Uncertainty and error in SARS-CoV-2 epidemiological parameters inferred from population-level epidemic models.

During the SARS-CoV-2 pandemic, epidemic models have been central to policy-making. Public health responses have been shaped by model-based projections and inferences, especially related to the impact of various non-pharmaceutical interventions. Accompanying this has been increased scrutiny over model performance, model assumptions, and the way that uncertainty is incorporated and presented. Here we consider a population-level model, focusing on how distributions representing host infectiousness and the infection-to-death times are modelled, and particularly on the impact of inferred epidemic characteristics if these distributions are mis-specified. We introduce an SIR-type model with the infected population structured by 'infected age', i.e. the number of days since first being infected, a formulation that enables distributions to be incorporated that are consistent with clinical data. We show that inference based on simpler models without infected age, which implicitly mis-specify these distributions, leads to substantial errors in inferred quantities relevant to policy-making, such as the reproduction number and the impact of interventions. We consider uncertainty quantification via a Bayesian approach, implementing this for both synthetic and real data focusing on UK data in the period 15 Feb-14 Jul 2020, and emphasising circumstances where it is misleading to neglect uncertainty. This manuscript was submitted as part of a theme issue on "Modelling COVID-19 and Preparedness for Future Pandemics".

A mathematical model for simulating the spread of a disease through a country divided into geographical regions with different population densities.

The SIR (susceptible-infectious-recovered) model is a well known method for predicting the number of people (or animals) in a population who become infected by and then recover from a disease. Modifications can include categories such people who have been exposed to the disease but are not yet infectious or those who die from the disease. However, the model has nearly always been applied to the entire population of a country or state but there is considerable observational evidence that diseases can spread at different rates in densely populated urban regions and sparsely populated rural areas. This work presents a new approach that applies a SIR type model to a country or state that has been divided into a number of geographical regions, and uses different infection rates in each region which depend on the population density in that region. Further, the model contains a simple matrix based method for simulating the movement of people between different regions. The model is applied to the spread of disease in the United Kingdom and the state of Rio Grande do Sul in Brazil.

Assessing the effect of school closures on the spread of COVID-19 in Zurich

The effect of school closure on the spread of COVID-19 has been discussed intensively in the literature and the news. To capture the interdependencies between children and adults, we consider daily age-stratified incidence data and contact patterns between age groups which change over time to reflect social distancing policy indicators. We fit a multivariate time-series endemic-epidemic model to such data from the Canton of Zurich, Switzerland and use the model to predict the age-specific incidence in a counterfactual approach (with and without school closures). The results indicate a 17% median increase of incidence in the youngest age group (0-14 year olds), whereas the relative increase in the other age groups drops to values between 2% and 3%. We argue that our approach is more informative to policy makers than summarising the effect of school closures with time-dependent effective reproduction numbers, which are difficult to estimate due to the sparsity of incidence counts within the relevant age groups.

Data-driven clustering of infectious disease incidence into age groups.

Understanding the patterns of infectious diseases spread in the population is an important element of mitigation and vaccination programs. A major and common characteristic of most infectious diseases is age-related heterogeneity in the transmission, which potentially can affect the dynamics of an epidemic as manifested by the pattern of disease incidence in different age groups. Currently there are no statistical criteria of how to partition the disease incidence data into clusters. We develop the first data-driven methodology for deciding on the best partition of incidence data into age-groups, in a well defined statistical sense. The method employs a top-down hierarchical partitioning algorithm, with a stopping criteria based on multiple hypotheses significance testing controlling the family wise error rate. The type one error and statistical power of the method are tested using simulations. The method is then applied to Covid-19 incidence data in Israel, in order to extract the significant age-group clusters in each wave of the epidemic.

A mixed methods analysis of participation in a social contact survey.

