Time variations in the transmissibility of pandemic influenza in Prussia, Germany, from 1918-19

Estimating the effective reproduction number of COVID-19 from population-wide wastewater data: An application in Kagawa, Japan

Although epidemiological surveillance of COVID-19 has been gradually downgraded globally, the transmission of COVID-19 continues. It is critical to quantify the transmission dynamics of COVID-19 using multiple datasets including wastewater virus concentration data. Herein, we propose a comprehensive method for estimating the effective reproduction number using wastewater data. The wastewater virus concentration data, which were collected twice a week, were analyzed using daily COVID-19 incidence data obtained from Takamatsu, Japan between January 2022 and September 2022. We estimated the shedding load distribution (SLD) as a function of time since the date of infection, using a model employing the delay distribution, which is assumed to follow a gamma distribution, multiplied by a scaling factor. We also examined models that accounted for the temporal smoothness of viral load measurement data. The model that smoothed temporal patterns of viral load was the best fit model (WAIC = 2795.8), which yielded a mean estimated distribution of SLD of 3.46 days (95% CrI: 3.01-3.95 days). Using this SLD, we reconstructed the daily incidence, which enabled computation of the effective reproduction number. Using the best fit posterior draws of parameters directly, or as a prior distribution for subsequent analyses, we first used a model that assumed temporal smoothness of viral load concentrations in wastewater, as well as infection counts by date of infection. In the subsequent approach, we examined models that also incorporated weekly reported case counts as a proxy for weekly incidence reporting. Both approaches enabled estimations of the epidemic curve as well as the effective reproduction number from twice-weekly wastewater viral load data. Adding weekly case count data reduced the uncertainty of the effective reproduction number. We conclude that wastewater data are still a valuable source of information for inferring the transmission dynamics of COVID-19, and that inferential performance is enhanced when those data are combined with weekly incidence data.

Time warping between main epidemic time series in epidemiological surveillance

The most common reported epidemic time series in epidemiological surveillance are the daily or weekly incidence of new cases, the hospital admission count, the ICU admission count, and the death toll, which played such a prominent role in the struggle to monitor the Covid-19 pandemic. We show that pairs of such curves are related to each other by a generalized renewal equation depending on a smooth time varying delay and a smooth ratio generalizing the reproduction number. Such a functional relation is also explored for pairs of simultaneous curves measuring the same indicator in two neighboring countries. Given two such simultaneous time series, we develop, based on a signal processing approach, an efficient numerical method for computing their time varying delay and ratio curves, and we verify that its results are consistent. Indeed, they experimentally verify symmetry and transitivity requirements and we also show, using realistic simulated data, that the method faithfully recovers time delays and ratios. We discuss several real examples where the method seems to display interpretable time delays and ratios. The proposed method generalizes and unifies many recent related attempts to take advantage of the plurality of these health data across regions or countries and time, providing a better understanding of the relationship between them. An implementation of the method is publicly available at the EpiInvert CRAN package.

SARS-CoV-2 Serology Across Scales: A Framework for Unbiased Estimation of Cumulative Incidence Incorporating Antibody Kinetics and Epidemic Recency

Serosurveys are a key resource for measuring severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) population exposure. A growing body of evidence suggests that asymptomatic and mild infections (together making up over 95% of all infections) are associated with lower antibody titers than severe infections. Antibody levels also peak a few weeks after infection and decay gradually. We developed a statistical approach to produce estimates of cumulative incidence from raw seroprevalence survey results that account for these sources of spectrum bias. We incorporate data on antibody responses on multiple assays from a postinfection longitudinal cohort, along with epidemic time series to account for the timing of a serosurvey relative to how recently individuals may have been infected. We applied this method to produce estimates of cumulative incidence from 5 large-scale SARS-CoV-2 serosurveys across different settings and study designs. We identified substantial differences between raw seroprevalence and cumulative incidence of over 2-fold in the results of some surveys, and we provide a tool for practitioners to generate cumulative incidence estimates with preset or custom parameter values. While unprecedented efforts have been launched to generate SARS-CoV-2 seroprevalence estimates over this past year, interpretation of results from these studies requires properly accounting for both population-level epidemiologic context and individual-level immune dynamics.

Learning from the past: A short term forecast method for the COVID-19 incidence curve

The COVID-19 pandemy has created a radically new situation where most countries provide raw measurements of their daily incidence and disclose them in real time. This enables new machine learning forecast strategies where the prediction might no longer be based just on the past values of the current incidence curve, but could take advantage of observations in many countries. We present such a simple global machine learning procedure using all past daily incidence trend curves. Each of the 27,418 COVID-19 incidence trend curves in our database contains the values of 56 consecutive days extracted from observed incidence curves across 61 world regions and countries. Given a current incidence trend curve observed over the past four weeks, its forecast in the next four weeks is computed by matching it with the first four weeks of all samples, and ranking them by their similarity to the query curve. Then the 28 days forecast is obtained by a statistical estimation combining the values of the 28 last observed days in those similar samples. Using comparison performed by the European Covid-19 Forecast Hub with the current state of the art forecast methods, we verify that the proposed global learning method, EpiLearn, compares favorably to methods forecasting from a single past curve.

Spectral Method in Epidemic Time Series: Application to COVID-19 Pandemic

BACKGROUND

The age of infection plays an important role in assessing an individual's daily level of contagiousness, quantified by the daily reproduction number. Then, we derive an autoregressive moving average model from a daily discrete-time epidemic model based on a difference equation involving the age of infection. Novelty: The article's main idea is to use a part of the spectrum associated with this difference equation to describe the data and the model.

RESULTS

We present some results of the parameters' identification of the model when all the eigenvalues are known. This method was applied to Japan's third epidemic wave of COVID-19 fails to preserve the positivity of daily reproduction. This problem forced us to develop an original truncated spectral method applied to Japanese data. We start by considering ten days and extend our analysis to one month.

CONCLUSION

We can identify the shape for a daily reproduction numbers curve throughout the contagion period using only a few eigenvalues to fit the data.

Modeling COVID-19 Incidence by the Renewal Equation after Removal of Administrative Bias and Noise

Simple Summary

In the past two years, the COVID-19 incidence curves and reproduction number Rt have been the main metrics used by policy makers and journalists to monitor the spread of this global pandemic. However, these metrics are not always reliable in the short term, because of a combination of delay in detection, administrative delays and random noise. In this article, we present a complete model of COVID-19 incidence, faithfully reconstructing the incidence curve and reproduction number from the renewal equation of the disease and precisely estimating the biases associated with periodic weekly bias, festive day bias and residual noise.

