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1.  Generation interval contraction and epidemic data analysis 
Mathematical biosciences  2008;213(1):71-79.
The generation interval is the time between the infection time of an infected person and the infection time of his or her infector. Probability density functions for generation intervals have been an important input for epidemic models and epidemic data analysis. In this paper, we specify a general stochastic SIR epidemic model and prove that the mean generation interval decreases when susceptible persons are at risk of infectious contact from multiple sources. The intuition behind this is that when a susceptible person has multiple potential infectors, there is a “race” to infect him or her in which only the first infectious contact leads to infection. In an epidemic, the mean generation interval contracts as the prevalence of infection increases. We call this global competition among potential infectors. When there is rapid transmission within clusters of contacts, generation interval contraction can be caused by a high local prevalence of infection even when the global prevalence is low. We call this local competition among potential infectors. Using simulations, we illustrate both types of competition. Finally, we show that hazards of infectious contact can be used instead of generation intervals to estimate the time course of the effective reproductive number in an epidemic. This approach leads naturally to partial likelihoods for epidemic data that are very similar to those that arise in survival analysis, opening a promising avenue of methodological research in infectious disease epidemiology.
PMCID: PMC2365921  PMID: 18394654
2.  Nonparametric survival analysis of infectious disease data 
This paper develops nonparametric methods based on contact intervals for the analysis of infectious disease data. The contact interval from person i to person j is the time between the onset of infectiousness in i and infectious contact from i to j, where we define infectious contact as a contact sufficient to infect a susceptible individual. The hazard function of the contact interval distribution equals the hazard of infectious contact from i to j, so it provides a summary of the evolution of infectiousness over time. When who-infects-whom is observed, the Nelson-Aalen estimator produces an unbiased estimate of the cumulative hazard function of the contact interval distribution. When who-infects-whom is not observed, we use an EM algorithm to average the Nelson-Aalen estimates from all possible combinations of who-infected-whom consistent with the observed data. This converges to a nonparametric maximum likelihood estimate of the cumulative hazard function that we call the marginal Nelson-Aalen estimate. We study the behavior of these methods in simulations and use them to analyze household surveillance data from the 2009 influenza A(H1N1) pandemic.
PMCID: PMC3681432  PMID: 23772180
Chain-binomial models; Contact intervals; Generation intervals; Infectious disease; Nonparametric methods; Survival analysis
3.  Statistical inference to advance network models in epidemiology 
Epidemics  2011;3(1):38-45.
Contact networks are playing an increasingly important role in the study of epidemiology. Most of the existing work in this area has focused on considering the effect of underlying network structure on epidemic dynamics by using tools from probability theory and computer simulation. This work has provided much insight on the role that heterogeneity in host contact patterns plays on infectious disease dynamics. Despite the important understanding afforded by the probability and simulation paradigm, this approach does not directly address important questions about the structure of contact networks such as what is the best network model for a particular mode of disease transmission, how parameter values of a given model should be estimated, or how precisely the data allow us to estimate these parameter values. We argue that these questions are best answered within a statistical framework and discuss the role of statistical inference in estimating contact networks from epidemiological data.
PMCID: PMC3090691  PMID: 21420658
4.  Estimating a Markovian Epidemic Model Using Household Serial Interval Data from the Early Phase of an Epidemic 
PLoS ONE  2013;8(8):e73420.
The clinical serial interval of an infectious disease is the time between date of symptom onset in an index case and the date of symptom onset in one of its secondary cases. It is a quantity which is commonly collected during a pandemic and is of fundamental importance to public health policy and mathematical modelling. In this paper we present a novel method for calculating the serial interval distribution for a Markovian model of household transmission dynamics. This allows the use of Bayesian MCMC methods, with explicit evaluation of the likelihood, to fit to serial interval data and infer parameters of the underlying model. We use simulated and real data to verify the accuracy of our methodology and illustrate the importance of accounting for household size. The output of our approach can be used to produce posterior distributions of population level epidemic characteristics.
