2.1 Dynamics of epidemic spreading and contact tracing

The probabilities that a node i is in each compartment at time t are denoted as , , , , , , and (Susceptible, Latent, Presymptomatic, Symptomatic, Asymptomatic, Detected, and Removed, respectively). The dynamics of spreading of epidemics and different types of contact tracing intervention can be captured by the discrete-time evolution of theses probabilities in the form of microscopic Markov chain approach (MMCA) [6]. The dynamics depend on both time and space, we capture this dependence through explicitly expressing these probabilities as a function of time t and including node identity i as a parameter. We devise a framework to compute these probabilities. The evolution of these probabilities is given by

Note that at each time t. In the equations above, is the probability that a susceptible node i is infected (and then transits to Latent state) at time t, which can be described as (1) where A = (A_jk) is the adjacency matrix of the original graphs G(V, E), in which |V| = N, and A_ij = 1 if there exists an edge between node i and node j, and A_ij = 0 otherwise. Nodes i and j are said to be direct neighbors of each other if A_ij = 1. Note that a node can be infected only through contact with his direct neighbors. The terms in the bracket […] represent the probability that the node i is not infected via contact with j. Thus, represents the probability that the node i is infected by at least one node.

Furthermore, is the probability that a node i in the state X is detected with testing at time t. We start with a sanity check. The above probability when there is no contact tracing and 1-hop contact tracing have been calculated in [7]; our equations in the absence of any contact tracing provide below the same expression as has been obtained by [7] the probability: (2)

In the absence of contact tracing, only symptomatic individuals can be detected, and that too with probability ω. Thus, is ω for symptomatic individuals while it is 0 for individuals who do not display symptoms.

We now proceed to utilize our framework to obtain the above probability when there is traditional 1-hop contact tracing. It is: (3)

The terms in the bracket […] represent the probability that the node i is not detected due to tracing from his direct neighbor j. Thus, represent the probability that the presymptomatic or asymptomatic node i is traced through 1-hop contact tracing. Similarly, represent the probability that the symptomatic node i is detected either through 1-hop contact tracing or because it shows symptoms.

We next compute the probabilities that a node i is detected under 2-hop contact tracing. For simplicity, we consider only 2-hop in the main body of this paper, but we provide the generalization to arbitrary k hops in S1 Text. The 1-hop neighborhood of node i is defined as the set of direct neighbors of a source node i. Denote as the sub-graph consisting of a set of nodes (the 1-hop neighborhood of node i as well as node i) and a set of edges (all the edges between them). The 2-hop neighborhood of node i is defined as the set of nodes that are reachable from the source node i in 2 hops or fewer. Denote as the sub-graph consisting of a set of nodes (the 2-hop neighborhood of node i as well as node i) and a set of edges (all the edges between them); see an example of in Fig 2a, a sub-graph consisting of 2-hop neighborhood from a node i = 1 and edges between them. The challenge is, if is not a tree-like graph, it is very hard to compute the true probability that a node i is detected under 2-hop contact tracing due to the complex correlations among nodes. Thus, in order to compute the approximate probability, we convert the undirected cyclic graph, , into an undirected acyclic graph, , through the following approach.

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Fig 2.

(a) An example sub-graph consisting of 2-hop neighborhood of the source node i = 1 and edges between them. (b) A new acyclic graph converted from the example sub-graph shown in (a). Shadow nodes are colored in blue. (c) Probability of detection for each node i in University Student Network under 2-hop contact tracing: experimental probability (black dots), theoretical probability under the acyclic graph assumption (blue circles), and the difference between the probabilities. Nodes are numbered in ascending order according to their degree. The theoretical probability, , given in Eq (4) closely approximates the experimental probability; the average of the differences is 0.00098.

