Lectures
Series of lectures
Vincenzo Capasso Rescaling stochastic systems of interacting elements; Their mean field approximations (3 lectures)
Abstract: Approximating a system of stochastic "individuals" by a deterministic continuous reaction diffusion system, has been the subject of a long lasting scientific investigations, mainly in Statistical Physics. During last decades a large interest has derived by models in Life Sciences, such as aggregation of individuals in swarms, crystallization of shells, vasculogenesis. An important application of such approximation concerns the analytical and computational affordability of continuous deterministic systems as opposed to large systems of interacting "particles" subject to stochastic fluctuations. As a by product we may have the so called hybrid models for multi-scale systems, which keep individual behaviour at the lower scale coupled with deterministic mean field behaviour at the larger scale.
This minicourse will be devoted to lay the mathematical foundations for a rigorous derivation of mean field approximations of systems of interacting "particles" subject to stochastic fluctuations, including
(i) repulsion models
(ii) aggregation-repulsion models
(iii) angiogenesis models
Ryszard Rudnicki Piecewise deterministic Markov processes and their applications in biology (3 lectures)
Abstract: The aim of these lectures is to give a short mathematical introduction to piecewise deterministic Markov processes (PDMPs) and to present biological models where they appear. In the first two lectures we will give some examples of biological phenomena such as gene activity and population growth leading to different type of PDMPs: continuous time Markov chains, deterministic processes with jumps, dynamical systems with random switching and stochastic billiards. Biological models introduced in these lectures will be birth-death processes, kangaroo movement, dispersal of insects, gene expression, cell cycle models and neural activity. We will also present mathematical tools useful in studing properties of these processes: Kolmogorov equations and stochastic semigroups. The last lecture will be devoted to the long-time behaviour (asymptotic stability and sweeping) of the stochastic semigroups induced by PDMPs. The lectures are based on the book:
R.Rudnicki, M.Tyran-Kamińska, Piecewise Deterministic Processes in Biological Models, SpringerBriefs in Applied Sciences and Technology, 2017.
Yuri Kondratiev Statistical dynamics in the continuum: from microscopic to kinetic description (2 lectures)
Abstract: We will discuss a notion of states evolution corresponding to Markov generators for interacting particle systems in the continuum. A mesoscopic scaling of such dynamics leads the kinetic hierarchy for correlation functions and to a non-linear Vlasov-type evolution equation for the density of the system.
Considering a random time change in the dynamics, we arrive in a non-Markov evolution of the system which may be described by means of a fractional in time evolution equations for the states. The kinetic behavior of the density in such non-Markov dynamics may be obtained from the original evolution via a subordination principle. We will show how fractional kinetics exhibit new surprising properties comparing with the original ones.
Mustapha Mokhtar-Kharroubi A short introduction to linear semigroup theory (2 lectures)
Abstract: The object if these short lectures is to provide a smooth introduction to operator semigroups. Linear semigroup theory deals with linear autonomous Cauchy problems
f'(t)=A(f(t)) (t≥0) f(0)=x∈X
in a Banach space X where A:D(A)⊂X→X is an unbounded linear operator. This theory is motivated by a wealth of problems arising in Probability, Mathematical Physics, Mathematical Biology etc. This Cauchy problem is easily solved by taking the exponential function f(t)=etAx if A is a bounded operator on X. However, the situation is much more involved when A is unbounded. In this case, the construction of an exponential function "etA" is obtained in a quite indirect way for a suitable class of unbounded linear operators A:D(A)⊂X→X. This the object of the classical Hille-Yoshida's theorem which solves the Cauchy problem by
f(t)=T(t)x (t≥0)
where T(t) (indexed by t≥0) are bounded linear operators on X satisfying the properties: (i) T(0) is the identity operator on X, (ii) the semigroup property T(t+s)=T(t)T(s), (iii) the continuity of the function [0,+∞)∋t→ T(t)x∈X for all x∈X. We will survey this key generation theorem and related results and some classical examples of applied interest. The understanding of the behaviour of T(t) as t goes to infinity is an important topic in semigroup theory. We will consider this issue when X is some L1 space and the semigroup (T(t))t≥0 is positive (i.e. T(t)f≥0 when f≥0) and has a "spectral gap"; an illustration by neutron transport equations on the torus is given. Since we don't have the time to prove all the results, precise references will be provided.
