Abstract
Our research is motivated by the rapidlyevolving outbreaks of rare and fatal infectious diseases, for example, the severe acute respiratory syndrome (SARS) and the Middle East respiratory syndrome. In many of these outbreaks, main transmission routes were healthcare facilityassociated and through persontoperson contact. While a majority of existing work on modelling of the spread of infectious diseases focuses on transmission processes at a community level, we propose a new methodology to model the outbreaks of healthcareassociated infections (HAIs), which must be considered at an individual level. Our work also contributes to a novel aspect of integrating realtime positioning technologies into the tracking and modelling framework for effective HAI outbreak control and prompt responses. Our proposed solution methodology is developed based on three key components – timevarying contact network construction, individuallevel transmission tracking and HAI parameter estimation – and aims to identify the hidden health state of each patient and worker within the healthcare facility. We conduct experiments with a fourmonth human tracking data set collected in a hospital, which bore a big nosocomial outbreak of the 2003 SARS in Hong Kong. The evaluation results demonstrate that our framework outperforms existing epidemic models for characterizing macrolevel phenomena such as the number of infected people and epidemic threshold.
Introduction
Nosocomial infections, or known as healthcareassociated infections (HAIs), are infections that are acquired in a healthcare setting, such as those caught during an inpatient hospital stay or developed among patients and healthcare staff within a healthcare facility. HAIs have become one of the greatest challenges in the modern world and a global threat to health security. Magill et al. [1] estimated that there were around 722,000 HAIs in U.S. acute care hospitals in 2011, and the [2] reported that around 75,000 patients deaths were related to HAIs. Among the different types of transmission routes of HAIs, the most important and common one is through persontoperson contact, or known as directcontact transmission. Persontoperson contact transmission takes place when there is a physical contact between an infected or colonized individual and a susceptible person such that diseasecausing microorganisms may be transfered. In many cases, due to the incubation period of the infectious disease or unawareness of the disease severity, the disease had already been spread through the hospital before the index patient was identified and quarantined.
A striking example of recent HAI outbreaks through persontoperson contact transmission is the spread of 2013 Middle East respiratory syndrome coronavirus (MERSCoV) in Saudi Arabia. Assiri et al. [3] reported that, of the 23 confirmed cases of MERSCoV infection identified in eastern province of Saudi Arabia, 21 cases were acquired by persontoperson transmission in different healthcare facilities including hemodialysis units, intensive care units (ICUs), and inpatient units. In the incident, Patient A with symptoms of dizziness and diaphoresis was admitted to a hospital. No MERSCoV test was performed and this patient was not suspected of carrying this deadly virus at the time of admission. Another patient, Patient C, was admitted on the next day to the room adjacent to Patient A to undergo hemodialysis. Later, MERSCoV infection was confirmed in nine other patients who were receiving hemodialysis treatments in the same hospital, six of whom had overlapping time with Patient C undergoing hemodialysis. MERSCoV infection also developed in a nurse administrator, who had once been present in the ICU where the treatment of Patient A was provided. From this incident, [3] concluded that persontoperson transmission of MERSCoV can be associated with considerable morbidity in healthcare facilities and suggested that surveillance and infectioncontrol measures are crucial to a global public health response.
Another classic example of large HAI outbreaks is the 2003 severe acute respiratory syndrome (SARS) epidemic in Hong Kong. The 2003 SARS outbreak resulted in 8,096 confirmed cases and 774 deaths in total around the globe [4], and 1755 confirmed cases and 299 deaths in Hong Kong alone (World Health Organization (WHO) 2003). The SARS outbreak began at the Prince of Wales Hospital (PWH) in Hong Kong. The index patient was admitted to PWH before WHO issued a high global alert about cases of this deadly pneumonia. Due to the unawareness of this previously unknown virus, the index patient was not treated as a carrier of a highly infectious and severe disease and SARS started to spread through the hospital among the patients, healthcare workers and visitors, resulting in at least 138 additional cases of SARS acquisition within the hospital (including at least 20 doctors, 34 nurses, 15 allied health workers, and 16 medical students) [5]. The disease was further spread to the community through the individuals leaving the hospital, e.g., leading to the outbreak in a housing estate Amoy Gardens with a total of 329 residents infected and 42 deaths [6], and became an epidemic threatening the world. During the outbreak, the governmental organizations (e.g., Department of Health and Hospital Authority) and public health workers put substantial efforts into contact tracing for effective infection control [6, 7]. It turns out that the contact tracing work was one of the key interventions that helped the Hong Kong Government successfully contained the SARS epidemic, e.g., through screening of symptoms and medical surveillance.
The lesson learned from the 2003 SARS outbreak suggests that a rapid contact tracing is critical for effective quarantine and epidemic management in serious disease outbreaks in hospital settings. A traditional way of conducting contact tracing relies on interviewing. However, conducting interviews may be ineffective because the patient is generally very sick to recall and talk about the past activities and contacts. Another way is to interview the healthcare workers to reconstruct the patients’ past activities. However, both ways cannot guarantee the information is exhaustive and accurate. More importantly, these procedures require lengthy investigations such that prompt isolations cannot be carried out to prevent the disease from spreading.
Our research was motivated by a project initiated at PWH after the SARS outbreak in Hong Kong. The project objectives were to mitigate HAI risk and to support rapid contingency responses in case of severe infectious disease outbreak. Our research team investigated how advanced indoor positioning technologies can be applied to enhance the persontoperson contact traceability for prompt contact tracing, and developed a radiofrequency identification (RFID) system for tracking people’s interactivities (including those among patients and ward staff) and tracing highrisk individuals when infectious disease outbreak occurs. A pilot study was carried out in two medical wards at PWH. Our research team developed an RFIDbased realtime locating system for this application. Figure 1 shows a screenshot of the realtime locating system. For the details of the RFID hardwares, we refer the reader to [8]. It is worth mentioning that the rare experience that the hospital management and staff members of PWH had gone through during the 2003 SARS outbreak and their suggestions were crucial when developing this platform. While our pilot study was enabled by RFID, other indoor positioning technologies are also viable. In our application, RFID was adopted because it is a mature technology and can be deployed at a affordable cost.
A visionbased system is an alternative technology capturing events of persontoperson contacts and monitoring activities of healthcare workers (HCWs) and patients. It can accurately review whether two individuals have been in close contact. Also, most hospitals have installed visionbased systems in the waiting area such as lobbies, escalators, etc., for surveillance purpose. However, such systems may raise privacy issues when installed in wards. In particular, patients’ and HCWs’ privacy is always a primary concern in Hong Kong. It is unlikely that HCWs and patients would accept an environment putting them under constant visual recording. Thus, a sensing method appeared to be more receptive in this study.
To mitigate the risk of HAIs, we characterize the contact patterns among the patients and healthcare workers, and study how an HAI is transmitted through persontoperson contact. From the onsite pilot project conducted at PWH, a large amount of positioning data of patients and healthcare workers were collected. This set of data provides the unique and necessary information to construct dynamic human networks to trace persontoperson contacts, and to develop a more comprehensive understanding of the ways individuals contact with each other in a healthcare environment. After the constructions of human networks within the healthcare facility, we investigate how an HAI is transmitted through persontoperson contact. This investigation requires the capability of tracking the transmission paths of HAIs at an individual level. In other words, we infer the health status of any person at any time and identify potential infected patients and healthcare workers.
