A Bayesian framework for parameter estimation in dynamical models

On the influence of prior information evaluated by fully Bayesian criteria in a personalized whole-brain model of epilepsy spread

Individualized anatomical information has been used as prior knowledge in Bayesian inference paradigms of whole-brain network models. However, the actual sensitivity to such personalized information in priors is still unknown. In this study, we introduce the use of fully Bayesian information criteria and leave-one-out cross-validation technique on the subject-specific information to assess different epileptogenicity hypotheses regarding the location of pathological brain areas based on a priori knowledge from dynamical system properties. The Bayesian Virtual Epileptic Patient (BVEP) model, which relies on the fusion of structural data of individuals, a generative model of epileptiform discharges, and a self-tuning Monte Carlo sampling algorithm, is used to infer the spatial map of epileptogenicity across different brain areas. Our results indicate that measuring the out-of-sample prediction accuracy of the BVEP model with informative priors enables reliable and efficient evaluation of potential hypotheses regarding the degree of epileptogenicity across different brain regions. In contrast, while using uninformative priors, the information criteria are unable to provide strong evidence about the epileptogenicity of brain areas. We also show that the fully Bayesian criteria correctly assess different hypotheses about both structural and functional components of whole-brain models that differ across individuals. The fully Bayesian information-theory based approach used in this study suggests a patient-specific strategy for epileptogenicity hypothesis testing in generative brain network models of epilepsy to improve surgical outcomes.

Data-driven modelling and spatial complexity supports heterogeneity-based integrative management for eliminating Simulium neavei-transmitted river blindness

Concern is emerging regarding the challenges posed by spatial complexity for modelling and managing the area-wide elimination of parasitic infections. While this has led to calls for applying heterogeneity-based approaches for addressing this complexity, questions related to spatial scale, the discovery of locally-relevant models, and its interaction with options for interrupting parasite transmission remain to be resolved. We used a data-driven modelling framework applied to infection data gathered from different monitoring sites to investigate these questions in the context of understanding the transmission dynamics and efforts to eliminate Simulium neavei- transmitted onchocerciasis, a macroparasitic disease that causes river blindness in Western Uganda and other regions of Africa. We demonstrate that our Bayesian-based data-model assimilation technique is able to discover onchocerciasis models that reflect local transmission conditions reliably. Key management variables such as infection breakpoints and required durations of drug interventions for achieving elimination varied spatially due to site-specific parameter constraining; however, this spatial effect was found to operate at the larger focus level, although intriguingly including vector control overcame this variability. These results show that data-driven modelling based on spatial datasets and model-data fusing methodologies will be critical to identifying both the scale-dependent models and heterogeneity-based options required for supporting the successful elimination of S. neavei-borne onchocerciasis.

A primer on model selection using the Akaike Information Criterion

A powerful investigative tool in biology is to consider not a single mathematical model but a collection of models designed to explore different working hypotheses and select the best model in that collection. In these lecture notes, the usual workflow of the use of mathematical models to investigate a biological problem is described and the use of a collection of model is motivated. Models depend on parameters that must be estimated using observations; and when a collection of models is considered, the best model has then to be identified based on available observations. Hence, model calibration and selection, which are intrinsically linked, are essential steps of the workflow. Here, some procedures for model calibration and a criterion, the Akaike Information Criterion, of model selection based on experimental data are described. Rough derivation, practical technique of computation and use of this criterion are detailed.

The Decade-Long Chinese Methadone Maintenance Therapy Yields Large Population and Economic Benefits for Drug Users in Reducing Harm, HIV and HCV Disease Burden

Objectives: We aimed to conduct a comprehensive evaluation of the population impact of methadone maintenance treatment (MMT) for its future program planning.

Methods: We conducted a literature review of the effects of MMT in China on HIV and HCV disease burden, injecting, and sexual behaviors and drug-related harm during 2004–2015. Data synthesis and analysis were conducted to obtain the pooled estimates of parameters for a mathematical model which was constructed to evaluate the effectiveness and cost-effectiveness of the program.

