Castilho, C.; Gondim, J.

The classical concept of Critical Community Size (CCS) as formulated by Bartlett defines the minimum host population required for a pathogen to persist endemically without stochastic extinction. While this framework successfully described directly transmitted childhood infections in relatively isolated populations, it is increasingly inadequate for modern urban systems characterized by strong connectivity between cities. Pathogens circulating in highly connected urban networks can repeatedly re-emerge through spatial reintroduction even when local transmission temporarily fades out. In such systems, persistence is inherently probabilistic and influenced simultaneously by population size, environmental suitability, and network connectivity. In this study, we develop a generalization of the CCS concept, the Empirical Persistence Threshold (EPT), and apply it to three of the main arboviruses circulating in Brazil--dengue, chikungunya, and Zika--over the period 2017-2024. The Empirical Persistence Threshold generalizes the classical notion of critical community size by replacing a single deterministic threshold with a probabilistic, data driven measure. Instead of asking for the minimum population at which persistence is guaranteed, EPT characterizes the lower tail of the population distribution among municipalities that empirically sustain transmission. Using weekly incidence data from thousands of municipalities, we transform temporal incidence series into binary sequences describing the presence or absence of reported transmission. For each municipality, we characterize persistence through the empirical distribution of run lengths of consecutive weeks with reported cases. Distances between run-length distributions are computed using the Wasserstein-1 metric, allowing a geometrically meaningful comparison between persistence profiles, and municipalities are grouped into epidemiological regimes using hierarchical clustering methods. Across all three arboviruses, we identify two robust regimes: one exhibiting sporadic and recurrent epidemic transmission, and the other exhibiting sustained persistent transmission. We then estimate the population scales associated with each persistence regime. The analysis is further extended to evaluate how persistence thresholds vary across climate regimes (Ko{diaeresis}ppen classification) and urban hierarchy levels (REGIC). This framework allows the estimation of probabilistic persistence thresholds analogous to CCS, but adapted to connected urban systems. We define the Empirical Persistence Threshold as lower quantiles of the population distribution among municipalities in the persistent regime, and additionally estimate persistence thresholds based on regime membership probabilities. Results reveal strong interactions between population size, climate, and urban connectivity. Dengue exhibits the lowest persistence thresholds, Zika intermediate thresholds, and chikungunya the highest thresholds. These findings demonstrate that pathogen persistence in modern urban systems cannot be described by a single deterministic population threshold. Instead, persistence emerges from the joint effects of demographic scale, environmental suitability, and network position within metapopulation systems.