Single lectures
Jacek Banasiak Structured population dynamics – patches and networks
Abstract: We consider structured population models in which the population is subdivided into states according to certain feature of the individuals. We consider various rules allowing individuals to move between the states – it may be physical migration between geographical patches or the change of the genotype by mutations during mitosis. We shall see that depending on the type of the migration rule the models can vary from a system of coupled McKendrick equations to a system of transport equations on a graph. We address the well-posedness of such problems, classical in the first case and more challenging in the second. The main interest, however, will be asymptotic state aggregation that, in the presence of significantly different time scales, allows for a significant simplification of the equations. Interestingly enough, the aggregated equations vary widely from scalar transport equations to systems of ordinary differential equations.
References
[1] J. Banasiak, A. Falkiewicz A singular limit for an age structured mutation problem, Mathematical Biosciences and Engineering, 14 (2017), no. 1, 17-30.
[2] J. Banasiak, A. Falkiewicz, P. Namayanja, Asymptotic state lumping in transport and diffusion problems on networks, Mathematical Models and Methods in Applied Sciences, 26 (2016), no. 2, 215-247.
[3] J. Banasiak, A. Goswami, Singularly perturbed population models with reducible migration matrix 1. Sova-Kurtz theorem and the convergence to the aggregated model, Discrete Contin. Dyn. Syst-B. 35 (2015), no. 2, 617–635.
Adam Bobrowski Robin-type boundary conditions in transition from reaction-diffusion equations in 3D domains to equations in 2D domains
Abstract: We consider a singular limit of diffusion equations in 3D domains of thickness converging to zero. In the 2D limit the resulting reaction-diffusion equation has a source term resulting from the Robin-type boundary conditions imposed on boundaries of the original 3D domain. The proposed approach can be applied to constructing approximate solutions of diffusion problems in thin planar, cylindrical, or spherical layers between two membranes. As an example we refer to the problem of activation of B lymphocytes, which typically have large nuclei and a thin cytoplasmic layer which can be considered as a spherical shell.
Joint work with Tomasz Lipniacki.
Jerzy Kozicki Fission-death dynamics with applications to cell division and tumor growth
Abstract: Nowadays, it is well established that the initiation and progression of tumor is related to cell division mechanisms. In particular, the initiation of tumor is related to (driver) mutations that may occur in the course of division. In the model which we consider, a random collection of entities (cells) undergoes a continuum time Markov evolution which amounts to two events: fission and death. The state of an entity is characterized a positive trait x – time to fission. The evolution is the drift in x towards zero that may be interrupted by a death occurring at random with intensity m(x). If the entity manages to stay alive until x reaches zero, it splits to produce some (random) amount of new entities with random x. A detailed analysis of this evolution will be done, and some of its therapeutic-relevant conclusions will be discussed.
Mirosław Lachowicz A class of individual based models - an example of partially integral Markov semigroups
Abstract: I am going to propose the a class of models at microscopic level — i.e. at the level of interacting agents of a population. Under (rather strong) assumptions this class leads to partially integral Markov semigroups (PIMS). Following the general theory of the PIMS — developed in [3] — I am going to discuss the asymptotic properties of these models e.g. the long–time behavior. In particular under some (again rather strong) assumption ([2]) it is shown that any, even non–factorized, initial probability density tends in the evolution to a factorized equilibrium probability density. Such a case one can refer to as annihilation of initial correlations. On the other hand if the given assumptions are not satisfied — a number of equilibrium states could be large and no annihilation is observed. A general basis gives Chapter 8 in [1].
References
[1] J. Banasiak, M. Lachowicz, Methods of small parameter in mathematical biology, Birkhauser/Springer, Cham, 2014.
[2] M. Lachowicz, A class of microscopic individual models corresponding to the macroscopic logistic growth, Math. Methods Appl. Sci., 2017, on-line, DOI:10.1002/mma.4680.
[3] K. Pichór, R. Rudnicki, Continuous Markov semigroups and stability of transport equations, J. Math. Analysis Appl., 249, 2000, 668-685, DOI:10.1006/jmaa.2000.6968.
Jacek Miękisz Mean-field approximation and phase transitions in models of interacting spins and players
Abstract: Many processes in natural and social sciences can be modeled by systems of interacting objects. It is usually very difficult to obtain analytic expressions describing time evolution and equilibrium behavior of such systems. We will present here a method of self-consistent mean-field approximation - the force exerted on a given object, coming from its neighbors, is replaced by an unknown mean force - a mean field.
We will illustrate the above approach in the ferromagnetic Ising model of interacting spins located on regular and Barabasi-Albert random graphs.
We will discuss differences between models of interacting particles in statistical physics and models of interacting individuals in evolutionary game theory. In particular, we will compare phase transitions in Ising-type models and Prisoner's Dilemma game on random graphs with costs of links.
