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Joel C. Miller

Publications and source records attributed to Joel C. Miller.

At least 19 recordsLinked to original sources
arXiv2025

Including frameworks of public health ethics in computational modelling of infectious disease interventions

Decisions on public health interventions to control infectious disease are often informed by computational models. Interpreting the predicted outcomes of a public health decision requires not only high-quality modelling, but also an ethical framework for assessing the benefits and harms associated with different options. The design and specification of ethical frameworks matured independently of computational modelling, so many values recognised as important for ethical decision-making are missing from computational models. We demonstrate a proof-of-concept approach to incorporate multiple public health values into the evaluation of a simple computational model for vaccination against a pathogen such as SARS-CoV-2. By examining a bounded space of alternative prioritisations of values (outcome equity and aggregate benefit) we identify value trade-offs, where the outcomes of optimal strategies differ depending on the ethical framework. This work demonstrates an approach to incorporating diverse values into decision criteria used to evaluate outcomes of models of infectious disease interventions.

physics.soc-ph↗
arXiv2024

A primer on the use of probability generating functions in infectious disease modeling

We explore the application of probability generating functions (PGFs) to invasive processes, focusing on infectious disease introduced into large populations. Our goal is to acquaint the reader with applications of PGFs, moreso than to derive new results. PGFs help predict a number of properties about early outbreak behavior while the population is still effectively infinite, including the probability of an epidemic, the size distribution after some number of generations, and the cumulative size distribution of non-epidemic outbreaks. We show how PGFs can be used in both discrete-time and continuous-time settings, and discuss how to use these results to infer disease parameters from observed outbreaks. In the large population limit for susceptible-infected-recovered (SIR) epidemics PGFs lead to survival-function based models that are equivalent the the usual mass-action SIR models but with fewer ODEs. We use these to explore properties such as the final size of epidemics or even the dynamics once stochastic effects are negligible. We target this tutorial to biologists and public health researchers who want to learn how to apply PGFs to invasive diseases, but it could also be used in an introductory mathematics course on PGFs. We include many exercises to help demonstrate concepts and to give practice applying the results. We summarize our main results in a few tables. Additionally we provide a small python package which performs many of the relevant calculations.

q-bio.PE↗
arXiv2022

Approximating quasi-stationary behaviour in network-based SIS dynamics

Deterministic approximations to stochastic Susceptible-Infectious-Susceptible models typically predict a stable endemic steady-state when above threshold. This can be hard to relate to the underlying stochastic dynamics, which has no endemic steady-state but can exhibit approximately stable behaviour. Here we relate the approximate models to the stochastic dynamics via the definition of the quasi-stationary distribution (QSD), which captures this approximately stable behaviour. We develop a system of ordinary differential equations that approximate the number of infected individuals in the QSD for arbitrary contact networks and parameter values. When the epidemic level is high, these QSD approximations coincide with the existing approximation methods. However, as we approach the epidemic threshold, the models deviate, with these models following the QSD and the existing methods approaching the all susceptible state. Through consistently approximating the QSD, the proposed methods provide a more robust link to the stochastic models.

q-bio.PE↗
arXiv2021

Questioning the use of global estimates of reproduction numbers, with implications for policy

The basic reproduction number, $R_0$ is an important and widely used concept in the study of infectious diseases. We briefly review the recent trend of calculating the average of various $R_0$ estimates in systematic reviews aimed at estimating the basic reproduction number of SARS-CoV-2, and discuss the drawbacks and implications of using such averaging methods. Additionally, we argue that even a theoretically grounded approach such as next generation matrix could have practical impediments in its use. More generally, the practice of associating an infectious disease with a single value of $R_0$ is problematic, when the disease can, in fact have different reproduction numbers in various populations.

q-bio.PE↗
arXiv2021

Concurrency measures in the era of temporal network epidemiology: A review

Diseases spread over temporal networks of interaction events between individuals. Structures of these temporal networks hold the keys to understanding epidemic propagation. One early concept of the literature to aid in discussing these structures is concurrency -- quantifying individuals' tendency to form time-overlapping "partnerships". Although conflicting evaluations and an overabundance of operational definitions have marred the history of concurrency, it remains important, especially in the area of sexually transmitted infections. Today, much of theoretical epidemiology uses more direct models of contact patterns, and there is an emerging body of literature trying to connect methods to the concurrency literature. In this review, we will cover the development of the concept of concurrency and these new approaches.

