Nasze seminarium zakładowe odbywa się w środy 14:15-15:45 w sali 4050
2016-05-04: Monika Piotrowska i Marek Bodnar
Tytuł: Hypoxia and brain tumours
Streszczenie: Based on the Hahnfeldt et al. and Poleszczuk et al. ideas we propose a preliminary model of vascular tumour growth that takes into account two distinguishable fractions of cells (proliferative and these suffering form hypoxia) that describes the tumour reaction to different therapies. Our main aim is to construct the model that reflects medical observations reported for the glioma tumours. We wish to verify hypothesis that vessel normalisation (due to anti-angiogenic treatment) is beneficial to patients (in the terms of survival time) when radiotherapy is delayed. We would also say few words about the available clinical data.
2016-04-13: Urszula Foryś
Tytuł: Some remarks on the Gottman, Murray et al. model of marital dissolution and time delays
Streszczenie: I consider mathematical model proposed by Gottman, Murray and collaborators to describe marital dissolution. This model is described in the framework of discrete dynamical system reflecting emotional states of wife and husband, which is, however, non-symmetric. To make the model symmetric, one need to assume that the husband reacts with delay. Following this idea I consider the influence of time delays in the reaction terms of wife or husband, and study possibility of the change of stability with increasing delay. Surprisingly, it occurs that the delay has no impact on the stability, that is the condition of stability proposed by Murray remains unchanged under some additional, not very restrictive, assumption.
2016-04-06: Kamila Łyczek
Tytuł: Resource management problem – cooperative incentive equilibrium
Streszczenie: In my talk I’m going to present the model, first defined by Ehtamo and Hamalainen, which is subject of my current study.
Dynamic games consist of stages, in which conditions change over time (discrete or continuous). In a repeated game, change is caused by the accumulation of information about history of the game. In a differential game, like this model, time is continuous and, besides players’ information, there is a natural state variable, which changes in response to players’ decisions, which is given by a differential equation.
This model is a two-country dynamic model of whaling. Each country is an independent decision maker. The action sets consists of the player’s admissible whaling effort measured by the number of vessels involved in whaling. It is assumed that at every instant the players have exact knowledge of each other’s actions, and that their strategies make use of this information. Each country maximises the revenue over finite time horizon. Stock of whales, which is our state variable, has intrinsic growth, but it is influenced by whaling. In this game we look for socially optimal and equilibrium strategy profiles.
In future, we are also interested in reconsidering the model in infinite time horizon, which requires using quite different tools.
THIS PRESENTATION WILL BE IN POLISH!
2016-03-23: Adam Korpusik (UWM Olsztyn)
Tytuł: A nonstandard finite difference scheme for a basic model of cellular immune response to viral infection
Streszczenie: I will present a numerical method designed for the simulation of the basic mathematical model of cellular immune response to viral infection (originally proposed by M.A. Nowak and C.R. Bangham). The proposed numerical method is a nonstandard (semi-implicit) finite difference scheme, which preserves the essential qualitative features of the continuous model independently of the chosen step-size of the simulation. Namely, the non-negativity, boundedness and equilibria of the original model (along with their stability conditions) are preserved. Our numerical scheme does not generate qualitative errors that can easily be obtained using a standard simulation approach.
2016-03-16: Rajani Singh
Tytuł: Numerical versus analytic calculation of optima and equilibria in Fish Wars model with finite time horizon.
Streszczenie: In our research, we analyse a model of Fish Wars, first introduced by Levhari and Mirman, restricted to finite time horizon. In this models, dynamic games of extraction of common fishery by n>2 countries is studied, with logarithmic instantaneous and terminal payoffs and exponential function of regeneration of the biomass. We are interested in Nash equilibria and a profile resulting from maximization of aggregate payoff. We study it both by analytic and numerical methods and compare results. Since for this model, analytic results can be easilty calculated, comparison of numerical and analytic results is possible. However, if the model is seriously modified, then analytic calculation of optima and equilibria ceases to be feasible. In such a case, only numerical methods can be used. Therefore, analysis of dynamic games of this type using numerical methods is really needed. Although we study a specific model, one of more general objectives of the presentation is to answer the question, whether using numerical methods in a dynamic game model with a singularity in payoff, which results from considering instantaneuous and terminal payoffs with logarithmic part, can result in reasonable oucomes. Suprisigly, in this study, the answer is positive.
