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Contact tracing is a mitigation strategy that aims at immediately detecting, testing, and treating the next-generation cases during the spreading of an infectious disease. Such local targeted control measure is very effective when the number of cases is limited, for example at the early stage of an outbreak. In 2014, West Africa experienced the most widespread Ebola epidemic in the history with more than 28,000 reported cases. Secondary infections were reported in several European countries and the United States. The Ebola virus is transmitted via physical contact with the infected individuals or their body fluids; the infected ones can transmit the virus to their contacts after becoming symptomatic [4]. In the case of the Ebola epidemic, the objective of contact tracing is to identify and monitor the individuals who have been exposed to the infectious ones for 21 days [27]. This procedure allows for a prompt isolation of the contacts of an infectious individual as soon as he/she becomes symptomatic. Contact tracing has shown effectiveness in several cases. In 2014, there was an Ebola virus disease (EVD) outbreak in Nigeria due to a traveler who returned from Liberia. An extensive contact tracing effort took place starting from day 3 and a total of 894 contacts were traced, all linked to the single index case [21]. Compared to other regions of West Africa, the outbreak of Nigeria was better contained with only 19 confirmed and 1 probable EVD cases out of which 8 died. The improved outcomes can be attributed to the early detection of the index case and effective isolation of infectious individuals due to contact tracing [6]. In September 2014, a person in Dallas - TX, returning from Liberia, was diagnosed with EVD. All of his contacts were traced and monitored for 21 days. Two healthcare workers who provided care for the index case were also diagnosed positive and one of them took a round trip to Cleveland, OH before detection. This prompted the Centers for Disease Control and Prevention (CDC) to trace all the passengers of the two flights. Another case was detected when a person returned to New York from Guinea in October 2014 and contacts of that individual were traced as well. A total of 458 contacts were traced in Texas, Ohio, and New York [26]. Contact tracing was an appropriate approach to stop the transmission of Ebola in the USA as the number of cases reported was quite small [5]. In this paper, we attempt to quantify the effect of contact tracing such early stages of the epidemic.
In general, contact tracing can be carried out using different protocols, depending on the characteristics of the pathogen transmission. Definition of contact, duration and frequency of monitoring, are some examples of variables to consider while implementing contact tracing. Additionally, delays in implementing the contact tracing process are possible in realistic scenarios. Predicting the effectiveness of contact tracing as a function of the disease characteristics, disease stage, and protocol characteristics is a challenging task. Eames and Keeling have proposed a formula to correlate the effectiveness of contact tracing and the basic reproductive ratio by using detailed pairwise equations for a susceptible-infected-removed (
Klinkenberg et al., evaluated the impact of time-related characteristics of the infection and the tracing process on the success of contact tracing [11]. They showed why contact tracing is effective for control of smallpox and SARS, only partially effective for foot-and-mouth disease and likely not effective for influenza [11]. The impact of contact pattern on the efficacy of contact tracing has also been studied. Researchers have concluded that not only the disease properties but also the contact network properties are crucial for the success of contact tracing. For example, Eames and Keeling indicated that contact tracing effectiveness increases with network clustering [5]. Kiss et al., showed that contact tracing is typically ineffective for random contact networks with high average node degrees and small clustering coefficients [10].
Researchers have also attempted to analyze the impact of intervention strategies on the recent Ebola epidemic using mean-field compartmental models, which can be either stochastic or deterministic in nature. Browne et al., used a deterministic version of the compartment models and separated the infected individuals into different compartments based on whether they are hospitalized or unreported. They evaluated the impact of relevant epidemiological properties of Ebola on contact tracing efficiency and presented a formula to determine the minimum number of contacts to be traced per identified infectious individual in order to bring down the effective reproductive ratio below one [1]. In [16], Rizzo et al., adopted a susceptible-exposed-infected-removed (SEIR) compartmental model with additional compartments for hospitalized and dead people who had traditional funerals. Then, they adopted activity driven networks (
Despite being a relatively new area of research, there have been several works involving ADNs. Starnini et al., used ADNs to study temporal percolation properties and showed how SIR models can be mapped to the percolation problem [22]. In another work, they explored the relation between network topology and activity potential distribution to obtain analytical expressions for several topological properties of the integrated social networks [23]. Mata et al., studied a power law distribution of activity potentials and found very slow relaxation dynamics and aging in random walks [13]. Sun et al., investigated ADNs with Markovian and non-Markovian dynamics and found that memory slows down the spreading process in SIR model and boosts the spreading process in SIS models [24]. Perra et al., experimented random walks in time-varying networks and found that results vary significantly. They concluded that the network dynamics should be considered to avoid misleading results in practical situations [15].