Our research is different from traditional epidemiology studies, which investigate the spread of infectious disease among people. Traditional epidemic models, for example, [9,10,11], assume homogeneous human connection (i.e., all individuals in the population are the same) and model the spread of infectious diseases with ordinary differential equations. As suggested by [12], individual differences should be considered when modelling the transmission and designing more effective strategies to reduce the propagation of disease. For HAIs, the assumption that all individuals are equally likely to be infected does not hold, and therefore estimates from communitybased models can be inaccurate [13]. Due to the importance of considering individual interactions, some approaches have been proposed to capture the impacts of individual differences using heterogeneous networks, such as the percolation methods [14] and the nonlinear dynamical system [15]. However, there is inadequate research on studying the problem of individuallevel tracking of HAIs, where most existing methods can hardly be applied to identifying individual infection status. The novelty and uniqueness of our research are as follows. (1) The contact pattern of patients and healthcare workers in a healthcare facility differs from the contact pattern of people in a large region, for example, a city, studied by a majority of the existing epidemic research. A healthcare facility is a relatively closed community with a highly hierarchical and modular structure, leading to dense human connectivity. On the other hand, a citylevel population is sparsely connected. (2) We construct timevarying networks to represent dynamic human interaction in a hospital, while many existing models assume static networks that treat human contact unchanged over time. (3) Individual differences in a healthcare setting are related to the roles of individuals in a hospital. As an example, a nurse generally has more interactions with other people than a patient does. (4) We aim to track transmission of HAIs at an individual level that infers the hidden health status for any person at any time, while traditional epidemic models focus on macrolevel phenomena such as the total number of infected individuals and epidemic thresholds.
The recent advancement of realtime indoor positioning technologies has provided us with a precious opportunity to study the spread of infectious disease from a new perspective. Given the locational data of patients and healthcare workers in a healthcare institution collected continuously to the system, if an outbreak of an HAI occurs in the institution among patients and healthcare workers, our goal is to track transmission of the HAI at an individual level through persontoperson contact over timevarying contact networks. To achieve the goal, we propose a framework with three key components: timevarying network construction, individuallevel transmission tracking and HAI parameter estimation. We realize individuallevel transmission tracking with the assumption that all input parameters for the HAI model (HAI parameters) are known. Then we discuss our proposed parameter estimation procedure for effective transmission tracking in a more general setting. Our work leverages a previous RFID tracking study in which we collected locational data of patients and healthcare workers for four months in two medical wards at PWH.
Literature review
Our research investigates the transmission of HAIs over timevarying human networks at an individual level. For an overview of the existing research on epidemic models, we refer the reader to [16,17,18,19,20].
Traditional epidemic models such as the SusceptibleInfectedRecovered (SIR) model and the SusceptibleInfectedSusceptible (SIS) model are based on differential equations [9, 11]. Pairwise models proposed by [21] also belong to this class; in addition, they leverage the local network structure such as the number of pairs or triples to substitute the simple infection number in standard epidemic models. These differential equationbased models have been widely used for analyzing the spread of various types of transmissible diseases. For example, [22] studied the spread of sexually transmitted diseases and considered network heterogeneities in their differential equationbased model. They showed that their predictive scheme was more accurate than models that assume homogeneous networks. Percolation models utilize network degree distributions and epidemic dynamics to examine outcomes of disease outbreaks in the steady state. [14] showed that a large class of standard SIR models could be solved exactly with the use of percolation theory. This approach is more realistic than the traditional approach as it allows heterogeneous and correlated infectiveness times and transmission probabilities. Meyers et al. [23] applied percolation theory to reduce timeconsuming calculations and derive different epidemic outcomes on a contact network for the city of Vancouver, British Columbia. Newman [24] and Karrer et al. [25] developed percolation models to study the transmission of two competing diseases based on probability generating functions. Volz [26] showed that SIR model can be represented by a system of ordinary differential equations with the use of probability generating functions. The result enables SIR dynamics to be modeled in random networks.
Another popular class is probabilistic models. This type of models incorporates uncertainty when modeling the disease spreading process. Larson [27] studied the use of “social distancing” (e.g., closing of schools, mandated minimum physical distances between coworkers) to control influenza progression by reducing the frequency and intensity of daily humantohuman contacts. He considered heterogeneous populations, distinguished by highactivity and lowactivity persons, in their probabilistic mixing model. Teytelman and Larson [28] considered heterogeneous population where the individuals are different regarding their social activities, pronenesses to infection, and pronenesses to shed virus and spread infection. These attributes determine the rate of human contacts per day and impact the probability that a susceptible individual becomes infected. Yaesoubi and Cohen [29] proposed a discretetime Markov chain approach to modeling the transmission of diseases. Dynamic optimization techniques can be integrated into their model to aid realtime selection and modification of public health interventions. The studies mentioned above, however, focused on populationlevel epidemics, which differs from an infectious disease outbreak in a healthcare setting.
As suggested by [30] and [12], human behaviors that might lead to transmission of disease differ significantly between individuals. Much research thus has been carried out to study human interaction pattern. The majority of work in this direction is the use of network analysis. For instance, [31] constructed a bipartite network to represent the connectivity of patients and caregivers in a psychiatric institution. Liljeros et al. [32] studied network properties such as transitivity, assortativity and variation based on a large database constructing from the contact records associated 295,108 inpatients over two years. They concluded that the risk and adverse consequences of epidemic outbreaks might be reduced if these network properties are taken into account when designing the intervention schemes. Ueno and Masuda [33] constructed a hierarchical and modular contact network from a hospital setting. They showed that healthcare workers are main transmitters of diseases and shall be vaccinated with a higher priority. Curtis et al. [34] modeled dynamic contact networks by deriving spatial distributions of healthcare workers and generating random walks to predict human movements in a hospital. Prakash et al. [15] constructed a timevarying network which follows an alternating connectivity behavior to model the daynight pattern of nurse shifts and derived a closedform equation for the epidemic threshold with their network. In these studies, their main focuses were on the constructions of networks to capture the effects of human interactions on disease transmission. In our work, we construct timevarying dynamic human contact networks from real data and study how these networks can be leveraged for prediction of health status of each individual in a healthcare facility.
Research on disease transmission in a healthcare setting is not adequately addressed in the existing literature. Meyers et al. [31] modelled the disease outbreak in a psychiatric institution, but their model did not utilize realworld human contact data and required some simplified assumptions, e.g., the connection degree of each object in the model follows a Poisson distribution. Prakash et al. [15] used a nonlinear dynamical system to model the spread of infections in a timevarying network which takes into account the shifts of nurses in a hospital. Dong et al. [35] modeled nosocomial infections in an MIT college dormitory using graphcoupled hidden Markov models and solved the problem by Gibbs sampling. However, these studies either are limited to describing macrolevel epidemic phenomena for HAIs, or did not attempt solutions to the individuallevel transmission tracking problem.
