arXiv subjects

David J. D. Earn

Publications and source records attributed to David J. D. Earn.

11 recordsLinked to original sources

arXiv2026

Scarlet Fever Dynamics in 19th and 20th Century London

Weekly scarlet fever (SF) mortality records for London, UK, from 1842 to 1939, together with notified case records from 1901 to 1939, reveal a strong annual epidemic pattern with peak prevalence in the autumn. In addition to the annual epidemic pattern, this long time series reveals a cyclical envelope with a period that lengthened over the decades. In particular, the period of the envelope increased from about four years to about eight years between 1880 and 1920, coinciding with a dramatic decline in pre-antibiotic-era SF deaths (from about 2300/yr to about 80/yr). We quantify the spectral features of the SF time series using a wavelet transform, and attempt to explain why the frequency structure changed over time, using a mechanistic mathematical model of SF transmission dynamics. We estimate the parameters of the model in part from the literature and in part by fitting mortality and incidence jointly. We use a mechanistic transition analysis (considering both attractors and transients in model solutions) to relate the observed changes in frequency structure to our estimated changes in model parameters. We find that almost all spectral evolution in the time series can be explained by changes in the effective reproduction number and the amplitude of seasonal forcing. These changes reflect observed variation in birth rates together with inferred changes in transmission, which may in part have arisen through pathogen evolution.

q-bio.PE↗

arXiv2026

Epidemic "momentum" and a conservation law for infectious disease dynamics

Infectious disease outbreaks have precipitated a profusion of mathematical models. Epidemic curves predicted by these models are typically qualitatively similar, despite distinct model assumptions, but there is no theoretical explanation for this similarity in terms of any recognised common structure. We introduce a unifying concept of "epidemic momentum"---prevalence weighted by potential to infect---which is more informative than prevalence, yet analytically tractable. Epidemic momentum reveals a common underlying geometry in which outbreak trajectories always follow contours of a conserved quantity. This previously unrecognised conservation law constrains how epidemics can unfold and provides the mathematical basis for disentangling transmissibility from prior immunity using a single epidemic time series. Epidemic momentum also exposes the true final size of an outbreak and a universal phase-plane description that links generic renewal models to the classical SIR system.

q-bio.PE↗

arXiv2026

Jointly estimating transmissibility and prior immunity from epidemic time series

Infectious disease time series are often used to estimate a pathogen's basic reproduction number, $R_0$. However, fits of epidemic models to time series conflate pathogen transmissibility with pre-existing population immunity, so only the *effective* reproduction number, $R_{eff}$, can be inferred. This composite parameter is the product of the underlying $R_0$ and the pre-epidemic susceptible fraction, $x^-$. We show that a conservation law associated with epidemic momentum---prevalence weighted by potential to infect---makes it possible to disentangle transmissibility from prior immunity and to infer $R_0$ and $x^-$ separately from a single epidemic time series. We test the methodology using stochastic epidemic simulations, and illustrate the approach with a reappraisal of influenza transmissibility during the 1918 pandemic, estimating rather than assuming the degree of prior population immunity. For the autumn wave in Philadelphia, USA, we find $R_0\approx2.7$ and $x^-\approx0.8$, implying that about 20% of the population was already immune before that wave, plausibly as a result of infection during the spring 1918 herald wave.

q-bio.PE↗

arXiv2025

Global stability of epidemic models with uniform susceptibility

Transmission dynamics of infectious diseases are often studied using compartmental mathematical models, which are commonly represented as systems of autonomous ordinary differential equations. A key step in the analysis of such models is to identify equilibria and find conditions for their stability. Local stability analysis reduces to a problem in linear algebra, but there is no general algorithm for establishing global stability properties. Substantial progress on global stability of epidemic models has been made in the last 20 years, primarily by successfully applying Lyapunov's method to specific systems. Here, we show that any compartmental epidemic model in which susceptible individuals cannot be distinguished and can be infected only once, has a globally asymptotically stable (GAS) equilibrium. If the basic reproduction number ${R}_0$ satisfies ${R}_0 > 1$, then the GAS fixed point is an endemic equilibrium (i.e., constant, positive disease prevalence). Alternatively, if ${R}_0 \le 1$, then the GAS equilibrium is disease-free. This theorem subsumes a large number of results published over the last century, strengthens most of them by establishing global rather than local stability, avoids the need for any stability analyses of these systems in the future, and settles the question of whether co-existing stable solutions or non-equilibrium attractors are possible in such models: they are not.

q-bio.PE↗

arXiv2023

Toward a comprehensive system for constructing compartmental epidemic models

Compartmental models are valuable tools for investigating infectious diseases. Researchers building such models typically begin with a simple structure where compartments correspond to individuals with different epidemiological statuses, e.g., the classic SIR model which splits the population into susceptible, infected, and recovered compartments. However, as more information about a specific pathogen is discovered, or as a means to investigate the effects of heterogeneities, it becomes useful to stratify models further -- for example by age, geographic location, or pathogen strain. The operation of constructing stratified compartmental models from a pair of simpler models resembles the Cartesian product used in graph theory, but several key differences complicate matters. In this article we give explicit mathematical definitions for several so-called ``model products'' and provide examples where each is suitable. We also provide examples of model stratification where no existing model product will generate the desired result.

