«Abstract Ebola is a deadly virus that causes frequent disease outbreaks in the human population. Here, we analyse its rate of new introductions, case ...»
Epidemiological Dynamics of Ebola Outbreaks
Ebola is a deadly virus that causes frequent disease outbreaks in the human population. Here,
we analyse its rate of new introductions, case fatality ratio, and potential to spread from person
to person. The analysis is performed for all completed outbreaks, and for a scenario where
these are augmented by a more severe outbreak of several thousand cases. The results show a
fast rate of new outbreaks, a high case fatality ratio, and an eﬀective reproductive ratio of just less than 1.
Introduction Ebola virus disease is an often-fatal disease of humans that is not vaccine-preventable and has no speciﬁc treatment. A total of 25 outbreaks, believed to have arisen due to zoonotic transmission from wild mammals, have occurred since the ﬁrst observed cases in humans in 1976 . The current epidemic is the largest to date . This gives particular urgency to quantitative estimation of epidemiological quantities relevant to Ebola such as case fatality ratio, timing of new outbreaks, and the strength of human-to-human transmission.
The most important epidemiological quantity to estimate for an infectious disease is typically the basic reproductive ratio, R0, deﬁned as the expected number of secondary cases produced per primary case early in the epidemic . When R0 is greater than 1, the expectation is that a new epidemic will eventually infect a signiﬁcant percentage of the population if it is not stopped by interventions or chance extinction; conversely, when R0 is less than 1, chance events may lead to a large number of cases, but these are always expected to be much less numerous than the total population size.
Previous attempts to estimate R0 for Ebola have found values between 1.34 and 3.65 by ﬁtting compartmental epidemic models to the incidence over time of the large outbreaks in the Democratic Republic of Congo in 1995 and Uganda in 2000 [4, 5, 6], with similar results obtained for the ongoing outbreak . This leads to the question of why all completed outbreaks numbered at most several hundreds, with the typical answer being that the medical and social response to an outbreak reduces transmission, leading to an eﬀective reproductive ratio Rt R0 [4, 6], although it is also important to note that heterogeneity in transmission can lead to extremely high probabilities of an outbreak becoming extinct even if Rt is slightly greater than 1 .
∗ Warwick Infectious Disease Epidemiology Research Centre (WIDER) and Warwick Mathematics Institute, University of Warwick, Coventry, CV4 7AL, UK. T.A.House@warwick.ac.uk Methods Description Here, a diﬀerent approach is taken based on using the time between outbreaks, number of deaths, and ﬁnal number of cases, for all 24 completed Ebola outbreaks reported by the World Health Organisation . Full mathematical details of the approach are given below.
First, we model the start of new outbreaks as a ‘memoryless’ Poisson process with a rate λ. Secondly, we assume that each new outbreak has a case fatality ratio (CFR – the probability that a case
will die) picked from a Beta distribution. Thirdly, the ﬁnal size model involves two components:
(i) A geometrically distributed number of cases, A, which includes cases arising from animal-tohuman and pre-control transmission; (ii) A branching process model of human-to-human transmission [9, 10], whose oﬀspring distribution has mean Rt, generating Z cases. The ﬁnal size is then K = A + Z|A. This quantity should be interpreted as arising from a combination of Rt, R0 and timing of interventions.
Bayesian MCMC with uninformative priors was used to ﬁt all models . Since doubts have been raised in the literature about the use of ﬁnal size data for emerging diseases , a simulation study was also performed to test identiﬁability, although a recent study by Blumberg and LloydSmith  of joint identiﬁability of two parameters in a related model is also highly relevant in this context.
Finally, the ﬁnal size data were augmented by an outbreak of unknown size in the range 1000with mathematical details given by Eq. (5) below) and the model was reﬁtted. Due to the signiﬁcant uncertainty in the severity of the current outbreak, this is not intended to be a realtime analysis, but rather to show how the modelling approach responds to such a scenario in general.
Technical details Transmission model
New outbreak model We model the start of new outbreaks in the human population as a Poisson process of rate λ. If the time period over which N outbreaks is observed is T years, then the likelihood is
We estimate α = 6.1[2.8, 11] and β = 3.1[1.5, 5.9], with posterior distributions given in Figure 1D,E.
Statistical methodology The MCMC methodology used was Random-walk Metropolis Hastings with thinning to produce 103 uncorrelated samples, with each posterior ultimately derived from one long chain. The parameter spaces involved are low-dimensional enough that large-scale sweeps can be performed to check for multimodality, which was not observed, and convergence of the chains was observed to be fast and independent of initial conditions.
For the simulation study, the real-time incidence curves of Figure 3A are produced by modelling the geometric distributions as arising from Poissonian transmission with exponentially distributed rates. The times between new introductions are not explicitly modelled or shown.
