Abstract: The collisional kinetic theory and the Boltzmann equation play a central role in the description of nonequilibrium processes in gas flows. They are relevant as soon as there are not enough gas particle collisions to maintain local equilibrium, and classical fluid dynamic models, such as the laws of Navier-Stokes and Fourier, lose validity. Insufficient collisions may occur either due to microscopic or rarefied setting, typically characterized by a large Knudsen number, which is given by the ratio between the mean free path and an observation length scale. The Boltzmann equation is known to describe the whole regime of the Knudsen number, but also to serve as the starting point to derive improved continuum models. While originally derived and studied for the description of monatomic gas particle collisions, nowadays the proper modeling and analysis of the systems that include collisions of polyatomic molecules and molecules that might belong to different gas components (mixtures) is an active field of research, especially because it is highly relevant in applications to step out from the monatomic framework. 

In this talk we will present recent advances in studying the system of Boltzmann-like equations describing a multi-component gas mixture composed of both monatomic and polyatomic species. 

In the space homogeneous setting, we will discuss well-posedness of the system of Boltzmann equations with cut-off and hard potentials-like kernels. The approach uses an abstract ordinary differential equation theory in Banach spaces. The Cauchy problem is resolved for the initial data with finite and strictly positive mixture mass and energy, and additionally finite 2+ mixture moment determined by the hard potential rates discrepancy. The obtained result is, in its generality, comparable to the classical Cauchy theory of the single monatomic homogeneous Boltzmann equation. Furthermore, we will discuss the possibility of exploiting differential inequality approach to study Lebesgue's integrability propagation using entropy-based estimates.

Abstract: Societal challenges like the climate crisis, pandemics, and political conflicts highlight the need for collective action on global scales. Yet, we often witness substantial parts of contemporary society clustering and polarizing into opposed factions. Understanding the dynamics of opinion formation in social networks is therefor crucial.

In this talk, I want to introduce you to network science, the theoretical framework to address such questions. Delving into the specific applications for opinion dynamics, I will introduce the so-called Hegselmann-Krause model that describes similarity-biased opinion interaction. Agent-based simulations for different network structures and insight into analytical results show the rich spectrum of behaviors that feature consensus, clustering and polarization. This system can be linked to models for reaction networks, and some aspects of convergence via entropy methods are discussed.

Abstract: During embryonic development, groups of cells are organized creating patterns to give rise to tissues and organs. Experimental discoveries have unveiled a plethora of mechanisms underlying pattern formation, ranging from molecular pathways to different molecular transport processes. Pattern formation has been extensively investigated theoretically using reaction-diffusion (R-D) equations (Turing model) and various models that go beyond diffusion-based mechanism.  

The experimental findings suggest that pattern formation is highly specific to each biological model. However, emerging evidence also suggest a common framework for pattern formation. For example, cellular channels known as cytonemes are prevalent across the main signaling pathways, delivering molecules regardless of biochemical properties. Employing a multidisciplinary approach, we conducted experiments and simulations to study cytoneme signaling during development. Our results showed that cytoneme signaling and free diffusion-based mechanisms can produce similar signaling patterns, implying that evolution focuses on the emergent pattern rather than the specific formation mechanism.

Abstract: Interacting particle systems (IPS) play an important role in many applied sciences, both as a modelling framework for social and biological phenomena and as a tool for statistical algorithms used in data science and uncertainty quantification.  These two aspects are often intertwined as, for example, certain algorithms are inspired by dynamics observed in nature. While the use of IPS is now classical in fields such as filtering, recent exciting developments at the interface with PDE and optimal transport theory on the one hand and data science and uncertainty quantification on the other, have initiated fast progress in the design and analysis of new algorithms. 

In this talk I will discuss how tools from PDE theory and optimal transport can be used to analyse and guide the behavior of IPS based algorithms.  One particular example of interest will be the performance of IPS based methods for inverse problems with expensive and noisy likelihoods.

Abstract: We study a chemotaxis system that includes two competitive prey and one predator species in a two-dimensional domain. In this system, the movement of prey (resp. predators) is driven by chemicals secreted by predators (resp. prey), called mutually repulsive and mutually attractive chemotaxis effects. Considering the biological scenario that the chemicals diffuse much faster than the individual diffusion of all species, equations for the chemical concentrations are approximately elliptic. We explain this approximation via the fast signal diffusion limit, where equations for chemical concentration include slow evolution. We first show the global existence of a unique classical solution to the system. Secondly, a rigorous analysis of the fast signal diffusion limit is presented, where all solution components strongly converge to those of the limiting system. Finally, the convergence rate is estimated uniformly in time, where the effect of the initial layer is indicated. The Hausdorff distance between the sets of all trajectories and a link to enzyme reaction with a no longer possible application of the Michaelis–Menten kinetics will be remarked.

Abstract: We consider a thermodynamically consistent framework for reaction-diffusion systems modeling the evolution of a mixture of charged species. This approach covers, in particular, a large class of inorganic semiconductor-type models. Here, thermodynamical consistency refers to charge and energy conservation and the production of entropy. This is achieved by formulating the model as a gradient flow system in Onsager form (for the concentrations and the internal energy) coupled to Poisson's equation (for the electrostatic potential). 

We will first focus on the structure of the Onsager operator and its relation to the electrostatic potential. Similarities and differences to other temperature-dependent semiconductor-type models shall be discussed as well. A major goal of the project is the construction of global solutions (either in a weak or a renormalized sense) to an appropriate class of electro-energy-reaction-diffusion systems. First results in this direction will be presented, while open problems shall be addressed as well. 

Another aspect is the well-posedness of the corresponding stationary system. By resorting to the Lagrange formalism, one can rewrite the entropy maximization problem under the constraints of charge and energy conservation as a convex minimization problem. We will see that this problem admits a unique solution, hence, a unique thermodynamic equilibrium exists. This project is joint work with Alexander Mielke. 

Abstract: We develop a general theory of solvability of linear inhomogeneous boundary-value problems for systems of ordinary differential equations of arbitrary order in Sobolev spaces. Boundary conditions are allowed to be overdetermined or underdetermined. They may contain derivatives, of the unknown vector-valued function, whose integer or fractional orders exceed the order of the differential equation. Similar problems arise naturally in various applications. The theory introduces the notion of a rectangular number characteristic matrix of the problem. The index and Fredholm numbers of this matrix coincide respectively with the index and Fredholm numbers of the inhomogeneous boundary-value problem. Unlike the index, the Fredholm numbers (ie the dimensions of the problem kernel and co-kernel) are unstable even for small (in the norm) finite-dimensional perturbations. We give examples in which the characteristic matrix can be explicitly found. We also prove a limit theorem for a sequence of characteristic matrices. Specifically, it follows from this theorem that the Fredholm numbers of the problems under investigation are semicontinuous in the strong operator topology. Such a property ceases to be valid in the general case.

In the case when the number of inhomogeneous boundary conditions coincides with the order of the differential equation, we obtained necessary and sufficient conditions for continuity in a parameter of solutions to the introduced boundary-value problems in the Sobolev spaces. Some applications of these results to the solutions of multipoint boundary-value problems with integer and fractional orders are also presented.

References:

1. Mikhailets, V., Atlasiuk, O. The solvability of inhomogeneous boundary-value problems in Sobolev spaces. Banach J. Math. Anal. 18, 12 (2024). 

2. Atlasiuk, OM, Mikhailets, VA Fredholm one-dimensional boundary-value problems with parameters in Sobolev spaces. Ukraine. Math. J. 70, 11 (2019). 

Abstract: We study the long-time behaviour of the solutions of the wave equation with damping a unbounded at infinity and defined in an open (possibly unbounded) set Ω ⊂ R^n. Using multiplier methods, Ikehata and Takeda (2017) proved polynomial decay rates for unbounded continuous damping when n ≥ 3. Our results significantly extend their findings by assuming minimal regularity conditions on a and placing no restriction on the space dimension n. We apply different methods that rely upon recent advances in the theory of operator semigroups on Hilbert spaces and the spectral analysis of the resolvent of the generator G of the equation. This presentation is based on joint work with Borbala Gerhat, Julien Royer and Petr Siegl.

Abstract: One of the main difficulties that arises in the analysis of fluid-structure interaction problems is the a-priori mismatch between parabolic and hyperbolic regularity. Possibilities to circumvent this issue are adding a structural damping term that regularizes the hyperbolic dynamics, using a finite dimensional approximation for the elasticity, or to consider very smooth data which yields local-in-time existence of smooth solutions but leads to a loss of regularity. Another possibility is establishing improved or hidden regularity results for the normal derivative of the hyperbolic solution which allow to show existence and regularity results without additional damping terms. The way how these hidden regularity results are established requires a restriction on the geometry of the domain. Particularly, the interface between the solid and fluid region needs to be flat which also requires periodic boundary conditions in order to handle the problem analytically on a bounded domain. In this talk, we present new improved regularity results for linear hyperbolic equations with Dirichlet data that has anisotropic regularity. These results can be used to extend existence and regularity results for the unsteady Navier-Stokes-Lamé system for geometrical configurations with flat interface to smooth domains. 