Our research studies the transmission of HAIs at an individual level. We propose a modeling framework to describe the interactions between individuals with a dynamic human contact network and tracks infections over this network. Furthermore, we utilize realtime indoor locating technology for tracking the interactions between individuals and constructing their timevarying contact networks. For our modeling contributions, we develop coupled hidden Markov models to solve the problem. Our method captures the difference of the underlying human connectivity where existing epidemic models are unable to address and contributes to more accurate estimations for predicting macrolevel outcomes of HAI outbreaks. Moreover, compared to the existing epidemic models, our proposed approach has the unique capability to estimate individual hidden health status and to infer HAI parameters for practical solutions.
Overview
In this section, we introduce the problem of individuallevel HAI tracking over dynamic human networks, and then provide an overview of the key components of our proposed solution framework.
Problem definition
Many HAIs are acquired through persontoperson transmission. Our research aims to leverage the use of advanced indoor positioning technologies, such as RFID, for human tracking in healthcare facilities. The availability of the large amounts of locational data about the individuals within the facility, including patients and healthcare workers, enables us to develop a groundbreaking and effective approach to modeling the transmission of disease in a healthcare setting. It provides a new opportunity to characterize persontoperson contacts in a hospital environment and systematically study the transmission pattern of HAIs. More specifically, we aim to develop a modeling framework to track the transmission of HAIs among patients and healthcare workers over timevarying contact networks.
Unlike traditional methods of contact tracing that involve manual processes such as interviewing, advanced positioning technologies collect timestamped locations of tracked objects automatically and continuously for timely construction of contact networks. In the following, we introduce the terminologies and framework used for our approach.
Definition 1 (Contact)
A pair of individuals is considered to constitute a contact if the distance between them is within a prespecified distance threshold for a duration that is longer than a prespecified time threshold.
This definition establishes the foundation that we model the persontoperson transmission of HAIs. A person is considered to have been exposed to the disease if (i) he or she has had any facetoface contact with an infectious patient, (ii) is in the same hospital room with an infectious patient for more than a certain amount of time, or (iii) is provided care by an infected healthcare worker [3]. With the definition of contact, we construct a persontoperson contact network, denoted by G = (V,E), with vertice set V consisting of individuals in the healthcare facility and edge set E representing the contact records between individuals, i.e., e_{ij} ∈ E indicating a contact between individual v_{i},v_{j} ∈ V. The concept of timevarying contact network can then be extended from the above setting:
Definition 2 (Timevarying contact network)
A timevarying contact network is a series of static contact networks indexed by time points, which is denoted by \(G_{0:T}= \{G_{t}\}_{t = 1}^{T} = \{(V_{t},E_{t})\}_{t = 1}^{T}\), where V _{t} and E_{t} are the sets of individuals and contacts at time t, respectively.
Without loss of generality, we can denote a timevarying contact network by \(G_{0:T} = \{(V,E_{t})\}_{t = 1}^{T}\) because any vertex v_{k} that exists but is not present at time t, that is, v_{k} with \(k \in \{\cup _{s = 1}^{T} V_{s} \}\setminus V_{t}\), can be viewed as an isolated vertex at time t in network G_{t}. Thus for simplicity, we use \(G_{0:T}=\{(V,E_{t})\}_{t = 1}^{T}\) in the rest of the paper.
In the context of disease, the terms “symptom” and “sign” are both used to refer to an indication of a certain set of medical characteristics that can reflect the presence of a disease, such as runny nose, coughing and fever in a case of influenza. Technically, symptoms and signs are different; a symptom is a feature observed by the patient whereas a sign is observable by the others, e.g., physicians. For consistency, we use observation to represent a symptom or a sign. The presence of a disease, in general, is difficult to be identified without the use of medical diagnostic tests, and therefore, the actual health state of a patient is often hidden. We define the terms observation and health state used for our model as follows.
Definition 3 (Observation)
An observation is a feature that reflects the presence of a disease, which can be a symptom or a sign, or both. \({o}_{t}^{i}\) denotes the observation of person v_{i} at time t. The observation vector \(O_{t}=({o}_{t}^{1},{o}_{t}^{2},\ldots ,{o}_{t}^{n}), n=V\) represents the collection of observations of all individuals at time t.
Definition 4 (Health state)
A health state denotes the (hidden) health status of an individual. A patient who is a host of an infectious disease can only be in one health state at a time. \({x}_{t}^{i}\) denotes the hidden health state of person v_{i} at time t. The state vector \(X_{t}=({x}_{t}^{1},{x}_{t}^{2},\ldots ,{x}_{t}^{n}), n=V\) denotes the combination of health states of all individuals at time t.
Our objective is to infer the hidden health states of each person at different time points. Observations and networks are necessary to achieve this goal. Observations provide the essential information for diagnoses of an infection, while networks give the information the set of individuals an infected person has contacted. On the other hand, they also impose practical challenges on the accurate identification of hidden health states of a person. Observations can be misleading since the same observation may appear in different health states. The networking effect complicates the problem in the sense that the health state of a person not only is dependent on his/her medical history but also can be affected by other “connected” patients. The challenges motivate us to consider the following overall problem, and the three subproblems required to tackle in our framework.
Overall problem
Given timestamped locations of the individuals in a healthcare facility, if an HAI outbreak takes place through persontoperson contact, how can we track the transmission of the HAI over the human contact network at an individual level?
Problem 1 (Timevarying contact network construction)
Given timestamped locations of the individuals in a healthcare facility, how do we establish contact between any pair of individuals and to construct a timevarying contact network G _{0: T} to characterize the human connection pattern?
Problem 2 (Individuallevel transmission tracking)
Given (1) a timevarying contact network G_{0:T} of the individuals in a healthcare facility, (2) observation vectors O_{0:T} of these people, and (3) the transmission parameter set 𝜃 of the HAI, if an HAI outbreak takes place, how do we identify the hidden health state\({x}_{t}^{i}\) for any person v_{i} at any time t ≥ 0?
Problem 3 (HAI parameter estimation)
Given (1) a timevarying contact network G_{0:T} of the individuals in a healthcare facility, (2) observation vectors O_{0:T} of these people, and (3) a subset 𝜃_{s} of the HAI parameter set 𝜃 with 𝜃_{s} ⊂𝜃, if an HAI outbreak takes place, how do we estimate the unknown parameter set𝜃 ∖𝜃_{s}?
Solution framework
We propose a threestage solution framework, consisting of the approaches to solving the three respective subproblems, for tackling the overall problem. An overview of our solution framework is as follows.
 Stage1 :

(Timevarying contact network construction) Persontoperson transmission is the primary mode of transmission of HAIs. We define a distance threshold and a time threshold to utilize the timestamped locations of individuals to construct the list of temporalspatial cooccurrence events for the establishment of persontoperson contacts. Individuals are divided into multiple groups according to their roles and attributes, and contacts are labeled with four types. We first generate static hierarchical networks based on the types of contacts, and then construct a timevarying contact network.
 Stage2 :

(Individuallevel transmission tracking) We formulate the problem of individuallevel transmission tracking with networkbased coupled HMMs, in which the SIS model describes the transmission dynamics of HAIs. HMMs are known to have the power to recover hidden pattern from observable information, and shown to be effective in inference of health states, e.g., [36, 37]. Solutions are obtained in three steps. First, we give basic solutions to a standard HMM by considering all individuals as a single vertex. Then we derive solutions at an individual level by factoring basic solutions according to SIS dynamics. Finally, we reduce the computational complexity of the solutions based on meanfield analysis to speed up computations for largescale problems.
