Schedule for: 25w5369 - Mathematical and Statistical Challenges in Post-Pandemic Epidemiology and Public Health

We study a variant of the classical Markovian logistic SIS epidemic model on a complete graph, which has the additional feature that healthy individuals can become infected without contacting an infected member of the population. This additional ``self-infection'' is used to model situations where there is an unknown source of infection or an external disease reservoir, such as an animal carrier population. In contrast to the classical logistic SIS epidemic model, the version with self-infection has a non-degenerate stationary distribution, and we derive precise asymptotics for the time to converge to stationarity (mixing time) as the population size becomes large. It turns out that the chain exhibits the cutoff phenomenon, which is a sharp transition in time from one to zero of the total variation distance to stationarity. We obtain the exact leading constant for the cutoff time, and show the window size is constant (optimal) order.

I will present an analysis of, and modelling based on, the Office for National Statistics’ COVID-19 Infection Survey (ONS CIS), a large community study of half a million individuals with a longitudinal household design. This allows the relative rates of within- and between-household transmission to be estimated over time, as well as the impacts of vaccination, strains, age and NPIs. The methods for such tasks are in constant development and I will also discuss some general mathematical challenges and results that arise from inference for modern household-stratified data.

In 1927, Kermack and McKendrick proposed a SIR model with age-of-infection dependent infectivity. Recently, R. Forien, G. Pang and E. P. proved in a paper published by SIAM J. Appl. Math. that this model is a law of large numbers limit, as the size of the population tends to infinity, of appropriate stochastic models. In 1932, Kermack and McKendrick added to their model the progressive waning of immunity. In a paper to appear in the AAP, R. Forien, G. Pang E. P. and A. B. Zotsa-Ngoufack show that this model is a particular case of a more general model, which is the LLN limit of appropriate stochastic models. All those stochastic models are non Markovian, and the limits are not ODE models but integral equation models (which can be turned into PDE models, where the age of infection (and possibly of recovery) is added as an additional variable). This means that those deterministic models, like true epidemics, have memory. We show that the widely used simplified ODE models have at least two two pitfalls. Concerning the SIR model, the Markovian/ODE model tends to underestimate the time needed for a declining epidemic to go extinct. Concerning the loss of immunity, the sudden total loss of immunity (which is present in most SIS or SIRS models) tends to underestimate the severity of the infection, and hence the necessary vaccination coverage, compared to models incorporating the more realistic progressive loss of immunity. This does not mean that one should abandon completely the ODE and memoryless models, but at lest that one should realize the bias which they lead to.

Epidemiological models must be consistent with observed data in order to generate reliable knowledge and evidence-based policy. Metapopulation systems, which consist of a network of connected sub-populations, pose technical challenges in statistical inference owing to nonlinear, stochastic interactions. Numerical difficulties encountered in conducting inference can obstruct the core scientific questions concerning the link between the mathematical models and the data. Recently, an algorithm has been proposed that enables computationally tractable likelihood-based inference for high-dimensional partially observed stochastic dynamic models of metapopulation systems. We use this algorithm to build a statistically principled data analysis workflow for metapopulation systems. Via a case study of COVID-19, we show how this workflow addresses the limitations of previous approaches. The COVID-19 pandemic provides a situation where mathematical models and their policy implications were widely visible, and we revisit an influential metapopulation model used to inform basic epidemiological understanding early in the pandemic. Our methods support self-critical data analysis, enabling us to identify and address model weaknesses, leading to a new model with substantially improved statistical fit and parameter identifiability. Our results suggest that the lockdown initiated on 23 January 2020 in China was more effective than previously thought.

Variation in fitness traits lends population studies prone to selective depletion biases. When an ageing cohort exhibits declining mortality, it could be individuals becoming healthier or selective depletion of the frail. In an epidemic, when growth in cumulative infections decelerates, it could be individuals cautiously changing behaviour or selective depletion of the most susceptible. In microbial populations, when an isogenic population is stressed by antimicrobial treatment and some cells survive, this could be due to individual cells switching between normal and persister phenotypes or the antibiotic selectively killing cells that divide faster. In each case, the first explanation invokes individuals changing (1), while the second posits selection on pre-existing variation (2). While explanations of type (1) are intuitive and widely adopted, those of type (2) are often neglected due to challenges in estimating all variation that matters. To overcome these selective depletion biases, we propose remodelling selection in study design and analysis. The procedure is being tested in systems where trait distributions can be inferred from population trends as well as reconstructed directly from individual measurements. Results of this ongoing research will be presented, and the wider applicability discussed.

