The Effective Reproduction Number as a Prelude to Statistical Estimation of Time-Dependent Epidemic Trends

rtestim: Time-varying reproduction number estimation with trend filtering

To understand the transmissibility and spread of infectious diseases, epidemiologists turn to estimates of the instantaneous reproduction number. While many estimation approaches exist, their utility may be limited. Challenges of surveillance data collection, model assumptions that are unverifiable with data alone, and computationally inefficient frameworks are critical limitations for many existing approaches. We propose a discrete spline-based approach that solves a convex optimization problem-Poisson trend filtering-using the proximal Newton method. It produces a locally adaptive estimator for instantaneous reproduction number estimation with heterogeneous smoothness. Our methodology remains accurate even under some process misspecifications and is computationally efficient, even for large-scale data. The implementation is easily accessible in a lightweight R package rtestim.

Physical Chemistry Lab for Data Analysis of COVID-19 Spreading Kinetics in Different Countries

The COVID-19 pandemic has passed. It gives us a real-world example of kinetic data analysis practice for our undergraduate physical chemistry laboratory class. It is a great example to connect this seemingly very different problem to the kinetic theories for chemical reactions that the students have learned in the lecture class. At the beginning of the spring 2023 semester, we obtained COVID-19 kinetic data from the "Our World in Data" database, which summarizes the World Health Organization (WHO) data reported from different countries. We analyzed the effective spreading kinetics based on the susceptible-infectious-recovered-vaccinated (SIR-V) model. We then compared the effective rate constants represented by the real-time reproduction numbers (R t ) underlining the reported data for these countries and discussed the results and the limitations of the model with the students.

Analysis of the impact of COVID-19 variants and vaccination on the time-varying reproduction number: statistical methods

Introduction

The COVID-19 pandemic has profoundly impacted global health systems, requiring the monitoring of infection waves and strategies to control transmission. Estimating the time-varying reproduction number is crucial for understanding the epidemic and guiding interventions.

Methods

Probability distributions of serial interval are estimated for Pre-Delta and Delta periods. We conducted a comparative analysis of time-varying reproduction numbers, taking into account population immunity and variant differences. We incorporated the regional heterogeneity and age distribution of the population, as well as the evolving variants and vaccination rates over time. COVID-19 transmission dynamics were analyzed with variants and vaccination.

Results

The reproduction number is computed with and without considering variant-based immunity. In addition, values of reproduction number significantly differed by variants, emphasizing immunity's importance. Enhanced vaccination efforts and stringent control measures were effective in reducing the transmission of the Delta variant. Conversely, Pre-Delta variant appeared less influenced by immunity levels, due to lower vaccination rates. Furthermore, during the Pre-Delta period, there was a significant difference between the region-specific and the non-region-specific reproduction numbers, with particularly distinct pattern differences observed in Gangwon, Gyeongbuk, and Jeju in Korea.

Discussion

This research elucidates the dynamics of COVID-19 transmission concerning the dominance of the Delta variant, the efficacy of vaccinations, and the influence of immunity levels. It highlights the necessity for targeted interventions and extensive vaccination coverage. This study makes a significant contribution to the understanding of disease transmission mechanisms and informs public health strategies.

Estimating time-varying epidemiological parameters and underreporting of Covid-19 cases in Brazil using a mathematical model with fuzzy transitions between epidemic periods

Our study conducts a comprehensive analysis of the Covid-19 pandemic in Brazil, spanning five waves over three years. We employed a novel Susceptible-Infected-Recovered-Dead-Susceptible (SIRDS) model with a fuzzy transition between epidemic periods to estimate time-varying parameters and evaluate case underreporting. The initial basic reproduction number (R0) is identified at 2.44 (95% Confidence Interval (CI): 2.42-2.46), decreasing to 1.00 (95% CI: 0.99-1.01) during the first wave. The model estimates an underreporting factor of 12.9 (95% CI: 12.5-13.2) more infections than officially reported by Brazilian health authorities, with an increasing factor of 5.8 (95% CI: 5.2-6.4), 12.9 (95% CI: 12.5-13.3), and 16.8 (95% CI: 15.8-17.5) in 2020, 2021, and 2022 respectively. Additionally, the Infection Fatality Rate (IFR) is initially 0.88% (95% CI: 0.81%-0.94%) during the initial phase but consistently reduces across subsequent outbreaks, reaching its lowest value of 0.018% (95% CI: 0.011-0.033) in the last outbreak. Regarding the immunity period, the observed uncertainty and low sensitivity indicate that inferring this parameter is particularly challenging. Brazil successfully reduced R0 during the first wave, coinciding with decreased human mobility. Ineffective public health measures during the second wave resulted in the highest mortality rates within the studied period. We attribute lower mortality rates in 2022 to increased vaccination coverage and the lower lethality of the Omicron variant. We demonstrate the model generalization by its application to other countries. Comparative analyses with serological research further validate the accuracy of the model. In forecasting analysis, our model provides reasonable outbreak predictions. In conclusion, our study provides a nuanced understanding of the Covid-19 pandemic in Brazil, employing a novel epidemiological model. The findings contribute to the broader discourse on pandemic dynamics, underreporting, and the effectiveness of health interventions.

A novel indicator in epidemic monitoring through a case study of Ebola in West Africa (2014-2016)

The E/S (exposed/susceptible) ratio is analyzed in the SEIR model. The ratio plays a key role in understanding epidemic dynamics during the 2014-2016 Ebola outbreak in Sierra Leone and Guinea. The maximum value of the ratio occurs immediately before or after the time-dependent reproduction number (Rt) equals 1, depending on the initial susceptible population (S(0)). It is demonstrated that transmission rate curves corresponding to various incubation periods intersect at a single point referred to as the Cross Point (CP). At this point, the E/S ratio reaches an extremum, signifying a critical shift in transmission dynamics and aligning with the time when Rt approaches 1. By plotting transmission rate curves, β(t), for any two arbitrary incubation periods and tracking their intersections, we can trace CP over time. CP serves as an indicator of epidemic status, especially when Rt is close to 1. It provides a practical means of monitoring epidemics without prior knowledge of the incubation period. Through a case study, we estimate the transmission rate and reproduction number, identifying CP and Rt = 1 while examining the E/S ratio across various values of S(0).

Comparing Contact Tracing Through Bluetooth and GPS Surveillance Data: Simulation-Driven Approach

BACKGROUND

Accurate and responsive epidemiological simulations of epidemic outbreaks inform decision-making to mitigate the impact of pandemics. These simulations must be grounded in quantities derived from measurements, among which the parameters associated with contacts between individuals are notoriously difficult to estimate. Digital contact tracing data, such as those provided by Bluetooth beaconing or GPS colocating, can provide more precise measures of contact than traditional methods based on direct observation or self-reporting. Both measurement modalities have shortcomings and are prone to false positives or negatives, as unmeasured environmental influences bias the data.

OBJECTIVE

We aim to compare GPS colocated versus Bluetooth beacon-derived proximity contact data for their impacts on transmission models' results under community and types of diseases.

METHODS

We examined the contact patterns derived from 3 data sets collected in 2016, with participants comprising students and staff from the University of Saskatchewan in Canada. Each of these 3 data sets used both Bluetooth beaconing and GPS localization on smartphones running the Ethica Data (Avicenna Research) app to collect sensor data about every 5 minutes over a month. We compared the structure of contact networks inferred from proximity contact data collected with the modalities of GPS colocating and Bluetooth beaconing. We assessed the impact of sensing modalities on the simulation results of transmission models informed by proximate contacts derived from sensing data. Specifically, we compared the incidence number, attack rate, and individual infection risks across simulation results of agent-based susceptible-exposed-infectious-removed transmission models of 4 different contagious diseases. We have demonstrated their differences with violin plots, 2-tailed t tests, and Kullback-Leibler divergence.