Abstract

The sanitary crisis of the past two years has focused the public’s attention on quantitative indicators of the spread of the COVID-19 pandemic. The daily reproduction number Rt, defined by the average number of new infections caused by a single infected individual at time t, is one of the best metrics for estimating the epidemic trend. In this paper, we provide a complete observation model for sampled epidemiological incidence signals obtained through periodic administrative measurements. The model is governed by the classic renewal equation using an empirical reproduction kernel, and subject to two perturbations: a time-varying gain with a weekly period and a white observation noise. We estimate this noise model and its parameters by extending a variational inversion of the model recovering its main driving variable Rt. Using Rt, a restored incidence curve, corrected of the weekly and festive day bias, can be deduced through the renewal equation. We verify experimentally on many countries that, once the weekly and festive days bias have been corrected, the difference between the incidence curve and its expected value is well approximated by an exponential distributed white noise multiplied by a power of the magnitude of the restored incidence curve.

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.

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 R_t 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 R_t 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 R_t But it estimates R_t with a significant temporal delay. Here, we propose a method, EpiInvert, that shows good agreement with EpiEstim, but that provides estimates of R_t several days in advance. We show that R_t can be estimated by inverting the renewal equation linking R_t with the observed incidence curve of new cases, i_t Our signal-processing approach to this problem yields both R_t and a restored i_t 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.

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.

Modelling the SARS-CoV-2 outbreak: Assessing the usefulness of protective measures to reduce the pandemic at population level

A new bioinspired computational model was developed for the SARS-CoV-2 pandemic using the available epidemiological information, high-resolution population density data, travel patterns, and the average number of contacts between people. The effectiveness of control measures such as contact reduction measures, closure of communities (lockdown), protective measures (social distancing, face mask wearing, and hand hygiene), and vaccination were modelled to examine possibilities for control of the disease under several protective vaccination levels in the population. Lockdown and contact reduction measures only delay the spread of the virus in the population because it resumes its previous dynamics as soon as the restrictions are lifted. Nevertheless, these measures are probably useful to avoid hospitals being overwhelmed in the short term. Our model predicted that 56% of the Spanish population would have been infected and subsequently recovered over a 130 day period if no protective measures were taken but this percentage would have been only 34% if protective measures had been put in place. Moreover, this percentage would have been further reduced to 41.7, 27.7, and 13.3% if 25, 50 and 75% of the population had been vaccinated, respectively. Finally, this percentage would have been even lower at 25.5, 12.1 and 7.9% if 25, 50 and 75% of the population had been vaccinated in combination with the application of protective measures, respectively. Therefore, a combination of protective measures and vaccination would be highly efficacious in decreasing not only the number of those who become infected and subsequently recover, but also the number of people who die from infection, which falls from 0.41% of the population over a 130 day period without protective measures to 0.15, 0.08 and 0.06% if 25, 50 and 75% of the population had been vaccinated in combination with protective measures at the same time, respectively.

SARS-CoV-2 serology across scales: a framework for unbiased seroprevalence estimation incorporating antibody kinetics and epidemic recency

Serosurveys are a key resource for measuring SARS-CoV-2 cumulative incidence. A growing body of evidence suggests that asymptomatic and mild infections (together making up over 95% of all infections) are associated with lower antibody titers than severe infections. Antibody levels also peak a few weeks after infection and decay gradually. We developed a statistical approach to produce adjusted estimates of seroprevalence from raw serosurvey results that account for these sources of spectrum bias. We incorporate data on antibody responses on multiple assays from a post-infection longitudinal cohort, along with epidemic time series to account for the timing of a serosurvey relative to how recently individuals may have been infected. We applied this method to produce adjusted seroprevalence estimates from five large-scale SARS-CoV-2 serosurveys across different settings and study designs. We identify substantial differences between reported and adjusted estimates of over two-fold in the results of some surveys, and provide a tool for practitioners to generate adjusted estimates with pre-set or custom parameter values. While unprecedented efforts have been launched to generate SARS-CoV-2 seroprevalence estimates over this past year, interpretation of results from these studies requires properly accounting for both population-level epidemiologic context and individual-level immune dynamics.

Optimizing Spatial Allocation of COVID-19 Vaccine by Agent-Based Spatiotemporal Simulations

Optimizing allocation of vaccine, a highly scarce resource, is an urgent and critical issue during fighting against on-going COVID-19 epidemic. Prior studies suggested that vaccine should be prioritized by age and risk groups, but few of them have considered the spatial prioritization strategy. This study aims to examine the spatial heterogeneity of COVID-19 transmission in the city naturally, and optimize vaccine distribution strategies considering spatial prioritization. We proposed an integrated spatial model of agent-based model and SEIR (susceptible-exposed-infected-recovered). It simulated spatiotemporal process of COVID-19 transmission in a realistic urban context. Individual movements were represented by trajectories of 8,146 randomly sampled mobile phone users on December 28, 2016 in Guangzhou, China, 90% of whom aged 18-60. Simulations were conducted under seven scenarios. Scenarios 1 and 2 examined natural spreading process of COVID-19 and its final state of herd immunity. Scenarios 3-6 applied four vaccination strategies (random strategy, age strategy, space strategy, and space & age strategy), and identified the optimal vaccine strategy. Scenario 7 assessed the most appropriate vaccine coverage. The results demonstrates herd immunity is heterogeneously distributed in space, thus, vaccine intervention strategies should be spatialized. Among four strategies, space & age strategy is substantially most efficient, with 7.7% fewer in attack rate and 44 days longer than random strategy under 20% vaccine uptake. Space & age strategy requires 30%-40% vaccine coverage to control the epidemic, while the coverage for a random strategy is 60%-70% as a comparison. The application of our research would greatly improves the effectiveness of the vaccine usability.

Statistical Estimation of the Reproductive Number From Case Notification Data

The reproductive number, or reproduction number, is a valuable metric in understanding infectious disease dynamics. There is a large body of literature related to its use and estimation. In the last 15 years, there has been tremendous progress in statistically estimating this number using case notification data. These approaches are appealing because they are relevant in an ongoing outbreak (e.g., for assessing the effectiveness of interventions) and do not require substantial modeling expertise to be implemented. In this article, we describe these methods and the extensions that have been developed. We provide insight into the distinct interpretations of the estimators proposed and provide real data examples to illustrate how they are implemented. Finally, we conclude with a discussion of available software and opportunities for future development.

