Elsevier

Mathematical Biosciences

Volume 191, Issue 1, September 2004, Pages 19-40
Mathematical Biosciences

Stochastic multitype epidemics in a community of households: estimation and form of optimal vaccination schemes

https://doi.org/10.1016/j.mbs.2004.05.001Get rights and content

Abstract

This paper treats a stochastic model for an SIR (susceptible  infective  removed) multitype household epidemic. The community is assumed to be closed, individuals are of different types and each individual belongs to a household. Previously obtained probabilistic and inferential results for the model are used to derive the optimal vaccination scheme. By this is meant the scheme that vaccinates the fewest among all vaccination schemes that reduce the threshold parameter below 1. This is done for the situation where all model parameters are known and also for the case where parameters are estimated from an outbreak in the community prior to vaccination. It is shown that the algorithm which chooses vaccines sequentially, at each step selecting the individual which reduces the threshold parameter the most, is not in general an optimal scheme. As a consequence, explicit characterisation of the optimal scheme is only possible in certain special cases. Two different types of vaccine responses, leaky and all-or-nothing, are considered and compared for the problems mentioned above. The methods are illustrated with some numerical examples.

Introduction

This paper is concerned with SIR (susceptible  infective  removed) epidemic models, describing the spread of an infectious disease in a closed finite community (see, for example, Lefèvre [1] and Andersson and Britton [2]). The effect that vaccination of part of the community has on the fundamental threshold parameter (often referred to as the basic reproduction number R0, see for example, Heesterbeek and Dietz [3]) is studied. Vaccination schemes which reduce this number to below its threshold value of 1 are said to be preventive, since major outbreaks cannot occur in the community once such a vaccination scheme has been launched. A vaccination scheme is said to be optimal if it vaccinates the fewest number of individuals among all preventive vaccination schemes. The main focus of the paper lies in deriving the structure of such optimal vaccination schemes. This is done for a fully stochastic model for a multitype community in which individuals reside in households. The different types of individual have different susceptibilities to the disease and/or different infectivities if infected, and could for example reflect different age-groups, sex and/or health status. The household structure reflects the fact that infection rates between individuals of the same household are higher than infection rates between individuals of different households.

Two models for vaccine response are considered. In the first model, a vaccinated individual is either rendered completely immune or the vaccine has no effect. In the second model, vaccinated individuals have a reduced probability of infection given exposure to infection. These models are defined in Smith et al. [4] and, following Halloran et al. [5], are referred to as all-or-nothing and leaky, respectively.

Ball and Lyne [6] studied the probabilistic behaviour of the multitype households model treated in this paper. In particular, they derived a threshold parameter R (the households model equivalent of R0) that determines whether or not a major outbreak can occur; see also Becker and Hall [7]. Statistical inference for model parameters, based on final outcome data (possibly only for a sample of households in the community) is considered by Ball and Lyne [8]. Ball et al. [9] treat inference procedures for the same kind of data, but now for the threshold parameter R, both before and after vaccination. It is shown that R cannot be estimated consistently. Instead, sharp upper and lower bounds for R are derived, both before and after vaccination, which can be estimated consistently from final outcome data. This investigation is continued here, by determining how to allocate vaccines in an optimal way, i.e. how to select which individuals to vaccinate. This is done both for the case where all model parameters, and hence also R, are known, and for the case where parameters are estimated from final size data. In the latter case, the vaccine allocation which reduces the upper bound of R down to 1 with minimum vaccine coverage is determined.

It is shown that a complex non-linear optimisation problem has to be solved in order to find the optimal vaccination scheme when all parameters are known, except when the between-household transmission parameters satisfy so-called proportionate mixing, in which case the optimal vaccination scheme may be found by solving a linear programming problem. When parameters are estimated, and the upper bound estimate of R must be reduced down to 1 for a vaccination scheme to surely be preventive, the derivation of the optimal vaccination scheme is also a linear programming problem. Thus the vaccination problem with parameter estimation proves simpler than the general known parameters case and can be used to provide bounds on the general problem.

A second observation is that the optimal vaccination scheme vopt, giving the smallest overall vaccination coverage cv, has no explicit form in general. This is in contrast to, for example, the single type household case with all-or-nothing vaccines. In this scenario it has been proven for some special cases, and conjectured to hold in general, that successive vaccinations within the same household yield diminishing reductions in the threshold parameter R, leading to simple characterisations of the optimal vaccination scheme (see Ball and Lyne [10]).

The paper is organised as follows. The stochastic multitype SIR households epidemic model is described in Section 2, where its threshold behaviour is outlined. The threshold parameters following a vaccination scheme, using the two models for vaccine response, are determined and compared. Optimal vaccination schemes and their form are considered in Sections 3, for the case when global mixing is proportionate and all infection rates are known, and in Section 4, for the case when the infection rates need to be estimated from final outcome data. Some numerical examples are given in Section 5 and the paper concludes with a brief discussion in Section 6.

Section snippets

Model

The model under consideration in this paper is that of Ball and Lyne [6] for the spread of an SIR epidemic among a closed, finite population that contains J classes of individuals, labelled 1,2,…,J, and is partitioned into households. Let J={1,2,…,J} and N0={n=(n1,n2,…,nJ)∈ZJ: nj⩾0 (j⩾J), |n|=∑j=1Jnj⩾1}. Suppose that, for nN0, the population contains mn households of category n, where a household of category n contains nj individuals of class j (j∈J). Let m=∑nN0mn denote the total number of

Optimal vaccination schemes, known infection rates

As noted in Section 2.3.1, the main aim of any vaccination scheme is to bring the threshold parameter below one, i.e. to ensure that R(v)⩽1. Therefore, for a given community and a given vaccine response, the vaccination scheme v is said to be preventive (written vP) if the induced threshold parameter satisfies R(v)⩽1. If the vaccine response, or efficacy, ϵ is not large enough, it could be that no vaccination scheme is preventive, i.e. that R(vfull)>1, where vfull corresponds to everyone in

Estimation of local and global infection parameters

When estimating the threshold parameter R(v) associated with any given vaccination scheme, and to design vaccination schemes that prevent global epidemics with minimal vaccination coverage, it is necessary to have estimates of the local and global infection parameters. In the present section these parameters are assumed to be unknown and are to be estimated from data on one previous outbreak in the population. Suppose that the data consists of the final outcome, for a sample of households, of

Numerical examples

The first example is similar to that used in Section 4.2.2. Here the parameters are assumed to be known and the global infection rates take the proportionate mixing form, i.e. λijG=ηiGκjG. The vaccine is perfect, i.e. ϵ=1, so that the leaky and all-or-nothing formulations coincide. There are two classes of individual, i.e. J=2, local mixing is uniform, so λijL=λ (i,j∈J), and the distribution of an infective's infectious period is constant and equal to the unit of time. The population consists

Discussion

This paper considers optimal vaccination schemes for an epidemic model allowing for observable (and hence classifiable) individual heterogeneities as well as mixing heterogeneities caused by the presence of households. In reality there are also unobservable individual heterogeneities and mixing heterogeneities due to other social structures, for example schools and workplaces, which affect the spread of an infectious disease. Still, it is believed that households, in combination with having

Acknowledgements

This research was supported by the UK Engineering and Physical Sciences Research Council (EPSRC), under research grants GR/N09091 and GR/R08292, and by the Royal Swedish Academy of Sciences and the Swedish Research Council.

References (19)

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