(joint work with Michael Ulbrich and Stefan Ulbrich)

Abstract: Biological and medical applications motivate reaction-diffusion models with nonlocal and heterogeneous reaction terms. The parameters of the nonlocal terms influence the stability of stationary states from stable behavior to instability. I present the first results of a joint project with Cinzia Soresina and Bao Quoc Tang on such instabilities induced by heterogeneous nonlocality in reaction-diffusion equations. The system under investigation provides new challenges compared to previously studied systems: The nonlocal term leads to a strongly coupled infinite dimensional system of ordinary differential equations for the eigenmodes. A combination of analytical and numerical results for the nonlinear, a linearized, and a truncated system shows the existence of instability depending on the nonlocality parameter.

Abstract:  The fermionic Boltzmann (Boltzmann-Fermi-Dirac or fermionic Nordheim) equation is a kinetic description of rarefied gases of fermions (e.g. electrons). The setting is similar to the classical Boltzmann equation, with a modification of the collision operator, in order to take into account the Pauli exclusion principle. As a result, the corresponding equilibrium distributions (Fermi distributions) and the relevant entropy (Fermi entropy) do also differ from their classical analogues (Maxwellian distribution and Boltzmann entropy).

Entropy methods are a at the core of quantitative studies on relaxation to equilibrium. For the classical Boltzmann equation, the quantitative decay of the relative entropy to equilibrium is provided by a relationship between the relative entropy to equilibrium and its dissipation in time. These relationships are called « Cercignani’s conjecture-type » inequalities.

In this talk, I present a method of « transfer » of inequalities, which establishes an (almost) equivalence, in terms of entropy inequalities, between the classical and the fermionic Boltzmann cases, hence providing a large class of such results for solutions to the fermionic Boltzmann equation, and therefore, quantitative rates of convergence towards equilibrium.

• Tuesday, 12.12.2023 - Maxime Payan (Ecole polytechnique) - SR 11.32 (12:00 - 13:00)

Title: Computer-assisted proofs for the many steady states of a chemotaxis model with local sensing

Abstract: We consider second-order ergodic Mean-Field Games systems in $\R^N$ with coercive potential and aggregating nonlocal coupling, defined in terms of a Riesz interaction kernel. Equilibria solve a system of PDEs where a Hamilton-Jacobi-Bellman equation is combined with a  Kolmogorov-Fokker-Planck equation for the mass distribution. Due to the interplay between the strength of the attractive term and the behavior of the diffusive part, we will obtain three different regimes for existence and nonexistence of classical solutions to the MFG system. In the Hardy-Littlewood-Sobolev-supercritical regime, by means of a Pohozaev-type identity, we prove nonexistence of regular solutions to the MFG system without potential term. On the other hand, in the Hardy-Littlewood-Sobolev-subcritical regime, using a fixed point argument, we show existence of classical solutions at least for masses smaller than a given threshold value. In the mass-subcritical regime, we show that actually this threshold can be taken to be $+\infty$. Finally, considering the MFG system with a small parameter $\varepsilon>0$ in front of the Laplacian, we study the behavior of solutions in the vanishing  viscosity limit, namely when the diffusion becomes negligible. 

This talk is based on a joint work with A. Cesaroni (Padova).

Abstract: Volume exclusion interactions play a key role in many biological systems. In particular, it seems to be the key to explaining spontaneous alignment of anisotropic particles, for example, alignment of myxobacteria or fibers in a network. Most individual-based models impose this type of alignment in their equations. Here we do not wish to impose this type of alignment, but to investigate how it might emerge from volume exclusion interactions. To carry this out, volume exclusion interactions will be modelled via a soft anisotropic repulsion potential (which are vastly used in the literature of liquid crystals). We will present an individual-based model based on this potential and derive the corresponding kinetic and macroscopic equations. This approach allows us to understand how alignment emerges from volume exclusion and how it also affects not only the orientation of the particles, but also their positions.

Abstract: Social-ecological systems (SESs) are complex systems par excellence. They link individual and collective human behaviour, institutions and economies with ecosystems at local, regional and global levels. Increasingly pathological behaviour in SESs, as evidenced by human transgression of 6 of 9 planetary boundaries in 2023, gives an imperative to better understand and navigate intertwined social-ecological dynamics. Of particular interest is the ’race of nonlinearities’ between social tipping processes that might, it is hoped, alleviate ecological pressures fast enough to avoid cascading collapse of the global climate system. The urgency and high complexity of these issues present modellers with various challenges. I will begin this talk with a big-picture overview of the field, survey methods currently used by SESs modellers and discuss their strengths and limitations. I will then present a stylized model that I co-developed to explore multiscale dynamics in SESs. Finally, I will discuss the challenge of modelling socio-metabolic transformations.

Abstract: We provide a rigorous mathematical framework to establish the limit of the nonlocal model of cell-cell adhesion introduced in [2] to a local model. When the parameter of the nonlocality goes to 0, the system tends to a Cahn-Hilliard system with degenerate mobility and cross interaction forces. The proof is based on the strategy developed in [1] for the single Cahn-Hilliard equation. Numerical simulations show that the latter model preserves the diversity of cell sorting patterns seen in experiments and previous nonlocal models. It also has the advantage of having explicit stationary solutions. 

References 

[1] C. Elbar and J. Skrzeczkowski, Degenerate Cahn-Hilliard equation: From nonlocal to local, 2022. 

[2] C. Falcó, R. E. Baker, and J. A. Carrillo, A local continuum model of cell-cell adhesion, To appear in SIAM J. Appl. Math.

Abstract: We present the latest results in studies modeling anomalous immune responses, which extend the work proposed in the literature [1]. These models provide a description of the dynamics over time of a large number of interacting cells within an autoimmune framework, utilizing the tools of the kinetic theory of active particles. Firstly, we propose the application of a new nonconservative kinetic framework, taking into account the influence of an external force field representing treatment strategies. Next, we describe a more realistic spatio-temporal model, considering the motion of immune cells stimulated by cytokines [2] and applying it to a specific case of autoimmune disease, Multiple Sclerosis. We derive macroscopic reaction- diffusion equations for the number densities of the constituents with a chemotaxis term. A natural progression is to study the system, exploring the formation of spatial patterns through a Turing instability analysis of the problem, and basing the discussion on microscopic parameters of the model. In particular, we observe spatial patterns that reproduce the brain lesions characteristic of the pathology during its different stages.

References:

• [1]  R. Della Marca, M. P. D. Machado Ramos, C. Ribeiro, A. J. Soares, Mathematical modelling of oscillating patterns for chronic autoimmune diseases. Math.l Meth. Appl. Sci., 45(11), 7144-7161 (2022)

• [2]  J. Oliveira, A. J. Soares, R. Travaglini, Kinetic models leading to pattern formation in the response of the immune system. Special Issue of Rivista di Matematica dell’Universit`a di Parma in memory of Giampiero Spiga. (Accepted for publication)

• [3]  J. M. S. Oliveira, R. Travaglini, Reaction-diffusion systems derived from kinetic models for Multiple Sclerosis. arXiv preprint arXiv:2309.05119

Abstract: We propose a model for fluid diffusion in partially saturated porous media taking into account hysteresis effects in the pressure-saturation relation. The resulting mathematical problem leads to a diffusion equation for the pressure in a bounded N-dimensional domain with a Preisach hysteresis operator under the time derivative. The problem is doubly degenerate in the sense that the saturation range is bounded, and no a priori control of the time derivative of the pressure is available. A convexification argument applied to the hysteresis operator makes it possible to prove the existence and uniqueness of a strong solution to the problem with standard initial and boundary data and study its long time behavior.  

This is a joint work with Chiara Gavioli from TU Wien. 

Abstract: In this presentation, we will delve into the world of complex contagions and their significance in the spread of various phenomena, including viral outbreaks, preventive measures, emotional contagions, and information diffusion. We will explore the intricacies of complex contagions, highlighting their differences from simple contagions, such as their reliance on shared local neighbourhoods, network reinforcement and the weakness of long ties.

What adds an intriguing dimension to our discussion is our exploration of treating human behaviour like entangled quantum states. We will draw parallels between these enigmatic quantum phenomena and the behaviour observed in complex contagions. This comparison may shed light on an unexpected aspect of human behaviour, suggesting that indeterminacy could play a crucial role in our understanding of complex contagions.