2016-03-09: Mariusz Bodzioch (doktorant UWM, stażysta KNOW-a)
Tytuł: Active case finding among homeless people as a means of reducing the incidence of pulmonary tuberculosis in general population.
Streszczenie: The incidence of tuberculosis (TB) declined more than two-fold, compared with the national average, in the northeastern region of Poland in the period of 2003–2012. During that time, four programs of active case finding of TB were conducted. The objective of the study was to find out whether the observed beneficial epidemiological trend could be a result of those programs. We addressed the issue of how the active case finding programs in the homeless community affected the TB incidence in the general population using a modified criss-cross SIS-type model which describes the dynamics of TB spread between the homeless and non-homeless populations. It is known that the community of homeless people is a natural reservoir of TB. Simulations show that effective preventive measures in this group considerably decreased the incidence of TB in the general population. Simulations also demonstrate that continued conduction of periodic active case finding programs in this group may further reduce the incidence of TB in the general population.
2016-03-09: Piotr Bajger (doktorant MISDoMP) i Mariusz Bodzioch
Tytuł: Role of cell competition in acquired chemotherapy resistance
Streszczenie: Acquired drug resistance poses a major obstacle in a design of effective chemotherapy protocols. A cytotoxic agent imposes a powerful force of selection on the tumour cells. As a result, an initially insignificant drug-resistant subpopulation may become dominant in the course of treatment. In this work, a mathematical model for tumour and vasculature growth proposed by Hahnfeldt et al. in 1999 is extended to account for this phenomenon. The malignant cell population is subdivided into n compartments with varying drug resistance. By means of numerical simulations, it is shown that acquired drug resistance may be explained solely on the basis of cell competition. The special case n=2 is investigated analytically in greater detail. As the current status is "work in progress", possible directions for further research are proposed.
2016-03-02: Wojciech Kaszyński (doktorant MIM)
Tytuł: Eastern Europe and Euro's "Catch-22", Study of the Region's Currency-pegging and inflation
Streszczenie: This presentation is based on my masters thesis submitted to the London School of Economics and Political Science in May of 2008. It is an empirical paper about inflation and currency-pegging in Eastern Europe, among the nations which joined the European Union on or after 1 May 2004.
As part of EU accession, all 12 nations agreed to eventually adopt the common currency Euro. To be eligible to do so, they had to fulfill a number of conditions, with two among them as follows:
1. For the two years prior to the adoption they had to be part of the European Exchange Rate Mechanism, ie. keeping their currency exchange rates within a small band versus the Euro (first +/2.25%, later changed to +/15%)
2. Having inflation no bigger than 1.5% above the average of the three EuroArea countries with lowest inflation
The thesis argues that these two conditions are mutually exclusive. The study looks at a microcosm of the economy, the Bread & Cereal market, to make its point. Bread & Cereal market was chosen, because its products are basicneed goods in Europe, easily traded and transported. This was crucial, because the argument rests on the Law of One Pricetwo similar goods should cost the same in all markets where there are no barriers to trade. That last conditions was satisfied for the EU countries once they joined the European Common Market. Also, Cyprus and Malta were dropped from the analysis, because, as islands, though formally they had no legal barriers to trade, the necessary overseas transport created one.
The study does indeed find that the gap between Bread & Cereal price in each new EU country and the average of that price in the old EU is a good predictor for inflation in the new EU country. This means that the bigger this difference versus the “Old EU”, the more the prices in the new EU country go up. This is consistent with the law of one price. Currencypegging is key in this, because unpegged currencies, instead of experiencing inflation, appreciate in value versus the Euro. Therefore, each new EU Member State either pegs its currency to Euro and experiences inflation or does not peg its currency, experiences lower inflation, but its currency appreciates a lot. Hence, the two conditions of Euroentry, low inflation and low variability in currency rate versus the Euro are, if not mutually exclusive, then at least really difficult to satisfy at the same time. The simple model in this study does indeed predict this phenomenon, with inflation higher by 11.5% on average in countries with the currency peg.
2016-01-20: Aleksandra Falkiewicz (doktorantka na Politechnice Łódzkiej)
Tytuł: Modeling of proliferation of gene mutations in aged structured populations
Streszczenie: One of the basic methods in modeling the proliferation of gene mutations is by the systems of ordinary differential equations expressing appropriate conservation principles. Another possibility is to use a model containing more information about the micro parameters such as the age population density. This approach is provided by the transport equations on network. We will prove the existence, find the form of solution to such problems and explain in what sense the transport model on a network can be regarded as a generalization of a model of population balance. Our work answers the question when it is reasonable to use a macro model for describing cells’ mutations and when the micro information essentially changes the long time behavior of the system.
