A Gaussian-process approximation to a spatial SIR process using moment closures and emulators.

The dynamics that govern disease spread are hard to model because infections are functions of both the underlying pathogen as well as human or animal behavior. This challenge is increased when modeling how diseases spread between different spatial locations. Many proposed spatial epidemiological models require trade-offs to fit, either by abstracting away theoretical spread dynamics, fitting a deterministic model, or by requiring large computational resources for many simulations. We propose an approach that approximates the complex spatial spread dynamics with a Gaussian process. We first propose a flexible spatial extension to the well-known SIR stochastic process, and then we derive a moment-closure approximation to this stochastic process. This moment-closure approximation yields ordinary differential equations for the evolution of the means and covariances of the susceptibles and infectious through time. Because these ODEs are a bottleneck to fitting our model by MCMC, we approximate them using a low-rank emulator. This approximation serves as the basis for our hierarchical model for noisy, underreported counts of new infections by spatial location and time. We demonstrate using our model to conduct inference on simulated infections from the underlying, true spatial SIR jump process. We then apply our method to model counts of new Zika infections in Brazil from late 2015 through early 2016.

Riding the waves from epidemic to endemic: Viral mutations, immunological change and policy responses.

Nonpharmaceutical interventions (NPI) are an important tool for countering pandemics such as COVID-19. Some are cheap; others disrupt economic, educational, and social activity. The latter force governments to balance the health benefits of reduced infection and death against broader lockdown-induced societal costs. A literature has developed modeling how to optimally adjust lockdown intensity as an epidemic evolves. This paper extends that literature by augmenting the classic SIR model with additional states and flows capturing decay over time in vaccine-conferred immunity, the possibility that mutations create variants that erode immunity, and that protection against infection erodes faster than protecting against severe illness. As in past models, we find that small changes in parameter values can tip the optimal response between very different solutions, but the extensions considered here create new types of solutions. In some instances, it can be optimal to incur perpetual epidemic waves even if the uncontrolled infection prevalence would settle down to a stable intermediate level.

Propagation properties in a multi-species SIR reaction-diffusion system.

We consider a multi-species reaction-diffusion system that arises in epidemiology to describe the spread of several strains, or variants, of a disease in a population. Our model is a natural spatial, multi-species, extension of the classical SIR model of Kermack and McKendrick. First, we study the long-time behavior of the solutions and show that there is a "selection via propagation" phenomenon: starting with N strains, only a subset of them - that we identify - propagates and invades space, with some given speeds that we compute. Then, we obtain some qualitative properties concerning the effects of the competition between the different strains on the outcome of the epidemic. In particular, we prove that the dynamics of the model is not characterized by the usual notion of basic reproduction number, which strongly differs from the classical case with one strain.

The hammer and the jab: Are COVID-19 lockdowns and vaccinations complements or substitutes?

The COVID-19 pandemic has devastated lives and economies around the world. Initially a primary response was locking down parts of the economy to reduce social interactions and, hence, the virus' spread. After vaccines have been developed and produced in sufficient quantity, they can largely replace broad lock downs. This paper explores how lockdown policies should be varied during the year or so gap between when a vaccine is approved and when all who wish have been vaccinated. Are vaccines and lockdowns substitutes during that crucial time, in the sense that lockdowns should be reduced as vaccination rates rise? Or might they be complementary with the prospect of imminent vaccination increasing the value of stricter lockdowns, since hospitalization and death averted then may be permanently prevented, not just delayed? We investigate this question with a simple dynamic optimization model that captures both epidemiological and economic considerations. In this model, increasing the rate of vaccine deployment may increase or reduce the optimal total lockdown intensity and duration, depending on the values of other model parameters. That vaccines and lockdowns can act as either substitutes or complements even in a relatively simple model casts doubt on whether in more complicated models or the real world one should expect them to always be just one or the other. Within our model, for parameter values reflecting conditions in developed countries, the typical finding is to ease lockdown intensity gradually after substantial shares of the population have been vaccinated, but other strategies can be optimal for other parameter values. Reserving vaccines for those who have not yet been infected barely outperforms simpler strategies that ignore prior infection status. For certain parameter combinations, there are instances in which two quite different policies can perform equally well, and sometimes very small increases in vaccine capacity can tip the optimal solution to one that involves much longer and more intense lockdowns.

