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. The method combines a semi-mechanistic model for disease transmission to predict cases with Gaussian process regression models to predict the fraction of reported cases that will require hospital treatment. We evaluate forecast performance against out-of-sample data over the period from 2 October 2022 to 23 July 2023. Our results show that forecast performance is reasonably good over a 1-3 week time horizon, although generally deteriorates as the time horizon is lengthened. The model has been operationalised to provide weekly national and regional forecasts in real-time. This study is an important step towards development of more sophisticated situational awareness and infectious disease forecasting tools in Aotearoa New Zealand.

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. In many biological applications, a typical approach is to assume the data are normally distributed about the solution of the mathematical model. Despite the widespread use of the standard additive Gaussian measurement error model, the assumptions inherent in this approach are rarely explicitly considered or compared with other options. Here, we interpret scratch assay data, involving migration, proliferation and delays in a population of cancer cells using a reaction-diffusion PDE model. We consider relating experimental measurements to the PDE solution using a standard additive Gaussian measurement error model alongside a comparison to a more biologically realistic binomial measurement error model. While estimates of model parameters are relatively insensitive to the choice of measurement error model, model predictions for data realisations are very sensitive. The standard additive Gaussian measurement error model leads to biologically inconsistent predictions, such as negative counts and counts that exceed the carrying capacity across a relatively large spatial region within the experiment. Furthermore, the standard additive Gaussian measurement error model requires estimating an additional parameter compared to the binomial measurement error model. In contrast, the binomial measurement error model leads to biologically plausible predictions and is simpler to implement. We provide open source Julia software on GitHub to replicate all calculations in this work, and we explain how to generalise our approach to deal with coupled PDE models with several dependent variables through a multinomial measurement error model, as well as pointing out other potential generalisations by linking our work with established practices in the field of generalised linear models.

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. Here we present a systematic, computationally-efficient workflow we call Profile-Wise Analysis (PWA) that addresses all three steps in a unified way. Recently-developed methods for constructing 'profile-wise' prediction intervals enable this workflow and provide the central linkage between different workflow components. These methods propagate profile-likelihood-based confidence sets for model parameters to predictions in a way that isolates how different parameter combinations affect model predictions. We show how to extend these profile-wise prediction intervals to two-dimensional interest parameters. We then demonstrate how to combine profile-wise prediction confidence sets to give an overall prediction confidence set that approximates the full likelihood-based prediction confidence set well. Our three case studies illustrate practical aspects of the workflow, focusing on ordinary differential equation (ODE) mechanistic models with both Gaussian and non-Gaussian noise models. While the case studies focus on ODE-based models, the workflow applies to other classes of mathematical models, including partial differential equations and simulation-based stochastic models. Open-source software on GitHub can be used to replicate the case studies.

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. In this work we apply a recent likelihood-based workflow, called Profile-Wise Analysis (PWA), to non-identifiable models for the first time. The PWA workflow addresses identifiability, parameter estimation, and prediction in a unified framework that is simple to implement and interpret. Previous implementations of the workflow have dealt with idealised identifiable problems only. In this study we illustrate how the PWA workflow can be applied to both structurally non-identifiable and practically non-identifiable models in the context of simple population growth models. Dealing with simple mathematical models allows us to present the PWA workflow in a didactic, self-contained document that can be studied together with relatively straightforward Julia code provided on GitHub . Working with simple mathematical models allows the PWA workflow prediction intervals to be compared with gold standard full likelihood prediction intervals. Together, our examples illustrate how the PWA workflow provides us with a systematic way of dealing with non-identifiability, especially compared to other approaches, such as seeking ad hoc parameter combinations, or simply setting parameter values to some arbitrary default value. Importantly, we show that the PWA workflow provides insight into the commonly-encountered situation where some parameters are identifiable and others are not, allowing us to explore how uncertainty in some parameters, and combinations of parameters, regardless of their identifiability status, influences model predictions in a way that is insightful and interpretable.

