The basic reproduction number R0 and effectiveness of reactive interventions during dengue epidemics: the 2002 dengue outbreak in Easter Island, Chile

We use a stochastic simulation model to explore the effect of reactive intervention strategies during the 2002 dengue outbreak in the small population of Easter Island, Chile. We quantified the effect of interventions on the transmission dynamics and epidemic size as a function of the simulated control intensity levels and the timing of initiation of control interventions. Because no dengue outbreaks had been reported prior to 2002 in Easter Island, the 2002 epidemic provided a unique opportunity to estimate the basic reproduction number R0 during the initial epidemic phase, prior to the start of control interventions. We estimated R0 at 27.2 (95%CI: 14.8, 49.3). We found that the final epidemic size is highly sensitive to the timing of start of interventions. However, even when the control interventions start several weeks after the epidemic onset, reactive intervention efforts can have a significant impact on the final epidemic size. Our results indicate that the rapid implementation of control interventions can have a significant effect in reducing the epidemic size of dengue epidemics.

Mathematical models of contact patterns between age groups for predicting the spread of infectious diseases

The spread of an infectious disease is sensitive to the contact patterns in the population and to precautions people take to reduce the transmission of the disease. We investigate the impact that different mixing assumptions have on the spread an infectious disease in an age-structured ordinary differential equation model. We consider the impact of heterogeneity in susceptibility and infectivity within the population on the disease transmission. We apply the analysis to the spread of a smallpox-like disease, derive the formula for the reproduction number, ~0, and based on this threshold parameter, show the level of human behavioral change required to control the epidemic. We analyze how different mixing patterns can affect the disease prevalence, the cumulative number of new infections, and the final epidemic size. Our analysis indicates that the combination of residual immunity and behavioral changes during a smallpox-like disease outbreak can play a key role in halting infectious disease spread; and that realistic mixing patterns must be included in the epidemic model for the predictions to accurately reflect reality.

Estimating the transmissibility of H5N1 and the effect of vaccination in Indonesia

The spread of H5N1 avian influenza continues to pose an economic burden and a public health risk worldwide. Despite this, estimates of the transmissibility of infection exist in only a handful of settings and vary considerably. Using final size methods and flock-level infection data from a field trial of mass vaccination, we obtained the first estimates of the transmissibility of infection between and within flocks in Indonesia. We also found that outbreaks in areas designated as vaccination zones were less transmissible than in non-vaccination zones. However, this reduction is only comparable with a limited degree of protective vaccination coverage. Quantifying the overall effect of vaccination in these zones remains challenging. However, this result would appear to imply that, although the interventions applied in vaccination zones were not sufficient to completely prevent transmission in all areas, when outbreaks occur, they are less transmissible than those in areas where vaccination was not applied. This could be either a direct or an indirect effect of vaccination. Given the dynamism of small-scale poultry production in Indonesia, more regular vaccination may be required to ensure that infection is fully controlled in vaccination zones.

Deterministic modelling for transmission of Human Papillomavirus 6/11: impact of vaccination

This paper is devoted to assess the impact of quadrivalent human papillomavirus (HPV) vaccine on the prevalence of non-oncogenic HPV 6/11 types in French males and females. For this purpose, a non-linear dynamic model of heterosexual transmission for HPV 6/11 types infection is developed, which accounts for immunity due to vaccination, in particular. The vaccinated reproduction number Rv is derived using the approach described by Diekmann et al. (2010) called the next generation operator approach. The model proposed is analysed, with regard to existence and uniqueness of the solution, steady-state stability. Precisely, the stability of the model is investigated depending on the sign of Rv-1. Prevalence data are used to fit a numerical HPV model, so as to assess infection rates. Our approach suggests that 10 years after introducing vaccination, the prevalence of HPV 6/11 types in females will be halved and that in males will be reduced by one-quarter, assuming a sustained vaccine coverage of 30% among females. Using the formula, we derived for the vaccinated reproduction number, we show that the non-oncogenic HPV 6/11 types would be eradicated if vaccine coverage in females is kept above 12%.

