No abstract

In the context of mathematical models applied to social sciences, we present and analyze a model based on differential equations for the intimate partner violence (IPV). Such a model describes the dynamics of a heterosexual romantic couple in which the man perpetrates violence against the woman. We focus on incorporating different key factors reported in the literature as causal or motivational factors to perpetrate IPV. Among the main factors included are the failures in self-regulation, the man’s need to control the woman, the social pressure on the woman to remain married, and empowerment programs. Another aspect that we include is periodic alcohol consumption for the man. The discussion of the model includes a stability analysis of its equilibrium points and the asymptotic behavior of its solutions. Also, the interpretation of results is presented in terms of IPV phenomenon. Finally, a brief review is given on different scales to quantify human behavioral traits and numerical simulations for some IPV scenarios.

No abstract

We study, from a probabilistic standpoint, first‐order impulsive linear differential equations, where all its parameters (initial condition and coefficients) are absolutely continuous random variables with a joint probability density function. We assume an infinite train of Dirac delta impulse applications at given time instants to control the model output. We take extensive advantage of the random variable transformation method to determine, first, an explicit expression for the probability density of the solution stochastic process, and secondly, of the random sequences for the maxima and minima. From these sequences, we determine the probability of stability of the solution stochastic process. This analysis is extended in the case that the application times are evenly spaced, via a period T, which is assumed to be a random variable. All the theoretical results are illustrated by means of several numerical examples, where we also perform a sensitivity analysis, via Sobol indexes, to highlight those model parameters that most explain the variability of the model response.

<p style='text-indent:20px;'>We study a full randomization of the complete linear differential equation subject to an infinite train of Dirac's delta functions applied at different time instants. The initial condition and coefficients of the differential equation are assumed to be absolutely continuous random variables, while the external or forcing term is a stochastic process. We first approximate the forcing term using the Karhunen-Loève expansion, and then we take advantage of the Random Variable Transformation method to construct a formal approximation of the first probability density function (1-p.d.f.) of the solution. By imposing mild conditions on the model parameters, we prove the convergence of the aforementioned approximation to the exact 1-p.d.f. of the solution. All the theoretical findings are illustrated by means of two examples, where different types of probability distributions are assumed to model parameters.</p>