Making Predictions Using Poorly Identified Mathematical Models.

Many commonly used mathematical models in the field of mathematical biology involve challenges of parameter non-identifiability. Practical non-identifiability, where the quality and quantity of data does not provide sufficiently precise parameter estimates is often encountered, even with relatively simple models. In particular, the situation where some parameters are identifiable and others are not is often encountered.

The impact of Covid-19 vaccination in Aotearoa New Zealand: A modelling study.

Aotearoa New Zealand implemented a Covid-19 elimination strategy in 2020 and 2021, which enabled a large majority of the population to be vaccinated before being exposed to the virus. This strategy delivered one of the lowest pandemic mortality rates in the world. However, quantitative estimates of the population-level health benefits of vaccination are lacking.

Implementing measurement error models with mechanistic mathematical models in a likelihood-based framework for estimation, identifiability analysis and prediction in the life sciences.

Throughout the life sciences, we routinely seek to interpret measurements and observations using parametrized mechanistic mathematical models. A fundamental and often overlooked choice in this approach involves relating the solution of a mathematical model with noisy and incomplete measurement data. This is often achieved by assuming that the data are noisy measurements of the solution of a deterministic mathematical model, and that measurement errors are additive and normally distributed.

Modelling count data with partial differential equation models in biology.

Partial differential equation (PDE) models are often used to study biological phenomena involving movement-birth-death processes, including ecological population dynamics and the invasion of populations of biological cells. Count data, by definition, is non-negative, and count data relating to biological populations is often bounded above by some carrying capacity that arises through biological competition for space or nutrients. Parameter estimation, parameter identifiability, and making model predictions usually involves working with a measurement error model that explicitly relating experimental measurements with the solution of a mathematical model.

Near-term forecasting of Covid-19 cases and hospitalisations in Aotearoa New Zealand.

Near-term forecasting of infectious disease incidence and consequent demand for acute healthcare services can support capacity planning and public health responses. Despite well-developed scenario modelling to support the Covid-19 response, Aotearoa New Zealand lacks advanced infectious disease forecasting capacity. We develop a model using Aotearoa New Zealand's unique Covid-19 data streams to predict reported Covid-19 cases, hospital admissions and hospital occupancy.

Profile-Wise Analysis: A profile likelihood-based workflow for identifiability analysis, estimation, and prediction with mechanistic mathematical models.

Interpreting data using mechanistic mathematical models provides a foundation for discovery and decision-making in all areas of science and engineering. Developing mechanistic insight by combining mathematical models and experimental data is especially critical in mathematical biology as new data and new types of data are collected and reported. Key steps in using mechanistic mathematical models to interpret data include: (i) identifiability analysis; (ii) parameter estimation; and (iii) model prediction.

Likelihood-based estimation and prediction for a measles outbreak in Samoa.

Prediction of the progression of an infectious disease outbreak is important for planning and coordinating a response. Differential equations are often used to model an epidemic outbreak's behaviour but are challenging to parameterise. Furthermore, these models can suffer from misspecification, which biases predictions and parameter estimates.

Modelling Aotearoa New Zealand's COVID-19 protection framework and the transition away from the elimination strategy.

For the first 18 months of the COVID-19 pandemic, New Zealand used an elimination strategy to suppress community transmission of SARS-CoV-2 to zero or very low levels. In late 2021, high vaccine coverage enabled the country to transition away from the elimination strategy to a mitigation strategy. However, given negligible levels of immunity from prior infection, this required careful planning and an effective public health response to avoid uncontrolled outbreaks and unmanageable health impacts.

Modelling the impact of the Omicron BA.5 subvariant in New Zealand.

New Zealand experienced a wave of the Omicron variant of SARS-CoV-2 in early 2022, which occurred against a backdrop of high two-dose vaccination rates, ongoing roll-out of boosters and paediatric doses, and negligible levels of prior infection. New Omicron subvariants have subsequently emerged with a significant growth advantage over the previously dominant BA.2.

Computationally efficient framework for diagnosing, understanding and predicting biphasic population growth.

Throughout the life sciences, biological populations undergo multiple phases of growth, often referred to as for the commonly encountered situation involving two phases. Biphasic population growth occurs over a massive range of spatial and temporal scales, ranging from microscopic growth of tumours over several days, to decades-long regrowth of corals in coral reefs that can extend for hundreds of kilometres. Different mathematical models and statistical methods are used to diagnose, understand and predict biphasic growth.

Profile likelihood-based parameter and predictive interval analysis guides model choice for ecological population dynamics.

Calibrating mathematical models to describe ecological data provides important insight via parameter estimation that is not possible from analysing data alone. When we undertake a mathematical modelling study of ecological or biological data, we must deal with the trade-off between data availability and model complexity. Dealing with the nexus between data availability and model complexity is an ongoing challenge in mathematical modelling, particularly in mathematical biology and mathematical ecology where data collection is often not standardised, and more broad questions about model selection remain relatively open.

Using mechanistic model-based inference to understand and project epidemic dynamics with time-varying contact and vaccination rates.

Epidemiological models range in complexity from relatively simple statistical models that make minimal assumptions about the variables driving epidemic dynamics to more mechanistic models that include effects such as vaccine-derived and infection-derived immunity, population structure and heterogeneity. The former are often fitted to data in real-time and used for short-term forecasting, while the latter are more suitable for comparing longer-term scenarios under differing assumptions about control measures or other factors. Here, we present a mechanistic model of intermediate complexity that can be fitted to data in real-time but is also suitable for investigating longer-term dynamics.

Modelling the dynamics of infection, waning of immunity and re-infection with the Omicron variant of SARS-CoV-2 in Aotearoa New Zealand.

