Avoidable errors in the modelling of outbreaks of emerging pathogens, with special reference to Ebola

As an emergent infectious disease outbreak unfolds, public health response is reliant on information on key epidemiological quantities, such as transmission potential and serial interval. Increasingly, transmission models fit to incidence data are used to estimate these parameters and guide policy. Some widely used modelling practices lead to potentially large errors in parameter estimates and, consequently, errors in model-based forecasts. Even more worryingly, in such situations, confidence in parameter estimates and forecasts can itself be far overestimated, leading to the potential for large errors that mask their own presence. Fortunately, straightforward and computationally inexpensive alternatives exist that avoid these problems. Here, we first use a simulation study to demonstrate potential pitfalls of the standard practice of fitting deterministic models to cumulative incidence data. Next, we demonstrate an alternative based on stochastic models fit to raw data from an early phase of 2014 West Africa Ebola virus disease outbreak. We show not only that bias is thereby reduced, but that uncertainty in estimates and forecasts is better quantified and that, critically, lack of model fit is more readily diagnosed. We conclude with a short list of principles to guide the modelling response to future infectious disease outbreaks.     

Endemic persistence or disease extinction: The effect of separation into sub-communities

Consider an infectious disease which is endemic in a population divided into several large sub-communities that interact. Our aim is to understand how the time to extinction is affected by the level of interaction between communities. We present two approximations of the expected time to extinction in a population consisting of a small number of large sub-communities. These approximations are described for an SIR epidemic model, with focus on diseases with short infectious period in relation to life length, such as childhood diseases. Both approximations are based on Markov jump processes. Simulations indicate that the time to extinction is increasing in the degree of interaction between communities. This behaviour can also be seen in our approximations in relevant regions of the parameter space.     

Bayesian inference for stochastic epidemic models with time-inhomogeneous removal rates

Stochastic compartmental models of the SEIR type are often used to make inferences on epidemic processes from partially observed data in which only removal times are available. For many epidemics, the assumption of constant removal rates is not plausible. We develop methods for models in which these rates are a time-dependent step function. A reversible jump MCMC algorithm is described that permits Bayesian inferences to be made on model parameters, particularly those associated with the step function. The method is applied to two datasets on outbreaks of smallpox and a respiratory disease. The analyses highlight the importance of allowing for time dependence by contrasting the predictive distributions for the removal times and comparing them with the observed data.      

Modeling the simple epidemic with deterministic differential equations and random initial conditions

In a simple epidemic the only transition in the population is from susceptible to infected and the total population size is fixed for all time. This paper investigates the effect of random initial conditions on the deterministic model for the simple epidemic. By assuming a Beta distribution on the initial proportion of susceptibles, we define a distribution that describes the proportion of susceptibles in a population at any time during an epidemic. The mean and variance for this distribution are derived as hypergeometric functions, and the behavior of these functions is investigated. Lastly, we define a distribution to describe the time until a given proportion of the population remains susceptible. A method for finding the quantiles of this distribution is developed and used to make confidence statements regarding the time until a given proportion of the population is susceptible.     

Phylogenetic assessment of filoviruses: how many lineages of Marburg virus?

Filoviruses have to date been considered as consisting of one diverse genus (Ebola viruses) and one undifferentiated genus (Marburg virus). We reconsider this idea by means of detailed phylogenetic analyses of sequence data available for the Filoviridae: using coalescent simulations, we ascertain that two Marburg isolates (termed the "RAVN" strain) represent a quite-distinct lineage that should be considered in studies of biogeography and host associations, and may merit recognition at the level of species. In contrast, filovirus isolates recently obtained from bat tissues are not distinct from previously known strains, and should be considered as drawn from the same population. Implications for understanding the transmission geography and host associations of these viruses are discussed.     

A new SEIR epidemic model with applications to the theory of eradication and control of diseases, and to the calculation of R0

We present a novel SEIR (susceptible-exposure-infective-recovered) model that is suitable for modeling the eradication of diseases by mass vaccination or control of diseases by case isolation combined with contact tracing, incorporating the vaccine efficacy or the control efficacy into the model. Moreover, relying on this novel SEIR model and some probabilistic arguments, we have found four formulas that are suitable for estimating the basic reproductive numbers R(0) in terms of the ratio of the mean infectious period to the mean latent period of a disease. The ranges of R(0) for most known diseases, that are calculated by our formulas, coincide very well with the values of R(0) estimated by the usual method of fitting the models to observed data.     

Managing the 2004/05 anthrax outbreak in Queen Elizabeth and Lake Mburo National Parks, Uganda

In August 2004, hippo mortality in the waters of Kazinga Channel, Lakes George and Edward within Queen Elizabeth National Park (QENP) was observed. Veterinary investigation confirmed the disease killing hippos to be anthrax, using clinical, postmortem and laboratory diagnosis, including the polymerase chain reaction technique. Anthrax is believed to have occurred in QENP in 1959, 1962 and 1991 amongst Hippopotamus amphibious but these was not as devastating as the outbreak of 2004–2005. During the outbreak, 306 hippopotami representing 11.63%, 63 zebras representing 1.47%, 60 buffaloes representing 0.9%, thirteen warthogs representing 0.69%, twelve kobs representing 0.07%, three waterbucks representing 0.09% and five elephants representing 0.02% died. A multisectoral National Taskforce was set up, to among other things contain the disease at source and halt its spread. Carcass disposal by burying and burning, decontamination of disposal sites by 10% formaldehyde, ring vaccination of cattle and sheep using blanthrax vaccine and community sensitization, were carried out by the taskforce. A surveillance programme is in place.     

Niemann-Pick C1 Heterogeneity of Bat Cells Controls Filovirus Tropism

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Stochastic epidemic models: A survey

This paper is a survey paper on stochastic epidemic models. A simple stochastic epidemic model is defined and exact and asymptotic (relying on a large community) properties are presented. The purpose of modelling is illustrated by studying effects of vaccination and also in terms of inference procedures for important parameters, such as the basic reproduction number and the critical vaccination coverage. Several generalizations towards realism, e.g. multitype and household epidemic models, are also presented, as is a model for endemic diseases.     

A new approach for designing disease intervention strategies in metapopulation models

We describe a new approach for investigating the control strategies of compartmental disease transmission models. The method rests on the construction of various alternative next-generation matrices, and makes use of the type reproduction number and the target reproduction number. A general metapopulation SIRS (susceptible–infected–recovered–susceptible) model is given to illustrate the application of the method. Such model is useful to study a wide variety of diseases where the population is distributed over geographically separated regions. Considering various control measures such as vaccination, social distancing, and travel restrictions, the procedure allows us to precisely describe in terms of the model parameters, how control methods should be implemented in the SIRS model to ensure disease elimination. In particular, we characterize cases where changing only the travel rates between the regions is sufficient to prevent an outbreak.