 Stage3 :

(HAI parameter estimation) The problem of HAI parameter estimation is formulated as Maximum Likelihood Estimation (MLE). An auxiliary function is introduced to transform the MLE problem to a computationally efficient optimization problem. By solving this optimization problem with Lagrangian multiplier method, we can obtain the BaumWelch reestimate and recover the original HAI parameters from this reestimate. This learning method improves the estimation of HAI parameters iteratively.
Table 1 summarizes the notation used throughout this paper.
Timevarying contact network construction
Contact establishment
We can establish a linkage between two persons in the contact network by utilizing the information about their temporalspatial cooccurrence. First, movement trajectories of individuals, e.g. trajectories of person v_{i} and person v_{j} shown in Fig. 2a, are extracted from the timestamped locational data captured from the indoor positioning infrastructure. The distance between person v_{i} and person v_{j} at any time is calculated; examples of the distances are represented by dashed lines in Fig. 2a. Let d_{ij}(t) be a function of time t that measures the distance between persons v_{i} and v_{j}. A contact between v_{i} and v_{j} is established if ∃t_{1},t_{2}, where t_{2} − t_{1} > ΔT_{th} such that
where D_{th} is the distance threshold and ΔT_{th} is the time threshold.
Static hierarchical network generation
A threelevel hierarchical network is generated to represent the human connectivity in a hospital environment. Individuals are labeled with their role classes and divided into a patient group and a caregiver group. The patient group is further divided according to the ward where a patient is staying. Correspondingly, there are four types of contacts, namely, intraward contacts, interward contacts, intratype contacts and intertype contacts. An example of a simple hierarchical contact network is shown in Fig. 3. The bottomlevel network consists of patients and they are linked according to intraward contacts; the middlelevel network consists of both patients and caregivers and they are linked with intratype and interward contacts; the toplevel network consists of intertype contacts between patientcaregiver pairs.
Timevarying contact network construction
A timevarying network can be represented as a series of static networks. We divide a continuous period into discrete time intervals and construct the timevarying contact network by combining the static networks constructed for each time period in sequence. In a hospital setting, it is common to divide a day into a daytime nighttime sessions [15], because most healthcare workers have fixed shift times and human connectivity has dissimilar structures in the two sessions.
Most healthcare facilities adopt a hierarchical and modular structure for the separation of wards and different roles of individuals. Interaction pattern differs between the intraward and interward contacts or between the intratype and intertype contacts. The timevarying hierarchical contact network naturally captures such characteristics in a hospital environment. Moreover, the hierarchical network is an extension of the bipartite patientcaregiver network in [31]. The bipartite patientcaregiver network assumes that only intertype contacts exist, but does not consider contacts within the same individual group. Undoubtedly, it would be more practical to consider contacts among caregivers as they are the main transmitters of HAIs.
Individuallevel transmission tracking
Transmission dynamics
The classical SIS model formulates transmission dynamics of HAIs. There are two health states in the SIS model, namely, susceptible and infected. A person can be either susceptible or infected at one time. A susceptible person v_{i} might get infected with an infection rate τ_{i} due to a contact with an infectious patient. An infected patient v_{j} gets recovered independently with a recovery rate μ_{j}, and turns from the infected state to the susceptible state immediately. Let health state \({x}_{t}^{i}\) be a binary variable such that \({x}_{t}^{i}= 1\) and 0 respectively indicate that person v_{i} is infected and suspectible at time t. Let \(\mathcal {N}_{t}(v_{i})\) be the neighbor set of v_{i} at time t, where a neighbor of v_{i} is defined to be the set of adjacent vertices of v_{i} on graph G_{t}. In the SIS model, the transmission through persontoperson contact is determined by the following set of equations.
Equation 1 represents the case that a susceptible person v_{i} remains susceptible at the next time period if all his/her infectious neighbors fail to transmit the infection to v_{i}. Equation 2 indicates that a susceptible person gets infected if one or more of the infectious neighbors transmit the infection to him/her successfully. Equations 3 and 4 respectively state that an infected person recovers and remains infected independently. The SIS dynamics is shown in Fig. 4. The links between vertices represent the persontoperson contacts. At time t, v_{1},v_{2} and v_{3} are infected while v_{4} and v_{5} are susceptible. At time t + 1, v_{4} remains susceptible as the infected neighbors v_{1} and v_{3} both fail to transmit the disease to v_{4}. v_{5} gets infected because v_{1} transmits the infection to v_{5} successfully. v_{2} recovers independently and becomes susceptible at time t + 1 , while v_{1} and v_{3} remain infected.
Detection, tracing and prediction
The problem of individuallevel transmission tracking is to identify the hidden health state of a person at any time, given the set of observations, parameters of the HAI model, and the contact networks. Suppose that the current time is t. Specifically, the problems of tracing, detection and prediction are respectively to infer the hidden state of an individual before, at and after time t, as illustrated in Fig. 5. Formally, given the observation set O_{0:t}, the timevarying contact network G_{0:t}, and the parameter set 𝜃 of an HAI, the problem of individuallevel transmission tracking is to determine a mapping function
We call the function f for detection if s = t, tracing if s < t, and prediction if s > t.
We propose networkbased coupled HMMs to tackle the individuallevel transmission tracking problem. As shown in Fig. 6, a standard HMM has two sequences of components: a sequence of hidden states of a Markov chain and a sequence of observations. An observation is dependent on its hidden state only, but not affected by other states and observations. HMMs have the power to reveal hidden pattern from observable information and have been widely applied in various fields such as speech recognition and social network analysis. The coupled HMMs incorporate several subHMMs together under a network structure, as illustrated in Fig. 7. This connection creates the interdependence of the hidden states of multiple subHMMs. For example, the hidden state of HMM2 at time t is not only determined by its own state at time t − 1, but also affected by the hidden states of HMM1 and HMM3 at time t − 1.
We propose a twophase approach to deriving solutions from the coupled HMMs. First, the coupled HMMs are regarded as a standard HMM to give basic solutions to the problems of detection, tracing and prediction. In the standard HMM, all individuals are considered as single vertices such that the combined state X_{t} is determined rather than individual \({x}_{t}^{i}\). The second phase is to derive solutions for individuallevel transmission tracking based on factorization and meanfield analysis.
For each person v_{i}, we denote the detection probability at time t by \({\pi }_{t}^{i}\), the tracing probability at time s by \({\pi }_{st}^{i}\), and the kstep ahead prediction probability at time t + k by \({\pi }_{t+kt}^{i}\). Correspondingly, π_{t}, π_{st} and π_{t + kt} respectively denote the detection probability at time t, the tracing probability at time k and the kstep ahead prediction probability at time t + k. Let h_{tt− 1} = P(X_{t}X_{t− 1}) be the state transition probability and ϕ_{t} = P(O_{t}X_{t}) be the observation probability. A standard HMM provides the detection, tracing and prediction probabilities in recursive forms by the following set of equations [38], respectively:
where \(\beta _{st} = {\sum }_{X_{s + 1}} \beta _{s + 1t} \phi _{s + 1} h_{s + 1s}\) can be computed with a backward approach. The basic solutions can be obtained by the analogies of detection, tracing and prediction to filtering, smoothing and prediction in a standard HMM.
