Estimating the transmission potential of supercritical processes based on the final size distribution of minor outbreaks

Inference of epidemic dynamics in the COVID-19 era and beyond

The COVID-19 pandemic demonstrated the key role that epidemiology and modelling play in analysing infectious threats and supporting decision making in real-time. Motivated by the unprecedented volume and breadth of data generated during the pandemic, we review modern opportunities for analysis to address questions that emerge during a major modern epidemic. Following the broad chronology of insights required - from understanding initial dynamics to retrospective evaluation of interventions, we describe the theoretical foundations of each approach and the underlying intuition. Through a series of case studies, we illustrate real life applications, and discuss implications for future work.

Estimating the reproduction number and transmission heterogeneity from the size distribution of clusters of identical pathogen sequences

Quantifying transmission intensity and heterogeneity is crucial to ascertain the threat posed by infectious diseases and inform the design of interventions. Methods that jointly estimate the reproduction number R and the dispersion parameter k have however mainly remained limited to the analysis of epidemiological clusters or contact tracing data, whose collection often proves difficult. Here, we show that clusters of identical sequences are imprinted by the pathogen offspring distribution, and we derive an analytical formula for the distribution of the size of these clusters. We develop and evaluate an inference framework to jointly estimate the reproduction number and the dispersion parameter from the size distribution of clusters of identical sequences. We then illustrate its application across a range of epidemiological situations. Finally, we develop a hypothesis testing framework relying on clusters of identical sequences to determine whether a given pathogen genetic subpopulation is associated with increased or reduced transmissibility. Our work provides tools to estimate the reproduction number and transmission heterogeneity from pathogen sequences without building a phylogenetic tree, thus making it easily scalable to large pathogen genome datasets.

A beta-Poisson model for infectious disease transmission

Outbreaks of emerging and zoonotic infections represent a substantial threat to human health and well-being. These outbreaks tend to be characterised by highly stochastic transmission dynamics with intense variation in transmission potential between cases. The negative binomial distribution is commonly used as a model for transmission in the early stages of an epidemic as it has a natural interpretation as the convolution of a Poisson contact process and a gamma-distributed infectivity. In this study we expand upon the negative binomial model by introducing a beta-Poisson mixture model in which infectious individuals make contacts at the points of a Poisson process and then transmit infection along these contacts with a beta-distributed probability. We show that the negative binomial distribution is a limit case of this model, as is the zero-inflated Poisson distribution obtained by combining a Poisson-distributed contact process with an additional failure probability. We assess the beta-Poisson model's applicability by fitting it to secondary case distributions (the distribution of the number of subsequent cases generated by a single case) estimated from outbreaks covering a range of pathogens and geographical settings. We find that while the beta-Poisson mixture can achieve a closer to fit to data than the negative binomial distribution, it is consistently outperformed by the negative binomial in terms of Akaike Information Criterion, making it a suboptimal choice on parsimonious grounds. The beta-Poisson performs similarly to the negative binomial model in its ability to capture features of the secondary case distribution such as overdispersion, prevalence of superspreaders, and the probability of a case generating zero subsequent cases. Despite this possible shortcoming, the beta-Poisson distribution may still be of interest in the context of intervention modelling since its structure allows for the simulation of measures which change contact structures while leaving individual-level infectivity unchanged, and vice-versa.

Transmission characteristics and inactivated vaccine effectiveness against transmission of the SARS-CoV-2 Omicron BA.2 variant in Shenzhen, China

We conducted a retrospective cohort study to evaluate the transmission risk of severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) Omicron BA.2 variant and the effectiveness of inactivated COVID-19 vaccine boosters in Shenzhen during a BA.2 outbreak period from 1 February to 21 April 2022. A total of 1,248 individuals were infected with the BA.2 variant, and 7,855 close contacts were carefully investigated. The risk factors for the high secondary attack rate of SARS-CoV-2 infection were household contacts [adjusted odds ratio (aOR): 1.748; 95% confidence interval (CI): 1.448, 2.110], younger individuals aged 0-17 years (aOR: 2.730; 95% CI: 2.118, 3.518), older persons aged ≥60 years (aOR: 1.342; 95% CI: 1.135, 1.588), women (aOR: 1.442; 95% CI: 1.210, 1.718), and the subjects exposed to the post-onset index cases (aOR: 8.546; 95% CI: 6.610, 11.050), respectively. Compared with the unvaccinated and partially vaccinated individuals, a relatively low risk of secondary attack was found for the individuals who received booster vaccination (aOR: 0.871; 95% CI: 0.761, 0.997). Moreover, a high transmission risk was found for the index cases aged ≥60 years (aOR: 1.359; 95% CI: 1.132, 1.632), whereas a relatively low transmission risk was observed for the index cases who received full vaccination (aOR: 0.642; 95% CI: 0.490, 0.841) and booster vaccination (aOR: 0.676; 95% CI: 0.594, 0.770). Compared with full vaccination, booster vaccination of inactivated COVID-19 vaccine showed an effectiveness of 24.0% (95% CI: 7.0%, 37.9%) against BA.2 transmission for the adults ≥18 years and 93.7% (95% CI: 72.4%, 98.6%) for the adults ≥60 years, whereas the effectiveness was 51.0% (95% CI: 21.9%, 69.3%) for the individuals of 14 days to 179 days after booster vaccination and 51.2% (95% CI: 37.5%, 61.9%) for the non-household contacts. The estimated mean values of the generation interval, serial interval, incubation period, latent period, and viral shedding period were 2.7 days, 3.2 days, 2.4 days, 2.1 days, and 17.9 days, respectively. In summary, our results confirmed that the main transmission route of Omicron BA.2 subvariant was household contact, and booster vaccination of the inactivated vaccines was relatively effective against BA.2 subvariant transmission in older people.

Transmission bottleneck size estimation from de novo viral genetic variation

Sequencing of viral infections has become increasingly common over the last decade. Deep sequencing data in particular have proven useful in characterizing the roles that genetic drift and natural selection play in shaping within-host viral populations. They have also been used to estimate transmission bottleneck sizes from identified donor-recipient pairs. These bottleneck sizes quantify the number of viral particles that establish genetic lineages in the recipient host and are important to estimate due to their impact on viral evolution. Current approaches for estimating bottleneck sizes exclusively consider the subset of viral sites that are observed as polymorphic in the donor individual. However, allele frequencies can change dramatically over the course of an individual's infection, such that sites that are polymorphic in the donor at the time of transmission may not be polymorphic in the donor at the time of sampling and allele frequencies at donor-polymorphic sites may change dramatically over the course of a recipient's infection. Because of this, transmission bottleneck sizes estimated using allele frequencies observed at a donor's polymorphic sites may be considerable underestimates of true bottleneck sizes. Here, we present a new statistical approach for instead estimating bottleneck sizes using patterns of viral genetic variation that arose de novo within a recipient individual. Specifically, our approach makes use of the number of clonal viral variants observed in a transmission pair, defined as the number of viral sites that are monomorphic in both the donor and the recipient but carry different alleles. We first test our approach on a simulated dataset and then apply it to both influenza A virus sequence data and SARS-CoV-2 sequence data from identified transmission pairs. Our results confirm the existence of extremely tight transmission bottlenecks for these two respiratory viruses, using an approach that does not tend to underestimate transmission bottleneck sizes.

