Applied Mathematics and Numerical Analysis Seminar

Past sessions

On the development of a numerical model for the simulation of air flow in the human airways

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).

Multiple Timescales in Microbial Interactions

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.

Learning stable nonlinear cross-diffusion models for image restoration

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.

Existence of strong solutions for a compressible viscous fluid and a wave equation interaction system

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.

Modelling, analysis, observability and identifiability of epidemic dynamics with reinfections

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.

Minimal time optimal control problems

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.

Dengue outbreak mitigation via instant releases

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).

Minimizing epidemic final size through social distancing

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

Predator-prey dynamics with hunger structure

We present, analyse and simulate a model for predator-prey interaction with hunger structure. The model consists of a nonlocal transport equation for the predator, coupled to an ODE for the prey. We deduce a system of 3 ODEs for some integral quantities of the transport equation, which generalises some classical Lotka-Volterra systems. By taking an asymptotic regime of fast hunger variation, we find that this system provides new interpretations and derivations of several variations of the classical Lotka--Volterra system, including the Holling-type functional responses. We next establish a well-posedness result for the nonlocal transport equation by means of a fixed-point method. Finally, we show that in the basin of attraction of the nontrivial equilibrium, the asymptotic behaviour of the original coupled PDE-ODE system is completely described by solutions of the ODE system [SIAM J. Appl. Math., 80(6), 2631-2656 (2020)].

Mathematical Models in Epidemiology. The COVID-19 case.

In this talk we overview the mathematical continuous and discrete models in use in Mathematical epidemiology. We analyse the evolution of COVI-19 in Portugal.

Some results on predator-prey and competitive population dynamics

Mathematical analysis is a useful tool to give insights in very different mathematical biology problems.

In this talk we will consider predator-prey and competition population dynamics models. We will give an overview of recent results in the case of seasonally forced models not entering in technical details.

First of all we consider predator-prey models with or without Allee effect and prove results on extinction or persistence. We will give some examples such as models including competition among predators, prey-mesopredator-superpredator models and Leslie-Gower systems. When Allee effect is considered, we deal with the cases of strong and weak Allee effect.

Then we consider competition models of two species giving conditions for the extinction of one or both species and for coexistence.

This talk is based in joint works with I. Coelho, M. Garrione, C. Soresina and E. Sovrano.

References:

[1] I. Coelho and C. Rebelo, Extinction or coexistence in periodic Kolmogorov systems of competitive type, submitted.

[2] M. Garrione and C. Rebelo, Persistence in seasonally varying predator-prey systems via the basic reproduction number, Nonlinear Analysis: Real World Applications, 30, (2016) 73-98.

[3] C. Rebelo and C. Soresina, Coexistence in seasonally varying predator-prey systems with Allee effect, Nonlinear Anal. Real World Appl. 55 (2020), 103140, 21 pp.

Understanding the dynamics of co-colonization systems with multiple strains

The high number and diversity of microbial strains circulating in host populations pose challenges to human health and have inspired extensive research on the mechanisms that maintain such biodiversity. While much of the theoretical work focuses on strain-specific and cross-immunity interactions, another less explored mode of pairwise interaction is via altered susceptibilities to co-colonization (co-infection) in hosts already colonized by one strain. Diversity in such interaction coefficients enables strains to create dynamically their niches for growth and persistence, and 'engineer' their common environment. How such a network of interactions with others mediates collective coexistence remains puzzling analytically and computationally difficult to simulate. Furthermore, the gradients modulating stability-complexity regimes in such multi-player endemic systems remain poorly understood.

In this seminar I will present results from an epidemiological study where we analyze mathematically such an interacting system and the eco-evolutionary dynamics that emerge. Adopting a slow-fast dynamic decomposition of the original SIS model, we obtain a model reduction coinciding with a version of the replicator equation from evolutionary game theory. This enables us to highlight the key coexistence principles and the critical shifts in multi-strain dynamics potentiated by mean-field gradients.

Gielis Transformations in mathematics, the natural sciences and technological applications

The Gielis Transformation (GT) defines measures and unit elements specific to the shape, extending Euclidean geometry and challenging current notions of curvature, complexity and entropy. Global anisotropies or (quasi-) periodic local deviations from isotropy or Euclidean perfection in many forms that occur in nature can be effectively dealt with by applying Gielis transformations to the basic forms that show up in Euclidean geometry, e.g. circle and spiral. Anisotropic versions of the classical constant mean curvature and minimal surfaces have been developed. In mathematical physics it has led to developing analytical solutions to a variety of boundary value problems with Fourier-like solutions for anisotropic domains.