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. METHODS: Using data from New Zealand, we estimate the time-varying sensitivity of SARS-CoV-2 RT-PCR under varying temporal, biological, and demographic factors. RESULTS: Sensitivity peaks 4-5 days postinfection at 92.7% (91.4%-94.0%) and remains over 88% between 5 and 14 days postinfection. After the peak, sensitivity declined more rapidly in vaccinated cases compared with unvaccinated, females compared with males, those aged under 40 compared with over 40s, and Pacific peoples compared with other ethnicities. CONCLUSIONS: Reverse transcription PCR remains a sensitive technique and has been an effective tool in New Zealand's border and postborder measures to control coronavirus disease 2019. Our results inform model parameters and decisions concerning routine testing frequency.

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. One approach for overcoming this computational limitation is to coarse-grain the stochastic model to provide an approximate continuum model that can be solved using far less computational effort. However, coarse-grained continuum models can be biased or inaccurate, particularly for certain parameter regimes. In this work, we combine stochastic and continuum mathematical models in the context of lattice-based models of two-dimensional cell biology experiments by demonstrating how to simulate two commonly used experiments: cell proliferation assays and barrier assays. Our approach involves building a simple statistical model of the discrepancy between the expensive stochastic model and the associated computationally inexpensive coarse-grained continuum model. We form this statistical model based on a limited number of expensive stochastic model evaluations at design points sampled from a user-chosen distribution, corresponding to a computer experiment design problem. With straightforward design point selection schemes, we show that using the statistical model of the discrepancy in tandem with the computationally inexpensive continuum model allows us to carry out prediction and inference while correcting for biases and inaccuracies due to the continuum approximation. We demonstrate this approach by simulating a proliferation assay, where the continuum limit model is the well-known logistic ordinary differential equation, as well as a barrier assay where the continuum limit model is closely related to the well-known Fisher-KPP partial differential equation. We construct an approximate likelihood function for parameter inference, both with and without discrepancy correction terms. Using maximum likelihood estimation, we provide point estimates of the unknown parameters, and use the profile likelihood to characterise the uncertainty in these estimates and form approximate confidence intervals. For the range of inference problems considered, working with the continuum limit model alone leads to biased parameter estimation and confidence intervals with poor coverage. In contrast, incorporating correction terms arising from the statistical model of the model discrepancy allows us to recover the parameters accurately with minimal computational overhead. The main tradeoff is that the associated confidence intervals are typically broader, reflecting the additional uncertainty introduced by the approximation process. All algorithms required to replicate the results in this work are written in the open source Julia language and are available at GitHub.

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. We use a profile-likelihood approach to explore practical parameter identifiability using data describing the re-growth of hard coral. With this approach, we explore the relationship between parameter identifiability and model misspecification, finding that the logistic growth model does not suffer identifiability issues for the type of data we consider whereas the Gompertz and Richards' models encounter practical non-identifiability issues. This analysis of parameter identifiability and model selection is important because different growth models are in biological modelling without necessarily considering whether parameters are identifiable. Standard practices that do not consider parameter identifiability can lead to unreliable or imprecise parameter estimates and potentially misleading mechanistic interpretations. For example, using the Gompertz model, the estimate of the time scale of coral re-growth is 625 days when we estimate the initial density from the data, whereas it is 1429 days using a more standard approach where variability in the initial density is ignored. While tools developed here focus on three standard sigmoid growth models only, our theoretical developments are applicable to any sigmoid growth model and any appropriate data set. MATLAB implementations of all software are available on GitHub.

