Estimating the Reproduction Number of Ebola Virus (EBOV) During the 2014 Outbreak in West Africa

The 2014 Ebola virus (EBOV) outbreak in West Africa is the largest outbreak of the genus Ebolavirus to date. To better understand the spread of infection in the affected countries, it is crucial to know the number of secondary cases generated by an infected index case in the absence and presence of control measures, i.e., the basic and effective reproduction number. In this study, I describe the EBOV epidemic using an SEIR (susceptible-exposed-infectious-recovered) model and fit the model to the most recent reported data of infected cases and deaths in Guinea, Sierra Leone and Liberia. The maximum likelihood estimates of the basic reproduction number are 1.51 (95% confidence interval [CI]: 1.50-1.52) for Guinea, 2.53 (95% CI: 2.41-2.67) for Sierra Leone and 1.59 (95% CI: 1.57-1.60) for Liberia. The model indicates that in Guinea and Sierra Leone the effective reproduction number might have dropped to around unity by the end of May and July 2014, respectively. In Liberia, however, the model estimates no decline in the effective reproduction number by end-August 2014. This suggests that control efforts in Liberia need to be improved substantially in order to stop the current outbreak.

On estimating the basic reproduction number in distinct stages of a contagious disease spreading

In epidemiology, the basic reproduction number R-0 is usually defined as the average number of new infections caused by a single infective individual introduced into a completely susceptible population. According to this definition. R-0 is related to the initial stage of the spreading of a contagious disease. However, from epidemiological models based on ordinary differential equations (ODE), R-0 is commonly derived from a linear stability analysis and interpreted as a bifurcation parameter: typically, when R-0 >1, the contagious disease tends to persist in the population because the endemic stationary solution is asymptotically stable: when R-0 <1, the corresponding pathogen tends to naturally disappear because the disease-free stationary solution is asymptotically stable. Here we intend to answer the following question: Do these two different approaches for calculating R-0 give the same numerical values? In other words, is the number of secondary infections caused by a unique sick individual equal to the threshold obtained from stability analysis of steady states of ODE? For finding the answer, we use a susceptibleinfective-recovered (SIR) model described in terms of ODE and also in terms of a probabilistic cellular automaton (PCA), where each individual (corresponding to a cell of the PCA lattice) is connected to others by a random network favoring local contacts. The values of R-0 obtained from both approaches are compared, showing good agreement. (C) 2012 Elsevier B.V. All rights reserved.

Vectorial capacity, basic reproduction number, force of infection and all that: formal notation to complete and adjust their classical concepts and equations

A dimensional analysis of the classical equations related to the dynamics of vector-borne infections is presented. It is provided a formal notation to complete the expressions for the Ross' threshold theorem, the Macdonald's basic reproduction "rate" and sporozoite "rate", Garret-Jones' vectorial capacity and Dietz-Molineaux-Thomas' force of infection. The analysis was intended to provide a formal notation that complete the classical equations proposed by these authors.

Rapid drop in the reproduction number during the Ebola outbreak in the Democratic Republic of Congo.

The Democratic Republic of Congo (DRC) experienced a confined rural outbreak of Ebola virus disease (EVD) with 69 reported cases from July to October 2014. Understanding the transmission dynamics during the outbreak can provide important information for anticipating and controlling future EVD epidemics. I fitted an EVD transmission model to previously published data of this outbreak and estimated the basic reproduction number R 0 = 5.2 (95% CI [4.0-6.7]). The model suggests that the net reproduction number Rt fell below unity 28 days (95% CI [25-34] days) after the onset of symptoms in the index case. This study adds to previous epidemiological descriptions of the 2014 EVD outbreak in DRC, and is consistent with the notion that a rapid implementation of control interventions helped reduce further spread.

Control dynamics of severe acute respiratory syndrome transmission

Severe acute respiratory syndrome (SARS) is a serious disease with many puzzling features. We present a simple, dynamic model to assess the epidemic potential of SARS and the effectiveness of control measures. With this model, we analysed the SARS epidemic data in Beijing. The data fitting gives the basic case reproduction number of 2.16 leading to the outbreak, and the variation of the effective reproduction number reflecting the control effect. Noticeably, our study shows that the response time and the strength of control measures have significant effects on the scale of the outbreak and the lasting time of the epidemic.