(References: A singular limit for an age structured mutation problem, J. Banasiak, A. Falkiewicz, submitted to MATHEMATICAL BIOSCIENCES AND ENGINEERING)
2016-01-13: Mateusz Dębowski (doktorant MIM)
Tytuł: Cell cycle model - continuation
Streszczenie: My presentation will be continuation about model of cell cycle which include influence of protein CDC6. In my first presentation I showed my primary approach with many unknown functions in model and now I will show several version of model with simulations and some analysis.
2015-12-16: Oskar Górniewicz (doktorant w projekcie A. Wiszniewskiej-Matyszkiel)
Tytuł: Verification and refinement of Fischer-Mirman fish wars model
Streszczenie: In 1992 the paper "Strategic Dynamic Interaction -- Fish wars" by Fischer and Mirman was published. In that paper authors described dynamic, discrete, two population fish model. It is considered three problems:
1) symbiosis
2) prey-predator
3) competition.
We use the Bellman equation (some extension of the B.E.) in order to find
optimal strategies for both players. However there appears some technical problems in 2) as prey and in 3) both populations.
We modify dynamics inputting some minimal level for populations below which population extinct (no matter what players do). This natural assumption let us proof that the Value function fulfills the B.E.
Note: The talk will be in Polish this time.
2015-12-09: Jan Poleszczuk (H. Lee Moffitt Cancer Center & Research Institute)
Tytuł: Abscopal benefits of localized radiotherapy depend on activated T cell trafficking and distribution between metastatic lesions
Streszczenie: It remains unclear how localized radiotherapy for cancer metastases can occasionally elicit a systemic antitumor effect, known as the abscopal effect, but historically it has been speculated to reflect the generation of a host immunotherapeutic response. The ability to purposefully and reliably induce abscopal effects in metastatic tumors could meet many unmet clinical needs.
Here, we describe a mathematical model that incorporates physiological information about T cell trafficking to estimate the distribution of focal therapy-activated T cells between metastatic lesions. We integrated a dynamic model of tumor-immune interactions with systemic T cell trafficking patterns to simulate the development of metastases.
In virtual case studies, we found that the dissemination of activated T cells among multiple metastatic sites is complex and not intuitively predictable. Furthermore, we show that not all metastatic sites participate in systemic immune surveillance equally, and therefore the success in triggering the abscopal effect depends, at least in part, on which metastatic site is selected for localized therapy. Moreover, simulations revealed that seeding new metastatic sites may accelerate the growth of the primary tumor because T cell responses are partially diverted to the developing metastases, but the removal of the primary tumor can also favor the rapid growth of pre-existing metastatic lesions. Collectively, our work provides the framework to prospectively identify anatomically-defined focal therapy targets that are most likely to trigger an immune-mediated abscopal response, and therefore may inform personalized treatment strategies in patients with metastatic disease.
2015-12-02: Roman Cherniha (Institute of Mathematics of NASU, Kyiv, Ukraine)
Tytuł: A simplified Keller-Segel model: construction of exact solutions for the Cauchy and Neumann problems
Streszczenie: A simplified Keller-Segel model is studied by means of Lie symmetry based approaches. It is shown that this (1+2)-dimensional nonlinear system is invariant with respect infinity-dimensional Lie algebra. The result is extended on the Cauchy and Neumann problems for this system. The Lie symmetries obtained are used for reduction of the problems in question to two-dimensional and, as a result, exact solutions of some two-dimensional problems are constructed. In particular, we have proved that the Cauchy problem for the (1+1)-dimensional Keller-Segel type system can be linearized and solved in an explicit form. Moreover, additional biologically motivated restrictions were established in order to obtain uniqueness of solution. An analogous result is also derived for the (1+1)-dimensional Neumann problem with the same governing system. This research is a natural continuation of the paper "Exact solutions of the simplified Keller-Segel model" published in Commun Nonlinear Sci Numer Simulat 2013; 18: 2960-2971. by Cherniha R. and Didovych M.
2015-11-25: Marcin Choiński (doktorant MIM)
Tytuł: Discrete models of epidemics
Streszczenie: Two discrete models of epidemics will be presented. In the first model there is assumption that there are two groups in the population – healthy and infected people. In the second model there is additional third group – immune people. We suppose that the population is constant, it means the number of individuals does not change. Each individual has the same number of contacts with other people in a given period of time, that results in an equal chance of getting infected with disease.