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. We collect highly detailed temporal measurements of the average cell density, tissue coverage, and tissue geometry. To quantify tissue growth in terms of the underlying cell proliferation and migration processes, we introduce and calibrate a mechanistic mathematical model based on the Porous-Fisher reaction-diffusion equation. Parameter estimates and uncertainty quantification through profile likelihood analysis reveal consistency in the rate of cell proliferation and steady-state cell density between pore sizes. This analysis also serves as an important model verification tool: while the use of reaction-diffusion models in biology is widespread, the appropriateness of these models to describe tissue growth in 3D scaffolds has yet to be explored. We find that the Porous-Fisher model is able to capture features relating to the cell density and tissue coverage, but is not able to capture geometric features relating to the circularity of the tissue interface. Our analysis identifies two distinct stages of tissue growth, suggests several areas for model refinement, and provides guidance for future experimental work that explores tissue growth in 3D printed scaffolds.

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. Some reports suggest that proliferation and migration may not be related while other studies support a direct relationship. We used established cell-tracking methods based on thymine analog cell labeling and developed tailored mathematical models to quantify cell proliferation and migration under normal conditions and when proliferation is reduced and when it is temporarily halted. We found that epithelial cell migration velocities along the villi are coupled to cell proliferation rates within the crypts in all conditions. Furthermore, halting and resuming proliferation results in the synchronized response of cell migration on the villi. We conclude that cell proliferation within the crypt is the primary force that drives cell migration along the villus. This methodology can be applied to interrogate intestinal epithelial dynamics and characterize situations in which processes involved in cell turnover become uncoupled, including pharmacological treatments and disease models.-Parker, A., Maclaren, O. J., Fletcher, A. G., Muraro, D., Kreuzaler, P. A., Byrne, H. M., Maini, P. K., Watson, A. J. M., Pin, C. Cell proliferation within small intestinal crypts is the principal driving force for cell migration on villi.

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. Second, in considering how to more rigorously process these data and interpret them using mathematical models, we use a probabilistic, hierarchical approach. This provides a best-practice approach for systematically modelling and understanding the uncertainties that can otherwise undermine the generation of reliable conclusions-uncertainties in experimental measurement and treatment, difficult-to-compare mathematical models of underlying mechanisms, and unknown or unobserved parameters. Both spatially discrete and continuous mechanistic models are considered and related via hierarchical conditional probability assumptions. We perform model checks on both in-sample and out-of-sample datasets and use them to show how to test possible model improvements and assess the robustness of our conclusions. We conclude, for the present set of experiments, that a primarily proliferation-driven model suffices to predict labelled cell dynamics over most time-scales.

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. While this assumption of additive Gaussian noise is extremely common and simple to implement and interpret, it is often unjustified and can lead to poor parameter estimates and non-physical predictions. One way to overcome this challenge is to implement a different measurement error model. In this review, we demonstrate how to implement a range of measurement error models in a likelihood-based framework for estimation, identifiability analysis and prediction, called profile-wise analysis. This frequentist approach to uncertainty quantification for mechanistic models leverages the profile likelihood for targeting parameters and understanding their influence on predictions. Case studies, motivated by simple caricature models routinely used in systems biology and mathematical biology literature, illustrate how the same ideas apply to different types of mathematical models. Open-source Julia code to reproduce results is available on GitHub.

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. Here, we use a validated mathematical model of Covid-19 in New Zealand to investigate counterfactual scenarios with differing levels of vaccine coverage in different age and ethnicity groups. The model builds on earlier research by adding age- and time-dependent case ascertainment, the effect of antiviral medications, improved hospitalisation rate estimates, and the impact of relaxing control measures. The model was used for scenario analysis and policy advice for the New Zealand Government in 2022 and 2023. We compare the number of Covid-19 hospitalisations, deaths, and years of life lost in each counterfactual scenario to a baseline scenario that is fitted to epidemiological data between January 2022 and June 2023. Our results estimate that vaccines saved 6650 (95% credible interval [4424, 10180]) lives, and prevented 74500 [51000, 115400] years of life lost and 45100 [34400, 55600] hospitalisations during this 18-month period. Making the same comparison before the benefit of antiviral medications is accounted for, the estimated number of lives saved by vaccines increases to 7604 [5080, 11942]. Due to inequities in the vaccine rollout, vaccination rates among Maori were lower than in people of European ethnicity. Our results show that, if vaccination rates had been equitable, an estimated 11%-26% of the 292 Maori Covid-19 deaths that were recorded in this time period could have been prevented. We conclude that Covid-19 vaccination greatly reduced health burden in New Zealand and that equity needs to be a key focus of future vaccination programmes.