https://doi.org/10.1371/journal.pone.0288394.g002

Let denote the set of direct neighbors of the source node i (i.e., where ). Let denote the set of nodes at exactly 2-hop from a source node i (i.e., ). We first traverse from the source node i to its direct neighbors, . For 2-hop contact tracing, we then traverse from each node of the set to reachable nodes in the set of . During this process, we can visit the previously visited nodes again. In the example of Fig 2a, we first traverse from the source node 1 to a set of direct neighbors . We then traverse from node to a set of reachable nodes {4, 5, 6, 7} ∈ (Δ⁽¹⁾∪Δ⁽²⁾). Since node 4 is a node that was already visited, we make a copy of node 4 (i.e., shadow node, colored in blue) and append it as a child of 2. We traverse from nodes 2 and 3 through the same process. The Fig 2b shows the acyclic graph converted from the original sub-graph . Note that and . Let denote the adjacency matrix of .

Recall that is the probability that a node i in the state X is detected at time t. The theoretical probability, , for 2-hop contact tracing can be calculated using the acyclic graph assumption: (4)

The terms in the bracket […] represent the probability that the node i is not detected by any node in the particular branch including node j (e.g., in the graph shown in Fig 2b, there are three branches in the network from the node i = 1, each containing nodes 2, 3 and 4, respectively). Thus represent the probability that the node i is detected by any of the nodes within radius 2 through 2-hop contact tracing. The second term in the bracket represents the probability that node i is detected by node j (1-hop neighbor). The third term represents the probability that node i is not detected by node j (1-hop neighbor) but is detected by any of the node’s 2-hop neighbors in the particular branch including node j (e.g., if i = 1 and j = 2, the third term represents the probability that node i = 1 is not detected by node j = 2 but is detected by any of the nodes 4, 5, 6, and 7 in the graph shown in Fig 2b).

To verify the accuracy of the approximate formulation, for each node i, we compare the output of the Eq (4) (under the assumption of acyclic graph, ) to the empirical detection probability obtained from simulation (under the original graph, , without the assumption of acyclic graph). We compute the probability of detection for each node i in a data-driven network (a total of 672 nodes and a set of 32, 689 edges) under 2-hop contact tracing scheme with f = 0.1 (refer to Section 3 for details on the data-driven network). We assume that a random 1% of the population is in Detected state. The experimental probability without the acyclic assumption is calculated by dividing the number of outcomes with detection by the total number of 100, 000 trials. Fig 2c shows that the output of Eq (4) closely approximates the experimental probability despite the assumption.

Next, we compute the probability that a node i is tested. The probability can be derived from an adaptation of the probability that a node i is detected. Individuals in symptomatic (I_s) state can be tested and detected either after showing symptoms or via contact tracing; thus, the probability that an individual in symptomatic (I_s) state is tested at time t is equivalent to the probability that the person is detected at time t. Moreover, individuals in asymptomatic (I_a) and presymptomatic (I_p) states can be tested and detected only via contact tracing, and individuals in Susceptible (S) and Latent (L) state are tested only if they are contact traced by their neighbors, but they do not transit to the detected state; thus, the probability that an individual in those states is tested at time t is equivalent to the probability that the person is detected via contact tracing at time t. Hence, the probability that a node i in state X is tested at time t is (5)

2.2 Evaluation metrics

Using the expressions for the probabilities of states and state transitions obtained in Section 2.1 as a function of time and neighborhood for each node, we define metrics that will be used to quantify the cost-benefit tradeoff of different contact tracing strategies. We first present expressions for the 1) expected number of nodes detected, 2) expected number of nodes detected as a result of contact tracing and expected number of nodes detected as a result of tests initiated because they show symptoms 3) expected number of tests, for the contact tracing strategies we consider. We subsequently obtain expressions for these attributes over time in the time horizon [1, …, τ] that we consider (Section 2.2.1). We use the above expressions to compare the outbreak sizes, cumulative number of nodes quarantined, tested and detected under the differing contact tracing strategies we consider. When we compare these attributes across contact tracing strategies we distinguish the attributes associated with different contact tracing strategies using the number of hops k of the contact tracing strategies in subscript (Section 2.2.2). Refer to Table 2 for the evaluation metrics.