RESULTS

Both network structure analyses show visually salient differences in proximity contact data collected between GPS colocating and Bluetooth beaconing, regardless of the underlying population. Significant differences were found for the estimated attack rate based on distance threshold, measurement modality, and simulated disease. This finding demonstrates that the sensor modality used to trace contact can have a significant impact on the expected propagation of a disease through a population. The violin plots of attack rate and Kullback-Leibler divergence of individual infection risks demonstrated discernible differences for different sensing modalities, regardless of the underlying population and diseases. The results of the t tests on attack rate between different sensing modalities were mostly significant (P<.001).

CONCLUSIONS

We show that the contact networks generated from these 2 measurement modalities are different and generate significantly different attack rates across multiple data sets and pathogens. While both modalities offer higher-resolution portraits of contact behavior than is possible with most traditional contact measures, the differential impact of measurement modality on the simulation outcome cannot be ignored and must be addressed in studies only using a single measure of contact in the future.

Scalable gradients enable Hamiltonian Monte Carlo sampling for phylodynamic inference under episodic birth-death-sampling models

Birth-death models play a key role in phylodynamic analysis for their interpretation in terms of key epidemiological parameters. In particular, models with piecewise-constant rates varying at different epochs in time, to which we refer as episodic birth-death-sampling (EBDS) models, are valuable for their reflection of changing transmission dynamics over time. A challenge, however, that persists with current time-varying model inference procedures is their lack of computational efficiency. This limitation hinders the full utilization of these models in large-scale phylodynamic analyses, especially when dealing with high-dimensional parameter vectors that exhibit strong correlations. We present here a linear-time algorithm to compute the gradient of the birth-death model sampling density with respect to all time-varying parameters, and we implement this algorithm within a gradient-based Hamiltonian Monte Carlo (HMC) sampler to alleviate the computational burden of conducting inference under a wide variety of structures of, as well as priors for, EBDS processes. We assess this approach using three different real world data examples, including the HIV epidemic in Odesa, Ukraine, seasonal influenza A/H3N2 virus dynamics in New York state, America, and Ebola outbreak in West Africa. HMC sampling exhibits a substantial efficiency boost, delivering a 10- to 200-fold increase in minimum effective sample size per unit-time, in comparison to a Metropolis-Hastings-based approach. Additionally, we show the robustness of our implementation in both allowing for flexible prior choices and in modeling the transmission dynamics of various pathogens by accurately capturing the changing trend of viral effective reproductive number.

Estimation of the Time-Varying Effective Reproductive Number of COVID-19 Based on Multivariate Time Series of Severe Health Outcomes

The time-varying effective reproduction number (Rt at time t) measures the transmissibility of SARS-CoV-2 and is conventionally based on daily case counts, which may suffer from time-varying ascertainment. We analyzed Rt estimates from case counts and severe COVID-19 (intensive care unit admissions, severe or critical cases, and mortality) across 2022 in Hong Kong's fifth and sixth waves of infection. Within the fifth wave, the severe disease-based Rt (3.5) was significantly higher than the case-based Rt (2.4) but not in the sixth wave. During periods with fluctuating underreporting, data based on severe diseases may provide more reliable Rt estimates.

GrowthPredict: A toolbox and tutorial-based primer for fitting and forecasting growth trajectories using phenomenological growth models

Simple dynamic modeling tools can help generate real-time short-term forecasts with quantified uncertainty of the trajectory of diverse growth processes unfolding in nature and society, including disease outbreaks. An easy-to-use and flexible toolbox for this purpose is lacking. This tutorial-based primer introduces and illustrates GrowthPredict, a user-friendly MATLAB toolbox for fitting and forecasting time-series trajectories using phenomenological dynamic growth models based on ordinary differential equations. This toolbox is accessible to a broad audience, including students training in mathematical biology, applied statistics, and infectious disease modeling, as well as researchers and policymakers who need to conduct short-term forecasts in real-time. The models included in the toolbox capture exponential and sub-exponential growth patterns that typically follow a rising pattern followed by a decline phase, a common feature of contagion processes. Models include the 1-parameter exponential growth model and the 2-parameter generalized-growth model, which have proven useful in characterizing and forecasting the ascending phase of epidemic outbreaks. It also includes the 2-parameter Gompertz model, the 3-parameter generalized logistic-growth model, and the 3-parameter Richards model, which have demonstrated competitive performance in forecasting single peak outbreaks. We provide detailed guidance on forecasting time-series trajectories and available software ( https://github.com/gchowell/forecasting_growthmodels ), including the full uncertainty distribution derived through parametric bootstrapping, which is needed to construct prediction intervals and evaluate their accuracy. Functions are available to assess forecasting performance across different models, estimation methods, error structures in the data, and forecasting horizons. The toolbox also includes functions to quantify forecasting performance using metrics that evaluate point and distributional forecasts, including the weighted interval score. This tutorial and toolbox can be broadly applied to characterizing and forecasting time-series data using simple phenomenological growth models. As a contagion process takes off, the tools presented in this tutorial can help create forecasts to guide policy regarding implementing control strategies and assess the impact of interventions. The toolbox functionality is demonstrated through various examples, including a tutorial video, and the examples use publicly available data on the monkeypox (mpox) epidemic in the USA.

Mathematical modeling of the infectious spread and outbreak dynamics of avian influenza with seasonality transmission for chicken farms

A compartmental model with a time-varying contact rate, the seasonality effect, and its corresponding nonautonomous model are investigated. The model is developed based on the six compartments: susceptible, latent, infected, asymptomatic, treated, and recovered individuals. We determine the effective reproduction number for this nonautonomous system, and analytic discussion shows that at least one positive periodic solution exists for R0>1. The model is simulated using the RK-45 numerical method, and the parameter values for the model are taken from the available literature. From the numerical results, we observe that the degree of seasonality and vaccine efficacy significantly impact the amplitude of the epidemic curve. The latent-infected phase plane shows that periodic solutions exhibit a period-doubling bifurcation as the amplitude of seasonality increases. Finally, the model outcome was compared with the actual field data and found to be consistent.

The effectiveness of vaccination, testing, and lockdown strategies against COVID-19

The ability of various policy activities to reduce the reproduction rate of the COVID-19 disease is widely discussed. Using a stringency index that comprises a variety of lockdown levels, such as school and workplace closures, we analyze the effectiveness of government restrictions. At the same time, we investigate the capacity of a range of lockdown measures to lower the reproduction rate by considering vaccination rates and testing strategies. By including all three components in an SIR (Susceptible, Infected, Recovery) model, we show that a general and comprehensive test strategy is instrumental in reducing the spread of COVID-19. The empirical study demonstrates that testing and isolation represent a highly effective and preferable approach towards overcoming the pandemic, in particular until vaccination rates have risen to the point of herd immunity.

