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

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

The state-reproduction number for a multistate class age structured epidemic system and its application to the asymptomatic transmission model

In this paper, we develop the theory of a state-reproduction number for a multistate class age structured epidemic system and apply it to examine the asymptomatic transmission model. We formulate a renewal integral equation system to describe the invasion of infectious diseases into a multistate class age structured host population. We define the state-reproduction number for a class age structured system, which is the net reproduction number of a specific host type and which plays an analogous role to the type-reproduction number [M.G. Roberts, J.A.P. Heesterbeek, A new method for estimating the effort required to control an infectious disease, Proc. R. Soc. Lond. B 270 (2003) 1359; J.A.P. Heesterbeek, M.G. Roberts, The type-reproduction number T in models for infectious disease control, Math. Biosci. 206 (2007) 3] in discussing the critical level of public health intervention. The renewal equation formulation permits computations not only of the state-reproduction number, but also of the generation time and the intrinsic growth rate of infectious diseases.Subsequently, the basic theory is applied to capture the dynamics of a directly transmitted disease within two types of infected populations, i.e., asymptomatic and symptomatic individuals, in which the symptomatic class is observable and hence a target host of the majority of interventions. The state-reproduction number of the symptomatic host is derived and expressed as a measurable quantity, leading to discussion on the critical level of case isolation. The serial interval and other epidemiologic indices are computed, clarifying the parameters on which these indices depend. As a practical example, we illustrate the eradication threshold for case isolation of smallpox. The generation time and serial interval are comparatively examined for pandemic influenza.     

Controlling emerging infectious diseases like SARS

To control emerging infectious diseases like SARS, it is necessary to resort to basic control measures that limit exposures to infectious individuals. These measures include isolating cases at diagnosis, quarantining household members and tracing contacts of diagnosed cases, providing the community with advice on how to reduce exposures, and closing schools. To justify such intervention it is important to understand how well each of these measures helps to limit transmission. In this paper, we determine the effect of a number of different interventions on the effective reproduction number and estimate requirements to achieve elimination of the infectious disease. We find that the strategy of tracing and quarantining contacts of diagnosed cases can be very successful in reducing transmission.     

Predicting and Evaluating the Epidemic Trend of Ebola Virus Disease in the 2014-2015 Outbreak and the Effects of Intervention Measures

We constructed dynamic Ebola virus disease (EVD) transmission models to predict epidemic trends and evaluate intervention measure efficacy following the 2014 EVD epidemic in West Africa. We estimated the effective vaccination rate for the population, with basic reproduction number (R₀) as the intermediate variable. Periodic EVD fluctuation was analyzed by solving a Jacobian matrix of differential equations based on a SIR (susceptible, infective, and removed) model. A comprehensive compartment model was constructed to fit and predict EVD transmission patterns, and to evaluate the effects of control and prevention measures. Effective EVD vaccination rates were estimated to be 42% (31–50%), 45% (42–48%), and 51% (44–56%) among susceptible individuals in Guinea, Liberia and Sierra Leone, respectively. In the absence of control measures, there would be rapid mortality in these three countries, and an EVD epidemic would be likely recur in 2035, and then again 8~9 years later. Oscillation intervals would shorten and outbreak severity would decrease until the periodicity reached ~5.3 years. Measures that reduced the spread of EVD included: early diagnosis, treatment in isolation, isolating/monitoring close contacts, timely corpse removal, post-recovery condom use, and preventing or quarantining imported cases. EVD may re-emerge within two decades without control and prevention measures. Mass vaccination campaigns and control and prevention measures should be instituted to prevent future EVD epidemics.     