Temporal Pattern of the Emergence of a Mutant Virus Escaping Cross-Immunity and Stochastic Extinction Within a Host

A high mutation rate of the RNA virus results in the emergence of novel mutants that may escape the immunity activated by the original (wild-type) strain. However, many of them go extinct because of the stochasticity due to the small initial number of infected cells. In a previous paper, we studied the probability of escaping stochastic extinction when the novel mutant has a faster rate of infection and when it is resistant to a drug that suppresses the wild-type virus. In this study, we examine the effect of escaping the immune reaction of the host. Based on a continuous-time branching process with time-dependent rates, we conclude the chance for a mutant strain to be established [Formula: see text] decreases with time [Formula: see text] since the wild-type infection when the mutant is produced. The number of novel mutants that can escape extinction risk has a peak soon after the wild-type infection. The number of novel escape mutations produced per patient in the early phase of host infection is small both for very strong and very weak immune responses, and it attains its maximum value when immune activity is of an intermediate strength.

Quantifying Mycobacterium tuberculosis Transmission Dynamics Across Global Settings: A Systematic Analysis

The degree to which individual heterogeneity in the production of secondary cases ("superspreading") affects tuberculosis (TB) transmission has not been systematically studied. We searched for population-based or surveillance studies in which whole genome sequencing was used to estimate TB transmission and in which the size distributions of putative TB transmission clusters were enumerated. We fitted cluster-size-distribution data to a negative binomial branching process model to jointly infer the transmission parameters $R$ (the reproduction number) and the dispersion parameter, $k$, which quantifies the propensity of superspreading in a population (generally, lower values of $k$ ($<1.0$) suggest increased heterogeneity). Of 4,796 citations identified in our initial search, 9 studies from 8 global settings met the inclusion criteria (n = 5 studies of all TB; n = 4 studies of drug-resistant TB). Estimated $R$ values (range, 0.10-0.73) were below 1.0, consistent with declining epidemics in the included settings; estimated $k$ values were well below 1.0 (range, 0.02-0.48), indicating the presence of substantial individual-level heterogeneity in transmission across all settings. We estimated that a minority of cases (range, 2%-31%) drive the majority (80%) of ongoing TB transmission at the population level. Identifying sources of heterogeneity and accounting for them in TB control may have a considerable impact on mitigating TB transmission.

Basic reproduction number of the COVID-19 Delta variant: Estimation from multiple transmission datasets

The basic reproduction number, $ R_0 $, plays a central role in measuring the transmissibility of an infectious disease, and it thus acts as the fundamental index for planning control strategies. In the present study, we apply a branching process model to meticulously observed contact tracing data from Wakayama Prefecture, Japan, obtained in early 2020 and mid-2021. This allows us to efficiently estimate $ R_0 $ and the dispersion parameter $ k $ of the wild-type COVID-19, as well as the relative transmissibility of the Delta variant and relative transmissibility among fully vaccinated individuals, from a very limited data. $ R_0 $ for the wild type of COVID-19 is estimated to be 3.78 (95% confidence interval [CI]: 3.72-3.83), with $ k = 0.236 $ (95% CI: 0.233-0.240). For the Delta variant, the relative transmissibility to the wild type is estimated to be 1.42 (95% CI: 0.94-1.90), which gives $ R_0 = 5.37 $ (95% CI: 3.55-7.21). Vaccine effectiveness, determined by the reduction in the number of secondary transmissions among fully vaccinated individuals, is estimated to be 91% (95% CI: 85%-97%). The present study highlights that basic reproduction numbers can be accurately estimated from the distribution of minor outbreak data, and these data can provide further insightful epidemiological estimates including the dispersion parameter and vaccine effectiveness regarding the prevention of transmission.

Quantifying the future risk of dengue under climate change in Japan

Background

In metropolitan Tokyo in 2014, Japan experienced its first domestic dengue outbreak since 1945. The objective of the present study was to quantitatively assess the future risk of dengue in Japan using climate change scenarios in a high-resolution geospatial environment by building on a solid theory as a baseline in consideration of future adaptation strategies.

Methods

Using climate change scenarios of the Model for Interdisciplinary Research on Climate version 6 (MIROC6), representative concentration pathway (RCP) 2.6, 4.5, and 8.5, we computed the daily average temperature and embedded this in the effective reproduction number of dengue, R(T), to calculate the extinction probability and interepidemic period across Japan.

Results

In June and October, the R(T) with daily average temperature T, was <1 as in 2022; however, an elevation in temperature increased the number of days with R(T) >1 during these months under RCP8.5. The time period with a risk of dengue transmission gradually extended to late spring (April–May) and autumn (October–November). Under the RCP8.5 scenario in 2100, the possibility of no dengue-free months was revealed in part of southernmost Okinawa Prefecture, and the epidemic risk extended to the entire part of northernmost Hokkaido Prefecture.

Conclusion

Each locality in Japan must formulate action plans in response to the presented scenarios. Our geographic analysis can help local governments to develop adaptation policies that include mosquito breeding site elimination, distribution of adulticides and larvicides, and elevated situation awareness to prevent transmission via bites from Aedes vectors.

Modeling scenarios for mitigating outbreaks in congregate settings

The explosive outbreaks of COVID-19 seen in congregate settings such as prisons and nursing homes, has highlighted a critical need for effective outbreak prevention and mitigation strategies for these settings. Here we consider how different types of control interventions impact the expected number of symptomatic infections due to outbreaks. Introduction of disease into the resident population from the community is modeled as a stochastic point process coupled to a branching process, while spread between residents is modeled via a deterministic compartmental model that accounts for depletion of susceptible individuals. Control is modeled as a proportional decrease in the number of susceptible residents, the reproduction number, and/or the proportion of symptomatic infections. This permits a range of assumptions about the density dependence of transmission and modes of protection by vaccination, depopulation and other types of control. We find that vaccination or depopulation can have a greater than linear effect on the expected number of cases. For example, assuming a reproduction number of 3.0 with density-dependent transmission, we find that preemptively reducing the size of the susceptible population by 20% reduced overall disease burden by 47%. In some circumstances, it may be possible to reduce the risk and burden of disease outbreaks by optimizing the way a group of residents are apportioned into distinct residential units. The optimal apportionment may be different depending on whether the goal is to reduce the probability of an outbreak occurring, or the expected number of cases from outbreak dynamics. In other circumstances there may be an opportunity to implement reactive disease control measures in which the number of susceptible individuals is rapidly reduced once an outbreak has been detected to occur. Reactive control is most effective when the reproduction number is not too high, and there is minimal delay in implementing control. We highlight the California state prison system as an example for how these findings provide a quantitative framework for understanding disease transmission in congregate settings. Our approach and accompanying interactive website (https://phoebelu.shinyapps.io/DepopulationModels/) provides a quantitative framework to evaluate the potential impact of policy decisions governing infection control in outbreak settings.

