Household Epidemics: Modelling Effects of Early Stage Vaccination

A Markovian susceptible → infectious → removed (SIR) epidemic model is considered in a community partitioned into households. A vaccination strategy, which is implemented during the early stages of the disease following the detection of infected individuals is proposed. In this strategy, the detection occurs while an individual is infectious and other susceptible household members are vaccinated without further delay. Expressions are derived for the influence on the reproduction numbers of this vaccination strategy for equal and unequal household sizes. We fit previously estimated parameters from influenza and use household distributions for Sweden and Tanzania census data. The results show that the reproduction number is much higher in Tanzania (6 compared with 2) due to larger households, and that infected individuals have to be detected (and household members vaccinated) after on average 5 days in Sweden and after 3.3 days in Tanzania, a much smaller difference.     

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.     

Can Antiviral Drugs Contain Pandemic Influenza Transmission?

Antiviral drugs dispensed during the 2009 influenza pandemic generally failed to contain transmission. This poses the question of whether preparedness for a future pandemic should include plans to use antiviral drugs to mitigate transmission.Simulations using a standard transmission model that allows for infected arrivals and delayed vaccination show that attempts to contain transmission require relatively few antiviral doses. In contrast, persistent use of antiviral drugs when the reproduction number remains above 1 use very many doses and are unlikely to reduce the eventual attack rate appreciably unless the stockpile is very large. A second model, in which the community has a household structure, shows that the effectiveness of a strategy of dispensing antiviral drugs to infected households decreases rapidly with time delays in dispensing the antivirals. Using characteristics of past pandemics it is estimated that at least 80% of primary household cases must present upon show of symptoms to have a chance of containing transmission by dispensing antiviral drugs to households. To determine data needs, household outbreaks were simulated with 50% receiving antiviral drugs early and 50% receiving antiviral drugs late. A test to compare the size of household outbreaks indicates that at least 100-200 household outbreaks need to be monitored to find evidence that antiviral drugs can mitigate transmission of the newly emerged virus.Use of antiviral drugs in an early attempt to contain transmission should be part of preparedness plans for a future influenza pandemic. Data on the incidence of the first 350 cases and the eventual attack rates of the first 200 hundred household outbreaks should be used to estimate the initial reproduction number R and the effectiveness of antiviral drugs to mitigate transmission. Use of antiviral drugs to mitigate general transmission should cease if these estimates indicate that containment of transmission is unlikely.     

Estimating vaccine effects on transmission of infection from household outbreak data

This article is concerned with a method for making inferences about various measures of vaccine efficacy. These measures describe reductions in susceptibility and in the potential to transmit infection. The method uses data on household outbreaks; it is based on a model that allows for transmission of infection both from within a household and from the outside. The use of household data is motivated by the hope that these are informative about vaccine-induced reduction of the potential to transmit infection, as household outbreaks contain some information about the possible source of infection. For illustration, the method is applied to observed data on household outbreaks of smallpox. These data are of the form needed and the number of households is of a size that can be managed in a vaccine trial. It is found that vaccine effects, such as the mean reduction in susceptibility and the mean reduction in the potential to infect others, per infectious contact, can be estimated with precision. However, a more specific parameter reflecting the reduction in infectivity for individuals partially responding to vaccination is not estimated well in the application. An evaluation of the method using artificial data shows that this parameter can be estimated with greater precision when we have outbreak data on a large number of small households.     

Analysis of a Dengue Model with Vertical Transmission and Application to the 2014 Dengue Outbreak in Guangdong Province,China

There is evidence showing that vertical transmission of dengue virus exists in Aedes mosquitoes. In this paper, we propose a deterministic dengue model with vertical transmission in mosquitoes by including aquatic mosquitoes (eggs, larvae and pupae), adult mosquitoes (susceptible, exposed and infectious) and human hosts (susceptible, exposed, infectious and recovered). We first analyze the existence and stability of disease-free equilibria, calculate the basic reproduction number and discuss the existence of the disease-endemic equilibrium. Then, we study the impact of vertical transmission of the virus in mosquitoes on the spread dynamics of dengue. We also use the model to simulate the reported infected human data from the 2014 dengue outbreak in Guangdong Province, China, carry out sensitivity analysis of the basic reproduction number in terms of the model parameters, and seek for effective control measures for the transmission of dengue virus.     