What do aquaporin knockout studies tell us about fluid transport in epithelia?

The investigation of near-isosmotic water transport in epithelia goes back over 100 years; however, debates over mechanism and pathway remain. Aquaporin (AQP) knockouts have been used by various research groups to test the hypothesis of an osmotic mechanism as well as to explore the paracellular versus transcellular pathway debate. Nonproportional reductions in the water permeability of a water-transporting epithelial cell (e.g., a reduction of around 80-90 %) compared to the reduction in overall water transport rate in the knockout animal (e.g., a reduction of 50-60 %) are commonly found. This nonproportionality has led to controversy over whether AQP knockout studies support or contradict the osmotic mechanism. Arguments raised for and against an interpretation supporting the osmotic mechanism typically have partially specified, implicit, or incorrect assumptions. We present a simple mathematical model of the osmotic mechanism with clear assumptions and, for models based on this mechanism, establish a baseline prediction of AQP knockout studies. We allow for deviations from isotonic/isosmotic conditions and utilize dimensional analysis to reduce the number of parameters that must be considered independently. This enables a single prediction curve to be used for multiple epithelial systems. We find that a simple, transcellular-only osmotic mechanism sufficiently predicts the results of knockout studies and find criticisms of this mechanism to be overstated. We note, however, that AQP knockout studies do not give sufficient information to definitively rule out an additional paracellular pathway.

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. Stochastic models can help with misspecification but are even more expensive to simulate and perform inference with. Here, we develop an explicitly likelihood-based variation of the generalised profiling method as a tool for prediction and inference under model misspecification. Our approach allows us to carry out identifiability analysis and uncertainty quantification using profile likelihood-based methods without the need for marginalisation. We provide justification for this approach by introducing a new interpretation of the model approximation component as a stochastic constraint. This preserves the rationale for using profiling rather than integration to remove nuisance parameters while also providing a link back to stochastic models. We applied an initial version of this method during an outbreak of measles in Samoa in 2019-2020 and found that it achieved relatively fast, accurate predictions. Here we present the most recent version of our method and its application to this measles outbreak, along with additional validation.

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. Therefore, choosing an appropriate model almost always requires case-by-case consideration. In this work we present a straightforward approach to quantitatively explore this trade-off using a case study exploring mathematical models of coral reef regrowth after some ecological disturbance, such as damage caused by a tropical cyclone. In particular, we compare a simple single species ordinary differential equation (ODE) model approach with a more complicated two-species coupled ODE model. Univariate profile likelihood analysis suggests that the both models are practically identifiable. To provide additional insight we construct and compare approximate prediction intervals using a new parameter-wise prediction approximation, confirming both the simple and complex models perform similarly with regard to making predictions. Our approximate parameter-wise prediction interval analysis provides explicit information about how each parameter affects the predictions of each model. Comparing our approximate prediction intervals with a more rigorous and computationally expensive evaluation of the full likelihood shows that the new approximations are reasonable in this case. All algorithms and software to support this work are freely available as jupyter notebooks on GitHub so that they can be adapted to deal with any other ODE-based models.

Efficiency of primary saliva secretion: an analysis of parameter dependence in dynamic single-cell and acinus models, with application to aquaporin knockout studies.