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Table 2. Evaluation metrics.

https://doi.org/10.1371/journal.pone.0288394.t002

3.1 Empirical validation

In this section, we validate our model through a comparison between results of Monte Carlo (MC) simulations and MMCA formulation. Considering both the data-driven network we used in Section 2, and synthetic networks (specifically Watts-Strogatz network [9] and scale-free network [10]), we show that the computations from our model equations relatively closely match the results of MC simulations. The data-driven network we consider is generated from data collected by smartphones of University students, as part of the Copenhagen Networks Study [11]. The smartphones were equipped with Bluetooth cards which recorded proximity between participating students at 5-minute resolution. According to the definition of close contact by CDC [12], we only used proximity events between individuals that lasted more than 15 minutes in a row in the same day to construct daily contact networks. We postulated that two individuals had a contact if there are at least three consecutive proximity events with a 5-minute resolution between them. The constructed contact network has 32, 689 contacts among 672 individuals. The advantage of this data set is that it provides actual proximity events of a moderate number of individuals. The synthetic networks we consider have N = 1, 000 nodes and average degrees of 〈k〉 = 8.

We now describe the synthetic networks we consider in this section. Watts-Strogatz network is a synthetic network generated as follows. We start with a regular ring lattice and each endpoint of each edge is rewired with a certain probability to another node chosen uniformly randomly over the rest of the ring. We choose the Watts-Strogatz as representative of synthetic networks because two important attributes that determine the rate of spread of infectious diseases in contact networks, namely average path lengths, and clustering coefficients, can be controlled in this class through the choice of a parameter, namely the rewiring probability. Accordingly, this class of networks has been utilized extensively for simulation studies in spread and control of infectious diseases over contact networks [13]. Note that the average path length is defined as follows for a given network. The distance between a pair of nodes is the length of the shortest path between them, i.e., the number of links in the path between them that has the minimum number of links among all paths between them. The average path length for a given network is the average of this distance over all pairs of nodes in it. Clustering coefficient is the average of C_i over all nodes i where C_i is defined as the ratio between the actual number of links between the neighbors of i and the maximum possible number of links between the neighbors of i [9]. As the rewiring probability increases from 0 to 1, 1) average path lengths range from linear to logarithmic functions of the number of nodes 2) clustering coefficients range from high to vanishingly small [9, 14]. The special case of the Watts-Strogatz model in which the rewiring probability is 1 corresponds to a variant of the Erdős Rényi random networks which has also been extensively utilized in the study of spread and control of infectious diseases [15, 16]; we consider this variant as well. Furthermore, we consider scale-free network (generated by Barabási-Albert model) to capture the existence of hubs with an excessive number of connections. Unlike scale-free network, the probability of the existence of the hubs in Watts–Strogatz network is low.

We first compare the total infection counts over the course of 1 year (i.e., τ = 365) obtained from MMCA formulation with an average of 200 runs of the MC simulations under the data-driven network and the Erdős Rényi random networks. We consider different values of the activation probability f under 1-hop and 2-hop contact tracing and focus on average and maximum percentage discrepancies or errors. We assume that a random 1% of the population is initially infected and is in a Latent state. Fig 3 represents the total infection count as a function of transmission probability β under 1-hop contact tracing for various values of f. This shows that MMCA formulation closely approximate the MC simulation results under 1-hop contact tracing. The trend of the variation of the total infection count as a function of β is the same for both MMCA and MC. We compare the results of MMCA and MC for a total of 201 values of β, varying from 0 to 1 with uniform interval 0.005 (with the exception of replacing 0 with 0.001). We compute the discrepancy, i.e., , for each value of β, and average the discrepancies over all the values of β. For data-driven network, the average discrepancy between the two is (a) 0.1% for f = 0.3, (b) 0.2% for f = 0.5, and (c) 0.4% for f = 0.9; for random network, the average discrepancy is (d) 1.5% for f = 0.3, (e) 1.5% for f = 0.5, and (f) 2.9% for f = 0.9; for scale-free network, the average discrepancy between the two is (g) 1.8% for f = 0.3, (h) 2.2% for f = 0.5, and (i) 3.4% for f = 0.9. The average discrepancy is least for data-driven network, next lowest for random network and highest for scale-free network. All the average discrepancies are nonetheless low, below 3.5%.

