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.     

Comparing sources of mobility for modelling the epidemic spread of Zika virus in Colombia

Timely, accurate, and comparative data on human mobility is of paramount importance for epidemic preparedness and response, but generally not available or easily accessible. Mobile phone metadata, typically in the form of Call Detail Records (CDRs), represents a powerful source of information on human movements at an unprecedented scale. In this work, we investigate the potential benefits of harnessing aggregated CDR-derived mobility to predict the 2015-2016 Zika virus (ZIKV) outbreak in Colombia, when compared to other traditional data sources. To simulate the spread of ZIKV at sub-national level in Colombia, we employ a stochastic metapopulation epidemic model for vector-borne diseases. Our model integrates detailed data on the key drivers of ZIKV spread, including the spatial heterogeneity of the mosquito abundance, and the exposure of the population to the virus due to environmental and socio-economic factors. Given the same modelling settings (i.e. initial conditions and epidemiological parameters), we perform in-silico simulations for each mobility network and assess their ability in reproducing the local outbreak as reported by the official surveillance data. We assess the performance of our epidemic modelling approach in capturing the ZIKV outbreak both nationally and sub-nationally. Our model estimates are strongly correlated with the surveillance data at the country level (Pearson’s r = 0.92 for the CDR-informed network). Moreover, we found strong performance of the model estimates generated by the CDR-informed mobility networks in reproducing the local outbreak observed at the sub-national level. Compared to the CDR-informed networks, the performance of the other mobility networks is either comparatively similar or substantially lower, with no added value in predicting the local epidemic. This suggests that mobile phone data captures a better picture of human mobility patterns. This work contributes to the ongoing discussion on the value of aggregated mobility estimates from CDRs data that, with appropriate data protection and privacy safeguards, can be used for social impact applications and humanitarian action.     

An Optimal Cost Effectiveness Study on Zimbabwe Cholera Seasonal Data from 2008–2011

Incidence of cholera outbreak is a serious issue in underdeveloped and developing countries. In Zimbabwe, after the massive outbreak in 2008–09, cholera cases and deaths are reported every year from some provinces. Substantial number of reported cholera cases in some provinces during and after the epidemic in 2008–09 indicates a plausible presence of seasonality in cholera incidence in those regions. We formulate a compartmental mathematical model with periodic slow-fast transmission rate to study such recurrent occurrences and fitted the model to cumulative cholera cases and deaths for different provinces of Zimbabwe from the beginning of cholera outbreak in 2008–09 to June 2011. Daily and weekly reported cholera incidence data were collected from Zimbabwe epidemiological bulletin, Zimbabwe Daily cholera updates and Office for the Coordination of Humanitarian Affairs Zimbabwe (OCHA, Zimbabwe). For each province, the basic reproduction number () in periodic environment is estimated. To the best of our knowledge, this is probably a pioneering attempt to estimate in periodic environment using real-life data set of cholera epidemic for Zimbabwe. Our estimates of agree with the previous estimate for some provinces but differ significantly for Bulawayo, Mashonaland West, Manicaland, Matabeleland South and Matabeleland North. Seasonal trend in cholera incidence is observed in Harare, Mashonaland West, Mashonaland East, Manicaland and Matabeleland South. Our result suggests that, slow transmission is a dominating factor for cholera transmission in most of these provinces. Our model projects cholera cases and cholera deaths during the end of the epidemic in 2008–09 to January 1, 2012. We also determine an optimal cost-effective control strategy among the four government undertaken interventions namely promoting hand-hygiene & clean water distribution, vaccination, treatment and sanitation for each province.     