PMCID: PMC3758268  PMID: 24023679
5.  Quantifying social distancing arising from pandemic influenza 
Local epidemic curves during the 1918–1919 influenza pandemic were often characterized by multiple epidemic waves. Identifying the underlying cause(s) of such waves may help manage future pandemics. We investigate the hypothesis that these waves were caused by people avoiding potentially infectious contacts—a behaviour termed ‘social distancing’. We estimate the effective disease reproduction number and from it infer the maximum degree of social distancing that occurred during the course of the multiple-wave epidemic in Sydney, Australia. We estimate that, on average across the city, people reduced their infectious contact rate by as much as 38%, and that this was sufficient to explain the multiple waves of this epidemic. The basic reproduction number, R0, was estimated to be in the range of 1.6–2.0 with a preferred estimate of 1.8, in line with other recent estimates for the 1918–1919 influenza pandemic. The data are also consistent with a high proportion (more than 90%) of the population being initially susceptible to clinical infection, and the proportion of infections that were asymptomatic (if this occurs) being no higher than approximately 9%. The observed clinical attack rate of 36.6% was substantially lower than the 59% expected based on the estimated value of R0, implying that approximately 22% of the population were spared from clinical infection. This reduction in the clinical attack rate translates to an estimated 260 per 100 000 lives having been saved, and suggests that social distancing interventions could play a major role in mitigating the public health impact of future influenza pandemics.
PMCID: PMC3226987  PMID: 17916550
disease reproduction number; R0; pandemic influenza; social distancing; epidemic attack rate; prior immunity
6.  A Hybrid EM and Monte Carlo EM Algorithm and Its Application to Analysis of Transmission of Infectious Diseases 
Biometrics  2012;68(4):1238-1249.
In epidemics of infectious diseases such as influenza, an individual may have one of four possible final states: prior immune, escaped from infection, infected with symptoms, and infected asymptomatically. The exact state is often not observed. In addition, the unobserved transmission times of asymptomatic infections further complicate analysis. Under the assumption of missing at random, data-augmentation techniques can be used to integrate out such uncertainties. We adapt an importance-sampling-based Monte Carlo EM (MCEM) algorithm to the setting of an infectious disease transmitted in close contact groups. Assuming the independence between close contact groups, we propose a hybrid EM-MCEM algorithm that applies the MCEM or the traditional EM algorithms to each close contact group depending on the dimension of missing data in that group, and discuss the variance estimation for this practice. In addition, we propose a bootstrap approach to assess the total Monte Carlo error and factor that error into the variance estimation. The proposed methods are evaluated using simulation studies. We use the hybrid EM-MCEM algorithm to analyze two influenza epidemics in the late 1970s to assess the effects of age and pre-season antibody levels on the transmissibility and pathogenicity of the viruses.
PMCID: PMC3402623  PMID: 22506893
Data augmentation; EM algorithm; Infectious disease; Missing data; Monte Carlo
7.  Estimating the Per-Contact Probability of Infection by Highly Pathogenic Avian Influenza (H7N7) Virus during the 2003 Epidemic in The Netherlands 
PLoS ONE  2012;7(7):e40929.
Estimates of the per-contact probability of transmission between farms of Highly Pathogenic Avian Influenza virus of H7N7 subtype during the 2003 epidemic in the Netherlands are important for the design of better control and biosecurity strategies. We used standardized data collected during the epidemic and a model to extract data for untraced contacts based on the daily number of infectious farms within a given distance of a susceptible farm. With these data, we used a maximum likelihood estimation approach to estimate the transmission probabilities by the individual contact types, both traced and untraced. The estimated conditional probabilities, conditional on the contact originating from an infectious farm, of virus transmission were: 0.000057 per infectious farm within 1 km per day, 0.000413 per infectious farm between 1 and 3 km per day, 0.0000895 per infectious farm between 3 and 10 km per day, 0.0011 per crisis organisation contact, 0.0414 per feed delivery contact, 0.308 per egg transport contact, 0.133 per other-professional contact and, 0.246 per rendering contact. We validate these outcomes against literature data on virus genetic sequences for outbreak farms. These estimates can be used to inform further studies on the role that improved biosecurity between contacts and/or contact frequency reduction can play in eliminating between-farm spread of the virus during future epidemics. The findings also highlight the need to; 1) understand the routes underlying the infections without traced contacts and, 2) to review whether the contact-tracing protocol is exhaustive in relation to all the farm’s day-to-day activities and practices.