Estimating the propagation of both reported and undocumented COVID-19 cases in Spain: a panel data frontier approximation of epidemiological models.

We use a stochastic frontier analysis (SFA) approach to model the propagation of the COVID-19 epidemic across geographical areas. The proposed models permit reported and undocumented cases to be estimated, which is important as case counts are overwhelmingly believed to be undercounted. The models can be estimated using only epidemic-type data but are flexible enough to permit these reporting rates to vary across geographical cross-section units of observation. We provide an empirical application of our models to Spanish data corresponding to the initial months of the original outbreak of the virus in early 2020. We find remarkable rates of under-reporting that might explain why the Spanish Government took its time to implement strict mitigation strategies. We also provide insights into the effectiveness of the national and regional lockdown measures and the influence of socio-economic factors in the propagation of the virus.

A simulation approach for COVID-19 pandemic assessment based on vaccine logistics, SARS-CoV-2 variants, and spread rate.

Despite advances in clinical care for the coronavirus (COVID-19) pandemic, population-wide interventions are vital to effectively manage the pandemic due to its rapid spread and the emergence of different variants. One of the most important interventions to control the spread of the disease is vaccination. In this study, an extended Susceptible-Infected Healed (SIR) model based on System Dynamics was designed, considering the factors affecting the rate of spread of the COVID-19 pandemic. The model predicts how long it will take to reach 70% herd immunity based on the number of vaccines administered. The designed simulation model is modeled in AnyLogic 8.7.2 program. The model was performed for three different vaccine supply scenarios and for Turkey with ~83 million population. The results show that, with a monthly supply of 15 million vaccines, social immunity reached the target value of 70% in 161 days, while this number was 117 days for 30 million vaccines and 98 days for 40 million vaccines.

Revisiting classical SIR modelling in light of the COVID-19 pandemic.

Background: Classical infectious disease models during epidemics have widespread usage, from predicting the probability of new infections to developing vaccination plans for informing policy decisions and public health responses. However, it is important to correctly classify reported data and understand how this impacts estimation of model parameters. The COVID-19 pandemic has provided an abundant amount of data that allow for thorough testing of disease modelling assumptions, as well as how we think about classical infectious disease modelling paradigms. Objective: We aim to assess the appropriateness of model parameter estimates and prediction results in classical infectious disease compartmental modelling frameworks given available data types (infected, active, quarantined, and recovered cases) for situations where just one data type is available to fit the model. Our main focus is on how model prediction results are dependent on data being assigned to the right model compartment. Methods: We first use simulated data to explore parameter reliability and prediction capability with three formulations of the classical Susceptible-Infected-Removed (SIR) modelling framework. We then explore two applications with reported data to assess which data and models are sufficient for reliable model parameter estimation and prediction accuracy: a classical influenza outbreak in a boarding school in England and COVID-19 data from the fall of 2020 in Missoula County, Montana, USA. Results: We demonstrated the magnitude of parameter estimation errors and subsequent prediction errors resulting from data misclassification to model compartments with simulated data. We showed that prediction accuracy in each formulation of the classical disease modelling framework was largely determined by correct data classification versus misclassification. Using a classical example of influenza epidemics in an England boarding school, we argue that the Susceptible-Infected-Quarantined-Recovered (SIQR) model is more appropriate than the commonly employed SIR model given the data collected (number of active cases). Similarly, we show in the COVID-19 disease model example that reported active cases could be used inappropriately in the SIR modelling framework if treated as infected. Conclusions: We demonstrate the role of misclassification of disease data and thus the importance of correctly classifying reported data to the proper compartment using both simulated and real data. For both a classical influenza data set and a COVID-19 case data set, we demonstrate the implications of using the "right" data in the "wrong" model. The importance of correctly classifying reported data will have downstream impacts on predictions of number of infections, as well as minimal vaccination requirements.

Mutation and SARS-CoV-2 strain competition under vaccination in a modified SIR model.