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

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

Estimation of Transmission Parameters of H5N1 Avian Influenza Virus in Chickens

Despite considerable research efforts, little is yet known about key epidemiological parameters of H5N1 highly pathogenic influenza viruses in their avian hosts. Here we show how these parameters can be estimated using a limited number of birds in experimental transmission studies. Our quantitative estimates, based on Bayesian methods of inference, reveal that (i) the period of latency of H5N1 influenza virus in unvaccinated chickens is short (mean: 0.24 days; 95% credible interval: 0.099–0.48 days); (ii) the infectious period of H5N1 virus in unvaccinated chickens is approximately 2 days (mean: 2.1 days; 95%CI: 1.8–2.3 days); (iii) the reproduction number of H5N1 virus in unvaccinated chickens need not be high (mean: 1.6; 95%CI: 0.90–2.5), although the virus is expected to spread rapidly because it has a short generation interval in unvaccinated chickens (mean: 1.3 days; 95%CI: 1.0–1.5 days); and (iv) vaccination with genetically and antigenically distant H5N2 vaccines can effectively halt transmission. Simulations based on the estimated parameters indicate that herd immunity may be obtained if at least 80% of chickens in a flock are vaccinated. We discuss the implications for the control of H5N1 avian influenza virus in areas where it is endemic.     

How generation intervals shape the relationship between growth rates and reproductive numbers

Mathematical models of transmission have become invaluable management tools in planning for the control of emerging infectious diseases. A key variable in such models is the reproductive number R. For new emerging infectious diseases, the value of the reproductive number can only be inferred indirectly from the observed exponential epidemic growth rate r. Such inference is ambiguous as several different equations exist that relate the reproductive number to the growth rate, and it is unclear which of these equations might apply to a new infection. Here, we show that these different equations differ only with respect to their assumed shape of the generation interval distribution. Therefore, the shape of the generation interval distribution determines which equation is appropriate for inferring the reproductive number from the observed growth rate. We show that by assuming all generation intervals to be equal to the mean, we obtain an upper bound to the range of possible values that the reproductive number may attain for a given growth rate. Furthermore, we show that by taking the generation interval distribution equal to the observed distribution, it is possible to obtain an empirical estimate of the reproductive number.     

Estimating individual and household reproduction numbers in an emerging epidemic

Reproduction numbers, defined as averages of the number of people infected by a typical case, play a central role in tracking infectious disease outbreaks. The aim of this paper is to develop methods for estimating reproduction numbers which are simple enough that they could be applied with limited data or in real time during an outbreak. I present a new estimator for the individual reproduction number, which describes the state of the epidemic at a point in time rather than tracking individuals over time, and discuss some potential benefits. Then, to capture more of the detail that micro-simulations have shown is important in outbreak dynamics, I analyse a model of transmission within and between households, and develop a method to estimate the household reproduction number, defined as the number of households infected by each infected household. This method is validated by numerical simulations of the spread of influenza and measles using historical data, and estimates are obtained for would-be emerging epidemics of these viruses. I argue that the household reproduction number is useful in assessing the impact of measures that target the household for isolation, quarantine, vaccination or prophylactic treatment, and measures such as social distancing and school or workplace closures which limit between-household transmission, all of which play a key role in current thinking on future infectious disease mitigation.     

Effect of the interval of serial sections of ovarian tissue in the tissue chopper on the number of isolated caprine preantral follicles.

The present work investigated the effect of the interval of serial sections of ovarian tissue on the number of isolated preantral follicles in the goat. Goat ovaries were cut in the tissue chopper at eight different intervals. The quality of isolated follicles were evaluated by histology and transmission electron microscopy. Best results were obtained when the ovaries were cut in the tissue chopper at intervals of 75.0 microm (9664 preantral follicles per ovary). Histochemical and ultrastructural analysis showed that the follicular morphology was preserved after mechanical isolation as demonstrated by the normality of oocytes and granulosa cells as well as by preservation of basement membrane. The percentages of isolated primordial, primary and secondary follicles were 96.3%, 2.5%, and 1.2% and their average diameters were 21.5, 34.7 and 65.3 microm, respectively. It was concluded that the interval of serial sections of ovarian tissue in the tissue chopper affects the number of isolated preantral follicles, and that the follicles remained intact after mechanical isolation in goats.     

