Inference for nonlinear dynamical systems

A two-stage approach for estimating the parameters of an age-group epidemic model from incidence data

Age-dependent dynamics is an important characteristic of many infectious diseases. Age-group epidemic models describe the infection dynamics in different age-groups by allowing to set distinct parameter values for each. However, such models are highly nonlinear and may have a large number of unknown parameters. Thus, parameter estimation of age-group models, while becoming a fundamental issue for both the scientific study and policy making in infectious diseases, is not a trivial task in practice. In this paper, we examine the estimation of the so-called next-generation matrix using incidence data of a single entire outbreak, and extend the approach to deal with recurring outbreaks. Unlike previous studies, we do not assume any constraints regarding the structure of the matrix. A novel two-stage approach is developed, which allows for efficient parameter estimation from both statistical and computational perspectives. Simulation studies corroborate the ability to estimate accurately the parameters of the model for several realistic scenarios. The model and estimation method are applied to real data of influenza-like-illness in Israel. The parameter estimates of the key relevant epidemiological parameters and the recovered structure of the estimated next-generation matrix are in line with results obtained in previous studies.

Recurring infection with ecologically distinct HPV types can explain high prevalence and diversity

The high prevalence of human papillomavirus (HPV), the most common sexually transmitted infection, arises from the coexistence of over 200 genetically distinct types. Accurately predicting the impact of vaccines that target multiple types requires understanding the factors that determine HPV diversity. The diversity of many pathogens is driven by type-specific or "homologous" immunity, which promotes the spread of variants to which hosts have little immunity. To test for homologous immunity and to identify mechanisms determining HPV transmission, we fitted nonlinear mechanistic models to longitudinal data on genital infections in unvaccinated men. Our results provide no evidence for homologous immunity, instead showing that infection with one HPV type strongly increases the risk of infection with that type for years afterward. For HPV16, the type responsible for most HPV-related cancers, an initial infection increases the 1-year probability of reinfection by 20-fold, and the probability of reinfection remains 14-fold higher 2 years later. This increased risk occurs in both sexually active and celibate men, suggesting that it arises from autoinoculation, episodic reactivation of latent virus, or both. Overall, our results suggest that high HPV prevalence and diversity can be explained by a combination of a lack of homologous immunity, frequent reinfections, weak competition between types, and variation in type fitness between host subpopulations. Because of the high risk of reinfection, vaccinating boys who have not yet been exposed may be crucial to reduce prevalence, but our results suggest that there may also be large benefits to vaccinating previously infected individuals.

A second-order iterated smoothing algorithm

Simulation-based inference for partially observed stochastic dynamic models is currently receiving much attention due to the fact that direct computation of the likelihood is not possible in many practical situations. Iterated filtering methodologies enable maximization of the likelihood function using simulation-based sequential Monte Carlo filters. Doucet et al. (2013) developed an approximation for the first and second derivatives of the log likelihood via simulation-based sequential Monte Carlo smoothing and proved that the approximation has some attractive theoretical properties. We investigated an iterated smoothing algorithm carrying out likelihood maximization using these derivative approximations. Further, we developed a new iterated smoothing algorithm, using a modification of these derivative estimates, for which we establish both theoretical results and effective practical performance. On benchmark computational challenges, this method beat the first-order iterated filtering algorithm. The method's performance was comparable to a recently developed iterated filtering algorithm based on an iterated Bayes map. Our iterated smoothing algorithm and its theoretical justification provide new directions for future developments in simulation-based inference for latent variable models such as partially observed Markov process models.

Modelling and parameter inference of predator-prey dynamics in heterogeneous environments using the direct integral approach

Most bacterial habitats are topographically complex in the micro scale. Important examples include the gastrointestinal and tracheal tracts, and the soil. Although there are myriad theoretical studies that explore the role of spatial structures on antagonistic interactions (predation, competition) among animals, there are many fewer experimental studies that have explored, validated and quantified their predictions. In this study, we experimentally monitored the temporal dynamic of the predatory bacterium Bdellovibrio bacteriovorus, and its prey, the bacterium Burkholderia stabilis in a structured habitat consisting of sand under various regimes of wetness. We constructed a dynamic model, and estimated its parameters by further developing the direct integral method, a novel estimation procedure that exploits the separability of the states and parameters in the model. We also verified that one of our parameter estimates was consistent with its known, directly measured value from the literature. The ability of the model to fit the data combined with realistic parameter estimates indicate that bacterial predation in the sand can be described by a relatively simple model, and stress the importance of prey refuge on predation dynamics in heterogeneous environments.

Counteracting structural errors in ensemble forecast of influenza outbreaks

For influenza forecasts generated using dynamical models, forecast inaccuracy is partly attributable to the nonlinear growth of error. As a consequence, quantification of the nonlinear error structure in current forecast models is needed so that this growth can be corrected and forecast skill improved. Here, we inspect the error growth of a compartmental influenza model and find that a robust error structure arises naturally from the nonlinear model dynamics. By counteracting these structural errors, diagnosed using error breeding, we develop a new forecast approach that combines dynamical error correction and statistical filtering techniques. In retrospective forecasts of historical influenza outbreaks for 95 US cities from 2003 to 2014, overall forecast accuracy for outbreak peak timing, peak intensity and attack rate, are substantially improved for predicted lead times up to 10 weeks. This error growth correction method can be generalized to improve the forecast accuracy of other infectious disease dynamical models.