BACKGROUND: Social contact survey data forms a core component of modern epidemic models: however, there has been little assessment of the potential biases in such data. METHODS: We conducted focus groups with university students who had (n = 13) and had never (n = 14) completed a social contact survey during the COVID-19 pandemic. Qualitative findings were explored quantitatively by analysing participation data. RESULTS: The opportunity to contribute to COVID-19 research, to be heard and feel useful were frequently reported motivators for participating in the contact survey. Reductions in survey engagement following lifting of COVID-19 restrictions may have occurred because the research was perceived to be less critical and/or because the participants were busier and had more contacts. Having a high number of contacts to report, uncertainty around how to report each contact, and concerns around confidentiality were identified as factors leading to inaccurate reporting. Focus groups participants thought that financial incentives or provision of study results would encourage participation. CONCLUSIONS: Incentives could improve engagement with social contact surveys. Qualitative research can inform the format, timing, and wording of surveys to optimise completion and accuracy.

Estimating the effect of non-pharmaceutical interventions on US SARS-CoV-2 infections in the first year of the pandemic.

SARS-CoV-2 emerged in late 2019 as a zoonotic infection of humans, and proceeded to cause a worldwide pandemic of historic magnitude. Here, we use a simple epidemiological model and consider the full range of initial estimates from published studies for infection and recovery rates, seasonality, changes in mobility, the effectiveness of masks and the fraction of people wearing them. Monte Carlo simulations are used to simulate the progression of possible pandemics and we show a match for the real progression of the pandemic during 2020 with an R 2 of 0.91. The results show that the combination of masks and changes in mobility avoided approximately 248.3 million (σ = 31.2 million) infections in the US before vaccinations became available.

Modelling preventive measures and their effect on generation times in emerging epidemics.

We present a stochastic epidemic model to study the effect of various preventive measures, such as uniform reduction of contacts and transmission, vaccination, isolation, screening and contact tracing, on a disease outbreak in a homogeneously mixing community. The model is based on an infectivity process, which we define through stochastic contact and infectiousness processes, so that each individual has an independent infectivity profile. In particular, we monitor variations of the reproduction number and of the distribution of generation times. We show that some interventions, i.e. uniform reduction and vaccination, affect the former while leaving the latter unchanged, whereas other interventions, i.e. isolation, screening and contact tracing, affect both quantities. We provide a theoretical analysis of the variation of these quantities, and we show that, in practice, the variation of the generation time distribution can be significant and that it can cause biases in the estimation of reproduction numbers. The framework, because of its general nature, captures the properties of many infectious diseases, but particular emphasis is on COVID-19, for which numerical results are provided.

Misinformation can prevent the suppression of epidemics.

The effectiveness of non-pharmaceutical interventions, such as mask-wearing and social distancing, as control measures for pandemic disease relies upon a conscientious and well-informed public who are aware of and prepared to follow advice. Unfortunately, public health messages can be undermined by competing misinformation and conspiracy theories, spread virally through communities that are already distrustful of expert opinion. In this article, we propose and analyse a simple model of the interaction between disease spread and awareness dynamics in a heterogeneous population composed of both trusting individuals who seek better quality information and will take precautionary measures, and distrusting individuals who reject better quality information and have overall riskier behaviour. We show that, as the density of the distrusting population increases, the model passes through a phase transition to a state in which major outbreaks cannot be suppressed. Our work highlights the urgent need for effective interventions to increase trust and inform the public.

Are epidemic growth rates more informative than reproduction numbers?

statistics, often derived from simplified models of epidemic spread, inform public health policy in real time. The instantaneous reproduction number, R t , is predominant among these statistics, measuring the average ability of an infection to multiply. However, R t encodes no temporal information and is sensitive to modelling assumptions. Consequently, some have proposed the epidemic growth rate, r t , that is, the rate of change of the log-transformed case incidence, as a more temporally meaningful and model-agnostic policy guide. We examine this assertion, identifying if and when estimates of r t are more informative than those of R t . We assess their relative strengths both for learning about pathogen transmission mechanisms and for guiding public health interventions in real time.

Source code and secondary data of the stochastic process based COVID-19 simulation model.