However, treating all individuals as a single vertex is incapable of tracking the health state of each individual. It is also impractical to apply the basic solutions directly due to the high computational complexity. For example, if we are to track the transmission of an SIStype HAI among n individuals, the computational complexities for the detection and tracing procedures are respectively O(2^{2n}) and O(2^{n}). Thus, we propose an integrated approach of factorization and meanfield analysis to reducing the complexities.
Detection
Since the basic solutions obtained from a standard HMM treat all individuals as a whole, we first factorize the basic solutions for individuallevel detection, and then reduce the computational complexity using meanfield analysis. In this way, we can solve the individuallevel detection problem and substantially improve the solvability for largescale problems.
Assumption 1 (Independence assumption)
The onestepahead prediction probability of a subHMM is independent of those of other subHMMs, or formally
where n is the number of subHMMs.
The state transition probability h_{tt− 1} is determined by SIS dynamics. By Eqs. 5, 6, and 7, the individual states are conditionally independent, that is,
Theorem 1
The solution to the problem of individuallevel detection is given by
and the basic solution to the detection problem can be factored in a product form
Factorization of the basic solution gives the detection probability for each individual. The complexity is reduced to O(2^{n}) after factorization, and the computational burden now becomes the calculation of individuallevel onestepahead prediction probability \(\pi _{tt1}^{i}\). We apply meanfield analysis for solving \( \pi _{tt1}^{i} \) to reduce the overall computational complexity further. The meanfield analysis studies the behavior of a large and complex system in view of simpler systems. Such system considers a large number of small individuals who interact with each other, where the effects of the other individuals on any given individual can be approximated by an averaged effect. With independence assumptions and decomposition, meanfield analysis reduces a multiplebody problem to a onebody problem.
Theorem 2
The solution to the problem of individuallevel onestepahead prediction is given by
where \( p_{tt1}^{i} = \prod \nolimits _{j:v_{j} \in \mathcal {N}_{t1} (v_{i})}\! \left (\pi _{t1}^{j} (1) \!\cdot (1  \tau _{i}) + \pi _{t1}^{j}(0) \right ) \) .
Theorems 1 and 2 enable us to recursively calculate the onestepahead prediction and detection probabilities for each individual. The computational complexity is, therefore, reduced to O(n^{2}) by meanfield analysis.
Algorithm 1 outlines the forward algorithm for solving individuallevel detection. Let detection vector \( \boldsymbol {\pi }_{t}^{i} = \left ({\pi _{t}^{i}}(0), {\pi _{t}^{i}}(1)\right ) \), observation vector \( \boldsymbol {\phi }_{t}^{i} = \left ({\phi }_{t}^{i}({O_{t}^{i}}0), {\phi }_{t}^{i}({O_{t}^{i}}1)\right ) \), and ρ_{i} be the initial distribution of hidden states of individual v_{i}. We initialize and calculate the detection probability for each individual at time 0. Then we recursively compute the onestepahead probability and the detection probability.
Tracing
The basic solution to the tracing problem in the standard HMM requires a backward computing procedure for β_{st},s < t. The procedure introduced in Subsection appears to be not applicable for tracing because β_{st} is not normalized. To derive the tracing probability at an individual level, we rewrite the basic solution based on the following formula [39, 40]:
Theorem 3
The solution to the problem of individuallevel tracing is provided by
and the basic solution to the tracing problem can be factored in the following product form:
where s ≤ t, \( P(x_{s + 1}^{i}= 0{\pi _{s}^{i}}= 1)=\mu _{i} \), and \( P(x_{s + 1}^{i}= 1{\pi _{s}^{i}}= 1)= 1\mu _{i} \).
Algorithm 2 outlines the forwardbackward algorithm for solving individuallevel tracing. We first implement the forward algorithm to obtain the detection probability and the onestepahead probability. Then we set the tracing probability equal to the detection probability at time T. Thus, the tracing probability with time earlier than T can be computed recursively in a backward fashion.
Prediction
We have provided the solution approach to the problem of onestepahead prediction in the previous subsection. Trivially, we can solve the problem of sstep ahead prediction by substituting the onestep transition probability with the sstep transition probability. Here we introduce the pure prediction probability \( \pi _{t0}^{i}(x) \). In pure prediction, no health observations are given, but the initial outbreak information is available. The health states of individuals are completely determined by the initial conditions and epidemic dynamics given by the SIS model. Intuitively, the effect of having no observation at all is equivalent to the setting that all individuals have the same observation at any time. Based on this intuition, we modify the computation for the detection probability \( {\pi _{t}^{i}}(x) \) to derive \( \pi _{t0}^{i}(x) \). We set the observation space to {0} and the observation probability ϕ(⋅,0) to one. By doing so, this tracking procedure becomes pure prediction. We calculate the tracking probability \( {\pi _{t}^{i}}(x) \) under this condition, and we have \( \pi _{t0}^{i}(x)={\pi _{t}^{i}}(x) \) with ϕ(⋅,0) = 1. Pure prediction is consistent with the nonlinear dynamical system (NLDS) discussed in [15]. The model reduces to NLDS when observable features of HAIs are unavailable. As observations provide useful information for more effective estimation of the health states, onestepahead prediction is expected to give better performance than the NLDS or pure prediction.
HAI parameter estimation
In “Individuallevel transmission tracking”, we discussed our modeling framework on tracking HAI transmission under the assumption that all the required parameters – the infection rate, the recovery rate, the initial state distribution and the observation probability matrix – are given. In practice, however, the real values of these parameters are very likely not known exactly. In general, at the beginning of an outbreak, making an initial guess is the only possible option as no prior information is available. In this section we present an estimation method to refine the guess in a stepbystep manner, thus guaranteeing the practicality of our approach for realworld problems. Let \( \boldsymbol {\theta } = (\{ \rho ({x_{0}^{i}}) \}, \{ \tau _{i} \}, \{ \mu _{i} \}, \{ {\phi _{x}^{o}} \}) \) be an HAI parameter configuration. The goal of HAI parameter estimation is to find the best 𝜃 that maximizes the likelihood function \(\mathcal {L}(\boldsymbol {\theta }) = P(O_{0:T}\boldsymbol {\theta }) \),
As analytical global optimal solutions are unlikely to exist, we use the BaumWelch method [41, 42] to solve the problem. Let λ = ({ρ(X)},{h(X,X^{′})},{ϕ(X,O)}). We first consider the coupled HMMs as a single HMM for estimating λ, and then recover the original parameter set 𝜃 from λ. We introduce the auxiliary function
Using Jensen’s inequality, we obtain
Let \( \boldsymbol {\lambda }^{*}=\arg \max \limits _{\bar {\boldsymbol {\lambda }}} Q(\boldsymbol {\lambda }, \bar {\boldsymbol {\lambda }}) \). we have
The above fact suggests that the likelihood of λ never exceeds the likelihood of λ^{∗} and that computing a new estimate \( \bar {\boldsymbol {\lambda }} \) by maximizing the auxiliary function \( Q(\boldsymbol {\lambda },\bar {\boldsymbol {\lambda }}) \) improves the likelihood. Thus, we can derive a maximum likelihood estimate by iteratively updating \( \bar {\boldsymbol {\lambda }} \) until convergence. In this way, the original intractable optimization problem is reduced to maximizing the auxiliary function \(Q(\boldsymbol {\lambda },\bar {\boldsymbol {\lambda }}) \), which can be solved with a Lagrangian multiplier approach efficiently.