PMCID: PMC3396644  PMID: 22808285
8.  Contact Heterogeneity and Phylodynamics: How Contact Networks Shape Parasite Evolutionary Trees 
The inference of population dynamics from molecular sequence data is becoming an important new method for the surveillance of infectious diseases. Here, we examine how heterogeneity in contact shapes the genealogies of parasitic agents. Using extensive simulations, we find that contact heterogeneity can have a strong effect on how the structure of genealogies reflects epidemiologically relevant quantities such as the proportion of a population that is infected. Comparing the simulations to BEAST reconstructions, we also find that contact heterogeneity can increase the number of sequence isolates required to estimate these quantities over the course of an epidemic. Our results suggest that data about contact-network structure will be required in addition to sequence data for accurate estimation of a parasitic agent's genealogy. We conclude that network models will be important for progress in this area.
PMCID: PMC2995904  PMID: 21151699
9.  Transmissibility of the Influenza Virus in the 1918 Pandemic 
PLoS ONE  2008;3(1):e1498.
With a heightened increase in concern for an influenza pandemic we sought to better understand the 1918 Influenza pandemic, the most devastating epidemic of the previous century.
Methodology/Principal Findings
We use data from several communities in Maryland, USA as well as two ships that experienced well-documented outbreaks of influenza in 1918. Using a likelihood-based method and a nonparametric method, we estimate the serial interval and reproductive number throughout the course of each outbreak. This analysis shows the basic reproductive number to be slightly lower in the Maryland communities (between 1.34 and 3.21) than for the enclosed populations on the ships (R0 = 4.97, SE = 3.31). Additionally the effective reproductive number declined to sub epidemic levels more quickly on the ships (within around 10 days) than in the communities (within 30–40 days). The mean serial interval for the ships was consistent (3.33, SE = 5.96 and 3.81, SE = 3.69), while the serial intervals in the communities varied substantially (between 2.83, SE = 0.53 and 8.28, SE = 951.95).
These results illustrate the importance of considering the population dynamics when making statements about the epidemiological parameters of Influenza. The methods that we employ for estimation of the reproductive numbers and the serial interval can be easily replicated in other populations and with other diseases.
PMCID: PMC2204055  PMID: 18231585
10.  A Bayesian MCMC approach to survival analysis with doubly-censored data 
Doubly-censored data refers to time to event data for which both the originating and failure times are censored. In studies involving AIDS incubation time or survival after dementia onset, for example, data are frequently doubly-censored because the date of the originating event is interval-censored and the date of the failure event usually is right-censored. The primary interest is in the distribution of elapsed times between the originating and failure events and its relationship to exposures and risk factors. The estimating equation approach [Sun, et al. 1999. Regression analysis of doubly censored failure time data with applications to AIDS studies. Biometrics 55, 909-914] and its extensions assume the same distribution of originating event times for all subjects. This paper demonstrates the importance of utilizing additional covariates to impute originating event times, i.e., more accurate estimation of originating event times may lead to less biased parameter estimates for elapsed time. The Bayesian MCMC method is shown to be a suitable approach for analyzing doubly-censored data and allows a rich class of survival models. The performance of the proposed estimation method is compared to that of other conventional methods through simulations. Two examples, an AIDS cohort study and a population-based dementia study, are used for illustration. Sample code is shown in the appendix.