Dynamic analysis of rabies transmission and elimination in mainland China

Rabies is an acute zoonotic infectious disease caused by rabies virus. In 2015, the World Health Organization proposed the goal of eliminating dog-induced human rabies by 2030. In response to this goal positively, China has been dedicated to the control and elimination of rabies mainly caused by dogs, for nearly 10 years. By applying infectious disease dynamics, in this paper, we establish a dog-human rabies transmission model to forecast future epidemic trends of rabies, assess whether the goal of eliminating dog-induced human rabies cases in China can be achieved in 2030, and further evaluate and suggest the follow-up sustained preventive measures after the elimination of human rabies. By analyzing and simulating above dynamic model, it is concluded that rabies has been well controlled in China in recent years, but dog-induced human rabies cannot be eliminated by 2030 according to current situation. In addition, we propose to improve rabies control efforts by increasing the immunization coverage rate of rural domestic dogs, controlling the number of stray dogs and preventing the import of rabies virus in wild animals. Immunization coverage rate of rural domestic dogs which is currently less than 10% is far from requirement, and it needs to reach 50%-60% to meet the goal of 2030. Since it is difficult to immunize stray dogs, we suggest to control the number of stray dogs below 15.27 million to achieve the goal. If the goal of eliminating human rabies is reached in 2030, the essential immunization coverage needs to be maintained for 18 years to reduce the number of canine rabies cases to zero. Lastly, to prevent transmission of rabies virus from wild animals to dogs, the thresholds of the number of dogs and the immunization coverage rate of dogs after eliminating canine rabies cases are also discussed.

Estimating effective reproduction number revisited

Accurately estimating the effective reproduction number is crucial for characterizing the transmissibility of infectious diseases to optimize interventions and responses during epidemic outbreaks. In this study, we improve the estimation of the effective reproduction number through two main approaches. First, we derive a discrete model to represent a time series of case counts and propose an estimation method based on this framework. We also conduct numerical experiments to demonstrate the effectiveness of the proposed discretization scheme. By doing so, we enhance the accuracy of approximating the underlying epidemic process compared to previous methods, even when the counting period is similar to the mean generation time of an infectious disease. Second, we employ a negative binomial distribution to model the variability of count data to accommodate overdispersion. Specifically, given that observed incidence counts follow a negative binomial distribution, the posterior distribution of secondary infections is obtained as a Dirichlet multinomial distribution. With this formulation, we establish posterior uncertainty bounds for the effective reproduction number. Finally, we demonstrate the effectiveness of the proposed method using incidence data from the COVID-19 pandemic.

Scalable gradients enable Hamiltonian Monte Carlo sampling for phylodynamic inference under episodic birth-death-sampling models

Birth-death models play a key role in phylodynamic analysis for their interpretation in terms of key epidemiological parameters. In particular, models with piecewise-constant rates varying at different epochs in time, to which we refer as episodic birth-death-sampling (EBDS) models, are valuable for their reflection of changing transmission dynamics over time. A challenge, however, that persists with current time-varying model inference procedures is their lack of computational efficiency. This limitation hinders the full utilization of these models in large-scale phylodynamic analyses, especially when dealing with high-dimensional parameter vectors that exhibit strong correlations. We present here a linear-time algorithm to compute the gradient of the birth-death model sampling density with respect to all time-varying parameters, and we implement this algorithm within a gradient-based Hamiltonian Monte Carlo (HMC) sampler to alleviate the computational burden of conducting inference under a wide variety of structures of, as well as priors for, EBDS processes. We assess this approach using three different real world data examples, including the HIV epidemic in Odesa, Ukraine, seasonal influenza A/H3N2 virus dynamics in New York state, America, and Ebola outbreak in West Africa. HMC sampling exhibits a substantial efficiency boost, delivering a 10- to 200-fold increase in minimum effective sample size per unit-time, in comparison to a Metropolis-Hastings-based approach. Additionally, we show the robustness of our implementation in both allowing for flexible prior choices and in modeling the transmission dynamics of various pathogens by accurately capturing the changing trend of viral effective reproductive number.

Monitoring the reproductive number of COVID-19 in France: Comparative estimates from three datasets

BACKGROUND

The effective reproduction number (Rt) quantifies the average number of secondary cases caused by one person with an infectious disease. Near-real-time monitoring of Rt during an outbreak is a major indicator used to monitor changes in disease transmission and assess the effectiveness of interventions. The estimation of Rt usually requires the identification of infected cases in the population, which can prove challenging with the available data, especially when asymptomatic people or with mild symptoms are not usually screened. The purpose of this study was to perform sensitivity analysis of Rt estimates for COVID-19 surveillance in France based on three data sources with different sensitivities and specificities for identifying infected cases.

METHODS

We applied a statistical method developed by Cori et al. to estimate Rt using (1) confirmed cases identified from positive virological tests in the population, (2) suspected cases recorded by a national network of emergency departments, and (3) COVID-19 hospital admissions recorded by a national administrative system to manage hospital organization.

RESULTS

Rt estimates in France from May 27, 2020, to August 12, 2022, showed similar temporal trends regardless of the dataset. Estimates based on the daily number of confirmed cases provided an earlier signal than the two other sources, with an average lag of 3 and 6 days for estimates based on emergency department visits and hospital admissions, respectively.

CONCLUSION

The COVID-19 experience confirmed that monitoring temporal changes in Rt was a key indicator to help the public health authorities control the outbreak in real time. However, gaining access to data on all infected people in the population in order to estimate Rt is not straightforward in practice. As this analysis has shown, the opportunity to use more readily available data to estimate Rt trends, provided that it is highly correlated with the spread of infection, provides a practical solution for monitoring the COVID-19 pandemic and indeed any other epidemic.

Multivariate epidemic count time series model

An infectious disease spreads not only over a single population or community but also across multiple and heterogeneous communities. Moreover, its transmissibility varies over time because of various factors such as seasonality and epidemic control, which results in strongly nonstationary behavior. In conventional methods for assessing transmissibility trends or changes, univariate time-varying reproduction numbers are calculated without taking into account transmission across multiple communities. In this paper, we propose a multivariate-count time series model for epidemics. We also propose a statistical method for estimating the transmission of infections across multiple communities and the time-varying reproduction numbers of each community simultaneously from a multivariate time series of case counts. We apply our method to incidence data for the novel coronavirus disease 2019 (COVID-19) pandemic to reveal the spatiotemporal heterogeneity of the epidemic process.

An integrated chance constraints approach for optimal vaccination strategies under uncertainty for COVID-19

Despite concerted efforts by health authorities worldwide to contain COVID-19, the SARS-CoV-2 virus has continued to spread and mutate into new variants with uncertain transmission characteristics. Therefore, there is a need for new data-driven models for determining optimal vaccination strategies that adapt to the new variants with their uncertain transmission characteristics. Motivated by this challenge, we derive an integrated chance constraints stochastic programming (ICC-SP) approach for finding vaccination strategies for epidemics that incorporates population demographics for any region of the world, uncertain disease transmission and vaccine efficacy. An optimal vaccination strategy specifies the proportion of individuals in a given household-type to vaccinate to bring the reproduction number to below one. The ICC-SP approach provides a quantitative method that allows to bound the expected excess of the reproduction number above one by an acceptable amount according to the decision-maker's level of risk. This new methodology involves a multi-community household based epidemiology model that uses census demographics data, vaccination status, age-related heterogeneity in disease susceptibility and infectivity, virus variants, and vaccine efficacy. The new methodology was tested on real data for seven neighboring counties in the United States state of Texas. The results are promising and show, among other findings, that vaccination strategies for controlling an outbreak should prioritize vaccinating certain household sizes as well as age groups with relatively high combined susceptibility and infectivity.