The influence of awareness on epidemic spreading on random networks

During an outbreak, the perceived infection risk of an individual affects his/her behavior during an epidemic to lower the risk. We incorporate the awareness of infection risk into the Volz-Miller SIR epidemic model, to study the effect of awareness on disease dynamics. We consider two levels of awareness, the local one represented by the prevalence among the contacts of an individual, and the global one represented by the prevalence in the population. We also consider two possible effects of awareness: reducing infection rate or breaking infectious edges. We use the next generation matrix method to obtain the basic reproduction number of our models, and show that awareness acquired during an epidemic does not affect the basic reproduction number. However, awareness acquired from outbreaks in other regions before the start of the local epidemic reduces the basic reproduction number. Awareness always reduces the final size of an epidemic. Breaking infectious edges causes a larger reduction than reducing the infection rate. If awareness reduces the infection rate, the reduction increases with both local and global awareness. However, if it breaks infectious edges, the reduction may not be monotonic. For the same awareness, the reduction may reach a maximum on some intermediate infection rates. Whether local or global awareness has a larger effect on reducing the final size depends on the network degree distribution and the infection rate.

Comparison of methods to Estimate Basic Reproduction Number (R 0) of influenza, Using Canada 2009 and 2017-18 A (H1N1) Data

Background

The basic reproduction number (R ₀) has a key role in epidemics and can be utilized for preventing epidemics. In this study, different methods are used for estimating R ₀'s and their vaccination coverage to find the formula with the best performance.

Materials and Methods

We estimated R ₀ for cumulative cases count data from April 18 to July 6, 2009 and 35-2017 to 34-2018 weeks in Canada: maximum likelihood (ML), exponential growth rate (EG), time-dependent reproduction numbers (TD), attack rate (AR), gamma-distributed generation time (GT), and the final size of the epidemic. Gamma distribution with mean and standard deviation 3.6 ± 1.4 is used as GT.

Results

The AR method obtained a R _{0 (}95% confidence interval [CI]) value of 1.116 (1.1163, 1.1165) and an EG (95%CI) value of 1.46 (1.41, 1.52). The R ₀ (95%CI) estimate was 1.42 (1.27, 1.57) for the obtained ML, 1.71 (1.12, 2.03) for the obtained TD, 1.49 (1.0, 1.97) for the gamma-distributed GT, and 1.00 (0.91, 1.09) for the final size of the epidemic. The minimum and maximum vaccination coverage were related to AR and TD methods, respectively, where the TD method has minimum mean squared error (MSE). Finally, the R ₀ (95%CI) for 2018 data was 1.52 (1.11, 1.94) by TD method, and vaccination coverage was estimated as 34.2%.

Conclusion

For the purposes of our study, the estimation of TD was the most useful tool for computing the R ₀, because it has the minimum MSE. The estimation R ₀ > 1 indicating that the epidemic has occurred. Thus, it is required to vaccinate at least 41.5% to prevent and control the next epidemic.

Impact of Public Health Responses During a Measles Outbreak in an Amish Community in Ohio: Modeling the Dynamics of Transmission

We quantified measles transmissibility during a measles outbreak in Ohio in 2014 to evaluate the impact of public health responses. Case incidence and the serial interval (time between symptom onset in primary cases and secondary cases) were used to assess trends in the effective reproduction number R (the average number of secondary cases generated per case). A mathematical model was parameterized using early R values to determine the size and duration of the outbreak that would have occurred if containment measures had not been initiated, as well as the impact of vaccination. As containment started, we found a 4-fold decline in R (from approximately 4 to 1) over the course of 2 weeks and maintenance of R < 1 as control measures continued. Under a conservative scenario, the model estimated 8,472 cases (90% confidence interval (CI): 8,447, 8,489) over 195 days (90% CI: 179, 223) without control efforts and 715 cases (90% CI: 103, 1,338) over 128 days (90% CI: 117, 139) when vaccination was included; 7,757 fewer cases (90% CI: 7,130, 8,365) and 67 fewer outbreak days (90% CI: 48, 98) were attributed to vaccination. Vaccination may not account entirely for transmission reductions, suggesting that changes in community behavior (social distancing) and other control efforts (isolation, quarantining) are important. Our findings highlight the benefits of measles outbreak response and of understanding behavior change dynamics.

A Bayesian inferential approach to quantify the transmission intensity of disease outbreak

BACKGROUND

Emergence of infectious diseases like influenza pandemic (H1N1) 2009 has become great concern, which posed new challenges to the health authorities worldwide. To control these diseases various studies have been developed in the field of mathematical modelling, which is useful tool for understanding the epidemiological dynamics and their dependence on social mixing patterns.

METHOD

We have used Bayesian approach to quantify the disease outbreak through key epidemiological parameter basic reproduction number (R0), using effective contacts, defined as sum of the product of incidence cases and probability of generation time distribution. We have estimated R0 from daily case incidence data for pandemic influenza A/H1N1 2009 in India, for the initial phase.

RESULT

The estimated R0 with 95% credible interval is consistent with several other studies on the same strain. Through sensitivity analysis our study indicates that infectiousness affects the estimate of R0.

CONCLUSION

Basic reproduction number R0 provides the useful information to the public health system to do some effort in controlling the disease by using mitigation strategies like vaccination, quarantine, and so forth.

Transmission dynamics and control of Ebola virus disease (EVD): a review

The complex and unprecedented Ebola epidemic ongoing in West Africa has highlighted the need to review the epidemiological characteristics of Ebola Virus Disease (EVD) as well as our current understanding of the transmission dynamics and the effect of control interventions against Ebola transmission. Here we review key epidemiological data from past Ebola outbreaks and carry out a comparative review of mathematical models of the spread and control of Ebola in the context of past outbreaks and the ongoing epidemic in West Africa. We show that mathematical modeling offers useful insights into the risk of a major epidemic of EVD and the assessment of the impact of basic public health measures on disease spread. We also discuss the critical need to collect detailed epidemiological data in real-time during the course of an ongoing epidemic, carry out further studies to estimate the effectiveness of interventions during past outbreaks and the ongoing epidemic, and develop large-scale modeling studies to study the spread and control of viral hemorrhagic fevers in the context of the highly heterogeneous economic reality of African countries.

Estimates of the reproduction number for seasonal, pandemic, and zoonotic influenza: a systematic review of the literature

BACKGROUND

The potential impact of an influenza pandemic can be assessed by calculating a set of transmissibility parameters, the most important being the reproduction number (R), which is defined as the average number of secondary cases generated per typical infectious case.

METHODS

We conducted a systematic review to summarize published estimates of R for pandemic or seasonal influenza and for novel influenza viruses (e.g. H5N1). We retained and summarized papers that estimated R for pandemic or seasonal influenza or for human infections with novel influenza viruses.

