Jérôme Coville (INRAE, Avignon)

How to construct travelling wave in homogeneous reaction dispersion equation

Travelling wave solutions are an essential tools for understanding the spreading properties of population dynamics models. The existence and properties of these solutions have been studied in various local and nonlocal semilinear equations.  In all these examples, the dispersal operator can be linked to the generator of a particular Levy process :Brownien, alpha-stable processes, compound Poisson processes, ... .  A natural question to ask is whether such solutions still exist when a generic generator is considered. I will present some recent work with E. Bouin that provides a methodology offering an almost complete answer to this question for symmetric generators of Levy processes.  I will detail our approach when the nonlinearity is bistable and the dispersal operator is purely a nonlocal with a diffuse Levy measure.

Karina Garcia-Martinez (Center of Molecular Immunology)

 How tumors dynamically evolve during interaction with T cells?: Implications for immunotherapies efficacy

 I will present a mathematical model developed in our group to study the evolution of tumors in interaction with the immune response. After a brief biological introduction, I will present the ODE system and the algebraic equations that allowed us to predict the different modes of tumor growth. We were able to characterize the dynamic properties of tumor clones during evolution. Finally, we predicted the impact of tumor evolution on the efficacy of different immunotherapies.

Victoria Hernández Mederos (Instituto de Cibernética, Matemática y Física)

Isogeometric approach to the study of focused ultrasound induced heating in biological tissues

Ultrasound medical treatments are non-invasive, repeatable, and non-toxic methods that can be used to treat muscle, bone, and skin tissue. Focused ultrasound systems for medical therapy must be carefully designed, implemented, and applied to avoid tissue damage. In this talk we discuss a study of the heating of a biological tissue induced by the ultrasonic pulse emitted by a curved transducer. The thermal diffusion effect of the ultrasound is computed solving Helmholtz and Pennes partial differential equations. Isogeometric analysis (IgA) is proposed as an effective tool to compute the acoustic pressure field, solution of Helmholtz equation. In combination with the method of lines, IgA is also used to compute the temperature field, solution of Pennes bioheat equation. Numerical experiments are performed with a biological tissue composed of three layers: skin, fat and muscle and a fixed value of the frequency of the pulse, in order to determine the intensity at which ultrasound effectively stimulates the tissues while avoiding tissue injury, according to the time-temperature combinations recommended in the literature.

Florence Hubert (Aix-Marseille Université)

 Growth Fragmentations Models

Growth fragmentation models are often used in structured population dynamics to model, for example, cell division and polymerization. The classical form of the equation is:

∂ₜρ(t,x) + ∂ₓ(g(x)ρ(t,x)) = -B(x)ρ(t,x) + ∫ₓ^∞ B(y)k(y,x)ρ(t,y) dy, t > 0, x > 0
ρ(t,0) = 0, t > 0
ρ(0,x) = ρ⁰(x), x > 0

The well-posedness of a globally defined solution to this equation as well as its asymptotic behaviour has been widely studied. We will first give some contexts where this model is used. We will then recall the main results (see Gabriel et al. 2021, Perthame). We will also propose extensions of such equations in the context of metastasis spreading and microtubule dynamical instabilities. We will highlight the properties of these models and the remaining challenges.

References

J. A. Cañizo, P. Gabriel, and H. Yoldaş. Spectral gap for the growth-fragmentation equation via Harris's theorem. SIAM J. Math. Anal., Vol.53, No.5, pp.5185-5214, (2021).

N. Hartung, S. Mollard, D. Barbolosi, A. Benabdallah, G. Chapuisat, G. Henry, S. Giacometti, A. Iliadis, J. Ciccolini, C. Faivre, F. Hubert. Mathematical Modeling of tumor growth and metastatic spreading: validation in tumor-bearing mice, Cancer Research 74, p. 6397-6407, 2014.

S. Honoré, F. Hubert, M. Tournus, D. White. A growth-fragmentation approach for modeling microtubule dynamic instability, Bulletin of Mathematical Biology, 81, p. 722–758 (2019).

B. Perthame. Transport equations in biology, Springer.

 

Laura Kanzler (LJLL - Sorbonne Université)

Modelling the evolution of the size-distribution in aquatic ecosystems

Trophic interactions between animals in the ocean were matter of interest since the 1960ies, where it was quickly discovered that the body size of individuals acts as ’master trait’ in food webs of animals, giving rise to emergent distributions of biomass, abundance and production of organisms. We propose and investigate a deterministic jump-growth model, which is given by a kinetic equation for coalescing particles, aiming to capture this emergence phenomenon in aquatic ecosystems. The equation of interest is derived from individual based dynamics governed by a stochastic process. Following the observation of the body mass being the crucial trait in these dynamics it is based on the assumption that binary interactions between individuals in the ecosystem take place: A predator feeding on a prey, which then results in growth of the predator with assimilating a certain (usually very small) amount of its prey’s mass as well as plankton production. Analytical results in various parameter regimes are discussed and numerical simulations underlying these observations are given.