For a realistic study of contact tracing effectiveness in the early-stage of an Ebola outbreak, stochastic and individual-level models are needed. When considering a small number of cases, the localized and highly-structured contacts of infected individuals prominently influence the numbers, the timings, and the locations of the future cases. In this scenario, the accuracy of meta-population models, characterized by high levels of aggregation, dramatically deteriorates. The challenge is that successful modeling approaches to evaluate the effectiveness of contact tracing need to take into account the highly structured network of contacts and data on the network of contacts is often not available or too large. The real world networks are not static. The set of people with whom a person remain in contact changes with time. Temporal networks can incorporate these changes and accurately represent real behaviors in human populations. Agent-based models are accurate but computationally expensive. However, ADNs provide a tractable way to produce accurate results [18]. Our work focuses on the microscopic processes that occur at the beginning of an Ebola outbreak. We use ADNs to capture the contact dynamics using a constant underlying stochastic process. We also estimate the basic reproductive number of the disease spread in different scenarios.
In this paper, we evaluate the effectiveness of contact tracing using a novel modeling framework. Our modeling framework has several features to characterize early-stage Ebola transmission:
Our model is built on susceptible-exposed-infected-hospitalized-removed compartments (
The remainder of this paper is organized as follows. In section 2, we propose a compartmental model for Ebola transmission incorporating contact tracing, discuss an overview of activity driven network (
Spreading of an infectious disease is a complex event with many interacting variables. One of the primary tools to analyze and predict the disease diffusion as well as the severity of infectious disease is the compartmental model. Compartmental models are the mathematical frameworks that can capture some major features of epidemic spreading such as pathogen transmission probabilities and host transition rates from one state to another [2]. In this work, we employ a discrete-time expression of the susceptible, exposed, infected, hospitalized and removed (
We classify transitions between different epidemiological compartments of the proposed model into two groups: node-based transition and edge-based transition. In a node-based transition, a node moves from one state to another individually and the transition does not depend on the states of the node's neighbors. Contrary to the nodal transition, the edge-based transition is dependent on the states of a node's neighbors. Based on these definitions, we describe the transition processes of our proposed model as follows:
● Edge-based transition: When a susceptible individual has contact with an infectious or hospitalized one, he/she moves to exposed state with probability
● Node-based transition: An exposed individual undergoes an average incubation period of
A schematic of the epidemiological transition processes of the proposed model is depicted in figure 1.
In our model, we ignore demography and since the population is much greater than the number of infected people, we assume that
Based on the proposed model, we develop a quantitative approach to measure the effectiveness of contact tracing implementations. To asses the impact of contact tracing protocols in Ebola disease spreading before the epidemic phase, we propose two measures: missed-detection probability and contact tracing cost.
Definition 2.1. To assess the risk detection capabilities of contact tracing efforts for Ebola, we introduce the missed-detection probability. Missed-detection probability denotes the probability that a secondary infected individual is not detected before transmitting the virus to others. Based on our model, we propose the missed-detection probability as follows:
Definition 2.2. The aim of contact tracing is to facilitate the detection of secondary infections from the contacts of an infected person. However, a large proportion of an infectious individual's contacts could remain uninfected (susceptible). We define contact tracing cost as the number of detected individuals who had contact with infections but were not infected. Based on the proposed model, the definition of contact tracing cost is:
Disease contagious process and network structure are two important elements which can have significant impacts on disease spreading [5]. Many intervention strategies such as contact tracing, target strategy and egocentric strategy aim at controlling the contagious process based on interactions between individuals in a social network [12] [1]. In particular, the contact tracing strategy or the identification of individuals who have contact with infections is fundamentally linked to potential transmission paths in the network [10] [5]. The goal of contact tracing is to identify all the potential routes in the network and isolate all the new infected individuals, before they become infectious [10]. Here, we implement activity driven network (
Activity driven network considers an activity firing rate
● At time
● Each node
● At time
We implement the activity driven network (
Algorithm 1: ADN for Ebola Contagion Process |
1: Set |
2: while |
3: if |
4: |
5: end if |
where |
6: 1. Network generation: |
Generate a network |
7: 2. Contact identification: |
8: if |
9: |
10: |
11: end if |
where |
12: 3. Contact tracing: |
13: if |
14: for all |
15: Update the states of node |
16: end for |
17: end if |
18: 4. Edge-based transition: Find susceptible nodes in contact with infectious nodes and update their state based on edge-based transition rule. Add all nodes that are neighbors of infected nodes to the set |
19: 5. Node-based transition: Update the states of nodes other than susceptible; |
20: 6. Tracing removal: Remove nodes belonging to |
21: |
22: end while |
The basic reproductive ratio,
Definition 3.1. The basic reproductive number (
$ R_0 = \dfrac{Total \ number \ of \ hosts \ infected \ by \ I_0}{Active \ subset \ of \ I_0} \label{R_0eq} $ | (1) |
where,
A receiver operating characteristic or
The major aim of contact tracing is to identify exposed individuals before they become infectious, in order to halt the chain of pathogen transmission. We define the positive samples as those secondary infectious individuals who had contact with infections during the disease evolution. The true positives in the context of contact tracing are defined as the secondary infectious individuals who were traced. Similarly, false positives are defined as those individuals who were not infected but traced as possible secondary infections. Therefore, based on the different epidemiological compartments in our proposed model, we define