RESULTS

The search yielded 567 papers. Ninety-one papers were retained, and an additional twenty papers were identified from the references of the retained papers. Twenty-four studies reported 51 R values for the 1918 pandemic. The median R value for 1918 was 1.80 (interquartile range [IQR]: 1.47-2.27). Six studies reported seven 1957 pandemic R values. The median R value for 1957 was 1.65 (IQR: 1.53-1.70). Four studies reported seven 1968 pandemic R values. The median R value for 1968 was 1.80 (IQR: 1.56-1.85). Fifty-seven studies reported 78 2009 pandemic R values. The median R value for 2009 was 1.46 (IQR: 1.30-1.70) and was similar across the two waves of illness: 1.46 for the first wave and 1.48 for the second wave. Twenty-four studies reported 47 seasonal epidemic R values. The median R value for seasonal influenza was 1.28 (IQR: 1.19-1.37). Four studies reported six novel influenza R values. Four out of six R values were <1.

CONCLUSIONS

These R values represent the difference between epidemics that are controllable and cause moderate illness and those causing a significant number of illnesses and requiring intensive mitigation strategies to control. Continued monitoring of R during seasonal and novel influenza outbreaks is needed to document its variation before the next pandemic.

The serial intervals of seasonal and pandemic influenza viruses in households in Bangkok, Thailand

The serial interval (SI) of human influenza virus infections is often described by a single distribution. Understanding sources of variation in the SI could provide valuable information for understanding influenza transmission dynamics. Using data from a randomized household study of nonpharmaceutical interventions to prevent influenza transmission in Bangkok, Thailand, over 34 months between 2008 and 2011, we estimated the influence of influenza virus type/subtype and other characteristics of 251 pediatric index cases and their 315 infected household contacts on estimates of household SI. The mean SI for all households was 3.3 days. Relative to influenza A(H1N1)pdm09 (3.1 days), the SI for influenza B (3.7 days) was 22% longer (95% confidence interval: 4, 43), or about half a day. The SIs for influenza viruses A(H1N1) and A(H3N2) were similar to that for A(H1N1)pdm09. SIs were shortest for older index cases (age 11-14 years) and for younger infected household contacts (age ≤15 years). Greater time spent in proximity to the index child was associated with shorter SIs. Differences in the SI might reflect differences in incubation period, viral shedding, contact, or susceptibility. These findings could improve parameterization of mathematical models to better predict the impact of epidemic or pandemic influenza mitigation strategies.

Tracking Epidemics With Google Flu Trends Data and a State-Space SEIR Model

In this article, we use Google Flu Trends data together with a sequential surveillance model based on state-space methodology to track the evolution of an epidemic process over time. We embed a classical mathematical epidemiology model [a susceptible-exposed-infected-recovered (SEIR) model] within the state-space framework, thereby extending the SEIR dynamics to allow changes through time. The implementation of this model is based on a particle filtering algorithm, which learns about the epidemic process sequentially through time and provides updated estimated odds of a pandemic with each new surveillance data point. We show how our approach, in combination with sequential Bayes factors, can serve as an online diagnostic tool for influenza pandemic. We take a close look at the Google Flu Trends data describing the spread of flu in the United States during 2003-2009 and in nine separate U.S. states chosen to represent a wide range of health care and emergency system strengths and weaknesses. This article has online supplementary materials.

The R0 package: a toolbox to estimate reproduction numbers for epidemic outbreaks

Background

Several generic methods have been proposed to estimate transmission parameters during an outbreak, especially the reproduction number. However, as of today, no dedicated software exists that implements these methods and allow comparisons.

Results

A review of generic methods used to estimate transmissibility parameters during outbreaks was carried out. Most methods used the epidemic curve and the generation time distribution. Two categories of methods were available: those estimating the initial reproduction number, and those estimating a time dependent reproduction number. We implemented five methods as an R library, developed sensitivity analysis tools for each method and provided numerical illustrations of their use. A comparison of the performance of the different methods on simulated datasets is reported.

Conclusions

This software package allows a standardized and extensible approach to the estimation of the reproduction number and generation interval distribution from epidemic curves.

Modeling rapidly disseminating infectious disease during mass gatherings

We discuss models for rapidly disseminating infectious diseases during mass gatherings (MGs), using influenza as a case study. Recent innovations in modeling and forecasting influenza transmission dynamics at local, regional, and global scales have made influenza a particularly attractive model scenario for MG. We discuss the behavioral, medical, and population factors for modeling MG disease transmission, review existing model formulations, and highlight key data and modeling gaps related to modeling MG disease transmission. We argue that the proposed improvements will help integrate infectious-disease models in MG health contingency plans in the near future, echoing modeling efforts that have helped shape influenza pandemic preparedness plans in recent years.

Mathematical assessment of Canada's pandemic influenza preparedness plan

OBJECTIVE

The presence of the highly pathogenic avian H5N1 virus in wild bird populations in several regions of the world, together with recurrent cases of H5N1 influenza arising primarily from direct contact with poultry, have highlighted the urgent need for prepared-ness and coordinated global strategies to effectively combat a potential influenza pandemic. The purpose of the present study was to evaluate the Canadian pandemic influenza preparedness plan.

PATIENTS AND METHODS

A mathematical model of the transmission dynamics of influenza was used to keep track of the population according to risk of infection (low or high) and infection status (susceptible, exposed or infectious). The model was parametrized using available Canadian demographic data. The model was then used to evaluate the key components outlined in the Canadian plan.

RESULTS

The results indicated that the number of cases, mortalities and hospitalizations estimated in the Canadian plan may have been underestimated; the use of antivirals, administered therapeutically, prophylactically or both, is the most effective single intervention followed by the use of a vaccine and basic public health measures; and the combined use of pharmaceutical interventions (antivirals and vaccine) can dramatically minimize the burden of the pending influenza pandemic in Canada. Based on increasing concerns of Oseltamivir resistance (wide-scale implementation), coupled with the expected unavailability of a suitable vaccine during the early stages of a pandemic, the present study evaluated the potential impact of non-pharmaceutical interventions (NPIs) which were not emphasized in the current Canadian plan. To this end, the findings suggest that the use of NPIs can drastically reduce the burden of a pandemic in Canada.

CONCLUSIONS

A deterministic model was designed and used to assess Canada's pandemic preparedness plan. The study showed that the estimates of pandemic influenza burden given in the Canada pandemic preparedness plan may be an underestimate, and that Canada needs to adopt NPIs to complement its preparedness plan.

