The most efficient critical vaccination coverage and its equivalence with maximizing the herd effect
In infectious disease epidemiology the severity of an infectious disease is often expressed in terms of the reproduction ratio R, related to the initial growth rate of infected individuals, and the final size (the eventual number of people that have become infected). Although the two measures are related, there is no obvious connection between minimization of the two. In this paper we establish a connection between these measures.
We show that for the threshold R = 1 the introduction of the disease in a population does not result in an outbreak. `Critical vaccination coverages' are vaccination allocations that result in this threshold. To find the most efficient critical vaccination coverage, we define the following optimization problem: minimize the required amount of vaccines to obtain R=1. We prove that this optimization problem is equivalent to maximizing the proportion of susceptibles that escape infection during an epidemic, i.e., maximizing the herd effect. This herd effect is directly related to the final size of an outbreak.
We propose an efficient general algorithm to solve these optimization problems based on Perron-Frobenius theory. We study two special cases that provide further insight into these optimization problems. Finally, we apply our solutions in a case study for pre-pandemic vaccination in the initial phase of an influenza pandemic. The results show that for the optimal allocation the critical vaccination coverage is achieved for a much smaller amount of vaccines as compared to allocations proposed previously.
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