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Fig 3. The total infection count I as a function of β ∈ [0.001, 1] for 1-hop contact tracing for various contact networks.

The first, second, and third rows respectively represent Data-Driven network, Erdős-Rényi Random Network, and Scale-Free Network. For each network topology, we consider various values of activation probability f = 0.3, 0.5, 0.9. The lines represent the MMCA formulation and the points represent MC simulation. The MMCA formulation closely approximates the MC simulation results under 1-hop contact tracing.

https://doi.org/10.1371/journal.pone.0288394.g003

Fig 4 shows a comparison between MMCA formulation and MC simulation under 2-hop contact tracing. MMCA still approximates the MC simulation reasonably well. Again the trend of the variation of the total infection count as a function of β is the same for both MMCA and MC. For data-driven network, the average discrepancy between the two is (a) 0.2% for f = 0.3, (b) 0.8% for f = 0.5, and (c) 1.5% for f = 0.9; for random network, the average discrepancy is (d) 1.5% for f = 0.3, (e) 2.1% for f = 0.5, and (f) 5.9% for f = 0.9; for scale-free network, the average discrepancy is (g) 2.6% for f = 0.3, (h) 4.4% for f = 0.5, and (i) 8.2% for f = 0.9. Again, the average discrepancy is the least for data-driven network, next lowest for random network and highest for scale-free network. All the average discrepancies are nonetheless low, below 4.5%, except in one case that of the combination of very high value of f and scale-free network. Even for the last combination, the average discrepancy is below 10% and therefore deemed low.

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Fig 4. The total infection count I as a function of β ∈ [0.001, 1] for 2-hop contact tracing for various contact networks.

The first, second, and third rows respectively represent Data-Driven network, Erdős-Rényi Random Network, and Scale-Free Network. For each network topology, we consider various values of activation probability f = 0.3, 0.5, 0.9. The lines represent the MMCA formulation and the points represent MC simulation. The MMCA formulation closely approximates the MC simulation results under 2-hop contact tracing.

https://doi.org/10.1371/journal.pone.0288394.g004

The mismatch that we observe arises because the model equations (refer to Eq (1)) were derived under simplifying assumptions and approximations. The assumption was that the infection status of different nodes constitute independent random variables, but in practice the infection statuses of nodes exhibit stochastic correlation and the correlation typically decreases with increase in distance between concerned nodes in the contact graph. This assumption is somewhat commonplace in works that model contact tracing and the spread of epidemic through contacts, and has for example been resorted to in [6]. For k-hop contact tracing where k > 1, there is an additional approximation: we approximate a cyclic contact graph as an acylic one through a specific construction. The results discussed above show that the discrepancy under 2-hop contact tracing is greater than the one under 1-hop contact tracing, which is likely because of this additional approximation. Furthermore, the results above shows that, for both 1-hop and 2-hop contact tracing, the discrepancy increases as the value of f increases. This is because as f increases greater number of contacts can be traced, which increases the correlations between the infection status of nodes by facilitating detection of infection in neighbors of detected nodes.

Next, we consider Watts-Strogatz networks with various values of rewiring probability r. Fig 5 represents the total infection count as a function of β for various values of r, 2-hop contact tracing and f = 0.3. This shows that average error decreases with increase in r; the error is (a) 2.4% for r = 0.1, (b) 1.6% for r = 0.5, and (c) 1.5% for r = 1.0. Fig 6 shows a comparison for 2-hop contact tracing and f = 0.7. It also confirms the phenomenon observed in Fig 5; the average error is (a) 5.3% for r = 0.1, (b) 4.1% for r = 0.5, and (c) 3.5% for r = 1.0.