Detecting Presymptomatic Infection Is Necessary to Forecast Major Epidemics in the Earliest Stages of Infectious Disease Outbreaks

We assess how presymptomatic infection affects predictability of infectious disease epidemics. We focus on whether or not a major outbreak (i.e. an epidemic that will go on to infect a large number of individuals) can be predicted reliably soon after initial cases of disease have appeared within a population. For emerging epidemics, significant time and effort is spent recording symptomatic cases. Scientific attention has often focused on improving statistical methodologies to estimate disease transmission parameters from these data. Here we show that, even if symptomatic cases are recorded perfectly, and disease spread parameters are estimated exactly, it is impossible to estimate the probability of a major outbreak without ambiguity. Our results therefore provide an upper bound on the accuracy of forecasts of major outbreaks that are constructed using data on symptomatic cases alone. Accurate prediction of whether or not an epidemic will occur requires records of symptomatic individuals to be supplemented with data concerning the true infection status of apparently uninfected individuals. To forecast likely future behavior in the earliest stages of an emerging outbreak, it is therefore vital to develop and deploy accurate diagnostic tests that can determine whether asymptomatic individuals are actually uninfected, or instead are infected but just do not yet show detectable symptoms.     

Using Combined Diagnostic Test Results to Hindcast Trends of Infection from Cross-Sectional Data

Infectious disease surveillance is key to limiting the consequences from infectious pathogens and maintaining animal and public health. Following the detection of a disease outbreak, a response in proportion to the severity of the outbreak is required. It is thus critical to obtain accurate information concerning the origin of the outbreak and its forward trajectory. However, there is often a lack of situational awareness that may lead to over- or under-reaction. There is a widening range of tests available for detecting pathogens, with typically different temporal characteristics, e.g. in terms of when peak test response occurs relative to time of exposure. We have developed a statistical framework that combines response level data from multiple diagnostic tests and is able to ‘hindcast’ (infer the historical trend of) an infectious disease epidemic. Assuming diagnostic test data from a cross-sectional sample of individuals infected with a pathogen during an outbreak, we use a Bayesian Markov Chain Monte Carlo (MCMC) approach to estimate time of exposure, and the overall epidemic trend in the population prior to the time of sampling. We evaluate the performance of this statistical framework on simulated data from epidemic trend curves and show that we can recover the parameter values of those trends. We also apply the framework to epidemic trend curves taken from two historical outbreaks: a bluetongue outbreak in cattle, and a whooping cough outbreak in humans. Together, these results show that hindcasting can estimate the time since infection for individuals and provide accurate estimates of epidemic trends, and can be used to distinguish whether an outbreak is increasing or past its peak. We conclude that if temporal characteristics of diagnostics are known, it is possible to recover epidemic trends of both human and animal pathogens from cross-sectional data collected at a single point in time.     

Epidemic modelling: Aspects where stochasticity matters

Epidemic models are always simplifications of real world epidemics. Which real world features to include, and which simplifications to make, depend both on the disease of interest and on the purpose of the modelling. In the present paper we discuss some such purposes for which a stochastic model is preferable to a deterministic counterpart. The two main examples illustrate the importance of allowing the infectious and latent periods to be random when focus lies on the probability of a large epidemic outbreak and/or on the initial speed, or growth rate, of the epidemic. A consequence of the latter is that estimation of the basic reproduction number R₀ is sensitive to assumptions about the distributions of the infectious and latent periods when using data from the early stages of an outbreak, which we illustrate with data from the H1N1 influenza A pandemic. Some further examples are also discussed as are some practical consequences related to these stochastic aspects.     

A Mathematical Model of Chikungunya Dynamics and Control: The Major Epidemic on Réunion Island

Chikungunya is a re-emerging arboviral disease transmitted by Aedes spp. mosquitoes. Although principally endemic to Africa and Asia, recent outbreaks have occurred in Europe following introductions by returning travellers. A particularly large outbreak occurred on Réunion Island in 2006, the published data from which forms the basis of the current study. A simple, deterministic mathematical model of the transmission of the virus between humans and mosquitoes was constructed and parameterised with the up-to-date literature on infection biology. The model is fitted to the large Réunion epidemic, resulting in an estimate of 4.1 for the type reproduction number of chikungunya. Although simplistic, the model provided a close approximation of both the peak incidence of the outbreak and the final epidemic size. Sensitivity analysis using Monte Carlo simulation demonstrated the strong influence that both the latent period of infection in humans and the pre-patent period have on these two epidemiological outcomes. We show why separating these variables, which are epidemiologically distinct in chikungunya infections, is not only necessary for accurate model fitting but also important in informing control.     