Results: Based on a review of 134 articles, this study demonstrated that MMT is highly effective in reducing crime-related, high risk sexual, and injecting behaviors. The model estimated US$1,037 m which was invested in MMT from 2004 to 2015 has prevented 29,463 (15,325–43,600) new HIV infections, 130,563 (91,580–169,546) new HCV infections, 10,783 (10,380–11,187) deaths related to HIV, HCV and drug-related harm, and 338,920.0 (334,596.2–343,243.7) disability-adjusted life years (DALYs). The costs for each prevented HIV infection, HCV infection, death, and DALY were $35,206.8 (33,594.8–36,981.4), $7,944.7 ($7,714.4–8,189.2), $96,193.4 (92,726.0–99,930.2), and $3,060.6 ($3,022.0–3,100.1) respectively.

Conclusion: The Chinese MMT program has been effective and cost-effective in reducing injecting, injecting-related risk behaviors and adversities due to HIV/HCV infection and drug-related harm among drug users.

Bayesian inference for parameter estimation in lactoferrin-mediated iron transport across blood-brain barrier

BACKGROUND

In neurodegenerative diseases such as Alzheimer's and Parkinson's, excessive irons as well as lactoferrin (Lf), but not transferrin (Tf), have been found in and around the affected regions of the brain. These evidences suggest that lactoferrin plays a critical role during neurodegenerative diseases, although Lf-mediated iron transport across blood-brain barrier (BBB) is negligible compared to that of transferrin in normal condition. However, the kinetics of lactoferrins and lactoferrin-mediated iron transport are still unknown.

METHOD

To determine the kinetic rate constants of lactoferrin-mediated iron transport through BBB, a mass-action based ordinary differential equation model has been presented. A Bayesian framework is developed to estimate the kinetic rate parameters from posterior probability density functions. The iron transport across BBB is studied by considering both Lf- and Tf-mediated pathways for both normal and pathologic conditions.

RESULTS

Using the point estimates of kinetic parameters, our model can effectively reproduce the experimental data of iron transport through BBB endothelial cells. The robustness of the model and parameter estimation process are further verified by perturbation of kinetic parameters. Our results show that surge in high-affinity receptor density increases lactoferrin as well as iron in the brain.

CONCLUSIONS

Due to the lack of a feedback loop such as iron regulatory proteins (IRPs) for lactoferrin, iron can transport to the brain continuously, which might increase brain iron to pathological levels and can contribute to neurodegeneration.

GENERAL SIGNIFICANCE

This study provides an improved understanding of presence of lactoferrin and iron in the brain during neurodegenerative diseases.

Sex-Specific Asymmetrical Attack Rates in Combined Sexual-Vectorial Transmission Epidemics

In 2015–2016, South America went through the largest Zika epidemic in recorded history. One important aspect of this epidemic was the importance of sexual transmission in combination with the usual vectorial transmission, with asymmetrical transmissibilities between sexual partners depending on the type of sexual contact; this asymmetry manifested itself in data as an increased risk to women. We propose a mathematical model for the transmission of the Zika virus including sexual transmission via all forms of sexual contact, as well as vector transmission, assuming a constant availability of mosquitoes. From this model, we derive an expression for R0, which is used to study and analyze the relative contributions of the male to female sexual transmission route vis-à-vis vectorial transmission. We also perform Bayesian inference of the model’s parameters using data from the 2016 Zika epidemic in Rio de Janeiro.

Modeling Approaches Toward Understanding Infectious Disease Transmission

Long-standing neglected diseases continue to challenge our global health infrastructure, and emerging pathogens pose new threats worldwide. To inform prevention and response efforts, mathematical models of infectious disease dynamics are being increasingly applied. Here we explain how models can be developed to enhance our understanding and predictive power over population-level disease trends, by capturing both fundamental aspects of transmission and also the effects of medical and behavioral interventions. We review advances in transdisciplinary approaches of disease modeling and illustrate these advances with applications including community-based initiatives undertaken during the Ebola epidemic in West Africa and age-targeting of influenza vaccination in the USA. We further discuss how modern statistical inference facilitates the incorporation of data from behavioral sciences and epidemiology into models, highlighting how data-driven models can constitute powerful tools to inform and improve public health strategies.