References
[1] J. Miękisz and P. Szymańska, On spins and genes, Mathematica Applicanda 40(1): 15-25 (2012)
[2] J. Miękisz, Evolutionary game theory and population dynamics, in: Multiscale Problems in the Life Sciences, From Microscopic to Macroscopic, V. Capasso and M. Lachowicz (eds.), Lecture Notes in Mathematics 1940: 269-316 (2008).
Zbigniew Peradzyński Calcium waves supported by stress activated ion channels
Abstract: In most papers dealing with calcium waves in living cells, the mechanism of propagation is based on autocatalytic release of calcium from the internal stores (e.g. endoplasmic reticulum) located inside the cell. According to L. Jaffe this cannot explain the speed of the second group of „fast waves” . Their speed can be by two orders higher. Such waves are also observed in cells not having internal stores of calcium. Therefore Jaffe suggested that this second group of fast calcium waves is supported by the influx of calcium through the stress activated channels located in the cell membrane. The lecture presents a mathematical model based on Jaffe suggestion.
Joint work with Bogdan Kaźmierczak.
Massimiliano Rosini Many-particle approximation of conservation laws in 1D and appplication to traffic flows
Abstract: In this talk we present recent results on the deterministic many-particle approximation of nonlinear Conservation Laws (CLs). The unique entropy solution to a scalar CL was rigorously approximated in [1, 2] by a discrete density constructed from the follow-the-leader particle system. Said result can be based on a discrete version of the classical Oleinik one-sided jump condition or on a BV contraction estimate for BV initial data. The initial-boundary value problem for a scalar CL has been considered in [4]. The results in [1] have been extended in [3] to the 2×2 system of conservation laws describing the multi-population vehicular traffic model by Aw, Rascle and Zhang. Finally, we present the extension of these techniques obtained in [5] for the one dimensional version of the Hughes model for pedestrian flows in a bounded interval with Dirichlet boundary conditions.
References
[1] M. Di Francesco, M. D. Rosini, Rigorous derivation of nonlinear scalar conservation laws from follow-the-leader type models via many particle limit. Archive for rational mechanics and analysis, 217(3):831–871, 2015
[2] M. Di Francesco, S. Fagioli, M. D. Rosini, Deterministic particle approximation of scalar conservation laws, Bollettino dell'Unione Matematica Italiana 10:487–501, 2017
[3] M. Di Francesco, S. Fagioli, M. D. Rosini, Many particle approximation of the Aw-Rascle-Zhang second order model for vehicular traffic. Mathematical Biosciences and Engineering, 14(1):127–141, 2017
[4] M. Di Francesco, S. Fagioli, M. D. Rosini, G. Russo, Follow-the-leader approximations of macroscopic models for vehicular and pedestrian flows. Active Particles, Volume 1, Springer International Publishing, Cham, pp. 333–378, 2017
[5] M. Di Francesco, S. Fagioli, M. D. Rosini, G. Russo, Deterministic particle approximation of the Hughes model in one space dimension. Kinetic and Related Models, 10(1):215–237, 2017
Aleksander Weron Mathematical models for dynamics of molecular processes in living biological cells. A single particle tracking approach.
Abstract: In this lecture we present a systematic methodology how to identify origins of fractional dynamics. We consider three models leading to it, namely fractional Brownian motion (FBM), fractional Levy stable motion (FLSM) and autoregressive fractionally integrated moving average (ARFIMA) process. The discrete-time ARFIMA process is stationary, and when aggregated, in the limit, it converges to either FBM or FLSM. In this sense it generalizes both models. We discuss also experimental data sets related to some molecular biology problems described by single particle tracking. They are successfully resolved by means of the universal ARFIMA time series model with various noises.
References
[1] G.Sikora, E.Kepten, A.Weron, M.Balcerek, K.Burnecki (2017) "An efficient algorithm for extracting the magnitude of the measurement error for fractional dynamics", Physical Chemistry Chemical Physics, 2017, 19, 26566-26581.
[2] T.Sungkaworn, M-L.Jobin, K.Burnecki, A.Weron, M.J.Lohse, D.Calebiro (2017) "Single-molecule imaging reveals receptor–G protein interactions at cell surface hot spots", Nature 550, 543–547.
[3] A.Weron, K.Burnecki, E.J.Akin, L.Sole, M.Balcerek, M.M.Tamkun, D.Krapf (2017) "Ergodicity breaking on the neuronal surface emerges from random switching between diffusive states", Scientific Reports 7, 5404.
Short lectures
Krzysztof Argasiński Interaction rates, vital rates and the real world. How parameters in continuous models are related to phenomena observable in real populations and individual based simulations.