Characterizing superspreading potential of infectious disease: Decomposition of individual transmissibility

In the context of infectious disease transmission, high heterogeneity in individual infectiousness indicates that a few index cases can generate large numbers of secondary cases, a phenomenon commonly known as superspreading. The potential of disease superspreading can be characterized by describing the distribution of secondary cases (of each seed case) as a negative binomial (NB) distribution with the dispersion parameter, k. Based on the feature of NB distribution, there must be a proportion of individuals with individual reproduction number of almost 0, which appears restricted and unrealistic. To overcome this limitation, we generalized the compound structure of a Poisson rate and included an additional parameter, and divided the reproduction number into independent and additive fixed and variable components. Then, the secondary cases followed a Delaporte distribution. We demonstrated that the Delaporte distribution was important for understanding the characteristics of disease transmission, which generated new insights distinct from the NB model. By using real-world dataset, the Delaporte distribution provides improvements in describing the distributions of COVID-19 and SARS cases compared to the NB distribution. The model selection yielded increasing statistical power with larger sample sizes as well as conservative type I error in detecting the improvement in fitting with the likelihood ratio (LR) test. Numerical simulation revealed that the control strategy-making process may benefit from monitoring the transmission characteristics under the Delaporte framework. Our findings highlighted that for the COVID-19 pandemic, population-wide interventions may control disease transmission on a general scale before recommending the high-risk-specific control strategies.

Superspreading is one of the key transmission features of many infectious diseases and is considered a consequence of the heterogeneity in infectiousness of individual cases. To characterize the superspreading potential, we divided individual infectiousness into two independent and additive components, including a fixed baseline and a variable part. Such decomposition produced an improvement in the fit of the model explaining the distribution of real-world datasets of COVID-19 and SARS that can be captured by the classic statistical tests. Disease control strategies may be developed by monitoring the characteristics of superspreading. For the COVID-19 pandemic, population-wide interventions are suggested first to limit the transmission at a scale of general population, and then high-risk-specific control strategies are recommended subsequently to lower the risk of superspreading.

An open-access database of infectious disease transmission trees to explore superspreader epidemiology

Historically, emerging and reemerging infectious diseases have caused large, deadly, and expensive multinational outbreaks. Often outbreak investigations aim to identify who infected whom by reconstructing the outbreak transmission tree, which visualizes transmission between individuals as a network with nodes representing individuals and branches representing transmission from person to person. We compiled a database, called OutbreakTrees, of 382 published, standardized transmission trees consisting of 16 directly transmitted diseases ranging in size from 2 to 286 cases. For each tree and disease, we calculated several key statistics, such as tree size, average number of secondary infections, the dispersion parameter, and the proportion of cases considered superspreaders, and examined how these statistics varied over the course of each outbreak and under different assumptions about the completeness of outbreak investigations. We demonstrated the potential utility of the database through 2 short analyses addressing questions about superspreader epidemiology for a variety of diseases, including Coronavirus Disease 2019 (COVID-19). First, we found that our transmission trees were consistent with theory predicting that intermediate dispersion parameters give rise to the highest proportion of cases causing superspreading events. Additionally, we investigated patterns in how superspreaders are infected. Across trees with more than 1 superspreader, we found preliminary support for the theory that superspreaders generate other superspreaders. In sum, our findings put the role of superspreading in COVID-19 transmission in perspective with that of other diseases and suggest an approach to further research regarding the generation of superspreaders. These data have been made openly available to encourage reuse and further scientific inquiry.

This study compiles and standardizes reported infectious disease transmission trees to analyze trends in superspreader frequency and generation; these data support theories that intermediate dispersion parameters give rise to the highest proportion of cases causing superspreading events, and that superspreaders generate other superspreaders.

Characterizing tuberculosis transmission dynamics in high-burden urban and rural settings

Mycobacterium tuberculosis transmission dynamics in high-burden settings are poorly understood. Growing evidence suggests transmission may be characterized by extensive individual heterogeneity in secondary cases (i.e., superspreading), yet the degree and influence of such heterogeneity is largely unknown and unmeasured in high burden-settings. We conducted a prospective, population-based molecular epidemiology study of TB transmission in both an urban and rural setting of Botswana, one of the highest TB burden countries in the world. We used these empirical data to fit two mathematical models (urban and rural) that jointly quantified both the effective reproductive number, [Formula: see text], and the propensity for superspreading in each population. We found both urban and rural populations were characterized by a high degree of individual heterogeneity, however such heterogeneity disproportionately impacted the rural population: 99% of secondary transmission was attributed to only 19% of infectious cases in the rural population compared to 60% in the urban population and the median number of incident cases until the first outbreak of 30 cases was only 32 for the rural model compared to 791 in the urban model. These findings suggest individual heterogeneity plays a critical role shaping local TB epidemiology within subpopulations.

A Cluster-based Method to Quantify Individual Heterogeneity in Tuberculosis Transmission

BACKGROUND

Recent evidence suggests transmission of Mycobacterium tuberculosis (Mtb) may be characterized by extreme individual heterogeneity in secondary cases (i.e., few cases account for the majority of transmission). Such heterogeneity implies outbreaks are rarer but more extensive and has profound implications in infectious disease control. However, discrete person-to-person transmission events in tuberculosis (TB) are often unobserved, precluding our ability to directly quantify individual heterogeneity in TB epidemiology.

METHODS

We used a modified negative binomial branching process model to quantify the extent of individual heterogeneity using only observed transmission cluster size distribution data (i.e., the simple sum of all cases in a transmission chain) without knowledge of individual-level transmission events. The negative binomial parameter k quantifies the extent of individual heterogeneity (generally, indicates extensive heterogeneity, and as transmission becomes more homogenous). We validated the robustness of the inference procedure considering common limitations affecting cluster size data. Finally, we demonstrate the epidemiologic utility of this method by applying it to aggregate US molecular surveillance data from the US Centers for Disease Control and Prevention.

RESULTS

The cluster-based method reliably inferred k using TB transmission cluster data despite a high degree of bias introduced into the model. We found that the TB transmission in the United States was characterized by a high propensity for extensive outbreaks (; 95% confidence interval = 0.09, 0.10).

CONCLUSIONS

The proposed method can accurately quantify critical parameters that govern TB transmission using simple, more easily obtainable cluster data to improve our understanding of TB epidemiology.

The Epidemic Risk of Dengue Fever in Japan: Climate Change and Seasonality

Dengue fever is a leading cause of illness and death in the tropics and subtropics, and the disease has become a threat to many nonendemic countries where the competent vectors such as Aedes albopictus and Aedes aegypti are abundant. The dengue epidemic in Tokyo, 2014, poses the critical importance to accurately model and predict the outbreak risk of dengue fever in nonendemic regions. Using climatological datasets and traveler volumes in Japan, where dengue was not seen for 70 years by 2014, we investigated the outbreak risk of dengue in 47 prefectures, employing the temperature-dependent basic reproduction number and a branching process model. Our results show that the effective reproduction number varies largely by season and by prefecture, and, moreover, the probability of outbreak if an untraced case is imported varies greatly with the calendar time of importation and location of destination. Combining the seasonally varying outbreak risk with time-dependent traveler volume data, the unconditional outbreak risk was calculated, illustrating different outbreak risks between southern coastal areas and northern tourist cities. As the main finding, the large travel volume with nonnegligible risk of outbreak explains the reason why a summer outbreak in Tokyo, 2014, was observed. Prefectures at high risk of future outbreak would be Tokyo again, Kanagawa or Osaka, and highly populated prefectures with large number of travelers.

