This expository paper is an introduction to the mathematical modelling of vaccine degradation and to its industrial applications, including the study of vaccine stability and the so called “WHO last mile” program.

We are interested in the system of ion channels present at the membrane of the human red blood cell. The cell, under specific experimental circumstances, presents important variations of its membrane potential coupled to variations of the main ions’ concentration ensuring its homeostasis.

In this collaborative work between biologists and mathematicians a simple mathematical model is designed to explain experimental measurements of membrane potential and ion concentrations. Its construction is presented, as well as illustrative simulations and a calibration of the model on real data measurements. A sensitivity analysis of the model parameters is performed. The impact of blood sample storage on ion permeabilities is discussed.

Contractile force in muscle tissue is produced by myosin molecular motors that bind and pull on specific sites located on surrounding actin filaments. The classical framework to model this active system was set by the landmark works of A.F. Huxley and T.L. Hill. This framework is built on the central assumption that the relevant quantity for the model parametrization is the myosin head reference position. In this paper, we present an alternative formulation that allows to take into account the current position of the myosin head as the main model parameter.

The actin-myosin system is described as a Markov process combining Langevin drift-diffusion and Poisson jumps dynamics. We show that the corresponding system of Stochastic Differential Equation is well-posed and derive its Partial Differential Equation analog in order to obtain the thermodynamic balance laws. We finally show that by applying standard elimination procedures, a modified version of the original Huxley–Hill framework can be obtained as a reduced version of our model. Theoretical results are supported by numerical simulations where the model outputs are compared to benchmark experimental data.

We describe microbial growth and production of value-added chemical compounds in a continuous bioreactor through a dynamical system and we study the local stability of the equilibrium of interest by means of the classical Routh–Hurwitz criterion. The mathematical model considers various biological and structural parameters related to the bioprocess (concentration of substrate inflow, constants of the microchemical reactions, steady-state mass fractions of intracellular proteins, etc.) and thus, the stability condition is given in terms of these parameters. This boils down to deciding the consistency of a system of polynomial inequalities over the reals, which is challenging to solve from an analytical perspective, and out of reach even for traditional computational software designed to solve such problems. We show how to adapt classical techniques for solving polynomial systems to cope with this problem within a few minutes by leveraging its structural properties, thus completing the stability analysis of our model. The paper is accompanied by a Maple worksheet available online.

We study the classic pure selection integrodifferential equation, stemming from adaptative dynamics, in a measure framework by mean of duality approach. After providing a well posedness result under fairly general assumptions, we focus on the asymptotic behaviour of various cases, illustrated by some numerical simulations.

In this paper, we introduce a type switching mechanism for the Contact Process. That is, we allow the individual particles/sites to switch between two (or more) types independently of one another, and the different types may exhibit specific infection and recovery dynamics. Such type switches can e.g. be motivated from biology, where “phenotypic switching” is common among micro-organisms. Our framework includes as special cases systems with switches between “active” and “dormant” states (the Contact Process with dormancy, CPD), and the Contact Process in a randomly evolving environment (CPREE) introduced by Broman (2007). The “standard” multi-type Contact Process (without type-switching) can also be recovered as a limiting case.

After constructing the process from a graphical representation, we first establish basic properties that are mostly analogous to the classical Contact Process. We then provide couplings between several variants of the system, obtaining sufficient conditions for the existence of a phase transition. Further, we investigate the effect of the switching parameters on the critical value of the system by providing rigorous bounds obtained from the coupling arguments as well as numerical and heuristic results. Finally, we investigate scaling limits for the process as the switching parameters tend to 0 (slow switching regime) resp. $\infty$ (fast switching regime). We conclude with a brief discussion of further model variants and questions for future research.

