Sufficient reproduction numbers to prevent recurrent epidemics

Abstract Current practice in the design and evaluation of control measures in disease ecology and epidemiology, including vaccination, is largely based on reproduction numbers (RNs), which represent prognostic indices of long‐term disease transmission, both in naïve populations (basic RN) and in the presence of prior exposure or infection containment interventions (effective RN). A standard control objective is to establish herd immunity, for example by immunizing enough susceptible individuals to achieve RN < 1. However, achieving this goal may not be sufficient to avoid transient subthreshold outbreaks. Starting from simple metrics originally designed to analyse transient dynamics in population ecology, we propose an analytical framework based on RNs to determine sufficient conditions to prevent transient epidemic dynamics in epidemiological and disease ecology models. Specifically, we consider a general SIR model with age‐of‐infection structure, here applied to respiratory viruses in humans, and a stage‐structured model for tick‐borne zoonoses. We show that preventing transient outbreaks requires stricter RN thresholds than simply maintaining RN < 1. For viral respiratory diseases, epidemicity‐curbing RN thresholds vary between 0.10 (rubella) and 0.51 (MERS). For tick‐borne infections, the RN threshold is <0.3 in each of the pathogen‐tick associations considered. The portion of the population that needs to be included in containment efforts to avoid short‐term outbreaks is considerably larger than herd immunity thresholds (HITs) based solely on the basic RN (e.g. 93% vs. 72% for ancestral SARS‐CoV‐2; 98% vs. 39% for tick‐borne encephalitis virus). The definition of sufficient RN thresholds to prevent transient outbreaks may have significant practical consequences for the design of control measures, as the weaker RN reductions and HITs associated with customary control targets may prove ineffective in preventing potentially recurrent flare‐ups that might degenerate into subthreshold epidemics. Our theoretical framework may thus have important implications for human and non‐human diseases caused by emerging pathogens. Additional applications may be found in population ecology, particularly for the containment of transient outbreaks of invasive species.