Abstract: The talk is related to the problem of parameterization of the models based on differential equations by measurable empirical parameters. The fundamental question is when the model is properly defined from the point of view of falsification and how it can be tested against collected empirical data. Therefore, we should distinguish between situations when models explain analyzed phenomena and when they only mimic their behavior by producing similar patterns. We will discuss how the cause-effect chains can be described by differential models in terms of instantaneous rates and how those rates can be estimated by collected data. This problem is important for example for correct derivation of the link between mathematical structure of evolutionary game, describing some particular type of interaction. and detailed population dynamics being shaped by different interactions of different types. Another problem which will be discussed and it is related to topic of the talk, is the mechanistic model of growth suppression driven by explicit limiting factor (such as the number of nest sites available for newborns).
Joanna Barańska A Widom-Rowlinson jump dynamics in the continuum
Abstract: I am going to present the dynamics of an in finite system of point particles of two types. They perform random jumps in Rd in the course of which particles of different types repel each other whereas those of the same type do not interact. The states of the system are probability measures on the corresponding configuration space, the global (in time) evolution of which is constructed by means of correlation functions. It is proved that for each initial sub-Poissonian state µ0, the states evolve from µ0 to µt in such a way that µt is sub-Poissonian for all t > 0. The mesoscopic (approximate) description of the evolution of states and corresponding kinetic equations are given.
Peter Boldog Herd immunity caused by a toxoid vaccine – the case study of diphtheria by dynamic models
Abstract: Herd immunity is the indirect protection of a population from an infectious disease when the proportion of immune individuals is sufficiently high. Vaccination is a very effective way of achieving herd immunity. However, there are some vaccines, so-called toxoid vaccines, which do not protect against the pathogen, but neutralize the toxins created by the pathogen, thus preventing serious disease. It is a puzzling observation that even toxoid vaccines can generate herd immunity. We consider the example of diphtheria, and explain this phenomenon by a mathematical model. We also fit our model to the times series of diphtheria in Romania after the introduction of the vaccine. This is a joint work with Gergely Röst.
Antoni Leon Dawidowicz On photosynthesis process in tree
Abstract: The process of photosynthesis depends on a water balance and the accessibility of water and mineral salts dissolved in it and on bioenergetic processes. The mathematical model of photosynthesis presented now takes into consideration both the water balance differential model of a plant and the dissolved mineral salts which get to the tissues assimilating carbon dioxide together with water, being used in the course and for regulation of metabolic processes as well as for building the anatomical structures of the plant, i.e. its biomass. Conduction of The lecture deals with the description of the mathematical model of photosynthesis process in two interacting (peripheral and shaded) leaves. The bioenergetic phenomena is described by six equations for surfaces, thicknesses and levels of substrates of photosynthesis contents in two types of leaves. The well-posedness of the problem and the uniqueness of its solution are proved.
Joint work with Anna Poskrobko
References
[1] A. L. Dawidowicz, A. Poskrobko, J. L. Zalasinski, A Mathematical Model of the Bioenergetic Processes in Green Plants. Mathematical Population Studies. 2014;21: 159–165.
{2] A. L. Dawidowicz, A. Poskrobko, J. L. Zalasinski Mathematical Model of Photosyn-thesis Process in Leaf, The interaction between two leaves. Proceedings of the XX National Conference Applications of Mathematics in Biology and Medicine . 2014; Łochów: 29–34.
[3] A. L. Dawidowicz, J. L. Zalasinski, On the Mathematical Model of Morphology of Leaves. Proceedings of the Eleventh National Conference Application of Mathemat- ics in Biology and Medicine . 2005; Zawoja: 97–100.
[4] A. L. Dawidowicz, A. Poskrobko, J. L. Zalasinski, A Model Participation of Autotrophic and Het- erotrophic Organisms in Phenomenon of Life. Proceedings of the Twelfth National Conference Application of Mathematics in Biology and Medicine . 2006: Koninki: 37–39.
[5] A. L. Dawidowicz, A. Poskrobko On photosynthesis process with the interaction between two types of leaves MMAS 2018 (to appear)
Urszula Foryś Perceptual decision-making: neuronal population approach.
Abstract: On the basis of simple time-delayed neuronal population model, we study if the delay in self-inhibition can explain impairments in a decision-making process, which often appear in neurodegenerative diseases. Analysis of the proposed model reveals that there can be up to three positive steady states, with the one having the lowest neuronal activity which is stable in the case without delay and loses stability with increasing delay. So-called psychometric function translates model predictions into probabilities that a decision is being made. It occurs that for small synaptic delays the decision-making process depends directly on the strength of supplied stimulus and the system correctly identifies to which neuronal population the stimulus was applied, but for large delays complex impairments in the decision-making process are observed and the system exhibits ambiguity in the decision-making.
