THE MOST EFFICIENT CRITICAL VACCINATION COVERAGE AND ITS EQUIVALENCE WITH MAXIMIZING THE HERD EFFECT

Purpose: In infectious disease epidemiology the potential of an infectious agent to cause an epidemic is often expressed in terms of the reproduction ratio R, related to the initial growth rate of infected individuals, and the final size (i.e., 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.

Method(s): We study `critical vaccination coverages', which are vaccination allocations that result in R=1. We show that for this threshold the introduction of the disease in a population does not result in an outbreak. In a population with interacting subpopulations there are many different critical vaccination coverages. To find the most efficient one, 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 the problem of 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.

Result(s): We propose an efficient general algorithm based on Perron-Frobenius theory to solve these optimization problems. We study two special cases that provide further insight into these optimization problems: the case of separable mixing and the case of n=2 populations. The case of separable mixing is often studied and assumes that upon transmission from one individual to another the two individuals involved influence transmission independently. For these two special cases we are able to characterize the optimal solution completely. The algorithm for separable mixing provides especially interesting insights: we show that vaccinating according to a very simple priority ordering based on population size and disease parameters results in the optimal allocation. We illustrate our approach to find the optimal allocation in a case study for pre-pandemic vaccination in the initial phase of an impending influenza pandemic.

Conclusion(s): The results of the case study show that using the optimal allocation determined with our solution methods can increase the herd effect by 9 to 26% compared to proposed allocations in literature. Equivalently, our optimal allocation is able to significantly reduce the required vaccine stockpile to attain a reproduction ratio of one.