Applied Mathematics and Numerical Analysis Seminar 

I will present some models in ecology and epidemiology using a transport equation approach, so called structured models. The first models are of predator-prey type and include a variable hunger structure. They take the form of nonlocal transport equations coupled to ODEs. Then, we use a similar approach in an epidemiological model including disease awareness and variable susceptibility. We show well-posedness results, asymptotic behavior, and numerical simulations. This is joint work with C. Rebelo, A. Margheri, and P. Lafargeas.

Modeling of the non-linear dynamics of virus in vivo, for example during primary infection or following drug treatment, has been used in the last two decades to study the biology of diverse viruses. I will discuss, with examples from HIV and hepatitis C virus (HCV) infection, the principles and approach of this methodology. I will also present recent examples of insights into the biology of these viruses and SARS-CoV-2 gained with viral dynamics.

Lymph nodes (LNs) are organs scattered throughout the lymphatic system which play a vital role in our immune response by breaking down bacteria, viruses, and waste; the interstitial fluid, called lymph once inside the lymphatic system, is of fundamental importance in this process as it transports these substances inside the lymph node. The main mechanical features of the lymph node include the presence of a porous bulk region (lymphoid compartment, LC), surrounded by a thin layer (subcapsular sinus, SCS) where the fluid can flow freely.

These nodes are vital for filtering and processing lymph, which contains immune cells, antigens, and other molecules. Understanding the fluid dynamics within lymph nodes is essential for elucidating immune mechanisms and developing therapies for lymphatic disorders. Despite its significance, few models in the literature attempt to describe lymph behavior from a mechanical perspective.

In this talk, we will introduce a mathematical model, derived using the asymptotic homogenization technique, to describe fluid flow within a lymph node, considering its multiscale nature. We will discuss how this model can elucidate flow patterns, pressure distribution, and shear stress within the node.

One of the key development areas in battery research is finding ways to use metallic anodes, like Zn and Mg, but avoiding lithium, which is pyrophoric and sourced only in potentially critical geopolitical areas. Unfortunately, use of post-Li batteries is impaired by poorly understood shape changes, responsible for various failure modes.

Over the past decade, in the framework of the RD-PDEs, a powerful mathematical approach has been developed in [1], able to capture the essential features of unstable material growth in electrochemical systems in terms of Turing pattern formation. Recharge instability problems in batteries with metal anodes are a special case of this phenomenon. On the other hand, the difficulty of studying materials in real-life battery context leads to a methodological gap between theory and experiments. For this reason, parameter identification in the above PDE modelling is crucial for advancement in this direction.

In this research, based on [2], we propose to apply Deep-Learning as a new approach for parameter estimation, instead of the more traditional PDE constrained optimization, as for example in [3]. In the seminar we will discuss the Convolutional Neural Network devised for our goals, trained with the numerical solutions of the morphochemical PDE model that is able to capture the essential features of unstable material growth in electrochemical systems.

We will show that the CNN carries out three tasks:

1. automatic partitioning of the parameter space associated to the PDE model, according to the types of patterns generated;
2. classification of simulated and experimental patterns;
3. identification of the model parameters for experimental electrode images.

References

1. B. Bozzini, D. Lacitignola, I. Sgura. Spatio-temporal organization in alloy electrodeposition: a morphochemical mathematical model and its experimental validation. J. Solid State Electrochem (2013).
2. I. Sgura, L. Mainetti, F. Negro, M. G. Quarta, B. Bozzini. Deep-Learning based parameter identification enables rationalization of battery material evolution in complex electrochemical systems. J. of Computational Science (2023).
3. I. Sgura, A. Lawless, B. Bozzini. Parameter estimation for a morphochemical reaction-diffusion model of electrochemical pattern formation. Inverse Probl. Sci. Eng. (2019).