Characterising model dynamics using sparse grid interpolation: Parameter estimation of cholera

Sparse grid interpolation is a popular numerical discretization technique for the treatment of high dimensional, multivariate problems. We consider the case of using time-series data to calibrate epidemiological models from both phenomenological and mechanistic perspectives using this computational tool. By capturing the dynamics underlying both global and local spaces, our algorithm identifies potentially optimal regions of the parameter space and directs computational effort towards resolving the dynamics and resulting fits of these regions. We demonstrate how sparse grid interpolants can be effectively deployed to fit available data and discriminate between competing hypotheses to explain the current cholera epidemic in Yemen.

Microbial communities as dynamical systems

Nowadays, microbial communities are frequently monitored over long periods of time and the interactions between their members are explored in vitro. This development has opened the way to apply mathematical models to characterize community structure and dynamics, to predict responses to perturbations and to explore general dynamical properties such as stability, alternative stable states and periodicity. Here, we highlight the role of dynamical systems theory in the exploration of microbial communities, with a special emphasis on the generalized Lotka-Volterra (gLV) equations. In particular, we discuss applications, assumptions and limitations of the gLV model, mention modifications to address these limitations and review stochastic extensions. The development of dynamical models, together with the generation of time series data, can improve the design and control of microbial communities.

Optimal control approach for establishing wMelPop Wolbachia infection among wild Aedes aegypti populations

Wolbachia-based biocontrol has recently emerged as a potential method for prevention and control of dengue and other vector-borne diseases. Major vector species, such as Aedes aegypti females, when deliberately infected with Wolbachia become less capable of getting viral infections and transmitting the virus to human hosts. In this paper, we propose an explicit sex-structured population model that describes an interaction of uninfected (wild) male and female mosquitoes and those deliberately infected with wMelPop strain of Wolbachia in the same locality. This particular strain of Wolbachia is regarded as the best blocker of dengue and other arboviral infections. However, wMelPop strain of Wolbachia also causes the loss of individual fitness in Aedes aegypti mosquitoes. Our model allows for natural introduction of the decision (or control) variable, and we apply the optimal control approach to simulate wMelPop Wolbachia infestation of wild Aedes aegypti populations. The control action consists in continuous periodic releases of mosquitoes previously infected with wMelPop strain of Wolbachia in laboratory conditions. The ultimate purpose of control is to find a tradeoff between reaching the population replacement in minimum time and with minimum cost of the control effort. This approach also allows us to estimate the number of Wolbachia-carrying mosquitoes to be released in day-by-day control action. The proposed method of biological control is safe to human health, does not contaminate the environment, does not make harm to non-target species, and preserves their interaction with mosquitoes in the ecosystem.

A Bayesian approach for the identification of patient-specific parameters in a dialysis kinetic model

Hemodialysis is the most common therapy to treat renal insufficiency. However, notwithstanding the recent improvements, hemodialysis is still associated with a non-negligible rate of comorbidities, which could be reduced by customizing the treatment. Many differential compartment models have been developed to describe the mass balance of blood electrolytes and catabolites during hemodialysis, with the goal of improving and controlling hemodialysis sessions. However, these models often refer to an average uremic patient, while on the contrary the clinical need for customization requires patient-specific models. In this work, we assume that the customization can be obtained by means of patient-specific model parameters. We propose and validate a Bayesian approach to estimate the patient-specific parameters of a multi-compartment model, and to predict the single patient's response to the treatment, in order to prevent intra-dialysis complications. The likelihood function is obtained by means of a discretized version of the multi-compartment model, where the discretization is in terms of a Runge-Kutta method to guarantee convergence, and the posterior densities of model parameters are obtained through Markov Chain Monte Carlo simulation. Results show fair estimations and the applicability in the clinical practice.