PMCID: PMC2877214  PMID: 20514348
AIDS; dementia; doubly censored data; incubation period; MCMC; midpoint imputation
11.  Inferring population-level contact heterogeneity from common epidemic data 
Models of infectious disease spread that incorporate contact heterogeneity through contact networks are an important tool for epidemiologists studying disease dynamics and assessing intervention strategies. One of the challenges of contact network epidemiology has been the difficulty of collecting individual and population-level data needed to develop an accurate representation of the underlying host population's contact structure. In this study, we evaluate the utility of common epidemiological measures (R0, epidemic peak size, duration and final size) for inferring the degree of heterogeneity in a population's unobserved contact structure through a Bayesian approach. We test the method using ground truth data and find that some of these epidemiological metrics are effective at classifying contact heterogeneity. The classification is also consistent across pathogen transmission probabilities, and so can be applied even when this characteristic is unknown. In particular, the reproductive number, R0, turns out to be a poor classifier of the degree heterogeneity, while, unexpectedly, final epidemic size is a powerful predictor of network structure across the range of heterogeneity. We also evaluate our framework on empirical epidemiological data from past and recent outbreaks to demonstrate its application in practice and to gather insights about the relevance of particular contact structures for both specific systems and general classes of infectious disease. We thus introduce a simple approach that can shed light on the unobserved connectivity of a host population given epidemic data. Our study has the potential to inform future data-collection efforts and study design by driving our understanding of germane epidemic measures, and highlights a general inferential approach to learning about host contact structure in contemporary or historic populations of humans and animals.
PMCID: PMC3565785  PMID: 23034353
network model; infectious disease; epidemic data; statistical inference; contact heterogeneity
Many epidemic models approximate social contact behavior by assuming random mixing within mixing groups (e.g., homes, schools, and workplaces). The effect of more realistic social network structure on estimates of epidemic parameters is an open area of exploration. We develop a detailed statistical model to estimate the social contact network within a high school using friendship network data and a survey of contact behavior. Our contact network model includes classroom structure, longer durations of contacts to friends than non-friends and more frequent contacts with friends, based on reports in the contact survey. We performed simulation studies to explore which network structures are relevant to influenza transmission. These studies yield two key findings. First, we found that the friendship network structure important to the transmission process can be adequately represented by a dyad-independent exponential random graph model (ERGM). This means that individual-level sampled data is sufficient to characterize the entire friendship network. Second, we found that contact behavior was adequately represented by a static rather than dynamic contact network. We then compare a targeted antiviral prophylaxis intervention strategy and a grade closure intervention strategy under random mixing and network-based mixing. We find that random mixing overestimates the effect of targeted antiviral prophylaxis on the probability of an epidemic when the probability of transmission in 10 minutes of contact is less than 0.004 and underestimates it when this transmission probability is greater than 0.004. We found the same pattern for the final size of an epidemic, with a threshold transmission probability of 0.005. We also find random mixing overestimates the effect of a grade closure intervention on the probability of an epidemic and final size for all transmission probabilities. Our findings have implications for policy recommendations based on models assuming random mixing, and can inform further development of network-based models.
PMCID: PMC3359895  PMID: 22639701
contact network; epidemic model; influenza; simulation model; social network
13.  Inferring Social Network Structure from Bacterial Sequence Data 
PLoS ONE  2011;6(8):e22685.
Using DNA sequence data from pathogens to infer transmission networks has traditionally been done in the context of epidemics and outbreaks. Sequence data could analogously be applied to cases of ubiquitous commensal bacteria; however, instead of inferring chains of transmission to track the spread of a pathogen, sequence data for bacteria circulating in an endemic equilibrium could be used to infer information about host contact networks. Here, we show—using simulated data—that multilocus DNA sequence data, based on multilocus sequence typing schemes (MLST), from isolates of commensal bacteria can be used to infer both local and global properties of the contact networks of the populations being sampled. Specifically, for MLST data simulated from small-world networks, the small world parameter controlling the degree of structure in the contact network can robustly be estimated. Moreover, we show that pairwise distances in the network—degrees of separation—correlate with genetic distances between isolates, so that how far apart two individuals in the network are can be inferred from MLST analysis of their commensal bacteria. This result has important consequences, and we show an example from epidemiology: how this result could be used to test for infectious origins of diseases of unknown etiology.
