High resolution finite difference schemes for a size structured coagulation-fragmentation model in the space of radon measures.

In this paper, we develop explicit and semi-implicit second-order high-resolution finite difference schemes for a structured coagulation-fragmentation model formulated on the space of Radon measures. We prove the convergence of each of the two schemes to the unique weak solution of the model. We perform numerical simulations to demonstrate that the second order accuracy in the Bounded-Lipschitz norm is achieved by both schemes.

Coupling Epidemiological Models with Social Dynamics.

In this work we study a Susceptible-Infected-Susceptible model coupled with a continuous opinion dynamics model. We assume that each individual can take measures to reduce the probability of contagion, and the level of effort each agent applies can change due to social interactions. We propose simple rules to model the propagation of behaviors that modify the level of effort, and analyze their impact on the dynamics of the disease. We derive a two dimensional set of ordinary differential equations describing the dynamic of the proportion of the number of infected individuals and the mean value of the effort parameter, and analyze the equilibria of the system. The stability of the endemic phase and disease free equilibria depends only on the mean value of the levels of efforts, and not on the initial distribution of levels of effort.

A model for the competition between political mono-polarization and bi-polarization.

We investigate the phenomena of political bi-polarization in a population of interacting agents by means of a generalized version of the model introduced by Vazquez et al. [Phys. Rev. E 101, 012101 (2020)] for the dynamics of voting intention. Each agent has a propensity p in [0,1] to vote for one of two political candidates. In an iteration step, two randomly chosen agents i and j with respective propensities pi and pj interact, and then pi either increases by an amount h>0 with a probability that is a nonlinear function of pi and pj or decreases by h with the complementary probability. We assume that each agent can interact with any other agent (all-to-all interactions). We study the behavior of the system under variations of a parameter q≥0 that measures the nonlinearity of the propensity update rule. We focus on the stability properties of the two distinct stationary states: mono-polarization in which all agents share the same extreme propensity (0 or 1), and bi-polarization where the population is divided into two groups with opposite and extreme propensities. We find that the bi-polarized state is stable for qqc, where qc(h) is a transition value that decreases as h decreases. We develop a rate equation approach whose stability analysis reveals that qc vanishes when h becomes infinitesimally small. This result is supported by the analysis of a transport equation derived in the continuum h→0 limit. We also show by Monte Carlo simulations that the mean time τ to reach mono-polarization in a system of size N scales as τ∼Nα at qc , where α is a nonuniversal exponent that depends on h.

Role of voting intention in public opinion polarization.

We introduce and study a simple model for the dynamics of voting intention in a population of agents that have to choose between two candidates. The level of indecision of a given agent is modeled by its propensity to vote for one of the two alternatives, represented by a variable p∈[0,1]. When an agent i interacts with another agent j with propensity p_{j}, then i either increases its propensity p_{i} by h with probability P_{ij}=ωp_{i}+(1-ω)p_{j}, or decreases p_{i} by h with probability 1-P_{ij}, where h is a fixed step. We assume that the interactions form a complete graph, where each agent can interact with any other agent. We analyze the system by a rate equation approach and contrast the results with Monte Carlo simulations. We find that the dynamics of propensities depends on the weight ω that an agent assigns to its own propensity. When all the weight is assigned to the interacting partner (ω=0), agents' propensities are quickly driven to one of the extreme values p=0 or p=1, until an extremist absorbing consensus is achieved. However, for ω>0 the system first reaches a quasistationary state of symmetric polarization where the distribution of propensities has the shape of an inverted Gaussian with a minimum at the center p=1/2 and two maxima at the extreme values p=0,1, until the symmetry is broken and the system is driven to an extremist consensus. A linear stability analysis shows that the lifetime of the polarized state, estimated by the mean consensus time τ, diverges as τ∼(1-ω)^{-2}lnN when ω approaches 1, where N is the system size. Finally, a continuous approximation allows us to derive a transport equation whose convection term is compatible with a drift of particles from the center toward the extremes.

Sensitivity equations for measure-valued solutions to transport equations.

We consider the following transport equation in the space of bounded, nonnegative Radon measures $\mathcal{M}^+(\mathbb{R})$:$$ ∂_t\mu_t + ∂_x(v(x) \mu_t) = 0.$$We study the sensitivity of the solution $\mu_t$ with respect to a perturbation in the vector field, $v(x)$. In particular, we replace the vector field $v$ with a perturbation of the form $v^h = v_0(x) + h v_1(x)$ and let $\mu^h_t$ be the solution of $$ ∂_t\mu^h_t + ∂_x(v^h(x)\mu^h_t) = 0.$$We derive a partial differential equation that is satisfied by the derivative of $\mu^h_t$ with respect to $h$, $∂artial_h(\mu_t^h)$. We show that this equation has a unique very weak solution on the space $Z$, being the closure of $\mathcal{M}(\mathbb{R})$ endowed with the dual norm $(C^{1,\alpha}(\mathbb{R}))^*$. We also extend the result to the nonlinear case where the vector field depends on $\mu_t$, i.e., $v=v[\mu_t](x)$.

Finite difference schemes for a structured population model in the space of measures.

We present two finite-difference methods for approximating solutions to a structured population model in the space of non-negative Radon Measures. The first method is a first-order upwind-based scheme and the second is high-resolution method of second-order. We prove that the two schemes converge to the solution in the Bounded-Lipschitz norm. Several numerical examples demonstrating the order of convergence and behavior of the schemes around singularities are provided. In particular, these numerical results show that for smooth solutions the upwind and high-resolution methods provide a first-order and a second-order approximation, respectively. Furthermore, for singular solutions the second-order high-resolution method is superior to the first-order method.

In situ synchrotron wide-angle X-ray diffraction investigation of fatigue cracks in natural rubber.

Natural rubber exhibits remarkable mechanical fatigue properties usually attributed to strain-induced crystallization. To investigate this phenomenon, an original experimental set-up that couples synchrotron radiation with a homemade fatigue machine has been developed. Diffraction-pattern recording is synchronized with cyclic loading in order to obtain spatial distributions of crystallinity in the sample at prescribed times of the mechanical cycles. Then, real-time measurement of crystallinity is permitted during uninterrupted fatigue experiments. First results demonstrate the relevance of the method: the set-up is successfully used to measure the crystallinity distribution around a fatigue crack tip in a carbon black filled natural rubber for different loading conditions.

Sensitivity analysis for the EEG forward problem.

Sensitivity analysis can provide useful information when one is interested in identifying the parameter θ of a system since it measures the variations of the output u when θ changes. In the literature two different sensitivity functions are frequently used: the traditional sensitivity functions (TSF) and the generalized sensitivity functions (GSF). They can help to determine the time instants where the output of a dynamical system has more information about the value of its parameters in order to carry on an estimation process. Both functions were considered by some authors who compared their results for different dynamical systems (see Banks and Bihari, 2001; Kappel and Batzel, 2006; Banks et al., 2008). In this work we apply the TSF and the GSF to analyze the sensitivity of the 3D Poisson-type equation with interfaces of the forward problem of electroencephalography. In a simple model where we consider the head as a volume consisting of nested homogeneous sets, we establish the differential equations that correspond to TSF with respect to the value of the conductivity of the different tissues and deduce the corresponding integral equations. Afterward we compute the GSF for the same model. We perform some numerical experiments for both types of sensitivity functions and compare the results.