The crisis caused by the COVID-19 outbreak around the globe raised an increasing concern about the ongoing emergence of variants of the virus that may evade the immune response provided by vaccines. New variants appear due to mutation, and as the cases accumulate, the probability of the emergence of a variant of concern increases. In this article, we propose a modified susceptible, infected, and recovered (SIR) model with waning immunity that captures the competition of two strain classes of an infectious disease under the effect of vaccination with a highly contagious and deadlier strain class emerging from a prior strain due to mutation. When these strains compete for a limited supply of susceptible individuals, changes in the efficiency of vaccines may affect the behaviour of the disease in a non-trivial way, resulting in complex outcomes. We characterise the parameter space including intrinsic parameters of the disease, and using the vaccine efficiencies as control variables. We find different types of transcritical bifurcations between endemic fixed points and a disease-free equilibrium and identify a region of strain competition where the two strain classes coexist during a transient period. We show that a strain can be extinguished either due to strain competition or vaccination, and we obtain the critical values of the efficiency of vaccines to eradicate the disease. Numerical studies using parameters estimated from publicly reported data agree with our theoretical results. Our mathematical model could be a tool to assess quantitatively the vaccination policies of competing and emerging strains using the dynamics in epidemics of infectious diseases.

Global sensitivity analysis in epidemiological modeling.

Operations researchers worldwide rely extensively on quantitative simulations to model alternative aspects of the COVID-19 pandemic. Proper uncertainty quantification and sensitivity analysis are fundamental to enrich the modeling process and communicate correctly informed insights to decision-makers. We develop a methodology to obtain insights on key uncertainty drivers, trend analysis and interaction quantification through an innovative combination of probabilistic sensitivity techniques and machine learning tools. We illustrate the approach by applying it to a representative of the family of susceptible-infectious-recovered (SIR) models recently used in the context of the COVID-19 pandemic. We focus on data of the early pandemic progression in Italy and the United States (the U.S.). We perform the analysis for both cases of correlated and uncorrelated inputs. Results show that quarantine rate and intervention time are the key uncertainty drivers, have opposite effects on the number of total infected individuals and are involved in the most relevant interactions.

Statistical Challenges in Tracking the Evolution of SARS-CoV-2.

Genomic surveillance of SARS-CoV-2 has been instrumental in tracking the spread and evolution of the virus during the pandemic. The availability of SARS-CoV-2 molecular sequences isolated from infected individuals, coupled with phylodynamic methods, have provided insights into the origin of the virus, its evolutionary rate, the timing of introductions, the patterns of transmission, and the rise of novel variants that have spread through populations. Despite enormous global efforts of governments, laboratories, and researchers to collect and sequence molecular data, many challenges remain in analyzing and interpreting the data collected. Here, we describe the models and methods currently used to monitor the spread of SARS-CoV-2, discuss long-standing and new statistical challenges, and propose a method for tracking the rise of novel variants during the epidemic.

Simulation applications to support teaching and research in epidemiological dynamics.

BACKGROUND: An understanding of epidemiological dynamics, once confined to mathematical epidemiologists and applied mathematicians, can be disseminated to a non-mathematical community of health care professionals and applied biologists through simple-to-use simulation applications. We used Numerus Model Builder RAMP Ⓡ (Runtime Alterable Model Platform) technology, to construct deterministic and stochastic versions of compartmental SIR (Susceptible, Infectious, Recovered with immunity) models as simple-to-use, freely available, epidemic simulation application programs. RESULTS: We take the reader through simulations used to demonstrate the following concepts: 1) disease prevalence curves of unmitigated outbreaks have a single peak and result in epidemics that 'burn' through the population to become extinguished when the proportion of the susceptible population drops below a critical level; 2) if immunity in recovered individuals wanes sufficiently fast then the disease persists indefinitely as an endemic state, with possible dampening oscillations following the initial outbreak phase; 3) the steepness and initial peak of the prevalence curve are influenced by the basic reproductive value R0, which must exceed 1 for an epidemic to occur; 4) the probability that a single infectious individual in a closed population (i.e. no migration) gives rise to an epidemic increases with the value of R0>1; 5) behavior that adaptively decreases the contact rate among individuals with increasing prevalence has major effects on the prevalence curve including dramatic flattening of the prevalence curve along with the generation of multiple prevalence peaks; 6) the impacts of treatment are complicated to model because they effect multiple processes including transmission, recovery and mortality; 7) the impacts of vaccination policies, constrained by a fixed number of vaccination regimens and by the rate and timing of delivery, are crucially important to maximizing the ability of vaccination programs to reduce mortality. CONCLUSION: Our presentation makes transparent the key assumptions underlying SIR epidemic models. Our RAMP simulators are meant to augment rather than replace classroom material when teaching epidemiological dynamics. They are sufficiently versatile to be used by students to address a range of research questions for term papers and even dissertations.