Predictive accuracy of particle filtering in dynamic models supporting outbreak projections

BACKGROUND

While a new generation of computational statistics algorithms and availability of data streams raises the potential for recurrently regrounding dynamic models with incoming observations, the effectiveness of such arrangements can be highly subject to specifics of the configuration (e.g., frequency of sampling and representation of behaviour change), and there has been little attempt to identify effective configurations.

METHODS

Combining dynamic models with particle filtering, we explored a solution focusing on creating quickly formulated models regrounded automatically and recurrently as new data becomes available. Given a latent underlying case count, we assumed that observed incident case counts followed a negative binomial distribution. In accordance with the condensation algorithm, each such observation led to updating of particle weights. We evaluated the effectiveness of various particle filtering configurations against each other and against an approach without particle filtering according to the accuracy of the model in predicting future prevalence, given data to a certain point and a norm-based discrepancy metric. We examined the effectiveness of particle filtering under varying times between observations, negative binomial dispersion parameters, and rates with which the contact rate could evolve.

RESULTS

We observed that more frequent observations of empirical data yielded super-linearly improved accuracy in model predictions. We further found that for the data studied here, the most favourable assumptions to make regarding the parameters associated with the negative binomial distribution and changes in contact rate were robust across observation frequency and the observation point in the outbreak.

CONCLUSION

Combining dynamic models with particle filtering can perform well in projecting future evolution of an outbreak. Most importantly, the remarkable improvements in predictive accuracy resulting from more frequent sampling suggest that investments to achieve efficient reporting mechanisms may be more than paid back by improved planning capacity. The robustness of the results on particle filter configuration in this case study suggests that it may be possible to formulate effective standard guidelines and regularized approaches for such techniques in particular epidemiological contexts. Most importantly, the work tentatively suggests potential for health decision makers to secure strong guidance when anticipating outbreak evolution for emerging infectious diseases by combining even very rough models with particle filtering method.

Mathematical Modeling of "Chronic" Infectious Diseases: Unpacking the Black Box

Background

Mathematical models are increasingly used to understand the dynamics of infectious diseases, including “chronic” infections with long generation times. Such models include features that are obscure to most clinicians and decision-makers.

Methods

Using a model of a hypothetical active case-finding intervention for tuberculosis in India as an example, we illustrate the effects on model results of different choices for model structure, input parameters, and calibration process.

Results

Using the same underlying data, different transmission models produced different estimates of the projected intervention impact on tuberculosis incidence by 2030 with different corresponding uncertainty ranges. We illustrate the reasons for these differences and present a simple guide for clinicians and decision-makers to evaluate models of infectious diseases.

Conclusions

Mathematical models of chronic infectious diseases must be understood to properly inform policy decisions. Improved communication between modelers and consumers is critical if model results are to improve the health of populations.

Estimating the Attack Ratio of Dengue Epidemics under Time-varying Force of Infection using Aggregated Notification Data

Quantifying the attack ratio of disease is key to epidemiological inference and public health planning. For multi-serotype pathogens, however, different levels of serotype-specific immunity make it difficult to assess the population at risk. In this paper we propose a Bayesian method for estimation of the attack ratio of an epidemic and the initial fraction of susceptibles using aggregated incidence data. We derive the probability distribution of the effective reproductive number, Rt, and use MCMC to obtain posterior distributions of the parameters of a single-strain SIR transmission model with time-varying force of infection. Our method is showcased in a data set consisting of 18 years of dengue incidence in the city of Rio de Janeiro, Brazil. We demonstrate that it is possible to learn about the initial fraction of susceptibles and the attack ratio even in the absence of serotype specific data. On the other hand, the information provided by this approach is limited, stressing the need for detailed serological surveys to characterise the distribution of serotype-specific immunity in the population.