For many years, voltage sensitive dye imaging (VSDI) has enabled the fruitful analysis of neuronal transmission by monitoring the spreading of neuronal signals. Although useful, the display of diffusion of neuronal depolarization provides insufficient information in the quest for a greater understanding of neuronal computation in brain function. Here, we propose the optimal mass transportation theory as a model to describe the dynamics of neuronal activity. More precisely, we use the solution of an $L^2$-Monge–Kantorovich problem to model VSDI data, to extract the velocity and overall orientation of depolarization spreading in anatomically defined brain areas. The main advantage of this approach over earlier models (e.g. optical flow) is that the solution does not rely on intrinsic approximations or on additional arbitrary parameters, as shown from simple signal propagation examples. As proof of concept application of our model, we found that in the mouse hippocampal CA1 network, increasing Schaffers collaterals stimulation intensity leads to an increased VSDI-recorded depolarization associated with dramatic decreases in velocity and divergence of signal spreading. In addition, the pharmacological activation of cannabinoid type 1 receptors (CB1) leads to slight but significant decreases in neuronal depolarization and velocity of signal spreading in a region-specific manner within the CA1, indicating the reliability of the approach to identify subtle changes in circuit activity. Overall, our study introduces a novel approach for the analysis of optical imaging data, potentially highlighting new region-specific features of neuronal networks dynamics.

We establish a connection between two population models by showing that one is the scaling limit of the other, as the population grows large. In the infinite population model, individuals are split into two subpopulations, carrying either a selective advantageous allele, or a disadvantageous one. The proportion of disadvantaged individuals in the population evolves according to the $\Lambda$-Wright–Fisher stochastic differential equation (SDE) with selection, and the genealogy is described by the so-called Bolthausen–Sznitman coalescent. This equation has appeared in the $\Lambda$-lookdown model with selection studied by Bah and Pardoux [1]. Schweinsberg in [16] showed that in a specific setting, due to the strong selection, the genealogy of the so-called Moran model with selection converges to the Bolthausen–Sznitman coalescent. By splitting the population into two adversarial subgroups and adding a weak selection mechanism, we show that the proportion of disadvantaged individuals in the Moran model with strong and weak selections converges to the solution of the $\Lambda$-Wright–Fisher SDE of [1].

The complex transverse water proton magnetization subject to diffusion-encoding magnetic field gradient pulses can be modeled by the Bloch-Torrey partial differential equation (PDE). The associated diffusion MRI signal is the spatial integral of the solution of the Bloch-Torrey PDE. In addition to the signal, the time-dependent apparent diffusion coefficient (ADC) can be obtained from the solution of another partial differential equation, called the HADC model, which was obtained using homogenization techniques.

In this paper, we analyze the Bloch-Torrey PDE and the HADC model in the context of geometrical deformations starting from a canonical configuration. To be more concrete, we focused on two analytically defined deformations: bending and twisting. We derived asymptotic models of the diffusion MRI signal and the ADC where the asymptotic parameter indicates the extent of the geometrical deformation. We compute numerically the first three terms of the asymptotic models and illustrate the effects of the deformations by comparing the diffusion MRI signal and the ADC from the canonical configuration with those of the deformed configuration.

The purpose of this work is to relate the diffusion MRI signal more directly with tissue geometrical parameters.

Sharp prediction of extinction times is needed in biodiversity monitoring and conservation management. The Galton–Watson process is a classical stochastic model for describing population dynamics. Its evolution is like the matrix population model where offspring numbers are random. Extinction probability, extinction time, abundance are well known and given by explicit formulas. In contrast with the deterministic model, it can be applied to small populations. Parameters of this model can be estimated through the Bayesian inference framework. This enables to consider various scenarios. We show how coupling Bayesian inference with the Galton–Watson model provides several features: (i) a flexible modelling approach with easily understandable parameters (ii) compatibility with the classical matrix population model (Leslie type model) (iii) An approach which leads to more information with less computing iv) inclusion of expert or previous knowledge...It can be seen to go one step further than the classical matrix population model for the viability problem. To illustrate these features, we provide analysis details for a real life example with French Pyrenean brown bears.

We introduce a model of parasite infection in a cell population, where cells can be infected, either at birth through maternal transmission, from a contact with the parasites reservoir, or because of the parasites released in the cell medium by infected cells. Inside the cells and between infection events, the quantity of parasites evolves as a general non linear branching process. We study the long time behaviour of the infection.