Transmission potential of chikungunya virus and control measures: the case of Italy

During summer 2007 Italy has experienced an epidemic caused by Chikungunya virus - the first large outbreak documented in a temperate climate country - with approximately 161 laboratory confirmed cases concentrated in two bordering villages in North-Eastern Italy comprising 3,968 inhabitants. The seroprevalence was recently estimated to be 10.2%. In this work we provide estimates of the transmission potential of the virus and we assess the efficacy of the measures undertaken by public health authorities to control the epidemic spread. To such aim, we developed a model describing the temporal dynamics of the competent vector, known as Aedes albopictus, explicitly depending on climatic factors, coupled to an epidemic transmission model describing the spread of the epidemic in both humans and mosquitoes. The cumulative number of notified cases predicted by the model was 185 on average (95% CI 117-278), in good agreement with observed data. The probability of observing a major outbreak after the introduction of an infective human case was estimated to be in the range of 32%-76%. We found that the basic reproduction number was in the range of 1.8-6 but it could have been even larger, depending on the density of mosquitoes, which in turn depends on seasonal meteorological effects, besides other local abiotic factors. These results confirm the increasing risk of tropical vector-borne diseases in temperate climate countries, as a consequence of globalization. However, our results show that an epidemic can be controlled by performing a timely intervention, even if the transmission potential of Chikungunya virus is sensibly high.     

Generation interval contraction and epidemic data analysis

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.     

The inference of stepwise changes in substitution rates using serial sequence samples.

It is frequently true that molecular sequences do not evolve in a strictly clocklike manner. Instead, substitution rate may vary for a number of reasons, including changes in selection pressure and effective population size, as well as changes in mean generation time. Here we present two new methods for estimating stepwise changes in substitution rates when serially sampled molecular sequences are available. These methods are based on multiple rates with dated tips (MRDT) models and allow different rates to be estimated for different intervals of time. These intervals may correspond to the sampling intervals or to a priori--defined intervals that are not coincident with the times the serial samples are obtained. Two methods for obtaining estimates of multiple rates are described. The first is an extension of the phylogeny-based maximum-likelihood estimation procedure introduced by Rambaut. The second is a new parameterization of the pairwise distance least-squares procedure used by Drummond and Rodrigo. The utility of these methods is demonstrated on a genealogy of HIV sequences obtained at five different sampling times from a single patient over a period of 34 months.     

Avian influenza H5N1 transmission in households, Indonesia

Background

Disease transmission patterns are needed to inform public health interventions, but remain largely unknown for avian influenza H5N1 virus infections. A recent study on the 139 outbreaks detected in Indonesia between 2005 and 2009 found that the type of exposure to sources of H5N1 virus for both the index case and their household members impacted the risk of additional cases in the household. This study describes the disease transmission patterns in those outbreak households.

Methodology/Principal Findings

We compared cases (n = 177) and contacts (n = 496) in the 113 sporadic and 26 cluster outbreaks detected between July 2005 and July 2009 to estimate attack rates and disease intervals. We used final size household models to fit transmission parameters to data on household size, cases and blood-related household contacts to assess the relative contribution of zoonotic and human-to-human transmission of the virus, as well as the reproduction number for human virus transmission. The overall household attack rate was 18.3% and secondary attack rate was 5.5%. Secondary attack rate remained stable as household size increased. The mean interval between onset of subsequent cases in outbreaks was 5.6 days. The transmission model found that human transmission was very rare, with a reproduction number between 0.1 and 0.25, and the upper confidence bounds below 0.4. Transmission model fit was best when the denominator population was restricted to blood-related household contacts of index cases.

Conclusions/Significance

The study only found strong support for human transmission of the virus when a single large cluster was included in the transmission model. The reproduction number was well below the threshold for sustained transmission. This study provides baseline information on the transmission dynamics for the current zoonotic virus and can be used to detect and define signatures of a virus with increasing capacity for human-to-human transmission.     