PMCID: PMC3148245  PMID: 21829645
14.  FluTE, a Publicly Available Stochastic Influenza Epidemic Simulation Model 
PLoS Computational Biology  2010;6(1):e1000656.
Mathematical and computer models of epidemics have contributed to our understanding of the spread of infectious disease and the measures needed to contain or mitigate them. To help prepare for future influenza seasonal epidemics or pandemics, we developed a new stochastic model of the spread of influenza across a large population. Individuals in this model have realistic social contact networks, and transmission and infections are based on the current state of knowledge of the natural history of influenza. The model has been calibrated so that outcomes are consistent with the 1957/1958 Asian A(H2N2) and 2009 pandemic A(H1N1) influenza viruses. We present examples of how this model can be used to study the dynamics of influenza epidemics in the United States and simulate how to mitigate or delay them using pharmaceutical interventions and social distancing measures. Computer simulation models play an essential role in informing public policy and evaluating pandemic preparedness plans. We have made the source code of this model publicly available to encourage its use and further development.
Author Summary
Computer simulations can provide valuable information to communities preparing for epidemics. These simulations can be used to investigate the effectiveness of various intervention strategies in reducing or delaying the peak of an epidemic. We have made a detailed influenza epidemic simulator for the United States publicly available so that others may use the software to inform public policy or adapt it to suit their needs.
PMCID: PMC2813259  PMID: 20126529
15.  Time variations in the transmissibility of pandemic influenza in Prussia, Germany, from 1918–19 
Time variations in transmission potential have rarely been examined with regard to pandemic influenza. This paper reanalyzes the temporal distribution of pandemic influenza in Prussia, Germany, from 1918–19 using the daily numbers of deaths, which totaled 8911 from 29 September 1918 to 1 February 1919, and the distribution of the time delay from onset to death in order to estimate the effective reproduction number, Rt, defined as the actual average number of secondary cases per primary case at a given time.
A discrete-time branching process was applied to back-calculated incidence data, assuming three different serial intervals (i.e. 1, 3 and 5 days). The estimated reproduction numbers exhibited a clear association between the estimates and choice of serial interval; i.e. the longer the assumed serial interval, the higher the reproduction number. Moreover, the estimated reproduction numbers did not decline monotonically with time, indicating that the patterns of secondary transmission varied with time. These tendencies are consistent with the differences in estimates of the reproduction number of pandemic influenza in recent studies; high estimates probably originate from a long serial interval and a model assumption about transmission rate that takes no account of time variation and is applied to the entire epidemic curve.
The present findings suggest that in order to offer robust assessments it is critically important to clarify in detail the natural history of a disease (e.g. including the serial interval) as well as heterogeneous patterns of transmission. In addition, given that human contact behavior probably influences transmissibility, individual countermeasures (e.g. household quarantine and mask-wearing) need to be explored to construct effective non-pharmaceutical interventions.
PMCID: PMC1892008  PMID: 17547753
16.  Unravelling transmission trees of infectious diseases by combining genetic and epidemiological data 
Knowledge on the transmission tree of an epidemic can provide valuable insights into disease dynamics. The transmission tree can be reconstructed by analysing either detailed epidemiological data (e.g. contact tracing) or, if sufficient genetic diversity accumulates over the course of the epidemic, genetic data of the pathogen. We present a likelihood-based framework to integrate these two data types, estimating probabilities of infection by taking weighted averages over the set of possible transmission trees. We test the approach by applying it to temporal, geographical and genetic data on the 241 poultry farms infected in an epidemic of avian influenza A (H7N7) in The Netherlands in 2003. We show that the combined approach estimates the transmission tree with higher correctness and resolution than analyses based on genetic or epidemiological data alone. Furthermore, the estimated tree reveals the relative infectiousness of farms of different types and sizes.
PMCID: PMC3234549  PMID: 21733899
transmission tree; molecular epidemiology; avian influenza
17.  Real-time forecasting of an epidemic using a discrete time stochastic model: a case study of pandemic influenza (H1N1-2009) 
Real-time forecasting of epidemics, especially those based on a likelihood-based approach, is understudied. This study aimed to develop a simple method that can be used for the real-time epidemic forecasting.
