Estimation of mixed-effects ordinary differential equation models linear in the parameters

We propose a general framework for estimating fixed and random effects in ordinary differential equation (ODE) models that are linear in the parameters, accommodating single-level, nested hierarchical, and crossed random-effect structures. The method-Direct Integral Mixed-Effects (DIME)-exploits the separability of parameters and states to reformulate the problem within a linear mixed-effects model framework. This enables the use of standard inference tools, including confidence intervals and model selection. We provide theoretical guarantees of consistency and asymptotic normality. By bridging nonlinear dynamics and linear mixed-effects model methodology, DIME extends the scope of mixed-effects ODE modeling to complex hierarchical data structures, enhancing accessibility of statistical inference for a broad class of dynamical systems. Monte Carlo simulations compare DIME to established nonlinear mixed-effects approaches implemented in nlme and nlmixr2 R packages. Across varied sample sizes, noise levels, and random-effect structures, DIME exhibits competitive bias, RMSE, and superior coverage probabilities in many scenarios, particularly for variance component estimation with limited data, and remains applicable when competing methods cannot be used. Applications to real-world datasets demonstrate the method's flexibility. For population growth data from 43 countries, DIME recovers exponential growth rates consistent with demographic studies. In modeling joint dynamics of atmospheric pressure and wind speed for 55 U.S. cities, it identifies an oscillatory relationship and supports hierarchical nesting of city effects within periods, outperforming alternative random-effect configurations.