Public health mitigation programs, including vaccination and non-pharmaceutical intervention programs, are used to mitigate infectious disease morbidity and mortality. Mathematical models that incorporate the effects of immunity and behvaiour change can be used to quantify the effects of such public health programs in isolation and in combination. In this talk, we will discuss our recent work in behaviour and immunity modelling. Model results associated with COVID-19 vaccination and non-pharmaceutical intervention programs will be highlighted.

Prisons present a unique environment for the spread of infectious diseases such as influenza, tuberculosis, and SARS-CoV-2: prison residents have generally poorer health than average individuals of similar age; are generally confined in close proximity of each other, with regulated movements and interaction with staff often physical in nature; and their turnover constantly replenishes the susceptible population. Prisons are therefore highly vulnerable to new introductions, large outbreaks and significant disease burden. Although logistically difficult to manage, non-pharmaceutical interventions have been used during the COVID-19 pandemic. Inspired by this, I will present an analysis of an intervention that has not been used in the UK, namely asymptomatic testing and isolation of the entire population very early on to interrupt an outbreak before it becomes large. I will compare it with the more common approach of isolating cases based on symptoms, highlighting differences in terms of effectiveness as well as logistical challenges involved. Authors: Brooks J, Pellis L, Scarabel F, Bakker P, UKHSA prison team, Hall I, Edge C, Fowler T

The effective reproductive number (Rt) is a key epidemiological metric that captures the temporal dynamics of an infectious disease outbreak and serves as a critical indicator of intervention effectiveness. Accurate and timely estimation of Rt is especially vital during the early stages of an epidemic. Traditional estimation approaches are often compromised by biases arising from misspecified generation or serial intervals, as well as incomplete or delayed case reporting. In this study, we propose a novel estimation framework that eliminates the need to estimate the generation or serial interval. We illustrate the approach using synthetically generated human sensor network data that records individual infection times, applying likelihood-based inference within a standard SEIR modeling framework to estimate Rt. The proposed method demonstrates improved robustness compared to conventional estimators, particularly in scenarios where classical assumptions do not hold.

Monday's panel session will feature the speakers from Monday morning, who presented on themes 1 (Multi-strain, multi-pathogen, and multi-level dynamics) and 6 (Informing rapid responses to emerging infections). We will be able to ask more detailed questions on the morning's presentations and discuss as a group topics related to themes 1 and 6. Panel chair: Wasiur

We will begin this first working group session with a short introduction in in Conference Room San Felipe

Social contact studies, investigating how many, and with whom, individuals tend to have close contact with, have been used to improve modelling of the potential for spread of infections in the community considered. The main output from such studies is an age-structured contact matrix $M$ describing the average number of contacts between individuals of different age-groups. $M$ often shows that social mixing is assortative with respect to age, and that younger age-groups have more contacts overall. What is lost when summarizing contact surveys with the matrix $M$ is the heterogeneity within age-groups. In the talk I will describe how this can be taken into account, and what the consequences are on the potential for the spread of infections. (Joint work with Frank Ball).

The COVID-19 pandemic is an example of an epidemic for which the incidence of cases was highly dependent on age. We propose several age-structured models with the goal of identifying the type of population heterogeneity (e.g., susceptibility to infection) underlying the epidemic dynamics. We employ data from COVID-19 cases in Mexico to parametrize and compare models. We discuss our findings on heterogeneity, as well as limitations and potential approaches to improve our analysis.

The contact process (SIS epidemic model) has long been studied as a model for the spread of infectious disease through a population. One important question concerns the long-term behavior of the epidemic--does it result in a large outbreak, or does the infection die out quickly? There is a large literature on the effects of the underlying population structure on this long-term behavior. However, the role of individual-level behavioral responses to the epidemic is less studied. In this talk, I will introduce the contact process, some key ideas used to analyze it, and a few notable results. I will then discuss my recent work on modified versions of the contact process that include individual-level behavioral responses to the epidemic. I will present some results on how individual-level behavioral responses can influence the long-term behavior of an epidemic and discuss why analyzing these models is mathematically challenging.