Bayesian analysis of severe acute respiratory syndrome: the 2003 Hong Kong epidemic

This paper analyzes data arising from a Severe Acute Respiratory Syndrome (SARS) epidemic in Hong Kong in 2003 involving 1755 cases. A discrete time stochastic model that uses a back-projection approach is proposed. Markov Chain Monte Carlo (MCMC) methods are developed for estimation of model parameters. The algorithm is further extended to integrate numerically over unobserved variables of the model. Applying the method to SARS data from Hong Kong, a value of 3.88 with a posterior standard deviation of 0.09 was estimated for the basic reproduction number. An estimate of the transmission parameter at the beginning of the epidemic was also obtained as 0.149 with a posterior standard deviation of 0.003. A reduction in the transmission parameter during the course of the epidemic forced the effective reproduction number to cross the threshold value of one, seven days after control interventions were introduced. At the end of the epidemic, the effective reproduction number was as low as 0.001 suggesting that the epidemic was brought under control by the intervention measures introduced.     

Evaluation of genetic variation in the international Brown Swiss population

The international Brown Swiss cattle population pedigree was studied to measure genetic variations and to identify the most influential animals. Twenty-two countries provided pedigree information on 71 497 Brown Swiss bulls used for artificial insemination (AI). The total number of animals with the pedigree is 181 094. The mean inbreeding coefficient for the pedigree population was 0.77%. There was, in most cases, an increase in the mean inbreeding coefficient, with the highest value at 2.89% during the last 5-year period (2000 to 2004). The mean average relatedness for the pedigree population was 1.1%. The effective population size in 2004 was 204. There was notable variation between average generation intervals for the four parental pathways. The longest average generation interval, at 8.73 years, was observed in the sire–son pathway. The average generation interval for the whole population was 6.53 years. Most genetically influential individuals were sires. The highest contributing founder was a sire with a 3.22% contribution, and the highest contributing founder dam made a contribution of 1.75%. The effective number of founders and the effective number of ancestors were 141 and 88, respectively. The study showed that genetic variation within the pedigree population has been decreasing over recent years. Increasing the number of AI bulls with a low individual coefficient of inbreeding could help to maintain a good level of genetic variation in the Brown Swiss population.     

The dilution effect of the domestic animal population on the transmission of P. vivax malaria

The diversion of disease carrying insect from humans to animals may reduce transmission of diseases such as malaria. The use of animals to mitigate mosquito bites on human is called ‘zooprophylaxis’. We introduce a mathematical model for Plasmodium vivax malaria transmission with two bloodmeal hosts (humans and domestic animals) to study the effect of zooprophylaxis. After computing the basic reproduction number from the proposed model, we explore how perturbations in the parameters, sensitive to the effects of control measures, affect its value. Zooprophylaxis is shown to determine whether a basic reproduction becomes bigger than an outbreak threshold value or not. Sensitivity analysis shows that increasing the relative animal population size works better in P. vivax malaria control than decreasing the mosquito population when the relative animal population size is larger than a threshold value.     

Cost-Effectiveness Analysis of Optimal Control Measures for Tuberculosis

We propose and analyze an optimal control problem where the control system is a mathematical model for tuberculosis that considers reinfection. The control functions represent the fraction of early latent and persistent latent individuals that are treated. Our aim was to study how these control measures should be implemented, for a certain time period, in order to reduce the number of active infected individuals, while minimizing the interventions implementation costs. The optimal intervention is compared along different epidemiological scenarios, by varying the transmission coefficient. The impact of variation of the risk of reinfection, as a result of acquired immunity to a previous infection for treated individuals on the optimal controls and associated solutions, is analyzed. A cost-effectiveness analysis is done, to compare the application of each one of the control measures, separately or in combination.     