The time to extinction for a stochastic SIS-household-epidemic model

We analyse a Markovian SIS epidemic amongst a finite population partitioned into households. Since the population is finite, the epidemic will eventually go extinct, i.e., have no more infectives in the population. We study the effects of population size and within household transmission upon the time to extinction. This is done through two approximations. The first approximation is suitable for all levels of within household transmission and is based upon an Ornstein-Uhlenbeck process approximation for the diseases fluctuations about an endemic level relying on a large population. The second approximation is suitable for high levels of within household transmission and approximates the number of infectious households by a simple homogeneously mixing SIS model with the households replaced by individuals. The analysis, supported by a simulation study, shows that the mean time to extinction is minimized by moderate levels of within household transmission.     

Modelling the proportion of influenza infections within households during pandemic and non-pandemic years

Background

The key epidemiological difference between pandemic and seasonal influenza is that the population is largely susceptible during a pandemic, whereas, during non-pandemic seasons a level of immunity exists. The population-level efficacy of household-based mitigation strategies depends on the proportion of infections that occur within households. In general, mitigation measures such as isolation and quarantine are more effective at the population level if the proportion of household transmission is low.

Methods/Results

We calculated the proportion of infections within households during pandemic years compared with non-pandemic years using a deterministic model of household transmission in which all combinations of household size and individual infection states were enumerated explicitly. We found that the proportion of infections that occur within households was only partially influenced by the hazard h of infection within household relative to the hazard of infection outside the household, especially for small basic reproductive numbers. During pandemics, the number of within-household infections was lower than one might expect for a given because many of the susceptible individuals were infected from the community and the number of susceptible individuals within household was thus depleted rapidly. In addition, we found that for the value of at which 30% of infections occur within households during non-pandemic years, a similar 31% of infections occur within households during pandemic years.

Interpretation

We suggest that a trade off between the community force of infection and the number of susceptible individuals in a household explains an apparent invariance in the proportion of infections that occur in households in our model. During a pandemic, although there are more susceptible individuals in a household, the community force of infection is very high. However, during non-pandemic years, the force of infection is much lower but there are fewer susceptible individuals within the household.     

Temporal Variability and Social Heterogeneity in Disease Transmission: The Case of SARS in Hong Kong

The extent to which self-adopted or intervention-related changes in behaviors affect the course of epidemics remains a key issue for outbreak control. This study attempted to quantify the effect of such changes on the risk of infection in different settings, i.e., the community and hospitals. The 2002–2003 severe acute respiratory syndrome (SARS) outbreak in Hong Kong, where 27% of cases were healthcare workers, was used as an example. A stochastic compartmental SEIR (susceptible-exposed-infectious-removed) model was used: the population was split into healthcare workers, hospitalized people and general population. Super spreading events (SSEs) were taken into account in the model. The temporal evolutions of the daily effective contact rates in the community and hospitals were modeled with smooth functions. Data augmentation techniques and Markov chain Monte Carlo (MCMC) methods were applied to estimate SARS epidemiological parameters. In particular, estimates of daily reproduction numbers were provided for each subpopulation. The average duration of the SARS infectious period was estimated to be 9.3 days (±0.3 days). The model was able to disentangle the impact of the two SSEs from background transmission rates. The effective contact rates, which were estimated on a daily basis, decreased with time, reaching zero inside hospitals. This observation suggests that public health measures and possible changes in individual behaviors effectively reduced transmission, especially in hospitals. The temporal patterns of reproduction numbers were similar for healthcare workers and the general population, indicating that on average, an infectious healthcare worker did not infect more people than any other infectious person. We provide a general method to estimate time dependence of parameters in structured epidemic models, which enables investigation of the impact of control measures and behavioral changes in different settings.     

Control of transmission with two types of infection

The effectiveness of a vaccination strategy to control transmission of an infectious disease depends on the way vaccine doses are distributed to individuals in a community of households. Here we show that this dependence is more complicated when acquisition and severity of illness are determined by the size of the infecting dose, as is thought to be the case for measles and varicella. Two alternative formulations for the way vaccination changes an individual's susceptibility and infectivity show that vaccination coverage, the nature of the vaccine response and the distribution of household size also have a big impact on which strategy is more effective. These judgements are made by comparing the post-vaccination reproduction numbers corresponding to different vaccination strategies.     

Deterministic epidemic models with explicit household structure

For a wide range of airborne infectious diseases, transmission within the family or household is a key mechanism for the spread and persistence of infection. In general, household-based transmission is relatively strong but only involves a limited number of individuals in contact with each infectious person. In contrast, transmission outside the household can be characterised by many contacts but a lower probability of transmission. Here we develop a relatively simple dynamical model that captures these two transmission regimes. We compare the dynamics of such models for a range of household sizes, whilst constraining all models to have equal early growth rate so that all models fit to the same early incidence observations of an epidemic. Finally we consider the use of prophylactic vaccination, responsive vaccination, or antivirals to combat epidemic spread and focus on whether it is optimal to target controls at entire households or to treat individuals independently.     

