Optimal surveillance network design: a value of information model

Surveillance and uncertainty

Increased movements of people, and expansion of international trade in food and other commodities, enhanced by social and environmental changes linked to urbanization, are all manifestations of the rapidly-changing nature of the world we live in. Such changes contribute to the rapid adaptation and movement of microorganisms, which has facilitated the return of old communicable diseases, the emergence of new ones, and the evolution of antimicrobial resistance. Food, is in many aspect the “connectome” of most issues related to infectious diseases and for this motivation experts are now talking about “emerging foodborne infectious diseases” whose complex dynamics embraces a larger spectrum of than the one traditionally considered for infectious diseases. The majority of the pathogens causing the significant foodborne disease burden - estimated as -1.9 million people annually at the global level - are now considered to be zoonotic (Schlundt et al. 2004). For this motivation a one-health approach is now embraced (NAS 2012). To put it more simply, food is in many cases involved in many infectious disease propagations as the vehicle of pathogen spread in the transmission chain (e.g., environment-animal-food-humans) leading to infections (e.g. for cholera and tuberculosis) beyond the classical foodborne illnesses where there is no human-human transmission. As for foodborne diseases, in USA result in 37.2 million illnesses, 228,744 hospitalizations, and 2,612 deaths each year (Batz et al. 2011; Hanson et al. 2012; NAS 2012; Scallan et al. 2011a, 2011b). Foodborne illnesses can be caused by a variety of microbial pathogens, chemicals, and parasites that contaminate food at different points in the food production and preparation process. Although most of these diarrheal deaths occur in poor countries, foodborne diseases are neither limited to developing countries nor to children. In general, infectious diseases other than foodborne are a huge problem also for creating susceptibility to chronic diseases. Because of the necessity to decrease such burden, surveillance is the primary public health strategy to emphasize, but a set of questions arise spontaneously.

What is the optimal set of surveillance nodes that best inform about the spread of an epidemic accurately? Is it possible to gauge the shape of transmission networks based on epidemic outbreaks? The accuracy in such estimations is a function of the source of information reliability. This means that a node, or a set of nodes performing surveillance, can report or not report whether a case of an infectious disease case is observed. Such information tremendously affects the estimation of outbreak path and sources. Thus, the functioning of surveillance networks is crucial for early detection and rapid response of epidemics which is possibly the only way to respond to emerging/re-emerging infectious diseases for which we are unprepared (Stachenko 2008). The function of surveillance networks is the ability to observe epidemic information (outbreak occurrence) moving along the mobility network. Therefore, the design of surveillance networks via the maximization of reporting accuracy (i.e., the accuracy in surveillance function) is a key process for saving lives of human and animal populations (Helbing et al. 2014). Economical and environmental outcomes are also associated to such optimal design: for instance industries and government costs related to commodity recalls and health care, and footprints on the environment related to pathogen and/or chemical spills can be highly reduced. To face such interconnected problems systemic risk approaches are welcome in order to design globally resilient systems (Haldane and May 2011; Helbing 2013; Helbing et al. 2014; Park et al. 2012).

For surveillance design, the schematization of surveillance as a network is very convenient and realistic (Figure 1). Nodes are sites where information about cases is observed and reported and these nodes may share information in real-time. The network scheme is related to the simplification of the system into a physical structure (the network) that is much simpler and controllable than the whole “3D domain” that is unnecessary to simulate. The mathematical problem of network search is easier than the one in which nodes are considered as separate nodes. The realism of the network scheme is related to the fact that surveillance - that is a network function - is occurring on a subset of the transmission network. Yet, surveillance and transmission networks are multiplex networks (Estrada and Gómez-Gardeñes 2014; Newman 2003) in which information and pathogens are spreading, respectively. The backbone of such networks are mobility networks of people (e.g., subway, train, airline networks, etc), food (e.g., supply chains), wild animals (e.g., migration routes), and environmental vehicle networks (e.g., watercourses and urban drainage networks) that connect portions of ecosystems and countries worldwide by contributing to the spread of infectious diseases (den Broeck et al. 2011; Goncalves et al. 2013; Knobler et al. 2006; On Effectiveness of National Biosurveillance Systems: BioWatch C, the Public Health System NRC 2011; Pastore Y Piontti et al. 2014). Thus, it is important to implement optimal surveillance on mobility networks (that can be transmission networks when a pathogen spread) via personnel and/or sensors whose reported information can be integrated into a cyber-infrastructure that supports public health authorities and industries.