The aim of the talk is numerical simulations of various physical problems, specifically a streamer propagation and a fluid flow, on unstructured dynamically adapted grids. The streamer simulations are aimed on a proper choice of an adaptation criterion, an influence of width of a computational domain, interactions among streamer filaments and a streamer branching. The fluid flow simulations are devoted to a compressible turbulent flow through a turbine cascade and a restrictor.

The aim of the talk is the existence of a solution to the Navier-Stokes-type problem in a more complicated geometry.
Specially in 2D cascade of profiles (a model of blade machine) and a 2D multiply connected domain, modelling a flow of an incompressible viscous fluid through a rotating radial turbine. The main goal is to consider artificial boundary condition of the ''do nothing'' type on the outflow part of the domain and prove the existence for an arbitrarily large flux through the inflow part of the domain.

Mathematical models of the heart have emerged as a powerful tool in understanding the mechanisms underlying proper heart function and dysfunction in cardiovascular diseases. One particular area of interest is the propagation of the action potential wave that precedes and synchronizes the heart's contraction, which is crucial for proper heart function. Disruptions in this process can lead to cardiac arrhythmia, a condition characterized by irregular heartbeats.
To better understand the mechanisms underlying cardiac arrhythmia, personalized patient-specific models of the heart can be generated by incorporating both personalized electrophysiological data, such as electrocardiogram (ECG), and geometric information from imaging techniques such as cardiac MRI. These models accurately capture the individualized anatomy and electrophysiology of the heart.
In this talk, we will highlight the benefits of patient-specific models for assessing the risk of arrhythmia and improving patient outcomes in the management of cardiac arrhythmia.

The aim of the talk is to give a brief presentation of novel results related to the body-fluid interaction problem.

The motion is described by a system of coupled differential equations: Newton’s second law and Navier-Stokes type equations. We shall discuss the global solvability result of weak solution of the problem, when the slippage is allowed at the boundaries of the rigid body and of the bounded domain, occupied by the fluid.

This result completely resolves a famous non-contact paradox between the rigid body and the domain boundary.

In this talk, we present some stability results in a classical model concerning population dynamics, the nonautonomous prey-predator Lotka-Volterra model under the assumption that the coefficients are $T$-periodic functions \begin{equation}\label{sysLV}
\left\lbrace \begin{array}{l} \dot{u}=
u(a(t) - b(t)\,u - c(t)\,v), \\ \dot{v}=
v(d(t) + e(t)\,u - f(t)\,v), \end{array} \right.
\end{equation} where $u\gt 0$, $v\gt 0$. The variables $u$ and $v$ represent the population of a prey and its predator, respectively. Some instances with this kind of dynamics can be: snowshoe hare and lynx canadensis, paramecium and didinium, fish population and fishermen, etc. The periodicity of this model takes into account changes of the environment in which the predation process takes place. For instance, seasonality or variations of the temperature in laboratory conditions.

In the system \eqref{sysLV} the coefficients $b(t)$, $c(t)$, $e(t)$ and $f(t)$ are positive. The coefficients $c(t)$ and $e(t)$ describe the interaction between $u$ and $v$; $a(t)$ and $b(t)$ describe the growth rate for the prey $u$; $d(t)$ and $f(t)$ represent the analogous for the predator $v$.

Solutions for the system \eqref{sysLV} with both components positive are called coexistence states and the necessary and sufficient conditions for their existence are well understood, see [2].

After reviewing those conditions, we present some results concerning the stability of a special kind of coexistence state, positive $T$-periodic solutions. In [3] the author gave a sufficient condition for the uniqueness and asymptotic stability of the positive $T$-periodic solution. This criterion is formulated in terms of the $L^1$ norm of the coefficients of a planar linear system associated to \eqref{sysLV}. On the other hand, in [1], assuming that the system \eqref{sysLV} has no sub-harmonic solutions of second order (periodic solutions with minimal period $2T$), the authors proved that there exists at least one asymptotically stable $T$-periodic solution. Here the result is formulated in terms of the $L^\infty$ norm. Our result, in [4], gives a $L^p$ criterion, building a bridge between the two previous results.