Consider the following optimization problem
where \( \mathcal {X} \) is the value set of state vector X.
By substituting
we can write the Lagrangian function as
where η,γ_{j} and ω_{j} are Lagrangian multipliers. The optimization problem thus becomes
which can be separated to three independent maximization problems. Solving each subproblem, we have
where π_{t,t+ 1T}(X,X^{′}) = P(X_{t} = X,X_{t+ 1} = X^{′}O_{0:T}) denotes the probability that the node is at state X at time t and at state X^{′} at time t + 1. As an example, we take the derivative of \( \bar {h}(X,X^{\prime }) \) to obtain the solution to the maximization problem. By letting \( L_{3} \! = \! \sum \limits _{X_{0:T}} P(X_{0:T}O_{0:T}, \boldsymbol {\lambda }) \sum \limits _{t = 0}^{T1} \log \bar {h}({X}_{t}, {X}_{t + 1}) + \sum \limits _{j = 1}^{\mathcal {X}} \omega _{j} \left (\sum \limits _{X^{\prime }\in \mathcal {X}} \bar {h}(X,X^{\prime })  1 \right ) \) and \( X_{0:T}^{\prime }=(X_{0:T}: X_{t}=X,X_{t + 1}=X^{\prime }) \), we have
By setting \(\frac {\partial L_{3}}{\partial \bar {h}(X,X^{\prime })} = 0\), we have
By setting \( \omega _{j} = {\sum }_{t = 0}^{T1} \pi _{tT}(X) \), the third constraint is satisfied, and Equality (20) holds.
Note that \( \bar {\rho }(X_{0}), \bar {\phi }(X,F) \text { and } \bar {h}(X,X^{\prime }) \) in Eqs. 18, 19 and 20 are the same with the BaumWelch reestimate. Iteratively updating \( \bar {\boldsymbol {\lambda }} \) in this manner keeps improving the estimate of λ until it converges. The subsequent step to recover the original parameter configuration \( \bar {\boldsymbol {\theta }} \) from \( \bar {\boldsymbol {\lambda }} \). As an example, we illustrate the recovery of SIS parameters τ_{i} and μ_{i}.
Lemma 1
The probability π_{t,t+ 1T}(X,X^{′}) can be derived from the detection probability π_{t}(X), the tracing probability π_{t+ 1T}(X^{′}), the onestepahead prediction probability π_{t+ 1t}(X^{′}), and the transition probability h(X,X^{′}) by the following formula
Based on the above lemma, we can recover the SIS parameters τ_{i} and μ_{i}, from h(X_{t},X_{t+ 1}).
Let \(\xi _{i} = \prod \limits _{j:v_{j}\in \mathcal {N}_{t}(v_{i})} \left ((1\tau _{i}){\pi _{t}^{j}}(1) + {\pi _{t}^{j}}(0) \right ) = \prod \limits _{j:v_{j}\in \mathcal {N}_{t}(v_{i})} \left (1\tau _{i}{\pi _{t}^{j}}(1) \right ) \approx 1\tau _{i} \cdot {\sum }_{j: v_{j}\in \mathcal {N}_{t}(v_{i})} {\pi _{t}^{j}}(1). \) Then we have
and
The proposed method provides an effective way to infer HAI parameters. Even if complete information on HAI parameters is not available initially, we can resort to this approach to improve estimation of the parameters with the updating observations and the tracked human contact networks. While theoretically the BaumWelch reestimate converges to a local maximum, our computational experiments to be presented in “Computational study” show that our proposed approach has a good performance in the sense that the estimated values are close to the actual ones.
Computational study
In this section, we carry out a computational study, based on a realworld healthcare setting and realworld human tracking data collected from the facility, for assessing the performance of our proposed solution framework and conducting a comparative analysis with other existing epidemic models.
Baseline algorithms
We compare our proposed methods of individuallevel transmission tracking approaches – detection (ILTTDT), tracing (ILTTTR), onestepahead prediction (ILTTPD1) and pure prediction (ILTTPD0) – with the following three baseline methods.

Ordinarydifferentialequationbased SIS model (ODESIS) assumes homogeneous populations. All individuals share the same infection rate τ and recovery rate μ. The initial number of infected individuals I_{0} is an input. The output is the number of infected individuals as a function of time determined by \( I(t) = {I_{\infty }}/\left ({1+\nu e^{(\tau \mu )(tt_{0})} }\right ) \), where ν = I_{∞}/I_{0} − 1 and I_{∞} = (τ − μ)n/τ.

Percolation method (Percolation) is based on probability generating functions and considers disease spread over a heterogeneous network [14]. It requires network degree distributions as input and returns the number of infected individuals at steady state.

Nonlinear dynamical system (NLDS) uses the probability of infection vector (p_{t}) to approximate the infection dynamics and model the evolution of epidemic outbreak over a timevarying network [15]. p_{t} is determined by p_{t+ 1} = g(p_{t}), where the nonlinear function g is defined by p_{i,t+ 1} = 1 − μp_{i,t} − (1 − p_{i,t})ξ_{t}(i), and \( \xi _{t}(i)={\prod }_{j\in \{1,\cdots ,n\}} (1\tau A_{t}(i,j)p_{j,t}) \). This approach requires an adjacency matrix A_{t} to represent the timevarying network at time t.
Setup of experiments
We leveraged the RFID human tracking data collected from two medical wards at PWH, which suffered from a nosocomial outbreak the 2003 SARS, for conducting our computational experiments. The data consists of timestamped locations of 56 patients and 70 healthcare workers in two medical wards over a period of four months. Indoor locations of the tracked objects were recorded every 3 seconds with a spatial resolution of 0.5 meter. We set the time threshold ΔT_{th} = 30 minutes and the distance threshold D_{th} = 1 meter. While these threshold distance and time were chosen to illustrate our idea, our algorithm allows the user to specify these threshold values. We also note that the threshold distance and time depend on the type of the HAI. We constructed static daytime networks and nighttime human contact networks for each time period based on the tracking data; a timevarying hierarchical contact network of 240 time periods was then obtained.
We considered the nosocomial outbreak of the 2013 MERSCoV in Saudi Arabia [3] for deriving practical HAI parameters.
From April 1 to May 23, 2013, thirtyfour individuals, including four healthcare workers, acquired MERSCoV in three healthcare institutions. Fever, cough, shortness of breath and gastrointestinal symptoms were respectively observed in 20, 20, 11 and 8 individuals. As of June 12, 2013 a total of 15 deaths were related to the disease. We prioritized the observations fever, shortness of breath, gastrointestinal symptoms and cough, in a descending order of priority (from 4 to 1). If multiple observations were found at the same time period for an individual, we consider the individual is at the observation state of the highest priority. There were only two health states; each individual either is susceptible (state 0) or infected (state 1). Table 2 provides the health stateobservation probability matrix. We include the observation state 0, which indicates that no symptom was found.