The novel coronavirus disease (COVID-19) culminated in a pandemic with many countries affected in varying stages. We aimed to develop a simulation environment for COVID-19 spread, taking environmental and social factors into account. This program consists of three main components; a stochastic process-based model for simulating epidemics, a basic reproduction number estimation unit and a graphics generator. The model can take a variety of environmental factors as input and simulate expected behaviours of the infection spread, enabling policymakers and the scientific community to test the effects of different mitigation strategies in a sandbox.

A numerical method to calculate multiple epidemic waves in COVID-19 with a realistic total number of people involved

We use the total number of individuals involved in the coronavirus disease-2019 (COVID-19), namely, N, inside a specific region as a parameter in the susceptible-infected-quarantined-recovery (SIQR) model of Odagaki. Public data on the number of newly detected individuals are fitted by the numerical results of the SIQR model with optimized parameters. As a result of the optimization, we can determine the total number of individuals involved in COVID-19 inside a specific region and call such an SIQR model with a realistic total number of people involved the SIQR-N model. We then propose two methods to simulate multiple epidemic waves (MEWs), which appear in the time evolution of the number of the newly detected individuals. One is a decomposition of MEWs into independent epidemic waves that can be approximated by multiple time-derivative logistic functions (MTLF). Once the decomposition of the MEWs is completed, we fit the solution of the SIQR-N model to each MTLF using optimized parameters. Finally, we superpose the solutions obtained by multiple SIQR-N (MSIQR-N) models with the optimized parameters to fit the MEWs. The other is a set of N in the SIQR-N model as a function of time, namely, N(t), now called the SIQR-N ( t ) model. Numerical results indicate that a logistic functional approximation of N(t) fits MEWs with good accuracy. Finally, we confirm the availability of the MSIQR-N model with effects of vaccination using the recent data in Israel.

Vaccination with partial transmission and social distancing on contact networks

We study the impact of vaccination on the risk of epidemics spreading through structured networks using the cavity method of statistical physics. We relax the assumption that vaccination prevents all transmission of a disease used in previous studies, such that vaccinated nodes have a small probability of transmission. To do so, we extend the cavity method to study networks where nodes have heterogeneous transmissibility. We find that vaccination with partial transmission still provides herd immunity and show how the herd immunity threshold depends upon the assortativity between nodes of different transmissibility. In addition, we study the impact of social distancing via bond percolation and show that percolation targeting links between nodes of high transmissibility can reduce the risk of an epidemic greater than targeting links between nodes of high degree. Finally, we extend recent methods to compute the distributional equations of risk in populations with heterogeneous transmissibility and show how targeted social distancing measures may reduce overall risk greater than untargeted vaccination campaigns, by comparing the effect of random and targeted strategies of node and link deletion on the risk distribution.

Effects of Vaccination Efficacy on Wealth Distribution in Kinetic Epidemic Models.

The spread of the COVID-19 pandemic has highlighted the close link between economics and health in the context of emergency management. A widespread vaccination campaign is considered the main tool to contain the economic consequences. This paper will focus, at the level of wealth distribution modeling, on the economic improvements induced by the vaccination campaign in terms of its effectiveness rate. The economic trend during the pandemic is evaluated, resorting to a mathematical model joining a classical compartmental model including vaccinated individuals with a kinetic model of wealth distribution based on binary wealth exchanges. The interplay between wealth exchanges and the progress of the infectious disease is realized by assuming, on the one hand, that individuals in different compartments act differently in the economic process and, on the other hand, that the epidemic affects risk in economic transactions. Using the mathematical tools of kinetic theory, it is possible to identify the equilibrium states of the system and the formation of inequalities due to the pandemic in the wealth distribution of the population. Numerical experiments highlight the importance of the vaccination campaign and its positive effects in reducing economic inequalities in the multi-agent society.

Calibrating COVID-19 susceptible-exposed-infected-removed models with time-varying effectivecontact rates.