A new method for the joint estimation of instantaneous reproductive number and serial interval during epidemics

Although some methods for estimating the instantaneous reproductive number during epidemics have been developed, the existing frameworks usually require information on the distribution of the serial interval and/or additional contact tracing data. However, in the case of outbreaks of emerging infectious diseases with an unknown natural history or undetermined characteristics, the serial interval and/or contact tracing data are often not available, resulting in inaccurate estimates for this quantity. In the present study, a new framework was specifically designed for joint estimates of the instantaneous reproductive number and serial interval. Concretely, a likelihood function for the two quantities was first introduced. Then, the instantaneous reproductive number and the serial interval were modeled parametrically as a function of time using the interpolation method and a known traditional distribution, respectively. Using the Bayesian information criterion and the Markov Chain Monte Carlo method, we ultimately obtained their estimates and distribution. The simulation study revealed that our estimates of the two quantities were consistent with the ground truth. Seven data sets of historical epidemics were considered and further verified the robust performance of our method. Therefore, to some extent, even if we know only the daily incidence, our method can accurately estimate the instantaneous reproductive number and serial interval to provide crucial information for policymakers to design appropriate prevention and control interventions during epidemics.

The Effects of Time Window-Averaged Mobility on Effective Reproduction Number of COVID-19 Viral Variants in Urban Cities

During epidemics, the estimation of the effective reproduction number (ERN) associated with infectious disease is a challenging topic for policy development and medical resource management. The emergence of new viral variants is common in widespread pandemics including the severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2). A simple approach is required toward an appropriate and timely policy decision for understanding the potential ERN of new variants is required for policy revision. We investigated time-averaged mobility at transit stations as a surrogate to correlate with the ERN using the data from three urban prefectures in Japan. The optimal time windows, i.e., latency and duration, for the mobility to relate with the ERN were investigated. The optimal latency and duration were 5-6 and 8 days, respectively (the Spearman's ρ was 0.109-0.512 in Tokyo, 0.365-0.607 in Osaka, and 0.317-0.631 in Aichi). The same linear correlation was confirmed in Singapore and London. The mobility-adjusted ERN of the Alpha variant was 15-30%, which was 20-40% higher than the original Wuhan strain in Osaka, Aichi, and London. Similarly, the mobility-adjusted ERN of the Delta variant was 20%-40% higher than that of the Wuhan strain in Osaka and Aichi. The proposed metric would be useful for the proper evaluation of the infectivity of different SARS-CoV-2 variants in terms of ERN as well as the design of the forecasting system.

An approximate Bayesian approach for estimation of the instantaneous reproduction number under misreported epidemic data

In epidemic models, the effective reproduction number is of central importance to assess the transmission dynamics of an infectious disease and to orient health intervention strategies. Publicly shared data during an outbreak often suffers from two sources of misreporting (underreporting and delay in reporting) that should not be overlooked when estimating epidemiological parameters. The main statistical challenge in models that intrinsically account for a misreporting process lies in the joint estimation of the time-varying reproduction number and the delay/underreporting parameters. Existing Bayesian approaches typically rely on Markov chain Monte Carlo algorithms that are extremely costly from a computational perspective. We propose a much faster alternative based on Laplacian-P-splines (LPS) that combines Bayesian penalized B-splines for flexible and smooth estimation of the instantaneous reproduction number and Laplace approximations to selected posterior distributions for fast computation. Assuming a known generation interval distribution, the incidence at a given calendar time is governed by the epidemic renewal equation and the delay structure is specified through a composite link framework. Laplace approximations to the conditional posterior of the spline vector are obtained from analytical versions of the gradient and Hessian of the log-likelihood, implying a drastic speed-up in the computation of posterior estimates. Furthermore, the proposed LPS approach can be used to obtain point estimates and approximate credible intervals for the delay and reporting probabilities. Simulation of epidemics with different combinations for the underreporting rate and delay structure (one-day, two-day, and weekend delays) show that the proposed LPS methodology delivers fast and accurate estimates outperforming existing methods that do not take into account underreporting and delay patterns. Finally, LPS is illustrated in two real case studies of epidemic outbreaks.

A Small Surge in Incidence of SARS-CoV-2 Omicron Variant in the "Dynamic Zero" Period

To describe the epidemiological characteristics and transmission dynamics of SARS-CoV-2 Omicron variant during "Dynamic Zero" period, we analyzed data on the 108 laboratory-confirmed SARS-CoV-2 cases during 14 to 30 May 2022 in Beichen district, Tianjin, China. We collected information on demographic characteristics, exposure history, and illness timelines of the 108 cases. We described characteristics of the patients and estimated the key epidemiological parameters, including serial interval and the time-dependent reproduction number of the Omicron variant, Rt. Among the 108 laboratory-confirmed patients, the median age was 38 years old, and 50.9% were females. Obvious symptoms were observed among 67.6% (73/108) of all cases, and major clinical manifestations included fever, sore throat, and cough, which occurred in 31.5%, 26.9%, and 19.4% of the 108 cases, respectively. The mean and standard deviation of the SI were estimated as 2.89 and 0.95 days, the Rt varied from 1.24 to 0.27 for a 7-day timelapse. The low reproduction number and the Omicron outbreak being suppressed within a short time marked the effectiveness of the implemented public health measures, such as nucleic acid screening, social distancing, masking, vaccination, medical treatment of patients, and isolation of close contacts. These measures play an important role in fulfilling the goal of controlling the spread of the disease.

An evaluation of the early impact of the COVID-19 pandemic on Zambia's routine immunization program

Implications of the COVID-19 pandemic for both populations and healthcare systems are vast. In addition to morbidity and mortality from COVID-19, the pandemic also disrupted local health systems, including reductions or delays in routine vaccination services and catch-up vaccination campaigns. These disruptions could lead to outbreaks of other infectious diseases that result in an additional burden of disease and strain on the healthcare system. We evaluated the impact of the COVID-19 pandemic on Zambia's routine childhood immunization program in 2020 using multiple sources of data. We relied on administrative vaccination data and Zambia's 2018 Demographic and Health Survey to project national disruptions to district-specific routine childhood vaccination coverage within the pandemic year 2020. Next, we leveraged a 2016 population-based serological survey to predict age-specific measles seroprevalence and assessed the impact of changes in vaccination coverage on measles outbreak risk in each district. We found minor disruptions to routine administration of measles-rubella and pentavalent vaccines in 2020. This was in part due to Zambia's Child Health Week held in June of 2020 which helped to reach children missed during the first six months of the year. We estimated that the two-month delay in a measles-rubella vaccination campaign, originally planned for September of 2020 but conducted in November of 2020 as a result of the pandemic, had little impact on modeled district-specific measles outbreak risks. This study estimated minimal increases in the number of children missed by vaccination services in Zambia during 2020. However, the ongoing SARS-CoV-2 transmission since our analysis concluded means efforts to maintain routine immunization services and minimize the risk of measles outbreaks will continue to be critical. The methodological framework developed in this analysis relied on routinely collected data to estimate disruptions of the COVID-19 pandemic to national routine vaccination program performance and its impact on children missed at the subnational level can be deployed in other countries or for other vaccines.

Factors related to human-vector contact that modify the likelihood of malaria transmission during a contained Plasmodium falciparum outbreak in Praia, Cabo Verde

Background

Determining the reproductive rate and how it varies over time and space (RT) provides important insight to understand transmission of a given disease and inform optimal strategies for controlling or eliminating it. Estimating RT for malaria is difficult partly due to the widespread use of interventions and immunity to disease masking incident infections. A malaria outbreak in Praia, Cabo Verde in 2017 provided a unique opportunity to estimate RT directly, providing a proxy for the intensity of vector-human contact and measure the impact of vector control measures.

