. Steady state and the Hopf bifurcation analysis are presented. The results are illustrated by computer simulations.
14
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Ananysis of a predator-prey model with disease in the predator species

64%
Radziński P., Foryś U.
Mathematica Applicanda
|
2018
|
tom 46
|
nr 1
EN
In the paper we analyse a diffusive predator-prey model with disease in predator species proposed by Qiao et al. (2014). In the original article there appears a mistake in the procedure of the model undimensionalisation. We make a correction in this procedure and show that some changes in the model analysis are necessary to obtain results similar to those presented by Qiao et al.We propose corrected conditions for global stability of one of existing equilibria -- disease free steady state and endemic state in the case without diffiusion as well as in the model with diffusion. On the basis of the corrected analysis we present new stability results.
15
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O pewnym ciekawym zastosowaniu modelu Lotki-Volterry

64%
Matejek P., Foryś U.
Matematyka Poglądowa
|
2014
|
nr 1
8-27
PL
-
16
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Some remarks on modelling of drug resistance for low grade gliomas

64%
Bodnar M., Foryś U.
Mathematica Applicanda
|
2019
|
tom 47
|
nr 2
PL
W artykule analizujemy nową wersję modelu opisującego efekt nabytej lekooporności, który zaproponowaliśmy w pracy Bodnar & Foryś (2017). Oryginalny model powstał w oparciu o idee przedstawione w artykule Pérez-García i in. (2015). W bieżącej pracy włączamy do modelu dodatkowy składnik opisujący bezpośrednią śmiertelność komórek uszkodzonych. Okazuje się, że dynamika tak zmienionego modelu jest analogiczna, jak w przypadku drugiego modelu rozważanego przez nas, który z kolei powstał w oparciu o idee Olliera i in. (2017). Dynamika modelu została przeanalizowana dla parametrów odzwierciedlających wzrost glejaka niskiego stopnia, przy czym analizowaliśmy wpływ zmian poszczególnych parametrów na tę dynamikę.
EN
In this paper we present a version of a simple mathematical model of acquiring drug resistance which was proposed in Bodnar and Foryś (2017). We based the original model on the idea coming from Pérez-García et al. (2015). Now, we include the explicit death term into the system and show that the dynamics of the new version of the model is the same as the dynamics of the second model considered by us and based on the idea of Ollier et al. (2017). We discuss the model dynamics and its dependence on the model parameters on the example of gliomas.
17
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Periodic dynamics in a model of immune system

64%
Bodnar M., Foryś U.
Applicationes Mathematicae
|
2000
|
tom 27
|
nr 1
113-126
EN
The aim of this paper is to study periodic solutions of Marchuk's model, i.e. the system of ordinary differential equations with time delay describing the immune reactions. The Hopf bifurcation theorem is used to show the existence of a periodic solution for some values of the delay. Periodic dynamics caused by periodic immune reactivity or periodic initial data functions are compared. Autocorrelation functions are used to check the periodicity or quasiperiodicity of behaviour.
18
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Complementary analysis of the initial value problem for a system of o.d.e. modelling the immune system after vaccinations

64%
Foryś U., Żołek N. S.
Applicationes Mathematicae
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2000
|
tom 27
|
nr 1
103-111
EN
Complementary analysis of a model of the human immune system after a series of vaccinations, proposed in [7] and studied in [6], is presented. It is shown that all coordinates of every solution have at most two extremal values. The theoretical results are compared with experimental data.
19
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A modified van der Pol equation with delay in a description of the heart action

52%
Zduniak B., Bodnar M., Foryś U.
International Journal of Applied Mathematics and Computer Science
|
2014
|
tom 24
|
nr 4
853-863
EN
In this paper, a modified van der Pol equation is considered as a description of the heart action. This model has a number of interesting properties allowing reconstruction of phenomena observed in physiological experiments as well as in Holter electrocardiographic recordings. Our aim is to study periodic solutions of the modified van der Pol equation and take into consideration the influence of feedback and delay which occur in the normal heart action mode as well as in pathological modes. Usage of certain values for feedback and delay parameters allows simulating the heart action when an accessory conducting pathway is present (Wolff-Parkinson-White syndrome).
20
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Model of AIDS-related tumour with time delay

52%
Bodnar M., Foryś U., Szymańska Z.
Applicationes Mathematicae
|
2009
|
tom 36
|
nr 3
263-278
EN
We present and compare two simple models of immune system and cancer cell interactions. The first model reflects simple cancer disease progression and serves as our "control" case. The second describes the progression of a cancer disease in the case of a patient infected with the HIV-1 virus.
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