The consideration of deep uncertainty in reported outbreaks is a very rarely addressed topic in epidemiology. Recently large attention has been placed on detecting outbreak sources (Convertino and Hedberg 2014a; Convertino and Hedberg 2014b; Pinto et al. 2012). However, in these studies uncertainty was neglected in reported outbreaks. Convertino and Hedberg (Convertino M, Hedberg C: Multisite outbreak detection: how many foodborne outbreak are attributable to farmerõs markets?, submitted) considered large variability in food trade network factors to capture the deep uncertainty related to attribution of foodborne outbreaks to different local food establishments (versus food imported globally), with particular focus on farmers’ markets. While this approach considered deep uncertainty, reported outbreaks were considered as “perfect information”. Other studies, such as (Pinto et al. 2012) considered random and non-random placement of surveillance observers which partially captures reporting uncertainty; nonetheless these studies did not formulate any optimization algorithm maximizing detection accuracy even in presence of uncertainty.

Moreover, outbreak uncertainty is certainly biased, thus (Pinto et al. 2012)’s and others’ approaches, based on regular schemes of surveillance underestimated such bias related to the anisotropic spreading of epidemics. In addition to such considerations, (Pinto et al. 2012)’s approach considered the problem of surveillance and outbreak source detection as coupled problems. In this study and in (Convertino and Hedberg 2014a) it is shown that such problem can be mathematically decoupled and the optimization of surveillance implies the optimal detection of outbreak sources but not vice versa. Pinto et al. (2012)’s model, analogous to the method used by telecommunication towers to pinpoint cell phone users, focused on arrival times of outbreaks - thought as information arrivals -that are correctly reproduced by the correct definition of outbreak sources. Here, we propose a simpler model, based on the original idea of (Brockmann and Helbing 2013) and subsequently developed by (Convertino and Hedberg 2014a; Convertino and Hedberg 2014b; (Convertino M, Hedberg C: Multisite outbreak detection: how many foodborne outbreak are attributable to farmerõs markets?, submitted)) and (Manitz et al. 2014), that considers the minimization of the error in the distance from real outbreak multi-sources with the most informative surveillance networks. All explored surveillance networks correspond to different reported outbreak patterns. Other approaches for designing surveillance have been developed in the past; for instance (Bajardi et al. 2012) developed a model for robust outbreak cluster detection based on network features. While such model has an optimization component, deep uncertainty in outbreak reporting (that is uncertainty related to report cases at a selected site and time) is not considered and the detection of clusters do not detect outbreak sources. The optimal surveillance network maximizes outbreak information, or equivalently minimizes the uncertainty related to the average distance between real and predicted outbreak sources.

As a side note we want to emphasize that outbreak sources are oftentimes confused with “onset areas” which are areas where first cases occur. Those areas are not necessarily where the first contamination of the environment and/or human infection occur considering environmental and social determinants that contribute to the spread of the pathogen in space. However, considering the typical short range of dispersal of most pathogens (Mundt et al. 2009), onset areas tend to coincide with outbreak sources. In some cases one of the onsets is the outbreak source and all other onsets are generated by epidemic propagation. For other outbreaks in which vehicle or vector (e.g, wild birds, food, and people via airplanes) of pathogens travel for long distances the proposed model works well because of the large resolution at which outbreaks are investigated and because of the uniqueness of the outbreak source. We argue that the algorithm of (Pinto et al. 2012) is more likely to fail in detecting real outbreak sources because it does not consider uncertainty in reported outbreaks, thus the implemented matching function of the observed delay in reported information detects onset areas. Certainly, the definition of outbreak sources is always dependent on the spatial and temporal scale of the system considered and care has to be placed when planning control strategies based on supposed outbreak sources.