We examine how behavioral changes in vaccinated people who do not develop immunity influence the dynamics of a directly transmitted disease and key indices such as the basic reproductive number and vaccine effectiveness. We propose a model that considers a vaccine with three facets of failure: “take”, “degree”, and “duration”. Additionally, the behavioral change of non-immune vaccinated individuals is modeled through a parameter that adjusts their contact rate based on compliance with mitigation measures. Our results allow us to visualize the role of behavioral change in various factors influencing disease transmission dynamics. First, we demonstrate the existence of a backward bifurcation common in models for not fully effective vaccines. Second, we define a behavioral index threshold, which serves as a key indicator for determining whether the disease persists due to behavioral effects. Finally, our results highlight that both the behavioral index and the initial value of the infected population can play a decisive role in determining whether vaccine effectiveness reaches negative values.

Open time for Working Groups

Open time for Working Groups

Most compartment-based infectious disease models tend to ignore social variables such as ethnicity and socioeconomic deprivation. However, these can be key determinants of disease transmission and impact in heterogeneous populations. Accounting for these effects is crucial in models that are designed to be used for policy advice and decision support. Agent-based models offer one approach to including these types of variables, but there are drawbacks to using agent-based models. Here, I will talk about some preliminary work to stratify compartment-based models by ethnicity in the New Zealand population. I will discuss some of the benefits and challenges of this approach, including how they can help disentangle the relative contributions of factors such as vaccination rates, contact rates and clinical severity to observed health inequities.

Malaria remains a public health burden affecting billions of people worldwide, and the transmission of malaria parasites from human hosts to mosquitoes is an essential step in the life cycle of the parasite. In this study, we developed a stochastic model of human-to-mosquito transmission which integrates intra-host parasite dynamics in both human and mosquito hosts and fit the model to data from a direct feeding assay in a human challenge study. Then, by embedding these intra-host dynamics and the transmission probability (per bite) in an epidemiological transmission dynamics framework, we explore the clinical and population level impacts of alternative treatment strategies. Authors: James M McCaw, Xiao Sun, Pengxing Cao

Tuesday's panel session will feature the speakers from Tuesday, who presented mostly on theme 2 (Behaviour and heterogeneity). We will be able to ask more detailed questions on the morning's presentations and discuss as a group topics related to theme 2. Panel chair: Joel

In this talk, I will discuss the problem of parameter inference in the context of epidemic modelling when we have access to only a sparse data set, which is almost always the case. I will discuss a new statistical methodology and its theoretical guarantees under different regimes of sparsity of data and probabilistic asymptotic properties of the model. No prior background in statistics is required.

Dynamical Survival Analysis (DSA) offers a novel framework for epidemic modeling by describing infection dynamics through survival functions governed by differential equations. This approach has proven effective for parameter inference in both network-based and mass-action epidemic models. In this talk, I will review the core ideas behind DSA and highlight recent developments in local SIR models, which incorporate heterogeneity in contact patterns and infectiousness rates. These models provide greater flexibility in capturing realistic transmission patterns and enhance both parameter estimation and predictive accuracy. I will illustrate the framework with examples that demonstrate its advantages in modeling and inference for complex epidemic systems.

We introduce a Bayesian sequential data assimilation and forecasting method for non-autonomous dynamical systems. We applied this method to the COVID-19 pandemic, assuming previously validated transmission, epidemic, and observation models. The transmission and epidemic models are encoded in a dynamical system, while the observation model—cast as a likelihood function—depends on state variables and parameters. Forecasts are sequentially updated over a sliding window as new data becomes available. Prior distributions for the next forecast are obtained by propagating the current posterior using the dynamical system. Since epidemic systems are non-autonomous, influenced by human behavior, viral evolution, and climate, long-term forecasting is unreliable. We illustrate our approach using a SEIR-type model and COVID-19 data from various Mexican localities, and discuss the insights it provides. Our method offers a balance between data fitting and dynamical system prediction.

Dynamical models based on ordinary differential equations (ODEs) play a fundamental role in understanding and forecasting the spread of infectious diseases. However, the reliability of these models hinges on the proper assessment of structural and practical identifiability, robust parameter estimation techniques, and the quantification of uncertainty in model forecasts. In this talk, I will present a systematic methodological workflow for parameter estimation and forecasting in epidemic contexts, emphasizing critical components such as model validation, identifiability analysis, and uncertainty propagation. A key focus will be on existing tools—including those developed in my lab—that facilitate rigorous evaluation of structural and practical identifiability to ensure parameters can be uniquely and reliably estimated from available data. These elements are crucial for generating forecasts that are both accurate and actionable in public health settings. I will illustrate these techniques using real epidemic data, demonstrating the iterative workflow for integrating epidemiological and molecular surveillance data into model calibration and validation. Specifically, I will highlight how incorporating multiple data streams enhances the practical identifiability of key epidemiological parameters, leading to more precise and reliable short-term forecasts.

