Inferring the dynamics of quasi-reaction systems via nonlinear local mean-field approximations

Parameter estimation of kinetic rates in stochastic quasi-reaction systems can be challenging, particularly when the time gap between consecutive measurements is large. Local linear approximation approaches account for the stochasticity in the system but fail to capture the intrinsically nonlinear nature of the mean dynamics of the process. Moreover, the mean dynamics of a quasi-reaction system can be described by a system of ODEs, which have an explicit solution only for simple unitary systems. An approximate analytical solution is derived for generic quasi-reaction systems via a first-order Taylor approximation of the hazard rate. This allows a nonlinear forward prediction of the future dynamics given the current state of the system. Predictions and corresponding observations are embedded in a nonlinear least-squares approach for parameter estimation. The performance of the algorithm is compared to existing methods via a simulation study. Besides the generality of the approach in the specification of the quasi-reaction system and the gains in computational efficiency, the results show an improvement in the kinetic rate estimation, particularly for data observed at large time intervals. Additionally, the availability of an explicit solution makes the method robust to stiffness, which is often present in biological systems. Application to Rhesus Macaque data illustrates the use of the method in the study of cell differentiation.