Definition 4.1. We define the true positive ratio (
$ TPR = \dfrac{Number \ of \ exposed \ hosts \ traced}{Total \ exposed \ hosts} \label{tpradn} $ | (2) |
Definition 4.2. We define the false positive ratio or
$ FPR = \dfrac{Number \ of \ usceptible \ hosts \ traced}{Total \ susceptible \ hosts \ with \ infectious \ neighbors} \label{fpradn} $ | (3) |
A point
To generate realizations for Ebola disease spreading without any immunization strategy, parameters of our proposed model for contagious process are given in table 1. The parameter values used in our proposed model were inspired from the works of Rizzo et al.[16]. The time unit is
Parameter | Value |
Transmission probability( |
|
Incubation rate ( |
|
Recovery/removal probability ( |
|
Hospitalization probability in existence of contact tracing ( |
|
Hospitalization probability ( |
|
Parameter | Value |
Density function exponent ( |
|
Links per active node ( |
|
Scaling factor for susceptible ( |
|
Scaling factor for infected ( |
|
Scaling factor for hospitalized ( |
|
To assess the effectiveness of contact tracing in the early stage of the epidemic, we assume three different implementation-time scenarios for contact tracing. The first one is when we implement contact tracing from the beginning (
To measure the effectiveness of contact tracing, we use the term epidemic attack rate (
In figure 3, we study the impact of hospitalization delay,
To evaluate the performance of contact tracing on the Ebola contagious process, we employ the
Using equation 1, we compute the basic reproductive number in a set of
In this paper, we have simulated contact tracing on a compartmental model of Ebola in an activity driven network (ADN). We have performed simulations to analyze the effects of contact tracing initiation delay, contact identification delay, and hospitalization delay. Our results suggest that it is critical to start contact tracing within a few days (
The contacts are usually traced up to the maximum incubation period for the disease [27]. The duration of incubation period has counteracting effects on contact tracing. The contacts need to be monitored longer for diseases with longer incubation periods, which is expensive in terms of resources required. When the incubation time is long, we do not gain much by allocating all resources on immediate identification and monitoring of contacts. On the other hand, longer incubation period provides the tracing agencies more time to identify the potential exposed contacts before they become infectious and start to spread pathogens to other people. On this regard, a long incubation period has a positive impact on the disease control. In our model, "identification" and "monitoring" of contacts happen simultaneously. However, these two processes can be separated. A long incubation time will help better identification. However, monitoring individuals for a longer time is expensive. Due to the long incubation period of Ebola, contact tracing could be inefficient. Hence, immediate hospitalization of the infected cases could be a crucial factor in disease control and infection containment.
A good collaboration between public health authorities and people can lead to rapid identification of secondary infections. Public health authorities should also keep the host population alert and run awareness campaigns to educate people about the disease. Since contact tracing protocols need to monitor all the contacts of the infected and hospitalized people, it cannot separate the exposed contacts from the healthy ones. Therefore, contact tracing might increase the financial burden on the public health authorities when many people are needed to be traced, as happened in the West African countries in 2014. Contact tracing is therefore, economically efficient during the early stage of the epidemic.
Our work can be extended in the future by modifying the activity potentials based on the states of nodes. Hence, the behavioral changes of individuals in response to the disease can be incorporated and properly modeled. It has been found that these changes have a controlling effect on the epidemic [17]. The model can be made even closer to the reality by using temporal networks with memory (non-Markovian link creation process). There is an alternative approach that can be used with ADNs. Instead of using continuous distribution of activity potential and discrete time-steps, we can use a discrete activity potential distribution with continuous time-steps. This can overcome some limitations of the continuous distribution discrete time ADNs and enhance our capabilities. For example, non-exponential inter-event times can be incorporated, and the partition of nodes in several classes based on their activity potentials can help studying the non mean-field dynamics [25].
When it comes to controlling invasion of a disease within a population, the best option is to contain the propagation processes in the early stage and at its source. Contact tracing is one of the important strategies towards this goal; and this applies to many diseases besides Ebola. Behavior of the early-stage dynamic process is very different from large scale outbreak. When number of cases are high, the infection process possesses a stable momentum for spreading which in turn makes modeling easier because mean-field assumption applies. When the number of cases are low, however, the infection process is characterized by extreme randomness and dynamic fluctuations. Furthermore, contact network dynamism is very influential. As a result, mean-field models or models based on quenched/averaged contact networks are not viable candidates. Our modeling effort in this paper and the use of ADN framework calls for further work on efficient and accurate modeling of epidemic processes in heterogeneous populations during early stages after introduction of the infection.
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Parameter | Value |
Transmission probability( |
|
Incubation rate ( |
|
Recovery/removal probability ( |
|
Hospitalization probability in existence of contact tracing ( |
|
Hospitalization probability ( |
|
Parameter | Value |
Density function exponent ( |
|
Links per active node ( |
|
Scaling factor for susceptible ( |
|
Scaling factor for infected ( |
|
Scaling factor for hospitalized ( |
|
Parameter | Value |
Transmission probability( |
|
Incubation rate ( |
|
Recovery/removal probability ( |
|
Hospitalization probability in existence of contact tracing ( |
|
Hospitalization probability ( |
|
Parameter | Value |
Density function exponent ( |
|
Links per active node ( |
|
Scaling factor for susceptible ( |
|
Scaling factor for infected ( |
|
Scaling factor for hospitalized ( |
|