Special Session 40: Applications of dynamical systems in medicine and biology

Cardiomyocyte Dynamics: From Low Dimensional To Realistic Models

Roberto Barrio University of Zaragoza Spain Co-Author(s):    R. Barrio, H. Kitajima, M.A. Martinez, S. Serrano

Abstract: Early afterdepolarizations (EADs) are abnormal behaviors that can lead to heart failure and even cardiac death. In this presentation, we review recent results and we mathematically investigate the occurrence and development of these phenomena in two realistic ventricular myocyte models: the rabbit model of Sato (2009) and the human model of O'Hara (2011). These models are of high dimension, 27 and 41 respectively, so a mix of techniques must be used in their study. We connect the results with a reduced low-dimensional model, the Luo-Rudy cardiomyocyte model (1991). The combined use of analytical and numerical techniques allows us to propose a global conjecture of a mathematical mechanism of EAD creation in low- and high-dimensional models. By examining the bifurcation structure of the model, we elucidate the dynamical elements associated with these patterns and their transitions. Using a fast-slow analysis, we explore the emergence and evolution of EAD in the low-dimensional model and develop new methodologies for fast-slow decomposition for the realistic high-dimensional O`Hara model. This decomposition has allowed us to propose some new theoretical techniques for the control of prearrhythmia situations.

Fast inference for multivariate evolutionary dynamics

Krzysztof J. Bartoszek Department of Computer and Information Science, Linkoping University Sweden Co-Author(s):    Krzysztof Bartoszek

Abstract: Co-adaptation is key to understanding species evolution as different traits have to be coordinated. We have been developing fast calculation R packages (bgphy, glinvci, mvSLOUCH, pcmabc, PCMBase, PCMFit) that are able to handle general, multivariate, Gaussian, branching evolutionary dynamics. I will discuss setting up different hypotheses, consequences of measurement error, errorsome model selection, and asymptotic results that allow for improved choice of seeds for numerical procedures. The talk will be illustrated with relationships between traits in ungulates, in the genus Ferula (Apiaceae), in angiosperms, and carnivorans.

A Mathematical Model of CAR T Cell Therapy: How B Cell Influx Shapes Leukemia Outcomes

Juan Belmonte Beitia UCLM Spain Co-Author(s):    Marek Bodnar, Monika J. Piotrowska, Sergio Perez Luque

Abstract: Chimeric Antigen Receptor (CAR) T cell therapy has shown remarkable success in treating B cell malignancies, yet the long term dynamics of CAR T cells, tumor cells, and healthy B cells remain incompletely understood. We present a mathematical model of CAR T cell therapy for leukemia that incorporates a constant influx of healthy B cells from the bone marrow, a source term previously neglected, as well as CAR T cell inactivation by tumor cells. The model is analyzed as a system of nonlinear ordinary differential equations. We characterize the existence, uniqueness, and local and global stability of steady states, revealing how the interplay between CAR T cell activation and tumor burden determines outcomes such as remission or relapse. Bifurcation analysis identifies critical thresholds for treatment success. For periodic treatment protocols, we prove the existence of periodic solutions and derive conditions for tumor eradication. Additionally, we analyze an impulsive treatment regimen and obtain criteria for local stability of tumor free periodic solutions. Our results highlight the dual role of healthy B cells in sustaining CAR T cell activity and provide a theoretical framework for optimizing treatment strategies.

An impact of B-cells dynamics on CAR-T cell therapy for glioblastoma

Marek Bodnar Institute of Applied Mathematics, University of Warsaw Poland Co-Author(s):    Monika J. Piotrowska, Eryk Werens, Juan Belmonte-Beitia

Abstract: CAR-T cell immunotherapy involves the genetic reprogramming of T-lymphocytes to recognize and engage cancer cells, thereby triggering an anti-tumour immune response. While this treatment has been approved for hematological malignancies, its application to solid tumours presents significant new obstacles. These challenges include the heterogeneity of antigen expression, the presence of antigen-positive non-tumoural cells, immune inhibitory molecules, and the limited trafficking of CAR-T cells within the complex tumour microenvironment. In this talk, we explore a modification of the model introduced by Bodnar et al. (Nonlinear Dynamics, 2025). The original model neglected the production of B-cells; here, we extend the framework by incorporating appropriate terms to account for their dynamics. The resulting system of two ordinary differential equations describes the interaction between CAR-T cells in the blood and B-cells, which in turn influences the dynamics of both tumour cells and CAR-T cells in the brain. We present stability analyses for the system under both constant and impulsive treatment regimens, accounting for the external influx of both CAR-T and B-cells.