Inaccuracy of influenza forecasts based on dynamical models is partly due to nonlinear error growth. Here the authors address the error structure of a compartmental influenza model, and develop a new improved forecast approach combining dynamical error correction and statistical filtering techniques.

tsiR: An R package for time-series Susceptible-Infected-Recovered models of epidemics

tsiR is an open source software package implemented in the R programming language designed to analyze infectious disease time-series data. The software extends a well-studied and widely-applied algorithm, the time-series Susceptible-Infected-Recovered (TSIR) model, to infer parameters from incidence data, such as contact seasonality, and to forward simulate the underlying mechanistic model. The tsiR package aggregates a number of different fitting features previously described in the literature in a user-friendly way, providing support for their broader adoption in infectious disease research. Also included in tsiR are a number of diagnostic tools to assess the fit of the TSIR model. This package should be useful for researchers analyzing incidence data for fully-immunizing infectious diseases.

An ensemble approach to predicting the impact of vaccination on rotavirus disease in Niger

Recently developed vaccines provide a new way of controlling rotavirus in sub-Saharan Africa. Models for the transmission dynamics of rotavirus are critical both for estimating current burden from imperfect surveillance and for assessing potential effects of vaccine intervention strategies. We examine rotavirus infection in the Maradi area in southern Niger using hospital surveillance data provided by Epicentre collected over two years. Additionally, a cluster survey of households in the region allows us to estimate the proportion of children with diarrhea who consulted at a health structure. Model fit and future projections are necessarily particular to a given model; thus, where there are competing models for the underlying epidemiology an ensemble approach can account for that uncertainty. We compare our results across several variants of Susceptible-Infectious-Recovered (SIR) compartmental models to quantify the impact of modeling assumptions on our estimates. Model-specific parameters are estimated by Bayesian inference using Markov chain Monte Carlo. We then use Bayesian model averaging to generate ensemble estimates of the current dynamics, including estimates of R0, the burden of infection in the region, as well as the impact of vaccination on both the short-term dynamics and the long-term reduction of rotavirus incidence under varying levels of coverage. The ensemble of models predicts that the current burden of severe rotavirus disease is 2.6–3.7% of the population each year and that a 2-dose vaccine schedule achieving 70% coverage could reduce burden by 39–42%.

Forecasting infectious disease emergence subject to seasonal forcing

BACKGROUND

Despite high vaccination coverage, many childhood infections pose a growing threat to human populations. Accurate disease forecasting would be of tremendous value to public health. Forecasting disease emergence using early warning signals (EWS) is possible in non-seasonal models of infectious diseases. Here, we assessed whether EWS also anticipate disease emergence in seasonal models.

METHODS

We simulated the dynamics of an immunizing infectious pathogen approaching the tipping point to disease endemicity. To explore the effect of seasonality on the reliability of early warning statistics, we varied the amplitude of fluctuations around the average transmission. We proposed and analyzed two new early warning signals based on the wavelet spectrum. We measured the reliability of the early warning signals depending on the strength of their trend preceding the tipping point and then calculated the Area Under the Curve (AUC) statistic.

RESULTS

Early warning signals were reliable when disease transmission was subject to seasonal forcing. Wavelet-based early warning signals were as reliable as other conventional early warning signals. We found that removing seasonal trends, prior to analysis, did not improve early warning statistics uniformly.

CONCLUSIONS

Early warning signals anticipate the onset of critical transitions for infectious diseases which are subject to seasonal forcing. Wavelet-based early warning statistics can also be used to forecast infectious disease.

Effects of reactive social distancing on the 1918 influenza pandemic

The 1918 influenza pandemic was characterized by multiple epidemic waves. We investigated reactive social distancing, a form of behavioral response where individuals avoid potentially infectious contacts in response to available information on an ongoing epidemic or pandemic. We modelled its effects on the three influenza waves in the United Kingdom. In previous studies, human behavioral response was modelled by a Power function of the proportion of recent influenza mortality in a population, and by a Hill function, which is a function of the number of recent influenza mortality. Using a simple epidemic model with a Power function and one common set of parameters, we provided a good model fit for the observed multiple epidemic waves in London boroughs, Birmingham and Liverpool. We further applied the model parameters from these three cities to all 334 administrative units in England and Wales and including the population sizes of individual administrative units. We computed the Pearson's correlation between the observed and simulated for each administrative unit. We found a median correlation of 0.636, indicating that our model predictions are performing reasonably well. Our modelling approach is an improvement from previous studies where separate models are fitted to each city. With the reduced number of model parameters used, we achieved computational efficiency gain without over-fitting the model. We also showed the importance of reactive behavioral distancing as a potential non-pharmaceutical intervention during an influenza pandemic. Our work has both scientific and public health significance.

Monte Carlo profile confidence intervals for dynamic systems

Monte Carlo methods to evaluate and maximize the likelihood function enable the construction of confidence intervals and hypothesis tests, facilitating scientific investigation using models for which the likelihood function is intractable. When Monte Carlo error can be made small, by sufficiently exhaustive computation, then the standard theory and practice of likelihood-based inference applies. As datasets become larger, and models more complex, situations arise where no reasonable amount of computation can render Monte Carlo error negligible. We develop profile likelihood methodology to provide frequentist inferences that take into account Monte Carlo uncertainty. We investigate the role of this methodology in facilitating inference for computationally challenging dynamic latent variable models. We present examples arising in the study of infectious disease transmission, demonstrating our methodology for inference on nonlinear dynamic models using genetic sequence data and panel time-series data. We also discuss applicability to nonlinear time-series and spatio-temporal data.

