Hardness of Herd Immunity and Success Probability of Quarantine Measures: A Branching Process Approach

Abstract

Herd immunity refers to the collective resistance of a population against the spreading of an infection as an epidemic. Understanding the dependencies of herd immunity on various epidemiological parameters is of immense importance for strategizing control measures against an infection in a population. Using an age-dependent branching process model of infection propagation, we obtain interesting functional dependencies of herd immunity on the incubation period of the contagion, contact rate, and the probability of disease transmission from an infected to a susceptible individual. We show that herd immunity is difficult to achieve in case of a high incubation period of the contagion. We derive a method to quantify the success probabilities of quarantine measures to mitigate infection from a population, before achieving herd immunity. We provide a mechanistic derivation of the distribution of generation time from basic principles, which is of central importance to estimate the reproduction number R0, but has been assumed in an ad hoc manner in epidemiological studies, by far. This derivation of the generation time distribution has the generality to be applied in the study of many other age-dependent branching processes, such as the growth of bacterial colonies, various problems in evolutionary and population biology etc.