Proposed approach

Case-study outbreaks

The choice about outbreaks considered in this study is related to the objective of capturing the salient links between epidemic patterns and processes regardless of disease etiology that is a result of peculiar epidemiological dynamics in terms of socio-environmental drivers, velocity, and geographical coverage (Convertino et al. 2009). This is with the aim of a general theory and cyber-infrastructure development for automated surveillance (Convertino and Hedberg 2014a). The goal is to show how observed outbreak patterns can be analyzed with the same physical-based models to detect outbreak sources and design surveillance networks considering the underlying transmission networks; thus, linking structure and functions of networks (Newman 2003) related to “invisible” outbreak dynamics over space and time. We chose among the fastest and high incidence rate epidemics: the 2004-2007 H5N1 pandemic (Kilpatrick et al. 2006), the 2010 cholera epidemic in Cameroon ((Convertino M, S L, M A, Morris S: Importance, Interaction, and Scale-dependence of Cholera Outbreak Drivers: Metamodeling Predictions, submitted); Guevart et al. 2006; Njoh 2010; Tatah et al. 2012), and the 2012 Salmonella epidemic in USA due to contaminated tuna (GATS 2014a). 34%, 27%, and 12% is the incidence rate for H5N1, cholera, and Salmonella, respectively. The estimated traveling velocity of these epidemics is 11, 22, and 122 km/month, respectively. The velocity is estimated using epidemic arrival times and distance covered over time. Additional file 1 report more information about the outbreak considered, and Figure 3 shows the observed outbreak patterns and transmission networks.

Mobility and surveillance networks

Mobility networks are the backbone of both transmission and surveillance networks that are both subnetworks of the mobility network. Mobility networks can be related to the mobility of vehicles, vectors, or pathogens themselves. For salmonella in tuna and H5N1 in humans we consider the food and human mobility network that are the primary networks responsible for the transmission of the pathogen. For cholera, previous studies such as (Pinto et al. 2012), have considered only the river network; however, considering the higher importance of human mobility in the transmission of cholera infections here we take into account human mobility and water-flow networks, simultaneously. However, only one network is enough for detecting outbreak sources. Surveillance networks can be though to coincide with human mobility networks rather than with environmental networks unless sensors are placed along those.

Human and food mobility network fluxes are estimated using the radiation model of (Simini et al. 2012) (Additional file 1) that is based on the population distribution. The population dataset is from the Web sites of the Gridded Population of the World and the Global Urban-Rural Mapping projects (CIESIN 2014; GRUMP 2014), which are run by the Socioeconomic Data and Application Center (SEDAC) of Columbia University. According to this dataset, the surface of the world is divided into a grid of cells that can have different resolution levels. Each of these cells has been assigned an estimated population value. Out of the possible resolutions, we have opted for cells of 15 × 15 minutes of arc to constitute the basis of our model. This corresponds to an area of each cell approximately equivalent to an area of 25 × 25 km ² along the Equator. The dataset comprises 823,680 cells, of which 250,206 are populated. Since the coordinates of each cell and those of the airports in the World Airport Network are known, the distance between the cells and the airports can be calculated.

For the H5N1 we consider only the estimated mobility for the World Airport Network (WAN) (Colizza et al. 2006; den Broeck et al. 2011). WAN is composed of 3362 commercial airports indexed by the International Air Transport Association (IATA) that are located in 220 different countries. The database contains the number of available seats per year for each direct connection between two of these airports. The coverage of the dataset is estimated to be 99% of the global commercial traffic. The WAN can be seen as a weighted graph comprising 16,846 edges whose weight, represents the passenger flow between airports. The network shows a high degree of heterogeneity both in the number of destinations per airport and in the number of passengers per connection.

The International Agro-Food Trade Network (IFTN) (Convertino and Liang 2014; Ercsey-Ravasz et al. 2012) for the USA is built using data of (ComTrade 2014) for the worldwide linkages among countries, and data from the United States Department of Agriculture, Foreign Agricultural Service’s Global Agricultural Trade System (GATS) (FAS 2014; GATS 2014b) for the trade network of imported food commodities in USA. Additional data is used from the Agricultural Marketing Service (AMS 2014). The use of both datasets for food allows one to cross check erroneous data and to complement missing data missing. Data of imported and produced food that is locally consumed is obtained from FAOSTAT food balance sheets (FAOSTAT 2014). Such information is useful for calibrating the model factor that determines how many individuals consume the contaminated food.

