How do diseases spread at the critical state?

Abstract

The transmission characteristics of infectious diseases near critical thresholds are essential for public health strategy formulation. This study employs reaction–diffusion SI with nonlinear and SIR models with saturated incidence rates, integrating optimal control theory to investigate epidemic propagation trends under critical conditions. The structural complexity of three epidemiological target states (extinction, quasi-uniform epidemic, and patterned epidemic) is quantitatively characterized using spatial entropy methods. A multi-indicator comparative analysis systematically reveals the evolutionary trends of epidemics in critical states from three dimensions: target attainability, average control intensity, and control complexity. The findings indicate that achieving a patterned epidemic state requires the lowest control intensity and spatial intervention complexity compared to extinction and quasi-uniform states, suggesting that epidemic systems in critical states are more inclined toward structured transmission patterns. The proposed framework for quantifying spatial structure and control complexity provides a theoretical basis and practical guidance for formulating spatial prevention and control strategies for infectious diseases in critical states.