A discrete time stochastic model, accounting for demographic stochasticity and conditional measurement, was developed and applied as a case study to the weekly incidence of pandemic influenza (H1N1-2009) in Japan. By imposing a branching process approximation and by assuming the linear growth of cases within each reporting interval, the epidemic curve is predicted using only two parameters. The uncertainty bounds of the forecasts are computed using chains of conditional offspring distributions.
The quality of the forecasts made before the epidemic peak appears largely to depend on obtaining valid parameter estimates. The forecasts of both weekly incidence and final epidemic size greatly improved at and after the epidemic peak with all the observed data points falling within the uncertainty bounds.
Real-time forecasting using the discrete time stochastic model with its simple computation of the uncertainty bounds was successful. Because of the simplistic model structure, the proposed model has the potential to additionally account for various types of heterogeneity, time-dependent transmission dynamics and epidemiological details. The impact of such complexities on forecasting should be explored when the data become available as part of the disease surveillance.
PMCID: PMC3045989  PMID: 21324153
18.  Predicting and controlling infectious disease epidemics using temporal networks 
F1000Prime Reports  2013;5:6.
Infectious diseases can be considered to spread over social networks of people or animals. Mainly owing to the development of data recording and analysis techniques, an increasing amount of social contact data with time stamps has been collected in the last decade. Such temporal data capture the dynamics of social networks on a timescale relevant to epidemic spreading and can potentially lead to better ways to analyze, forecast, and prevent epidemics. However, they also call for extended analysis tools for network epidemiology, which has, to date, mostly viewed networks as static entities. We review recent results of network epidemiology for such temporal network data and discuss future developments.
PMCID: PMC3590785  PMID: 23513178
19.  Mining social mixing patterns for infectious disease models based on a two-day population survey in Belgium 
Until recently, mathematical models of person to person infectious diseases transmission had to make assumptions on transmissions enabled by personal contacts by estimating the so-called WAIFW-matrix. In order to better inform such estimates, a population based contact survey has been carried out in Belgium over the period March-May 2006. In contrast to other European surveys conducted simultaneously, each respondent recorded contacts over two days. Special attention was given to holiday periods, and respondents with large numbers of professional contacts.
Participants kept a paper diary with information on their contacts over two different days. A contact was defined as a two-way conversation of at least three words in each others proximity. The contact information included the age of the contact, gender, location, duration, frequency, and whether or not touching was involved.
For data analysis, we used association rules and classification trees. Weighted generalized estimating equations were used to analyze contact frequency while accounting for the correlation between contacts reported on the two different days.
A contact surface, expressing the average number of contacts between persons of different ages was obtained by a bivariate smoothing approach and the relation to the so-called next-generation matrix was established.
People mostly mixed with people of similar age, or with their offspring, their parents and their grandparents. By imputing professional contacts, the average number of daily contacts increased from 11.84 to 15.70. The number of reported contacts depended heavily on the household size, class size for children and number of professional contacts for adults. Adults living with children had on average 2 daily contacts more than adults living without children. In the holiday period, the daily contact frequency for children and adolescents decreased with about 19% while a similar observation is made for adults in the weekend. These findings can be used to estimate the impact of school closure.
We conducted a diary based contact survey in Belgium to gain insights in social interactions relevant to the spread of infectious diseases. The resulting contact patterns are useful to improve estimating crucial parameters for infectious disease transmission models.
PMCID: PMC2656518  PMID: 19154612
20.  Inferring Epidemic Network Topology from Surveillance Data 
PLoS ONE  2014;9(6):e100661.