Aotearoa New Zealand experienced a wave of the Omicron variant of SARS-CoV-2 in 2022 with around 200 confirmed cases per 1000 people between January and May. Waning of infection-derived immunity means people become increasingly susceptible to re-infection with SARS-CoV-2 over time. We investigated a model that included waning of vaccine-derived and infection-derived immunity under scenarios representing different levels of behavioural change relative to the first Omicron wave.

Real-time estimation of the effective reproduction number of SARS-CoV-2 in Aotearoa New Zealand.

During an epidemic, real-time estimation of the effective reproduction number supports decision makers to introduce timely and effective public health measures. We estimate the time-varying effective reproduction number, , during Aotearoa New Zealand's August 2021 outbreak of the Delta variant of SARS-CoV-2, by fitting the publicly available EpiNow2 model to New Zealand case data. While we do not explicitly model non-pharmaceutical interventions or vaccination coverage, these two factors were the leading drivers of variation in transmission in this period and we describe how changes in these factors coincided with changes in .

Sensitivity of Reverse Transcription Polymerase Chain Reaction Tests for Severe Acute Respiratory Syndrome Coronavirus 2 Through Time.

Background: Reverse transcription polymerase chain reaction (RT-PCR) tests are the gold standard for detecting recent infection with severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). Reverse transcription PCR sensitivity varies over the course of an individual's infection, related to changes in viral load. Differences in testing methods, and individual-level variables such as age, may also affect sensitivity.

Reliable and efficient parameter estimation using approximate continuum limit descriptions of stochastic models.

Stochastic individual-based mathematical models are attractive for modelling biological phenomena because they naturally capture the stochasticity and variability that is often evident in biological data. Such models also allow us to track the motion of individuals within the population of interest. Unfortunately, capturing this microscopic detail means that simulation and parameter inference can become computationally expensive.

An assessment of the potential impact of the Omicron variant of SARS-CoV-2 in Aotearoa New Zealand.

New Zealand delayed the introduction of the Omicron variant of SARS-CoV-2 into the community by the continued use of strict border controls through to January 2022. This allowed time for vaccination rates to increase and the roll out of third doses of the vaccine (boosters) to begin. It also meant more data on the characteristics of Omicron became available prior to the first cases of community transmission.

Parameter identifiability and model selection for sigmoid population growth models.

Sigmoid growth models, such as the logistic, Gompertz and Richards' models, are widely used to study population dynamics ranging from microscopic populations of cancer cells, to continental-scale human populations. Fundamental questions about model selection and parameter estimation are critical if these models are to be used to make practical inferences. However, the question of parameter identifiability - whether a data set contains sufficient information to give unique or sufficiently precise parameter estimates - is often overlooked.

Model-based data analysis of tissue growth in thin 3D printed scaffolds.

Tissue growth in three-dimensional (3D) printed scaffolds enables exploration and control of cell behaviour in more biologically realistic geometries than that allowed by traditional 2D cell culture. Cell proliferation and migration in these experiments have yet to be explicitly characterised, limiting the ability of experimentalists to determine the effects of various experimental conditions, such as scaffold geometry, on cell behaviour. We consider tissue growth by osteoblastic cells in melt electro-written scaffolds that comprise thin square pores with sizes that were deliberately increased between experiments.

Profile likelihood analysis for a stochastic model of diffusion in heterogeneous media.

We compute profile likelihoods for a stochastic model of diffusive transport motivated by experimental observations of heat conduction in layered skin tissues. This process is modelled as a random walk in a layered one-dimensional material, where each layer has a distinct particle hopping rate. Particles are released at some location, and the duration of time taken for each particle to reach an absorbing boundary is recorded.

Practical parameter identifiability for spatio-temporal models of cell invasion.

We examine the practical identifiability of parameters in a spatio-temporal reaction-diffusion model of a scratch assay. Experimental data involve fluorescent cell cycle labels, providing spatial information about cell position and temporal information about the cell cycle phase. Cell cycle labelling is incorporated into the reaction-diffusion model by treating the total population as two interacting subpopulations.

A hierarchical Bayesian model for understanding the spatiotemporal dynamics of the intestinal epithelium.

Our work addresses two key challenges, one biological and one methodological. First, we aim to understand how proliferation and cell migration rates in the intestinal epithelium are related under healthy, damaged (Ara-C treated) and recovering conditions, and how these relations can be used to identify mechanisms of repair and regeneration. We analyse new data, presented in more detail in a companion paper, in which BrdU/IdU cell-labelling experiments were performed under these respective conditions.

Cell proliferation within small intestinal crypts is the principal driving force for cell migration on villi.

The functional integrity of the intestinal epithelial barrier relies on tight coordination of cell proliferation and migration, with failure to regulate these processes resulting in disease. It is not known whether cell proliferation is sufficient to drive epithelial cell migration during homoeostatic turnover of the epithelium. Nor is it known precisely how villus cell migration is affected when proliferation is perturbed.

Where next for the reproducibility agenda in computational biology?

Background: The concept of reproducibility is a foundation of the scientific method. With the arrival of fast and powerful computers over the last few decades, there has been an explosion of results based on complex computational analyses and simulations. The reproducibility of these results has been addressed mainly in terms of exact replicability or numerical equivalence, ignoring the wider issue of the reproducibility of conclusions through equivalent, extended or alternative methods.

Models, measurement and inference in epithelial tissue dynamics.

The majority of solid tumours arise in epithelia and therefore much research effort has gone into investigating the growth, renewal and regulation of these tissues. Here we review different mathematical and computational approaches that have been used to model epithelia. We compare different models and describe future challenges that need to be overcome in order to fully exploit new data which present, for the first time, the real possibility for detailed model validation and comparison.