physics.soc-ph↗
arXiv2020

The impact of network properties and mixing on control measures and disease-induced herd immunity in epidemic models: a mean-field model perspective

The contact structure of a population plays an important role in transmission of infection. Many ``structured models'' capture aspects of the contact structure through an underlying network or a mixing matrix. An important observation in such models, is that once a fraction $1-1/\mathcal{R}_0$ has been infected, the residual susceptible population can no longer sustain an epidemic. A recent observation of some structured models is that this threshold can be crossed with a smaller fraction of infected individuals, because the disease acts like a targeted vaccine, preferentially immunizing higher-risk individuals who play a greater role in transmission. Therefore, a limited ``first wave'' may leave behind a residual population that cannot support a second wave once interventions are lifted. In this paper, we systematically analyse a number of mean-field models for networks and other structured populations to address issues relevant to the Covid-19 pandemic. In particular, we consider herd-immunity under several scenarios. We confirm that, in networks with high degree heterogeneity, the first wave confers herd-immunity with significantly fewer infections than equivalent models with lower degree heterogeneity. However, if modelling the intervention as a change in the contact network, then this effect might become more subtle. Indeed, modifying the structure can shield highly connected nodes from becoming infected during the first wave and make the second wave more substantial. We confirm this finding by using an age-structured compartmental model parameterised with real data and comparing lockdown periods implemented either as a global scaling of the mixing matrix or age-specific structural changes. We find that results regarding herd immunity levels are strongly dependent on the model, the duration of lockdown and how lockdown is implemented.

q-bio.PE↗
arXiv2020

A Data-Driven Network Model for the Emerging COVID-19 Epidemics in Wuhan, Toronto and Italy

The ongoing Coronavirus Disease 2019 (COVID-19) pandemic threatens the health of humans and causes great economic losses. Predictive modelling and forecasting the epidemic trends are essential for developing countermeasures to mitigate this pandemic. We develop a network model, where each node represents an individual and the edges represent contacts between individuals where the infection can spread. The individuals are classified based on the number of contacts they have each day (their node degrees) and their infection status. The transmission network model was respectively fitted to the reported data for the COVID-19 epidemic in Wuhan (China), Toronto (Canada), and the Italian Republic using a Markov Chain Monte Carlo (MCMC) optimization algorithm. Our model fits all three regions well with narrow confidence intervals and could be adapted to simulate other megacities or regions. The model projections on the role of containment strategies can help inform public health authorities to plan control measures.

q-bio.PE↗
arXiv2020

Stochasticity and heterogeneity in the transmission dynamics of SARS-CoV-2

SARS-CoV-2 causing COVID-19 disease has moved rapidly around the globe, infecting millions and killing hundreds of thousands. The basic reproduction number, which has been widely used and misused to characterize the transmissibility of the virus, hides the fact that transmission is stochastic, is dominated by a small number of individuals, and is driven by super-spreading events (SSEs). The distinct transmission features, such as high stochasticity under low prevalence, and the central role played by SSEs on transmission dynamics, should not be overlooked. Many explosive SSEs have occurred in indoor settings stoking the pandemic and shaping its spread, such as long-term care facilities, prisons, meat-packing plants, fish factories, cruise ships, family gatherings, parties and night clubs. These SSEs demonstrate the urgent need to understand routes of transmission, while posing an opportunity that outbreak can be effectively contained with targeted interventions to eliminate SSEs. Here, we describe the potential types of SSEs, how they influence transmission, and give recommendations for control of SARS-CoV-2.

q-bio.PE↗
arXiv2018

A monotonic relationship between the variability of the infectious period and final size in pairwise epidemic modelling

For a recently derived pairwise model of network epidemics with non-Markovian recovery, we prove that under some mild technical conditions on the distribution of the infectious periods, smaller variance in the recovery time leads to higher reproduction number, and consequently to a larger epidemic outbreak, when the mean infectious period is fixed. We discuss how this result is related to various stochastic orderings of the distributions of infectious periods. The results are illustrated by a number of explicit stochastic simulations, suggesting that their validity goes beyond regular networks.