Impacts of Natural Environmental Factors and Prevalence of Airway Symptoms on the Local Spread of COVID-19: A Time-Series Analysis in Regional COVID-19 Epidemics

Coronavirus disease 2019 (COVID-19), caused by severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2), is the world's largest public health concern in 2021. This study evaluated the associations of the prevalence of airway symptoms among the tested individuals and data regarding the natural environmental factors with the weekly number of newly diagnosed COVID-19 patients in Sendai City (N_t). For the derivatives of the screening test results, data from individuals with a contact history who underwent nasopharyngeal swab reverse transcription-polymerase chain reaction (RT-PCR) testing between July 2020 and April 2021 (6,156 participants, including 550 test-positive patients) were used. The value of N_t correlated with the weekly RT-PCR test-positive rate after close contact, prevalence of cough symptoms in test-positive individuals or in test-negative individuals, lower air temperature, lower air humidity, and higher wind speed. The weekly test-positive rate correlated with lower air humidity and higher wind speed. In cross-correlation analyses, natural environmental factors correlated with the regional epidemic status on a scale of months, whereas the airway symptoms among non-COVID-19 population affected on a scale of weeks. When applying an autoregression model to the serial data of N_t, large-scale movements of people were suggested to be another factor to influence the local epidemics on a scale of days. In conclusion, the prevalence of cough symptoms in the local population, lower air humidity or higher wind speed, and large-scale movements of people in the locality would jointly influence the local epidemic status of COVID-19.

Beyond R0: heterogeneity in secondary infections and probabilistic epidemic forecasting

The basic reproductive number, R0, is one of the most common and most commonly misapplied numbers in public health. Often used to compare outbreaks and forecast pandemic risk, this single number belies the complexity that different epidemics can exhibit, even when they have the same R0. Here, we reformulate and extend a classic result from random network theory to forecast the size of an epidemic using estimates of the distribution of secondary infections, leveraging both its average R0 and the underlying heterogeneity. Importantly, epidemics with lower R0 can be larger if they spread more homogeneously (and are therefore more robust to stochastic fluctuations). We illustrate the potential of this approach using different real epidemics with known estimates for R0, heterogeneity and epidemic size in the absence of significant intervention. Further, we discuss the different ways in which this framework can be implemented in the data-scarce reality of emerging pathogens. Lastly, we demonstrate that without data on the heterogeneity in secondary infections for emerging infectious diseases like COVID-19 the uncertainty in outbreak size ranges dramatically. Taken together, our work highlights the critical need for contact tracing during emerging infectious disease outbreaks and the need to look beyond R0.

An exact and implementable computation of the final outbreak size distribution under Erlang distributed infectious period

This paper deals with a stochastic SIR (Susceptible-Infected-Recovered) model with Erlang(k,μ) distributed infectious period commonly referred as SI_kR model. We show that using the total number of remaining Erlang stages as the state variable, we do not need to keep track of the stages of individual infections, and can employ a first step analysis to efficiently obtain quantities of interest. We study the distribution of the total number of recovered individuals and the distribution of the maximum number of individuals who are simultaneously infected until the end of the disease. In the literature, final outbreak size is calculated only for a small population size exactly and derivations of approximate analytic solutions from asymptotic results are suggested for larger population sizes. We numerically demonstrate that our methods are implementable on large size problem instances.

Genotype-Specific Measles Transmissibility: A Branching Process Analysis

Background

Substantial heterogeneity in measles outbreak sizes may be due to genotype-specific transmissibility. Using a branching process analysis, we characterize differences in measles transmission by estimating the association between genotype and the reproduction number R among postelimination California measles cases during 2000-2015 (400 cases, 165 outbreaks).

Methods

Assuming a negative binomial secondary case distribution, we fit a branching process model to the distribution of outbreak sizes using maximum likelihood and estimated the reproduction number R for a multigenotype model.

Results

Genotype B3 is found to be significantly more transmissible than other genotypes (P = .01) with an R of 0.64 (95% confidence interval [CI], .48-.71), while the R for all other genotypes combined is 0.43 (95% CI, .28-.54). This result is robust to excluding the 2014-2015 outbreak linked to Disneyland theme parks (referred to as "outbreak A" for conciseness and clarity) (P = .04) and modeling genotype as a random effect (P = .004 including outbreak A and P = .02 excluding outbreak A). This result was not accounted for by season of introduction, age of index case, or vaccination of the index case. The R for outbreaks with a school-aged index case is 0.69 (95% CI, .52-.78), while the R for outbreaks with a non-school-aged index case is 0.28 (95% CI, .19-.35), but this cannot account for differences between genotypes.

Conclusions

Variability in measles transmissibility may have important implications for measles control; the vaccination threshold required for elimination may not be the same for all genotypes or age groups.

Sub- or supercritical transmissibilities in a finite disease outbreak: Symmetry in outbreak properties of a disease conditioned on extinction

This work is concerned with the transmissibility of a disease, on observation of an outbreak of limited size. When such an outbreak occurs, an accurate estimate of the transmissibility of the responsible pathogen is essential for an appropriate response to future outbreaks. Transmissibility is usually characterised in terms of the reproduction number, R, which is the mean number of new cases of infection produced by a single infectious individual. A subcritical reproduction number (R < 1) guarantees that an outbreak will eventually die out of its own accord. By contrast, a supercritical reproduction number (R > 1) does not guarantee spread of the disease, since even with appreciable transmissibility, an outbreak may become extinct due to stochastic effects associated with a small number of infected individuals. Once the number of infectious individuals is conditioned on extinction, we show that an exact symmetry of the underlying theory ensures two distinct values of R, one larger than unity, the other smaller than unity, for which all outbreak properties are identical, with no signature of difference. Therefore a disease with a subcritical R, or its supercritical counterpart, when conditioned on extinction, have, for a given outbreak, identical individual likelihoods. In the full likelihood, this symmetry is lost, since the individual likelihood for a subcritical R is weighted by an extinction probability of unity, but the individual likelihood of a supercritical R is weighted by a sub-unity extinction probability. However, the inference can still benefit from the underlying symmetry, since it yields a mapping of all supercritical reproduction numbers onto the subcritical domain (R < 1), thereby speeding up evaluation of the likelihood profile. The symmetry holds in the standard situation, where the distribution of secondary cases is Poisson, as well as where this distribution has a negative binomial form and super-spreading can occur.