In the experiments, we simulated the health states and observations based on the above setup.
Marcolevel phenomena of hospital outbreaks
Most existing epidemic models focus on the macrolevel phenomena of epidemic thresholds and the infected population. Figure 8 shows the fraction of infections at the steady state for different values of τ/μ on three static networks of snapshots at different time periods, G_{1}, G_{41} and G_{62}, respectively. G_{1} and G_{41} have the same degree distribution whereas G_{62} differs from them and is more sparse. The mean degrees of three networks are 15, 15 and 4, respectively. Although G_{1} and G_{41} have the same degree distribution, the underlying networks are not equivalent. As we observe from Fig. 8a and b, the simulated threshold effects of G_{1} and G_{41} are different: the curves of G_{1} “take off” at around τ/μ = 0.5 while the ones of G_{41} “take off” at around τ/μ = 0.2. Percolation gives the same threshold, τ/μ = 0.4, for G_{1} and G_{41} because it considers only the degree distribution of a network but ignores other network properties. ODESIS gives the same threshold for three different networks because it assumes a homogeneous connection. NLDS captures the difference between the networks but deviates much from the simulation result. Our ILTTPD1 predicts the thresholds to be 0.5, 0.2 and 1.2 for G_{1}, G_{41} and G_{62}, respectively, which are more consistent with the simulated results compared with the other approaches.
Figure 9 shows the plot of infection fraction at different time points resulting from the models for an HAI with infection rate τ = 0.4 and recovery rate μ = 0.3. Since Percolation is not applicable for investigating transient states over time, we compare only ILTTPD1, NLDS and ODESIS with the simulated result. The plots of infection fraction resulting from ILTTPD1, NLDS and Simulation all increase rapidly at t = 3, whereas ODESIS increases at a significantly lower rate. As expected, ODESIS provides the same prediction for three networks because it does not capture the network structures. The simulated infection fraction at steady state ranges from 0.03 to 0.26. Our ILTTPD1 provides a similar prediction to the simulated results (from 0.03 to 0.30), while NLDS has a significantly deviated prediction ranging from 0.33 to 0.54.
Among the baseline algorithms, only NLDS applies to timevarying networks. Figure 10 shows the comparison of NLDS, ILTTPD1, ILTTPD0 and simulated results on the timevarying network G_{0:T}. As shown in Fig. 10a, ILTTPD1 predicts the “takeoff” of the outbreak size at a threshold τ/μ = 0.4 while NLDS and ILTTPD0 predict a lower threshold of 0.3. Figure 10b shows the plot of infection fractions above and below the threshold at different time points. Above the threshold the infection reaches a steady state much higher than the starting point, and below the threshold the infection decays and dies out. Note that ILTTPD0 and NLDS give almost the same prediction in both figures. The reason is that our model reduces to NLDS when observations of HAIs are not available. Taking advantage of observable information improves the accuracy of prediction, which leads to a better performance of ILTTPD1 than ILTTPD0.
Individuallevel transmission tracking
Our proposed method has the capability to track the transmission of HAIs at an individual level. In other words, it infers the hidden health state of any person at any time. In this subsection, we compare the identification results obtained from ILTTPD1, ILTTDT, ILTTTR and NLDS with fixed infection rate τ = 0.03 and recovery rate μ = 0.02. Outcomes resulting from ILTTPD0 are identical to NLDS. Figure 11 shows the estimation of the illness evolution of person v_{1}, who is the patient zero of an HAI. This person stayed infected (at state 1) until time period 103 and remains susceptible (at state 0) from that time onwards. ILTTDT, ILTTTR, ILTTPD1 all capture the state transition close to time step 103 whereas NLDS gives a smooth curve with no clear indication of such change in state. ILTTTR identifies hidden states with the highest accuracy, and the ILTTDT performs better than ILTTPD1.
The Receiver Operating Characteristic (ROC) curves in Fig. 12 exhibit a consistent trend with the results shown in Fig. 11. We observe that ILTTTR demonstrates an advantage over the other approaches and ILTTDT performs slightly better than ILTTPD1 whereas NLDS’s performance is the worst. Their difference in performance is due to the fact that the approaches utilize different degrees of observable information. ILTTTR makes use of all available observable information to obtain the estimation of the hidden Markov processes, while NLDS utilizes no observations at all. ILTTPD1 and ILTTDT both use past observations, but ILTTDT performs slightly better because it also captures observable information at present.
When an HAI outbreak takes place, the healthcare organization conducts contact tracing to identify the index case and construct epidemiological links with a manual approach. If this patient zero has had frequent contacts with the others, the transmission path can only be estimated based on experience [3, 5]. Our method provides an effective tool to construct the transmission network automatically and accurately. For example, if we consider a person as being infected if his/her tracing probability is greater than 0.5, we can draw a transmission map of the hospital outbreak for the first month, as shown in Fig. 13. 27 individuals in total got infected in the first month. The patient zero v_{1} transmitted the disease to 4 other individuals and patient v_{3} is a “superspreader” who infected 7 people. This transmission map is similar to the one reported in [3], a realworld hospital outbreak of the MERSCoV.
HAI parameter estimation
In general, the exact values of HAI parameters were not known exactly. Existing models, which are highly sensitive to the precision of HAI parameters, might produce predictions much deviated from the actual situation if parameters are not determined correctly. Our solution framework is capable of refining the estimate of HAI parameters in a stepbystep manner based on available information of observable features and human contacts. As discussed in “HAI parameter estimation”, our solution framework learns the infection rate τ_{i} and the recovery rate μ_{i} for any individual v_{i}. Without loss of generality, we set τ_{p} and μ_{p} for the patient group, and τ_{c} and μ_{c} for the caregiver group. The following experiment illustrates the estimation of parameters of SIS dynamics.
Figure 14a and b show the estimation of infection rates and recovery rates for patients and caregivers. The dashed lines indicate true parameter values while the solid lines represent the estimated values at each step of estimation. In Fig. 14a, the true value of τ_{p} is 0.3 and the estimate \( \bar {\tau }_{p} \) converges to 0.32 after 6 runs using the proposed method. The true value of τ_{c} is 0.03 and \( \bar {\tau }_{c} \) reaches 0.01 after 4 runs. From Fig. 14b, we observe that \( \bar {\mu }_{p} \) and \( \bar {\mu }_{c} \) converge very quickly to their true values as well.
Figure 14c and d show the average gap between an estimate and the true value for the observation matrix and the initial state distribution. For an m × n matrix B, we define the average gap as
where \( \bar {e}_{i,j} \) is an estimate of element e_{i,j}. In Fig. 14 (c), three sets of estimated observation probability matrices are given with initial average gaps AveGap\(_{0} (\bar {\phi }_{1})= 0.12 \) , AveGap\(_{0} (\bar {\phi }_{2})= 0.18 \) , and AveGap\(_{0} (\bar {\phi }_{3})= 0.31 \) , respectively, where \( \bar {\phi }_{3} \) is generated randomly.
Even though we start with these initial guesses with fairly large average gaps, they converge to zero after only a few iterations. In contrast, Fig. 14d shows that the performance of the inference method is sensitive to the initial guess of the initial state distribution ρ. As the initial guess deviates from the true value gradually from \( \bar {\rho }_{1} \) to \( \bar {\rho }_{3} \), the gaps between the final estimates and the true value become larger. For \( \bar {\rho }_{3} \), there are oscillations and the estimation converges rather slowly but the estimate is far away from the true value. This indicates the importance of an appropriate initial guess of the initial state distribution.