Forecasting the 2001 Foot-and-Mouth Disease Epidemic in the UK

Near real-time epidemic forecasting approaches are needed to respond to the increasing number of infectious disease outbreaks. In this paper, we retrospectively assess the performance of simple phenomenological models that incorporate early sub-exponential growth dynamics to generate short-term forecasts of the 2001 foot-and-mouth disease epidemic in the UK. For this purpose, we employed the generalized-growth model (GGM) for pre-peak predictions and the generalized-Richards model (GRM) for post-peak predictions. The epidemic exhibits a growth-decelerating pattern as the relative growth rate declines inversely with time. The uncertainty of the parameter estimates \( (r{\text{ and }}p) \) narrows down and becomes more precise using an increasing amount of data of the epidemic growth phase. Indeed, using only the first 10–15 days of the epidemic, the scaling of growth parameter (p) displays wide uncertainty with the confidence interval for p ranging from values ~ 0.5 to 1.0, indicating that less than 15 epidemic days of data are not sufficient to discriminate between sub-exponential (i.e., p < 1) and exponential growth dynamics (i.e., p = 1). By contrast, using 20, 25, or 30 days of epidemic data, it is possible to recover estimates of p around 0.6 and the confidence interval is substantially below the exponential growth regime. Local and national bans on the movement of livestock and a nationwide cull of infected and contiguous premises likely contributed to the decelerating trajectory of the epidemic. The GGM and GRM provided useful 10-day forecasts of the epidemic before and after the peak of the epidemic, respectively. Short-term forecasts improved as the model was calibrated with an increasing length of the epidemic growth phase. Phenomenological models incorporating generalized-growth dynamics are useful tools to generate short-term forecasts of epidemic growth in near real time, particularly in the context of limited epidemiological data as well as information about transmission mechanisms and the effects of control interventions.     

Inferring the causes of the three waves of the 1918 influenza pandemic in England and Wales

Past influenza pandemics appear to be characterized by multiple waves of incidence, but the mechanisms that account for this phenomenon remain unclear. We propose a simple epidemic model, which incorporates three factors that might contribute to the generation of multiple waves: (i) schools opening and closing, (ii) temperature changes during the outbreak, and (iii) changes in human behaviour in response to the outbreak. We fit this model to the reported influenza mortality during the 1918 pandemic in 334 UK administrative units and estimate the epidemiological parameters. We then use information criteria to evaluate how well these three factors explain the observed patterns of mortality. Our results indicate that all three factors are important but that behavioural responses had the largest effect. The parameter values that produce the best fit are biologically reasonable and yield epidemiological dynamics that match the observed data well.     

Epidemic growth rate and household reproduction number in communities of households,schools and workplaces

In this paper we present a novel and coherent modelling framework for the characterisation of the real-time growth rate in SIR models of epidemic spread in populations with social structures of increasing complexity. Known results about homogeneous mixing and multitype models are included in the framework, which is then extended to models with households and models with households and schools/workplaces. Efficient methods for the exact computation of the real-time growth rate are presented for the standard SIR model with constant infection and recovery rates (Markovian case). Approximate methods are described for a large class of models with time-varying infection rates (non-Markovian case). The quality of the approximation is assessed via comparison with results from individual-based stochastic simulations. The methodology is then applied to the case of influenza in models with households and schools/workplaces, to provide an estimate of a household-to-household reproduction number and thus asses the effort required to prevent an outbreak by targeting control policies at the level of households. The results highlight the risk of underestimating such effort when the additional presence of schools/workplaces is neglected. Our framework increases the applicability of models of epidemic spread in socially structured population by linking earlier theoretical results, mainly focused on time-independent key epidemiological parameters (e.g. reproduction numbers, critical vaccination coverage, epidemic final size) to new results on the epidemic dynamics.     

