Sensitivity of Model-Based Epidemiological Parameter Estimation to Model Assumptions

Anchoring the mean generation time in the SEIR to mitigate biases in ℜ0 estimates due to uncertainty in the distribution of the epidemiological delays

The basic reproduction number, ℜ 0 , is of paramount importance in the study of infectious disease dynamics. Primarily, ℜ 0 serves as an indicator of the transmission potential of an emerging infectious disease and the effort required to control the invading pathogen. However, its estimates from compartmental models are strongly conditioned by assumptions in the model structure, such as the distributions of the latent and infectious periods (epidemiological delays). To further complicate matters, models with dissimilar delay structures produce equivalent incidence dynamics. Following a simulation study, we reveal that the nature of such equivalency stems from a linear relationship between ℜ 0 and the mean generation time, along with adjustments to other parameters in the model. Leveraging this knowledge, we propose and successfully test an alternative parametrization of the SEIR model that produces accurate ℜ 0 estimates regardless of the distribution of the epidemiological delays, at the expense of biases in other quantities deemed of lesser importance. We further explore this approach's robustness by testing various transmissibility levels, generation times and data fidelity (overdispersion). Finally, we apply the proposed approach to data from the 1918 influenza pandemic. We anticipate that this work will mitigate biases in estimating ℜ 0 .

Novel model prediction time-to-event analysis: data validation and estimation of 200 million cases in the global COVID-19 epidemic

OBJECTIVES

Assessment of recuperation and death times of a population inflicted by an epidemic has only been feasible through studying a sample of individuals via time-to-event analysis, which requires identified participants. Therefore, we aimed to introduce an original model to estimate the average recovery/death times of infected population of contagious diseases without the need to undertake survival analysis and just through the data of unidentified infected, recovered and dead cases.

DESIGN

Cross-sectional study.

SETTING

An internet source that asserted from official sources of each government. The model includes two techniques-curve fitting and optimisation problems. First, in the curve fitting process, the data of the three classes are simultaneously fitted to functions with defined constraints to derive the average times. In the optimisation problems, data are directly fed to the technique to achieve the average times. Further, the model is applied to the available data of COVID-19 of 200 million people throughout the globe.

RESULTS

The average times obtained by the two techniques indicated conformity with one another showing p values of 0.69, 0.51, 0.48 and 0.13 with one, two, three and four surges in our timespan, respectively. Two types of irregularity are detectable in the data, significant difference between the infected population and the sum of the recovered and deceased population (discrepancy) and abrupt increase in the cumulative distributions (step). Two indices, discrepancy index (DI) and error of fit index (EI), are developed to quantify these irregularities and correlate them with the conformity of the time averages obtained by the two techniques. The correlations between DI and EI and the quantified conformity of the results were -0.74 and -0.93, respectively.

CONCLUSION

The results of statistical analyses point out that the proposed model is suitable to estimate the average times between recovery and death.

Predicting epidemics and the impact of interventions in heterogeneous settings: standard SEIR models are too pessimistic

The basic reproduction number (R0) is an established concept to describe the potential for an infectious disease to cause an epidemic and to derive estimates of the required effect of interventions for successful control. Calculating R0 from simple deterministic transmission models may result in biased estimates when important sources of heterogeneity related to transmission and control are ignored. Using stochastic simulations with a geographically stratified individual-based SEIR (susceptible, exposed, infectious, recovered) model, we illustrate that if heterogeneity is ignored (i.e. no or too little assumed interindividual variation or assortative mixing) this may substantially overestimate the transmission rate and the potential course of the epidemic. Consequently, predictions for the impact of interventions then become relatively pessimistic. However, should such an intervention be suspended, then the potential for a consecutive epidemic wave will depend strongly on assumptions about heterogeneity, with more heterogeneity resulting in lower remaining epidemic potential, due to selection and depletion of high-risk individuals during the early stages of the epidemic. These phenomena have likely also affected current model predictions regarding COVID-19, as most transmission models assume homogeneous mixing or at most employ a simple age stratification, thereby leading to overcautious predictions of durations of lockdowns and required vaccine coverage levels.

Analysis of multi-strain infection of vaccinated and recovered population through epidemic model: Application to COVID-19

In this work, an innovative multi-strain SV EAIR epidemic model is developed for the study of the spread of a multi-strain infectious disease in a population infected by mutations of the disease. The population is assumed to be completely susceptible to n different variants of the disease, and those who are vaccinated and recovered from a specific strain k (k ≤ n) are immune to previous and present strains j = 1, 2, ⋯, k, but can still be infected by newer emerging strains j = k + 1, k + 2, ⋯, n. The model is designed to simulate the emergence and dissemination of viral strains. All the equilibrium points of the system are calculated and the conditions for existence and global stability of these points are investigated and used to answer the question as to whether it is possible for the population to have an endemic with more than one strain. An interesting result that shows that a strain with a reproduction number greater than one can still die out on the long run if a newer emerging strain has a greater reproduction number is verified numerically. The effect of vaccines on the population is also analyzed and a bound for the herd immunity threshold is calculated. The validity of the work done is verified through numerical simulations by applying the proposed model and strategy to analyze the multi-strains of the COVID-19 virus, in particular, the Delta and the Omicron variants, in the United State.

