Simulation and Parametric Inference of a Mixed Effects Model with Stochastic Differential Equations Using the Fokker-Planck Equation Solution

This chapter is concerned with estimation method for multidimensional and nonlinear dynamical models including stochastic differential equations containing random effects (random parameters). This type of model has proved useful for describing continuous random processes, for distinguishing intra- and interindividual variability as well as for accounting for uncertainty in the dynamic model itself. Pharmacokinetic/pharmacodynamic modeling often involves repeated measurements on a series of experimental units, and random effects are incorporated into the model to simulate the individual behavior in the entire population. Unfortunately, the estimation of this kind of models could involve some difficulties, because in most cases, the transition density of the diffusion process given the random effects is not available. In this work, we focus on the approximation of the transition density of a such process in a closed form in order to obtain parameter estimates in this kind of model, using the Fokker-Planck equation and the Risken approximation. In addition, the chapter discusses a simulation study using Markov Chain Monte Carlo simulation, to provide results of the proposed methodology and to illustrate an application of mixed effects models with SDEs in the epidemiology using the minimal model describing glucose-insulin kinetics