Abstract: In this talk, we will present stability of stochastic singular systems with delay. These systems are models arising in some applications such as electrical circuit simulation, multibody systems, control theory, economics, etc. Firstly, we introduce the index concept and establish a formula of solution for singular systems described by stochastic differential-algebraic equations (SDAEs) and stochastic singular difference equations (SSDEs). Stability and robust stability of SDAEs and SSDEs is studied. Secondly, we shall deal with SDAEs and SSDEs with constant coefficient matrices under stochastic delay perturbations.  By using the method of Lyapunov functions and comparison principle, the asymptotically mean square stability and the exponentially mean square stability of these equations are investigated.

Abstract: I review some results about the (non-)convergence (to strategy profiles or sets of strategy profiles) of evolutionary game dynamics such as the replicator dynamics and best reply dynamics in general normal form games.

Abstract (Nguyen): In this talk, we will discuss the questions of existence, nonexistence, uniqueness, and multiplicity concerning a class of semilinear elliptic equations driven by nonlocal diffusion. Our approach relies on a meticulous analysis of the Green operator, commonly referred to as the inverse of the diffusion operator.

Abstract (Mezache): In order to provide an explanation for the damped oscillations surprisingly observed in Prion depolymerization experiments, a bi-monomeric variant of the seminal Becker-Doring seminal system is proposed. In this talk, we look in detail at the mechanisms leading to these oscillations. We characterize the dynamics of the system in different kinetic phases: from the initial phase of high amplitude oscillations to progressive damping and convergence towards the unique stationary solution. This result is based on quantitative approximations of the main quantities of interest: the period of oscillations, the damping of oscillations corresponding to an energy loss and the number of oscillation cycles characterizing each kinetic phase.

Abstract: The classical assumption in the literature on numerical approximation of stochastic differential equations (SDEs) is global Lipschitz continuity of the coefficients of the equation. However, many SDEs arising in applications fail to have globally Lipschitz continuous coefficients.

In the last decade an intensive study of numerical approximation of SDEs with non-globally Lipschitz continuous coefficients has begun. In particular, strong approximation of   SDEs  with a drift coefficient that is discontinuous in space has recently gained a lot of interest. Such SDEs arise e.g. in mathematical finance, insurance  and stochastic control problems. Classical techniques of error analysis are not  applicable to such SDEs and well known convergence results for standard methods do not carry over in general. 

In this talk I will present   recent results on strong approximation of such SDEs.  

The talk is based on joint work with Arnulf Jentzen (University of Münster) and Thomas Müller-Gronbach (University of Passau).

Abstract: 

In this talk I will first discuss the mathematical models of microwave heating and induction heating in materials. The focus will be on what the role of the electric conductivity in the media is played in the model during the heating process. Then I will give a short survey about the well-posedness for the model problem. Finally, I will discuss open problems and challenges for the nonlinear model.

Abstract: In many biological systems, it is essential for individuals to gain information from their local environment before making decisions. In particular, through sight, hearing or smell, animals detect the presence of other individuals and adjust their behavior accordingly. Interestingly, this feature is not only restricted to higher level species, such as animals, but is also found in cells. For example, some human immune cells are able to interact non- locally by extending long thin protrusions to detect the presence of chemicals or signaling  molecules. Indeed, the process of gaining information about the surrounding environment is intrinsically non-local and mathematically this leads to non-local advection terms in continuum models.

In this seminar, I will focus on a class of nonlocal advection-diffusion equations modeling population movements generated by inter and intra-species interactions. I will show that the model supports a great variety of complex spatio-temporal patterns, including stationary aggregations, segregations, oscillatory patterns, and irregular spatio-temporal solutions.

However, if populations respond to each other in a symmetric fashion, the system admits an energy functional that is decreasing and bounded below, suggesting that patterns will be asymptotically stable. I will describe novel techniques for using this functional to gain insight into the analytic structure of the stable steady state solutions. This process reveals a range of possible stationary strongly modulated patterns, including regions of multi-stability. Through a nonlinear analysis, I will classify the bifurcations occurring at the onset of an instability and show the coexistence between small amplitude patterns and strongly modulated solutions. These will be validated via comparison with numerical simulations.

Abstract: 

Understanding the intermediate asymptotic and computing convergence rates towards equilibria are among the major problems in the study of parabolic equations. Convergence rates depend on the tail behaviour of solutions. This observation raised the following question: how can we understand the tail behaviour of solutions from the tail behaviour of the initial datum?

In this talk, I will discuss the asymptotic behaviour of solutions to the fast diffusion equation. It is well known that non-negative solutions behave for large times as the Barenblatt (or fundamental) solution, which has an explicit expression. In this setting, I will introduce the Global Harnack Principle (GHP), precise global pointwise upper and lower estimates of non-negative solutions in terms of the Barenblatt profile. I will characterize the maximal (hence optimal) class of initial data such that the GHP holds by means of an integral tail condition. As a consequence, I will provide rates of convergence towards the Barenblatt profile in entropy and in stronger norms such as the uniform relative error.

Abstract: Boehringer Ingelheim is working on breakthrough therapies that transform lives, today and for generations to come. Bringing new drugs to patients is often a lengthy and costly process. The use of mathematical modelling techniques can aid in the selection of promising drug compounds, as well as the prediction of their efficacy, toxicity, and pharmacokinetic properties, thus saves time and resources. In this talk I will share two examples of how mathematical modelling is used in the drug development.

Abstract: 

Nonlinear partial differential equations appear naturally in many biological or chemical systems. E.g., activator-inhibitor systems play a role in morphogenesis and may generate different patterns. Noisy random fluctuations are ubiquitous in the real world. The randomness leads to various new phenomena and may have a non–trivial impact on the behaviour of the solution. The presence of the stochastic term (or noise) in the model often leads to qualitatively new types of behaviour, which helps to understand the real processes and is also often more realistic. Due to the interplay of noise and nonlinearity, noise-induced transitions, stochastic resonance, metastability, or noise-induced chaos may appear. Noise in stochastic Turing patterns expands the range of parameters in which Turing patterns appears. 

The topic of the talk is a nonlinear partial differential equation disturbed by stochastic noise. Here, we will present recent results about the existence of martingale solutions using a stochastic version of a Tychanoff-Schauder type Theorem.  In particular, we will introduce the stochastic Klausmeier system, a system which is not monotone, nor do they satisfy a maximum principle. So, the existence of a solution can only be shown using compactness arguments. 

In the talk, we first introduce the System. Secondly, we will introduce the notion of martingale solutions and present our main result. Finally, we will outline the proof of our main result, i.e., the proof of the existence of martingale solutions.

Abstract: 

In the talk I present new results on a continuous version of the ELO rating, the widely used rating in chess. We extend previous works to include a factor of performance-fluctuation. Numerical simulations on both scales show good agreement between the microscopic and mesoscopic scales. We  use the latter to show analytically that a proper choice of parameters leads to a convergence of the rating $R$ to the expected strength of $\rho$.

Abstract:

In this talk, we introduce a mathematical description of the impact of social heterogeneity in the spread of infectious diseases by integrating an epidemiological compartmentalization with a kinetic model for population-based social features. The resulting set of Boltzmann-type equations models the evolution over time of the densities of social contacts of compartmental models. Furthermore, to discuss the mathematical interface of this class of models with available data, we derive the evolution of observable quantities based on suitable macroscopic limits of classical kinetic theory [1, 3, 4]. Finally, we analyze the effects of introducing an optimal control strategy into the system, to limit selectively the mean number of contacts and, consequently, to reduce the number of infected cases. Thanks to a data-driven approach, we show that this new mathematical framework permits to assess the effects of social restrictions [2].

[1] G. Dimarco, B. Perthame, G. Toscani, M. Zanella. Kinetic models for epidemic dynamics with social heterogeneity. J. Math. Biol., 83:4, 2021.

[2] G. Dimarco, G. Toscani, M. Zanella. Optimal control of epidemic spreading in presence of social heterogeneity. Phil. Trans. R. Soc. A, 380:20210160, 2022.

[3] J. Franceschi, A. Medaglia, M. Zanella. On the optimal control of kinetic epidemic models with uncertain social features. Preprint arXiv:2210.09201, 2022.

[4] M. Zanella. Kinetic models for epidemic dynamics in the presence of opinion polarization. Bull. Math. Biol., 85:36, 2023.

Abstract:

Cross-diffusion systems are non-linear parabolic systems with relevant applications in biology and ecology. In this talk, we study the existence of  strong solutions for a class of triangular cross-diffusion system with reaction terms that include the Lotka-Volterra type. The main idea consists in introducing a convenient change of variable that gives rise to a system in a non-divergence form. Then, we regularize the obtained system, we prove the existence of strong solutions by a fixed-point theorem and we pass to the limit. Moreover, we also investigate the regularity and the uniqueness of the solution. More precisely, we prove that the solution is bounded in L^∞((0, T)× Ω), with T > 0 and the space domain Ω ⊂ R^N , provided that N ≤ 3, and the solution is unique if N ≤ 2.