Cholera forecast for Dhaka, Bangladesh, with the 2015-2016 El Niño: Lessons learned

A substantial body of work supports a teleconnection between the El Niño-Southern Oscillation (ENSO) and cholera incidence in Bangladesh. In particular, high positive anomalies during the winter (Dec-Feb) in sea surface temperatures (SST) in the tropical Pacific have been shown to exacerbate the seasonal outbreak of cholera following the monsoons from August to November. Climate studies have indicated a role of regional precipitation over Bangladesh in mediating this long-distance effect. Motivated by this previous evidence, we took advantage of the strong 2015-2016 El Niño event to evaluate the predictability of cholera dynamics for the city in recent times based on two transmission models that incorporate SST anomalies and are fitted to the earlier surveillance records starting in 1995. We implemented a mechanistic temporal model that incorporates both epidemiological processes and the effect of ENSO, as well as a previously published statistical model that resolves space at the level of districts (thanas). Prediction accuracy was evaluated with "out-of-fit" data from the same surveillance efforts (post 2008 and 2010 for the two models respectively), by comparing the total number of cholera cases observed for the season to those predicted by model simulations eight to twelve months ahead, starting in January each year. Although forecasts were accurate for the low cholera risk observed for the years preceding the 2015-2016 El Niño, the models also predicted a high probability of observing a large outbreak in fall 2016. Observed cholera cases up to Oct 2016 did not show evidence of an anomalous season. We discuss these predictions in the context of regional and local climate conditions, which show that despite positive regional rainfall anomalies, rainfall and inundation in Dhaka remained low. Possible explanations for these patterns are given together with future implications for cholera dynamics and directions to improve their prediction for the city.

Time series modeling of pathogen-specific disease probabilities with subsampled data

Many diseases arise due to exposure to one of multiple possible pathogens. We consider the situation in which disease counts are available over time from a study region, along with a measure of clinical disease severity, for example, mild or severe. In addition, we suppose a subset of the cases are lab tested in order to determine the pathogen responsible for disease. In such a context, we focus interest on modeling the probabilities of disease incidence given pathogen type. The time course of these probabilities is of great interest as is the association with time-varying covariates such as meteorological variables. In this set up, a natural Bayesian approach would be based on imputation of the unsampled pathogen information using Markov Chain Monte Carlo but this is computationally challenging. We describe a practical approach to inference that is easy to implement. We use an empirical Bayes procedure in a first step to estimate summary statistics. We then treat these summary statistics as the observed data and develop a Bayesian generalized additive model. We analyze data on hand, foot, and mouth disease (HFMD) in China in which there are two pathogens of primary interest, enterovirus 71 (EV71) and Coxackie A16 (CA16). We find that both EV71 and CA16 are associated with temperature, relative humidity, and wind speed, with reasonably similar functional forms for both pathogens. The important issue of confounding by time is modeled using a penalized B-spline model with a random effects representation. The level of smoothing is addressed by a careful choice of the prior on the tuning variance.

A Likelihood Approach for Real-Time Calibration of Stochastic Compartmental Epidemic Models

Stochastic transmission dynamic models are especially useful for studying the early emergence of novel pathogens given the importance of chance events when the number of infectious individuals is small. However, methods for parameter estimation and prediction for these types of stochastic models remain limited. In this manuscript, we describe a calibration and prediction framework for stochastic compartmental transmission models of epidemics. The proposed method, Multiple Shooting for Stochastic systems (MSS), applies a linear noise approximation to describe the size of the fluctuations, and uses each new surveillance observation to update the belief about the true epidemic state. Using simulated outbreaks of a novel viral pathogen, we evaluate the accuracy of MSS for real-time parameter estimation and prediction during epidemics. We assume that weekly counts for the number of new diagnosed cases are available and serve as an imperfect proxy of incidence. We show that MSS produces accurate estimates of key epidemic parameters (i.e. mean duration of infectiousness, R0, and Reff) and can provide an accurate estimate of the unobserved number of infectious individuals during the course of an epidemic. MSS also allows for accurate prediction of the number and timing of future hospitalizations and the overall attack rate. We compare the performance of MSS to three state-of-the-art benchmark methods: 1) a likelihood approximation with an assumption of independent Poisson observations; 2) a particle filtering method; and 3) an ensemble Kalman filter method. We find that MSS significantly outperforms each of these three benchmark methods in the majority of epidemic scenarios tested. In summary, MSS is a promising method that may improve on current approaches for calibration and prediction using stochastic models of epidemics.

Estimating enhanced prevaccination measles transmission hotspots in the context of cross-scale dynamics

A key question in clarifying human-environment interactions is how dynamic complexity develops across integrative scales from molecular to population and global levels. Apart from its public health importance, measles is an excellent test bed for such an analysis. Simple mechanistic models have successfully illuminated measles dynamics at the city and country levels, revealing seasonal forcing of transmission as a major driver of long-term epidemic behavior. Seasonal forcing ties closely to patterns of school aggregation at the individual and community levels, but there are few explicit estimates of school transmission due to the relative lack of epidemic data at this scale. Here, we use data from a 1904 measles outbreak in schools in Woolwich, London, coupled with a stochastic Susceptible-Infected-Recovered model to analyze measles incidence data. Our results indicate that transmission within schools and age classes is higher than previous population-level serological data would suggest. This analysis sheds quantitative light on the role of school-aged children in measles cross-scale dynamics, as we illustrate with references to the contemporary vaccination landscape.

The pertussis enigma: reconciling epidemiology, immunology and evolution

Pertussis, a highly contagious respiratory infection, remains a public health priority despite the availability of vaccines for 70 years. Still a leading cause of mortality in developing countries, pertussis has re-emerged in several developed countries with high vaccination coverage. Resurgence of pertussis in these countries has routinely been attributed to increased awareness of the disease, imperfect vaccinal protection or high infection rates in adults. In this review, we first present 1980-2012 incidence data from 63 countries and show that pertussis resurgence is not universal. We further argue that the large geographical variation in trends probably precludes a simple explanation, such as the transition from whole-cell to acellular pertussis vaccines. Reviewing available evidence, we then propose that prevailing views on pertussis epidemiology are inconsistent with both historical and contemporary data. Indeed, we summarize epidemiological evidence showing that natural infection and vaccination both appear to provide long-term protection against transmission and disease, so that previously infected or vaccinated adults contribute little to overall transmission at a population level. Finally, we identify several promising avenues that may lead to a consistent explanation of global pertussis epidemiology and to more effective control strategies.