2015-11-18: Piotr Bajger (MISDoMP)
Tytuł: Mathematical models for the corrosion of magnesium and its alloys
Streszczenie: Biodegradable materials have been extensively studied in recent years due to their potential to revolutionise the use of orthopaedic implants. In this context, one of the most promising candidates for a biomaterial is magnesium. When designing bioimplants, it is crucial to achieve the required mechanical properties while keeping the degradation rate at very low levels. In order to aid in the design of bioimplants, mathematical models for the degradation of magnesium have been developed. I will describe current approach to the modelling, presented by Gastaldi, et al. (Journal of the Mechanical Behavior of Biomedical Materials, 2011) and Grogan, et al. (Acta Biomaterialia, 2011) basing on the continuum damage theory. I will then describe a model we have recently developed which uses the level-set method and a system of partial differential equations to represent the chemical processes occurring at the interface between the implant and the biological medium.
2015-11-4: Magdalena Bogdańska (MIMUW)
Tytuł: A data-motivated density-dependent diffusion model of in vitro glioblastoma growth
Streszczenie: I will present very recent article by Stepien, Rutter and Kuang in Mathematical Biosciences and Engeneering, 2015. The research concerns modelling glioblastoma multiforme, which is is an extremely fatal brain cancer. It is characterized by both proliferation and large amounts of migration, which contributes to the difficulty of treatment. Previous models of this type of cancer growth often include two separate equations to model proliferation or migration. Authors propose a single equation which uses density-dependent diffusion to capture the behavior of both proliferation and migration. The model is analyzed in order to determine the existence of traveling wave solutions. The viability of the density-dependent diffusion functionchosen has been done by comparison of model with in vitro experimental data.
2015-10-28: Vladimir Mityushev (Department of Computer Sciences and Computer Methods, Pedagogical University Krakow)
Tytuł: Pattern formations and optimal packing
Streszczenie: Patterns of different symmetries may arise after solution to reaction-diffusion equations. Hexagonal arrays, layers and their perturbations are observed in different models after numerical solution to the corresponding initial-boundary value problems. We demonstrate an intimate connection between pattern formations and optimal random packing on the plane. The main study is based on the following two points. First, the diffusive flux in reaction-diffusion systems is approximated by piecewise linear functions in the framework of structural approximations. This leads to a discrete network approximation of the considered continuous problem. Second, the discrete energy minimization yields optimal random packing of the domains (disks) in the representative cell.
Therefore, the general problem of pattern formations based on the reaction-diffusion equations is reduced to the geometric problem of random packing. It is demonstrated that all random packings can be divided onto classes associated with classes of isomorphic graphs obtained form the Delaunay triangulation. The unique optimal solution is constructed in each class of the random packings. If the number of disks per representative cell is finite, the number of classes of isomorphic graphs, hence, the number of optimal packings is also finite.
2015-10-21: Jan Karbowski
Tytuł: Optymalizacja połączeń w mózgu
Streszczenie: Seminarium będzie poświęcone strukturze mózgu ssaków, w szczególności korze mózgowej (kluczowej dla procesów kognitywnych). W przeszłości wysuwane były sugestie, że struktura mózgu jest w dużej części konsekwencją ewolucyjnej minimalizacji uzwojeń neuronowych (dendryty i aksony), jako że są one kosztowne metabolicznie i biofizycznie. Ostatnio jednak pojawiło się parę prac, które poddają w wątpliwość tą hipotezę. Na seminarium opowiem o swoim wkładzie w ten problem, i o własnej alternatywnej "zasadzie" na której może być oparta struktura kory mózgowej ssaków. Zasada ta, zwana przeze mnie ekonomiczną maksymalizacją połączeń neuronowych ("spine economy maximization"), wyjaśnia empiryczną hierarchię w strukturze kory oraz posiada potencjał do wyjasnienia pewnych faktów z wielko-skalowej struktury mózgu. Całość stanowi wstęp do teoretycznych badań na temat ludzkiego konektomu ("human connectome"), co jest jednoczesnie ostatnio bardzo ważne i modne w neuronauce.
Seminarium bedzie oparte w dużej mierze na mojej pracy:
http://journals.plos.org/ploscompbiol/article?id=10.1371/journal.pcbi.1004532
Karbowski J (2015) "Cortical composition hierarchy driven by spine proportion economical maximization or wire volume minimization",
PLoS Comput Biol 11(10): e1004532.
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