This is a Joint work with Carlota Rebelo (Departmento de Matemática and CEMAT, Faculdade de Ciências, Universidade de Lisboa, Portugal).

Acknowledgements: This work was partially supported by the Spanish Ministerio de Universidades and Next Generation Funds of the European Union.

References

1. Z. Amine, R. Ortega, A periodic prey-predator system, Journal of Mathematical Analysis and Applications,185(2): 477-489, 1994.
2. J. López-Gómez, R. Ortega and A. Tineo, The periodic predator-prey Lotka-Volterra model, Adv. Differential Equations, 1(3): 403-423, 1996.
3. R. Ortega, Variations on Lyapunov's stability criterion and periodic prey-predator systems, Electronic Research Archive, 29(6): 3995-4008, 2021.
4. V. Ortega, C. Rebelo, A $L^p$ stability criterion in the periodic prey-predator Lotka-Volterra model, In preparation, 2023.

In this presentation will be discussed recent advances of Boundary Element Method (BEM) in Computational Fluid Dynamics (CFD). Unlike other methods, BEM is a a multi-angle numerical technique, that permits the approach to a partial differential equation (PDE) in completely different ways. In Navier-Stokes equations in particular, many different test functions can be used in the weak form, as the Laplace, the Stokeslet, the convective parabolic-diffusion or other convective fundamental solutions, among others. Apart from that, it is found recently that hypersingular BEM in Navier-Stokes equations have a broad area of applicability, as they provide the gradients of the field. These gradients can further be applied to numerous cases as impovement of system's condition number, enforcing continuity, computation of wall quantities such as wall vorticities, strain and stress tensors, and pressure calculation, among others. However, derivation of such equations is not always simple since they are accompanied with extra terms, mainly in convection. Another important finding is that hypersingular equations can permit the use of constant elements simplifying immensely the preparation of the computational model. Another part of the talk will be dedicated to the transformation of the BEM system to Finite Element (FEM) or Finite Volume (FVM) equivalent in terms of sparsity. A system produced by BEM with domain unknowns cannot be solved efficiently, but with proper transformations it can be changed to a sparse system, which can be solved remarkably faster. Other accelerating techniques like hypersingular BEM/ Fast multipole (FMM) and meshless Local Boundary integral equation (LBIE) will be discussed.

Most studies on fluid dynamics have been devoted to Newtonian fluids, which are characterized by the classical Newton’s law of viscosity. However, there exist many real fluids with nonlinear viscoelastic behavior that does not obey Newton’s law of viscosity. My aim is to discuss two problems related to a class of non-Newtonian fluids of differential type. Namely, the optimal control of incompressible third-grade fluids in 2D, via Pontryagin’s maximum principle and the strong well-posedness, in PDEs and probabilistic senses, of the 3D stochastic third-grade fluids in the presence of multiplicative noise driven by a Q-Wiener process.

The talk is based on recent works with Fernanda Cipriano (CMA, Univ. NOVA de Lisboa).

The main motivation for this study is the air flow in the human respiratory system, although similar problems are also common in other areas of biomedical, environmental or industrial fluid mechanics. The detailed experimental studies of respiratory system in humans and animals are very challenging and even impossible in many cases due to various medical, technical or ethical reasons. This leads to the development of more and more realistic mathematical and numerical models of the flow in airways including the complex geometry of the problem, but also various fluid- and bio-mechanics features. The main difficulties are not just in the geometrical complexity of the computational domain with several levels of branching, but also in the need to prescribe mathematically suitable, but yet sufficiently realistic boundary conditions for the computational model. This leads to a complex multiscale problem, whose solution requires large amount of complicated and time-consuming numerical calculations.

