Optimal prophylactic vaccination in segregated populations: When can we improve on the equalising strategy?

Optimised prophylactic vaccination in metapopulations

A highly effective method for controlling the spread of an infectious disease is vaccination. However, there are many situations where vaccines are in limited supply. The ability to determine, under this constraint, a vaccination strategy which minimises the number of people that become infected over the course of a potential epidemic is essential. Two questions naturally arise: when is it best to allocate vaccines, and to whom should they be allocated? We address these questions in the context of metapopulation models of disease spread. We discover that in practice it is generally optimal to distribute all vaccines prophylactically, rather than withholding until infection is introduced. For small metapopulations, we provide a method for determining the optimal prophylactic allocation. As the optimal strategy becomes computationally intensive to obtain when the population size increases, we detail an approximation method to determine an approximately optimal vaccination scheme. We find that our approximate strategy is consistently at least as good as three strategies reported in the literature across a wide range of parameter values.

Incorporating household structure and demography into models of endemic disease

The spread of infectious diseases is intimately linked with the strength and type of contact between individuals. Multiple observational and modelling studies have highlighted the importance of two forms of social mixing: age structure, where the likelihood of interaction between two individuals is determined by their ages; and household structure, which recognizes the much stronger contacts and hence transmission potential within the family setting. Age structure has been ubiquitous in predictive models of both endemic and epidemic infections, in part due to the ease of assessing someone’s age. By contrast, although household structure is potentially the dominant heterogeneity, it has received less attention, in part due to an absence of the necessary methodology. Here, we develop the modelling framework necessary to predict the behaviour of endemic infections (which necessitates capturing demographic processes) in populations that possess both household and age structure. We compare two childhood infections, with measles-like and mumps-like parameters, and two populations with UK-like and Kenya-like characteristics, which allows us to disentangle the impact of epidemiology and demography. For this high-dimensional model, we predict complex nonlinear dynamics, where the dynamics of within-household outbreaks are tempered by historical waves of infection and the immunity of older individuals.

Dose-Optimal Vaccine Allocation over Multiple Populations

Vaccination is an effective way to prevent an epidemic. It results in immunity for the vaccinated individuals, but it also reduces the infection pressure for unvaccinated people. Thus people may actually escape infection without being vaccinated: the so-called "herd effect." We analytically study the relation between the herd effect and the vaccination fraction for the seminal SIR compartmental model, which consists of a set of differential equations describing the time course of an epidemic. We prove that the herd effect is in general convex-concave in the vaccination fraction and give precise conditions on the epidemic for the convex part to arise. We derive the significant consequences of these structural insights for allocating a limited vaccine stockpile to multiple non-interacting populations. We identify for each population a unique vaccination fraction that is most efficient per dose of vaccine: our dose-optimal coverage. We characterize the solution of the vaccine allocation problem and we show the crucial importance of the dose-optimal coverage. A single dose of vaccine may be a drop in the ocean, but multiple doses together can save a population. To benefit from this, policy makers should select a subset of populations to which the vaccines are allocated. Focusing on a limited number of populations can make a significant difference, whereas allocating equally to all populations would be substantially less effective.

Evaluation of vaccination strategies for SIR epidemics on random networks incorporating household structure

This paper is concerned with the analysis of vaccination strategies in a stochastic susceptible \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rightarrow $$\end{document}→ infected \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\rightarrow $$\end{document}→ removed model for the spread of an epidemic amongst a population of individuals with a random network of social contacts that is also partitioned into households. Under various vaccine action models, we consider both household-based vaccination schemes, in which the way in which individuals are chosen for vaccination depends on the size of the households in which they reside, and acquaintance vaccination, which targets individuals of high degree in the social network. For both types of vaccination scheme, assuming a large population with few initial infectives, we derive a threshold parameter which determines whether or not a large outbreak can occur and also the probability of a large outbreak and the fraction of the population infected by a large outbreak. The performance of these schemes is studied numerically, focusing on the influence of the household size distribution and the degree distribution of the social network. We find that acquaintance vaccination can significantly outperform the best household-based scheme if the degree distribution of the social network is heavy-tailed. For household-based schemes, when the vaccine coverage is insufficient to prevent a major outbreak and the vaccine is imperfect, we find situations in which both the probability and size of a major outbreak under the scheme which minimises the threshold parameter are larger than in the scheme which maximises the threshold parameter.

The most efficient critical vaccination coverage and its equivalence with maximizing the herd effect

'Critical vaccination coverages' are vaccination allocations that result in an effective reproduction ratio of one. In a population with interacting subpopulations there are many different critical vaccination coverages. To find the most efficient critical vaccination coverage, we define the following optimization problem: minimize the required amount of vaccines to obtain an effective reproduction ratio of exactly one. 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). We propose an efficient general approach to solve these optimization problems based on Perron-Frobenius theory. We study two special cases that provide further insight into these optimization problems. First, we derive an efficient algorithm for the case of multiple populations that interact according to separable mixing. In this algorithm the subpopulations are ordered by their ratio of population size to reproduction ratio. Allocating vaccines based on this priority order results in an optimal allocation. Second, we derive an explicit analytic solution for the case of two interacting populations. We apply our solutions in a case study for pre-pandemic vaccination in the initial phase of an influenza pandemic where the entire population is susceptible to the new influenza virus. 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.

Choice of Antiviral Allocation Scheme for Pandemic Influenza Depends on Strain Transmissibility, Delivery Delay and Stockpile Size

Recently, pandemic response has involved the use of antivirals. These antivirals are often allocated to households dynamically throughout the pandemic with the aim being to retard the spread of infection. A drawback of this is that there is a delay until infection is confirmed and antivirals are delivered. Here an alternative allocation scheme is considered, where antivirals are instead preallocated to households at the start of a pandemic, thus reducing this delay. To compare these two schemes, a deterministic approximation to a novel stochastic household model is derived, which allows efficient computation of key quantities such as the expected epidemic final size, expected early growth rate, expected peak size and expected peak time of the epidemic. It is found that the theoretical best choice of allocation scheme depends on strain transmissibility, the delay in delivering antivirals under a dynamic allocation scheme and the stockpile size. A broad summary is that for realistic stockpile sizes, a dynamic allocation scheme is superior with the important exception of the epidemic final size under a severe pandemic scenario. Our results, viewed in conjunction with the practical considerations of implementing a preallocation scheme, support a focus on attempting to reduce the delay in delivering antivirals under a dynamic allocation scheme during a future pandemic.