Methods

Out of 442 confirmed malaria cases reported in 2017 in Praia, 321 (73%) were geolocated and informed this analysis. RT was calculated using the joint likelihood of transmission between two cases, based on the time (serial interval) and physical distance (spatial interval) between them. Log-linear regression was used to estimate factors associated with changes in RT, including the impact of vector control interventions. A geostatistical model was developed to highlight areas receptive to transmission where vector control activities could be focused in future to prevent or interrupt transmission.

Results

The RT from individual cases ranged between 0 and 11 with a median serial- and spatial-interval of 34 days [interquartile range (IQR): 17-52] and 1,347 m (IQR: 832-1,985 m), respectively. The number of households receiving indoor residual spraying (IRS) 4 weeks prior was associated with a reduction in RT by 0.84 [95% confidence interval (CI) 0.80-0.89; p-value <0.001] in the peak-and post-epidemic compared to the pre-epidemic period.

Conclusions

Identifying the effect of reduced human-vector contact through IRS is essential to determining optimal intervention strategies that modify the likelihood of malaria transmission and can inform optimal intervention strategies to accelerate time to elimination. The distance within which two cases are plausibly linked is important for the potential scale of any reactive interventions as well as classifying infections as imported or introduced and confirming malaria elimination.

Heterogeneity in the onwards transmission risk between local and imported cases affects practical estimates of the time-dependent reproduction number

During infectious disease outbreaks, inference of summary statistics characterizing transmission is essential for planning interventions. An important metric is the time-dependent reproduction number (Rt), which represents the expected number of secondary cases generated by each infected individual over the course of their infectious period. The value of Rt varies during an outbreak due to factors such as varying population immunity and changes to interventions, including those that affect individuals' contact networks. While it is possible to estimate a single population-wide Rt, this may belie differences in transmission between subgroups within the population. Here, we explore the effects of this heterogeneity on Rt estimates. Specifically, we consider two groups of infected hosts: those infected outside the local population (imported cases), and those infected locally (local cases). We use a Bayesian approach to estimate Rt, made available for others to use via an online tool, that accounts for differences in the onwards transmission risk from individuals in these groups. Using COVID-19 data from different regions worldwide, we show that different assumptions about the relative transmission risk between imported and local cases affect Rt estimates significantly, with implications for interventions. This highlights the need to collect data during outbreaks describing heterogeneities in transmission between different infected hosts, and to account for these heterogeneities in methods used to estimate Rt. This article is part of the theme issue 'Technical challenges of modelling real-life epidemics and examples of overcoming these'.

Visualization for epidemiological modelling: challenges, solutions, reflections and recommendations

We report on an ongoing collaboration between epidemiological modellers and visualization researchers by documenting and reflecting upon knowledge constructs-a series of ideas, approaches and methods taken from existing visualization research and practice-deployed and developed to support modelling of the COVID-19 pandemic. Structured independent commentary on these efforts is synthesized through iterative reflection to develop: evidence of the effectiveness and value of visualization in this context; open problems upon which the research communities may focus; guidance for future activity of this type and recommendations to safeguard the achievements and promote, advance, secure and prepare for future collaborations of this kind. In describing and comparing a series of related projects that were undertaken in unprecedented conditions, our hope is that this unique report, and its rich interactive supplementary materials, will guide the scientific community in embracing visualization in its observation, analysis and modelling of data as well as in disseminating findings. Equally we hope to encourage the visualization community to engage with impactful science in addressing its emerging data challenges. If we are successful, this showcase of activity may stimulate mutually beneficial engagement between communities with complementary expertise to address problems of significance in epidemiology and beyond. See https://ramp-vis.github.io/RAMPVIS-PhilTransA-Supplement/. This article is part of the theme issue 'Technical challenges of modelling real-life epidemics and examples of overcoming these'.

Estimating the effective reproduction number for heterogeneous models using incidence data

The effective reproduction number, R ( t ) , plays a key role in the study of infectious diseases, indicating the current average number of new infections caused by an infected individual in an epidemic process. Estimation methods for the time evolution of R ( t ) , using incidence data, rely on the generation interval distribution, g(τ), which is usually obtained from empirical data or theoretical studies using simple epidemic models. However, for systems that present heterogeneity, either on the host population or in the expression of the disease, there is a lack of data and of a suitable general methodology to obtain g(τ). In this work, we use mathematical models to bridge this gap. We present a general methodology for obtaining explicit expressions of the reproduction numbers and the generation interval distributions, within and between model sub-compartments provided by an arbitrary compartmental model. Additionally, we present the appropriate expressions to evaluate those reproduction numbers using incidence data. To highlight the relevance of such methodology, we apply it to the spread of COVID-19 in municipalities of the state of Rio de Janeiro, Brazil. Using two meta-population models, we estimate the reproduction numbers and the contributions of each municipality in the generation of cases in all others.

Estimating the effective reproduction number for heterogeneous models using incidence data

The effective reproduction number, R ( t ) , plays a key role in the study of infectious diseases, indicating the current average number of new infections caused by an infected individual in an epidemic process. Estimation methods for the time evolution of R ( t ) , using incidence data, rely on the generation interval distribution, g(τ), which is usually obtained from empirical data or theoretical studies using simple epidemic models. However, for systems that present heterogeneity, either on the host population or in the expression of the disease, there is a lack of data and of a suitable general methodology to obtain g(τ). In this work, we use mathematical models to bridge this gap. We present a general methodology for obtaining explicit expressions of the reproduction numbers and the generation interval distributions, within and between model sub-compartments provided by an arbitrary compartmental model. Additionally, we present the appropriate expressions to evaluate those reproduction numbers using incidence data. To highlight the relevance of such methodology, we apply it to the spread of COVID-19 in municipalities of the state of Rio de Janeiro, Brazil. Using two meta-population models, we estimate the reproduction numbers and the contributions of each municipality in the generation of cases in all others.

Estimating the effective reproduction number for heterogeneous models using incidence data

The effective reproduction number, R ( t ) , plays a key role in the study of infectious diseases, indicating the current average number of new infections caused by an infected individual in an epidemic process. Estimation methods for the time evolution of R ( t ) , using incidence data, rely on the generation interval distribution, g(τ), which is usually obtained from empirical data or theoretical studies using simple epidemic models. However, for systems that present heterogeneity, either on the host population or in the expression of the disease, there is a lack of data and of a suitable general methodology to obtain g(τ). In this work, we use mathematical models to bridge this gap. We present a general methodology for obtaining explicit expressions of the reproduction numbers and the generation interval distributions, within and between model sub-compartments provided by an arbitrary compartmental model. Additionally, we present the appropriate expressions to evaluate those reproduction numbers using incidence data. To highlight the relevance of such methodology, we apply it to the spread of COVID-19 in municipalities of the state of Rio de Janeiro, Brazil. Using two meta-population models, we estimate the reproduction numbers and the contributions of each municipality in the generation of cases in all others.

Forecasting Covid-19 in the United Kingdom: A dynamic SIRD model

Making use of a state space framework, we present a stochastic generalization of the SIRD model, where the mortality, infection, and underreporting rates change over time. A new format to the errors in the Susceptible-Infected-Recovered-Dead compartments is also presented, that permits reinfection. The estimated trajectories and (out-of-sample) forecasts of all these variables are presented with their confidence intervals. The model only uses as inputs the number of reported cases and deaths, and was applied for the UK from April, 2020 to Sep, 2021 (daily data). The estimated infection rate has shown a trajectory in waves very compatible with the emergence of new variants and adopted social measures. The estimated mortality rate has shown a significant descendant behaviour in 2021, which we attribute to the vaccination program, and the estimated underreporting rate has been considerably volatile, with a downward tendency, implying that, on average, more people are testing than in the beginning of the pandemic. The evolution of the proportions of the population divided into susceptible, infected, recovered and dead groups are also shown with their confidence intervals and forecast, along with an estimation of the amount of reinfection that, according to our model, has become quite significant in 2021. Finally, the estimated trajectory of the effective reproduction rate has proven to be very compatible with the real number of cases and deaths. Its forecasts with confident intervals are also presented.