q-bio.PE↗
arXiv2018

Fast variables determine the epidemic threshold in the pairwise model with an improved closure

Pairwise models are used widely to model epidemic spread on networks. These include the modelling of susceptible-infected-removed (SIR) epidemics on regular networks and extensions to SIS dynamics and contact tracing on more exotic networks exhibiting degree heterogeneity, directed and/or weighted links and clustering. However, extra features of the disease dynamics or of the network lead to an increase in system size and analytical tractability becomes problematic. Various `closures' can be used to keep the system tractable. Focusing on SIR epidemics on regular but clustered networks, we show that even for the most complex closure we can determine the epidemic threshold as an asymptotic expansion in terms of the clustering coefficient.We do this by exploiting the presence of a system of fast variables, specified by the correlation structure of the epidemic, whose steady state determines the epidemic threshold. While we do not find the steady state analytically, we create an elegant asymptotic expansion of it. We validate this new threshold by comparing it to the numerical solution of the full system and find excellent agreement over a wide range of values of the clustering coefficient, transmission rate and average degree of the network. The technique carries over to pairwise models with other closures [1] and we note that the epidemic threshold will be model dependent. This emphasises the importance of model choice when dealing with realistic outbreaks.

q-bio.PE↗
arXiv2017

Saturation Effects and the Concurrency Hypothesis: Insights from an Analytic Model

Sexual partnerships that overlap in time (concurrent relationships) may play a significant role in the HIV epidemic, but the precise effect is unclear. We derive edge-based compartmental models of disease spread in idealized dynamic populations with and without concurrency to allow for an investigation of its effects. Our models assume that partnerships change in time and individuals enter and leave the at-risk population. Infected individuals transmit at a constant per-partnership rate to their susceptible partners. In our idealized populations we find regions of parameter space where the existence of concurrent partnerships leads to substantially faster growth and higher equilibrium levels, but also regions in which the existence of concurrent partnerships has very little impact on the growth or the equilibrium. Additionally we find mixed regimes in which concurrency significantly increases the early growth, but has little effect on the ultimate equilibrium level. Guided by model predictions, we discuss general conditions under which concurrent relationships would be expected to have large or small effects in real-world settings. Our observation that the impact of concurrency saturates suggests that concurrency-reducing interventions may be most effective in populations with low to moderate concurrency.

q-bio.PE↗
arXiv2017

An elementary proof of the total progeny size of a birth-death process, with application to network component sizes

We revisit the size distribution of finite components in infinite Configuration Model networks. We provide an elementary combinatorial proof about the sizes of birth-death trees which is more intuitive than previous proofs. We use this to rederive the component size distribution for Configuration Model networks. Our derivation provides a more intuitive interpretation of the formula as contrasted with the previous derivation based on contour integrations. We demonstrate that the formula performs well, even on networks with heavy tails which violate assumptions of the derivation. We explain why the result should remain robust for these networks.

q-bio.QM↗
arXiv2017

Random Spatial Networks: Small Worlds without Clustering, Traveling Waves, and Hop-and-Spread Disease Dynamics

Random network models play a prominent role in modeling, analyzing and understanding complex phenomena on real-life networks. However, a key property of networks is often neglected: many real-world networks exhibit spatial structure, the tendency of a node to select neighbors with a probability depending on physical distance. Here, we introduce a class of random spatial networks (RSNs) which generalizes many existing random network models but adds spatial structure. In these networks, nodes are placed randomly in space and joined in edges with a probability depending on their distance and their individual expected degrees, in a manner that crucially remains analytically tractable. We use this network class to propose a new generalization of small-world networks, where the average shortest path lengths in the graph are small, as in classical Watts-Strogatz small-world networks, but with close spatial proximity of nodes that are neighbors in the network playing the role of large clustering. Small-world effects are demonstrated on these spatial small-world networks without clustering. We are able to derive partial integro-differential equations governing susceptible-infectious-recovered disease spreading through an RSN, and we demonstrate the existence of traveling wave solutions. If the distance kernel governing edge placement decays slower than exponential, the population-scale dynamics are dominated by long-range hops followed by local spread of traveling waves. This provides a theoretical modeling framework for recent observations of how epidemics like Ebola evolve in modern connected societies, with long-range connections seeding new focal points from which the epidemic locally spreads in a wavelike manner.