Bayesian calibration of simulation models for supporting management of the elimination of the macroparasitic disease, Lymphatic Filariasis

Background

Mathematical models of parasite transmission can help integrate a large body of information into a consistent framework, which can then be used for gaining mechanistic insights and making predictions. However, uncertainty, spatial variability and complexity, can hamper the use of such models for decision making in parasite management programs.

Methods

We have adapted a Bayesian melding framework for calibrating simulation models to address the need for robust modelling tools that can effectively support management of lymphatic filariasis (LF) elimination in diverse endemic settings. We applied this methodology to LF infection and vector biting data from sites across the major LF endemic regions in order to quantify model parameters, and generate reliable predictions of infection dynamics along with credible intervals for modelled output variables. We used the locally calibrated models to estimate breakpoint values for various indicators of parasite transmission, and simulate timelines to parasite extinction as a function of local variations in infection dynamics and breakpoints, and effects of various currently applied and proposed LF intervention strategies.

Results

We demonstrate that as a result of parameter constraining by local data, breakpoint values for all the major indicators of LF transmission varied significantly between the sites investigated. Intervention simulations using the fitted models showed that as a result of heterogeneity in local transmission and extinction dynamics, timelines to parasite elimination in response to the current Mass Drug Administration (MDA) and various proposed MDA with vector control strategies also varied significantly between the study sites. Including vector control, however, markedly reduced the duration of interventions required to achieve elimination as well as decreased the risk of recrudescence following stopping of MDA.

Conclusions

We have demonstrated how a Bayesian data-model assimilation framework can enhance the use of transmission models for supporting reliable decision making in the management of LF elimination. Extending this framework for delivering predictions in settings either lacking or with only sparse data to inform the modelling process, however, will require development of procedures to estimate and use spatio-temporal variations in model parameters and inputs directly, and forms the next stage of the work reported here.

Electronic supplementary material

The online version of this article (doi:10.1186/s13071-015-1132-7) contains supplementary material, which is available to authorized users.

Bayesian Melding Approach to Estimate the Reproduction Number for Tuberculosis Transmission in Indian States and Union Territories

Tuberculosis (TB) is one of the most common infectious diseases and a leading cause of death in the world. Despite the full implementation of Revised National Tuberculosis Control Programme, the disease continues to be a leading cause of morality and economic burden in India. The basic reproduction is a fundamental key parameter that quantifies the spread of a disease. In this article, we present a Bayesian melding approach to estimate the basic reproduction number using a deterministic model of TB. We present a point estimate of the basic reproduction number of 35 states and union territories of India during 2006 to 2011. The basic reproduction number of TB for India is computed to be 0.92, which indicates the slow elimination of TB in India during 2006 to 2011.

Modelling lymphatic filariasis transmission and control: modelling frameworks, lessons learned and future directions

Mathematical modelling provides a useful tool for policy making and planning in lymphatic filariasis control programmes, by providing trend forecasts based on sound scientific knowledge and principles. This is now especially true, in view of the ambitious target to eliminate lymphatic filariasis as a public health problem globally by the year 2020 and the short remaining timeline to achieve this. To meet this target, elimination programmes need to be accelerated, requiring further optimization of strategies and tailoring to local circumstances. Insights from epidemiological transmission models provide a useful basis. Two general models of lymphatic filariasis transmission and control are nowadays in use to support decision-making, namely a population-based deterministic model (EPIFIL) and an individual-based stochastic model (LYMFASIM). Model predictions confirm that lymphatic filariasis transmission can be interrupted by annual mass drug administration (MDA), but this may need to be continued much longer than the initially suggested 4-6 years in areas with high transmission intensity or poor treatment coverage. However, the models have not been validated against longitudinal data describing the impact of MDA programmes. Some critical issues remain to be incorporated in one or both of the models to make predictions on elimination more realistic, including the possible occurrence of systematic noncompliance, the risk of emerging parasite resistance to anthelmintic drugs, and spatial heterogeneities. Rapid advances are needed to maximize the utility of models in decision-making for the ongoing ambitious lymphatic filariasis elimination programmes.