Conclusion
In this work, we propose a solution methodology integrating the techniques of network epidemiology and coupled hidden Markov models to infer the health state of any person at any time in a healthcare setting. We utilize advanced realtime positioning technologies for tracing persontoperson contacts among individuals, including patients and healthcare workers, in the healthcare facility and construct a timevarying human contact network. We also develop the algorithms for transmission tracking of individuals, with a given set of HAI parameters. We finally propose an estimation procedure to infer unknown HAI parameters to tackle the practical problem that the parameters are not completely known.
We conduct experiments based on fourmonth human tracking data collected from two medical wards at PWH, which suffered from the 2003 SARS nosocomial outbreak. Computational results show that our framework provides more accurate results for predicting macrolevel phenomena such as the number of infected individuals and epidemic threshold, compared to existing epidemic models.
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Acknowledgments
We would like to thank patients and staff of Ward 11A and 11C of Prince of Wales Hospital for taking part in this study. Further we express our gratitude to Research Grants Council (RGC) of Hong Kong for providing financial support through GRF to this work (Project No. 14201314 and 14209416). The research of the second author was partially supported by RGC GRF (Project No. 14202115) and Health and Medical Research Fund, Food and Health Bureau, the Hong Kong SAR Government (Project No. 14151771).
Funding
This study was funded by Research Grants Council (RGC) of Hong Kong (Project No. 14201314 and 14209416). The research of the second author was partially supported by RGC GRF (Project No. 14202115) and Health and Medical Research Fund, Food and Health Bureau, the Hong Kong SAR Government (Project No. 14151771).
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Conflict of interests
CH Cheng declares that he has no conflict of interest. YH Kuo declares that he has no conflict of interest. Z Zhou declares that he has no conflict of interest.
Ethical Approval
All procedures performed in studies involving human participants were in accordance with the ethical standards of the institutional and/or national research committee and with the 1964 Helsinki declaration and its later amendments or comparable ethical standards. This article does not contain any studies with animals performed by any of the authors.
Informed Consent
Informed consent was obtained from all individual participants included in the study.
Additional information
This article is part of the Topical Collection on Mobile & Wireless Health
Appendix: Proofs
Appendix: Proofs
Theorem 1
The solution to the problem of individuallevel detection is given by
and the basic solution to the detection problem can be factored in a product form
Proof
By the definition of the onestepahead prediction probability, stated in Eq. 7, and the independence assumption, stated in Eq. 8, we have
The denominator of the basic solution (5) can be rewritten as
The third equality follows from the Markov property. In coupled HMMs, observations are independent of each other and an observation is only affected by its own state. Thus we have
By Eqs. 5, 9, 23, 24 and 25, we have \( \pi _{t} = \prod \nolimits _{i = 1} {\pi _{t}^{i}} \) and \( {\pi _{t}^{i}} = \frac {{\phi _{t}^{i}} \pi _{tt1}^{i}}{\sum \nolimits _{{x_{t}^{i}}} {\phi _{t}^{i}} \pi _{tt1}^{i}} \). □
Theorem 2
The solution to the problem of individuallevel onestepahead prediction is given by
where \( p_{tt1}^{i} = {\prod }_{j:v_{j} \in \mathcal {N}_{t1} (v_{i})} \left (\pi _{t1}^{j} (1) \!\cdot \! (1  \tau _{i}) + \pi _{t1}^{j}(0) \right ) \).
Proof
\(\pi _{tt1}^{i} (0)\) is the estimate that person v_{i} is not infected at time t based on the information about the persontoperson contact history and features of all individuals by time t − 1. We have
According to the SIS dynamics, if \( x_{t1}^{i}= 1 \), \( {x_{t}^{i}} \) is independent of \( x_{t1}^{j} \) for all j≠i. That is,
If \( x_{t1}^{i}= 0 \), \( {x_{t}^{i}} \) is influenced by other \( x_{t1}^{j}, j \neq i \). Let \( X_{t1}^{\sim i} \) be a state vector excluding \( x_{t1}^{i} \), or formally, \( X_{t1}^{\sim i} = (x_{t1}^{1},x_{t1}^{2},..., x_{t1}^{i1},x_{t1}^{i + 1},...,x_{t1}^{n})\). Let
We apply the meanfield analysis approach to decompose the effect of \( X_{t1}^{\sim i} \) on \( {x_{t}^{i}} \) into single average effects. A neighbor v_{j} of v_{i} is infected at time t − 1 with probability \( \pi _{t1}^{j}(1) \), and is susceptible with probability \( \pi _{t1}^{j}(0) \). The average effect of v_{j} on v_{i} for which v_{i} stays susceptible is \( \pi _{t1}^{j}(1) \cdot (1\tau _{i}) + \pi _{t1}^{j}(0) \cdot 1 \). Thus we have
and this completes the proof. □
Theorem 3
The solution to the problem of individuallevel tracing is provided by
and the basic solution to the tracing problem can be factored in the following product form:
where s ≤ t, \( P(x_{s + 1}^{i}= 0{\pi _{s}^{i}}= 1)=\mu _{i} \), and \( P(x_{s + 1}^{i}= 1{\pi _{s}^{i}}= 1)= 1\mu _{i} \).
Proof
We first prove the claim that \( \pi _{st} = \prod \nolimits _{i} \pi _{st}^{i} \) by backward induction. When s = t, this equality holds since \( \pi _{st} = \pi _{t} = \prod \nolimits _{i} {\pi _{t}^{i}} \). Now suppose \( \pi _{s + 1t} = {\prod }_{i} \pi _{s + 1t}^{i} \) for s ≤ t − 1. We have
The second equality holds because X_{s+ 1} is a combination of \( x_{s + 1}^{i} \). From the above equation and the tracing formula, we have \( \pi _{st} = \prod \nolimits _{i} \pi _{st}^{i} \) for s ≤ t − 1, and the claim holds.
As \( h_{s + 1s}^{i} = P(x_{s + 1}^{i}X_{s}) \), we aim to decompose the coupling variable X_{s}. Note that an infected person is not affected by other individuals according to SIS dynamics. Thus we only need to consider the case of \( {x_{s}^{i}}= 1 \). We have
and it completes the proof. □
Lemma 1
The probability π_{t,t+ 1T}(X,X^{′}) can be derived from the detection probability π_{t}(X), the tracing probability π_{t+ 1T}(X^{′}), the onestepahead prediction probability π_{t+ 1t}(X^{′}), and the transition probability h(X,X^{′}) by the following formula
Proof
Let α_{t} and β_{t} be the forward and backward variables respectively as in standard HMMs. Note that
By the above relationships, we have
□
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Cheng, C., Kuo, Y. & Zhou, Z. Tracking Nosocomial Diseases at Individual Level with a RealTime Indoor Positioning System. J Med Syst 42, 222 (2018). https://doi.org/10.1007/s1091601810854
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Keywords
 Healthcareassociated infections
 Disease outbreak
 Tracking
 Traceability
 Persontoperson contact analytics