A nonnegative tensor has nonnegative rank at most 2 if and only if it is supermodular and has flattening rank at most 2. We prove this result, then explore the semialgebraic geometry of the general Markov model on phylogenetic trees with binary states, and comment on possible extensions to tensors of higher rank.

Tensors of Nonnegative Rank Two

We give the basic definitions and some theoretical results about hyperdeterminants, introduced by A Cayley in 1845. We prove integrability (understood as 4D consistency) of a nonlinear difference equation defined by the 2 × 2 × 2-hyperdeterminant. This result gives rise to the following hypothesis: the difference equations defined by hyperdeterminants of any size are integrable. We show that this hypothesis already fails in the case of the 2 × 2 × 2 × 2-hyperdeterminant.

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Hyperdeterminants as integrable discrete systems

Tensors of nonnegative rank two

INTRODUCTION Vaccine coverage data are typically collected through vaccine registers and retrospective surveys. Alternatively, cross-sectional serosurveys enable direct estimation of vaccine coverage from antibody prevalence by exploiting correlated seropositivity for multi-antigen vaccines. Here, we extend previous methods by accounting for temporal antibody decline in estimating vaccine coverage for measles-mumps-rubella vaccine using serial serosurvey data. METHODS We introduce a Markovian cohort model of antibody waning and boosting applied to dichotomous seropositivity data for measles, mumps, and rubella. Simulation studies are used to test model identifiability and to explore bias induced by previous methods that ignore waning. The cohort model is then fitted to three Australian serosurveys, entailing estimates of vaccine coverage from routine and catch-up vaccination as well as waning rates for each antigen. RESULTS The simulation results show that the cohort model is identifiable and qualitatively captures the decline in seropositivity observed in older children. When fitted to all three Australian surveys, the estimated seroconversion and waning parameters are similar to estimates based on recent meta-analyses, whereas the coverage estimates appear consistent with previous Australian survey-based estimates. DISCUSSION We show that previous methods of estimating coverage from serological data can be improved by fitting a cohort model with waning and boosting processes to serial serosurvey data, furthermore yielding estimates of more parameters of interest such as rates of waning. In settings where serial serosurvey data is available, our method could be duplicated or applied to related questions such as coverage in routine two-dose schedules or from other combination vaccines.

Estimating vaccine coverage from serial trivariate serologic data in the presence of waning immunity.

The effectiveness of childhood immunization programs depends on the vaccination coverage actually achieved. Routinely collected coverage data are not always available, and comparability between countries is often compromised because of different data collection methods. In 2000, Gay developed a method to estimate trivalent vaccination coverage from readily available trivariate serological data on the basis of parametric assumptions related to the rate of seroconversion for each vaccine component and probabilities of natural exposure to infection. Gay's work was indirectly published in a paper by Altmann and Altmann, who derived exact solutions for the parameters on the basis of Gay's modeling equations. In this paper, we propose a general likelihood-based marginal model framework to extend Gay's model by relaxing two of its main assumptions. We use the Bahadur model for trivariate binary data to explicitly account for an association between the disease-specific exposure probabilities. We fit several correlation structures to measles, mumps, and rubella serology from Belgium and Ireland. For both countries, we estimate a small positive pairwise exposure correlation, which improves the fit to the data. However, the effect on the estimated vaccination coverage and its associated variability is fairly moderate. For both Belgium and Ireland, all models reveal that the vaccination coverage achieved during the first 15 years since the introduction of measles, mumps, and rubella immunization is insufficient to eliminate measles. Copyright © 2012 John Wiley & Sons, Ltd.

Estimating vaccination coverage for the trivalent measles-mumps-rubella vaccine from trivariate serological data.

The approach of N. Gay for estimating the coverage of a multivalent vaccine from antibody prevalence data in certain age cohorts is improved by using computer aided elimination theory of variables. Hereby, Gay's usage of numerical approximation can be replaced by exact formulas which are surprisingly nice, too.

Estimating Vaccine Coverage by Using Computer Algebra

The approach of N Gay for estimating the coverage of a multivalent vaccine from antibody prevalence data in certain age cohorts is complemented by using computer aided elimination theory of variables. Hereby, Gay's usage of numerical approximation can be replaced by exact formulae which are surprisingly nice, too.

Klaus Altmann

Papers

Estimating Vaccine Coverage by Using Computer Algebra

Estimating vaccine coverage by using computer algebra

Estimating vaccine coverage by using computer algebra