Generalized SIR (GSIR) epidemic model: An improved framework for the predictive monitoring of COVID-19 pandemic.

Novel coronavirus respiratory disease COVID-19 has caused havoc in many countries across the globe. In order to contain infection of this highly contagious disease, most of the world population is constrained to live in a complete or partial lockdown for months together with a minimal human-to-human interaction having far reaching consequences on countries' economy and mental well-being of their citizens. Hence, there is a need for a good predictive model for the health advisory bodies and decision makers for taking calculated proactive measures to contain the pandemic and maintain a healthy economy. This paper extends the mathematical theory of the classical Susceptible-Infected-Removed (SIR) epidemic model and proposes a Generalized SIR (GSIR) model that is an integrative model encompassing multiple waves of daily reported cases. Existing growth function models of epidemic have been shown as the special cases of the GSIR model. Dynamic modeling of the parameters reflect the impact of policy decisions, social awareness, and the availability of medication during the pandemic. GSIR framework can be utilized to find a good fit or predictive model for any pandemic. The study is performed on the COVID-19 data for various countries with detailed results for India, Brazil, United States of America (USA), and World. The peak infection, total expected number of COVID-19 cases and thereof deaths, time-varying reproduction number, and various other parameters are estimated from the available data using the proposed methodology. The proposed GSIR model advances the existing theory and yields promising results for continuous predictive monitoring of COVID-19 pandemic.

Forecasting for COVID-19 has failed.

Epidemic forecasting has a dubious track-record, and its failures became more prominent with COVID-19. Poor data input, wrong modeling assumptions, high sensitivity of estimates, lack of incorporation of epidemiological features, poor past evidence on effects of available interventions, lack of transparency, errors, lack of determinacy, consideration of only one or a few dimensions of the problem at hand, lack of expertise in crucial disciplines, groupthink and bandwagon effects, and selective reporting are some of the causes of these failures. Nevertheless, epidemic forecasting is unlikely to be abandoned. Some (but not all) of these problems can be fixed. Careful modeling of predictive distributions rather than focusing on point estimates, considering multiple dimensions of impact, and continuously reappraising models based on their validated performance may help. If extreme values are considered, extremes should be considered for the consequences of multiple dimensions of impact so as to continuously calibrate predictive insights and decision-making. When major decisions (e.g. draconian lockdowns) are based on forecasts, the harms (in terms of health, economy, and society at large) and the asymmetry of risks need to be approached in a holistic fashion, considering the totality of the evidence.

Road to recovery: Managing an epidemic.

Without widespread immunization, the road to recovery from the current COVID-19 lockdowns will optimally follow a path that finds the difficult balance between the social and economic benefits of liberty and the toll from the disease. We provide an approach that combines epidemiology and economic models, taking as given that the maximum capacity of the healthcare system imposes a constraint that must not be exceeded. Treating the transmission rate as a decreasing function of the severity of the lockdown, we first determine the minimal lockdown that satisfies this constraint using an epidemiology model with a homogeneous population to predict future demand for healthcare. Allowing for a heterogeneous population, we then derive the optimal lockdown policy under the assumption of homogeneous mixing and show that it is characterized by a bang-bang solution. Possibilities such as the capacity of the healthcare system increasing or a vaccine arriving at some point in the future do not substantively impact the dynamically optimal policy until such an event actually occurs.

Effect estimates of COVID-19 non-pharmaceutical interventions are non-robust and highly model-dependent.