q-bio.QM↗
arXiv2016

Mathematical models of SIR disease spread with combined non-sexual and sexual transmission routes

The emergence of diseases such as Zika and Ebola has highlighted the need to understand the role of sexual transmission in the spread of diseases with a primarily non-sexual transmission route. In this paper we develop a number of low-dimensional models which are appropriate for a range of assumptions for how a disease will spread if it has sexual transmission through a sexual contact network combined with some other transmission mechanism, such as direct contact or vectors. The equations derived provide exact predictions for the dynamics of the corresponding simulations in the large population limit.

q-bio.PE↗
arXiv2016

Equivalence of several generalized percolation models on networks

In recent years, many variants of percolation have been used to study network structure and the behavior of processes spreading on networks. These include bond percolation, site percolation, $k$-core percolation, bootstrap percolation, the generalized epidemic process, and the Watts Threshold Model (WTM). We show that --- except for bond percolation --- each of these processes arises as a special case of the WTM and bond percolation arises from a small modification. In fact "heterogeneous $k$-core percolation", a corresponding "heterogeneous bootstrap percolation" model, and the generalized epidemic process are completely equivalent to one another and the WTM. We further show that a natural generalization of the WTM in which individuals "transmit" or "send a message" to their neighbors with some probability less than $1$ can be reformulated in terms of the WTM, and so this apparent generalization is in fact not more general. Finally, we show that in bond percolation, finding the set of nodes in the component containing a given node is equivalent to finding the set of nodes activated if that node is initially activated and the node thresholds are chosen from the appropriate distribution. A consequence of these results is that mathematical techniques developed for the WTM apply to these other models as well, and techniques that were developed for some particular case may in fact apply much more generally.

physics.soc-ph↗
arXiv2015

Complex Contagions and hybrid phase transitions in unclustered and clustered random networks

A complex contagion is an infectious process in which individuals may require multiple transmissions before changing state. These are used to model behaviors if an individual only adopts a particular behavior after perceiving a consensus among others. We may think of individuals as beginning inactive and becoming active after contact with a sufficient number of active partners. These have been studied in a number of cases, but analytic models for the dynamic spread of complex contagions are typically complex. Here we study the dynamics of the Watts Threshold Model (WTM) assuming transmission occurs in continuous time as a Poisson process, or in discrete time where individuals transmit to all partners in the time step following their activation. We adapt techniques developed for infectious disease modeling to develop and analyze analytic models for the dynamics of the WTM in Configuration Model networks and a class of random clustered (triangle-based) networks. The resulting model is relatively simple and compact. We use it to gain insights into the contagion dynamics. In the infinite population limit, we derive conditions under which cascades happen with an arbitrarily small initial proportion active, confirming a hypothesis of Watts for this case. We also observe hybrid phase transitions when cascades are not possible for small initial conditions, but occur for large enough initial conditions. We derive sufficient conditions for this hybrid phase transition to occur. We show that in many cases, if the hybrid phase transition occurs, all individuals eventually become active. Finally, we discuss the role clustering plays in facilitating or impeding the spread and find that the hypothesis of Watts that was confirmed in Configuration Model networks does not hold in general. This approach allows us to unify many existing disparate observations and derive new results.

physics.soc-ph↗
arXiv2015

Optimizing Hybrid Spreading in Metapopulations

Epidemic spreading phenomena are ubiquitous in nature and society. Examples include the spreading of diseases, information, and computer viruses. Epidemics can spread by local spreading, where infected nodes can only infect a limited set of direct target nodes and global spreading, where an infected node can infect every other node. In reality, many epidemics spread using a hybrid mixture of both types of spreading. In this study we develop a theoretical framework for studying hybrid epidemics, and examine the optimum balance between spreading mechanisms in terms of achieving the maximum outbreak size. We show the existence of critically hybrid epidemics where neither spreading mechanism alone can cause a noticeable spread but a combination of the two spreading mechanisms would produce an enormous outbreak. Our results provide new strategies for maximising beneficial epidemics and estimating the worst outcome of damaging hybrid epidemics.

physics.soc-ph↗