 

Alejandro Lage Castellanos (Universidad de la Habana) 

Inference of hidden processes in first waves of COVID in Cuba

I'll discuss two researches carried in our group at the early onset of COVID in Cuba. They both deal with the possibility of inferring what lies behind the surface of the public data of new cases.

First I'll discuss the inference of the number of undetected cases in Cuba's first waves of COVID. This research, published in Revista Cubana de Matematica, https://arxiv.org/pdf/2008.03332, was inspired by some similar work done in France (L. Roques[1]). The idea is Not to fix SIR models to the public data of cases. Instead, understand that such reports are a stochastic sample from the real epidemic, that evolves as SIR but is not detected. So, we fit two models, one stochastic for the detection process and one mechanistic for the underlying epidemic, to match the public data.

Second, I'll discuss how to treat the  contact tracing of cases in an epidemic as a cascade process. Cuba's public report of cases separated the primary from secondary cases. Notably, there was a much higher number of secondary cases than primary cases. We built a cascade model of contact tracing to try to fit this data and understand the efficiency of the Cuban health system in detecting cases. This approach is similar to results from the group of Ferguson in the early 2000's, and somehow is a rediscovery of their results. 

Stéphane Mischler (Université Paris-Dauphine)

Semigroup techniques and applications to evolution PDE

I will give a short overview about more or less recent semigroup techniques such as the change of the functional space technique, the (principal) spectral mapping theorem, the Weyl's Theorem, the stability under small perturbations theorem, the Doblin-Harris and the Krein-Rutman theorems. I will then show some possible applications to evolution PDEs encountered in a biological context such as the growth-fragmentation equation for proteins dynamics, the Keller-Segel equation and the the runs and tumbles equation for bacteria dynamics and the time elapsed equation, the FitzHugh-Nagumo equation and theVoltage-Conductance kinetic equation for neuron network.

Cinzia Soresina (University of Trento)

Derivation of cross-diffusion models in population dynamics: dichotomy, time-scales, and fast-reaction

In population dynamics, cross-diffusion describes the influence of one species on the diffusion of another. A benchmark problem is the cross-diffusion SKT model, proposed in the context of competing species to account for stable inhomogeneous steady states exhibiting spatial segregation. Even though the reaction part does not present the activator-inhibitor structure, the cross-diffusion terms are the key ingredient for the appearance of spatial patterns [1]. From the modelling perspective, cross-diffusion terms naturally appear in the fast-reaction limit of a ``microscopic'' model (in terms of time scales) presenting only standard diffusion and fast-reaction terms, thus incorporating processes occurring on different time scales [4]. In this talk, recent applications of this approach will be presented, e.g., predator-prey [2] and mutualistic interactions, plant dynamics with autotoxicity effects [3], and epidemiology.

[1] Breden, M., Kuehn, C., Soresina, C. (2021). On the influence of cross-diffusion in pattern formation. Journal of Computational Dynamics 8(2):213--240.
https://doi.org/10.3934/jcd.2021010

[2] Desvillettes, L., Soresina, C. (2019) Non-triangular cross-diffusion systems with predator-prey reaction terms. Ricerche di Matematica 68(1):295--314.
https://doi.org/10.1007/s11587-018-0403-y

[3] Giannino, F., Iuorio, A., Soresina,  C. (in preparation). The effect of auto-toxicity in plant-growth dynamics: a cross-diffusion model.

[4] Kuehn, C., Soresina, C. (2020). Numerical continuation for a fast-reaction system and its cross-diffusion limit. SN Partial Differential Equations and Applications 1:7.
https://doi.org/10.1007/s42985-020-0008-7

Ariane Trescases (CNRS & Institut de Mathématiques de Toulouse)

A viscous multi-tissue model for vertebrate embryo growth

During elongation of the vertebrate embryo, live imaging reveals cellular turbulence in the embryonic tissues. We propose a 2D mechanical model for tissue growth during embryo elongation, which recovers these turbulent movements and addresses the question of the mechanism of segregation between tissues. Having (formally) determined the incompressible limit, we study the qualitative behavior at the limit and discuss a ghost effect, then deduce new biological hypotheses.