OBJECTIVE: To compare the inference regarding the effectiveness of the various non-pharmaceutical interventions (NPIs) for COVID-19 obtained from different SIR models. STUDY DESIGN AND SETTING: We explored two models developed by Imperial College that considered only NPIs without accounting for mobility (model 1) or only mobility (model 2), and a model accounting for the combination of mobility and NPIs (model 3). Imperial College applied models 1 and 2 to 11 European countries and to the USA, respectively. We applied these models to 14 European countries (original 11 plus another 3), over two different time horizons. RESULTS: While model 1 found that lockdown was the most effective measure in the original 11 countries, model 2 showed that lockdown had little or no benefit as it was typically introduced at a point when the time-varying reproduction number was already very low. Model 3 found that the simple banning of public events was beneficial, while lockdown had no consistent impact. Based on Bayesian metrics, model 2 was better supported by the data than either model 1 or model 3 for both time horizons. CONCLUSION: Inferences on effects of NPIs are non-robust and highly sensitive to model specification. In the SIR modeling framework, the impacts of lockdown are uncertain and highly model-dependent.

Comment on "Poorly known aspects of flattening the curve of COVID-19″.

This short research note describes and summarizes several recent peer-reviewed and non-peer-reviewed studies on the concept of flattening-the-curve (FTC) in the context of the COVID-19 pandemic. This note also highlights contradictory findings of these studies in terms of the effect of FTC on the total number of infections (the final epidemic size), and poses a research problem for future studies.

Simple SIR models with Markovian control.

We consider a random dynamical system, where the deterministic dynamics are driven by a finite-state space Markov chain. We provide a comprehensive introduction to the required mathematical apparatus and then turn to a special focus on the susceptible-infected-recovered epidemiological model with random steering. Through simulations we visualize the behaviour of the system and the effect of the high-frequency limit of the driving Markov chain. We formulate some questions and conjectures of a purely theoretical nature.

Law of mass action and saturation in SIR model with application to Coronavirus modelling.

When using SIR and related models, it is common to assume that the infection rate is proportional to the product of susceptible and infected individuals. While this assumption works at the onset of the outbreak, the infection force saturates as the outbreak progresses, even in the absence of any interventions. We use a simple agent-based model to illustrate this saturation effect. Its continuum limit leads a modified SIR model with exponential saturation. The derivation is based on first principles incorporating the spread radius and population density. We use the data for coronavirus outbreak for the period from March to June, to show that using SIR model with saturation is sufficient to capture the disease dynamics for many jurstictions, including the overall world-wide disease curve progression. Our model suggests the R 0 value of above 8 at the onset of infection, but with infection quickly "flattening out", leading to a long-term sustained sub-exponential spread.

A time-varying SIRD model for the COVID-19 contagion in Italy.

The purpose of this work is to give a contribution to the understanding of the COVID-19 contagion in Italy. To this end, we developed a modified Susceptible-Infected-Recovered-Deceased (SIRD) model for the contagion, and we used official data of the pandemic for identifying the parameters of this model. Our approach features two main non-standard aspects. The first one is that model parameters can be time-varying, allowing us to capture possible changes of the epidemic behavior, due for example to containment measures enforced by authorities or modifications of the epidemic characteristics and to the effect of advanced antiviral treatments. The time-varying parameters are written as linear combinations of basis functions and are then inferred from data using sparse identification techniques. The second non-standard aspect resides in the fact that we consider as model parameters also the initial number of susceptible individuals, as well as the proportionality factor relating the detected number of positives with the actual (and unknown) number of infected individuals. Identifying the model parameters amounts to a non-convex identification problem that we solve by means of a nested approach, consisting in a one-dimensional grid search in the outer loop, with a Lasso optimization problem in the inner step.

How fast is the linear chain trick? A rigorous analysis in the context of behavioral epidemiology.

A prototype SIR model with vaccination at birth is analyzed in terms of the stability of its endemic equilibrium. The information available on the disease influences the parents' decision on whether vaccinate or not. This information is modeled with a delay according to the Erlang distribution. The latter includes the degenerate case of fading memory as well as the limiting case of concentrated memory. The linear chain trick is the essential tool used to investigate the general case. Besides its novel analysis and that of the concentrated case, it is showed that through the linear chain trick a distributed delay approaches a discrete delay at a linear rate. A rigorous proof is given in terms of the eigenvalues of the associated linearized problems and extension to general models is also provided. The work is completed with several computations and relevant experimental results.