Secretion from the salivary glands is driven by osmosis following the establishment of osmotic gradients between the lumen, the cell and the interstitium by active ion transport. We consider a dynamic model of osmotically driven primary saliva secretion and use singular perturbation approaches and scaling assumptions to reduce the model. Our analysis shows that isosmotic secretion is the most efficient secretion regime and that this holds for single isolated cells and for multiple cells assembled into an acinus. For typical parameter variations, we rule out any significant synergistic effect on total water secretion of an acinar arrangement of cells about a single shared lumen. Conditions for the attainment of isosmotic secretion are considered, and we derive an expression for how the concentration gradient between the interstitium and the lumen scales with water- and chloride-transport parameters. Aquaporin knockout studies are interpreted in the context of our analysis and further investigated using simulations of transport efficiency with different membrane water permeabilities. We conclude that recent claims that aquaporin knockout studies can be interpreted as evidence against a simple osmotic mechanism are not supported by our work. Many of the results that we obtain are independent of specific transporter details, and our analysis can be easily extended to apply to models that use other proposed ionic mechanisms of saliva secretion.

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. Because the durability of infection-derived immunity is a key uncertainty in epidemiological models, we investigated outcomes under different assumptions about the speed of waning. The model was used to provide scenarios to the New Zealand Government, helping to inform policy response and healthcare system preparedness ahead of the winter respiratory illness season. In all scenarios investigated, a second Omicron wave was projected to occur in the second half of 2022. The timing of the peak depended primarily on the speed of waning and was typically between August and November. The peak number of daily infections in the second Omicron wave was smaller than in the first Omicron wave. Peak hospital occupancy was also generally lower than in the first wave but was sensitive to the age distribution of infections. A scenario with increased contact rates in older groups had higher peak hospital occupancy than the first wave. Scenarios with relatively high transmission, whether a result of relaxation of control measures or voluntary behaviour change, did not necessarily lead to higher peaks. However, they generally resulted in more sustained healthcare demand (>250 hospital beds throughout the winter period). The estimated health burden of Covid-19 in the medium term is sensitive to the strength and durability of infection-derived and hybrid immunity against reinfection and severe illness, which are uncertain.

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. Our approach provides a bridge between primarily empirical approaches to forecasting and assumption-driven scenario models. The model was developed as a policy advice tool for New Zealand's 2021 outbreak of the Delta variant of SARS-CoV-2 and includes the effects of age structure, non-pharmaceutical interventions, and the ongoing vaccine rollout occurring during the time period studied. We use an approximate Bayesian computation approach to infer the time-varying transmission coefficient from real-time data on reported cases. We then compare projections of the model with future, out-of-sample data. We find that this approach produces a good fit with in-sample data and reasonable forward projections given the inherent limitations of predicting epidemic dynamics during periods of rapidly changing policy and behaviour. Results from the model helped inform the New Zealand Government's policy response throughout the outbreak.

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 biphasic growth 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. Common approaches can lead to inaccurate predictions of future growth that may result in inappropriate management and intervention strategies being implemented. Here, we develop a very general computationally efficient framework, based on profile likelihood analysis, for diagnosing, understanding and predicting biphasic population growth. The two key components of the framework are as follows: (i) an efficient method to form approximate confidence intervals for the change point of the growth dynamics and model parameters and (ii) parameter-wise profile predictions that systematically reveal the influence of individual model parameters on predictions. To illustrate our framework we explore real-world case studies across the life sciences.

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, Rt , 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 Rt . Alert Level 4, New Zealand's most stringent restriction setting which includes stay-at-home measures, was initially effective at reducing the median Rt to 0.6 (90% CrI 0.4, 0.8) on 29 August 2021. As New Zealand eased certain restrictions and switched from an elimination strategy to a suppression strategy, Rt subsequently increased to a median 1.3 (1.2, 1.4). Increasing vaccination coverage along with regional restrictions were eventually sufficient to reduce Rt below 1. The outbreak peaked at an estimated 198 (172, 229) new infected cases on 10 November, after which cases declined until January 2022. We continue to update Rt estimates in real time as new case data become available to inform New Zealand's ongoing pandemic response.