A primer on the use of probability generating functions in infectious disease modeling

We explore the application of probability generating functions (PGFs) to invasive processes, focusing on infectious disease introduced into large populations. Our goal is to acquaint the reader with applications of PGFs, moreso than to derive new results. PGFs help predict a number of properties about early outbreak behavior while the population is still effectively infinite, including the probability of an epidemic, the size distribution after some number of generations, and the cumulative size distribution of non-epidemic outbreaks. We show how PGFs can be used in both discrete-time and continuous-time settings, and discuss how to use these results to infer disease parameters from observed outbreaks. In the large population limit for susceptible-infected-recovered (SIR) epidemics PGFs lead to survival-function based models that are equivalent to the usual mass-action SIR models but with fewer ODEs. We use these to explore properties such as the final size of epidemics or even the dynamics once stochastic effects are negligible. We target this primer at biologists and public health researchers with mathematical modeling experience who want to learn how to apply PGFs to invasive diseases, but it could also be used in an applications-based mathematics course on PGFs. We include many exercises to help demonstrate concepts and to give practice applying the results. We summarize our main results in a few tables. Additionally we provide a small python package which performs many of the relevant calculations.

Dynamics of the pneumonic plague epidemic in Madagascar, August to October 2017

Transmission potential and severity of pneumonic plague in Madagascar were assessed. Accounting for reporting delay, the reproduction number was estimated at 1.73. The case fatality risk was estimated as 5.5%. Expected numbers of exported cases from Madagascar were estimated across the world and all estimates were below 1 person from August to October, 2017.

Genomic epidemiology reveals multiple introductions of Zika virus into the United States

Zika virus (ZIKV) is causing an unprecedented epidemic linked to severe congenital abnormalities. In July 2016, mosquito-borne ZIKV transmission was reported in the continental United States; since then, hundreds of locally acquired infections have been reported in Florida. To gain insights into the timing, source, and likely route(s) of ZIKV introduction, we tracked the virus from its first detection in Florida by sequencing ZIKV genomes from infected patients and Aedes aegypti mosquitoes. We show that at least 4 introductions, but potentially as many as 40, contributed to the outbreak in Florida and that local transmission is likely to have started in the spring of 2016-several months before its initial detection. By analysing surveillance and genetic data, we show that ZIKV moved among transmission zones in Miami. Our analyses show that most introductions were linked to the Caribbean, a finding corroborated by the high incidence rates and traffic volumes from the region into the Miami area. Our study provides an understanding of how ZIKV initiates transmission in new regions.

Inferring epidemiological dynamics of infectious diseases using Tajima's D statistic on nucleotide sequences of pathogens

The estimation of the basic reproduction number is essential to understand epidemic dynamics, and time series data of infected individuals are usually used for the estimation. However, such data are not always available. Methods to estimate the basic reproduction number using genealogy constructed from nucleotide sequences of pathogens have been proposed so far. Here, we propose a new method to estimate epidemiological parameters of outbreaks using the time series change of Tajima's D statistic on the nucleotide sequences of pathogens. To relate the time evolution of Tajima's D to the number of infected individuals, we constructed a parsimonious mathematical model describing both the transmission process of pathogens among hosts and the evolutionary process of the pathogens. As a case study we applied this method to the field data of nucleotide sequences of pandemic influenza A (H1N1) 2009 viruses collected in Argentina. The Tajima's D-based method estimated basic reproduction number to be 1.55 with 95% highest posterior density (HPD) between 1.31 and 2.05, and the date of epidemic peak to be 10th July with 95% HPD between 22nd June and 9th August. The estimated basic reproduction number was consistent with estimation by birth-death skyline plot and estimation using the time series of the number of infected individuals. These results suggested that Tajima's D statistic on nucleotide sequences of pathogens could be useful to estimate epidemiological parameters of outbreaks.

Assessing the transmission dynamics of measles in Japan, 2016

OBJECTIVES

Despite the verification of measles elimination, Japan experienced multiple generations of measles transmission following importation events in 2016. The purpose of the present study was to analyze the transmission dynamics of measles in Japan, 2016, estimating the transmission potential in the partially vaccinated population.

METHODS

All diagnosed measles cases were notified to the government, and the present study analyzed two pieces of datasets independently, i.e., the transmission tree of the largest outbreak in Osaka (n=49) and the final size distribution of all importation events (n=23 events). Branching process model was employed to estimate the effective reproduction number R_v, and the analysis of transmission tree in Osaka enabled us to account for the timing of introducing contact tracing and case isolation.

RESULTS

Employing a negative binomial distribution for the offspring distribution, the model with time-dependent decline in R_v due to interventions appeared to best fit to the transmission tree data with R_v of 9.20 (95% CI (confidence interval): 2.08, 150.68) and the dispersion parameter 0.32 (95% CI: 0.07, 1.17) before interventions were introduced. The relative transmissibility in the presence of interventions from week 34 was estimated at 0.005. Analyzing the final size distribution, models for subcritical and supercritical processes fitted equally well to the observed data, and the estimated reproduction number from both models did not exclude the possibility that R_v>1.

CONCLUSIONS

Our results likely reflect the highly contagious nature of measles, indicating that Japan is at risk of observing multiple generations of measles transmission given imported cases. Considering that importation events may continue in the future, supplementary vaccination of adults needs to be considered.

The transmissibility of noroviruses: Statistical modeling of outbreak events with known route of transmission in Japan

In Japan, the fraction of norovirus outbreaks attributable to human-to-human transmission has increased with time, and the timing of the increased fraction has coincided with the increase in the observed fraction of genogroup II genotype 4 (GII.4). The present study aimed to estimate the time-dependent changes in the transmissibility of noroviruses. The effective reproduction number (R_y), for year y, was estimated by analyzing the time series surveillance data for outbreak events from 2000 to 2016. R_y was estimated by using the fraction of outbreak events that were attributable to human-to-human transmission and by employing three different statistical models that are considered to mechanistically capture the possible data-generating process in different ways. The R_y estimates ranged from 0.14 to 4.15 in value, revealing an overall increasing trend (p<0.05 for all three models). The proportion of outbreaks caused by GII and GII.4 viruses among the total events also increased with time, and positive correlations were identified between transmissibility and these proportions. Parametric modeling of R_y indicated that the time-dependent patterns of R_y were better described by a step function plus linear trend rather than the step function alone that reflects the widespread use of reverse transcriptase PCR (RT-PCR) in and after 2007 for laboratory diagnosis. Accordingly, we conclude that norovirus transmissibility has increased over the past 16 years in Japan. The change is at least partially explained by the time-dependent domination of the contagious GII genogroup (e.g., GII.4), indicating that noroviruses better fitted to humans have selectively caused the human-to-human transmissions, thereby altering the epidemiology of this pathogen.

