Estimating the distribution of parameters in differential equations with repeated cross-sectional data

Differential equations are pivotal in modeling and understanding the dynamics of various systems, as they offer insights into their future states through parameter estimation fitted to time series data. In fields such as economy, politics, and biology, the observation data points in the time series are often independently obtained (i.e., Repeated Cross-Sectional (RCS) data). RCS data showed that traditional methods for parameter estimation in differential equations, such as using mean values of RCS data over time, Gaussian Process-based trajectory generation, and Bayesian-based methods, have limitations in estimating the shape of parameter distributions, leading to a significant loss of data information. To address this issue, this study proposes a novel method called Estimation of Parameter Distribution (EPD) that provides accurate distribution of parameters without loss of data information. EPD operates in three main steps: generating synthetic time trajectories by randomly selecting observed values at each time point, estimating parameters of a differential equation that minimizes the discrepancy between these trajectories and the true solution of the equation, and selecting the parameters depending on the scale of discrepancy. We then evaluated the performance of EPD across several models, including exponential growth, logistic population models, and target cell-limited models with delayed virus production, thereby demonstrating the ability of the proposed method in capturing the shape of parameter distributions. Furthermore, we applied EPD to real-world datasets, capturing various shapes of parameter distributions over a normal distribution. These results address the heterogeneity within systems, marking a substantial progression in accurately modeling systems using RCS data. Therefore, EPD marks a significant advancement in accurately modeling systems with RCS data, realizing a deeper understanding of system dynamics and parameter variability.Author summary: Observation data points in biological experiments are often obtained from independent objects over time (i.e., Repeated Cross-Sectional (RCS) data). One method to obtain biological information from RCS data is to design a mathematical model and determine the range of parameters in the model that fits the data correctly. Here, the shape of parameter distributions provides data information, such as heterogeneity in biological phenomena. However, we found that classical estimation methods fail to determine the correct shape of parameter distributions, leading to a significant loss of data information. To address this, we propose a novel method, Estimating Parameter Distribution (EPD), to catch the accurate shape of parameter distributions originating from RCS data. Specifically, EPD involves three steps: First, we generated several artificial time series data from the RCS data. Second, we evaluated the suitability of these artificial data by fitting them to the mathematical model. Finally, we determined the best artificial data that matched the RCS data. The results showed that EPD provided more accurate estimations of distributions for the growth rate of Amyloid beta peptides and the half-saturation constant of the viral population compared to previous research results. Specifically, EPD revealed correct parameter distribution with heterogeneity in parameters.