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?:abstract
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We model the propagation of an infection, in a population, as a simplified age-dependent branching process. We analytically estimate the fraction of population, needed to be infected or immuned, to achieve herd immunity for an infection. We calculate this estimation as a function of the incubation period of the contagion, contact probability among the infected and susceptible population, and the probability of disease transmission from an infected to a susceptible individual. We show how herd immunity is strongly dependent on the incubation period, and it may be extremely difficult to achieve herd immunity in case of large incubation period. We derive the distribution of generation time from basic principles, which, by far, has been assumed in an ad hoc manner in epidemiological studies. We quantify the success probability of quarantine measures before achieving herd immunity, and discuss a novel method for designing effective quarantine measures in the absence of any pharmaceutical interventions. We also compare the effectiveness of an early imposition against a delayed imposition of lockdown, of the same duration, in mitigating infection from a population.
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