A review of back-calculation techniques and their potential to inform mitigation strategies with application to non-transmissible acute infectious diseases

Back-calculation is a process whereby generally unobservable features of an event leading to a disease outbreak can be inferred either in real-time or shortly after the end of the outbreak. These features might include the time when persons were exposed and the source of the outbreak. Such inferences are important as they can help to guide the targeting of mitigation strategies and to evaluate the potential effectiveness of such strategies. This article reviews the process of back-calculation with a particular emphasis on more recent applications concerning deliberate and naturally occurring aerosolized releases. The techniques can be broadly split into two themes: the simpler temporal models and the more sophisticated spatio-temporal models. The former require input data in the form of cases' symptom onset times, whereas the latter require additional spatial information such as the cases' home and work locations. A key aspect in the back-calculation process is the incubation period distribution, which forms the initial topic for consideration. Links between atmospheric dispersion modelling, within-host dynamics and back-calculation are outlined in detail. An example of how back-calculation can inform mitigation strategies completes the review by providing improved estimates of the duration of antibiotic prophylaxis that would be required in the response to an inhalational anthrax outbreak.

Estimates of Outbreak Risk from New Introductions of Ebola with Immediate and Delayed Transmission Control

While the ongoing Ebola outbreak continues in the West Africa countries of Guinea, Sierra Leone, and Liberia, health officials elsewhere prepare for new introductions of Ebola from infected evacuees or travelers. We analyzed transmission data from patients (i.e., evacuees, international travelers, and those with locally acquired illness) in countries other than the 3 with continuing Ebola epidemics and quantitatively assessed the outbreak risk from new introductions by using different assumptions for transmission control (i.e., immediate and delayed). Results showed that, even in countries that can quickly limit expected number of transmissions per case to <1, the probability that a single introduction will lead to a substantial number of transmissions is not negligible, particularly if transmission variability is high. Identifying incoming infected travelers before symptom onset can decrease worst-case outbreak sizes more than reducing transmissions from patients with locally acquired cases, but performing both actions can have a synergistic effect.

Assessing the risk of observing multiple generations of Middle East respiratory syndrome (MERS) cases given an imported case

To guide risk assessment, expected numbers of cases and generations were estimated, assuming a case importation of Middle East respiratory syndrome (MERS). Our analysis of 36 importation events yielded the risk of observing secondary transmission events at 22.7% (95% confidence interval: 19.3–25.1). The risks of observing generations 2, 3 and 4 were estimated at 10.5%, 6.1% and 3.9%, respectively. Countries at risk should be ready for highly variable outcomes following an importation of MERS.

The role of superspreading in Middle East respiratory syndrome coronavirus (MERS-CoV) transmission

As at 15 June 2015, a large transmission cluster of Middle East respiratory syndrome coronavirus (MERSCoV)was ongoing in South Korea. To examine the potential for such events, we estimated the level of heterogeneity in MERS-CoV transmission by analyzing data on cluster size distributions. We found substantial potential for superspreading; even though it is likely that R0 < 1 overall, our analysis indicates that cluster sizes of over 150 cases are not unexpected forMERS-CoV infection.

The role of vaccination coverage, individual behaviors, and the public health response in the control of measles epidemics: an agent-based simulation for California

BACKGROUND

Measles cases continue to occur among susceptible individuals despite the elimination of endemic measles transmission in the United States. Clustering of disease susceptibility can threaten herd immunity and impact the likelihood of disease outbreaks in a highly vaccinated population. Previous studies have examined the role of contact tracing to control infectious diseases among clustered populations, but have not explicitly modeled the public health response using an agent-based model.

METHODS

We developed an agent-based simulation model of measles transmission using the Framework for Reconstructing Epidemiological Dynamics (FRED) and the Synthetic Population Database maintained by RTI International. The simulation of measles transmission was based on interactions among individuals in different places: households, schools, daycares, workplaces, and neighborhoods. The model simulated different levels of immunity clustering, vaccination coverage, and contact investigations with delays caused by individuals' behaviors and/or the delay in a health department's response. We examined the effects of these characteristics on the probability of uncontrolled measles outbreaks and the outbreak size in 365 days after the introduction of one index case into a synthetic population.

RESULTS

We found that large measles outbreaks can be prevented with contact investigations and moderate contact rates by having (1) a very high vaccination coverage (≥ 95%) with a moderate to low level of immunity clustering (≤ 0.5) for individuals aged less than or equal to 18 years, or (2) a moderate vaccination coverage (85% or 90%) with no immunity clustering for individuals (≤ 18 years of age), a short intervention delay, and a high probability that a contact can be traced. Without contact investigations, measles outbreaks may be prevented by the highest vaccination coverage with no immunity clustering for individuals (≤ 18 years of age) with moderate contact rates; but for the highest contact rates, even the highest coverage with no immunity clustering for individuals (≤ 18 years of age) cannot completely prevent measles outbreaks.

CONCLUSIONS

The simulation results demonstrated the importance of vaccination coverage, clustering of immunity, and contact investigations in preventing uncontrolled measles outbreaks.

Characterizing the transmission potential of zoonotic infections from minor outbreaks

The transmission potential of a novel infection depends on both the inherent transmissibility of a pathogen, and the level of susceptibility in the host population. However, distinguishing between these pathogen- and population-specific properties typically requires detailed serological studies, which are rarely available in the early stages of an outbreak. Using a simple transmission model that incorporates age-stratified social mixing patterns, we present a novel method for characterizing the transmission potential of subcritical infections, which have effective reproduction number R<1, from readily available data on the size of outbreaks. We show that the model can identify the extent to which outbreaks are driven by inherent pathogen transmissibility and pre-existing population immunity, and can generate unbiased estimates of the effective reproduction number. Applying the method to real-life infections, we obtained accurate estimates for the degree of age-specific immunity against monkeypox, influenza A(H5N1) and A(H7N9), and refined existing estimates of the reproduction number. Our results also suggest minimal pre-existing immunity to MERS-CoV in humans. The approach we describe can therefore provide crucial information about novel infections before serological surveys and other detailed analyses are available. The methods would also be applicable to data stratified by factors such as profession or location, which would make it possible to measure the transmission potential of emerging infections in a wide range of settings.

The transmission potential of a new infection depends on both the transmissibility of the pathogen and the level of immunity in the host population. However, it can be difficult to measure these properties if there are limited experimental studies of population immunity. By incorporating social contact patterns into a mathematical model of disease transmission, we show that it is possible to estimate both pathogen transmissibility and pre-existing immunity from available data on the size of outbreaks. When an infection does not transmit efficiently between humans, estimates often have to be made using case data from a limited number of small outbreaks. We find that, even with limited data, our technique can accurately evaluate the transmission potential of ‘stuttering’ chains of infection. We use the method to characterise transmission of four real infections: monkeypox, influenza A(H5N1) and A(H7N9) and MERS-CoV.

