arXiv subjects

Cathal Mills

Publications and source records attributed to Cathal Mills.

4 recordsLinked to original sources

arXiv2026

Multi-scale measures of time-varying epidemic spread on human mobility networks

Human movement drives the spatial spread and persistence of many infectious diseases, yet existing theory and real-time operational tools for inferring the instantaneous reproduction number R(t) often assume static and/or homogeneously mixing populations and cannot describe how individuals generate and acquire infections heterogeneously based on their movement patterns within a day. Renewal equations underpin many such popular estimators of R(t), and here, we develop a network-based modelling framework from which we derive new mechanism-led renewal equations and control indicators for outbreaks of infectious diseases. These equations directly integrate within-day human movement to rigorously define a family of instantaneous reproduction numbers; inward, outward, and type R(t) for individual locations, R(t) between locations, R(t) at meeting locations, and R(t) for the entire mobility network. These quantities correct for the unsuitability of existing location-specific R(t) estimators that operate in closed, static populations. Applying our framework to epidemics on diverse types of networks alongside mobile phone data, we demonstrate how our new framework's outputs provide new, multi-scale control indicators at the network, location, and transmission corridor scales, and can be used to design targeted disease control interventions including the strength, type, and length of intervention required across space and time. We capture the biasing effects of different existing ways to measure location-specific and network-level transmission potential without capturing within-day human movements. This generalisable framework redefines reproduction numbers in real-world outbreaks that are shaped by individuals moving across connected locations, enabling more spatially and temporally precise interventions.

q-bio.QM↗

arXiv2026

A note on conditional densities, Bayes' rule, and recent criticisms of Bayesian inference

When performing Bayesian inference, we frequently need to work with conditional probability densities. For example, the posterior function is the conditional density of the parameters given the data. Some might worry that conditional densities are ill-defined, considering that for a continuous random variable $Y$, the event $\{Y=y\}$ has probability zero, meaning the formula $\mathbb{P}(A|B)=\mathbb{P}(A\cap B)/\mathbb{P}(B)$ is inapplicable. In reality, when we work with conditional densities, we never condition directly on the zero-probability event $\{Y=y\}$; rather, we first condition on the random variable $Y$, and then we may plug in an observed value $y$. The first purpose of our article is to provide an exposition on conditional densities that elaborates on this point. While we have aimed to make this explanation accessible, we follow it with a roadmap of the measure theory needed to make it rigorous. A recent preprint (arXiv:2411.13570) has expressed the concern that probability densities are ill-defined and that as a result Bayes' theorem cannot be used, and they provide examples that allegedly demonstrate inconsistencies in the Bayesian framework. The second purpose of our article is to investigate their claims. We contend that the examples given in their work do not demonstrate any inconsistencies; we find that there are mathematical errors and that they deviate significantly from the Bayesian framework.

stat.ME↗

arXiv2025

A decision-theoretic framework for uncertainty quantification in epidemiological modelling

Estimating, understanding, and communicating uncertainty is fundamental to statistical epidemiology, where model-based estimates regularly inform real-world decisions. However, sources of uncertainty are rarely formalised, and existing classifications are often defined inconsistently. This lack of structure hampers interpretation, model comparison, and targeted data collection. Connecting ideas from machine learning, information theory, experimental design, and health economics, we present a first-principles decision-theoretic framework that defines uncertainty as the expected loss incurred by making an estimate based on incomplete information, arguing that this is a highly useful and practically relevant definition for epidemiology. We show how reasoning about future data leads to a notion of expected uncertainty reduction, which induces formal definitions of reducible and irreducible uncertainty. We demonstrate our approach using a case study of SARS-CoV-2 wastewater surveillance in Aotearoa New Zealand, estimating the uncertainty reduction if wastewater surveillance were expanded to the full population. We then connect our framework to relevant literature from adjacent fields, showing how it unifies and extends many of these ideas and how it allows these ideas to be applied to a wider range of models. Altogether, our framework provides a foundation for more reliable, consistent, and policy-relevant uncertainty quantification in infectious disease epidemiology.

stat.ME↗

arXiv2024

Renewal equations for vector-borne diseases

During infectious disease outbreaks, estimates of time-varying pathogen transmissibility, such as the instantaneous reproduction number R(t) or epidemic growth rate r(t), are used to inform decision-making by public health authorities. For directly transmitted infectious diseases, the renewal equation framework is a widely used method for measuring time-varying transmissibility. The framework uses information on the typical time elapsing between an infection and the offspring infections (quantified by the generation time distribution), and R(t), to describe the rate at which currently infected individuals generate new infections. For diseases with transmission cycles involving hosts and vectors, however, renewal equation models have been far less used. This is likely due to difficulties in mechanistically defining generation times that can capture the complexity of multi-stage, human-vector relationships. Here, using dengue as an example, we provide general renewal equations that are derived from first principles using age-structured systems of coupled partial differential equations across human and vector sub-populations. Our framework tracks the multi-stage transmission cycle over calendar time and across stage-specific ages, resulting in governing renewal equations that quantify how the rate at which new infections are generated from existing infections depends on stage-specific processes. The framework provides a foundation on which to base inferential frameworks for estimating R(t) and r(t) for infectious diseases with multiple stages in the transmission cycle

q-bio.PE↗