Estimation and worldwide monitoring of the effective reproductive number of SARS-CoV-2

The effective reproductive number R_e is a key indicator of the growth of an epidemic. Since the start of the SARS-CoV-2 pandemic, many methods and online dashboards have sprung up to monitor this number through time. However, these methods are not always thoroughly tested, correctly placed in time, or are overly confident during high incidence periods. Here, we present a method for timely estimation of R_e, applied to COVID-19 epidemic data from 170 countries. We thoroughly evaluate the method on simulated data, and present an intuitive web interface for interactive data exploration. We show that, in early 2020, in the majority of countries the estimated R_e dropped below 1 only after the introduction of major non-pharmaceutical interventions. For Europe the implementation of non-pharmaceutical interventions was broadly associated with reductions in the estimated R_e. Globally though, relaxing non-pharmaceutical interventions had more varied effects on subsequent R_e estimates. Our framework is useful to inform governments and the general public on the status of epidemics in their country, and is used as the official source of R_e estimates for SARS-CoV-2 in Switzerland. It further allows detailed comparison between countries and in relation to covariates such as implemented public health policies, mobility, behaviour, or weather data.

Estimation and worldwide monitoring of the effective reproductive number of SARS-CoV-2

The effective reproductive number Re is a key indicator of the growth of an epidemic. Since the start of the SARS-CoV-2 pandemic, many methods and online dashboards have sprung up to monitor this number through time. However, these methods are not always thoroughly tested, correctly placed in time, or are overly confident during high incidence periods. Here, we present a method for timely estimation of Re, applied to COVID-19 epidemic data from 170 countries. We thoroughly evaluate the method on simulated data, and present an intuitive web interface for interactive data exploration. We show that, in early 2020, in the majority of countries the estimated Re dropped below 1 only after the introduction of major non-pharmaceutical interventions. For Europe the implementation of non-pharmaceutical interventions was broadly associated with reductions in the estimated Re. Globally though, relaxing non-pharmaceutical interventions had more varied effects on subsequent Re estimates. Our framework is useful to inform governments and the general public on the status of epidemics in their country, and is used as the official source of Re estimates for SARS-CoV-2 in Switzerland. It further allows detailed comparison between countries and in relation to covariates such as implemented public health policies, mobility, behaviour, or weather data.

An assessment of transmission dynamics via time-varying reproduction number of the second wave of the COVID-19 epidemic in Fiji

This study involves the estimation of a key epidemiological parameter for evaluating and monitoring the transmissibility of a disease. The time-varying reproduction number is the index for quantifying the transmissibility of infectious diseases. Accurate and timely estimation of the time-varying reproduction number is essential for optimizing non-pharmacological interventions and movement control orders during epidemics. The time-varying reproduction number for the second wave of the pandemic in Fiji is estimated using the popular EpiEstim R package and the publicly available COVID-19 data from 19 April 2021 to 1 December 2021. Our findings show that the non-pharmacological interventions and movement control orders introduced and enforced by the Fijian Government had a significant impact in preventing the spread of COVID-19. Moreover, the results show that many restrictions were either relaxed or eased when the time-varying reproduction number was below the threshold value of 1. The results have provided some information on the second wave of the COVID-19 pandemic that could be used in the future as a guide for public health policymakers in Fiji. Estimation of time-varying reproduction numbers would be helpful for continuous monitoring of the effectiveness of the current public health policies that are being implemented in Fiji.

COVID-19 vaccination policies under uncertain transmission characteristics using stochastic programming

We develop a new stochastic programming methodology for determining optimal vaccination policies for a multi-community heterogeneous population. An optimal policy provides the minimum number of vaccinations required to drive post-vaccination reproduction number to below one at a desired reliability level. To generate a vaccination policy, the new method considers the uncertainty in COVID-19 related parameters such as efficacy of vaccines, age-related variation in susceptibility and infectivity to SARS-CoV-2, distribution of household composition in a community, and variation in human interactions. We report on a computational study of the new methodology on a set of neighboring U.S. counties to generate vaccination policies based on vaccine availability. The results show that to control outbreaks at least a certain percentage of the population should be vaccinated in each community based on pre-determined reliability levels. The study also reveals the vaccine sharing capability of the proposed approach among counties under limited vaccine availability. This work contributes a decision-making tool to aid public health agencies worldwide in the allocation of limited vaccines under uncertainty towards controlling epidemics through vaccinations.

Adding a reaction-restoration type transmission rate dynamic-law to the basic SEIR COVID-19 model

The classical SEIR model, being an autonomous system of differential equations, has important limitations when representing a pandemic situation. Particularly, the geometric unimodal shape of the epidemic curve is not what is generally observed. This work introduces the βSEIR model, which adds to the classical SEIR model a differential law to model the variation in the transmission rate. It considers two opposite thrives generally found in a population: first, reaction to disease presence that may be linked to mitigation strategies, which tends to decrease transmission, and second, the urge to return to normal conditions that pulls to restore the initial value of the transmission rate. Our results open a wide spectrum of dynamic variabilities in the curve of new infected, which are justified by reaction and restoration thrives that affect disease transmission over time. Some of these dynamics have been observed in the existing COVID-19 disease data. In particular and to further exemplify the potential of the model proposed in this article, we show its capability of capturing the evolution of the number of new confirmed cases of Chile and Italy for several months after epidemic onset, while incorporating a reaction to disease presence with decreasing adherence to mitigation strategies, as well as a seasonal effect on the restoration of the initial transmissibility conditions.

Tracking the molecular evolution and transmission patterns of SARS-CoV-2 lineage B.1.466.2 in Indonesia based on genomic surveillance data

BACKGROUND

As a new epi-center of COVID-19 in Asia and a densely populated developing country, Indonesia is facing unprecedented challenges in public health. SARS-CoV-2 lineage B.1.466.2 was reported to be an indigenous dominant strain in Indonesia (once second only to the Delta variant). However, it remains unclear how this variant evolved and spread within such an archipelagic nation.

METHODS

For statistical description, the spatiotemporal distributions of the B.1.466.2 variant were plotted using the publicly accessible metadata in GISAID. A total of 1302 complete genome sequences of Indonesian B.1.466.2 strains with high coverage were downloaded from the GISAID's EpiCoV database on 28 August 2021. To determine the molecular evolutionary characteristics, we performed a time-scaled phylogenetic analysis using the maximum likelihood algorithm and called the single nucleotide variants taking the Wuhan-Hu-1 sequence as reference. To investigate the spatiotemporal transmission patterns, we estimated two dynamic parameters (effective population size and effective reproduction number) and reconstructed the phylogeography among different islands.

RESULTS

As of the end of August 2021, nearly 85% of the global SARS-CoV-2 lineage B.1.466.2 sequences (including the first one) were obtained from Indonesia. This variant was estimated to account for over 50% of Indonesia's daily infections during the period of March-May 2021. The time-scaled phylogeny suggested that SARS-CoV-2 lineage B.1.466.2 circulating in Indonesia might have originated from Java Island in mid-June 2020 and had evolved into two disproportional and distinct sub-lineages. High-frequency non-synonymous mutations were mostly found in the spike and NSP3; the S-D614G/N439K/P681R co-mutations were identified in its larger sub-lineage. The demographic history was inferred to have experienced four phases, with an exponential growth from October 2020 to February 2021. The effective reproduction number was estimated to have reached its peak (11.18) in late December 2020 and dropped to be less than one after early May 2021. The relevant phylogeography showed that Java and Sumatra might successively act as epi-centers and form a stable transmission loop. Additionally, several long-distance transmission links across seas were revealed.

