Maximum likelihood inference for multivariate delay differential equation models

The maximum likelihood inference framework for delay differential equation models in the multivariate settings is developed. The number of delay parameters is assumed to be one or more. This study does not make any restrictive assumptions on the form of the underlying delay differential equations which was one of the limitations of some of the previous work. Thus, the maximum likelihood inference framework can be applied to general delay differential equation models with multiple delay parameters. To obtain the maximum likelihood estimator and estimate of the information matrix, two numerical algorithms are developed: (i) the adaptive grid and (ii) the gradient descent algorithms. Two examples of multivariate delay differential equation models related to the epidemic and pharmacokinetic models, respectively, are presented in this paper. For the unknown parameters, standard errors and confidence intervals are constructed, and formulas and techniques for producing the information matrix are developed. The code and computations are developed with the help of the mathematical software MATLAB.