Abstract
This study primarily addresses the analytical problem of the mathematical mechanism underlying the associative impact between online searches and vaccine uptake, a relationship that has become increasingly relevant in the context of public health management. As internet search behaviors reflect public interest and sentiment, understanding their impact on vaccination trends is crucial for real-time health decision-making. A Logistic model is constructed to observe the fundamental evolutionary patterns between online searches and vaccine uptake. To explore their mutual influence, an impact function is defined, and the common structural factors with the highest fitness are determined through data fitting. Subsequently, a dynamic detection model of the associative impact between online data and societal objects, based on the mathematical mechanism, is established. Using this model, dynamic predictions are conducted to verify its predictive capability at certain stages. Through research, a symbiotic effect between online searches and vaccine uptake is identified, revealing a nonlinear correlation between the two. The model demonstrates the ability to predict vaccine uptake trends based on online search data, with certain prediction windows showing high accuracy. This research not only clarifies the mathematical mechanism underlying this relationship but also demonstrates the advantage of integrated analysis and prediction. It provides a new method for predicting online searches and vaccine uptake, offering theoretical and empirical support for public health and social science research.