The transmission of infectious diseases can be affected by many or even hidden factors, making it difficult to accurately predict when and where outbreaks may emerge. One approach at the moment is to develop and deploy surveillance systems in an effort to detect outbreaks as timely as possible. This enables policy makers to modify and implement strategies for the control of the transmission. The accumulated surveillance data including temporal, spatial, clinical, and demographic information, can provide valuable information with which to infer the underlying epidemic networks. Such networks can be quite informative and insightful as they characterize how infectious diseases transmit from one location to another. The aim of this work is to develop a computational model that allows inferences to be made regarding epidemic network topology in heterogeneous populations. We apply our model on the surveillance data from the 2009 H1N1 pandemic in Hong Kong. The inferred epidemic network displays significant effect on the propagation of infectious diseases.
PMCID: PMC4076216  PMID: 24979215
21.  The Impact of Case Diagnosis Coverage and Diagnosis Delays on the Effectiveness of Antiviral Strategies in Mitigating Pandemic Influenza A/H1N1 2009 
PLoS ONE  2010;5(11):e13797.
Neuraminidase inhibitors were used to reduce the transmission of pandemic influenza A/H1N1 2009 at the early stages of the 2009/2010 pandemic. Policies for diagnosis of influenza for the purposes of antiviral intervention differed markedly between and within countries, leading to differences in the timing and scale of antiviral usage.
Methodology/Principal Findings
The impact of the percentage of symptomatic infected individuals who were diagnosed, and of delays to diagnosis, for three antiviral intervention strategies (each with and without school closure) were determined using a simulation model of an Australian community. Epidemic characteristics were based on actual data from the A/H1N1 2009 pandemic including reproduction number, serial interval and age-specific infection rate profile. In the absence of intervention an illness attack rate (AR) of 24.5% was determined from an estimated R0 of 1.5; this was reduced to 21%, 16.5% or 13% by treatment-only, treatment plus household prophylaxis, or treatment plus household plus extended prophylaxis antiviral interventions respectively, assuming that diagnosis occurred 24 hours after symptoms arose and that 50% of symptomatic cases were diagnosed. If diagnosis occurred without delay, ARs decreased to 17%, 12.2% or 8.8% respectively. If 90% of symptomatic cases were diagnosed (with a 24 hour delay), ARs decreased to 17.8%, 11.1% and 7.6%, respectively.
The ability to rapidly diagnose symptomatic cases and to diagnose a high proportion of cases was shown to improve the effectiveness of all three antiviral strategies. For epidemics with R0< = 1.5 our results suggest that when the case diagnosis coverage exceeds ∼70% the size of the antiviral stockpile required to implement the extended prophylactic strategy decreases. The addition of at least four weeks of school closure was found to further reduce cumulative and peak attack rates and the size of the required antiviral stockpile.
PMCID: PMC2972206  PMID: 21072188
22.  Change Point Estimation in Monitoring Survival Time 
PLoS ONE  2012;7(3):e33630.
Precise identification of the time when a change in a hospital outcome has occurred enables clinical experts to search for a potential special cause more effectively. In this paper, we develop change point estimation methods for survival time of a clinical procedure in the presence of patient mix in a Bayesian framework. We apply Bayesian hierarchical models to formulate the change point where there exists a step change in the mean survival time of patients who underwent cardiac surgery. The data are right censored since the monitoring is conducted over a limited follow-up period. We capture the effect of risk factors prior to the surgery using a Weibull accelerated failure time regression model. Markov Chain Monte Carlo is used to obtain posterior distributions of the change point parameters including location and magnitude of changes and also corresponding probabilistic intervals and inferences. The performance of the Bayesian estimator is investigated through simulations and the result shows that precise estimates can be obtained when they are used in conjunction with the risk-adjusted survival time CUSUM control charts for different magnitude scenarios. The proposed estimator shows a better performance where a longer follow-up period, censoring time, is applied. In comparison with the alternative built-in CUSUM estimator, more accurate and precise estimates are obtained by the Bayesian estimator. These superiorities are enhanced when probability quantification, flexibility and generalizability of the Bayesian change point detection model are also considered.
PMCID: PMC3306432  PMID: 22438969
23.  Bursts of Vertex Activation and Epidemics in Evolving Networks 
PLoS Computational Biology  2013;9(3):e1002974.