Data-driven outbreak forecasting with a simple nonlinear growth model

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We present EpiGro, a simple data-driven method to forecast the scope of an ongoing outbreak.

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We provide general hypotheses for expected model validity and also discuss model limitations.

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We propose an automated parameter estimation method that can be used for forecasting.

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We test our approach on 9 different outbreaks and show robustness over multiple systems and over noisy data sets.

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In the absence of other information or in conjunction with other models, EpiGro may be useful to public health responders.

Recent events have thrown the spotlight on infectious disease outbreak response. We developed a data-driven method, EpiGro, which can be applied to cumulative case reports to estimate the order of magnitude of the duration, peak and ultimate size of an ongoing outbreak. It is based on a surprisingly simple mathematical property of many epidemiological data sets, does not require knowledge or estimation of disease transmission parameters, is robust to noise and to small data sets, and runs quickly due to its mathematical simplicity. Using data from historic and ongoing epidemics, we present the model. We also provide modeling considerations that justify this approach and discuss its limitations. In the absence of other information or in conjunction with other models, EpiGro may be useful to public health responders.

Bayesian non-parametric inference for stochastic epidemic models using Gaussian Processes

This paper considers novel Bayesian non-parametric methods for stochastic epidemic models. Many standard modeling and data analysis methods use underlying assumptions (e.g. concerning the rate at which new cases of disease will occur) which are rarely challenged or tested in practice. To relax these assumptions, we develop a Bayesian non-parametric approach using Gaussian Processes, specifically to estimate the infection process. The methods are illustrated with both simulated and real data sets, the former illustrating that the methods can recover the true infection process quite well in practice, and the latter illustrating that the methods can be successfully applied in different settings.

Stochasticity and the limits to confidence when estimating R0 of Ebola and other emerging infectious diseases

Dynamic models - often deterministic in nature - were used to estimate the basic reproductive number, R0, of the 2014-5 Ebola virus disease (EVD) epidemic outbreak in West Africa. Estimates of R0 were then used to project the likelihood for large outbreak sizes, e.g., exceeding hundreds of thousands of cases. Yet fitting deterministic models can lead to over-confidence in the confidence intervals of the fitted R0, and, in turn, the type and scope of necessary interventions. In this manuscript we propose a hybrid stochastic-deterministic method to estimate R0 and associated confidence intervals (CIs). The core idea is that stochastic realizations of an underlying deterministic model can be used to evaluate the compatibility of candidate values of R0 with observed epidemic curves. The compatibility is based on comparing the distribution of expected epidemic growth rates with the observed epidemic growth rate given "process noise", i.e., arising due to stochastic transmission, recovery and death events. By applying our method to reported EVD case counts from Guinea, Liberia and Sierra Leone, we show that prior estimates of R0 based on deterministic fits appear to be more confident than analysis of stochastic trajectories suggests should be possible. Moving forward, we recommend including process noise among other sources of noise when estimating R0 CIs of emerging epidemics. Our hybrid procedure represents an adaptable and easy-to-implement approach for such estimation.

Fast and accurate dynamic estimation of field effectiveness of meningococcal vaccines

BACKGROUND

Estimating the effectiveness of meningococcal vaccines with high accuracy and precision can be challenging due to the low incidence of the invasive disease, which ranges between 0.5 and 1 cases per 100,000 in Europe and North America. Vaccine effectiveness (VE) is usually estimated with a screening method that combines in one formula the proportion of meningococcal disease cases that have been vaccinated and the proportion of vaccinated in the overall population. Due to the small number of cases, initial point estimates are affected by large uncertainties and several years may be required to estimate VE with a small confidence interval.

METHODS

We used a Monte Carlo maximum likelihood (MCML) approach to estimate the effectiveness of meningococcal vaccines, based on stochastic simulations of a dynamic model for meningococcal transmission and vaccination. We calibrated the model to describe two immunization campaigns: the campaign against MenC in England and the Bexsero campaign that started in the UK in September 2015. First, the MCML method provided estimates for both the direct and indirect effects of the MenC vaccine that were validated against results published in the literature. Then, we assessed the performance of the MCML method in terms of time gain with respect to the screening method under different assumptions of VE for Bexsero.

RESULTS

MCML estimates of VE for the MenC immunization campaign are in good agreement with results based on the screening method and carriage studies, yet characterized by smaller confidence intervals and obtained using only incidence data collected within 2 years of scheduled vaccination. Also, we show that the MCML method could provide a fast and accurate estimate of the effectiveness of Bexsero, with a time gain, with respect to the screening method, that could range from 2 to 15 years, depending on the value of VE measured from field data.

CONCLUSIONS

Results indicate that inference methods based on dynamic computational models can be successfully used to quantify in near real time the effectiveness of immunization campaigns against Neisseria meningitidis. Such an approach could represent an important tool to complement and support traditional observational studies, in the initial phase of a campaign.

Prevention and Control of Zika as a Mosquito-Borne and Sexually Transmitted Disease: A Mathematical Modeling Analysis

The ongoing Zika virus (ZIKV) epidemic in the Americas poses a major global public health emergency. While ZIKV is transmitted from human to human by bites of Aedes mosquitoes, recent evidence indicates that ZIKV can also be transmitted via sexual contact with cases of sexually transmitted ZIKV reported in Argentina, Canada, Chile, France, Italy, New Zealand, Peru, Portugal, and the USA. Yet, the role of sexual transmission on the spread and control of ZIKV infection is not well-understood. We introduce a mathematical model to investigate the impact of mosquito-borne and sexual transmission on the spread and control of ZIKV and calibrate the model to ZIKV epidemic data from Brazil, Colombia, and El Salvador. Parameter estimates yielded a basic reproduction number ₀ = 2.055 (95% CI: 0.523–6.300), in which the percentage contribution of sexual transmission is 3.044% (95% CI: 0.123–45.73). Our sensitivity analyses indicate that ₀ is most sensitive to the biting rate and mortality rate of mosquitoes while sexual transmission increases the risk of infection and epidemic size and prolongs the outbreak. Prevention and control efforts against ZIKV should target both the mosquito-borne and sexual transmission routes.