CONCLUSIONS

SARS-CoV-2 variants circulating in the tropical archipelago may follow unique patterns of evolution and transmission. Continuous, extensive and targeted genomic surveillance is essential.

Modeling the COVID-19 pandemic: a primer and overview of mathematical epidemiology

Since the start of the still ongoing COVID-19 pandemic, there have been many modeling efforts to assess several issues of importance to public health. In this work, we review the theory behind some important mathematical models that have been used to answer questions raised by the development of the pandemic. We start revisiting the basic properties of simple Kermack-McKendrick type models. Then, we discuss extensions of such models and important epidemiological quantities applied to investigate the role of heterogeneity in disease transmission e.g. mixing functions and superspreading events, the impact of non-pharmaceutical interventions in the control of the pandemic, vaccine deployment, herd-immunity, viral evolution and the possibility of vaccine escape. From the perspective of mathematical epidemiology, we highlight the important properties, findings, and, of course, deficiencies, that all these models have.

Spatio-temporal small area surveillance of the COVID-19 pandemic

The emergence of COVID-19 requires new effective tools for epidemiological surveillance. Spatio-temporal disease mapping models, which allow dealing with small units of analysis, are a priority in this context. These models provide geographically detailed and temporally updated overviews of the current state of the pandemic, making public health interventions more effective. These models also allow estimating epidemiological indicators highly demanded for COVID-19 surveillance, such as the instantaneous reproduction number R t , even for small areas. In this paper, we propose a new spatio-temporal spline model particularly suited for COVID-19 surveillance, which allows estimating and monitoring R t for small areas. We illustrate our proposal on the study of the disease pandemic in two Spanish regions. As a result, we show how tourism flows have shaped the spatial distribution of the disease in these regions. In these case studies, we also develop new epidemiological tools to be used by regional public health services for small area surveillance.

Individual risk-aversion responses tune epidemics to critical transmissibility (R = 1)

Changes in human behaviour are a major determinant of epidemic dynamics. Collective activity can be modified through imposed control measures, but spontaneous changes can also arise as a result of uncoordinated individual responses to the perceived risk of contagion. Here, we introduce a stochastic epidemic model implementing population responses driven by individual time-varying risk aversion. The model reveals an emergent mechanism for the generation of multiple infection waves of decreasing amplitude that progressively tune the effective reproduction number to its critical value R = 1. In successive waves, individuals with gradually lower risk propensity are infected. The overall mechanism shapes well-defined risk-aversion profiles over the whole population as the epidemic progresses. We conclude that uncoordinated changes in human behaviour can by themselves explain major qualitative and quantitative features of the epidemic process, like the emergence of multiple waves and the tendency to remain around R = 1 observed worldwide after the first few waves of COVID-19.

Association of anti-contagion policies with the spread of COVID-19 in United States

BACKGROUND

The outbreak of a novel coronavirus, known as severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), or COVID-19, raised worldwide concern. The present study investigates the association between anti-contagion policies and the spread of COVID-19 across the United States.

DESIGN AND METHODS

We selected the most frequently implemented COVID-19 anti-contagion policies in all the U.S. states issued from 29 February 2020. Accordingly, we modified an epidemiological model and combined it with a comprehensive statistical analysis to evaluate the policies' individual and overall likely impact.

RESULTS

For the first time, a novel index, evaluates the associations between policy implementation and COVID-19 spread at both statewide and national levels. Our results indicate that governmental policies requiring mask use, businesses social distancing, and quarantining travelers may be most effective for controlling COVID-19 spread. Simultaneously, widespread orders like school closure and safer-at-home that can be particularly disruptive to the economy and social fabric of society may be unnecessary given their lack of association with reducing infection.

CONCLUSIONS

The absence of any COVID-19 vaccines during the first several months of its pandemic necessitated using governmental policies to help stop the spread of this disease. Our index showed the association between implemented policies and COVID-19 spread, highlighting the specific policies with the greatest association - mandatory quarantine upon entering a state, businesses implementing social distancing, and mandatory mask use - and those with less association like school closure and safer-at-home orders. This study provided evidence to inform policy choices for the current global crisis and future pandemics.

COVID-19, measles, and yellow fever: The need to reinforce vaccination in the Democratic Republic of Congo

The immunization programs have been jeopardized all over the world due to the stay-at-home constraints imposed, to mitigate the ongoing COVID-19 pandemic. This has directly or indirectly placed the global health care system in peril, resulting in an overlapping public health crisis. With this commentary, we aim to accentuate the need to reinforce vaccination in the Democratic Republic of Congo, in lieu of the intersecting COVID-19, measles, and yellow fever outbreak, besides, providing recommendations so as to help alleviate the situation.

Fine-scale estimation of effective reproduction numbers for dengue surveillance

The effective reproduction number Rt is an epidemiological quantity that provides an instantaneous measure of transmission potential of an infectious disease. While dengue is an increasingly important vector-borne disease, few have used Rt as a measure to inform public health operations and policy for dengue. This study demonstrates the utility of Rt for real time dengue surveillance. Using nationally representative, geo-located dengue case data from Singapore over 2010-2020, we estimated Rt by modifying methods from Bayesian (EpiEstim) and filtering (EpiFilter) approaches, at both the national and local levels. We conducted model assessment of Rt from each proposed method and determined exogenous temporal and spatial drivers for Rt in relation to a wide range of environmental and anthropogenic factors. At the national level, both methods achieved satisfactory model performance (R2EpiEstim = 0.95, R2EpiFilter = 0.97), but disparities in performance were large at finer spatial scales when case counts are low (MASE EpiEstim = 1.23, MASEEpiFilter = 0.59). Impervious surfaces and vegetation with structure dominated by human management (without tree canopy) were positively associated with increased transmission intensity. Vegetation with structure dominated by human management (with tree canopy), on the other hand, was associated with lower dengue transmission intensity. We showed that dengue outbreaks were preceded by sustained periods of high transmissibility, demonstrating the potential of Rt as a dengue surveillance tool for detecting large rises in dengue cases. Real time estimation of Rt at the fine scale can assist public health agencies in identifying high transmission risk areas and facilitating localised outbreak preparedness and response.

Computing the daily reproduction number of COVID-19 by inverting the renewal equation using a variational technique

The COVID-19 pandemic has undergone frequent and rapid changes in its local and global infection rates, driven by governmental measures or the emergence of new viral variants. The reproduction number Rt indicates the average number of cases generated by an infected person at time t and is a key indicator of the spread of an epidemic. A timely estimation of Rt is a crucial tool to enable governmental organizations to adapt quickly to these changes and assess the consequences of their policies. The EpiEstim method is the most widely accepted method for estimating Rt But it estimates Rt with a significant temporal delay. Here, we propose a method, EpiInvert, that shows good agreement with EpiEstim, but that provides estimates of Rt several days in advance. We show that Rt can be estimated by inverting the renewal equation linking Rt with the observed incidence curve of new cases, it Our signal-processing approach to this problem yields both Rt and a restored it corrected for the "weekend effect" by applying a deconvolution and denoising procedure. The implementations of the EpiInvert and EpiEstim methods are fully open source and can be run in real time on every country in the world and every US state.

