The frequency-dependent Wright-Fisher model: diffusive and non-diffusive approximations

We study a class of processes that are akin to the Wright-Fisher model, with transition probabilities weighted in terms of the frequency-dependent fitness of the population types. By considering an approximate weak formulation of the discrete problem, we are able to derive a corresponding continuous weak formulation for the probability density. Therefore, we obtain a family of partial differential equations (PDE) for the evolution of the probability density, and which will be an approximation of the discrete process in the joint large population, small time-steps and weak selection limit. If the fitness functions are sufficiently regular, we can recast the weak formulation in a more standard formulation, without any boundary conditions, but supplemented by a number of conservation laws. The equations in this family can be purely diffusive, purely hyperbolic or of convection-diffusion type, with frequency dependent convection. The particular outcome will depend on the assumed scalings. The diffusive equations are of the degenerate type; using a duality approach, we also obtain a frequency dependent version of the Kimura equation without any further assumptions. We also show that the convective approximation is related to the replicator dynamics and provide some estimate of how accurate is the convective approximation, with respect to the convective-diffusion approximation. In particular, we show that the mode, but not the expected value, of the probability distribution is modelled by the replicator dynamics. Some numerical simulations that illustrate the results are also presented

Reward and Punishment Sensitivity in Children with ADHD: Validating the Sensitivity to Punishment and Sensitivity to Reward Questionnaire for Children (SPSRQ-C)

This study validates the Sensitivity to Punishment and Sensitivity to Reward Questionnaire for children (SPSRQ-C), using a Dutch sample of 1234 children between 6–13 years old. Factor analysis determined that a 4-factor and a 5-factor solution were best fitting, explaining 41% and 50% of the variance respectively. The 4-factor model was highly similar to the original SPSRQ factors found in adults (Punishment Sensitivity, Reward Responsivity, Impulsivity/Fun-Seeking, and Drive). The 5-factor model was similar to the 4-factor model, with the exception of a subdivision of the Punishment Sensitivity factor into a factor with ‘social-fear’ items and a factor with ‘anxiety’ items. To determine external validity, scores of three groups of children with attention deficit hyperactivity disorder (ADHD) were compared on the EFA models: ADHD-only (n = 34), ADHD and autism spectrum disorder (ADHD+ASD; n = 22), ADHD and oppositional defiant disorder (ADHD+ODD; n = 22). All ADHD groups scored higher than typical controls on Reward Responsivity and on the ‘anxiety’ factor (n = 75). The ADHD-only and ADHD+ODD group scored higher than other groups on Impulsivity/Fun-Seeking and Drive, while the ADHD+ASD group scored higher on Punishment Sensitivity. The findings emphasize the value of the SPSRQ-C to quickly and reliably assess a child’s sensitivity to reinforcement, with the aim to provide individually-tailored behavioral interventions that utilize reward and reprimands