Effective distance and traceback model

The model is based on the optimal inference of epidemiological factors that better explain the spatio-temporal occurrence of foodborne outbreaks for any potential outbreak location to which different food trade paths correspond. In this study, differently from (Convertino and Hedberg 2014a), Manitz et al. (2014), and (Brockmann and Helbing 2013), deep uncertainty in reported outbreaks is introduced. The possible outbreak sources are determined by perturbing the reported outbreaks considering different surveillance networks. All feasible outbreak sources are tested considering their likelihood to satisfy the relationship between outbreak arrival times and velocity at any time step of the epidemic and for all infected communities simultaneously.

We define the effective distance (Brockmann and Helbing 2013) of directly connected nodes as a function of the most probable food trajectories defined by the radiation model (Section S2.1 in Additional file 1) that can be derived from the connectivity matrix P that specifies which nodes are connected; specifically d _mn =1−logP _mn where for any couple m−n there are multiple distances (Brockmann and Helbing 2013). For any candidate outbreak source we use the definition of “shortest path” effective distance D _mn (Brockmann and Helbing 2013) from an arbitrary reference node n to another node m in the network (not necessarily directed connected) as the length of the shortest path from n to m, D _mn =min _Γ λ(Γ), where λ(Γ) is the directed length of an ordered path Γ=n ₁,…,n _L as the sum of effective lengths along the path Brockmann and Helbing (2013). The average shortest path tree is calculated among all effective distances that are associated to potential candidate outbreak sources. Thus, both average effective distance (d _mn ) and average shortest path tree (that is the shortest path among many effective distances) responsible of outbreak spreading (D _mn ) depend only on the static mobility matrix weighted by the food fluxes. There is a family of effective distances and there one of shortest path distances between community m and n. Theoretically, there can be more than one shortest path distance but the dimension of the ensemble of shortest path distances is always smaller (or equal) than the ensemble of effective distances.

This equation states that effective outbreak distances D _e can be computed with high fidelity based on outbreak arrival times on and effective spreading speed v _e , and that each factor depends on different factors of the dynamical system considered (Brockmann and Helbing 2013). Note that here we indicate D _e as D _mn previously defined. The epidemiological factors associated with the classical SIR model determine the effective speed, whereas effective distance depends only on the topological features of the static underlying network, i.e., the matrix P. Because, T _a and v _e are know and estimated from data (that can be generated in real-time), respectively, it is easy to estimate D _e of an outbreak considering also the uncertainty in epidemiological dynamics and food trade network. Thus, Eq. 1 can be used as a test of hypotheses where hypotheses are about all candidate outbreak sources responsible for the observed outbreak patterns. The most likely geographical distance can be assessed after determining the effective distance and considering all potential food trade paths. The SIR dynamics (Section “Epidemiological dynamics and information spreading: predictive metacommunity model”) is calibrated on the range of the outbreak velocity and arrival times, and is simulated by maximizing the prediction accuracy of the observed outbreak patterns in terms of cumulative outbreaks. Such accuracy is maximized when the uncertainty about the effective distance from the real outbreak source is minimized, that determines the maximization of the concentricity of outbreak spreading waves (Section “Outbreak detection: linking patterns and processes”). The concentricity is maximized only when the accuracy in the reported outbreaks is maximized that is equivalent to the maximum of the value of information (Section “Value of information and Pareto optimization”) considering any uncertainty source in outbreak patterns, supply chain and disease dynamics (Section “Global sensitivity and uncertainty analyses”).

Epidemiological dynamics and information spreading: predictive metacommunity model

where D=l ² γ is the diffusion coefficient that is related to a classic form of a Fisher equation which has been deployed in mathematical epidemiology for describing wave spreading in reaction-diffusion system such as for epidemics (Belik et al. 2011; Campos and Mendez 2005). Note that \(s_{n}=s_{n}(\hat {s}(O),t)\) and \(j_{n}=j_{n}(\hat {s}(O),t)\) where \((\hat {s}(O),t)\) is the surveillance network in Equation 4.