This is a joint work with L. Desvillettes and H. Dietert.

Abstract:

Systems of interacting particles are widely used to establish different mathematical models describing collective behaviors of organisms and social aggregations. In this talk, we focus on collective dynamics over domains with boundaries. Such domains are commonly involved in realistic physical settings. For example, the boundary can be an obstacle in the environment, such as a river or the ground. Our research contributions include the mean-field limit for the particle system with reflecting boundary condition and the zero-diffusion limit from the aggregation-diffusion model to the plain aggregation model.

Abstract: 

In this work, we investigate an initial-boundary-value problem of a reaction-diffusion equation in a bounded domain with a Robin boundary condition and introduce some particular parameters to consider the non-zero flux on the boundary. This problem arises in the study of mosquito populations under the intervention of the population replacement method, where the boundary condition takes into account the inflow and outflow of individuals through the boundary. Using phase-plane analysis, the present paper studies the existence and properties of spatially inhomogeneous steady-state solutions depending on several parameters. Then, we use the principle of linearized stability to prove some sufficient conditions for their stability. We show that the long-time efficiency of this biological control method depends strongly on the size of the treated zone and the migration rate. To illustrate these theoretical results, we provide some numerical simulations in the framework of mosquito population control.

References: 

[1] Ouyang, T., and Shi, J. Exact Multiplicity of Positive Solutions for a Class of Semilinear Problems. Journal of Differential Equations 146, 1 (June 1998), 121–156.

[2] Smoller, J., and Wasserman, A. Global bifurcation of steady-state solutions. Journal of Differential Equations 39, 2 (1981), 269–290.

[3] Goddard, J., and Shivaji, R. Stability analysis for positive solutions for classes of semilinear elliptic boundary-value problems with nonlinear boundary conditions. Proceedings of the Royal Society of Edinburgh: Section A Mathematics 147, 5 (2017), 1019–1040.

[4] Strugarek, M., and Vauchelet, N. Reduction to a single closed equation for 2 by 2 reaction-diffusion systems of Lotka-Volterra type. SIAM Journal on Applied Mathematics 76, 5 (2016), 2060–2080.

Abstract: 

Critical transitions can be conceptualized as abrupt shifts in the state of a system typically induced by changes in the system’s critical parameter. They have been observed in a variety of systems across many scientific disciplines including physics, ecology, and social science. Because critical transitions are important to such a diverse set of systems it is crucial to understand what parts of a system drive and shape the transition. The underlying network structure plays an important role in this regard. In this paper, we investigate how changes in a network’s degree sequence impact the resilience of a networked system. We find that critical transitions in degree mixed networks occur in general sooner than in their degree homogeneous counterparts of equal average degree. This relationship can be expressed with parabolic curves that describe how the tipping point changes when the nodes of an initially homogeneous degree network composed only of nodes with degree k1 are replaced by nodes of a different degree k2. These curves mark clear tipping boundaries for a given degree mixed network and thus allow the identification of possible tipping intersections and forbidden tipping regions when comparing networks with different degree sequences.

Abstract: 

The goal of a posteriori validation methods is to get a quantitative and rigorous description of some specific solutions of nonlinear dynamical systems, based on numerical simulations, which helps shed some light on the global dynamics. The general strategy consists in combining a priori and a posteriori error estimates, interval arithmetic, and a fixed point theorem applied to a quasi-Newton operator. Starting from a numerically computed approximate solution, one can then prove the existence of a true solution in a small and explicit neighborhood of the numerical approximation. In this talk I will present the main ideas behind these techniques, describe a rather general framework in which they can be applied, and showcase their interest by presenting examples of application related to population dynamics (the SKT system) and fluid dynamics (the Navier-Stokes equations).

Abstract:

In this talk, we present the amplitude dynamics of n-dimensional (nD) solitons in a fast collision under a framework of coupled perturbed (n + 1)D nonlinear Schrödinger equations with a saturable nonlinearity and competing nonlinearities for n = 2,3. We propose a new perturbative method which is based on the calculations on the energy balance of perturbed solitons and the analysis of the collision-induced change in the envelope of the perturbed soliton. These results measure the abrupt energy dropdown due to a fast collision of two optical beams (n = 2) or two light bullets (n = 3) in perturbed nonlinear media. Our analytic calculations are confirmed by the extensive simulations of the coupled (2 + 1)D and of the coupled (3 + 1)D nonlinear Schrödinger model in the presence of the cubic loss and in the presence of the quintic loss. The simulations are implemented by using the accelerated imaginary-time evolution scheme and the split-step Fourier method.

Abstract: 

We establish the local well-posedness of the Cauchy problem for a dispersion generalized Camassa-Holm equation by using Kato’s semigroup approach for quasi-linear evolution equations. We show that for initial data in the Sobolev space Hs(R) with s > (7/2)+ p, the Cauchy problem is locally well-posed, where p is a positive real number determined by the order of the differential operator L corresponding to the dispersive effect added to the Camassa-Holm equation. In this talk, first I will explain Kato’s semigroup approach on the Camassa-Holm equation briefly, and then I will give the proofs for the dispersion generalized Camassa-Holm equation. Finally, I will compare the results of both equations and propose open problems related to the dispersion generalized Camassa-Holm equation.

This talk is based on my master's thesis, supervised by Asst. Prof. Dr. Nilay Duruk Mutlubaş.

Abstract:

I present a model family with linear and nonlinear reaction-diffusion models, stationary models and reduced space-independent models describing the interactions of the virus and the cells of the immune system during a liver infection. Depending on the extension of the domain and parameters like the reaction change rate and the diffusion strength, we find solutions tending to zero (healing infection courses) or solutions tending towards a stationary spatially inhomogeneous state (chronic infection courses). Different models of the model family provide insight to different aspects of the system, for example the relevance of chemotactic effects or the tendency towards inhomogeneous states. The model finds application in analyzing different treatments and their effect on the organism.

References: 

• C. Reisch: Modelling health impacts of hepatitis - model selection and treatment plans, Math Comput Model Dyn Syst, 2022, doi: 10.1080/13873954.2021.2020296.

• C. Reisch, D. Langemann: Entropy functional for finding requirements in hierarchical reaction-diffusion models for inflammations, Math Meth App Sci, 2020, doi: 10.1002/mma.6682.

• C. Reisch, D. Langemann: Chemotactic Effects in Reaction-Diffusion Equations for inflammation, J Bio Phys, 2019, doi: 10.1007/s10867-019-09527-3.

Abstract: 

In this talk, we investigate the long-time behavior of solutions to a nonlinear coupled reaction-diffusion system with detailed balance on the real line. By assuming that the solutions are in equilibria at infinity, we study the convergence towards a self-similar profile, which is a generalized steady-state in parabolic scaling variables. With this convergence, we answer how the solutions to the system mix the two stable asymptotic boundary values when time increases. Our strategy is to use an entropy approach with the relative Boltzmann entropy functional. This is a standard method for reaction-diffusion systems of mass-action type on bounded domains and a meaningful alternative to the linearization around an equilibrium. While this approach is well-studied on bounded domains, things become more complicated on the whole real line.

This research is joint work with Alexander Mielke.

Abstract: 

We provide a new decomposition of the graph Laplacian (for strongly connected, labeled directed graphs), involving an invertible core Laplacian matrix based on an order on the vertices.

As an application, we further clarify the binomial structure of (weakly reversible) mass-action systems. In particular, using the new decomposition of the graph Laplacian and monomial evaluation orders (defining polyhedral cones), we give a "geometric" proof of classical results, namely the asymptotic stability of positive complex-balanced equilibria and the non-existence of other steady states. Moreover, we embed complex-balanced mass-action systems in binomial differential inclusions and extend the stability result.

Reference: https://arxiv.org/abs/2205.11210

Abstract: 

We introduce the notion of permeable sets and functions with permeable graphs, and we provide a couple of examples to demonstrate that the notion is indeed interesting. Next, we consider a method for constructing a continuous function which has impermeable graph. We show that the so constructed function can be Hölder or, on the other hand, it can be so wild that the Hausdorff dimension of its intersection set with every function f, that has total variation smaller than 1 and satisfies f(0)=0, is equal to 1. 

We will use the example to answer a question from metric topology, i.e., whether every function which is continuous on the R^d and intrinsic Lipschitz on the complement of a d-1 dimensional Hölder submanifold is Lipschitz.

Finally, we will present some recent results and open questions. 

References

[1] Z. Buczolich, G. Leobacher, and A. Steinicke. Continuous functions with impermeable graphs. submitted, 2022.