PREDICTIVE MODELING OF CHOLERA OUTBREAKS IN BANGLADESH

Despite seasonal cholera outbreaks in Bangladesh, little is known about the relationship between environmental conditions and cholera cases. We seek to develop a predictive model for cholera outbreaks in Bangladesh based on environmental predictors. To do this, we estimate the contribution of environmental variables, such as water depth and water temperature, to cholera outbreaks in the context of a disease transmission model. We implement a method which simultaneously accounts for disease dynamics and environmental variables in a Susceptible-Infected-Recovered-Susceptible (SIRS) model. The entire system is treated as a continuous-time hidden Markov model, where the hidden Markov states are the numbers of people who are susceptible, infected, or recovered at each time point, and the observed states are the numbers of cholera cases reported. We use a Bayesian framework to fit this hidden SIRS model, implementing particle Markov chain Monte Carlo methods to sample from the posterior distribution of the environmental and transmission parameters given the observed data. We test this method using both simulation and data from Mathbaria, Bangladesh. Parameter estimates are used to make short-term predictions that capture the formation and decline of epidemic peaks. We demonstrate that our model can successfully predict an increase in the number of infected individuals in the population weeks before the observed number of cholera cases increases, which could allow for early notification of an epidemic and timely allocation of resources.

Revealing the Micro-scale Signature of Endemic Zoonotic Disease Transmission in an African Urban Setting

The development of novel approaches that combine epidemiological and genomic data provides new opportunities to reveal the spatiotemporal dynamics of infectious diseases and determine the processes responsible for their spread and maintenance. Taking advantage of detailed epidemiological time series and viral sequence data from more than 20 years reported by the National Reference Centre for Rabies of Bangui, the capital city of Central African Republic, we used a combination of mathematical modeling and phylogenetic analysis to determine the spatiotemporal dynamics of rabies in domestic dogs as well as the frequency of extinction and introduction events in an African city. We show that although dog rabies virus (RABV) appears to be endemic in Bangui, its epidemiology is in fact shaped by the regular extinction of local chains of transmission coupled with the introduction of new lineages, generating successive waves of spread. Notably, the effective reproduction number during each wave was rarely above the critical value of 1, such that rabies is not self-sustaining in Bangui. In turn, this suggests that rabies at local geographic scales is driven by human-mediated dispersal of RABV among sparsely connected peri-urban and rural areas as opposed to dispersion in a relatively large homogenous urban dog population. This combined epidemiological and genomic approach enables development of a comprehensive framework for understanding disease persistence and informing control measures, indicating that control measures are probably best targeted towards areas neighbouring the city that appear as the source of frequent incursions seeding outbreaks in Bangui.

Differential and enhanced response to climate forcing in diarrheal disease due to rotavirus across a megacity of the developing world

The role of climate forcing in the population dynamics of infectious diseases has typically been revealed via retrospective analyses of incidence records aggregated across space and, in particular, over whole cities. Here, we focus on the transmission dynamics of rotavirus, the main diarrheal disease in infants and young children, within the megacity of Dhaka, Bangladesh. We identify two zones, the densely urbanized core and the more rural periphery, that respond differentially to flooding. Moreover, disease seasonality differs substantially between these regions, spanning variation comparable to the variation from tropical to temperate regions. By combining process-based models with an extensive disease surveillance record, we show that the response to climate forcing is mainly seasonal in the core, where a more endemic transmission resulting from an asymptomatic reservoir facilitates the response to the monsoons. The force of infection in this monsoon peak can be an order of magnitude larger than the force of infection in the more epidemic periphery, which exhibits little or no postmonsoon outbreak in a pattern typical of nearby rural areas. A typically smaller peak during the monsoon season nevertheless shows sensitivity to interannual variability in flooding. High human density in the core is one explanation for enhanced transmission during troughs and an associated seasonal monsoon response in this diarrheal disease, which unlike cholera, has not been widely viewed as climate-sensitive. Spatial demographic, socioeconomic, and environmental heterogeneity can create reservoirs of infection and enhance the sensitivity of disease systems to climate forcing, especially in the populated cities of the developing world.

Seasonality of Influenza A(H7N9) Virus in China-Fitting Simple Epidemic Models to Human Cases

Background

Three epidemic waves of influenza A(H7N9) (hereafter ‘H7N9’) human cases have occurred between March 2013 and July 2015 in China. However, the underlying transmission mechanism remains unclear. Our main objective is to use mathematical models to study how seasonality, secular changes and environmental transmission play a role in the spread of H7N9 in China.

Methods

Data on human cases and chicken cases of H7N9 infection were downloaded from the EMPRES-i Global Animal Disease Information System. We modelled on chicken-to-chicken transmission, assuming a constant ratio of 10⁻⁶ human case per chicken case, and compared the model fit with the observed human cases. We developed three different modified Susceptible-Exposed-Infectious-Recovered-Susceptible models: (i) a non-periodic transmission rate model with an environmental class, (ii) a non-periodic transmission rate model without an environmental class, and (iii) a periodic transmission rate model with an environmental class. We then estimated the key epidemiological parameters and compared the model fit using Akaike Information Criterion and Bayesian Information Criterion.

Results

Our results showed that a non-periodic transmission rate model with an environmental class provided the best model fit to the observed human cases in China during the study period. The estimated parameter values were within biologically plausible ranges.

Conclusions

This study highlighted the importance of considering secular changes and environmental transmission in the modelling of human H7N9 cases. Secular changes were most likely due to control measures such as Live Poultry Markets closures that were implemented during the initial phase of the outbreaks in China. Our results suggested that environmental transmission via viral shedding of infected chickens had contributed to the spread of H7N9 human cases in China.