A two-strain dengue model with a constant recruitment rate

Marcin Choinski Institute of Information Technology, Warsaw University of Life Sciences - SGGW Poland Co-Author(s):    Joanna Renclawowicz, Urszula Skwara

Abstract: We propose a two--strain dengue model that captures the interactions between human and mosquito populations. It incorporates vertical transmission of the dengue virus from adult mosquitoes to their offspring, along with a constant recruitment rate of susceptible larval mosquitoes. We discuss the existence and local stability of the disease-free and one--strain equilibria. Moreover, we indicate stationary states with two strains. Analytical outcomes are complemented with numerical simulations.

Mathematical modelling of epithelial-mesenchymal transitions

Urszula Forys University of Warsaw, Faculty of Mathematics, Informatics and Mechanics Poland Co-Author(s):    Magdalena Szafra\`{n}ska-\L{}\c{e}czycka

Abstract: Epithelial-to-Mesenchymal Transition and its reverse, Mesenchymal-to-Epithelial Transition, are important biological processes influencing cancer progression, in particular tumor invasiveness and metastatasis. Based on Celi\`{a}-Terrassa et al. (2018), we propose a mathematical model describing interactions between main components of these transitions, taking into account: initiation of TGF-$\beta$ signaling, the miR-200/ZEB feedback loop, and E-cadherin expression, which is a marker of the epithelial phenotype. In my talk I will present the scheme of the model and the basic dynamics of its two main subsystems: the first one describing TGF-$\beta$ in its free and bound forms, and the other one describing production and degradation of miR-200, ZEB and E-cadherin. The main feature of the second subsystem is the possibility of bistability and hysteresis, which is dependent on the level of TGF-$\beta$ flowing there. I will focus mainly on that aspect on the presented model.

Impact of Aging on Pacemaker Cell Electrophysiology: A Maltsev-Lakatta Modeling Study

Beata Jackowska-Zduniak Gda\`nsk University of Technology Poland Co-Author(s):

Abstract: Aging of the sinoatrial node (SAN) is associated with progressive alterations in ionic currents and intracellular calcium dynamics, leading to impaired cardiac automaticity. In this study, we investigate the effects of aging at the level of a single pacemaker cell using the Maltsev--Lakatta model, in which rhythmic activity emerges from the interaction between the membrane clock and the calcium clock. Aging is represented as a gradual modulation of system parameters, in particular a reduction of the $I_f$ current, attenuation of the L-type calcium current ($I_{\mathrm{CaL}}$), and weakening of the coupling between intracellular calcium cycling and the $\mathrm{Na^+}$--$\mathrm{Ca^{2+}}$ exchanger (NCX). Numerical simulations indicate that age-related parameter drift leads to a prolongation of the cycle length, reduced stability of oscillations, and increased sensitivity of the system to perturbations. The proposed approach provides a foundation for future multiscale analyses.

Regulatory Networks In The Life Sciences: Rule-Based And Data-Driven Modeling

Markus A Kirkilionis University of Warwick British Virgin Islands Co-Author(s):    n/a

Abstract: Life itself is a kind of wise tautology, it is there because it can be. The matter our life consists of, as created by the evolution of the universe, just has the right elements and molecules that with their constant recombination, can take over higher-order functions, like delivering a memory established in the genetic code, or a metabolism counteracting the overall increase of entropy in the wider cosmic environment. Once memory and activity achieved by energy storage are possible, the evolution of life on this planet was possible. New higher order functions are invented by evolution, creating new complexities, like functioning, i.e. relatively stable ecosystems thriving on stable matter and energy availability. Eventually human evolution created new regulatory networks, societies with their economies. Coupling all these regulatory networks together leads eventually to the planetary system, which also can be healthy or not, like single organisms or ecosystems, depending on their stability properties. In this contribution we ask ourselves whether there are universal principles such regulatory networks must possess in order to achieve the function they are either possessing by ancient evolutionary trials, or by novel, perhaps artificial construction. We must first be able to describe such systems. Here the suggestion is to look at rule-based systems, based on reaction kinetics, as a universal approach. This will eventually include descriptions as dynamical systems, but also as stochastic processes if necessary. Next we ask whether there are theories which from the set of all possible regulatory networks can determine the ones which are stable according to some stability criterion. Note that equilibrium, or total balance, is just one such stability concept, but perhaps the most important one. Then we are asking whether, once we have a language to describe all regulatory networks, whether there is a way to automatically select a network which best fits measured data. This is the domain of data-driven modelling, at which we will look at in an overview.