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.     

Quantification of Bird-to-Bird and Bird-to-Human Infections during 2013 Novel H7N9 Avian Influenza Outbreak in China

From February to May, 2013, 132 human avian influenza H7N9 cases were identified in China resulting in 37 deaths. We developed a novel, simple and effective compartmental modeling framework for transmissions among (wild and domestic) birds as well as from birds to human, to infer important epidemiological quantifiers, such as basic reproduction number for bird epidemic, bird-to-human infection rate and turning points of the epidemics, for the epidemic via human H7N9 case onset data and to acquire useful information regarding the bird-to-human transmission dynamics. Estimated basic reproduction number for infections among birds is 4.10 and the mean daily number of human infections per infected bird is 3.16*10⁻⁵ [3.08*10⁻⁵, 3.23*10⁻⁵]. The turning point of 2013 H7N9 epidemic is pinpointed at April 16 for bird infections and at April 9 for bird-to-human transmissions. Our result reveals very low level of bird-to-human infections, thus indicating minimal risk of widespread bird-to-human infections of H7N9 virus during the outbreak. Moreover, the turning point of the human epidemic, pinpointed at shortly after the implementation of full-scale control and intervention measures initiated in early April, further highlights the impact of timely actions on ending the outbreak. This is the first study where both the bird and human components of an avian influenza epidemic can be quantified using only the human case data.     

A mechanistic and data-driven reconstruction of the time-varying reproduction number: Application to the COVID-19 epidemic

The effective reproduction number R_eff is a critical epidemiological parameter that characterizes the transmissibility of a pathogen. However, this parameter is difficult to estimate in the presence of silent transmission and/or significant temporal variation in case reporting. This variation can occur due to the lack of timely or appropriate testing, public health interventions and/or changes in human behavior during an epidemic. This is exactly the situation we are confronted with during this COVID-19 pandemic. In this work, we propose to estimate R_eff for the SARS-CoV-2 (the etiological agent of the COVID-19), based on a model of its propagation considering a time-varying transmission rate. This rate is modeled by a Brownian diffusion process embedded in a stochastic model. The model is then fitted by Bayesian inference (particle Markov Chain Monte Carlo method) using multiple well-documented hospital datasets from several regions in France and in Ireland. This mechanistic modeling framework enables us to reconstruct the temporal evolution of the transmission rate of the COVID-19 based only on the available data. Except for the specific model structure, it is non-specifically assumed that the transmission rate follows a basic stochastic process constrained by the observations. This approach allows us to follow both the course of the COVID-19 epidemic and the temporal evolution of its R_eff(t). Besides, it allows to assess and to interpret the evolution of transmission with respect to the mitigation strategies implemented to control the epidemic waves in France and in Ireland. We can thus estimate a reduction of more than 80% for the first wave in all the studied regions but a smaller reduction for the second wave when the epidemic was less active, around 45% in France but just 20% in Ireland. For the third wave in Ireland the reduction was again significant (>70%).     

