A hybrid stochastic-deterministic operator framework for early detection of severe respiratory diseases with pandemic potential

We formulate the early detection of severe respiratory diseases with pandemic potential as a hybrid stochastic-deterministic operator problem. The central object is a transition operator T = D + S, where D is a deterministic component driven by confirmed clinical data and S is a stochastic component driven by probabilistic propagation patterns. We classify twelve early-warning algorithms as discrete realizations of T and evaluate them under the alarm-characterization framework of Daluwatte et al. Using a single sample of sixteen early warnings, we introduce a weighted alert functional f(x_k) = (k - 1) + x_k that maps the alert distribution into a scalar score. The results show that only the hybrid algorithms H1 and H3, together with the deterministic algorithm D4, attain high scores in the "early" category, while no purely stochastic algorithm produces early warnings. We then construct a hybrid operator for a real-time implementation using a transition matrix P derived from sociodemographic risk and symptom severity, alongside a cluster estimator. The work provides a unified operator formalism and a constructive hybrid operator consistent with theoretical predictions. The system is implemented in Python and deployed on mobile platforms, offering a reproducible tool for resource-limited settings.