[2] G. Leobacher and A. Steinicke. Exception sets of intrinsic and piecewise Lipschitz functions. Journal of Geometric Analysis, 2022.

[3] G. Leobacher, A. Steinicke, and T. Rajala. Complementary results on permeable sets. Working paper, 2022.

Abstract: 

Reactions with enzymes are critical in biochemistry, where the enzymes act as catalysis in the process. A simplest enzyme reaction has a substrate reacting with an enzyme to synthesize a complex that is then transformed into the enzyme and a product. Diagrammatically, E + S <--> C <--> E + P. One of the most appropriate mechanisms for modeling enzyme-catalyzed reactions is the Michaelis-Menten kinetic, which was first introduced in 1913 by applying a quasi-stead-state approximation (QSSA). 

For spatially homogeneous concentrations, such as in a test tube with shakers, QSSA is well-known and has been proved rigorously. However, in many contexts, chemical concentrations are spatially inhomogeneous, such as reactions in a vessel or inhomogeneous media, and in those cases, it is natural to consider reaction-diffusion systems. Formal QSSA deriving the Michaelis-Menten kinetic in the presence of diffusion has been shown, but, to our knowledge, no rigorous convergence has been given. 

In this talk, under the basic QSS hypothesis is that the initial concentration of enzyme is small compared with the initial substrate concentration, we will prove the convergence from a mass-action system to one with Michaelis-Menten kinetics in the presence of diffusion. Our proof is based on the maximal regularity for parabolic equations, an improved duality argument, and a modified energy estimate.

Cilia and flagella are active biological structures that are widespread in nature. Various microorganisms (spermatozoa, bacteria, algae) or organs (lungs, brain) are equipped with cilia or flagella. At the micrometer scale, the use of these appendages is a very efficient transport mechanism that is found, for example, in the swimming phenomena of certain bacteria or in the transport of bronchial fluids in the lungs. From a mathematical point of view, this system can be modeled by a fluid-structure interaction problem involving a viscous fluid with a low Reynolds number on the one hand and active elastic structures on the other. In this talk we will first discuss the modeling of these active biological structures in the framework of continuum mechanics. Then, we will study the coupling with the surrounding fluid and present some numerical results. 

Reference: Florian Gauer, Christoph Kuzmics, Cognitive empathy in conflict situations, International Economic Review, 61 (4) 2020, 1659-1678

https://onlinelibrary.wiley.com/doi/10.1111/iere.12471

Quantification of multi-dimensional image data obtained from microscopic techniques such as confocal laser scanning microscopy has become a fundamental approach for probing cellular events at the single-cell level. Single-cell analyses complement in vitro approaches and can overcome their inherent limitations. Commercial microscope systems and implemented image processing tools are now user-friendly and allow their operation and application without significant expertise. In addition, a variety of open-source tools covering most functions to process a typical imaging-informatics pipeline are accessible. Consequently, biological optical imaging and quantification of generated image data have become routine work in the life sciences. However, experts in imaging-informatics increasingly point out the limitations of such quantitative analyses and the resulting consequences for the interpretation of biological processes. 

In this talk, the discrepancy between routine application and in-depth experimental research is demonstrated in the framework of a current study in lipid droplet biology addressing the lipodystrophy protein ‘seipin’. Challenges of putative simple image-based quantification approaches such as the determination of the precise number and size of lipid droplets in yeast cell populations or of the application of advanced imaging methods such as fluorescence recovery after photobleaching (FRAP) are demonstrated. Finally, putative synergies between my current research focus and mathematicians are discussed. 

In this talk, we consider first-order consensus systems with time-constant interaction coefficients, both in the finite and in the infinite-dimensional cases. For symmetric coefficients, convergence to consensus is classically established by proving, for instance, that the usual variance is an exponentially decreasing Lyapunov function. We investigate here the convergence to consensus in the non-symmetric case. We identify a positive weight which allows to define a weighted mean corresponding to the consensus, and obtain exponential convergence towards consensus. Moreover, we compute the sharp exponential decay rate.

The results presented in this talk have been obtained in collaboration with Laurent Boudin and Emmanuel Trélat.

In this talk I will discuss the Kolmogorov’s entropy of uniform attractors for non-autonomous dissipative PDEs. The main attention is payed to the case where the external forces are not translation-compact. We attempt to present a new general scheme which allows us to give the upper bounds of this entropy for various classes of external forces through the entropy of proper projections of their hulls to the space of translation-compact functions. 

Vegetation patterns are a ubiquitous feature of semi-arid regions and are a prime example of a self-organisation principle in ecology. In this talk, I present bifurcation analyses of two PDE models to (i) investigate the effects of nonlocal seed dispersal, and (ii) identify a mechanism that enables species coexistence despite competition for a limiting resource.

First, I present a nonlocal model in which plant dispersal is modelled by a nonlocal convolution term, motivated by empirical data. Asymptotic analysis of the model is possible due to a scale difference between plant dispersal and water transport. I show that a condition for pattern onset in the model can be derived analytically, which indicates that long-range seed dispersal inhibits the onset of spatial patterns. Results on pattern existence and stability, obtained via a numerical continuation method, further show a change in the type of stability boundaries in the pattern's stability regions as dispersal distance is varied. This suggests increased resilience of patterns to reductions in precipitation due to long dispersal distances. Stability results further propose a resolution of a mismatch between previous mathematical models predicting movement of vegetation patterns and some field studies reporting stationary patterns.

Second, I reveal that the vegetation’s self-organisation principle also acts as a coexistence mechanism. I present a multispecies model for two plant species that interact with a sole limiting resource. A stability analysis of the system's single-species patterns, performed through a calculation of their essential spectra, provides an insight into the onset of coexistence states. I show that coexistence solution branches bifurcate off single-species solution branches as the single-species states lose their stability to the introduction of a second species. Moreover, I present a comprehensive existence and stability analysis to establish key conditions, including a balance between the species' local competitive abilities and their colonisation abilities, for species coexistence in the model.

The application of granular biofilms in engineered systems for wastewater treatment and valorisation has significantly increased over the past years. Granular biofilms have a regular, dense structure and allow the coexistence of a high number of microbial trophic groups. A multiscale model is presented to describe the de novo granulation, and the evolution of multispecies granular biofilms in a continuously fed bioreactor. The granule is modelled as a spherical free boundary domain with radial symmetry. Main phenomena involved in the process are considered: initial attachment by pioneer planktonic cells, biomass growth and decay, substrates conversion and diffusion, invasion by planktonic cells, and detachment. Specifically, non-linear hyperbolic PDEs model the biomass distribution, and quasi-linear parabolic PDEs model the transport of substrates and planktonic species within the biofilm granule. Nonlinear ODEs govern the evolution of soluble substrates and planktonic biomass within the bulk liquid. The free boundary evolution is governed by an ordinary differential equation which accounts for attachment and detachment. The model is applied to cases of biological and engineering interest. Numerical simulations are performed to test its qualitative behaviour and explore the main aspects of de novo anaerobic granulation: ecology, biomass distribution, dimensional evolution of the granules, soluble substrates and planktonic biomass dynamics within the bioreactor.

I present a model family with linear and nonlinear reaction-diffusion models, stationary models and reduced space-independent models describing the interactions of the virus and the cells of the immune system during a liver infection. Depending on the extension of the domain and parameters like the reaction change rate and the diffusion strength, we find solutions tending to zero (healing infection courses) or solutions tending towards a stationary spatially inhomogeneous state (chronic infection courses). Different models of the model family provide insight to different aspects of the system, for example the relevance of chemotactic effects or the tendency towards inhomogeneous states. The model finds application in analyzing different treatments and their effect on the organism.

References: 

• C. Reisch: Modelling health impacts of hepatitis - model selection and treatment plans, Math Comput Model Dyn Syst, 2022, doi: 10.1080/13873954.2021.2020296.

• C. Reisch, D. Langemann: Entropy functional for finding requirements in hierarchical reaction-diffusion models for inflammations, Math Meth App Sci, 2020, doi: 10.1002/mma.6682.

• C. Reisch, D. Langemann: Chemotactic Effects in Reaction-Diffusion Equations for inflammation, J Bio Phys, 2019, doi: 10.1007/s10867-019-09527-3.

The ability to locally degrade the Extracellular Matrix (ECM) and invade in the local tissue is a key process distinguishing cancer from normal cells, and is a critical step in the tumour metastasis. The tissue invasion involves the coordinated action of the cancer cells, the ECM, the Matrix Degrading Enzymes, and the Epithelial-to-Mesenchymal transition (EMT); a cellular (re-)programming process during which cancer cells switch from an Epithelial-like proliferative phenotype to acquire Mesenchymal-like invasive properties.