q-bio.PE↗

arXiv2022

Evaluating undercounts in epidemics: response to Maruotti et al. 2022

Maruotti et al. 2022 used a mark-recapture approach to estimate bounds on the true number of monkeypox infections in various countries. These approaches are fundamentally flawed; it is impossible to estimate undercounting based solely on a single stream of reported cases. Simulations based on a Richards curve for cumulative incidence show that, for reasonable epidemic parameters, the proposed methods estimate bounds on the ascertainment ratio of $\approx 0.2-0.5$ roughly independently of the true ascertainment ratio. These methods should not be used.

q-bio.PE↗

arXiv2021

Testing and Isolation Efficacy: Insights from a Simple Epidemic Model

Testing individuals for pathogens can affect the spread of epidemics. Understanding how individual-level processes of sampling and reporting test results can affect community- or population-level spread is a dynamical modeling question. The effect of testing processes on epidemic dynamics depends on factors underlying implementation, particularly testing intensity and on whom testing is focused. Here, we use a simple model to explore how the individual-level effects of testing might directly impact population-level spread. Our model development was motivated by the COVID-19 epidemic, but has generic epidemiological and testing structures. To the classic SIR framework we have added a per capita testing intensity, and compartment-specific testing weights, which can be adjusted to reflect different testing emphases -- surveillance, diagnosis, or control. We derive an analytic expression for the relative reduction in the basic reproductive number due to testing, test-reporting and related isolation behaviours. Intensive testing and fast test reporting are expected to be beneficial at the community level because they can provide a rapid assessment of the situation, identify hot spots, and may enable rapid contact-tracing. Direct effects of fast testing at the individual level are less clear, and may depend on how individuals' behaviour is affected by testing information. Our simple model shows that under some circumstances both increased testing intensity and faster test reporting can reduce the effectiveness of control, and allows us to explore the conditions under which this occurs. Conversely, we find that focusing testing on infected individuals always acts to increase effectiveness of control.

q-bio.PE↗

arXiv2019

Invariant predictions of epidemic patterns from radically different forms of seasonal forcing

Seasonal variation in environmental variables, and in rates of contact among individuals, are fundamental drivers of infectious disease dynamics. Unlike most periodically-forced physical systems, for which the precise pattern of forcing is typically known, underlying patterns of seasonal variation in transmission rates can be estimated approximately at best, and only the period of forcing is accurately known. Yet solutions of epidemic models depend strongly on the forcing function, so dynamical predictions---such as changes in epidemic patterns that can be induced by demographic transitions or mass vaccination---are always subject to the objection that the underlying patterns of seasonality are poorly specified. Here, we demonstrate that the key bifurcations of the standard epidemic model are invariant to the shape of seasonal forcing if the amplitude of forcing is appropriately adjusted. Consequently, analyses applicable to real disease dynamics can be conducted with a smooth, idealized sinusoidal forcing function, and qualitative changes in epidemic patterns can be predicted without precise knowledge of the underlying forcing pattern. We find similar invariance in a seasonally forced predator-prey model, and conjecture that this phenomenon---and the associated robustness of predictions---might be a feature of many other periodically forced dynamical systems.

q-bio.PE↗

arXiv2012

Early Real-time Estimation of Infectious Disease Reproduction Number

When an infectious disease strikes a population, the number of newly reported cases is often the only available information that one can obtain during early stages of the outbreak. An important goal of early outbreak analysis is to obtain a reliable estimate for the basic reproduction number, $R_{0}$, from the limited information available. We present a novel method that enables us to make a reliable real-time estimate of the reproduction number at a much earlier stage compared to other available methods. Our method takes into account the possibility that a disease has a wide distribution of infectious period and that the degree distribution of the contact network is heterogeneous. We validate our analytical framework with numerical simulations.

q-bio.QM↗

arXiv1996

Potential-Density Basis Sets for Galactic Disks

A class of complete potential-density basis sets in cylindrical (R,phi,z) coordinates is presented. This class is suitable for stability studies of galactic disks in three dimensions and includes basis sets tailored for disks with vertical density profiles that are exponential (exp(-|z|/\zn)), Gaussian (exp(-(z/\zn)^2) or locally isothermal (sech^2(z/\zn)). The basis sets are non-discrete and non-biorthogonal; however, the extra numerical computations required (compared with discrete biorthogonal sets) are explained and constitute a small overhead. The method of construction (and proof of completeness) is simple and can be used to construct basis sets for other density distributions that are best described in circular or elliptic cylindrical coordinates. When combined with a basis set designed for spheroidal systems, the basis sets presented here can be used to study the stability of realistic disks embedded in massive halos.

astro-ph↗

arXiv1995

THE OPTIMAL N-BODY METHOD FOR STABILITY STUDIES OF GALAXIES

The stability of a galaxy model is most easily assessed through N-body simulation. Particle-mesh codes have been widely used for this purpose, since they enable the largest numbers of particles to be employed. We show that the functional expansion technique, originally proposed by Clutton-Brock for other simulation problems, is in fact superior for stability work. For simulations of linear evolution it is not much slower than grid methods using the same number of particles, and reproduces analytical results with much greater accuracy. This success rests on its ability to represent global modes with a modest number of basis functions; grid methods may be more effective for other applications, however. Our conclusions are based on implementations of functional expansion and grid algorithms for disk galaxies.

astro-ph↗