Inference of Epidemiological Dynamics Based on Simulated Phylogenies Using Birth-Death and Coalescent Models

Quantifying epidemiological dynamics is crucial for understanding and forecasting the spread of an epidemic. The coalescent and the birth-death model are used interchangeably to infer epidemiological parameters from the genealogical relationships of the pathogen population under study, which in turn are inferred from the pathogen genetic sequencing data. To compare the performance of these widely applied models, we performed a simulation study. We simulated phylogenetic trees under the constant rate birth-death model and the coalescent model with a deterministic exponentially growing infected population. For each tree, we re-estimated the epidemiological parameters using both a birth-death and a coalescent based method, implemented as an MCMC procedure in BEAST v2.0. In our analyses that estimate the growth rate of an epidemic based on simulated birth-death trees, the point estimates such as the maximum a posteriori/maximum likelihood estimates are not very different. However, the estimates of uncertainty are very different. The birth-death model had a higher coverage than the coalescent model, i.e. contained the true value in the highest posterior density (HPD) interval more often (2–13% vs. 31–75% error). The coverage of the coalescent decreases with decreasing basic reproductive ratio and increasing sampling probability of infecteds. We hypothesize that the biases in the coalescent are due to the assumption of deterministic rather than stochastic population size changes. Both methods performed reasonably well when analyzing trees simulated under the coalescent. The methods can also identify other key epidemiological parameters as long as one of the parameters is fixed to its true value. In summary, when using genetic data to estimate epidemic dynamics, our results suggest that the birth-death method will be less sensitive to population fluctuations of early outbreaks than the coalescent method that assumes a deterministic exponentially growing infected population.     

Near real-time surveillance of the SARS-CoV-2 epidemic with incomplete data

When responding to infectious disease outbreaks, rapid and accurate estimation of the epidemic trajectory is critical. However, two common data collection problems affect the reliability of the epidemiological data in real time: missing information on the time of first symptoms, and retrospective revision of historical information, including right censoring. Here, we propose an approach to construct epidemic curves in near real time that addresses these two challenges by 1) imputation of dates of symptom onset for reported cases using a dynamically-estimated “backward” reporting delay conditional distribution, and 2) adjustment for right censoring using the NobBS software package to nowcast cases by date of symptom onset. This process allows us to obtain an approximation of the time-varying reproduction number (R_t) in real time. We apply this approach to characterize the early SARS-CoV-2 outbreak in two Spanish regions between March and April 2020. We evaluate how these real-time estimates compare with more complete epidemiological data that became available later. We explore the impact of the different assumptions on the estimates, and compare our estimates with those obtained from commonly used surveillance approaches. Our framework can help improve accuracy, quantify uncertainty, and evaluate frequently unstated assumptions when recovering the epidemic curves from limited data obtained from public health systems in other locations.     

Spatiotemporal Evolution of Ebola Virus Disease at Sub-National Level during the 2014 West Africa Epidemic: Model Scrutiny and Data Meagreness

Background

The Ebola outbreak in West Africa has infected at least 27,443 individuals and killed 11,207, based on data until 24 June, 2015, released by the World Health Organization (WHO). This outbreak has been characterised by extensive geographic spread across the affected countries Guinea, Liberia and Sierra Leone, and by localized hotspots within these countries. The rapid recognition and quantitative assessment of localised areas of higher transmission can inform the optimal deployment of public health resources.

Methods

A variety of mathematical models have been used to estimate the evolution of this epidemic, and some have pointed out the importance of the spatial heterogeneity apparent from incidence maps. However, little is known about the district-level transmission. Given that many response decisions are taken at sub-national level, the current study aimed to investigate the spatial heterogeneity by using a different modelling framework, built on publicly available data at district level. Furthermore, we assessed whether this model could quantify the effect of intervention measures and provide predictions at a local level to guide public health action. We used a two-stage modelling approach: a) a flexible spatiotemporal growth model across all affected districts and b) a deterministic SEIR compartmental model per district whenever deemed appropriate.

Findings

Our estimates show substantial differences in the evolution of the outbreak in the various regions of Guinea, Liberia and Sierra Leone, illustrating the importance of monitoring the outbreak at district level. We also provide an estimate of the time-dependent district-specific effective reproduction number, as a quantitative measure to compare transmission between different districts and give input for informed decisions on control measures and resource allocation. Prediction and assessing the impact of control measures proved to be difficult without more accurate data. In conclusion, this study provides us a useful tool at district level for public health, and illustrates the importance of collecting and sharing data.     