Quantifying Uncertainty in Mechanistic Models of Infectious Disease

This primer describes the statistical uncertainty in mechanistic models and provides R code to quantify it. We begin with an overview of mechanistic models for infectious disease, and then describe the sources of statistical uncertainty in the context of a case study on SARS-CoV-2. We describe the statistical uncertainty as belonging to three categories: data uncertainty, stochastic uncertainty, and structural uncertainty. We demonstrate how to account for each of these via statistical uncertainty measures and sensitivity analyses broadly, as well as in a specific case study on estimating the basic reproductive number, \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{upgreek} \usepackage{mathrsfs} \setlength{\oddsidemargin}{-69pt} \begin{document} }{}${R}_0$\end{document}, for SARS-CoV-2.

Building mean field ODE models using the generalized linear chain trick & Markov chain theory

The well known linear chain trick (LCT) allows modellers to derive mean field ODEs that assume gamma (Erlang) distributed passage times, by transitioning individuals sequentially through a chain of sub-states. The time spent in these sub-states is the sum of k exponentially distributed random variables, and is thus gamma distributed. The generalized linear chain trick (GLCT) extends this technique to the broader phase-type family of distributions, which includes exponential, Erlang, hypoexponential, and Coxian distributions. Phase-type distributions are the family of matrix exponential distributions on [0,∞) that represent the absorption time distributions for finite-state, continuous time Markov chains (CTMCs). Here we review CTMCs and phase-type distributions, then illustrate how to use the GLCT to efficiently build ODE models from underlying stochastic model assumptions. We introduce two novel model families by using the GLCT to generalize the Rosenzweig-MacArthur predator-prey model, and the SEIR model. We illustrate the kinds of complexity that can be captured by such models through multiple examples. We also show the benefits of using a GLCT-based model formulation to speed up the computation of numerical solutions to such models. These results highlight the intuitive nature, and utility, of using the GLCT to derive ODE models from first principles.

A theoretical framework for transitioning from patient-level to population-scale epidemiological dynamics: influenza A as a case study

Multi-scale epidemic forecasting models have been used to inform population-scale predictions with within-host models and/or infection data collected in longitudinal cohort studies. However, most multi-scale models are complex and require significant modelling expertise to run. We formulate an alternative multi-scale modelling framework using a compartmental model with multiple infected stages. In the large-compartment limit, our easy-to-use framework generates identical results compared to previous more complicated approaches. We apply our framework to the case study of influenza A in humans. By using a viral dynamics model to generate synthetic patient-level data, we explore the effects of limited and inaccurate patient data on the accuracy of population-scale forecasts. If infection data are collected daily, we find that a cohort of at least 40 patients is required for a mean population-scale forecasting error below 10%. Forecasting errors may be reduced by including more patients in future cohort studies or by increasing the frequency of observations for each patient. Our work, therefore, provides not only an accessible epidemiological modelling framework but also an insight into the data required for accurate forecasting using multi-scale models.

Generalizations of the 'Linear Chain Trick': incorporating more flexible dwell time distributions into mean field ODE models

In this paper we generalize the Linear Chain Trick (LCT; aka the Gamma Chain Trick) to help provide modelers more flexibility to incorporate appropriate dwell time assumptions into mean field ODEs, and help clarify connections between individual-level stochastic model assumptions and the structure of corresponding mean field ODEs. The LCT is a technique used to construct mean field ODE models from continuous-time stochastic state transition models where the time an individual spends in a given state (i.e., the dwell time) is Erlang distributed (i.e., gamma distributed with integer shape parameter). Despite the LCT’s widespread use, we lack general theory to facilitate the easy application of this technique, especially for complex models. Modelers must therefore choose between constructing ODE models using heuristics with oversimplified dwell time assumptions, using time consuming derivations from first principles, or to instead use non-ODE models (like integro-differential or delay differential equations) which can be cumbersome to derive and analyze. Here, we provide analytical results that enable modelers to more efficiently construct ODE models using the LCT or related extensions. Specifically, we provide (1) novel LCT extensions for various scenarios found in applications, including conditional dwell time distributions; (2) formulations of these LCT extensions that bypass the need to derive ODEs from integral equations; and (3) a novel Generalized Linear Chain Trick (GLCT) framework that extends the LCT to a much broader set of possible dwell time distribution assumptions, including the flexible phase-type distributions which can approximate distributions on \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${\mathbb {R}}^+$$\end{document}R+ and can be fit to data.

Assessing inference of the basic reproduction number in an SIR model incorporating a growth-scaling parameter

The standard mass action, which assumes that infectious disease transmission occurs in well-mixed populations, is popular for formulating compartmental epidemic models. Compartmental epidemic models often follow standard mass action for simplicity and to gain insight into transmission dynamics as it often performs well at reproducing disease dynamics in large populations. In this work, we formulate discrete time stochastic susceptible-infected-removed models with linear (standard) and nonlinear mass action structures to mimic varying mixing levels. Using simulations and real epidemic data, we demonstrate the sensitivity of the basic reproduction number to these mathematical structures of the force of infection. Our results suggest the need to consider nonlinear mass action in order to generate more accurate estimates of the basic reproduction number although its uncertainty increases due to the addition of one growth scaling parameter.

Equations of the End: Teaching Mathematical Modeling Using the Zombie Apocalypse

Mathematical models of infectious diseases are a valuable tool in understanding the mechanisms and patterns of disease transmission. It is, however, a difficult subject to teach, requiring both mathematical expertise and extensive subject-matter knowledge of a variety of disease systems. In this article, we explore several uses of zombie epidemics to make mathematical modeling and infectious disease epidemiology more accessible to public health professionals, students, and the general public. We further introduce a web-based simulation, White Zed (http://cartwrig.ht/apps/whitezed/), that can be deployed in classrooms to allow students to explore models before implementing them. In our experience, zombie epidemics are familiar, approachable, flexible, and an ideal way to introduce basic concepts of infectious disease epidemiology.