Toward unbiased assessment of treatment and prevention: modeling household transmission of pandemic influenza

Providing valid and reliable estimates of the transmissibility and severity of pandemic influenza in real time is key to guide public health policymaking. In particular, early estimates of the transmissibility are indispensable for determining the type and intensity of interventions. A recent study by House and colleagues in BMC Medicine devised a stochastic transmission model to estimate the unbiased risk of transmission within households, applying the method to datasets of the 2009 A/H1N1 influenza pandemic. Here, we discuss future challenges in household transmission studies and underscore the need to systematically collect epidemiological data to decipher the household transmission dynamics. We emphasize the need to consider three critical issues for future improvements: (i) capturing age-dependent heterogeneity within households calls for intensive modeling efforts, (ii) the timeline of observation during the course of an epidemic and the length of follow-up should be aligned with study objectives, and (iii) the use of laboratory methods, especially molecular techniques, is encouraged to distinguish household transmissions from those arising in the community.See related article: http://www.biomedcentral.com/1741-7015/10/117.

Epidemic Models

Communicable diseases such as measles, influenza, and tuberculosis are a fact of life. We will be concerned with both epidemics, which are sudden outbreaks of a disease, and endemic situations, in which a disease is always present. The AIDS epidemic, the recent SARS epidemic, recurring influenza pandemics, and outbursts of diseases such as the Ebola virus are events of concern and interest to many people. The prevalence and effects of many diseases in less-developed countries are probably not as well known but may be of even more importance. Every year millions, of people die of measles, respiratory infections, diarrhea, and other diseases that are easily treated and not considered dangerous in the Western world. Diseases such as malaria, typhus, cholera, schistosomiasis, and sleeping sickness are endemic in many parts of the world. The effects of high disease mortality on mean life span and of disease debilitation and mortality on the economy in afflicted countries are considerable.

Mathematical Models in Infectious Disease Epidemiology

The idea that transmission and spread of infectious diseases follows laws that can be formulated in mathematical language is old. In 1766 Daniel Bernoulli published an article where he described the effects of smallpox variolation (a precursor of vaccination) on life expectancy using mathematical life table analysis (Dietz and Heesterbeek 2000). However, it was only in the twentieth century that the nonlinear dynamics of infectious disease transmission was really understood. In the beginning of that century there was much discussion about why an epidemic ended before all susceptibles were infected with hypotheses about changing virulence of the pathogen during the epidemic.

Enhanced hygiene measures and norovirus transmission during an outbreak

Control of norovirus outbreaks relies on enhanced hygiene measures, such as handwashing, surface cleaning, using disposable paper towels, and using separate toilets for sick and well persons. However, little is known about their effectiveness in limiting further spread of norovirus infections. We analyzed norovirus outbreaks in 7 camps at an international scouting jamboree in the Netherlands during 2004. Implementation of hygiene measures coincided with an 84.8% (95% predictive interval 81.2%-86.6%) reduction in reproduction number. This reduction was unexpectedly large but still below the reduction needed to contain a norovirus outbreak. Even more stringent control measures are required to break the chain of transmission of norovirus.

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

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

Spatial Structure: Patch Models

Discrete spatial heterogenity is introduced into disease transmission models, resulting in large systems of ordinary differential equations. Such metapopulation models describe disease spread on a number of spatial patches. In the first model considered, there is no explicit movement of individuals; rather infectives can pass the disease to susceptibles in other patches. The second type of model explicitly includes rates of travel between patches and also takes account of the resident patch as well as the current patch of individuals. A formula for and useful bounds on the basic reproduction number of the system are determined. Brief descriptions of application of this type of metapopulation model are given to investigate the spread of bovine tuberculosis and the effect of quarantine on the spread of influenza.

Compartmental Models in Epidemiology

We describe and analyze compartmental models for disease transmission. We begin with models for epidemics, showing how to calculate the basic reproduction number and the final size of the epidemic. We also study models with multiple compartments, including treatment or isolation of infectives. We then consider models including births and deaths in which there may be an endemic equilibrium and study the asymptotic stability of equilibria. We conclude by studying age of infection models which give a unifying framework for more complicated compartmental models.

Simulating the SARS outbreak in Beijing with limited data

We propose a mathematical model to simulate the SARS outbreak in Beijing. The model consists of six subpopulations, namely susceptible, exposed, quarantined, suspect, probable and removed, as China started to report SARS cases as suspect and probable separately from April 27 and cases transferred from suspect class to probable class from May 2. By simplifying the model to a two-compartment suspect-probable model and a single-compartment probable model and using limited data, we are able to simulate the SARS outbreak in Beijing. We estimate that the reproduction number varies from 1.0698 to 3.2524 and obtain certain important epidemiological parameters.