In this work we are considering simplified simulations in a two-dimensional rigid channel coupled with a one-dimensional extended flow model derived from a 3D fluid-structure interaction (FSI) model under certain conditions. For this purpose we built a simple test code employing an immersed boundary method and a finite difference discretization. At this stage the air flow in human airways is considered as incompressible, described by the Navier-Stokes equations. This simple code was developed with the aim of testing and improving boundary conditions using reduced order models. The incompressible model will later be replaced by a compressible one, to be able to evaluate the impact of intensive pressure changes in human airways while using realistic, patient specific airways geometry. The main idea is to use different dimensional models, 3D(2D), 1D and 0D, with different levels of complexity and accuracy and to couple them into a single working model.

In the present talk, first results of the 2D-1D coupled toy model will be presented, focusing on the main features of the computational setup, coupling strategy and parameter sensitivity. In addition, some long term outlook of the more complex 3D-1D(-0D) model will be discussed.

Acknowledgment: Center for Computational and Stochastic Mathematics - CEMAT (UIDP/04621/2022 IST-ID).

The purpose of this work is the theoretical and numerical study of an epidemiological model of multi-strain co-infection. Depending on the situation, the model is written as ordinary differential equations or reaction-advection-diffusion equations. In all cases, the model is written at the host population level on the basis of a classical susceptible-infected-susceptible system (SIS).

The infecting agent is structured into N strains, which differ according to 5 traits: transmissibility, clearance rate of single infections, clearance rate of double infections, probability of transmission of strains, and co-infection rates. The resulting system is a large system ($N^2 + N + 1$ equations) whose complete theoretical study is generally inaccessible. This work is therefore based on a simplifying assumption of trait similarity - the so-called quasi-neutrality assumption. In this framework, it is then possible to implement Tikhonov-type time scale separation methods. The system is thus decomposed into two simpler subsystems. The first one is a so-called neutral system - i.e., the value of the traits of all the strains are equal - which supports a detailed mathematical analysis and whose dynamics turn out to be quite simple. The second one is a ”replication equation” type system that describes the frequency dynamics of the strains and contains all the complexity of the interactions between strains induced by the small variations in the trait values.

The first part explicitly determines the slow system in an a spatial framework for N strains using a system of ordinary differential equations and justifies that this system describes the complete system well. This system is a replication system that can be described using the $N(N −1)$ fitnesses of interaction between the pairs of strains. It is shown that these fitnesses are a weighted average of the perturbations of each trait.

The second part consists in using explicit expressions of these fitnesses to describe the dynamics of the pairs (i.e. the case $N = 2$) exhaustively. This part is illustrated with many simulations, and applications on vaccination are discussed.

The last part consists in using this approach in a spatialized framework. The SIS model is then a reaction-diffusion system in which the coefficients are spatially heterogeneous. Two limiting cases are considered: the case of an asymptotically small diffusion coefficient and the case of an asymptotically large diffusion coefficient. In the case of slow diffusion, we show that the slow system is a system of type ”replication equations”, describing again the temporal but also spatial evolution of the frequencies of the strains. This system is of the reaction-advection-diffusion type, the additional advection term explicitly involving the heterogeneity of the associated neutral system. In the case of fast diffusion, classical methods of aggregation of variables are used to reduce the spatialized SIS problem to a homogenized SIS system on which we can directly apply the previous results.

Image restoration is one of the major concerns in image processing with many interesting applications. In the last decades there has been intensive research around the topic and hence new approaches are constantly emerging. Partial differential equation based models, namely of non-linear diffusion type, are well-known and widely used for image noise removal. In this seminar we will start with a concise introduction about diffusion and cross-diffusion models for image restoration. Then, we will discuss a flexible learning framework in order to optimize the parameters of the models improving the quality of the denoising process. This is based on joint work with Diogo Lobo.

In this talk, we consider a fluid-structure interaction system where the fluid is viscous and compressible and where the structure is a part of the boundary of the fluid domain and is deformable. The reference configuration for the fluid domain is a rectangular cuboid with the elastic structure being the top face. The fluid is governed by the barotropic compressible Navier–Stokes system, whereas the structure displacement is described by a wave equation. We show that the corresponding coupled system admits a unique, locally-in-time strong solution for an initial fluid density and an initial fluid velocity in $H^3$ and for an initial deformation and an initial deformation velocity in $H^4$ and $H^3$ respectively.

