Sufficient reproduction numbers to prevent recurrent epidemics

Current practice in the design and evaluation of epidemic control measures, including vaccination, is largely based on reproduction numbers (RNs), which represent prognostic indexes of long-term disease transmission, both in naive populations (basic RN) and in the presence of prior exposure or interventions (effective RN). A standard control objective is to establish herd immunity, e.g., by immunizing enough susceptible individuals to achieve RN<1. However, attaining this goal is not sufficient to avoid transient outbreaks that, in the short term, might revamp epidemics by coalescence of subthreshold flare-ups. Using reactivity analysis applied to a discrete SIR model with age-of-infection structure, we determine sufficient conditions to prevent transient epidemic dynamics and recurrent, non-periodic outbreaks due to imported cases. These conditions are based on fundamental infection characteristics, namely the average infectiousness clearance rate, the generation time distribution, and the RN. We show that preventing subthreshold epidemicity requires stricter RN thresholds than simply maintaining RN<1. Taking into account a wide spectrum of respiratory viral infections, epidemicity-curbing RN thresholds vary between 0.10 (rubella) and 0.51 (MERS), with a median of 0.26 close to the estimate of 0.24 for the ancestral SARS-CoV-2 virus. The portion of the population that needs to be included in containment efforts to avoid short-term outbreaks is considerably higher than herd immunity thresholds (HITs) based solely on the basic RN (e.g., 93% vs. 72% for ancestral SARS-CoV-2). We also find that subthreshold epidemicity is harder to prevent for pathogens with a longer mean generation time, smaller standard deviation of the generation time distribution, longer duration of infection, and higher RN. Determining sufficient RN thresholds to prevent transient outbreaks is a key challenge in disease ecology, with 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. Due to its modest data requirements, our modeling framework may also have important implications for human and non-human diseases caused by emerging pathogens.