The dynamic nature of contact patterns creates diverse temporal structures. In particular, empirical studies have shown that contact patterns follow heterogeneous inter-event time intervals, meaning that periods of high activity are followed by long periods of inactivity. To investigate the impact of these heterogeneities in the spread of infection from a theoretical perspective, we propose a stochastic model to generate temporal networks where vertices make instantaneous contacts following heterogeneous inter-event intervals, and may leave and enter the system. We study how these properties affect the prevalence of an infection and estimate , the number of secondary infections of an infectious individual in a completely susceptible population, by modeling simulated infections (SI and SIR) that co-evolve with the network structure. We find that heterogeneous contact patterns cause earlier and larger epidemics in the SIR model in comparison to homogeneous scenarios for a vast range of parameter values, while smaller epidemics may happen in some combinations of parameters. In the case of SI and heterogeneous patterns, the epidemics develop faster in the earlier stages followed by a slowdown in the asymptotic limit. For increasing vertex turnover rates, heterogeneous patterns generally cause higher prevalence in comparison to homogeneous scenarios with the same average inter-event interval. We find that is generally higher for heterogeneous patterns, except for sufficiently large infection duration and transmission probability.
Author Summary
Networks of sexual contacts and of spatial proximity are of interest for the understanding of epidemics because they define potential pathways by which sexual and airborne infections spread. These networks are not static but vary, with both vertices and links appearing and disappearing at different times. One of the temporal properties observed across systems is that the time lapse between two contacts is irregular, which means that high activity is followed by long intervals of idleness. In this article, by using a theoretical model of a dynamic network co-evolving with a simulated infection, we show that such heterogeneity leads to earlier epidemic outbreaks and increased prevalence of infections for a range of parameters, in comparison to scenarios of regular activity, which is the current modeling paradigm in mathematical epidemiology. We also include a turnover rate to model individuals entering and leaving the system, and we show that if turnover is high, the relative difference in the prevalence of heterogeneous and homogeneous contact patterns increases due to the continuous influx of susceptible individuals. These heterogeneities also increase the expected number of secondary infections produced by a single infected vertex in a completely susceptible population.
PMCID: PMC3605099  PMID: 23555211
24.  Quantile Regression for Doubly Censored Data 
Biometrics  2011;68(1):101-112.
Double censoring often occurs in registry studies when left censoring is present in addition to right censoring. In this work, we propose a new analysis strategy for such doubly censored data by adopting a quantile regression model. We develop computationally simple estimation and inference procedures by appropriately using the embedded martingale structure. Asymptotic properties, including the uniform consistency and weak convergence, are established for the resulting estimators. Moreover, we propose conditional inference to address the special identifiability issues attached to the doubly censoring setting. We further show that the proposed method can be readily adapted to handle left truncation. Simulation studies demonstrate good finite-sample performance of the new inferential procedures. The practical utility of our method is illustrated by an analysis of the onset of the most commonly investigated respiratory infection, Pseudomonas aeruginosa, in children with cystic fibrosis through the use of the US Cystic Fibrosis Registry.
PMCID: PMC3312995  PMID: 21950348
Conditional inference; Double censoring; Empirical process; Martingale; Regression quantile; Truncation
25.  Ongoing Estimation of the Epidemic Parameters of a Stochastic, Spatial, Discrete-Time Model for a 1983–84 Avian Influenza Epidemic 
Avian diseases  2011;55(1):35-42.
We formulate a stochastic, spatial, discrete-time model of viral “Susceptible, Exposed, Infectious, Recovered” animal epidemics and apply it to an avian influenza epidemic in Pennsylvania in 1983–84. Using weekly data for the number of newly infectious cases collected during the epidemic, we find estimates for the latent period of the virus and the values of two parameters within the transmission kernel of the model. These data are then jackknifed on a progressive weekly basis to show how our estimates can be applied to an ongoing epidemic to generate continually improving values of certain epidemic parameters.
PMCID: PMC3615557  PMID: 21500633
epidemics; estimators; avian influenza; parameter estimation; mathematical models

Results 1-25 (842139)