An Efficient Moments-Based Inference Method for Within-Host Bacterial Infection Dynamics

Abstract

Over the last ten years, isogenic tagging (IT) has revolutionised the study of bacterial infection dynamics in laboratory animal models. However, quantitative analysis of IT data has been hindered by the piecemeal development of relevant statistical models. The most promising approach relies on stochastic Markovian models of bacterial population dynamics within and among organs. Here we present an efficient numerical method to fit such stochastic dynamic models to in vivo experimental IT data. A common approach to statistical inference with stochastic dynamic models relies on producing large numbers of simulations, but this remains a slow and inefficient method for all but simple problems, especially when tracking bacteria in multiple locations simultaneously. Instead, we derive and solve the systems of ordinary differential equations for the two lower-order moments of the stochastic variables (mean, variance and covariance). For any given model structure, and assuming linear dynamic rates, we demonstrate how the model parameters can be efficiently and accurately estimated by divergence minimisation. We then apply our method to an experimental dataset and compare the estimates and goodness-of-fit to those obtained by maximum likelihood estimation. While both sets of parameter estimates had overlapping confidence regions, the new method produced lower values for the division and death rates of bacteria: these improved the goodness-of-fit at the second time point at the expense of that of the first time point. This flexible framework can easily be applied to a range of experimental systems. Its computational efficiency paves the way for model comparison and optimal experimental design.\n\nAuthor SummaryRecent advancements in technology have meant that microbiologists are producing vast amounts of experimental data. However, statistical methods by which we can analyse that data, draw informative inference, and test relevant hypotheses, are much needed. Here, we present a new, efficient inference tool for estimating parameters of stochastic models, with a particular focus on models of within-host bacterial dynamics. The method relies on matching the two lower-order moments of the experimental data (i.e., mean, variance and covariance), to the moments from the mathematical model. The method is verified, and particular choices justified, through a number of simulation studies. We then use this method to estimate models that have been previously estimated using a \"gold-standard\" maximum likelihood procedure.\n\nList of symbolsO_LIA: number of animals\nC_LIO_LIT: number of tagged strains\nC_LIO_LIn: number of organs\nC_LIO_LINi: number of bacteria in organ i\nC_LIO_LImij: migration rate from organ i to organ j\nC_LIO_LIki: killing rate in organ i\nC_LIO_LIri: replication rate in organ i\nC_LIO_LI{tau}i: observation time i\nC_LIO_LIA, B, C: matrices\nC_LIO_LI{lambda}: vector of transition rates\nC_LIO_LIB: Number of bootstrap samples\nC_LIO_LI{theta}*: MDE parameter estimate\nC_LI\n\nAbbreviationsO_LIABC: approximate Bayesian computation\nC_LIO_LIIT: isogenic tagging\nC_LIO_LILV: live vaccine\nC_LIO_LIMARE: mean absolute relative error\nC_LIO_LIMDE: minimum divergence estimate\nC_LIO_LIMLE: maximum likelihood estimate\nC_LIO_LIqPCR: quantitative polymerase chain reaction\nC_LIO_LIWITS: wildtype isogenic tagged strain\nC_LI