We describe the population-based susceptible-exposed-infected-removed (SEIR) model developed by the Irish Epidemiological Modelling Advisory Group (IEMAG), which advises the Irish government on COVID-19 responses. The model assumes a time-varying effective contact rate (equivalently, a time-varying reproduction number) to model the effect of non-pharmaceutical interventions. A crucial technical challenge in applying such models is their accurate calibration to observed data, e.g. to the daily number of confirmed new cases, as the history of the disease strongly affects predictions of future scenarios. We demonstrate an approach based on inversion of the SEIR equations in conjunction with statistical modelling and spline-fitting of the data to produce a robust methodology for calibration of a wide class of models of this type. This article is part of the theme issue 'Data science approaches to infectious disease surveillance'.

The impact of multiple non-pharmaceutical interventions on controlling COVID-19 outbreak without lockdown in Hong Kong: A modelling study.

BACKGROUND: The 'third wave' of COVID-19 in Hong Kong, China was suppressed by non-pharmaceutical interventions (NPIs). Although social distancing regulations were quickly strengthened, the outbreak continued to grow, causing increasing delays in tracing and testing. Further regulations were introduced, plus 'targeted testing' services for at-risk groups. Estimating the impact of individual NPIs could provide lessons about how outbreaks can be controlled without radical lockdown. However, the changing delays in confirmation time challenge current modelling methods. We used a novel approach aimed at disentangling and quantifying the effects of individual interventions. METHODS: We incorporated the causes of delays in tracing and testing (i.e. load-efficiency relationship) and the consequences from such delays (i.e. the proportion of un-traced cases and the proportion of traced-cases with confirmation delay) into a deterministic transmission model, which was fitted to the daily number of cases with and without an epi­link (an indication of being contact-traced). The effect of each NPI was then calculated. FINDINGS: The model estimated that after earlier relaxation of regulations, Re rose from 0.7 to 3.2. Restoration of social distancing to the previous state only reduced Re to 1.3, because of increased delay in confirmation caused by load on the contact-tracing system. However, Re decreased by 20.3% after the introduction of targeted testing and by 17.5% after extension of face-mask rules, reducing Re to 0.9 and suppressing the outbreak. The output of the model without incorporation of delay failed to capture important features of transmission and Re. INTERPRETATION: Changing delay in confirmation has a significant impact on disease transmission and estimation of transmissibility. This leads to a clear recommendation that delay should be monitored and mitigated during outbreaks, and that delay dynamics should be incorporated into models to assess the effects of NPIs. FUNDING: City University of Hong Kong and Health and Medical Research Fund.

On some fundamental challenges in monitoring epidemics.

Epidemic models often reflect characteristic features of infectious spreading processes by coupled nonlinear differential equations considering different states of health (such as susceptible, infectious or recovered). This compartmental modelling approach, however, delivers an incomplete picture of the dynamics of epidemics, as it neglects stochastic and network effects, and the role of the measurement process, on which the estimation of epidemiological parameters and incidence values relies. In order to study the related issues, we combine established epidemiological spreading models with a measurement model of the testing process, considering the problems of false positives and false negatives as well as biased sampling. Studying a model-generated ground truth in conjunction with simulated observation processes (virtual measurements) allows one to gain insights into the fundamental limitations of purely data-driven methods when assessing the epidemic situation. We conclude that epidemic monitoring, simulation, and forecasting are wicked problems, as applying a conventional data-driven approach to a complex system with nonlinear dynamics, network effects and uncertainty can be misleading. Nevertheless, some of the errors can be corrected for, using scientific knowledge of the spreading dynamics and the measurement process. We conclude that such corrections should generally be part of epidemic monitoring, modelling and forecasting efforts. This article is part of the theme issue 'Data science approaches to infectious disease surveillance'.

Modelling lockdown measures in epidemic outbreaks using selective socio-economic containment with uncertainty.