Bayesian parameter inference for dynamic infectious disease modelling: rotavirus in Germany

Understanding infectious disease dynamics using epidemic models based on ordinary differential equations requires the calibration of model parameters from data. A commonly used approach in practice to simplify this task is to fix many parameters on the basis of expert or literature information. However, this not only leaves the corresponding uncertainty unexamined but often also leads to biased inference for the remaining parameters because of dependence structures inherent in any given model. In the present work, we develop a Bayesian inference framework that lessens the reliance on such external parameter quantifications by pursuing a more data-driven calibration approach. This includes a novel focus on residual autocorrelation combined with model averaging techniques in order to reduce these estimates' dependence on the underlying model structure. We applied our methods to the modelling of age-stratified weekly rotavirus incidence data in Germany from 2001 to 2008 using a complex susceptible-infectious-susceptible-type model complemented by the stochastic reporting of new cases. As a result, we found the detection rate in the eastern federal states to be more than four times higher compared with that of the western federal states (19.0% vs 4.3%), and also the infectiousness of symptomatically infected individuals was estimated to be more than 10 times higher than that of asymptomatically infected individuals (95% credibility interval: 8.1–19.6). Not only do these findings give valuable epidemiological insight into the transmission processes, we were also able to examine the considerable impact on the model-predicted transmission dynamics when fixing parameters beforehand.

On the uniqueness of epidemic models fitting a normalized curve of removed individuals

The susceptible-infected-removed (SIR) and the susceptible-exposed-infected-removed (SEIR) epidemic models with constant parameters are adequate for describing the time evolution of seasonal diseases for which available data usually consist of fatality reports. The problems associated with the determination of system parameters starts with the inference of the number of removed individuals from fatality data, because the infection to death period may depend on health care factors. Then, one encounters numerical sensitivity problems for the determination of the system parameters from a correct but noisy representative of the number of removed individuals. Finally as the available data is necessarily a normalized one, the models fitting this data may not be unique. We prove that the parameters of the (SEIR) model cannot be determined from the knowledge of a normalized curve of "Removed" individuals and we show that the proportion of removed individuals, [Formula: see text], is invariant under the interchange of the incubation and infection periods and corresponding scalings of the contact rate. On the other hand we prove that the SIR model fitting a normalized curve of removed individuals is unique and we give an implicit relation for the system parameters in terms of the values of [Formula: see text] and [Formula: see text], where [Formula: see text] is the steady state value of [Formula: see text] and [Formula: see text] and [Formula: see text] are the values of [Formula: see text] and its derivative at the inflection point [Formula: see text] of [Formula: see text]. We use these implicit relations to provide a robust method for the estimation of the system parameters and we apply this procedure to the fatality data for the H1N1 epidemic in the Czech Republic during 2009. We finally discuss the inference of the number of removed individuals from observational data, using a clinical survey conducted at major hospitals in Istanbul, Turkey, during 2009 H1N1 epidemic.

The relative importance of frequency of contacts and duration of exposure for the spread of directly transmitted infections

The recent availability of survey data on social contact patterns has made possible important advances in the understanding of the social determinants of the spread of close-contact infections, and of the importance of long-lasting contacts for effective transmission to occur. Still, little is known about the relationship between two of the most critical identified factors (frequency of contacts and duration of exposure) and how this relationship applies to different types of infections. By integrating data from two independently collected social surveys (Polymod and time use), we propose a model that combines these two transmission determinants into a new epidemiologically relevant measure of contacts: the number of "suitable" contacts, which is the number of contacts that involve a sufficiently long exposure time to allow for transmission. The validity of this new epidemiological measure is tested against Italian serological data for varicella and parvovirus-B19, with uncertainty evaluated using the Bayesian melding technique. The model performs quite well, indicating that the interplay between time of exposure and contacts is critical for varicella transmission, while for B19 it is the duration of exposure that matters for transmission.