The correlation between infectivity and incubation period of measles, estimated from households with two cases

The generation time of an infectious disease is the time between infection of a primary case and infection of a secondary case by the primary case. Its distribution plays a key role in understanding the dynamics of infectious diseases in populations, e.g. in estimating the basic reproduction number. Moreover, the generation time and incubation period distributions together characterize the effectiveness of control by isolation and quarantine. In modelling studies, a relation between the two is often not made specific, but a correlation is biologically plausible. However, it is difficult to establish such correlation, because of the unobservable nature of infection events. We have quantified a joint distribution of generation time and incubation period by a novel estimation method for household data with two susceptible individuals, consisting of time intervals between disease onsets of two measles cases. We used two such datasets, and a separate incubation period dataset. Results indicate that the mean incubation period and the generation time of measles are positively correlated, and that both lie in the range of 11-12 days, suggesting that infectiousness of measles cases increases significantly around the time of symptom onset. The correlation between times from infection to secondary transmission and to symptom onset could critically affect the predicted effectiveness of isolation and quarantine.

Transmission parameters of the A/H1N1 (2009) influenza virus pandemic: a review

Please cite this paper as: Boëlle P‐Y et al. (2011) Transmission parameters of the A/H1N1 (2009) influenza virus pandemic: a review. Influenza and Other Respiratory Viruses 5(5), 306–316.

Background  The new influenza virus A/H1N1 (2009), identified in mid‐2009, rapidly spread over the world. Estimating the transmissibility of this new virus was a public health priority.

Methods  We reviewed all studies presenting estimates of the serial interval or generation time and the reproduction number of the A/H1N1 (2009) virus infection.

Results  Thirteen studies documented the serial interval from household or close‐contact studies, with overall mean 3 days (95% CI: 2·4, 3·6); taking into account tertiary transmission reduced this estimate to 2·6 days. Model‐based estimates were more variable, from 1·9 to 6 days. Twenty‐four studies reported reproduction numbers for community‐based epidemics at the town or country level. The range was 1·2–3·1, with larger estimates reported at the beginning of the pandemic. Accounting for under‐reporting in the early period of the pandemic and limiting variation because of the choice of the generation time interval, the reproduction number was between 1·2 and 2·3 with median 1·5.

Discussion  The serial interval of A/H1N1 (2009) flu was typically short, with mean value similar to the seasonal flu. The estimates of the reproduction number were more variable. Compared with past influenza pandemics, the median reproduction number was similar (1968) or slightly smaller (1889, 1918, 1957).

Optimal control and sensitivity analysis of an influenza model with treatment and vaccination

We formulate and analyze the dynamics of an influenza pandemic model with vaccination and treatment using two preventive scenarios: increase and decrease in vaccine uptake. Due to the seasonality of the influenza pandemic, the dynamics is studied in a finite time interval. We focus primarily on controlling the disease with a possible minimal cost and side effects using control theory which is therefore applied via the Pontryagin's maximum principle, and it is observed that full treatment effort should be given while increasing vaccination at the onset of the outbreak. Next, sensitivity analysis and simulations (using the fourth order Runge-Kutta scheme) are carried out in order to determine the relative importance of different factors responsible for disease transmission and prevalence. The most sensitive parameter of the various reproductive numbers apart from the death rate is the inflow rate, while the proportion of new recruits and the vaccine efficacy are the most sensitive parameters for the endemic equilibrium point.

Theoretical basis to measure the impact of short-lasting control of an infectious disease on the epidemic peak

Background

While many pandemic preparedness plans have promoted disease control effort to lower and delay an epidemic peak, analytical methods for determining the required control effort and making statistical inferences have yet to be sought. As a first step to address this issue, we present a theoretical basis on which to assess the impact of an early intervention on the epidemic peak, employing a simple epidemic model.

Methods

We focus on estimating the impact of an early control effort (e.g. unsuccessful containment), assuming that the transmission rate abruptly increases when control is discontinued. We provide analytical expressions for magnitude and time of the epidemic peak, employing approximate logistic and logarithmic-form solutions for the latter. Empirical influenza data (H1N1-2009) in Japan are analyzed to estimate the effect of the summer holiday period in lowering and delaying the peak in 2009.

Results

Our model estimates that the epidemic peak of the 2009 pandemic was delayed for 21 days due to summer holiday. Decline in peak appears to be a nonlinear function of control-associated reduction in the reproduction number. Peak delay is shown to critically depend on the fraction of initially immune individuals.

Conclusions

The proposed modeling approaches offer methodological avenues to assess empirical data and to objectively estimate required control effort to lower and delay an epidemic peak. Analytical findings support a critical need to conduct population-wide serological survey as a prior requirement for estimating the time of peak.

Mortality patterns associated with the 1918 influenza pandemic in Mexico: evidence for a spring herald wave and lack of preexisting immunity in older populations

BACKGROUND

Although the mortality burden of the devastating 1918 influenza pandemic has been carefully quantified in the United States, Japan, and European countries, little is known about the pandemic experience elsewhere. Here, we compiled extensive archival records to quantify the pandemic mortality patterns in 2 Mexican cities, Mexico City and Toluca.

METHODS

We applied seasonal excess mortality models to age-specific respiratory mortality rates for 1915-1920 and quantified the reproduction number from daily data.

RESULTS

We identified 3 pandemic waves in Mexico City in spring 1918, autumn 1918, and winter 1920, which were characterized by unusual excess mortality among people 25-44 years old. Toluca experienced 2-fold higher excess mortality rates than Mexico City but did not experience a substantial third wave. All age groups, including that of people 65 years old, experienced excess mortality during 1918-1920. Reproduction number estimates were <2.5, assuming a 3-d generation interval.

CONCLUSION

Mexico experienced a herald pandemic wave with elevated young adult mortality in spring 1918, similar to the United States and Europe. In contrast to the United States and Europe, there was no mortality sparing among Mexican seniors 65 years old, highlighting potential geographical differences in preexisting immunity to the 1918 virus. We discuss the relevance of our findings to the 2009 pandemic mortality patterns.

Modelling the influence of human behaviour on the spread of infectious diseases: a review

Human behaviour plays an important role in the spread of infectious diseases, and understanding the influence of behaviour on the spread of diseases can be key to improving control efforts. While behavioural responses to the spread of a disease have often been reported anecdotally, there has been relatively little systematic investigation into how behavioural changes can affect disease dynamics. Mathematical models for the spread of infectious diseases are an important tool for investigating and quantifying such effects, not least because the spread of a disease among humans is not amenable to direct experimental study. Here, we review recent efforts to incorporate human behaviour into disease models, and propose that such models can be broadly classified according to the type and source of information which individuals are assumed to base their behaviour on, and according to the assumed effects of such behaviour. We highlight recent advances as well as gaps in our understanding of the interplay between infectious disease dynamics and human behaviour, and suggest what kind of data taking efforts would be helpful in filling these gaps.