While genomic epidemiological models are now routinely used to track the emergence of novel pathogens and strains, their use in forecasting is still developing. In this talk, I will present a statistical modelling framework that forecasts the size of an upcoming COVID-19 wave, such as that driven by a new variant. This framework combines diverse global data: bringing together COVID-19 genomic sequences with epidemiological, clinical and demographic features. We assess which predictors were more or less influential on wave size, and how this varied between pandemic waves. Focusing on the Omicron BA.1 and BA.2 waves, we found that local genomic landscapes and demographic features were impactful on wave sizes around the world, and the importance of predictors changed markedly between waves, reflecting ongoing changes in underlying epidemiology and our public health response.

Phylodynamic analyses can be heavily biased by the choice of tree prior. A large class of phylodynamic methods are based on the birth-death process, however empirical phylogenies across different fields tend to be ill-characterized by this model. For example, in macroevolution, species trees tend to be more imbalanced than expected under a birth-death model, while trees in developmental biology can be highly synchronous and much more balanced than expected. We will here present a phylodynamic method which generalizes the traditional birth-death likelihood and can capture a broader range of tree shapes. We present an efficient and scalable approximation to the exact tree density which allows us to perform inference in a Bayesian setting.

Pairwise regression models can simultaneously estimate the effects of covariates on susceptibility to infection and infectiousness, giving them the potential to produce detailed epidemiological insights that can inform public health interventions. We will show how they were used at Ohio State to estimate secondary attack risks in classrooms and dormitories, to estimate the effects of quarantine and isolation policies on campus, and to inform decisions about limits on the size of in-person classes. These models can be extended in several ways. Pathogen genetic sequences can be incorporated to reduce bias and variance in parameter estimates by providing partial information about who-infected-whom. We propose a pruning algorithm to develop efficient proposals for transmission trees. We also summarize ongoing work to develop model diagnostics similar to those for standard regression models from survival analysis. Finally, we outline the need to develop graphical and statistical methods for causal inference in pairwise survival analysis.

Pathogen genomes can provide deep insight into transmission, including transmission that had occurred long before surveillance began and potentially in other locations. However, current phylodynamic methods for gaining this insight often have a considerable computational cost and are often constrained by their simplifying assumptions. (Both of which are difficulties that plague many epidemiological models.) Neural estimation is a simulation-based inference method that can overcome these limitations. We will show how a simple recursive neural network, trained on simulated data, can take a viral phylogeny and then predict the effective reproduction number, prevalence, and cumulative infections through time.

Accurately modeling the dynamics of infectious disease transmission remains a challenge, as changes in human behavior, viral mutations, and public health interventions can significantly alter the trajectory of an epidemic. In this work, we propose a hybrid framework that combines a mechanistic Susceptible-Infected-Recovered (SIR) model with machine learning techniques to estimate the time-varying reproduction rate, which implicitly captures these variations. This framework also provides a method to estimate the time-varying reproduction number, offering valuable information on whether an epidemic is contracting or expanding. We apply our approach retrospectively to COVID incidence data at different state levels in the USA, enabling the model to fit more effectively across multiple epidemic waves. Our methodology offers insights into how variations in the effective reproduction number reflect the impact of behavioral and epidemiological changes over time.

The SIR model is widely used for modeling epidemic dynamics. Despite its widespread use, parameter inference in the presence of missing data is challenging due to the intractability of the likelihood for such compartmental models. To address this, we develop a closed-form likelihood for incidence data using the dynamical survival analysis (DSA) method, which offers flexibility and computational efficiency. Through simulation, we compare its performance in parameter estimation with other methods that rely on the exact posterior of the SIR model. To further demonstrate the adaptability of our approach, we extend the likelihood to frailty models, illustrating how it can be modified to incorporate individual heterogeneity. Finally, we apply our method to real-world data, demonstrating its practical utility for epidemic inference with limited observations.

Thursday's panel session will feature the speakers from Wednesday and Thursday, who presented on themes 3 (Linking epidemiology to genomics, phylogenetics and phylodynamics), 4 (Data opportunities and limitations) and 5 (Identifiability and Scalability). We will be able to ask more detailed questions on these speakers' presentations and discuss as a group topics related to theme 3, 4 and 5. Panel chair: Greg

This session will be dedicated to professional development and navigating an academic career. Topics to include finding an academic position. We will have some short presentations/reflections, followed by open time for discussion and peer mentorship. Session chair: Jessica

Open time for Working Groups

Open time for Working Groups

Open time for Working Groups