Evaluation of Argos Telemetry Accuracy in the High-Arctic and Implications for the Estimation of Home-Range Size

Animal tracking through Argos satellite telemetry has enormous potential to test hypotheses in animal behavior, evolutionary ecology, or conservation biology. Yet the applicability of this technique cannot be fully assessed because no clear picture exists as to the conditions influencing the accuracy of Argos locations. Latitude, type of environment, and transmitter movement are among the main candidate factors affecting accuracy. A posteriori data filtering can remove “bad” locations, but again testing is still needed to refine filters. First, we evaluate experimentally the accuracy of Argos locations in a polar terrestrial environment (Nunavut, Canada), with both static and mobile transmitters transported by humans and coupled to GPS transmitters. We report static errors among the lowest published. However, the 68^th error percentiles of mobile transmitters were 1.7 to 3.8 times greater than those of static transmitters. Second, we test how different filtering methods influence the quality of Argos location datasets. Accuracy of location datasets was best improved when filtering in locations of the best classes (LC3 and 2), while the Douglas Argos filter and a homemade speed filter yielded similar performance while retaining more locations. All filters effectively reduced the 68^th error percentiles. Finally, we assess how location error impacted, at six spatial scales, two common estimators of home-range size (a proxy of animal space use behavior synthetizing movements), the minimum convex polygon and the fixed kernel estimator. Location error led to a sometimes dramatic overestimation of home-range size, especially at very local scales. We conclude that Argos telemetry is appropriate to study medium-size terrestrial animals in polar environments, but recommend that location errors are always measured and evaluated against research hypotheses, and that data are always filtered before analysis. How movement speed of transmitters affects location error needs additional research.

Predictability of epidemic malaria under non-stationary conditions with process-based models combining epidemiological updates and climate variability

Background

Previous studies have demonstrated the feasibility of early-warning systems for epidemic malaria informed by climate variability. Whereas modelling approaches typically assume stationary conditions, epidemiological systems are characterized by changes in intervention measures over time, at scales typically longer than inter-epidemic periods. These trends in control efforts preclude simple application of early-warning systems validated by retrospective surveillance data; their effects are also difficult to distinguish from those of climate variability itself.

Methods

Rainfall-driven transmission models for falciparum and vivax malaria are fitted to long-term retrospective surveillance data from four districts in northwest India. Maximum-likelihood estimates (MLEs) of model parameters are obtained for each district via a recently introduced iterated filtering method for partially observed Markov processes. The resulting MLE model is then used to generate simulated yearly forecasts in two different ways, and these forecasts are compared with more recent (out-of-fit) data. In the first approach, initial conditions for generating the predictions are repeatedly updated on a yearly basis, based on the new epidemiological data and the inference method that naturally lends itself to this purpose, given its time-sequential application. In the second approach, the transmission parameters themselves are also updated by refitting the model over a moving window of time.

Results

Application of these two approaches to examine the predictability of epidemic malaria in the different districts reveals differences in the effectiveness of intervention for the two parasites, and illustrates how the ‘failure’ of predictions can be informative to evaluate and quantify the effect of control efforts in the context of climate variability. The first approach performs adequately, and sometimes even better than the second one, when the climate remains the major driver of malaria dynamics, as found for Plasmodium vivax for which an effective clinical intervention is lacking. The second approach offers more skillful forecasts when the dynamics shift over time, as is the case of Plasmodium falciparum in recent years with declining incidence under improved control.

Conclusions

Predictive systems for infectious diseases such as malaria, based on process-based models and climate variables, can be informative and applicable under non-stationary conditions.

Electronic supplementary material

The online version of this article (doi:10.1186/s12936-015-0937-3) contains supplementary material, which is available to authorized users.

The role of influenza in the epidemiology of pneumonia

Interactions arising from sequential viral and bacterial infections play important roles in the epidemiological outcome of many respiratory pathogens. Influenza virus has been implicated in the pathogenesis of several respiratory bacterial pathogens commonly associated with pneumonia. Though clinical evidence supporting this interaction is unambiguous, its population-level effects-magnitude, epidemiological impact and variation during pandemic and seasonal outbreaks-remain unclear. To address these unknowns, we used longitudinal influenza and pneumonia incidence data, at different spatial resolutions and across different epidemiological periods, to infer the nature, timing and the intensity of influenza-pneumonia interaction. We used a mechanistic transmission model within a likelihood-based inference framework to carry out formal hypothesis testing. Irrespective of the source of data examined, we found that influenza infection increases the risk of pneumonia by ~100-fold. We found no support for enhanced transmission or severity impact of the interaction. For model-validation, we challenged our fitted model to make out-of-sample pneumonia predictions during pandemic and non-pandemic periods. The consistency in our inference tests carried out on several distinct datasets, and the predictive skill of our model increase confidence in our overall conclusion that influenza infection substantially enhances the risk of pneumonia, though only for a short period.

Measuring the impact of Ebola control measures in Sierra Leone

Between September 2014 and February 2015, the number of Ebola virus disease (EVD) cases reported in Sierra Leone declined in many districts. During this period, a major international response was put in place, with thousands of treatment beds introduced alongside other infection control measures. However, assessing the impact of the response is challenging, as several factors could have influenced the decline in infections, including behavior changes and other community interventions. We developed a mathematical model of EVD transmission, and measured how transmission changed over time in the 12 districts of Sierra Leone with sustained transmission between June 2014 and February 2015. We used the model to estimate how many cases were averted as a result of the introduction of additional treatment beds in each area. Examining epidemic dynamics at the district level, we estimated that 56,600 (95% credible interval: 48,300-84,500) Ebola cases (both reported and unreported) were averted in Sierra Leone up to February 2, 2015 as a direct result of additional treatment beds being introduced. We also found that if beds had been introduced 1 month earlier, a further 12,500 cases could have been averted. Our results suggest the unprecedented local and international response led to a substantial decline in EVD transmission during 2014-2015. In particular, the introduction of beds had a direct impact on reducing EVD cases in Sierra Leone, although the effect varied considerably between districts.