GT have been used in over 100 widely different applications in science, education and technology. In the field of design and engineering they have been used, among others, for the optimization of wind turbine blades, antennas, metamaterials, nanoparticles and lasers.

Modelling the transmission dynamics of SARS-CoV-2 in Portugal

In March 11th of 2020, the World Health Organization declared the COVID-19 global public health emergency a pandemic [1]. Since the appearance of the first cases in Wuhan, China, several countries have employed the use of mathematical and statistical techniques to ascertain the course of the disease spread. The most common mathematical tool available to model such phenomena are systems of differential equations. The most notable are the SIR and SEIR model first developed by Kermack and McKendrick (1927). These models have been used to study an array of different epidemic questions. At the start of the pandemic, these models were employed to nowcast and forecast the national spread of SARS-CoV-2 in China. In [2] the authors create scenarios of transmissibility reduction and mobility reduction associated with the measures employed by the Chinese government. Similar models were also used to estimate the proportion of susceptible individuals in a population, i.e. how much is the case ascertainment in a given country [3]. This topic is very important since it has been shown that a high percentage of infected individuals do not develop symptoms [4] but are still able to infect others [5]. The main purpose of these modelling techniques has been to evaluate the impact of contagion mitigation measures, such as the closure of schools and lockdowns [6].

In Portugal, the team at the department of epidemiology Instituto Nacional de Saúde Doutor Ricardo Jorge, has been, since the start of the epidemic developing reports with an array of different statistical and mathematical procedures [7], in order to present a clear picture of the evolution of the epidemic, with the objective of supporting public health policy making. Part of this work involved building a SEIR-type model with heterogeneous mixing among age groups. This model was key to provide some evidence on the impact of the lockdown in Portugal from March 22th until May 4th. Using data from google mobility reports [8], the model showed that a decrease in transmission was expect after the implementation of the lockdown, which was not yet noticeable due to the delay between infection and case notification.

With the increase, as of late, of the daily incidence of COVID-19 cases and with the opening of schools, public health decision makers need to know what will be the expected impact on the Portuguese health system, and what non-pharmaceutical-interventions (NPI) can be adapted in order to compensate for such increase. Several epidemiologist state that higher and faster contact tracing might be the best and most efficient measure to compensate for such increase. The team is currently developing a new model that takes into account several NPIs, such as contact tracing, case ascertainment, mask usage, shielding of vulnerable (elderly) individuals, and closure/opening of schools, among others. The main objective is to provide possible scenarios for the magnitude of the impact of these measures.

Joint work with:

• Maria Luísa Morgado, Departamento de Matemática, UTAD & CEMAT IST
• Paula Patrício, Centro de Matemática e Aplicações & Departamento de Matemática Faculdade de Ciências e Tecnologia, Universidade Nova de Lisboa
• Baltazar Nunes, Instituto Nacional de Saúde Doutor Ricardo Jorge

References

1. ECDC: Event Background-COVID-19.
2. Wu, J. T., Leung, K., & Leung, G. M. (2020). Nowcasting and forecasting the potential domestic and international spread of the 2019-nCoV outbreak originating in Wuhan, China: a modelling study. The Lancet, 395(10225), 689–697. doi: 10.1016/s0140-6736(20)30260-9
3. Maugeri A, Barchitta M, Battiato S, Agodi A. Estimation of Unreported Novel Coronavirus (SARS-CoV-2) Infections from Reported Deaths: A Susceptible-Exposed-Infectious-Recovered-Dead Model. J Clin Med. 2020;9(5):1350. Published 2020 May 5. doi:10.3390/jcm9051350
4. Instituto Nacional de Saúde Dr. Ricardo Jorge (2020). Relatório de Apresentação dos Resultados Preliminares do Primeiro Inquérito Serológico Nacional COVID-19. Available: (acesso a 25/08/2020)
5. Huang L-S, Li L, Dunn L, He M. Taking. Account of Asymptomatic Infections in Modeling the Transmission Potential of the COVID-19 Outbreak on the Diamond Princess Cruise Ship. medRxiv. 2020:2020.04.22.20074286.
6. Prem, K., Liu, Y., Russell, T., Kucharski, A. J., Eggo, R. M., Davies, N., Jit, M. Klepac, P. (2020). The effect of control strategies that reduce social mixing on outcomes of the COVID-19 epidemic in Wuhan, China. The Lancet Public Health. doi: 10.1101/2020.03.09.20033050
7. Nunes B, Caetano C, Antunes L, et al. Evolução do número de casos de COVID-19 em Portugal. Relatório de nowcasting. Inst. Nac. Saúde Doutor Ricardo Jorge. 2020;
8. Relatórios de mobilidade da comunidade da COVID19.