Transmission Potential of Rift Valley Fever virus over the course of the 2010 epidemic in South Africa

A Rift Valley fever (RVF) epidemic affecting animals on domestic livestock farms was reported in South Africa during January-August 2010. The first cases occurred after heavy rainfall, and the virus subsequently spread countrywide. To determine the possible effect of environmental conditions and vaccination on RVF virus transmissibility, we estimated the effective reproduction number (R) for the virus over the course of the epidemic by extending the Wallinga and Teunis algorithm with spatial information. Re reached its highest value in mid-February and fell below unity around mid-March, when vaccination coverage was 7.5%-45.7% and vector-suitable environmental conditions were maintained. The epidemic fade-out likely resulted first from the immunization of animals following natural infection or vaccination. The decline in vector-suitable environmental conditions from April onwards and further vaccination helped maintain R below unity. Increased availability of vaccine use data would enable evaluation of the effect of RVF vaccination campaigns.

Modelling the dynamics of dengue real epidemics

Conselho Nacional de Desenvolvimento Científico e Tecnológico (CNPq)

Electron interaction with gel dosimeters : effective atomic numbers for collisional, radiative and total interaction processes

The effective atomic number is widely employed in radiation studies, particularly for the characterisation of interaction processes in dosimeters, biological tissues and substitute materials. Gel dosimeters are unique in that they comprise both the phantom and dosimeter material. In this work, effective atomic numbers for total and partial electron interaction processes have been calculated for the first time for a Fricke gel dosimeter, five hypoxic and nine normoxic polymer gel dosimeters. A range of biological materials are also presented for comparison. The spectrum of energies studied spans 10 keV to 100 MeV, over which the effective atomic number varies by 30 %. The effective atomic numbers of gels match those of soft tissue closely over the full energy range studied; greater disparities exist at higher energies but are typically within 4 %.

A dimensionless number for understanding the evolutionary dynamics of antigenically variable RNA viruses.

Antigenically variable RNA viruses are significant contributors to the burden of infectious disease worldwide. One reason for their ubiquity is their ability to escape herd immunity through rapid antigenic evolution and thereby to reinfect previously infected hosts. However, the ways in which these viruses evolve antigenically are highly diverse. Some have only limited diversity in the long-run, with every emergence of a new antigenic variant coupled with a replacement of the older variant. Other viruses rapidly accumulate antigenic diversity over time. Others still exhibit dynamics that can be considered evolutionary intermediates between these two extremes. Here, we present a theoretical framework that aims to understand these differences in evolutionary patterns by considering a virus's epidemiological dynamics in a given host population. Our framework, based on a dimensionless number, probabilistically anticipates patterns of viral antigenic diversification and thereby quantifies a virus's evolutionary potential. It is therefore similar in spirit to the basic reproduction number, the well-known dimensionless number which quantifies a pathogen's reproductive potential. We further outline how our theoretical framework can be applied to empirical viral systems, using influenza A/H3N2 as a case study. We end with predictions of our framework and work that remains to be done to further integrate viral evolutionary dynamics with disease ecology.

A comparative analysis of the relative efficacy of vector-control strategies against dengue fever

Dengue is considered one of the most important vector-borne infection, affecting almost half of the world population with 50 to 100 million cases every year. In this paper, we present one of the simplest models that can encapsulate all the important variables related to vector control of dengue fever. The model considers the human population, the adult mosquito population and the population of immature stages, which includes eggs, larvae and pupae. The model also considers the vertical transmission of dengue in the mosquitoes and the seasonal variation in the mosquito population. From this basic model describing the dynamics of dengue infection, we deduce thresholds for avoiding the introduction of the disease and for the elimination of the disease. In particular, we deduce a Basic Reproduction Number for dengue that includes parameters related to the immature stages of the mosquito. By neglecting seasonal variation, we calculate the equilibrium values of the model’s variables. We also present a sensitivity analysis of the impact of four vector-control strategies on the Basic Reproduction Number, on the Force of Infection and on the human prevalence of dengue. Each of the strategies was studied separately from the others. The analysis presented allows us to conclude that of the available vector control strategies, adulticide application is the most effective, followed by the reduction of the exposure to mosquito bites, locating and destroying breeding places and, finally, larvicides. Current vector-control methods are concentrated on mechanical destruction of mosquitoes’ breeding places. Our results suggest that reducing the contact between vector and hosts (biting rates) is as efficient as the logistically difficult but very efficient adult mosquito’s control.