Sequential modelling of the effects of mass drug treatments on anopheline-mediated lymphatic filariasis infection in Papua New Guinea

BACKGROUND

Lymphatic filariasis (LF) has been targeted by the WHO for global eradication leading to the implementation of large scale intervention programs based on annual mass drug administrations (MDA) worldwide. Recent work has indicated that locality-specific bio-ecological complexities affecting parasite transmission may complicate the prediction of LF extinction endpoints, casting uncertainty on the achievement of this initiative. One source of difficulty is the limited quantity and quality of data used to parameterize models of parasite transmission, implying the important need to update initially-derived parameter values. Sequential analysis of longitudinal data following annual MDAs will also be important to gaining new understanding of the persistence dynamics of LF. Here, we apply a Bayesian statistical-dynamical modelling framework that enables assimilation of information in human infection data recorded from communities in Papua New Guinea that underwent annual MDAs, into our previously developed model of parasite transmission, in order to examine these questions in LF ecology and control.

RESULTS

Biological parameters underlying transmission obtained by fitting the model to longitudinal data remained stable throughout the study period. This enabled us to reliably reconstruct the observed baseline data in each community. Endpoint estimates also showed little variation. However, the updating procedure showed a shift towards higher and less variable values for worm kill but not for any other drug-related parameters. An intriguing finding is that the stability in key biological parameters could be disrupted by a significant reduction in the vector biting rate prevailing in a locality.

CONCLUSIONS

Temporal invariance of biological parameters in the face of intervention perturbations indicates a robust adaptation of LF transmission to local ecological conditions. The results imply that understanding the mechanisms that underlie locally adapted transmission dynamics will be integral to identifying points of system fragility, and thus countermeasures to reliably facilitate LF extinction both locally and globally.

Modeling and statistical analysis of the spatio-temporal patterns of seasonal influenza in Israel

Background

Seasonal influenza outbreaks are a serious burden for public health worldwide and cause morbidity to millions of people each year. In the temperate zone influenza is predominantly seasonal, with epidemics occurring every winter, but the severity of the outbreaks vary substantially between years. In this study we used a highly detailed database, which gave us both temporal and spatial information of influenza dynamics in Israel in the years 1998–2009. We use a discrete-time stochastic epidemic SIR model to find estimates and credible confidence intervals of key epidemiological parameters.

Findings

Despite the biological complexity of the disease we found that a simple SIR-type model can be fitted successfully to the seasonal influenza data. This was true at both the national levels and at the scale of single cities.The effective reproductive number R_e varies between the different years both nationally and among Israeli cities. However, we did not find differences in R_e between different Israeli cities within a year. R_e was positively correlated to the strength of the spatial synchronization in Israel. For those years in which the disease was more “infectious”, then outbreaks in different cities tended to occur with smaller time lags. Our spatial analysis demonstrates that both the timing and the strength of the outbreak within a year are highly synchronized between the Israeli cities. We extend the spatial analysis to demonstrate the existence of high synchrony between Israeli and French influenza outbreaks.

Conclusions

The data analysis combined with mathematical modeling provided a better understanding of the spatio-temporal and synchronization dynamics of influenza in Israel and between Israel and France. Altogether, we show that despite major differences in demography and weather conditions intra-annual influenza epidemics are tightly synchronized in both their timing and magnitude, while they may vary greatly between years. The predominance of a similar main strain of influenza, combined with population mixing serve to enhance local and global influenza synchronization within an influenza season.

Towards uncertainty quantification and inference in the stochastic SIR epidemic model

In this paper we address the problem of estimating the parameters of Markov jump processes modeling epidemics and introduce a novel method to conduct inference when data consists on partial observations in one of the state variables. We take the classical stochastic SIR model as a case study. Using the inverse-size expansion of van Kampen we obtain approximations for the first and second moments of the state variables. These approximate moments are in turn matched to the moments of an inputed Generic Discrete distribution aimed at generating an approximate likelihood that is valid both for low count or high count data. We conduct a full Bayesian inference using informative priors. Estimations and predictions are obtained both in a synthetic data scenario and in two Dengue fever case studies.