In this talk, we present a chain of 2- and 3D mathematical models that describe the growth of the Epithelial-like (ECs) and the invasion strategy of the Mesenchymal-like cancer cells (MCs). We start with the simpler model versions where we discuss the existence of classical solutions, and proceed with a more elaborate multiscale and hybrid SDE-PDE model and its predictive capacity of realistic experimental situations. We conclude with some findings from our most recent model extensions to multi-organ conformation and our first steps in the formation of a virtual cancer patient.

The material of this presentation is based on joint works with: M. Chaplain, A. Madzvamuse, L. Franssen, T. Williams, A. Wilson, L. Fu, D. Katsaounis, J. Giesselman, Chr. Surulescu and N. Kolbe.

In this talk, we study semi-linear $\sigma$-evolution equations with double damping including frictional and visco-elastic damping for any $\sigma\ge 1$. We would like to investigate not only higher order asymptotic expansions of solutions but also diffusion phenomena in the $L^p-L^q$ framework, with $1\le p\le q\le \infty$, to the corresponding linear equations. By assuming additional $L^m$ regularity on the initial data, with $1\le m< 2$, we prove the global (in time) existence of small data energy solutions and indicate the large time behavior of global obtained solutions as well as semi-linear equations. Moreover, we also find out the so-called critical exponent when $\sigma$ is considered as any fractional number.

Hilbert’s sixth problem, stated in 1900 during the International Congress of Mathematicians, consists in the axiomatization of physics. In the case of fluid dynamics, this issue reduces to the derivation of hydrodynamic equations (a macroscopic description) from kinetic equations (a mesoscopic description), which would be themselves derived from Newton’s laws of motion applied to the particles making up the fluid (a microscopic description). In the special case of a gas close to a global thermodynamic equilibrium with constant density, temperature and velocity, the fluctuations of these two last quantities are driven by the Navier-Stokes equations. The problem of deriving this hydrodynamic model from this kinetic model is still partially open for strong solutions (the link between weak solutions being well understood thanks to the works of C. Bardos, F. Golse, D. Levermore and L. Saint- Raymond between 1989 and 2003). In most of the strong theory of hydrodynamic limits, the initial statistical distribution is required to have Gaussian decay, although the ideal decay assumption, suggested by physical a priori bounds, would be an algebraic one. The so called Enlargement Theory (of functional spaces), initiated by C. Mouhot and developed with M.P. Gualdani and S. Mischler between 2005 and 2017, allowed to get rid of this restriction for several kinetic models.

In this talk, I will explain how this theory can be used to study the hydrodynamic modes of the linearized Boltzmann operator, then present how to combine it with previous approaches (à la Bardos-Ukai or Gallagher-Tristani) to construct solutions to the Boltzmann equation for any initial distribution with algebraic decay.

Reference: 

[1] P. Gervais, Spectral study of the linearized Boltzmann operator in L2 spaces with polynomial and Gaussian weights, Kinet. Relat. Models, 14, 752-774 (2021).

We investigate certain two species ODE and PDE Lotka–Volterra competition models, where one of the competitors could potentially go extinct in finite time. We show that in this setting, various novel dynamics are possible. In particular, competitive exclusion can be avoided, and the slower diffusing competitor may not win. Numerical simulations are performed to corroborate our analytical findings.

Reference: 

[1] Rana D. Parshad, Kwadwo Antwi-Fordjour, and Eric M. Takyi, Some Novel Results in Two Species Competition, SIAM Journal on Applied Mathematics, 2021, Vol. 81, No. 5: pp. 1847-1869

Processes involving multicomponent diffusion and chemical reactions appear in many applications in fluid mechanics and chemistry. The diffusive behavior of the species is well described by the equations introduced by Maxwell and Stefan, which provide a more general and appropriate framework than the standard Fickian approach. In this talk, we consider a chemically reactive mixture described in the frame of the Boltzmann equation and study the reaction-diffusion limit of the kinetic system of equations. Under certain assumptions, we formally derive a reaction-diffusion system of Maxwell-Stefan type. The emphasis is on the contributions resulting from the chemical reaction.

References:  

[1] B. Anwasia, P. Gonçalves, A.J. Soares, From the simple reacting sphere kinetic model to the reaction-diffusion system of Maxwell-Stefan type, Commun. Math. Sci., 17, 507–538 (2019).

[2] B. Anwasia, P. Gonçalves, A.J. Soares, On the formal derivation of the reactive Maxwell-Stefan equations from the kinetic theory, Europhys. Lett. EPL, 129, 40005, 1–7 (2020).

[3] B. Anwasia, M. Bisi, F. Salvarani, A.J. Soares, On the Maxwell-Stefan diffusion limit for a reactive mixture of polyatomic gases in non-isothermal setting, Kinet. Relat. Models, 13, 63–95 (2020).

Abstract: In this talk, we will discuss the convergence of global classical solutions of reaction-diffusion systems to equilibria. The corresponding results are applicable to reaction-diffusion systems with analytic reaction terms which are only space and state-dependent. The core of achieving suitable convergence results will be provided by the Lojasiewicz-Simon inequality based on its finite-dimensional counterpart, the Lojasiewicz inequality. We will discuss how strong convergence to equilibria may be inferred and also how the convergence rate, at least for the \(L^2-\)norm, depends on the exponent occurring in the Lojasiewicz-Simon inequality.

Throughout the talk, we will regard numerical simulations in view of the Chafee-Infante problem and finally, discuss an implicit way to derive the convergence rate of the \(L^2-\)norm to the zero state. In general, the main preconditions are that one has to assure global existence of solutions to both the reaction-diffusion system and the corresponding time-independent problem. Apart from that, precompactness of the orbit is required.

This talk is based on my seminar thesis supervised by Dr. Bao Tang.

References:

Zheng, Songmu "Nonlinear Evolution Equations" (2004)

                        Jendoubi, Mohamed Ali "A Simple Unified Approach to Some Convergence Theorems of L. Simon" (1998)

Although several molecular markers have been described to predict stem cell potential, whether there exists a general definition of stemness in terms of biochemical properties of cells or not remains an open question. Recent results point to the interesting hypothesis that collective cell dynamics itself, far for being a consequence of individual cell properties, could play a key role in the emergence of the stem cell region in an organ. In that frame, stemness would be a context-dependent property emerging from the global dynamics of the tissue. In this talk, I will show a general framework based on stochastic dynamics of competition for space from which one can predict the robust emergence of a region made of functional stem cells, without referring to their biochemical identity. The mathematical framework provided predictions that have been confronted with data obtained from intravital live-imaging experiments in mammary gland development, kidney development, and from the self-renewal dynamics of the intestinal crypt. In addition, the proposed framework identifies key differences in terms of functionality that are not visible using the standard approach based on molecular marker identification.

References:

https://www.pnas.org/content/117/29/16969.short

https://www.cell.com/action/showPdf?pii=S1934-5909%2819%2930300-5

https://www.science.org/doi/10.1126/science.1196236

Compartmental models are popular in the mathematics of epidemiology for their simplicity and wide range of applications. Although they are typically solved as initial value problems for a system of ordinary differential equations, the observed data is typically akin of a boundary value type problem: we observe some of the dependent variables at given times, but we do not know the initial conditions. In this talk, I will present a reformulation of the classical SIR in terms of the number of detected positive infected cases at different times, and then the existence and uniqueness of a solution to the derived boundary value problem. Some results regarding parameter identification given infected data from Sussex region (UK) together with some numerical results will be presented at the end.

This is a joint work with E. Campillo-Funollet, H. Wragg, J. Van Yperen and A. Madzvamuse.

Preprint available here.

In ecology, "stability" describes the manner in which the abundances of interacting systems of species vary over time and in response to disturbances. To quantify ecological stability, theoretical studies often draw on mathematical concepts such as Eigenvalues, attractor blocks, and Lyapunov functions. However, in practice, these methods often run into trouble, either because there is insufficient data to measure necessary statistics with any accuracy, or because the scope of available data fails to capture important aspects of dynamics (e.g. boundary conditions). Several recent studies have begun to address these problems by borrowing concepts from linear stochastic dynamical systems theory, which has allowed the derivation of some important links between empirically measurable aspects of ecological systems (e.g. variability), and certain classes of theoretical stability metrics (e.g. asymptotic return rates). The next frontier will be to extend these methods to better address more complex types of systems, such as those that include oscillatory dynamics, nonstationary equilibria, or high levels of observation error. In this talk, I will discuss the history and progress of efforts to overcome these challenges and describe some current projects of mine that aim to overcome the linearity constraints of current methods.