Estimating the immunity coverage required to prevent epidemics in a community of households

An estimation of the immunity coverage needed to prevent future outbreaks of an infectious disease is considered for a community of households. Data on outbreak size in a sample of households from one epidemic are used to derive maximum likelihood estimates and confidence bounds for parameters of a stochastic model for disease transmission in a community of households. These parameter estimates induce estimates and confidence bounds for the basic reproduction number and the critical immunity coverage, which are the parameters of main interest when aiming at preventing major outbreaks in the future. The case when individuals are homogeneous, apart from the size of their household, is considered in detail. The generalization to the case with variable infectivity, susceptibility and/or mixing behaviour is discussed more briefly. The methods are illustrated with an application to data on influenza in Tecumseh, Michigan.     

Clutch size versus clutch interval: life history strategies in the colour-polymorphic pygmy grasshopper Tetrix subulata

Theory posits that reproduction carries a cost in terms either of future fecundity, growth or survival. Different life history strategies may evolve in response to different external sources of mortality. In ectothermic organisms, such as insects and reptiles, reproductive characteristics may also vary due to effects of differences in body temperature on activity and physiological performance. In this study, female pygmy grasshoppers [Tetrix subulata (L.) Orthoptera: Tetrigidae] belonging to four different colour morphs were maintained under two different temperatures, and data on reproductive life history traits were used to test for costs of reproduction, plasticity of reproductive characteristics in response to temperature and variation among colour morphs in reproductive strategies. The results revealed that average clutch size decreased progressively from the first to third clutch, and that females producing relatively large clutches displayed a greater reduction (in both absolute and relative terms) in the number of eggs to the following clutch, as expected from the hypothesis that present reproduction negatively affects future fecundity. Great expenditure on present reproduction also negatively influenced the time to next clutch:the decrease in mean clutch size with clutch number was associated with a reduction in inter-clutch interval, and clutch interval increased with clutch size across individuals within colour morphs. Females maintained in a warm environment were more likely to oviposit, laid their first clutch earlier, produced more clutches and had shorter intervals between sequential clutchesthan females in a cold environment, suggesting that differences in body temperature may contribute to variation in reproductive performance within and among natural populations. A comparison among colour morphs maintained under identical conditions suggested that females belonging to certain morphs produce relatively large clutches at the expense of fewer clutches per unit time. However, experimental data revealed no difference in relative fat content between dark and pale individuals maintained either in sun-exposed outdoor enclosures (where they were unable to increase their body temperature by basking) or in shaded enclosures. This suggest that the divergence in life history strategies among colour morphs may reflect a response to morph-specific differences in adult survival imposed by visually searching predators, rather than being due to the effects of differences in body temperature.     

Use of historical temperature data for timing insecticide applications of the Nantucket pine tip moth (Lepidoptera: Tortricidae): evaluation of damage and volume increment efficacy

• 1 The effectiveness of optimal spray period intervals based on mean daily temperatures were evaluated as a spray‐timing tool to control high density populations of the Nantucket pine tip moth Rhyacionia frustrana (Comstock).
• 2 Initial tree growth realized from first generation R. frustrana control was compared to that from conventional applications of one insecticide treatment scheduled for each of three annual generations.
• 3 The optimal spray period intervals provided by Fettig et al. (2000a) were highly effective for controlling R. frustrana infestations. The control group averaged 47.0 ± 2.2% whole tree damage for all sites and generations as compared to 0.6 ± 0.2% for the treated group.
• 4 Volume gains attributable to R. frustrana control averaged 16.9%, 46.4% and 98.6% for first generation control, and 46.6%, 72.7% and 146.3% for conventional applications of one insecticide treatment scheduled for each of three annual R. frustrana generations at Northampton, Halifax I and Halifax II, respectively. Growth returns increased as mean damage estimates increased for both treatments, suggesting that returns realized from a single, first generation application are likely to increase with population density.