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.     

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.     

Estimating and interpreting secondary attack risk: Binomial considered biased

The household secondary attack risk (SAR), often called the secondary attack rate or secondary infection risk, is the probability of infectious contact from an infectious household member A to a given household member B, where we define infectious contact to be a contact sufficient to infect B if he or she is susceptible. Estimation of the SAR is an important part of understanding and controlling the transmission of infectious diseases. In practice, it is most often estimated using binomial models such as logistic regression, which implicitly attribute all secondary infections in a household to the primary case. In the simplest case, the number of secondary infections in a household with m susceptibles and a single primary case is modeled as a binomial(m, p) random variable where p is the SAR. Although it has long been understood that transmission within households is not binomial, it is thought that multiple generations of transmission can be neglected safely when p is small. We use probability generating functions and simulations to show that this is a mistake. The proportion of susceptible household members infected can be substantially larger than the SAR even when p is small. As a result, binomial estimates of the SAR are biased upward and their confidence intervals have poor coverage probabilities even if adjusted for clustering. Accurate point and interval estimates of the SAR can be obtained using longitudinal chain binomial models or pairwise survival analysis, which account for multiple generations of transmission within households, the ongoing risk of infection from outside the household, and incomplete follow-up. We illustrate the practical implications of these results in an analysis of household surveillance data collected by the Los Angeles County Department of Public Health during the 2009 influenza A (H1N1) pandemic.     

Influenza transmission in a community during a seasonal influenza A(H3N2) outbreak (2010-2011) in Mongolia: a community-based prospective cohort study

Background

Knowledge of how influenza viruses spread in a community is important for planning and implementation of effective interventions, including social distancing measures. Households and schools are implicated as the major sites for influenza virus transmission. However, the overall picture of community transmission is not well defined during actual outbreaks. We conducted a community-based prospective cohort study to describe the transmission characteristics of influenza in Mongolia.

Methods and Findings

A total of 5,655 residents in 1,343 households were included in this cohort study. An active search for cases of influenza-like illness (ILI) was performed between October 2010 and April 2011. Data collected during a community outbreak of influenza A(H3N2) were analyzed. Total 282 ILI cases occurred during this period, and 73% of the subjects were aged <15 years. The highest attack rate (20.4%) was in those aged 1–4 years, whereas the attack rate in those aged 5–9 years was 10.8%. Fifty-one secondary cases occurred among 900 household contacts from 43 households (43 index cases), giving an overall crude household secondary attack rate (SAR) of 5.7%. SAR was significantly higher in younger household contacts (relative risk for those aged <1 year: 9.90, 1–4 years: 5.59, and 5–9 years: 6.43). We analyzed the transmission patterns among households and a community and repeated transmissions were detected between households, preschools, and schools. Children aged 1–4 years played an important role in influenza transmission in households and in the community at large. Working-age adults were also a source of influenza in households, whereas elderly cases (aged ≥65 years) had no link with household transmission.

Conclusions

Repeated transmissions between households, preschools, and schools were observed during an influenza A(H3N2) outbreak period in Mongolia, where subjects aged 1–4 years played an important role in influenza transmission.     

Transmission parameters of the 2001 foot and mouth epidemic in Great Britain

Despite intensive ongoing research, key aspects of the spatial-temporal evolution of the 2001 foot and mouth disease (FMD) epidemic in Great Britain (GB) remain unexplained. Here we develop a Markov Chain Monte Carlo (MCMC) method for estimating epidemiological parameters of the 2001 outbreak for a range of simple transmission models. We make the simplifying assumption that infectious farms were completely observed in 2001, equivalent to assuming that farms that were proactively culled but not diagnosed with FMD were not infectious, even if some were infected. We estimate how transmission parameters varied through time, highlighting the impact of the control measures on the progression of the epidemic. We demonstrate statistically significant evidence for assortative contact patterns between animals of the same species. Predictive risk maps of the transmission potential in different geographic areas of GB are presented for the fitted models.     

