Critical Thresholds in Non-Pharmaceutical Interventions for Epidemic Control

Non-pharmaceutical interventions, such as contact tracing and social distancing, are critical for controlling epidemic outbreaks, yet their dynamic interactions remain underexplored. We introduce a probabilistic framework to analyze the synergy between contact tracing speed, quantified by the contact tracing period $\tau$, and the average number of close contacts, $\bar{k}_+$, reflecting social distancing measures. We identify critical thresholds ($R=1$) that separate pandemic and contained phases in the $\bar{k}_{+}-\tau$ plane, validated using high-resolution data from Shenzhen's 2022 Omicron outbreak (1,187 cases, 86,451 contacts). Our findings show that contact tracing alone can contain diseases with $R_0 < 2.12$ (95% CI 2.07-2.16), covering 43.33% of major infectious diseases, while combining with social distancing extends control to $R_0 < 7.82$ (95% CI 7.70-7.93), encompassing 86.67% of pathogens. These results, supported by empirical data, highlight the efficacy of rapid tracing and targeted social distancing as alternatives to mass PCR testing. Our framework offers actionable insights for optimizing NPI strategies, though challenges in scaling to regions with higher tracing miss rates or weaker infrastructure underscore the need for adaptive, data-driven policies.