Resurgence of HIV infection among men who have sex with men in Switzerland: mathematical modelling study

Background New HIV infections in men who have sex with men (MSM) have increased in Switzerland since 2000 despite combination antiretroviral therapy (cART). The objectives of this mathematical modelling study were: to describe the dynamics of the HIV epidemic in MSM in Switzerland using national data; to explore the effects of hypothetical prevention scenarios; and to conduct a multivariate sensitivity analysis. Methodology/Principal Findings The model describes HIV transmission, progression and the effects of cART using differential equations. The model was fitted to Swiss HIV and AIDS surveillance data and twelve unknown parameters were estimated. Predicted numbers of diagnosed HIV infections and AIDS cases fitted the observed data well. By the end of 2010, an estimated 13.5% (95% CI 12.5, 14.6%) of all HIV-infected MSM were undiagnosed and accounted for 81.8% (95% CI 81.1, 82.4%) of new HIV infections. The transmission rate was at its lowest from 1995–1999, with a nadir of 46 incident HIV infections in 1999, but increased from 2000. The estimated number of new infections continued to increase to more than 250 in 2010, although the reproduction number was still below the epidemic threshold. Prevention scenarios included temporary reductions in risk behaviour, annual test and treat, and reduction in risk behaviour to levels observed earlier in the epidemic. These led to predicted reductions in new infections from 2 to 26% by 2020. Parameters related to disease progression and relative infectiousness at different HIV stages had the greatest influence on estimates of the net transmission rate. Conclusions/Significance The model outputs suggest that the increase in HIV transmission amongst MSM in Switzerland is the result of continuing risky sexual behaviour, particularly by those unaware of their infection status. Long term reductions in the incidence of HIV infection in MSM in Switzerland will require increased and sustained uptake of effective interventions.

Case and partnership reproduction numbers for a curable sexually transmitted infection

Sexually transmitted infections (STIs) are, by definition, transmitted between sexual partners. For curable STIs an infected index case can potentially re-infect the same partner multiple times. Thus, R0, the average number of secondary infections one typical infected individual will produce during his or her infectious period is not necessarily the same as the average number of secondary cases (infected persons). Here we introduce the new concept of the case reproduction number (Rc). In addition, we define the partnership reproduction number (Rp) as the average number of secondary partnerships consisting of two infected individuals one typical infected individual will produce over his or her infectious lifetime. Rp takes into account clearance and re-infection within partnerships, which results in a prolongation of the duration of the infectious period. The two new reproduction numbers were derived for a deterministic pair model with serial monogamous partnerships using infection parameters for Chlamydia trachomatis, an example of a curable STI. We showed that re-infection within partnerships means that curable STIs can be sustained endemically even when the average number of secondary cases a person produces during his or her infectious period is below one.

On the time to reach a critical number of infections in epidemic models with infective and susceptible immigrants.

In this paper we examine the time T to reach a critical number K0 of infections during an outbreak in an epidemic model with infective and susceptible immigrants. The underlying process X, which was first introduced by Ridler-Rowe (1967), is related to recurrent diseases and it appears to be analytically intractable. We present an approximating model inspired from the use of extreme values, and we derive formulae for the Laplace-Stieltjes transform of T and its moments, which are evaluated by using an iterative procedure. Numerical examples are presented to illustrate the effects of the contact and removal rates on the expected values of T and the threshold K0, when the initial time instant corresponds to an invasion time. We also study the exact reproduction number Rexact,0 and the population transmission number Rp, which are random versions of the basic reproduction number R0.

A simple model for the establishment of tick-borne pathogens of Ixodes scapularis : a global sensitivity analysis of R0

The basic reproduction number of a pathogen, R 0, determines whether a pathogen will spread (R0>1R 0>1), when introduced into a fully susceptible population or fade out (R0<1R 0<1), because infected hosts do not, on average, replace themselves. In this paper we develop a simple mechanistic model for the basic reproduction number for a group of tick-borne pathogens that wholly, or almost wholly, depend on horizontal transmission to and from vertebrate hosts. This group includes the causative agent of Lyme disease, Borrelia burgdorferi, and the causative agent of human babesiosis, Babesia microti, for which transmission between co-feeding ticks and vertical transmission from adult female ticks are both negligible. The model has only 19 parameters, all of which have a clear biological interpretation and can be estimated from laboratory or field data. The model takes into account the transmission efficiency from the vertebrate host as a function of the days since infection, in part because of the potential for this dynamic to interact with tick phenology, which is also included in the model. This sets the model apart from previous, similar models for R0 for tick-borne pathogens. We then define parameter ranges for the 19 parameters using estimates from the literature, as well as laboratory and field data, and perform a global sensitivity analysis of the model. This enables us to rank the importance of the parameters in terms of their contribution to the observed variation in R0. We conclude that the transmission efficiency from the vertebrate host to Ixodes scapularis ticks, the survival rate of Ixodes scapularis from fed larva to feeding nymph, and the fraction of nymphs finding a competent host, are the most influential factors for R0. This contrasts with other vector borne pathogens where it is usually the abundance of the vector or host, or the vector-to-host ratio, that determine conditions for emergence. These results are a step towards a better understanding of the geographical expansion of currently emerging horizontally transmitted tick-borne pathogens such as Babesia microti, as well as providing a firmer scientific basis for targeted use of acaricide or the application of wildlife vaccines that are currently in development.