On the dynamics of SEIRS epidemic model with transport-related infection

Transportation amongst cities is found as one of the main factors which affect the outbreak of diseases. To understand the effect of transport-related infection on disease spread, an SEIRS (Susceptible, Exposed, Infectious, Recovered) epidemic model for two cities is formulated and analyzed. The epidemiological threshold, known as the basic reproduction number, of the model is derived. If the basic reproduction number is below unity, the disease-free equilibrium is locally asymptotically stable. Thus, the disease can be eradicated from the community. There exists an endemic equilibrium which is locally asymptotically stable if the reproduction number is larger than unity. This means that the disease will persist within the community. The results show that transportation among regions will change the disease dynamics and break infection out even if infectious diseases will go to extinction in each isolated region without transport-related infection. In addition, the result shows that transport-related infection intensifies the disease spread if infectious diseases break out to cause an endemic situation in each region, in the sense of that both the absolute and relative size of patients increase. Further, the formulated model is applied to the real data of SARS outbreak in 2003 to study the transmission of disease during the movement between two regions. The results show that the transport-related infection is effected to the number of infected individuals and the duration of outbreak in such the way that the disease becomes more endemic due to the movement between two cities. This study can be helpful in providing the information to public health authorities and policy maker to reduce spreading disease when its occurs.     

The role of health care workers and antiviral drugs in the control of pandemic influenza

Until a vaccine against the new strain becomes available, the response to newly emerged pandemic influenza will consist of the use of antiviral drugs and measures that limit exposure to infectious individuals. These first-line defence measures include isolating cases upon diagnosis, reducing close contacts, the use of personal protective equipment and hygiene, and using antiviral drugs for treatment and prophylaxis. There are significant 'costs' associated with control measures, so to justify such interventions it is important to assess their potential to reduce transmission. In this paper, we determine the effect that a number of different antiviral interventions have on the reproduction number of infectives and the probability that an imported infection fades out, and determine parameter scenarios for which these interventions are able to eliminate an emerging pandemic of influenza. We also assess the role that health care workers play in transmission and the extent to which providing them with antiviral prophylaxis and personal protective equipment modifies this role. Our results indicate that this class requires protection to avoid a greatly disproportionate contribution to early infective numbers, and for the maintenance of a stable health care system. Further, we show that the role children play in increasing transmission is moderate, in spite of closer mixing with other children.     

Two-phase importance sampling for inference about transmission trees

There has been growing interest in the statistics community to develop methods for inferring transmission pathways of infectious pathogens from molecular sequence data. For many datasets, the computational challenge lies in the huge dimension of the missing data. Here, we introduce an importance sampling scheme in which the transmission trees and phylogenies of pathogens are both sampled from reasonable importance distributions, alleviating the inference. Using this approach, arbitrary models of transmission could be considered, contrary to many earlier proposed methods. We illustrate the scheme by analysing transmissions of Streptococcus pneumoniae from household to household within a refugee camp, using data in which only a fraction of hosts is observed, but which is still rich enough to unravel the within-household transmission dynamics and pairs of households between whom transmission is plausible. We observe that while probability of direct transmission is low even for the most prominent cases of transmission, still those pairs of households are geographically much closer to each other than expected under random proximity.     

Extending the type reproduction number to infectious disease control targeting contacts between types

A new quantity called the target reproduction number is defined to measure control strategies for infectious diseases with multiple host types such as waterborne, vector-borne and zoonotic diseases. The target reproduction number includes as a special case and extends the type reproduction number to allow disease control targeting contacts between types. Relationships among the basic, type and target reproduction numbers are established. Examples of infectious disease models from the literature are given to illustrate the use of the target reproduction number.     

Predicting the Herd Immunity Threshold during an Outbreak: A Recursive Approach

Background

The objective was to develop a novel algorithm that can predict, based on field survey data, the minimum vaccination coverage required to reduce the mean number of infections per infectious individual to less than one (the Outbreak Response Immunization Threshold or ORIT) from up to six days in the advance.

Methodology/Principal Findings

First, the relationship between the rate of immunization and the ORIT was analyzed to establish a link. This relationship served as the basis for the development of a recursive algorithm that predicts the ORIT using survey data from two consecutive days. The algorithm was tested using data from two actual measles outbreaks. The prediction day difference (PDD) was defined as the number of days between the second day of data input and the day of the prediction. The effects of different PDDs on the prediction error were analyzed, and it was found that a PDD of 5 minimized the error in the prediction. In addition, I developed a model demonstrating the relationship between changes in the vaccination coverage and changes in the individual reproduction number.

Conclusions/Significance

The predictive algorithm for the ORIT generates a viable prediction of the minimum number of vaccines required to stop an outbreak in real time. With this knowledge, the outbreak control agency may plan to expend the lowest amount of funds required stop an outbreak, allowing the diversion of the funds saved to other areas of medical need.