Transmissibility of the 1918 pandemic influenza in Montreal and Winnipeg of Canada

Background  The threat of 2009 pandemic influenza A (H1N1) is still causing widespread public concern. A comprehensive understanding of the epidemiology of 1918 pandemic influenza commonly referred to as the Spanish flu may be helpful in offering insight into control strategies for the new pandemic.

Objective  We explore how the preparedness for a pandemic at the community and individual level impacts the spread of the virus by comparing the transmissibility of the 1918 Spanish flu in two Canadian cities: Montreal and Winnipeg, bearing in mind that each pandemic is unique and the current one may not follow the pattern of the 1918 outbreak.

Methods  The historical epidemiological data obtained for Montreal and Winnipeg in Canada is analyzed to estimate the basic reproduction number which is the most important summary measure of transmission potential of the pandemic.

Results  The transmissibility of the 1918 pandemic influenza virus in Winnipeg in the fall of 1918 was found to be much lower than in Montreal based on the estimated reproduction number obtained assuming different serial intervals which are the time between onsets of symptoms in an index case and a secondary case.

Conclusion  The early preparedness and public health control measures could suggest an explanation for the fact that the number of secondary cases generated by a primary case was significantly reduced in Winnipeg comparing to it in Montreal.

Pros and cons of estimating the reproduction number from early epidemic growth rate of influenza A (H1N1) 2009

Background

In many parts of the world, the exponential growth rate of infections during the initial epidemic phase has been used to make statistical inferences on the reproduction number, R, a summary measure of the transmission potential for the novel influenza A (H1N1) 2009. The growth rate at the initial stage of the epidemic in Japan led to estimates for R in the range 2.0 to 2.6, capturing the intensity of the initial outbreak among school-age children in May 2009.

Methods

An updated estimate of R that takes into account the epidemic data from 29 May to 14 July is provided. An age-structured renewal process is employed to capture the age-dependent transmission dynamics, jointly estimating the reproduction number, the age-dependent susceptibility and the relative contribution of imported cases to secondary transmission. Pitfalls in estimating epidemic growth rates are identified and used for scrutinizing and re-assessing the results of our earlier estimate of R.

Results

Maximum likelihood estimates of R using the data from 29 May to 14 July ranged from 1.21 to 1.35. The next-generation matrix, based on our age-structured model, predicts that only 17.5% of the population will experience infection by the end of the first pandemic wave. Our earlier estimate of R did not fully capture the population-wide epidemic in quantifying the next-generation matrix from the estimated growth rate during the initial stage of the pandemic in Japan.

Conclusions

In order to quantify R from the growth rate of cases, it is essential that the selected model captures the underlying transmission dynamics embedded in the data. Exploring additional epidemiological information will be useful for assessing the temporal dynamics. Although the simple concept of R is more easily grasped by the general public than that of the next-generation matrix, the matrix incorporating detailed information (e.g., age-specificity) is essential for reducing the levels of uncertainty in predictions and for assisting public health policymaking. Model-based prediction and policymaking are best described by sharing fundamental notions of heterogeneous risks of infection and death with non-experts to avoid potential confusion and/or possible misuse of modelling results.

Adaptive vaccination strategies to mitigate pandemic influenza: Mexico as a case study

BACKGROUND

We explore vaccination strategies against pandemic influenza in Mexico using an age-structured transmission model calibrated against local epidemiological data from the Spring 2009 A(H1N1) pandemic.

METHODS AND FINDINGS

In the context of limited vaccine supplies, we evaluate age-targeted allocation strategies that either prioritize youngest children and persons over 65 years of age, as for seasonal influenza, or adaptively prioritize age groups based on the age patterns of hospitalization and death monitored in real-time during the early stages of the pandemic. Overall the adaptive vaccination strategy outperformed the seasonal influenza vaccination allocation strategy for a wide range of disease and vaccine coverage parameters.

CONCLUSIONS

This modeling approach could inform policies for Mexico and other countries with similar demographic features and vaccine resources issues, with regard to the mitigation of the S-OIV pandemic. We also discuss logistical issues associated with the implementation of adaptive vaccination strategies in the context of past and future influenza pandemics.

Early epidemiological assessment of the virulence of emerging infectious diseases: a case study of an influenza pandemic

Background

The case fatality ratio (CFR), the ratio of deaths from an infectious disease to the number of cases, provides an assessment of virulence. Calculation of the ratio of the cumulative number of deaths to cases during the course of an epidemic tends to result in a biased CFR. The present study develops a simple method to obtain an unbiased estimate of confirmed CFR (cCFR), using only the confirmed cases as the denominator, at an early stage of epidemic, even when there have been only a few deaths.

Methodology/Principal Findings

Our method adjusts the biased cCFR by a factor of underestimation which is informed by the time from symptom onset to death. We first examine the approach by analyzing an outbreak of severe acute respiratory syndrome in Hong Kong (2003) with known unbiased cCFR estimate, and then investigate published epidemiological datasets of novel swine-origin influenza A (H1N1) virus infection in the USA and Canada (2009). Because observation of a few deaths alone does not permit estimating the distribution of the time from onset to death, the uncertainty is addressed by means of sensitivity analysis. The maximum likelihood estimate of the unbiased cCFR for influenza may lie in the range of 0.16–4.48% within the assumed parameter space for a factor of underestimation. The estimates for influenza suggest that the virulence is comparable to the early estimate in Mexico. Even when there have been no deaths, our model permits estimating a conservative upper bound of the cCFR.

Conclusions

Although one has to keep in mind that the cCFR for an entire population is vulnerable to its variations among sub-populations and underdiagnosis, our method is useful for assessing virulence at the early stage of an epidemic and for informing policy makers and the public.

Estimation of the serial interval of influenza

BACKGROUND

: Estimates of the clinical-onset serial interval of human influenza infection (time between onset of symptoms in an index case and a secondary case) are used to inform public health policy and to construct mathematical models of influenza transmission. We estimate the serial interval of laboratory-confirmed influenza transmission in households.

METHODS

: Index cases were recruited after reporting to a primary healthcare center with symptoms. Members of their households were followed-up with repeated home visits.

RESULTS

: Assuming a Weibull model and accounting for selection bias inherent in our field study design, we used symptom-onset times from 14 pairs of infector/infectee to estimate a mean serial interval of 3.6 days (95% confidence interval = 2.9-4.3 days), with standard deviation 1.6 days.

CONCLUSION

: The household serial interval of influenza may be longer than previously estimated. Studies of the complete serial interval, based on transmission in all community contexts, are a priority.

