215. Application of differential equation models in the study of progression patterns of schizophrenia

Abstract Background The course of schizophrenia exhibits complex nonlinear dynamic progression characteristics, making it difficult to accurately capture the interaction patterns over time through traditional cross-sectional studies or static descriptions. Differential equations, as a powerful mathematical tool for describing continuous dynamical systems, offer a new approach to quantifying such dynamic processes. Although they have been applied in other disease domains, their use in refined modeling of the longitudinal course of schizophrenia remains insufficient, particularly in integrating multi-dimensional clinical indicators into dynamic models. Therefore, this study aims to develop a multi-dimensional differential equation model that integrates clinical indicators to uncover the interaction mechanisms among symptoms and simulate individualized disease trajectories, thereby providing a novel pathway for exploring disease progression patterns and enabling early intervention. Methods The study employed a longitudinal design combining retrospective and prospective elements. First, multi-timepoint follow-up data spanning three years were collected from 325 patients with schizophrenia across two specialized psychiatric hospitals. Core variables included Positive and Negative Syndrome Scale (PANSS) scores, MATRICS Consensus Cognitive Battery (MCCB) scores, and Personal and Social Performance scale (PSP) scores. Based on this data, a nonlinear dynamical system model was constructed. This model incorporated three core state variables (positive symptom severity, negative symptom severity, and cognitive function level) and characterized their mutual inhibitory or facilitative relationships through coupling terms. Model parameters were individually fitted using the maximum likelihood estimation method. Subsequently, data from the first two years were used to train the model, and data from the third year were used for external validation of the model's predictive efficacy. Model goodness-of-fit was evaluated using Root Mean Square Error (RMSE) and the coefficient of determination (R2). Results The differential equation model successfully fitted the longitudinal data for most patients. On the training set, the model achieved average R2 values of 0.78, 0.72, and 0.69 for the PANSS total score, negative symptom subscore, and MCCB total score, respectively. In predictive validation, the model's RMSE for PSP scores at 6 months and 12 months in the future were 8.4 points and 11.2 points, respectively. Its predictive accuracy was significantly superior to that of traditional linear regression models (p<.01). Parameter analysis revealed that the parameter θ, representing the "inhibitory strength of negative symptoms on cognitive function," showed a significant negative correlation with patients' final social functional outcomes (r = -0.53, p<.001). This suggests that this parameter may serve as a potential biomarker for predicting the risk of long-term functional decline. Discussion The study demonstrates that differential equation models can effectively simulate the dynamic process of multi-dimensional symptom interactions in schizophrenia and provide short-term disease course predictions with reasonable accuracy. This model translates complex clinical observations into a quantifiable dynamic system, aiding in identifying common dynamical principles underlying different progression patterns. It provides a theoretical framework for model-based individualized prognosis assessment and intervention timing selection. Future research needs to further validate the model's robustness in larger samples and over longer time spans, and explore the integration of neurobiological indicators to construct a more comprehensive "pathophysiological-clinical phenotype" computational model.