Detecting differential transmissibilities that affect the size of self-limited outbreaks

Our ability to respond appropriately to infectious diseases is enhanced by identifying differences in the potential for transmitting infection between individuals. Here, we identify epidemiological traits of self-limited infections (i.e. infections with an effective reproduction number satisfying [0 < R eff < 1) that correlate with transmissibility. Our analysis is based on a branching process model that permits statistical comparison of both the strength and heterogeneity of transmission for two distinct types of cases. Our approach provides insight into a variety of scenarios, including the transmission of Middle East Respiratory Syndrome Coronavirus (MERS-CoV) in the Arabian peninsula, measles in North America, pre-eradication smallpox in Europe, and human monkeypox in the Democratic Republic of the Congo. When applied to chain size data for MERS-CoV transmission before 2014, our method indicates that despite an apparent trend towards improved control, there is not enough statistical evidence to indicate that R eff has declined with time. Meanwhile, chain size data for measles in the United States and Canada reveal statistically significant geographic variation in R eff, suggesting that the timing and coverage of national vaccination programs, as well as contact tracing procedures, may shape the size distribution of observed infection clusters. Infection source data for smallpox suggests that primary cases transmitted more than secondary cases, and provides a quantitative assessment of the effectiveness of control interventions. Human monkeypox, on the other hand, does not show evidence of differential transmission between animals in contact with humans, primary cases, or secondary cases, which assuages the concern that social mixing can amplify transmission by secondary cases. Lastly, we evaluate surveillance requirements for detecting a change in the human-to-human transmission of monkeypox since the cessation of cross-protective smallpox vaccination. Our studies lay the foundation for future investigations regarding how infection source, vaccination status or other putative transmissibility traits may affect self-limited transmission.

Estimating the basic reproduction number from surveillance data on past epidemics

In this paper, we consider the basic reproduction number, R0, a parameter that characterizes the transmission potential of an epidemic, and explore a novel way for estimating it. We introduce a stochastic process which takes as starting points the classical SIR (susceptibles-infected-removed) models, deterministic and stochastic. The estimation method rests on an extremum property of the deterministic SIR model, and could be applied to past surveillance data on epidemic outbreaks, data gathered at different locations or in different years. Our estimators take into account some practical limitations, in particular the fact that data are collected at preassigned times. We derive asymptotic properties of the estimators and perform a simulation study to assess their small sample behavior. We illustrate the method on real data (from the USA Centers for Disease Control and Prevention site) and we point to various extensions to our approach, as well as practical implementation issues.

Identifying postelimination trends for the introduction and transmissibility of measles in the United States

The continued elimination of measles requires accurate assessment of its epidemiology and a critical evaluation of how its incidence is changing with time. National surveillance of measles in the United States between 2001 and 2011 provides data on the number of measles introductions and the size of the resulting transmission chains. These data allow inference of the effective reproduction number, Reff, and the probability of an outbreak occurring. Our estimate of 0.52 (95% confidence interval: 0.44, 0.60) for Reff is smaller than prior results. Our findings are relatively insensitive to the possibility that as few as 75% of cases were detected. Although we confirm that measles remains eliminated, we identify an increasing trend in the number of measles cases with time. We show that this trend is likely attributable to an increase in the number of disease introductions rather than a change in the transmissibility of measles. However, we find that transmissibility may increase substantially if vaccine coverage drops by as little as 1%. Our general approach of characterizing the case burden of measles is applicable to the epidemiologic assessment of other weakly transmitting or vaccine-controlled pathogens that are either at risk of emerging or on the brink of elimination.

Epidemic risk from cholera introductions into Mexico

Stemming from the 2010 cholera outbreak in Haiti, cholera transmission in Hispaniola continues with over 40,000 cases in 2013. The presence of an ongoing cholera outbreak in the region poses substantial risks to countries throughout the Americas, particularly in areas with poor infrastructure. Since September 9, 2013 nearly 200 cholera cases have been reported in Mexico, as a result of introductions from Hispaniola or Cuba. There appear to have been multiple introductions into Mexico resulting in outbreaks of 2 to over 150 people. Using publicly available data, we attempt to estimate the reproductive number (R) of cholera in Mexico, and thereby assess the potential of continued introductions to establish a sustained epidemic. We estimate R for cholera in Mexico to be between 0.8 to 1.1, depending on the number of introductions, with the confidence intervals for the most plausible estimates crossing 1. These results suggest that the efficiency of cholera transmission in some regions of Mexico is near that necessary for a large epidemic. Intensive surveillance, evaluation of water and sanitation infrastructure, and planning for rapid response are warranted steps to avoid potential large epidemics in the region.

Comparing methods for estimating R0 from the size distribution of subcritical transmission chains

Many diseases exhibit subcritical transmission (i.e. 0<R0<1) so that infections occur as self-limited 'stuttering chains'. Given an ensemble of stuttering chains, information about the number of cases in each chain can be used to infer R0, which is of crucial importance for monitoring the risk that a disease will emerge to establish endemic circulation. However, the challenge of imperfect case detection has led authors to adopt a variety of work-around measures when inferring R0, such as discarding data on isolated cases or aggregating intermediate-sized chains together. Each of these methods has the potential to introduce bias, but a quantitative comparison of these approaches has not been reported. By adapting a model based on a negative binomial offspring distribution that permits a variable degree of transmission heterogeneity, we present a unified analysis of existing R0 estimation methods. Simulation studies show that the degree of transmission heterogeneity, when improperly modeled, can significantly impact the bias of R0 estimation methods designed for imperfect observation. These studies also highlight the importance of isolated cases in assessing whether an estimation technique is consistent with observed data. Analysis of data from measles outbreaks shows that likelihood scores are highest for models that allow a flexible degree of transmission heterogeneity. Aggregating intermediate sized chains often has similar performance to analyzing a complete chain size distribution. However, truncating isolated cases is beneficial only when surveillance systems clearly favor full observation of large chains but not small chains. Meanwhile, if data on the type and proportion of cases that are unobserved were known, we demonstrate that maximum likelihood inference of R0 could be adjusted accordingly. This motivates the need for future empirical and theoretical work to quantify observation error and incorporate relevant mechanisms into stuttering chain models used to estimate transmission parameters.

How to interpret the transmissibility of novel influenza A(H7N9): an analysis of initial epidemiological data of human cases from China

Background

As the human infections with novel influenza A(H7N9) virus have been reported from several different provinces in China, the pandemic potential of the virus has been questioned. The presence of human-to-human transmission has not been demonstrated, but the absence of demonstration does not guarantee that there is no such transmission.

Methods

A mathematical model of cluster size distribution is devised without imposing an assumption of subcriticality of the reproduction number and accounting for right censoring of new clusters. The proportion of cases with a history of bird contact is analytically derived, permitting us to fit the model to the observed data of confirmed cases. Using contact history with bird among confirmed cases (n = 129), we estimate the reproduction number of the novel influenza A(H7N9) from human to human.

Results

Analysing twenty confirmed cases with known exposure, the reproduction number for human-to-human transmission was estimated at 0.28 (95% CI: 0.11, 0.45). Sensitivity analysis indicated that the reproduction number is substantially below unity.

Conclusions

It is unlikely to observe an immediate pandemic of novel influenza A(H7N9) virus with human to human transmission. Continued monitoring of cases and animals would be the key to elucidate additional epidemiological characteristics of the virus.