In order to understand if counting the number of reinfections may provide supplementary information on the evolution of an epidemic, we consider in this paper a general SEIRS model describing the dynamics of an infectious disease including latency, waning immunity and infection-induced mortality. We derive an infinite system of differential equations that provides an image of the same infection process, but counting also the reinfections. Well-posedness is established in a suitable space of sequence valued functions, and the asymptotic behavior of the solutions is characterized, according to the value of the basic reproduction number. This allows to determine several mean numbers of reinfections related to the population at endemic equilibrium. We then show how using jointly measurement of the number of infected individuals and of the number of primo-infected provides observability and identifiability to a simple SIS model for which none of these two measures is sufficient to ensure on its own the same properties.

This is a joint work with Marcel Fang. More details may be found in the report https://arxiv.org/abs/2011.12202.

This talk is devoted to the theoretical and numerical analysis of some minimal time optimal and control problems associated to linear and nonlinear differential equations. We start by studying simple cases concerning linear and nonlinear ODEs. Then, we deal with the heat equation. In all these situations, we analyze the existence of solution, we deduce optimality results and we present several algorithms for the computation of optimal controls. Finally, we illustrate the results with several numerical experiments.

In the fight against arboviruses, the endosymbiotic bacterium Wolbachia has become in recent years a promising tool as it has been shown to prevent the transmission of some of these viruses between mosquitoes and humans. This method offers an alternative strategy to the more traditional sterile insect technique, which aims at reducing or suppressing entirely the population instead of replacing it.

In this presentation I will present an epidemiological model including mosquitoes and humans. I will discuss optimal ways to mitigate a Dengue outbreak using instant releases, comparing the use of mosquitoes carrying Wolbachia and that of sterile mosquitoes.

This is a joint work with Luis Almeida (Laboratoire Jacques-Louis Lions), Yannick Privat (Université de Strasbourg) and Carlota Rebelo (Universidade de Lisboa).

How to apply partial or total containment measures during a given finite time interval, in order to minimize the final size of an epidemic - that is the cumulative number of cases infected during its course? We provide here a complete answer to this question for the SIR epidemic model. Existence and uniqueness of an optimal strategy is proved for the infinite-horizon problem corresponding to control on an interval $[0,T]$, $T\gt 0$ (1st problem), and then on any interval of length $T$ (2nd problem). For both problems, the best policy consists in applying the maximal allowed social distancing effort until the end of the interval $[0,T]$ (1st problem), or during a whole interval of length $T$ (2nd problem), starting at a date that is not systematically the closest date and that may be computed by a simple algorithm. These optimal interventions have to begin before the proportion of susceptible individuals crosses the herd immunity level, and lead to limit values of that proportion smaller than this threshold. More precisely, among all policies that stop at a given distance from the threshold, the optimal policies are the ones that realize this task with the minimal containment duration. Numerical results are exposed that provide the best possible performance for a large set of basic reproduction numbers and lockdown durations and intensities.

Details and proofs of the results are available in [BDPV,BD].

This is a joint work with Michel Duprez (Inria), Yannick Privat (Université de Strasbourg) and Nicolas Vauchelet (Université Sorbonne Paris Nord).

[BDPV] Bliman, P.-A., Duprez, M., Privat, Y., and Vauchelet, N. (2020). Optimal immunity control by social distancing for the SIR epidemic model. Journal of Optimization Theory and Applications. https://link.springer.com/article/10.1007/s10957-021-01830-1

[BD] Bliman, P. A., and Duprez, M. (2021). How best can finite-time social distancing reduce epidemic final size?. Journal of Theoretical Biology 511, 110557. https://www.sciencedirect.com/science/article/pii/S0022519320304124