An age-structured SEIR model for COVID-19 incidence in Dublin, Ireland with framework for evaluating health intervention cost

Strategies adopted globally to mitigate the threat of COVID-19 have primarily involved lockdown measures with substantial economic and social costs with varying degrees of success. Morbidity patterns of COVID-19 variants have a strong association with age, while restrictive lockdown measures have association with negative mental health outcomes in some age groups. Reduced economic prospects may also afflict some age cohorts more than others. Motivated by this, we propose a model to describe COVID-19 community spread incorporating the role of age-specific social interactions. Through a flexible parameterisation of an age-structured deterministic Susceptible Exposed Infectious Removed (SEIR) model, we provide a means for characterising different forms of lockdown which may impact specific age groups differently. Social interactions are represented through age group to age group contact matrices, which can be trained using available data and are thus locally adapted. This framework is easy to interpret and suitable for describing counterfactual scenarios, which could assist policy makers with regard to minimising morbidity balanced with the costs of prospective suppression strategies. Our work originates from an Irish context and we use disease monitoring data from February 29th 2020 to January 31st 2021 gathered by Irish governmental agencies. We demonstrate how Irish lockdown scenarios can be constructed using the proposed model formulation and show results of retrospective fitting to incidence rates and forward planning with relevant "what if / instead of" lockdown counterfactuals. Uncertainty quantification for the predictive approaches is described. Our formulation is agnostic to a specific locale, in that lockdown strategies in other regions can be straightforwardly encoded using this model.

On the use of aggregated human mobility data to estimate the reproduction number

The reproduction number of an infectious disease, such as CoViD-19, can be described through a modified version of the susceptible-infected-recovered (SIR) model with time-dependent contact rate, where mobility data are used as proxy of average movement trends and interpersonal distances. We introduce a theoretical framework to explain and predict changes in the reproduction number of SARS-CoV-2 in terms of aggregated individual mobility and interpersonal proximity (alongside other epidemiological and environmental variables) during and after the lockdown period. We use an infection-age structured model described by a renewal equation. The model predicts the evolution of the reproduction number up to a week ahead of well-established estimates used in the literature. We show how lockdown policies, via reduction of proximity and mobility, reduce the impact of CoViD-19 and mitigate the risk of disease resurgence. We validate our theoretical framework using data from Google, Voxel51, Unacast, The CoViD-19 Mobility Data Network, and Analisi Distribuzione Aiuti.

On the reproduction number in epidemics

This note provides an elementary derivation of the basic reproduction number and the effective reproduction number from the discrete Kermack-McKendrick epidemic model. The derived formulae match those derived from the continuous version of the model; however, the derivation from discrete model is a bit more intuitive. The MATLAB functions for its calculation are given. A real case example is considered and the results are compared with those obtained by the R0 and the EpiEstim software packages.

Mutual relationships between SARS-CoV-2 test numbers, fatality and morbidity rates

BACKGROUND

The number of SARS-CoV-2 tests conversely to other factors, such as age of population or comorbidities, influencing SARS-CoV-2 morbidity and fatality rates, can be increased or decreased by decision makers depending on the development of the pandemic, operational capacity, and financial restraints. The key objective of this study is to identify and describe, within the probabilistic approach, the relationships between SARS-CoV-2 test numbers and the mortality and morbidity rates.

METHODS

The study is based on a statistical analysis of 1058 monthly observations relating to 107 countries, from six different continents, in an 11-month period from March 2020 to January 2021. The variable utilised can be defined as the number of tests performed in a given country in 1 month, to the number of cases reported in a prior month and morbidities and mortalities per 1 million population. The probabilities of different mortality and morbidity rates for different test numbers were determined by moving percentiles and fitted by the power law and by the three-segment piecewise-linear approximation based on Theil Sen trend lines.

RESULTS

We have identified that for a given probability the dependence of mortality and morbidity rates on SARS-CoV-2 test rates follows a power law and it is well approximated by the three Theil Sen trend lines in the three test rate ranges. In all these ranges Spearman rho and Kendall tau-b rank correlation coefficients of test numbers and morbidity with fatality rates have values between - 0.5 and - 0.12 with p-values below 0.002.

CONCLUSIONS

According to the ABC classification: the most important, moderately important, and relatively unimportant ranges of test numbers for managing and control have been indicated based on the value of the Theil Sen trend line slope in the three SARS-CoV-2 test rate ranges identified. Recommendations for SARS-CoV-2 testing strategy are provided.

Improved estimation of time-varying reproduction numbers at low case incidence and between epidemic waves

We construct a recursive Bayesian smoother, termed EpiFilter, for estimating the effective reproduction number, R, from the incidence of an infectious disease in real time and retrospectively. Our approach borrows from Kalman filtering theory, is quick and easy to compute, generalisable, deterministic and unlike many current methods, requires no change-point or window size assumptions. We model R as a flexible, hidden Markov state process and exactly solve forward-backward algorithms, to derive R estimates that incorporate all available incidence information. This unifies and extends two popular methods, EpiEstim, which considers past incidence, and the Wallinga-Teunis method, which looks forward in time. We find that this combination of maximising information and minimising assumptions significantly reduces the bias and variance of R estimates. Moreover, these properties make EpiFilter more statistically robust in periods of low incidence, where several existing methods can become destabilised. As a result, EpiFilter offers improved inference of time-varying transmission patterns that are advantageous for assessing the risk of upcoming waves of infection or the influence of interventions, in real time and at various spatial scales.

Structure of epidemic models: toward further applications in economics

In this paper, we review the structure of various epidemic models in mathematical epidemiology for the future applications in economics. The heterogeneity of population and the generalization of nonlinear terms play important roles in making more elaborate and realistic models. The basic, effective, control and type reproduction numbers have been used to estimate the intensity of epidemic, to evaluate the effectiveness of interventions and to design appropriate interventions. The advanced epidemic models includes the age structure, seasonality, spatial diffusion, mutation and reinfection, and the theory of reproduction numbers has been generalized to them. In particular, the existence of sustained periodic solutions has attracted much interest because they can explain the recurrent waves of epidemic. Although the theory of epidemic models has been developed in decades and the development has been accelerated through COVID-19, it is still difficult to completely answer the uncertainty problem of epidemic models. We would have to mind that there is no single model that can solve all questions and build a scientific attitude to comprehensively understand the results obtained by various researchers from different backgrounds.

Fundamental Identifiability Limits in Molecular Epidemiology

Viral phylogenies provide crucial information on the spread of infectious diseases, and many studies fit mathematical models to phylogenetic data to estimate epidemiological parameters such as the effective reproduction ratio (Re) over time. Such phylodynamic inferences often complement or even substitute for conventional surveillance data, particularly when sampling is poor or delayed. It remains generally unknown, however, how robust phylodynamic epidemiological inferences are, especially when there is uncertainty regarding pathogen prevalence and sampling intensity. Here, we use recently developed mathematical techniques to fully characterize the information that can possibly be extracted from serially collected viral phylogenetic data, in the context of the commonly used birth-death-sampling model. We show that for any candidate epidemiological scenario, there exists a myriad of alternative, markedly different, and yet plausible "congruent" scenarios that cannot be distinguished using phylogenetic data alone, no matter how large the data set. In the absence of strong constraints or rate priors across the entire study period, neither maximum-likelihood fitting nor Bayesian inference can reliably reconstruct the true epidemiological dynamics from phylogenetic data alone; rather, estimators can only converge to the "congruence class" of the true dynamics. We propose concrete and feasible strategies for making more robust epidemiological inferences from viral phylogenetic data.