N.I.M.R.O.D.

# Model and problem definition

The mathematical definition of the problem is briefly described here.

• M. Prague, D. Commenges, J. Drylewicz and R. Thiebaut, Treatment monitoring of HIV infected patients based on mechanistic models (Biometrics to appear 2012).
• J. Guedj, R. Thiebaut and D. Commenges, Maximum likelihood estimation in dynamical models of HIV, 2007, Biometrics 63(4) 1198-1206.
• M. Prague, J. Guedj, J. drylewicz, R. thiebaut and D. Commenges, NIMROD: A Program for Inference via Normal Approximation of the Posterior in Models with Random effects based on Ordinary Differential Equations, (to appear one day somewhere).

For more information to apply NIMROD to specific problems see page How to implement one's own problems ?

Let us consider an ODE model for a population of subjects. For subject , with , this can be written:

with

where is a vector of individual parameters which appear naturally in the ODE system and have a biological interpretation; is the vector of the K state variables (or compartments). We let to underline that completely determines the trajectories . We assume that and are possibly non-linear functions, twice differentiable with respect to .

Reparametrization of the system allows us to take constraints into account: we introduce one-to-one functions and defined transformed parameters . For instance, biological parameters such as rates can be parametrized using a logarithmic transformation to ensure positivity, or parameters between 0 and 1 (such as probabilities or bioavailability) can be parametrized using a logistic transformation. A mixed effect model for the allows introducing covariates and taking into account the between-subject variations. In this approach, random effects give each subject a different value for a subset of the biological parameters:

where is the intercept and is a vector of (possibly time-dependent) explanatory variables associated with the fixed effects of the biological parameter; is a vector of regression coefficients; $u^i$ is the individual vector of random effects with for .

In practice, we do not observe directly, but we have discrete-time observations of some functions of . We assume that there are known link functions , leading to an additive measurement error model. For , and (the number of observation time for subject ), we observe:

This observation scheme can be complicated by left-censoring due to detection limits We denote by the parameter vector to estimate of length :