Chemical signaling is a ubiquitous component of biochemical systems and also used for their study in the form of microfluidics. Often such chemical signaling is considered a form of communication—known as molecular communication—where information is carried by molecules in a fluid rather than, for example, electromagnetic waves. In this talk, I will discuss how techniques from information theory connect with molecular communication in biochemical systems. As a key component of molecular communication is the dynamics of information-carrying molecules, stochastic reaction-diffusion models are critical for understanding the success or failure of the communication. An important aspect is therefore the study of underlying reaction-diffusion PDE models, due to the computational complexity of simulation and difficulty in obtaining tractable solutions of the Fokker-Planck equation for the stochastic model. I will highlight in particular the importance of convergence to equilibrium for molecular communications. 

A brief overview of historical results is presented, along with open questions and current progress from collaborations on a variety of projects.

We rigorously prove the passage from a weak competitive reaction-diffusion system towards a reaction-cross-diffusion system, in the fast reaction limit. The modelled ecosystem is composed of two competing species, one of which can choose a different diet with respect to the other species. The resulting limit system shows a starvation driven cross-diffusion term. The main ingredients used to prove the existence of global solutions are a family of energy functionals and compactness arguments. 

However, we also investigate the linear stability of homogeneous steady states of those systems and rule out the possibility of Turing instability. Then, no pattern formations occur. To conclude, numerical simulations are included, proving the compatibility with the theoretical results.

This talk is based on: ''Evolution of dietary diversity and a starvation driven cross-diffusion system as its singular limit'', E.Brocchieri, L.Corrias, H.Dietert, Y-J. Kim. Submitted for publication (2020). https://arxiv.org/abs/2011.10304

We consider a model for non-isothermal and incompressible flows that consist of two phases. The phase separation is described by an order parameter through the Cahn-Hilliard equation. This equation is coupled through the latent heat by a convection-diffusion equation for the normalized temperature. On the other hand, the evolution of the mean velocity is given by the Navier-Stokes equation accounting for capillary stress due to surface tension. A distributed optimal control for this system will be discussed. We establish the existence of optimal controls, the differentiability of the control-to-state operator, and the necessary and sufficient optimality conditions. For initial and target data with finite energy norms, a limited space-time regularity of the adjoint states arises due to convection and surface tension. 

G Peralta Distributed Optimal Control of the 2D Cahn–Hilliard–Oberbeck–Boussinesq System for Nonisothermal Viscous Two-Phase Flows - Applied Mathematics & Optimization, 2021 - Springer

We consider a population that is structured by a phenotypic trait and a spatial variable. We will consider three strains (ie three possible trait values): resistance to a treatment A, resistance to a treatment B and non-resistance (with a fitness cost for that last one). The treatments A and B will be alternated in space, and analyse the effect of this spatial heterogeneity on the propagation of the pathogen population. Several propagation strategies can be used by the pathogen to propagate in space: the propagation can for instance be driven by a specialist type, or by the multi-resistant. The main outcomes of this study are the following: 

• Through the use of propagation speed argument, we are able to discuss the emergence of multi-resistant strains at the edge of an epidemics, and show that propagation situations are more favourable than steady scenario for the development of this dangerous pathogen.  

• We show that a good understanding of the dynamics of simple population models can provide useful tools to consider complex situations. 

Finally, I will discuss some theoretical developments on stochastic propagation models that provide exciting perspectives on these questions. The biology aspects of this presentation is part of a work done in collaboration with Matthieu Alfaro, Sylvain Gandon and Quentin Griette. The theoretical work is part of the PhD of Julie Tourniaire, co-advised with Pascal Maillard.

The term cancer covers a multitude of bodily diseases, broadly categorised by having cells that do not behave normally. Cancer cells can arise from any type of cell in the body; cancers can grow in or around any tissue or organ making the disease highly complex. My research is focused on understanding the specific mechanisms that occur in the tumour microenvironment via mathematical and computational modeling. In this talk, I shall present a 3D individual-based force-based model for tumour growth in which we simulate the behaviour of, and spatio-temporal interactions between, cells, extracellular matrix fibres and blood vessels. Each agent (a single cell, for example) is fully realised within the model and interactions are primarily governed by mechanical forces between elements. However, as well as the mechanical interactions we also consider chemical interactions, for example, by coupling the code to a finite element solver to model the diffusion of oxygen from blood vessels to cells. The current state-of-the-art of the model allows us to simulate tumour growth around an arbitrary blood-vessel network or along the striations of fibrous tissue. 

Lipid hydrolysis, also called lipolysis, is a three-step coordinated process of triglyceride degradation generating fatty acids and glycerol, catalysed by three different enzymes. Triglycerides are a form of fat stored in lipid droplets found in our body.

In this presentation, the process of triglyceride degradation is discussed in details. A test model using the Michaelis-Menten substrate-enzyme kinetics for the evolution of the involved concentrations is proposed. Finally, an explicit radially symmetric stationary state solution in the case of high substrate and low substrate concentrations is solved.

In this talk, I will discuss recent advances of global existence and boundedness for reaction-diffusion systems (RDS). Two of the most common properties of RDS arising from chemistry, biology or population dynamics are (i) preservation of non-negativity of initial data and (ii) control of a total mass. In the most generality, these two assumptions are not sufficient to prevent solutions from a blow-up in finite time. Recent results show that if the nonlinearities are at most quadratic, then global bounded solutions can be proved. The case of nonlinearities with higher growth rates is largely open. When an additional assumption called intermediate sum condition is satisfied, one can allow arbitrarily high growth rates of nonlinearities at the price of a good cancellation of terms. Duality- and L^p-energy methods are introduced to utilise this new structure, where the former is elegant and efficient in the case of smooth diffusion coefficients, while the latter is more flexible to deal with discontinuous diffusion or coupled volume-surface systems.

This is based on joint works with Klemens Fellner, William Fitzgibbon, Jeff Morgan, and Hong-Ming Yin.

Mathematical modeling is the art of translation. First, you translate physiological or biological principles, mechanisms, and hypothesis into mathematical models. Then, after rigorously analysing these models within the mathematical framework and without the bias of intuition, the results are translated back into biology and physiology resulting either in the corroboration of current theories, the formulation of new and testable hypothesis, or guidance for new research directions. Some results might even find their way into clinical routines.

In this talk, I will focus on the translation of pathophysiological processes in patients suffering from chronic kidney disease into mathematical models. Chronic kidney disease is a major global health care problem, and the decline of kidney function triggers a cascade of complex and interwoven pathologies. We developed a mathematical model which allowed us to analyse these pathologies in detail. The translation to the clinic resulted in the implementation of a model-based drug administration strategy in thousands of hemodialysis patients in the US. 

Fast reaction limit or singular limit is widely used to simplify mathematical models. In this talk, we apply this idea to mathematical modeling in ecology. In particular, functional responses arising in prey-predator systems are discussed. Consequently, we can give another interpretation of functional responses through fast reaction limit.

In this talk, I will suggest that cancer is fundamentally a disease of the control on cell differentiation in multicellular organisms, uncontrolled cell proliferation being a mere consequence of blockade, or unbalance, of cell differentiation. Cancer cell populations, that can reverse the sense of differentiations, are extremely plastic and able to adapt without mutations their phenotypes to transiently resist drug insults, which is likely due to the reactivation of ancient, normally silenced, genes. Stepping from mathematical models of non-genetic plasticity in cancer cell populations and questions they raise, I will propose an evolutionary biology approach to shed light on this problem both from a theoretical viewpoint by a description of multicellular organisms in terms of multi-level structures, which integrate function and matter from lower to upper levels, and from a practical point of view oriented towards cancer therapeutics, as cancer is primarily a failure of multicellularity in animals and humans. This approach resorts to the emergent field of knowledge named philosophy of cancer.

A range of mathematical models has been used to gain a more in-depth theoretical understanding of different aspects of cancer dynamics. In this talk, deterministic, continuum models formulated as partial differential equations will be considered. The first part of the talk will focus on partial integro-differential equations modelling the eco-evolutionary dynamics of cancer cells in vascularised tumours. In the second part of the talk, attention will turn to models of avascular tumour growth that comprise coupled systems of nonlinear partial differential equations, which reflect the heterogeneous cellular composition of the tumour micro-environment. Analytical and numerical results summarising the behaviour of the solutions to the model equations will be presented and the biological insight generated by these results will be discussed.

We consider some biological and ecological systems with a certain dichotomy property. This dichotomy property divides a studied entity, e.g., cancer cells, insect, people, into two groups being opposed to each other, both groups evolving in space and time. If we allow individuals to switch between the groups, such switching may lead to interesting effects such as aggregation of the whole entity and population density pressure on the spread of the entity. We look at some examples of such systems with dichotomy phenomenon, formulate them in form of reaction-diffusion equations and consider a special case of unlimited switching known as singular or fast reaction limit of the reaction-diffusion equations.