Are All Hosts Created Equal? Partitioning Host Species Contributions to Parasite Persistence in Multihost Communities

Many parasites circulate endemically within communities of multiple host species. To understand disease persistence within these communities, it is essential to know the contribution each host species makes to parasite transmission and maintenance. However, quantifying those contributions is challenging. We present a conceptual framework for classifying multihost sharing, based on key thresholds for parasite persistence. We then develop a generalized technique to quantify each species' contribution to parasite persistence, allowing natural systems to be located within the framework. We illustrate this approach using data on gastrointestinal parasites circulating within rodent communities and show that, although many parasites infect several host species, parasite persistence is often driven by just one host species. In some cases, however, parasites require multiple host species for maintenance. Our approach provides a quantitative method for differentiating these cases using minimal reliance on system-specific parameters, enabling informed decisions about parasite management within poorly understood multihost communities.

Models for managing wildlife disease

Modelling wildlife disease poses some unique challenges. Wildlife disease systems are data poor in comparison with human or livestock disease systems, and the impact of disease on population size is often the key question of interest. This review concentrates specifically on the application of dynamic models to evaluate and guide management strategies. Models have proved useful particularly in two areas. They have been widely used to evaluate vaccination strategies, both for protecting endangered species and for preventing spillover from wildlife to humans or livestock. They have also been extensively used to evaluate culling strategies, again both for diseases in species of conservation interest and to prevent spillover. In addition, models are important to evaluate the potential of parasites and pathogens as biological control agents. The review concludes by identifying some key research gaps, which are further development of models of macroparasites, deciding on appropriate levels of complexity, modelling genetic management and connecting models to data.

Dynamic Variation in Sexual Contact Rates in a Cohort of HIV-Negative Gay Men

Human immunodeficiency virus (HIV) transmission models that include variability in sexual behavior over time have shown increased incidence, prevalence, and acute-state transmission rates for a given population risk profile. This raises the question of whether dynamic variation in individual sexual behavior is a real phenomenon that can be observed and measured. To study this dynamic variation, we developed a model incorporating heterogeneity in both between-person and within-person sexual contact patterns. Using novel methodology that we call iterated filtering for longitudinal data, we fitted this model by maximum likelihood to longitudinal survey data from the Centers for Disease Control and Prevention's Collaborative HIV Seroincidence Study (1992-1995). We found evidence for individual heterogeneity in sexual behavior over time. We simulated an epidemic process and found that inclusion of empirically measured levels of dynamic variation in individual-level sexual behavior brought the theoretical predictions of HIV incidence into closer alignment with reality given the measured per-act probabilities of transmission. The methods developed here provide a framework for quantifying variation in sexual behaviors that helps in understanding the HIV epidemic among gay men.

Interaction of Vaccination and Reduction of Antibiotic Use Drives Unexpected Increase of Pneumococcal Meningitis

Antibiotic-use policies may affect pneumococcal conjugate-vaccine effectiveness. The reported increase of pneumococcal meningitis from 2001 to 2009 in France, where a national campaign to reduce antibiotic use was implemented in parallel to the introduction of the 7-valent conjugate vaccine, provides unique data to assess these effects. We constructed a mechanistic pneumococcal transmission model and used likelihood to assess the ability of competing hypotheses to explain that increase. We find that a model integrating a fitness cost of penicillin resistance successfully explains the overall and age-stratified pattern of serotype replacement. By simulating counterfactual scenarios of public health interventions in France, we propose that this fitness cost caused a gradual and pernicious interaction between the two interventions by increasing the spread of nonvaccine, penicillin-susceptible strains. More generally, our results indicate that reductions of antibiotic use may counteract the benefits of conjugate vaccines introduced into countries with low vaccine-serotype coverages and high-resistance frequencies. Our findings highlight the key role of antibiotic use in vaccine-induced serotype replacement and suggest the need for more integrated approaches to control pneumococcal infections.

Quantifying Immune Response to Influenza Virus Infection via Multivariate Nonlinear ODE Models with Partially Observed State Variables and Time-Varying Parameters

Influenza A virus (IAV) infection continues to be a global health threat, as evidenced by the outbreak of the novel A/California/7/2009 IAV strain. Previous flu vaccines have proven less effective than hoped for emerging IAV strains, indicating a more thorough understanding of immune responses to primary infection is needed. One issue is the difficulty in directly measuring many key parameters and variables of the immune response. To address these issues, we considered a comprehensive workflow for statistical inference for ordinary differential question (ODE) models with partially observed variables and time-varying parameters, including identifiability analysis, two-stage and NLS estimation, and model selection etc‥ In particular, we proposed a novel one-step method to verify parameter identifiability and formulate estimating equations simultaneously. Thus, the pseudo-LS method can now deal with general ODE models with partially observed state variables for the first time. Using this workflow, we verified the relative significance of various immune factors to virus control, including target epithelial cells, cytotoxic T-lymphocyte (CD8+) cells and IAV specific antibodies (IgG and IgM). Factors other than cytotoxic T-lymphocyte (CTL) killing contributed the most to the loss of infected epithelial cells, though the effects of CTL are still significant. IgM antibody was found to be the major contributor to neutralization of free infectious viral particles. Also, the maximum viral load, which correlates well with mortality, was found to depend more on viral replication rates than infectivity. In contrast to current hypotheses, the results obtained via our methods suggest that IgM antibody and viral replication rates may be worth of further explorations in vaccine development.