For sufficiently localized initial conditions this systems exhibits traveling waves with speed \(c=2\sqrt {\alpha D(1-\beta /\alpha)} \sim \sqrt {\gamma }\) (Belik et al. 2011). Such velocity can consider bias transport as in (Bertuzzo et al. 2007) in case the bias is known and elevated in the transport process. The telegraph model introduced by (Holmes et al. 1993) and simulated by (Bertuzzo et al. 2007) for cholera possesses both diffusion and wave motions. The Fisher equation of diffusion works well when using the effective distance, while the telegraph model equation works well when using the geographical distance of the transmission network. GSUA is used here to consider uncertainty in the traceback model and to assess the relative importance and interactions of factors - epidemiological and mobility network factors (Section “Epidemiological dynamics and information spreading: predictive metacommunity model” and Additional file 1: Section S2.1, respectively) - leading to outbreaks. Interactions of causal factors are typically neglected as well as the determination of the causality of such factors. The traceback model is run multiple times according to the Sobol scheme (Additional file 1: Section S1.2) (Figure 2) sampling all model factors along their probability distributions (Additional file 1: Figure S4) estimated from data or assumed as uniform according to a maximum entropy principle (Convertino et al. 2014a).

Outbreak detection: linking patterns and processes

The simplest method to determine which food and which outbreak source are the most likely to determine the observed outbreak patterns is to test which feasible candidate source has the lowest variability in terms of mean and variance assessed from Eq. 1. This is essentially a portfolio problem (Convertino and Valverde 2013) in which a large set of outbreak source alternatives exist and the whole set is evaluated considering average risk and variance, where in this case the “risk” is related to the inability to predict the correct outbreak patterns in space and time. Thus, for each potential candidate outbreak location the model computes the effective distance to the subset of nodes in which an outbreak is observed. This is done for any time step of the epidemic. On the basis of this set of effective distances (denoted by different outbreak sources), we compute the mean μ(D _e ) and standard deviation σ(D _e ) of the effective distance. As shown by (Brockmann and Helbing 2013), concentricity of epidemic waves increases with a combined minimization of mean and standard deviation of the estimated effective distance. In other words, such approach minimizes the deviation from the expected relationship arrival times-distance, or velocity-distance, equivalently. Here we perform a minimization of the sum \(\sqrt {\mu (D_{e})^{2} + \sigma (D_{e})^{2}}\) as in a portfolio model where the objective function is the Euclidian distance built with the variables to minimize (Convertino and Valverde 2013). The correct outbreak source is the one that satisfies the minimum of the aforementioned sum for the whole epidemic duration. It is possible to visualize the dynamics of epidemic spread and concentricity of outbreak waves by plotting outbreak sources at the center of a network where all other nodes are placed around the central node at a radial distance equal to the effective distance. Correct outbreak sources determine cohesively evolving outbreak waves from the center to the nodes at the boundary from the beginning to the end of the epidemic.

Value of information and Pareto optimization

where C(t) and \(d(\hat {s}(O),t)\) is evaluated for all possible networks n (among all topologies (Additional file 1: Figure S1)), communities i, and time steps t; M is the total number of considered networks. ∝ stands for “proportional to”. Hence, the VoI is the difference of \(d(\hat {s}(O))\) estimated for different networks and all communities of the system. The VoI (Eq. 4) can be theoretically defined by the difference between two estimates of \(d(\hat {s}(O))\) for different surveillance networks (Eq. 4) (i.e. a different set of observer nodes). VoI is always taken equal or greater than zero and a rational decision maker always chooses more valuable information. Note that here we do not consider any cost function in the VoI because no information is available; however, that information is easily included as function detracted to the predictive benefits in Eq. 4. In presence of a physical-based model reproducing outbreaks in space and time the VoI can also be measured as accuracy in predicting prevalence patterns. In fact, errors in detecting outbreak sources translate into errors in initial conditions of the model that reduce model predictive accuracy. Such predictions generated by models that take in input surveillance data are used for public health decision making and system design.