Spatial spread of an epidemic through public transportation systems with a hub

This article investigates an epidemic spreading among several locations through a transportation system, with a hub connecting these locations. Public transportation is not only a bridge through which infections travel from one location to another but also a place where infections occur since individuals are typically in close proximity to each other due to the limited space in these systems. A mathematical model is constructed to study the spread of an infectious disease through such systems. A variant of the next generation method is proposed and used to provide upper and lower bounds of the basic reproduction number for the model. Our investigation indicates that increasing transportation efficiency, and improving sanitation and ventilation of the public transportation system decrease the chance of an outbreak occurring. Moreover, discouraging unnecessary travel during an epidemic also decreases the chance of an outbreak. However, reducing travel by infectives while allowing susceptibles to travel may not be enough to avoid an outbreak.     

An explosive outbreak of respiratory streptococcal infection with a droplet transmission mechanism

The data obtained in the study of an explosive outbreak of acute respiratory diseases, tonsillitis and scarlet fever in one of schools in Moscow have made it possible to exclude the alimentary mechanism of its development and to demonstrate the role of the droplet mechanism of transmission in the appearance of its outbreak. The epidemiological analysis of the outbreak has permitted the formulation of the hypothesis on the conditions of the formation and spread of the epidemic variant of the infective agent; this hypothesis corresponds to the available data in literature on the qualitative changes of the infective agent in the course of the epidemic process. The study has shown that the prophylaxis of the explosive outbreaks of respiratory streptococcal infections must be ensured by the system of epidemiological surveillance with timely intervention into the epidemiological process at its early stages.     

Disease Prevention versus Data Privacy: Using Landcover Maps to Inform Spatial Epidemic Models

The availability of epidemiological data in the early stages of an outbreak of an infectious disease is vital for modelers to make accurate predictions regarding the likely spread of disease and preferred intervention strategies. However, in some countries, the necessary demographic data are only available at an aggregate scale. We investigated the ability of models of livestock infectious diseases to predict epidemic spread and obtain optimal control policies in the event of imperfect, aggregated data. Taking a geographic information approach, we used land cover data to predict UK farm locations and investigated the influence of using these synthetic location data sets upon epidemiological predictions in the event of an outbreak of foot-and-mouth disease. When broadly classified land cover data were used to create synthetic farm locations, model predictions deviated significantly from those simulated on true data. However, when more resolved subclass land use data were used, moderate to highly accurate predictions of epidemic size, duration and optimal vaccination and ring culling strategies were obtained. This suggests that a geographic information approach may be useful where individual farm-level data are not available, to allow predictive analyses to be carried out regarding the likely spread of disease. This method can also be used for contingency planning in collaboration with policy makers to determine preferred control strategies in the event of a future outbreak of infectious disease in livestock.     

Distinguishing epidemic waves from disease spillover in a wildlife population

Serengeti lions frequently experience viral outbreaks. In 1994, one-third of Serengeti lions died from canine distemper virus (CDV). Based on the limited epidemiological data available from this period, it has been unclear whether the 1994 outbreak was propagated by lion-to-lion transmission alone or involved multiple introductions from other sympatric carnivore species. More broadly, we do not know whether contacts between lions allow any pathogen with a relatively short infectious period to percolate through the population (i.e. reach epidemic proportions). We built one of the most realistic contact network models for a wildlife population to date, based on detailed behavioural and movement data from a long-term lion study population. The model allowed us to identify previously unrecognized biases in the sparse data from the 1994 outbreak and develop methods for judiciously inferring disease dynamics from typical wildlife samples. Our analysis of the model in light of the 1994 outbreak data strongly suggest that, although lions are sufficiently well connected to sustain epidemics of CDV-like diseases, the 1994 epidemic was fuelled by multiple spillovers from other carnivore species, such as jackals and hyenas.