Inference of R(0) and transmission heterogeneity from the size distribution of stuttering chains

For many infectious disease processes such as emerging zoonoses and vaccine-preventable diseases, [Formula: see text] and infections occur as self-limited stuttering transmission chains. A mechanistic understanding of transmission is essential for characterizing the risk of emerging diseases and monitoring spatio-temporal dynamics. Thus methods for inferring [Formula: see text] and the degree of heterogeneity in transmission from stuttering chain data have important applications in disease surveillance and management. Previous researchers have used chain size distributions to infer [Formula: see text], but estimation of the degree of individual-level variation in infectiousness (as quantified by the dispersion parameter, [Formula: see text]) has typically required contact tracing data. Utilizing branching process theory along with a negative binomial offspring distribution, we demonstrate how maximum likelihood estimation can be applied to chain size data to infer both [Formula: see text] and the dispersion parameter that characterizes heterogeneity. While the maximum likelihood value for [Formula: see text] is a simple function of the average chain size, the associated confidence intervals are dependent on the inferred degree of transmission heterogeneity. As demonstrated for monkeypox data from the Democratic Republic of Congo, this impacts when a statistically significant change in [Formula: see text] is detectable. In addition, by allowing for superspreading events, inference of [Formula: see text] shifts the threshold above which a transmission chain should be considered anomalously large for a given value of [Formula: see text] (thus reducing the probability of false alarms about pathogen adaptation). Our analysis of monkeypox also clarifies the various ways that imperfect observation can impact inference of transmission parameters, and highlights the need to quantitatively evaluate whether observation is likely to significantly bias results.

Sample size considerations for one-to-one animal transmission studies of the influenza A viruses

Background

Animal transmission studies can provide important insights into host, viral and environmental factors affecting transmission of viruses including influenza A. The basic unit of analysis in typical animal transmission experiments is the presence or absence of transmission from an infectious animal to a susceptible animal. In studies comparing two groups (e.g. two host genetic variants, two virus strains, or two arrangements of animal cages), differences between groups are evaluated by comparing the proportion of pairs with successful transmission in each group. The present study aimed to discuss the significance and power to estimate transmissibility and identify differences in the transmissibility based on one-to-one trials. The analyses are illustrated on transmission studies of influenza A viruses in the ferret model.

Methodology/Principal Findings

Employing the stochastic general epidemic model, the basic reproduction number, R₀, is derived from the final state of an epidemic and is related to the probability of successful transmission during each one-to-one trial. In studies to estimate transmissibility, we show that 3 pairs of infectious/susceptible animals cannot demonstrate a significantly higher transmissibility than R₀ = 1, even if infection occurs in all three pairs. In comparisons between two groups, at least 4 pairs of infectious/susceptible animals are required in each group to ensure high power to identify significant differences in transmissibility between the groups.

Conclusions

These results inform the appropriate sample sizes for animal transmission experiments, while relating the observed proportion of infected pairs to R₀, an interpretable epidemiological measure of transmissibility. In addition to the hypothesis testing results, the wide confidence intervals of R₀ with small sample sizes also imply that the objective demonstration of difference or similarity should rest on firmly calculated sample size.

Using routine surveillance data to estimate the epidemic potential of emerging zoonoses: application to the emergence of US swine origin influenza A H3N2v virus

BACKGROUND

Prior to emergence in human populations, zoonoses such as SARS cause occasional infections in human populations exposed to reservoir species. The risk of widespread epidemics in humans can be assessed by monitoring the reproduction number R (average number of persons infected by a human case). However, until now, estimating R required detailed outbreak investigations of human clusters, for which resources and expertise are not always available. Additionally, existing methods do not correct for important selection and under-ascertainment biases. Here, we present simple estimation methods that overcome many of these limitations.

METHODS AND FINDINGS

Our approach is based on a parsimonious mathematical model of disease transmission and only requires data collected through routine surveillance and standard case investigations. We apply it to assess the transmissibility of swine-origin influenza A H3N2v-M virus in the US, Nipah virus in Malaysia and Bangladesh, and also present a non-zoonotic example (cholera in the Dominican Republic). Estimation is based on two simple summary statistics, the proportion infected by the natural reservoir among detected cases (G) and among the subset of the first detected cases in each cluster (F). If detection of a case does not affect detection of other cases from the same cluster, we find that R can be estimated by 1-G; otherwise R can be estimated by 1-F when the case detection rate is low. In more general cases, bounds on R can still be derived.

CONCLUSIONS

We have developed a simple approach with limited data requirements that enables robust assessment of the risks posed by emerging zoonoses. We illustrate this by deriving transmissibility estimates for the H3N2v-M virus, an important step in evaluating the possible pandemic threat posed by this virus. Please see later in the article for the Editors' Summary.

Real-time investigation of measles epidemics with estimate of vaccine efficacy

As part of measles elimination effort, evaluation of the vaccination program and real-time assessment of the epidemic dynamics constitute two important tasks to improve and strengthen the control. The present study aimed to develop an epidemiological modeling method which can be applied to estimating the vaccine efficacy at an individual level while conducting the timely investigation of the epidemic. The multivariate renewal process model was employed to describe the temporal evolution of infection by vaccination history, jointly estimating the time-dependent reproduction number and the vaccine efficacy. Analyzing the enhanced surveillance data of measles in Aichi prefecture, Japan from 2007-08, the vaccine efficacy was estimated at 96.7% (95% confidence interval: 95.8, 97.4). Using an age structured model, the vaccine efficacy among those aged from 5-19 years was shown to be smaller than that among those from 0-4 years. The age-dependent vaccine efficacy estimate informs the age-groups to be targeted for revaccination. Because the estimation method can rest on readily available epidemiological data, the proposed model has a potential to be integrated with routine surveillance.

The feasibility of age-specific travel restrictions during influenza pandemics

Background

Epidemiological studies have shown that imposing travel restrictions to prevent or delay an influenza pandemic may not be feasible. To delay an epidemic substantially, an extremely high proportion of trips (~99%) would have to be restricted in a homogeneously mixing population. Influenza is, however, strongly influenced by age-dependent transmission dynamics, and the effectiveness of age-specific travel restrictions, such as the selective restriction of travel by children, has yet to be examined.

Methods

A simple stochastic model was developed to describe the importation of infectious cases into a population and to model local chains of transmission seeded by imported cases. The probability of a local epidemic, and the time period until a major epidemic takes off, were used as outcome measures, and travel restriction policies in which children or adults were preferentially restricted were compared to age-blind restriction policies using an age-dependent next generation matrix parameterized for influenza H1N1-2009.

Results

Restricting children from travelling would yield greater reductions to the short-term risk of the epidemic being established locally than other policy options considered, and potentially could delay an epidemic for a few weeks. However, given a scenario with a total of 500 imported cases over a period of a few months, a substantial reduction in the probability of an epidemic in this time period is possible only if the transmission potential were low and assortativity (i.e. the proportion of contacts within-group) were unrealistically high. In all other scenarios considered, age-structured travel restrictions would not prevent an epidemic and would not delay the epidemic for longer than a few weeks.

Conclusions

Selectively restricting children from traveling overseas during a pandemic may potentially delay its arrival for a few weeks, depending on the characteristics of the pandemic strain, but could have less of an impact on the economy compared to restricting adult travelers. However, as long as adults have at least a moderate potential to trigger an epidemic, selectively restricting the higher risk group (children) may not be a practical option to delay the arrival of an epidemic substantially.