Global sensitivity and uncertainty analyses

GSUA involves five steps: (1) the probability distribution functions (pdfs) for each input factor are selected (epidemiological, mobility, and network topology factors); (2) sample points are generated on the input factor distributions using the Sobol method; (3) multiple model execution using each of the sample points and a set of outputs is generated; (4) global sensitivity analysis is performed (i.e., assessment of factor interaction and importance); and, (5) the important input factors are selected. Probability distributions of mobility network factors (link length and clustering coefficient) are derived from data as in (Convertino M, S L, M A, Morris S: Importance, Interaction, and Scale-dependence of Cholera Outbreak Drivers: Metamodeling Predictions, submitted). Epidemiological factors (α, and β) and contamination factors (σ an ε) are assumed to be normally distributed within a normalized range [0,1]. γ is related to the range of variability of the total food and human mobility flux. As for human mobility, more than 60 million people travel billions of miles on more than 2 million international flights each week. GSUA is extremely important in complex systems that are dominated by factor interactions rather than an additive contribution of each single factor. In fact, classical sensitivity analysis fails to account for factor interactions and does not offer a full probabilistic investigation of system states; yet, classical one factor sensitivity analysis provides misleading conclusions. GSUA is rarely performed and classical one-factor at a time sensitivity approaches are used to evaluated uncertainty. GSUA first assign uncertainty and that drives sensitivity analysis via variability of model output that accounts for non-linearity among model input factors and those and model output. The Sobol method (Sobol 1993, 2001) is a variance-based method that performs a quantitative analysis of model sensitivity based on the principle of variance decomposition. According to this principle, the full variance of the model output is given by the sum of the variances of all input factors (Saltelli et al. 2008). The Sobol method has the capacity to quantify the influence of the full range of variation of each input factor as well as the interaction effects among the input factors on the model output (Saltelli et al. 2008). The Sobol method estimates sensitivity measures which summarize the models behavior. The most common measure of sensitivity is the first-order sensitivity index, S _i , that represents the main effect (direct contribution) of each input factor to the variance of the output. It is expressed as S _i =V _i /V where V _i is the part of the variance due to the input factor X _i , and V is the total variance of the model output. Another measure of sensitivity is the total factor sensitivity that includes the interactions. The total effect index, \(S_{T_{i}}\), that is the result of the variance decomposition, accounts for the total contribution to the output variation due to factor X _i , i.e., its first-order effect plus all the higher order effects due to interactions (Saltelli et al. 2008). Thus, the total effect index of factor X _i can be expressed as \(S_{T_{i}} = S_{i} + S_{i,i+1} + S_{i,i+1,i+2} + S_{i,i+1,i+2,i+3} + S_{i,i+1,i+2,i+3+\ldots +N_{\textit {IF}}} = S_{i} + S_{I_{i}}\), where N _IF is the number of input factors and \(S_{I_{i}}\) is the sum of all the interaction indexes. The Sobol pairwise interaction between factors can be calculated and we identify this with the notation \(S_{I_{\textit {ij}}}\) where i and j are the factors that are considered. As an example, in the case of three input factors \(S_{T_{1}}\) is the total sensitivity index of X ₁, S ₁ is the main effect of X ₁, \(S_{I_{12}}\) is the interaction effect between X ₁ and X ₂, and \(S_{I_{123}}\) is the interaction effect between X ₁, X ₂, and X ₃. Considering the previous expression, \(S_{T_{1}} - S_{1}\) provides a measure of how much X ₁ is involved in interactions with all other input factors (Saltelli et al. 2008). The sum of all S _i is equal to one for additive models and less than one for non-additive models. The difference 1−S _i can be used as an indicator of the presence of interactions in the model and we indicate that as S _I . The number of simulations required for the Sobol method for a two-index analysis (i.e., for first order and total indices) is given as N=M(2k+2) where M is the sample size of each index (typically taken between 500-1000) and k is the number of uncertain input factors after the Morris screening. In this study because k=7, and we take M= 600 we have 9,600 Sobol simulations. Since the Sobol method uses a pseudo-randomized multivariate sampling procedure, it can be used as a basis for a global uncertainty evaluation by constructing the pdfs and cumulative distribution functions for each of the selected outputs.