On the predictive ability of mechanistic models for the Haitian cholera epidemic

Predictive models of epidemic cholera need to resolve at suitable aggregation levels spatial data pertaining to local communities, epidemiological records, hydrologic drivers, waterways, patterns of human mobility and proxies of exposure rates. We address the above issue in a formal model comparison framework and provide a quantitative assessment of the explanatory and predictive abilities of various model settings with different spatial aggregation levels and coupling mechanisms. Reference is made to records of the recent Haiti cholera epidemics. Our intensive computations and objective model comparisons show that spatially explicit models accounting for spatial connections have better explanatory power than spatially disconnected ones for short-to-intermediate calibration windows, while parsimonious, spatially disconnected models perform better with long training sets. On average, spatially connected models show better predictive ability than disconnected ones. We suggest limits and validity of the various approaches and discuss the pathway towards the development of case-specific predictive tools in the context of emergency management.

Inference of seasonal and pandemic influenza transmission dynamics

The inference of key infectious disease epidemiological parameters is critical for characterizing disease spread and devising prevention and containment measures. The recent emergence of surveillance records mined from big data such as health-related online queries and social media, as well as model inference methods, permits the development of new methodologies for more comprehensive estimation of these parameters. We use such data in conjunction with Bayesian inference methods to study the transmission dynamics of influenza. We simultaneously estimate key epidemiological parameters, including population susceptibility, the basic reproductive number, attack rate, and infectious period, for 115 cities during the 2003-2004 through 2012-2013 seasons, including the 2009 pandemic. These estimates discriminate key differences in the epidemiological characteristics of these outbreaks across 10 y, as well as spatial variations of influenza transmission dynamics among subpopulations in the United States. In addition, the inference methods appear to compensate for observational biases and underreporting inherent in the surveillance data.

Bayesian history matching of complex infectious disease models using emulation: a tutorial and a case study on HIV in Uganda

Advances in scientific computing have allowed the development of complex models that are being routinely applied to problems in disease epidemiology, public health and decision making. The utility of these models depends in part on how well they can reproduce empirical data. However, fitting such models to real world data is greatly hindered both by large numbers of input and output parameters, and by long run times, such that many modelling studies lack a formal calibration methodology. We present a novel method that has the potential to improve the calibration of complex infectious disease models (hereafter called simulators). We present this in the form of a tutorial and a case study where we history match a dynamic, event-driven, individual-based stochastic HIV simulator, using extensive demographic, behavioural and epidemiological data available from Uganda. The tutorial describes history matching and emulation. History matching is an iterative procedure that reduces the simulator's input space by identifying and discarding areas that are unlikely to provide a good match to the empirical data. History matching relies on the computational efficiency of a Bayesian representation of the simulator, known as an emulator. Emulators mimic the simulator's behaviour, but are often several orders of magnitude faster to evaluate. In the case study, we use a 22 input simulator, fitting its 18 outputs simultaneously. After 9 iterations of history matching, a non-implausible region of the simulator input space was identified that was times smaller than the original input space. Simulator evaluations made within this region were found to have a 65% probability of fitting all 18 outputs. History matching and emulation are useful additions to the toolbox of infectious disease modellers. Further research is required to explicitly address the stochastic nature of the simulator as well as to account for correlations between outputs.

An increasing number of scientific disciplines, and biology in particular, rely on complex computational models. The utility of these models depends on how well they are fitted to empirical data. Fitting is achieved by searching for suitable values for the models' input parameters, in a process known as calibration. Modern computer models typically have a large number of input and output parameters, and long running times, a consequence of their increasing computational complexity. The above two things hinder the calibration process. In this work, we propose a method that can help the calibration of models with long running times and several inputs and outputs. We apply this method on an individual based, dynamic and stochastic HIV model, using HIV data from Uganda. The final system has a 65% probability of selecting an input parameter set that fits all 18 model outputs.

Inference for dynamic and latent variable models via iterated, perturbed Bayes maps

Iterated filtering algorithms are stochastic optimization procedures for latent variable models that recursively combine parameter perturbations with latent variable reconstruction. Previously, theoretical support for these algorithms has been based on the use of conditional moments of perturbed parameters to approximate derivatives of the log likelihood function. Here, a theoretical approach is introduced based on the convergence of an iterated Bayes map. An algorithm supported by this theory displays substantial numerical improvement on the computational challenge of inferring parameters of a partially observed Markov process.

Inferring the temperature dependence of population parameters: the effects of experimental design and inference algorithm

Understanding and quantifying the temperature dependence of population parameters, such as intrinsic growth rate and carrying capacity, is critical for predicting the ecological responses to environmental change. Many studies provide empirical estimates of such temperature dependencies, but a thorough investigation of the methods used to infer them has not been performed yet. We created artificial population time series using a stochastic logistic model parameterized with the Arrhenius equation, so that activation energy drives the temperature dependence of population parameters. We simulated different experimental designs and used different inference methods, varying the likelihood functions and other aspects of the parameter estimation methods. Finally, we applied the best performing inference methods to real data for the species Paramecium caudatum. The relative error of the estimates of activation energy varied between 5% and 30%. The fraction of habitat sampled played the most important role in determining the relative error; sampling at least 1% of the habitat kept it below 50%. We found that methods that simultaneously use all time series data (direct methods) and methods that estimate population parameters separately for each temperature (indirect methods) are complementary. Indirect methods provide a clearer insight into the shape of the functional form describing the temperature dependence of population parameters; direct methods enable a more accurate estimation of the parameters of such functional forms. Using both methods, we found that growth rate and carrying capacity of Paramecium caudatum scale with temperature according to different activation energies. Our study shows how careful choice of experimental design and inference methods can increase the accuracy of the inferred relationships between temperature and population parameters. The comparison of estimation methods provided here can increase the accuracy of model predictions, with important implications in understanding and predicting the effects of temperature on the dynamics of populations.