After the introduction of drastic containment measures aimed at stopping the epidemic contagion from SARS-CoV2, many governments have adopted a strategy based on a periodic relaxation of such measures in the face of a severe economic crisis caused by lockdowns. Assessing the impact of such openings in relation to the risk of a resumption of the spread of the disease is an extremely difficult problem due to the many unknowns concerning the actual number of people infected, the actual reproduction number and infection fatality rate of the disease. In this work, starting from a SEIRD compartmental model with a social structure based on the age of individuals and stochastic inputs that account for data uncertainty, the effects of containment measures are introduced via an optimal control problem dependent on specific social activities, such as home, work, school, etc. Through a short time horizon approximation, we derive models with multiple feedback controls depending on social activities that allow us to assess the impact of selective relaxation of containment measures in the presence of uncertain data. After analyzing the effects of the various controls, results from different scenarios concerning the first wave of the epidemic in some major countries, including Germany, France, Italy, Spain, the United Kingdom and the United States, are presented and discussed. Specific contact patterns in the home, work, school and other locations have been considered for each country. Numerical simulations show that a careful strategy of progressive relaxation of containment measures, such as that adopted by some governments, may be able to keep the epidemic under control by restarting various productive activities.

Dynamical Analysis of Universal Masking on the Pandemic.

We investigate the impact of the delay in compulsory mask wearing on the spread of COVID-19 in the community, set in the Singapore context. By using modified SEIR-based compartmental models, we focus on macroscopic population-level analysis of the relationships between the delay in compulsory mask wearing and the maximum infection, through a series of scenario-based analysis. Our analysis suggests that collective masking can meaningfully reduce the transmission of COVID-19 in the community, but only if implemented within a critical time window of approximately before 80-100 days delay after the first infection is detected, coupled with strict enforcement to ensure compliance throughout the duration. We also identify a delay threshold of about 100 days that results in masking enforcement having little significant impact on the Maximum Infected Values. The results therefore highlight the necessity for rapid implementation of compulsory mask wearing to curb the spread of the pandemic.

SARS-CoV-2 infection in UK university students: lessons from September-December 2020 and modelling insights for future student return.

In this paper, we present work on SARS-CoV-2 transmission in UK higher education settings using multiple approaches to assess the extent of university outbreaks, how much those outbreaks may have led to spillover in the community, and the expected effects of control measures. Firstly, we found that the distribution of outbreaks in universities in late 2020 was consistent with the expected importation of infection from arriving students. Considering outbreaks at one university, larger halls of residence posed higher risks for transmission. The dynamics of transmission from university outbreaks to wider communities is complex, and while sometimes spillover does occur, occasionally even large outbreaks do not give any detectable signal of spillover to the local population. Secondly, we explored proposed control measures for reopening and keeping open universities. We found the proposal of staggering the return of students to university residence is of limited value in terms of reducing transmission. We show that student adherence to testing and self-isolation is likely to be much more important for reducing transmission during term time. Finally, we explored strategies for testing students in the context of a more transmissible variant and found that frequent testing would be necessary to prevent a major outbreak.

A hospital-related outbreak of SARS-CoV-2 associated with variant Epsilon (B.1.429) in Taiwan: transmission potential and outbreak containment under intensified contact tracing, January-February 2021.

OBJECTIVES: A hospital-related cluster of 22 cases of coronavirus disease 2019 (COVID-19) occurred in Taiwan in January-February 2021. Rigorous control measures were introduced and could only be relaxed once the outbreak was declared over. Each day after the apparent outbreak end, we estimated the risk of future cases occurring in order to inform decision-making. METHODS: Probabilistic transmission networks were reconstructed, and transmission parameters (the reproduction number R and overdispersion parameter k) were estimated. The reporting delay during the outbreak was estimated (Scenario 1). In addition, a counterfactual scenario with less effective interventions characterized by a longer reporting delay was considered (Scenario 2). Each day, the risk of future cases was estimated under both scenarios. RESULTS: The values of R and k were estimated to be 1.30 ((95% credible interval (CI) 0.57-3.80) and 0.38 (95% CI 0.12-1.20), respectively. The mean reporting delays considered were 2.5 days (Scenario 1) and 7.8 days (Scenario 2). Following the final case, ttthe inferred probability of future cases occurring declined more quickly in Scenario 1 than Scenario 2. CONCLUSIONS: Rigorous control measures allowed the outbreak to be declared over quickly following outbreak containment. This highlights the need for effective interventions, not only to reduce cases during outbreaks but also to allow outbreaks to be declared over with confidence.