Oscillatory behavior of a mixed type difference equation with variable coefficients

In this talk, we present a study on the oscillatory behaviour of the mixed type difference equation with variable coefficients$$\Delta x(n) = \sum_{i=1}^l p_i(n)x(\tau_i(n)) + \sum_{j=1}^m q_j(n)x(\sigma_j(n)),\, n \geq n_0.$$

An alternative stabilization in numerical simulations of Oldrod-B type fluids

The numerical simulation of non-Newtonian viscoelastic fluids flow is a challenging problem. One of the approaches being often adopted to stablize the numerical simulations is based on addition of stress diffusion term into the transport equations for viscoelastic stress tensor. The additional term affect the solution of the problem and special care should be taken to keep the modified model consistent with the original problem.

In this work it was analyzed in detail the influence of numerical stabilization using artificial stress diffusion and it was presented a new arternative. Instead of the classical addition of artificial stress diffusion term it was used the modified additional term which is only present during the transient phase and should vanish in when approaching the stationary case. The steady solution is not affected by such vanishing artificial term, however the stability of the numerical method is improved.

This is joint work with Tomás Bodnár (Institute of Mathematics, Czech Academy of Sciences and Faculty of Mechanical Engineering, Czech Technical University in Prague, Czech Republic).

Exact solution for a Benney-Lin equation type (Gain in Regularity)

In this seminar we will show a exact solution for a Benney-Lin equation type using mainly the Ince Transformation. We establish exact traveling waves solution to the nonlinear evolution equation Benney-Lin type.

Very high-order finite volume schemes with curved domains

Finite volume methods of third or higher order require a specific treatment of the boundary conditions when dealing with a non-polygonal domain that does not exactly fit with the mesh. We also face a similar situation with internal smooth interfaces sharing two subdomains. To address this issue, several technologies have been developed since the 90's such as the isoparamatric elements, the (ghost cells) immersed boundary and the inverse Lax-Wendroff boundary treatment among others. We propose a quick overview of the traditional methods and introduce the new Reconstruction of Off-site Data (ROD) method. Basically, the idea consists, first in definitively distinguishing the computational domain (cells or nodes where the solution is computed) to the physical one and, secondly, in "transporting" the boundary conditions prescribed on the real boundary to the computing domain. To this end, specific local polynomial reconstructions that contains a fingerprint of the boundary conditions are proposed and used to schemes that achieve up to sixth-order of accuracy. Several applications will be proposed in the context of the finite volume (flux reconstruction) and finite difference (ghost cells) for the convection diffusion equation and the Euler system.

Renormalized transport of inertial particles

We study how an imposed fluid flow — laminar or turbulent — modifies the transport properties of inertial particles (e.g. aerosols, droplets or bubbles), namely their terminal velocity, effective diffusivity, and concentration following a point-source emission.

Such quantities are investigated by means of analytical and numerical computations, as functions of the control parameters of both flow and particle; i.e., density ratio, inertia, Brownian diffusivity, gravity (or other external forces), turbulence intensity, compressibility degree, space dimension, and geometric/temporal properties.

The complex interplay between these parameters leads to the following conclusion of interest in the realm of applications: any attempt to model dispersion and sedimentation processes — or equivalently the wind-driven surface transport of floaters — cannot avoid taking into account the full details of the flow field and of the inertial particle.

On the existence of a solution of a class of non-stationary free boundary problems

We consider a class of parabolic free boundary problems with heterogeneous coefficients. We establish existence of a solution for this problem. We use a regularized problem for which we prove existence of a solution by applying the Tychonoff fixed point theorem. Then we pass to the limit to get a solution of our problem.

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