Figure 1 shows the results of ﬁtting to times between outbreaks, with 1A showing the empirical distribution of times between outbreaks together with the ﬁtted model distribution that has mean 1.49[1.02, 2.24] years between outbreaks, and 1C showing the posterior for the rate parameter.
Figure 1 also shows the results of ﬁtting CFR to number of deaths and ﬁnal size, with 1B showing empirical CFRs for diﬀerent outbreaks together with the ﬁtted model distribution. Other plots in Figure 1 (D,E) show the posteriors for the Beta distribution parameters.
Figure 2 shows the results of ﬁtting to completed outbreaks, with 2A,B giving the ﬁtted distribution against data, 2C showing the posterior for the reproductive ratio, which is estimated to be Rt = 0.88[0.64, 0.96]. 2D shows the posterior for the geometric parameter, which is estimated to be p = 0.089[0.029, 0.19].
While the model is designed not to depend explicitly on the temporal dynamics of Ebola virus disease, Figure 3A shows a set of 24 outbreaks simulated from a continuous-time Markov chain with the same probability distribution for ﬁnal size as the estimated model. These show behaviour that is typical of near-critical branching processes, which often becoming extinct early, but also often grow to signiﬁcant size before extinction. Figure 3B plots the likelihood surface for these simulated data showing parameter identiﬁability.
Figure 4 shows the results of ﬁtting to completed outbreak ﬁnal sizes augmented by an outbreak of uncertain size in the range 1000-5000. Here 4A gives the ﬁtted distribution against data, and 4B shows the posterior for the probability of the additional outbreak, which is estimated to be 0.023[0.0015, 0.088]. 4C shows the posterior for the reproductive ratio, which is estimated to be Rt = 0.94[0.87, 0.99], and 4D shows the posterior for the geometric parameter, which is estimated to be p = 0.11[0.054, 0.21].
The results obtained point to the following conclusions about Ebola transmission dynamics. (i) The rate of new epidemics and CFR are both high, but with signiﬁcant variability from outbreak to outbreak. (ii) The eﬀective reproductive ratio Rt for person-to-person transmission is just below 1. (iii) There is extremely large variability in the ﬁnal size of outbreaks.
It is also important to consider the sensitivity of these conclusions. A larger ﬁnal size for the current outbreak (but still signiﬁcantly less than the population size of a country) as suggested by the analysis above will tend to lead to a narrower posterior about a value of Rt closer to 1; this can be understood from general properties of branching processes . Such a ﬁnely tuned constant value of Rt would, however, become increasingly diﬃcult to interpret as a fundamental property of the outbreak and a modelling approach in which Rt was allowed to vary in time – along with the public health and behavioural responses – would be preferred.
Also, it is possible that a number of small outbreaks were not recorded by the WHO. This could be addressed through incorporation of additional variability into the model through introduction of explicit overdispersal parameters as in [8, 13], although for the data currently available there was no strong evidence for overdispersal beyond that implied by the geometric distributions.
All of these conclusions suggest no reason for complacency and give support to appeals for greater resources to respond to the ongoing epidemic .
Acknowledgements Work supported by the UK Engineering and Physical Sciences Research Council. I would like to thank Deirdre Hollingsworth, Matt Keeling and Graham Medley for helpful discussions, and the Editors and Reviewers for helpful comments and suggestions.
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Figure 1: Analysis of rate of new outbreaks and case fatality ratio. (A) shows empirical data and 95% CI (black lines) together with ﬁtted distribution and 95% CI (red lines) for rate of new outbreaks. (B) shows empirical data and 95% CI (black lines) together with ﬁtted distribution and 95% CI (red lines) for case fatality ratio. (C) shows the posterior density for rate of new outbreaks λ, while (D,E) show the posterior density for the Beta distribution parameters α and β respectively.
Figure 2: Analysis of transmission dynamics for completed outbreaks. (A, B) Model (solid red line) and 95% CI (dash-dot red line) versus data (black circles) and 95% CI (solid black lines) for diﬀerent axis scales. (C) Posterior for values of the reproductive ratio Rt. (D) Posterior for the geometric parameter p.
A B Figure 3: Simulation study. (A) Real-time model simulations, with change in colour denoting a new outbreak. (B) Likelihood contours (black lines and values multiplied by an unimportant constant) together with parameters used to simulate (red cross) showing that the parameters are identiﬁable from such data.
Figure 4: Analysis of transmission dynamics for completed outbreaks, plus one outbreak of size 1000-5000. (A) Model (solid red line) and 95% CI (dash-dot red line) versus data (black circles) and 95% CI (solid black lines). (B) Posterior for the probability of the large uncertain outbreak.