Models of epidemics: when contact repetition and clustering should be included

BACKGROUND

The spread of infectious disease is determined by biological factors, e.g. the duration of the infectious period, and social factors, e.g. the arrangement of potentially contagious contacts. Repetitiveness and clustering of contacts are known to be relevant factors influencing the transmission of droplet or contact transmitted diseases. However, we do not yet completely know under what conditions repetitiveness and clustering should be included for realistically modelling disease spread.

METHODS

We compare two different types of individual-based models: One assumes random mixing without repetition of contacts, whereas the other assumes that the same contacts repeat day-by-day. The latter exists in two variants, with and without clustering. We systematically test and compare how the total size of an outbreak differs between these model types depending on the key parameters transmission probability, number of contacts per day, duration of the infectious period, different levels of clustering and varying proportions of repetitive contacts.

RESULTS

The simulation runs under different parameter constellations provide the following results: The difference between both model types is highest for low numbers of contacts per day and low transmission probabilities. The number of contacts and the transmission probability have a higher influence on this difference than the duration of the infectious period. Even when only minor parts of the daily contacts are repetitive and clustered can there be relevant differences compared to a purely random mixing model.

CONCLUSION

We show that random mixing models provide acceptable estimates of the total outbreak size if the number of contacts per day is high or if the per-contact transmission probability is high, as seen in typical childhood diseases such as measles. In the case of very short infectious periods, for instance, as in Norovirus, models assuming repeating contacts will also behave similarly as random mixing models. If the number of daily contacts or the transmission probability is low, as assumed for MRSA or Ebola, particular consideration should be given to the actual structure of potentially contagious contacts when designing the model.

Controlling malaria: competition, seasonality and 'slingshotting' transgenic mosquitoes into natural populations

Forty years after the World Health Organization abandoned its eradication campaign, malaria remains a public health problem of the first magnitude with worldwide infection rates on the order of 300 million souls. The present paper reviews potential control strategies from the viewpoint of mathematical epidemiology. Following MacDonald and others, we argue in Section 1 that the use of imagicides, i.e., killing, or at least repelling, adult mosquitoes, is inherently the most effective way of combating the pandemic. In Section 2, we model competition between wild-type (WT) and plasmodium-resistant, genetically modified (GM) mosquitoes. Under the assumptions of inherent cost and prevalence-dependant benefit to transgenics, GM introduction can never eradicate malaria save by stochastic extinction of WTs. Moreover, alternative interventions that reduce prevalence have the undesirable consequence of reducing the likelihood of successful GM introduction. Section 3 considers the possibility of using seasonal fluctuations in mosquito abundance and disease prevalence to 'slingshot' GM mosquitoes into natural populations. By introducing GM mosquitoes when natural populations are about to expand, one can 'piggyback' on the yearly cycle. Importantly, this effect is only significant when transgene cost is small, in which case the non-trivial equilibrium is a focus (damped oscillations), and piggybacking is amplified by the system's inherent tendency to oscillate. By way of contrast, when transgene cost is large, the equilibrium is a node and no such amplification is obtained.

Smallpox and season: reanalysis of historical data

Seasonal variation in smallpox transmission is one of the most pressing ecological questions and is relevant to bioterrorism preparedness. The present study reanalyzed 7 historical datasets which recorded monthly cases or deaths. In addition to time series analyses of reported data, an estimation and spectral analysis of the effective reproduction number at calendar time t, R(t), were made. Meteorological variables were extracted from a report in India from 1890-1921 and compared with smallpox mortality as well as R(t). Annual cycles of smallpox transmission were clearly shown not only in monthly reports but also in the estimates of R(t). Even short-term epidemic data clearly exhibited an annual peak every January. Both mortality and R(t) revealed significant negative association (P < .01) and correlation (P < .01), respectively, with humidity. These findings suggest that smallpox transmission greatly varies with season and is most likely enhanced by dry weather.

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

Although the basic reproduction number, R₀, is useful for understanding the transmissibility of a disease and designing various intervention strategies, the classic threshold quantity theoretically assumes that the epidemic first occurs in a fully susceptible population, and hence, R₀ is essentially a mathematically defined quantity. In many instances, it is of practical importance to evaluate time-dependent variations in the transmission potential of infectious diseases. Explanation of the time course of an epidemic can be partly achieved by estimating the effective reproduction number, R(t), defined as the actual average number of secondary cases per primary case at calendar time t (for t >0). R(t) shows time-dependent variation due to the decline in susceptible individuals (intrinsic factors) and the implementation of control measures (extrinsic factors). If R(t)<1, it suggests that the epidemic is in decline and may be regarded as being under control at time t (vice versa, if R(t)>1). This chapter describes the primer of mathematics and statistics of R(t) and discusses other similar markers of transmissibility as a function of time.

Deaths from bacterial pneumonia during 1918-19 influenza pandemic

A sequential-infection hypothesis is consistent with characteristics of this pandemic.

Deaths during the 1918–19 influenza pandemic have been attributed to a hypervirulent influenza strain. Hence, preparations for the next pandemic focus almost exclusively on vaccine prevention and antiviral treatment for infections with a novel influenza strain. However, we hypothesize that infections with the pandemic strain generally caused self-limited (rarely fatal) illnesses that enabled colonizing strains of bacteria to produce highly lethal pneumonias. This sequential-infection hypothesis is consistent with characteristics of the 1918–19 pandemic, contemporaneous expert opinion, and current knowledge regarding the pathophysiologic effects of influenza viruses and their interactions with respiratory bacteria. This hypothesis suggests opportunities for prevention and treatment during the next pandemic (e.g., with bacterial vaccines and antimicrobial drugs), particularly if a pandemic strain–specific vaccine is unavailable or inaccessible to isolated, crowded, or medically underserved populations.

Extracting key information from historical data to quantify the transmission dynamics of smallpox

Background

Quantification of the transmission dynamics of smallpox is crucial for optimizing intervention strategies in the event of a bioterrorist attack. This article reviews basic methods and findings in mathematical and statistical studies of smallpox which estimate key transmission parameters from historical data.

Main findings

First, critically important aspects in extracting key information from historical data are briefly summarized. We mention different sources of heterogeneity and potential pitfalls in utilizing historical records. Second, we discuss how smallpox spreads in the absence of interventions and how the optimal timing of quarantine and isolation measures can be determined. Case studies demonstrate the following. (1) The upper confidence limit of the 99th percentile of the incubation period is 22.2 days, suggesting that quarantine should last 23 days. (2) The highest frequency (61.8%) of secondary transmissions occurs 3–5 days after onset of fever so that infected individuals should be isolated before the appearance of rash. (3) The U-shaped age-specific case fatality implies a vulnerability of infants and elderly among non-immune individuals. Estimates of the transmission potential are subsequently reviewed, followed by an assessment of vaccination effects and of the expected effectiveness of interventions.