On Observability of Limit Cycles in the Chemostat

Torsten A Lindstr\"{o}m Linnaeus University Sweden Co-Author(s):    Torsten Lindstr\{o}m

Abstract: Ever since May (Science, 1974) pointed out that the dynamics of equations describing commonly appearing equations in ecology contains the seeds of chaos and nonlinear dynamics, it has been a debate whether nonlinear dynamics and chaos actually can be observed in ecological systems. The chemostat is a very basic experimental setting describing of an ecosystem with an explicitly modelled resource flow. Presence of autonomous limit cycles is the simplest nonlinear dynamical phenomenon that can be studied and the Rosenzweig and MacArthur (Am. Nat. 1963) criterion divides the parameter space of the deterministic two-species chemostat into a one regime describing global stability and one with autonomous limit cycles. Deterministic systems of differential equations can, however, only be viewed only as approximations of the Markov processes describing the ecological phenomena (Renshaw, Stochastic Population Processes, 2011). It is likely that deterministic stable equilibria describing sufficiently large populations are described by approximately normal distributions and this can indeed, be proved in the univariate case. The objective of this talk is to study in what sense limit cycles approximate the quasi-stationary distributions arising in the Markov process describing the two-species chemostat.

Distances on DNA sequences

Magdalena Nowak Jan Kochanowski University of Kielce Poland Co-Author(s):    Taras Banakh, Judyta B\k{a}k, Grzegorz Czerwonka, Joanna Garbuli\`{n}ska-W\k{e}grzyn, Piotr Lisicki, Micha\l{} Pop\l{}awski

Abstract: We introduce several distances based on the Chaos Game Representation method. We use the pseudometric $d_k$ counting the differences in the number of corresponding $k$-mers in two DNA sequences $g,g`$, to define a metric $d(g,g`)=\sum_{k=1}^{\infty} \frac{d_k(g,g`)}{2^k}$ which combines the analysis of the frequency of occurrence of all possible $k$-mers. We compare this approach with known genetic distances such as Levenshtein edit distance, Average Nucleotide Identity or Mash distance. We also introduce distances invariant under the operations of inversion or nucleotides complementarity.

Mathematical Tools Combined with Neural CDE for prediction of Alzheimer diseas

Andrzej Nowakowski University of Lodz Poland Co-Author(s):    Anita Krawczyk

Abstract: The existing AI agents can do many actions, however, it seems that we still have problems with control and prediction next steps in time series. Our aim in this article, being in the spirit of machine learning, extends the applicability of neural ordinary differential equation (Neural CDE), described in \cite{FLN} for neural network modeled by an ordinary differential equations to a case of a mathematical model given by a system of partial differential equations (PDE). That requires new approach for prediction the next step in the given time series. In order to apply Neural CDE methodology we transform our system of PDE (modeling mathematically Alzheimer) to a new system of ODE. It is possible because we can treat an ANN as a function which in turn for a given set of weights determines a point in this space, and learning algorithms differ in the way they traverse it. We replace this space with a family of functions (family of solutions) which conforms well defined mathematical conditions. This allows us to develop an optimal control approach, parameterized by a set of controls and defined as neural controlled differential equations (Neural CDE). Thus we turn the difficult to explain artificial neural network learning process into a well-posed mathematical problem and apply to it a dual dynamic programming idea to formulate an optimization problem subject to Neural CDE. Turning the problem into optimal control problem allows one to define it in a rigorous way; however, it does not say much about the way it may be solved. To overcome theoretical difficulties and solve it in satisfactory and applicable way (our goal is to propose general methodology which is both theoretically correct and practically applicable to various tasks), we follow the methodology of the dual dynamic programming to derive the sufficient optimality conditions in the form of verification theorem and that simplify all practical considerations. We provide and prove verification conditions that should correctly predict the next step in time series. To show applicability of the presented theory we adopt state-of-the-art mathematical model of the Alzheimer disease for Neural CDE to predict state of the disease after 10, 20 years.

