Niels G. Becker

The effect of household distribution on transmission and control of highly infectious diseases

Two epidemic threshold parameters are derived for the spread of a highly infectious disease in a community of households, where a household is any group whose members have frequent contacts with each other. It is assumed that the infection of any member of a household results in the infection of all susceptible members of that household. The threshold parameters have simple expressions in terms of the mean household size and the mean and variance of the number of susceptibles per household. They provide a basic reproduction number R0 for the spread of infection from individual to individual and a basic reproduction number RH0 for the spread of infection from household to household. The threshold parameters are used to derive the levels of immunity required for the prevention of major epidemics in the community. They are also used to evaluate various vaccination strategies having the same vaccination coverage. For a community with households of equal size, it is found that random vaccination of individuals is better than immunizing all members of a corresponding fraction of households. In contrast, when households have varying sizes, immunizing all members of large households can be better than a corresponding vaccination coverage of randomly selected individuals. It is illustrated that these threshold parameters can also be used for a community of households with schools or day care centers. In particular, the effectiveness of immunizing all members of a school is quantified.

Statistical studies of infectious disease incidence

Methods for the analysis of data on the incidence of an infectious disease are reviewed, with an emphasis on important objectives that such analyses should address and identifying areas where further work is required. Recent statistical work has adapted methods for constructing estimating functions from martingale theory, methods of data augmentation and methods developed for studying the human immunodeficiency virus–acquired immune deficiency syndrome epidemic. Infectious disease data seem particularly suited to analysis by Markov chain Monte Carlo methods. Epidemic modellers have recently made substantial progress in allowing for community structure and heterogeneity among individuals when studying the requirements for preventing major epidemics. This has stimulated interest in making statistical inferences about crucial parameters from infectious disease data for such community settings.

Optimal vaccination strategies for a community of households

The effectiveness of a vaccination program depends on how the vaccinations are spread over the households of the community. Here we formulate the optimal allocation of vaccinations as a linear programming problem, when the objective is to prevent epidemics with the minimum vaccination coverage. A vaccine efficacy of less than 100%, as is usual in practice, is allowed for. Optimal vaccine allocations attempt to leave the same number of susceptibles in every household if the disease has a very high transmission rate within households. This means that proportionately more individuals need to be vaccinated in larger households if the vaccine efficacy is < 100%. The linear programming formulation can accommodate heterogeneity among individuals of the proportionate mixing form and can also minimize the initial reproduction number for a given achievable vaccination coverage.

Waning Immunity and its effects on vaccination schedules

A relatively comprehensive age-specific transmission model is used to determine the effect of various factors on the optimal vaccination ages in one-dose and two-dose vaccination schedules. Motivated by the situation for measles, the model allows the duration of immunity of newborns to depend on the level of immunity of the mother at the time of the birth and allows for waning immunity as well as boosting of immunity by exposure to the disease. It is found that a significant amount of waning of disease-acquired immunity is plausible when boosting occurs but this is not an important factor in determining optimal vaccination schedules. On the other hand, plausible rates of loss of vaccine-induced immunity can have a substantial effect on the optimal vaccination schedule, particularly when there is no boosting of immunity. For two-dose schedules the optimal vaccination ages depend significantly on the level of vaccination coverage achieved. In the presence of plausible rates of loss of vaccine-induced immunity for measles, it is found that the vaccination coverage required to eradicate the disease is substantially higher than previously suggested.

The effect of heterogeneity on the spread of disease

In the formulations of standard epidemic models it is usually assumed that, as far as the spread of the disease is concerned, the community consists of homogeneous individuals who mix uniformly with one another. This is a simplifying assumption which helps to make the mathematics tractable. Empirical evidence suggests that in real world epidemics there is often variability among individuals. It is therefore important to determine how the introduction of heterogeneity among individuals is likely to affect any conclusions arrived at from consideration of the standard epidemic models.

The effect of random vaccine response on the vaccination coverage required to prevent epidemics.

The response people have to vaccination varies because their immune systems differ and vaccine failures occur. Here we consider the effect that a random response, independent for each vaccinee, has on the vaccination coverage required to prevent epidemics in a large community. For a community of uniformly mixing individuals an explicit expression is found for the critical vaccination coverage (CVC) and the effect of the vaccine response is determined entirely by the mean E(AB), where A and B, respectively, reflect the infectivity and susceptibility of a vaccinated individual. This result shows that the usual concept of vaccine efficacy, which focuses on the amount of protection the vaccine provides the vaccinee against infection, is not adequate to describe the requirements for preventing epidemics when vaccination affect infectivity. The estimation of E(AB) poses a problem because A and B refer to the vaccine response of the same individual. Similar results are found when there are different types of individual, but now the mean E(AB) may differ between types. However, for a community made up of households it is shown that the CVC also depends on other characteristics of the vaccine response distribution. In practice this means that estimating a single measure of vaccine effectiveness is generally not enough to determine the CVC. For a specific community of households it is found that the vaccination coverage required to prevent epidemics decreases as the variation in the vaccine response increases.

Immunization levels for preventing epidemics in a community of households made up of individuals of various types

A method is proposed for computing an epidemic threshold parameter for the spread of a communicable disease in a community of households in which individuals are of p different types. The threshold parameter is the largest eigenvalue of a p x p matrix whose elements depend on the rates of disease transmission between types and the distribution of the household size. More explicit expressions are given for diseases that are highly infectious within households, to the point that the infection of any member of a household results in the infection of all susceptible members of that household. For a variety of vaccination strategies it is described how this approach can be used to determine the level of immunity required to prevent epidemics. A numerical example illustrates the results.

The effect of community structure on the immunity coverage required to prevent epidemics

Estimation of the immunity coverage required to effectively control disease transmission is an important public health problem. Using data on the eventual size of a major epidemic, we compare estimates based on the simplifying assumption that the community consists of uniformly mixing individuals with estimates obtained when the more complex community structure is acknowledged. The alternative community structures considered include households and localities that are quite separate. Several inequalities are established for estimates of the critical immunity coverage. For several settings, the coverage estimated by assuming an oversimplified community structure is found to actually be an underestimate. A serious consequence of this finding is that we may be misled into believing that we have estimated an immunity coverage that can prevent epidemics when it in fact cannot. The conclusion is that the heterogeneity in the community must be taken into account when estimating the critical immunity coverage.

Estimation in Epidemics with Incomplete Observations

The construction of estimating equations by martingale methods is generalized to yield estimators with explicit expressions for the parameters of the birth-and-death and the general epidemic processes when only partial observations are available. (For the birth-and-death process the death process is observed but the number of births is observed only at the end and for the general epidemic process only the removal process is observed.) For large populations, the use of the martingale central limit theorem yields asymptotic confidence regions for the parameters. Explicit expressions are derived for estimators of the variances of the large sample distributions. The range of validity and usefulness of the new estimators is determined by simulation.

On a general stochastic epidemic model.

Abstract A non-Markovian epidemic model is proposed for which a stochastic epidemic threshold theorem, like that by Whittle (1955) for a simpler model, is shown to hold. The threshold theorem can be a useful guide in determining public health measures aimed at preventing major outbreaks of a communicable disease. Bounds are obtained for the mean size of minor epidemics. A parameter of the model, whose value is crucial to applications of these results, is the product of the infection rate and the mean duration of the infectious period. Estimators are suggested for this parameter by identifying martingales associated with counting processes of the epidemic model and by constructing other martingales which can be written as stochastic integrals.