Paweł Klimasara Randomly switching distributions
Abstract: We consider a system of two different Liouville equations. Their solutions are given by two flows of densities. We switch between them according to a two-state Markov process. Resulting process satisfies the so-called stochastic Liouville equation. We study such process by analyzing its statistics.
Dominika Jasińska The Markov dynamics of a spatial individual-based birth and death model with age structure
Abstract: The evolution of a continuum infi nite system of particles with an age structure is described by the Markov generator approach and the analysis based on correlation functions. The underlying space of the model is the space of marked confi gurations where the evolution of states of the system is determined by appropriate Fokker-Planck equation. The birth rate at a given point x∈Rd depends on the distance to other points and their age. The exact solution to the equations for for correlation functions of fi rst and second order are presented.
Krzysztof Pilorz Coalescing random jumps in continuum: evolution of states and mesoscopic scaling
Abstract: An individual-based model of possibly infinitely many jumping and merging particles will be introduced. States of the model are probability measures on configuration space. The existence and uniqueness of the evolution of states will be discussed for the initial state being a sub-Poissonian measure. The result is achieved with help of correlation functions. Additionally, a Vlasov-type scaling will be presented, resulting in kinetic equation and therefore giving the mesoscopic description of the evolution.
Agnieszka Tanaś The introduction to the lectures of dynamical systems. An individual-based model of fragmentation.
Abstract: Individual-based models for large systems of interacting particles simulate populations and communities by following individuals and their properties. In such systems, each entity is characterized by its spatial location in Rd, d≥1. In the approach we consider, the phase space is the set of all locally finite subsets of Rd and the system’s states are probability measures on the phase space. The evolution of such states is described by means of the equation for corresponding correlation functions derived from the Fokker-Planck equation for measures. We discuss an example of individual-based model of an infinite system in which, each particle at random produces a finite number of new particles and disappears afterwards.
Andrzej Tomski Mathematical models of stochastic gene expression: formulation, results, new directions
Abstract: In this talk, we will present a model [1] of stochastic gene expression. In this model, stochastic effects originate from random fluctuations in gene activity status, while we precede typically considered mRNA production by the formation of pre-mRNA, which enriches classical transcription phase. We obtain a stochastically regulated system of ordinary differential equations (ODEs) describing evolution of pre-mRNA, mRNA and protein levels. Long-time behaviour of this stochastic process, identified as a piece-wise deterministic Markov process (PDMP) is investigated. We show some numerical simulations for the distributions of all three types of the particles. Moreover, we investigate the deterministic (adiabatic) limit state of the process, when depending on parameters it can exhibit two specific types of behavior: bistability and the existence of the limit cycle. The latter one is not present when only two kinds of gene expression products are considered [3]. In the end, we will mention new problems being investigated, like a slightly modified version of the model of subtilin production [4]).
References
[1] R. Rudnicki, A. Tomski, On a stochastic gene expression with pre-mRNA, mRNA and protein contribution. J. Theor. Biol. 387, p. 54-67, 2015.
[2] T. Lipniacki, P. Paszek, A. Marciniak-Czochra, A.R. Brasier, M. Kimmel, Transcriptional stochasticity in gene expression_. J. Theor. Biol. 238, p. 348-367, 2006.
[3] J. Jaruszewicz, P.J. Żuk, T. Lipniacki, Type of noise defines global attractors in bistable molecular regulatory systems. J. Theor. Biol. 317, p. 140-151, 2013.
[4] J. Hu, W.C. Wu, S.C. Sastry, Modelling subtilin production in bacillus subtilis using stochastic hybrid systems. In Hybrid Systems: Computation and Control, R. Alur and G.J. Pappas (eds.), LNCS vol.2993, Springer-Verlag, Berliin, p. 417-431, 2004.
Radosław Wieczorek Stochastic approaches to fragmentation with shattering
Abstract: Fragmentation can be observed commonly in many physical, industrial and biological processes, including grinding and crashing of such materials as ore, stone or flour, fragmentation of organisms, proliferation of cells or dissolving, etc. Fragmentation has been mathematically described with various methods both stochastic and deterministic. If the breakup is fast for small particles, some solutions for fragmentation equations, which are formally conservative, do not in fact preserve the total mass of particles. The phenomenon is called shattering, and interpreted as the mass loss to zero-size particles or dust. The aim of the talk is to present, among different approches to this fragmentations, an individual-based fragmentation model with infinitely many particles, in which shattering appears.