Targeted Control of Multidrug-Resistant Bacteria in Healthcare Networks

Monika J Piotrowska University of Warsaw Poland Co-Author(s):    Monika Joanna Piotrowska, Konrad Sakowski, Agata Lonc, Mirjam E. Kretzschmar, Andre Karch, Johannes Horn, Rafael Mikolajczyk

Abstract: We present a mathematical framework for modelling patient colonisation by multidrug-resistant bacteria across a healthcare system structured as a network of care units. The model incorporates patient movements between units, including direct hospital-to-hospital transfers and indirect transfers via home stays. Patients are stratified into three groups: low-risk, high-risk, and rare one-time visitors. Using this framework, we evaluate the effectiveness of targeted intervention strategies designed to reduce colonisation levels within the network. Interventions can be directed at specific patient groups or applied selectively in chosen care units. Each strategy is also assessed in terms of the costs it generates, allowing comparison of both epidemiological impact and resource use. The model is informed and calibrated using real-world patient movement data, enabling realistic evaluation of potential outcomes. By accounting for patient heterogeneity, movement patterns, and cost-effectiveness, the framework provides a flexible and practical tool for optimising infection control strategies in interconnected healthcare settings. This approach supports data-driven decision-making for managing multidrug-resistant bacteria while balancing intervention efficiency and resource allocation.

Co-infection model of airborne disease with two-strain dengue.

Joanna Renc{\\l}awowicz Institute of Mathematics and Cryptology, Cybernetics Faculty, Military University of Technology Poland Co-Author(s):    Marcin Choi\`{n}ski

Abstract: In this talk we propose a model describing the co-infection spread of airborne disease and dengue. We introduce an unusual approach that combines airborne disease with a relatively deep insight into dengue dynamics. We consider two distinct dengue strains that cause primary and secondary infections, which can yield antibody-dependent enhancement. Moreover, we incorporate vertical transmission into a mosquito population. Such an assumption provides a complex system for which we can provide analytical results. Thus, we prove the existence of stationary states across various epidemic scenarios and determine local stability for some states. Additionally, we conduct numerical simulations related to a real case of airborne disease and dengue spread in Latin America.The simulations confirm our established mathematical outcomes.

Dynamics of 2D hybrid non-autonomous neuron models with dynamical threshold

Justyna Signerska Gdansk University of Technology Poland Co-Author(s):    Piotr Bartlomiejczyk, Juan Belmonte Beitia

Abstract: We investigate a class of two-dimensional hybrid non-autonomous neuron models with a dynamical threshold, focusing on the interplay between continuous subthreshold dynamics and discrete reset mechanisms. The model consists of a linear voltage equation coupled with a threshold variable evolving according to a nonlinear function of the voltage, and incorporates periodic external input in the form of either pulse or rectified sinusoidal currents. In particular, we study periodic solutions of the system and prove the existence and uniqueness of a globally attracting periodic orbit in the non-hybrid case and show that the hybrid system admits at most one reset-free periodic orbit, while numerical evidence suggests the possible coexistence of reset-free and resetting periodic attractors. These results provide a rigorous mathematical framework for understanding dynamical threshold mechanisms in neuron models and contribute to the theoretical characterization of spiking versus non-spiking behavior under time-dependent forcing.

A little bit about math models for drug concentration in the body

Jacson Simsen Institute of Mathematics and Computing, Federal University of Itajub\`a, Brazil Brazil Co-Author(s):    Jacson Simsen

Abstract: Title: A little bit about math models for drug concentration in the body Jacson Simsen-IMC-UNIFEI - Brazil e-mail: jacson@unifei.edu.br Abstract: In this talk we will see an application of dynamical systems in medicine. A theoretical approach is done to propose math models to describe the drug concentration inside a human body in a medical treatment. Considering that the eliminitation could be nonlinear and the inaccuracy to determine the constant of elimination rate in the autonomous model, it is proposed that the elimination rate should be described by a bounded function and so the model becomes a nonlinear nonautonomous equation. Reference: J. Simsen, Nonlinear nonautonomous math models for drug concentration in the body, Mathematics in Engineering, Science and Aerospace, 2024, 15 (2), 579-589. Acknowledgements: Thanks to FAPEMIG for supporting my research.

Time Delay in a Mathematical Model of Epithelial--Mesenchymal Transitions and Implications for Cancer Treatment

Magdalena Szafranska-Leczycka Doctoral School of Exact and Natural Sciences, University of Warsaw, Warsaw, Poland Poland Co-Author(s):    K. Nec, Z. Szyma\`{n}ska, U. Fory\`{s}

Abstract: In this presentation, we investigate the role of time delay in regulating epithelial--mesenchymal transition within the model based on Celi\`{a}-Terrassa et al. (2018). From the perspective of cancer treatment, such temporal effects may influence the effectiveness of therapeutic strategies targeting TGF-$\beta$ pathways. Sensitivity analysis indicates that parameters associated with TGF-$\beta$ degradation and autocrine production play a key role in controlling E-cadherin levels and, consequently, cellular phenotype. Moreover, following Herbertz et al. (2015), we simulate treatment strategies aimed at inhibiting EMT--MET transitions. The results of these simulations will be presented during the presentation.