We consider the Boltzmann equation modelling monatomic gaseous mixtures. In a standard diffusive scaling where the Mach and Knudsen numbers approach zero at the same rate, we investigate the behaviour of particular solutions expanded around a non-equilibrium Maxwellian state of the mixture whose physical observables solve the Maxwell-Stefan diffusion equations. We make use of the hypocoercive formalism to prove uniform (with respect to the Mach and Knudsen numbers) well-posedness of such solutions, providing a first rigorous derivation of the Maxwell-Stefan model from a kinetic formulation. 

Reference:  A. Bondesan and M. Briant, Stability of the Maxwell-Stefan system in the diffusion asymptotics of the Boltzmann multi-species equation, Commun. Math. Phys., 382, 381–440 (2021).

This talk is concerned with a reaction-diffusion system modeling the fixation and the invasion in a population of a gene drive (an allele biasing inheritance, increasing its own transmission to offspring). In our model, the gene drive has a negative effect on the fitness of individuals carrying it and is therefore susceptible of decreasing the total carrying capacity of the population locally in space. This tends to generate an opposing demographic advection that the gene drive has to overcome in order to invade. While previous reaction-diffusion models neglected this aspect, here we focus on it and try to predict the sign of the traveling wave speed. It turns out to be an analytical challenge, only partial results being within reach, and we complete our theoretical analysis by numerical simulations. Our results indicate that taking into account the interplay between population dynamics and population genetics might actually be crucial, as it can effectively reverse the direction of the invasion and lead to failure. Our findings can be extended to other bistable systems, such as the spread of cytoplasmic incompatibilities caused by Wolbachia.

A scintillator crystal is a material which converts ionizing radiations into photons in the frequency range of visible light, hence its name. Scintillation is well understood at the microscopic level but scintillating crystals are massive object. Hence we model inorganic scintillating crystals with the tools of continuum mechanics. We show how the charged particles evolution within the crystal and the recombination into photons are described by a Reaction- Diffusion-Drift equation coupled with the Laplace equation of electrostatic and the Heat equation.  These equations generalise the two most used phenomenological models nowadays used for scintillators, namely the Kinetic and Diffusive ones. 

For the general boundary value problem we give a brief survey of the existence and asymptotic decays results, the latter being a direct estimate of the scintillation decay time. We show how a recent result of Fellner and Kniely, when applied to scintillators, lead to a very good estimate of the experimentally measured decay time for some specific scintillators.

The quasistatic limit is a convenient approximation in the modelling of several (suitable) mechanical systems when the evolution occurs at a sufficiently slow time-scale. One of the situations where the quasistatic limit is usually adopted is to study locomotion in earthworms, inchworms, leeches and other limbless terrestrial animals, as well as in an increasing number of soft-robotic devices mimicking such strategies. In this seminar, we discuss the meaning of the quasistatic assumption in soft-bodied crawlers by a mathematical perspective. The first part of the talk is about modelling. We show how to build a basic but effective family of models for soft crawlers and survey the relevance (or lack thereof) of inertia in some locomotion strategies. In the second part of the talk, we provide a mathematically precise description of this quasistatic limit. We formulate our models in the framework of finite-dimensional rate-independent systems and discuss the uniform convergence of dynamical solutions to the quasistatic one, via a slow-actuation asymptotic analysis.

Joint work with Filippo Riva (University of Pavia)

[1] P. Gidoni and F. Riva, A vanishing inertia analysis for finite dimensional rate-independent systems with nonautonomous dissipation, and an application to soft crawlers, submitted (2020).

[2] P. Gidoni, Rate–independent soft crawlers, Quarterly Journal of Mechanics and Applied Mathematics  71, pp. 369–409 (2018).

Glioblastoma (GB) growth and migration inside the brain tissue is driven by specific signaling pathways and by several interactions between tumor cells and their extracellular microenvironment. Recent studies on tumor cell activity have highlighted the important role of the cell membrane protrusions located at the tumor invasion front in the process of tumor progression. In this work, our main interest is focused on the study of the tumor cell membrane dynamics involved in cell transport and in the cell interactions with different molecular substrates during tumor evolution. Starting from biological experiments in a Drosophila model of GB, we formulate a nonlinear macroscopic model based on flux-saturated mechanisms [1, 2]. We exploit the ability of the mathematical model to partially guide the biological experiments on protein signaling distribution. We numerically analyze the evolution of the tumor propagation front and the emergence of a coordination between self-organized collective processes characterizing the different agents involved in tumor progression.

Joint work with J. Soler (University of Granada), and S. Casas Tint (Instituto Cajal -CSIC).

[1] Verbeni, M., Sánchez, O., Mollica, E., Siegl-Cachedenier, I., Carleton, A., Guerrero, I., ... & Soler, J. (2013). Morphogenetic action through flux-limited spreading. Physics of Life Reviews, 10(4), 457-475.

[2] Calvo, J., Campos, J., Caselles, V., Sánchez, O., & Soler, J. (2016). Pattern formation in a flux limited reaction-diffusion equation of porous media type. Inventiones Mathematicae, 206(1), 57-108.

Inflammation is the body's response to outside threats. Although it is a protective mechanism, a derangement of this biological process can impair physiological functions, leading to a significant number of severe autoimmune diseases. The complex dynamics of the inflammatory process are not yet fully known and thorough knowledge of these mechanisms is the key to control the onset and the evolution of inflammatory diseases. Recently, several mathematical models have been proposed to test biological hypotheses and improve the knowledge of the inflammatory processes.

In this talk, I shall present a study on a Reaction-Diffusion-Chemotaxis model aiming to explore the mechanisms of the inflammatory response. It is a recently proposed model, which captures key mechanisms of inflammation and is able to reproduce different clinical scenarios.

After a brief presentation of the system, in which I will focus on the most relevant modeling aspects, I shall present a study on the onset of both stationary and oscillating primary instabilities. I shall show that using numerical values of the parameters taken from the experimental literature, the resulting patterns are able to reproduce different clinical scenarios. I will also focus on some relevant mathematical properties, such as the emergence of oscillatory patterns generated by subharmonic resonances and the occurrence of irregular spatio-temporal solutions. Finally, I will discuss how these instabilities can be studied.

This talk discusses a space-time nonlocal parabolic equation. By the temporal nonlocality, the families of solution operators are not semigroups. Consequently, we cannot apply the $L^\infty_tL^\infty_x$ bootstrap argument, the Caffarelli-Silvestre extension problem, etc., which are well-known as powerful tools relying on semigroup structure. By considering solutions in Lebesgue spaces, Sobolev spaces of Bessel and Riesz potentials, the contributions include four main results

• Locally, globally well-posedness and regularity in Lebesgue spaces;

• Some criteria for blowup at finite time;  

• Globally well-posedness in Sobolev spaces of Bessel potentials; 

• Connection between space-time nonlocal superlinear parabolic equation and heat equation  

We formulate our study for the model with respect to the superlinear term $u^p$, but the methods of our proofs can be applied to more general parabolic problems. 

In this talk we discuss partial Hölder-regularity for bounded solutions of a certain class of cross-diffusion systems, which are strongly coupled, degenerate quasilinear parabolic systems. The cross-diffusion systems that we consider have a formal gradient flow structure, in the sense that they are formally identical to the gradient flow of a convex entropy functional. Furthermore, we assume that the cross-diffusion systems are not volume-filling.  The setting covers the famous two component Shigesada-Kawasaki-Teramoto (SKT) model for population dynamics. The main novel tool that we introduce in this contribution is a ``glued entropy density,'' which allows us to emulate the classical theory of partial Hölder regularity for nonlinear parabolic systems by Giaquinta and Struwe within this new setting.

This talk is based on:  Braukhoff, Raithel, Zamponi. Partial Hölder Regularity for Bounded Solutions of a Class of Cross-Diffusion Systems with Entropy Structure. Submitted for publication. arXiv:2007.03561 (2020)

The Bhatnagar-Gross-Krook model, or the BGK model, is a linear variant of the Boltzmann equation which models collision processes in rarefied gasses. The equations that describe this model cover a wide range of cases - including a simple 2-velocity case known as the Goldstein-Taylor model.  While the Goldstein-Taylor model exhibit many properties one would expect from such “relaxation” process, such as convergence to an equilibrium, it is far from trivial to explicitly understand the long time behaviour of solutions to its equation when the relaxation function that is associated to the system is non-constant. 

In our talk, we will present recent work with Anton Arnold, Beatrice Signorello and Tobias Wöhrer, where we have constructed a robust, though not optimal, method to find an appropriate Lyapunov functional to the Goldstein-Taylor model, with which we were able to attain an explicit rate of convergence to equilibrium. This functional is of a Pseudo-differential nature and is motivated by a modal analysis of the equations in the tractable case where the relaxation function is constant. 

We will also discuss possible improvements to our method, compare it to the optimal rate to the equation obtained by Bernard and Salvarani, and consider higher finite velocity models.

Reference:  https://arxiv.org/abs/2007.11792