Conclusion

Current debates on bio-terrorism preparedness indicate that public health decision making must account for the complex interplay and balance between vaccination strategies and other public health measures (e.g. case isolation and contact tracing) taking into account the frequency of adverse events to vaccination. In this review, we summarize what has already been clarified and point out needs to analyze previous smallpox outbreaks systematically.

A new SEIR epidemic model with applications to the theory of eradication and control of diseases, and to the calculation of R0

We present a novel SEIR (susceptible-exposure-infective-recovered) model that is suitable for modeling the eradication of diseases by mass vaccination or control of diseases by case isolation combined with contact tracing, incorporating the vaccine efficacy or the control efficacy into the model. Moreover, relying on this novel SEIR model and some probabilistic arguments, we have found four formulas that are suitable for estimating the basic reproductive numbers R(0) in terms of the ratio of the mean infectious period to the mean latent period of a disease. The ranges of R(0) for most known diseases, that are calculated by our formulas, coincide very well with the values of R(0) estimated by the usual method of fitting the models to observed data.

The 1918-1919 influenza pandemic in England and Wales: spatial patterns in transmissibility and mortality impact

Spatial variations in disease patterns of the 1918-1919 influenza pandemic remain poorly studied. We explored the association between influenza death rates, transmissibility and several geographical and demographic indicators for the autumn and winter waves of the 1918-1919 pandemic in cities, towns and rural areas of England and Wales. Average measures of transmissibility, estimated by the reproduction number, ranged between 1.3 and 1.9, depending on model assumptions and pandemic wave and showed little spatial variation. Death rates varied markedly with urbanization, with 30-40% higher rates in cities and towns compared with rural areas. In addition, death rates varied with population size across rural settings, where low population areas fared worse. By contrast, we found no association between transmissibility, death rates and indicators of population density and residential crowding. Further studies of the geographical mortality patterns associated with the 1918-1919 influenza pandemic may be useful for pandemic planning.

Quantifying the transmission potential of pandemic influenza

This article reviews quantitative methods to estimate the basic reproduction number of pandemic influenza, a key threshold quantity to help determine the intensity of interventions required to control the disease. Although it is difficult to assess the transmission potential of a probable future pandemic, historical epidemiologic data is readily available from previous pandemics, and as a reference quantity for future pandemic planning, mathematical and statistical analyses of historical data are crucial. In particular, because many historical records tend to document only the temporal distribution of cases or deaths (i.e. epidemic curve), our review focuses on methods to maximize the utility of time-evolution data and to clarify the detailed mechanisms of the spread of influenza.

First, we highlight structured epidemic models and their parameter estimation method which can quantify the detailed disease dynamics including those we cannot observe directly. Duration-structured epidemic systems are subsequently presented, offering firm understanding of the definition of the basic and effective reproduction numbers. When the initial growth phase of an epidemic is investigated, the distribution of the generation time is key statistical information to appropriately estimate the transmission potential using the intrinsic growth rate. Applications of stochastic processes are also highlighted to estimate the transmission potential using similar data. Critically important characteristics of influenza data are subsequently summarized, followed by our conclusions to suggest potential future methodological improvements.

Epidemiologic characterization of the 1918 influenza pandemic summer wave in Copenhagen: implications for pandemic control strategies

BACKGROUND

The 1918-1919 A/H1N1 influenza pandemic killed approximately 50 million people worldwide. Historical records suggest that an early pandemic wave struck Europe during the summer of 1918.

METHODS

We obtained surveillance data that were compiled weekly, during 1910-1919, in Copenhagen, Denmark; the records included medically treated influenza-like illnesses (ILIs), hospitalizations, and deaths by age. We used a Serfling seasonal regression model to quantify excess morbidity and mortality, and we estimated the reproductive number (R) for the summer, fall, and winter pandemic waves.

RESULTS

A large epidemic occurred in Copenhagen during the summer of 1918; the age distribution of deaths was characteristic of the 1918-1919 A/H1N1 pandemic overall. That summer wave accounted for 29%-34% of all excess ILIs and hospitalizations during 1918, whereas the case-fatality rate (0.3%) was many-fold lower than that of the fall wave (2.3%). Similar patterns were observed in 3 other Scandinavian cities. R was substantially higher in summer (2.0-5.4) than in fall (1.2-1.6) in all cities.

CONCLUSIONS

The Copenhagen summer wave may have been caused by a precursor A/H1N1 pandemic virus that transmitted efficiently but lacked extreme virulence. The R measured in the summer wave is likely a better approximation of transmissibility in a fully susceptible population and is substantially higher than that found in previous US studies. The summer wave may have provided partial protection against the lethal fall wave.

A biological model for influenza transmission: pandemic planning implications of asymptomatic infection and immunity

Background

The clinical attack rate of influenza is influenced by prior immunity and mixing patterns in the host population, and also by the proportion of infections that are asymptomatic. This complexity makes it difficult to directly estimate R₀ from the attack rate, contributing to uncertainty in epidemiological models to guide pandemic planning. We have modelled multiple wave outbreaks of influenza from different populations to allow for changing immunity and asymptomatic infection and to make inferences about R₀.

Data and Methods

On the island of Tristan da Cunha (TdC), 96% of residents reported illness during an H3N2 outbreak in 1971, compared with only 25% of RAF personnel in military camps during the 1918 H1N1 pandemic. Monte Carlo Markov Chain (MCMC) methods were used to estimate model parameter distributions.

Findings

We estimated that most islanders on TdC were non-immune (susceptible) before the first wave, and that almost all exposures of susceptible persons caused symptoms. The median R₀ of 6.4 (95% credibility interval 3.7–10.7) implied that most islanders were exposed twice, although only a minority became ill in the second wave because of temporary protection following the first wave. In contrast, only 51% of RAF personnel were susceptible before the first wave, and only 38% of exposed susceptibles reported symptoms. R₀ in this population was also lower [2.9 (2.3–4.3)], suggesting reduced viral transmission in a partially immune population.

Interpretation

Our model implies that the RAF population was partially protected before the summer pandemic wave of 1918, arguably because of prior exposure to interpandemic influenza. Without such protection, each symptomatic case of influenza would transmit to between 2 and 10 new cases, with incidence initially doubling every 1–2 days. Containment of a novel virus could be more difficult than hitherto supposed.