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Fig 5. The total infection count I as a function of β under 2-hop contact tracing with f = 0.3 for various values of rewiring probability (a) r = 0.1, (b) r = 0.5, and (c) r = 1.0 for Watts-Strogatz networks.

https://doi.org/10.1371/journal.pone.0288394.g005

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Fig 6. The total infection count I as a function of β under 2-hop contact tracing with f = 0.7 for various values of rewiring probability (a) r = 0.1, (b) r = 0.5, and (c) r = 1.0 for Watts-Strogatz networks.

https://doi.org/10.1371/journal.pone.0288394.g006

3.2 Use of the model for cost-benefit analysis

The Markovian equations enable us to explore the effectiveness of epidemic control via different types of contact tracing strategies on any contact network. Using the data-driven network, we evaluate the impact of different types of contact tracing strategies on benefit (i.e., reduction in outbreak size) and costs (i.e., the number of quarantines and tests). In this section, we show that the cost-benefit tradeoff can be enhanced through an implementation of the multi-hop contact tracing. We here assume that a random 1% of the population is initially infected and is in a Latent state.

3.2.1 Effectiveness of epidemic control via contact tracing.

We first compare the outbreak size (i.e., the total number of infections) over the course of 1 year (i.e., τ = 365) for different types of contact tracing strategies. We evaluate the ratio RI_k-hop of cases infected via k-hop contact tracing by time τ to the cases infected when no contact tracing is performed, with respect to the activation probability f and attack rate. RI_k-hop is less (greater, respectively) than 1 if k-hop contact tracing results in fewer (greater, respectively) number of infections than 0-hop contact tracing. The attack rate is defined as the proportion of individuals in a population who has been infected during the course of an epidemic by time τ when no contact tracing is performed, i.e., I_0-hop/N, which characterizes the intrinsic speed of virus spread in the absence of contact tracing. We consider f in the range of [0.1, 1.0] with uniform interval 0.1, and also consider the attack rate in the range of [0.2, 0.9] by varying β with uniform interval 0.0001. Recall that greater the value of f greater is the number of contacts available for tracing and more effective contact tracing is expected to be. Fig 7a shows that RI_1-hop is less than 1 in the range of all values of the variables (i.e., f and attack rate) we consider, i.e., 1-hop contact tracing enables the reduction in the number of infections across all the ranges. As f increases and attack rate decreases, RI_1-hop decreases; in an ideal scenario of the highest f, i.e., f = 1 and the lowest attack rate 0.2, RI_1-hop is lower bounded by 0.41. Fig 7b shows that RI_2-hop is also less than 1 regardless of the values of f and attack rate we consider, and more importantly it shows that RI_2-hop is significantly lower than RI_1-hop; RI_2-hop is lower bounded by 0.04. This shows that a slightly more aggressive policy of simply adding one more hop, 2-hop contact tracing, can significantly contain the spread of epidemics. In particular, even in an undesirable case with a high attack rate, a slight increase in f under 2-hop contact tracing scheme can lead to a dramatic decrease in outbreak size.

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Fig 7. The ratios (a) RI_1-hop and (b) RI_2-hop are expressed in color as a function of attack rate (x-axis) and activation probability f (y-axis).

We choose different color codes to represent different ranges of values of the ratios (i.e., RI_1-hop and RI_2-hop) corresponding to values of f and attack rate. We use the data-driven network for this figure.

https://doi.org/10.1371/journal.pone.0288394.g007

We now try to understand why 2-hop contact tracing has a significantly lower infection count as compared to 1-hop contact tracing. For this we plot the total number of infections (I), the total number of detected infections (D), and undetected infections (U) over the course of 1 year (i.e., τ = 365) for the two contact tracing strategies (Fig 8). We plot I, D, U as functions of activation probability f when attack rate is 0.8. The number of infections slowly decreases under 1-hop contact tracing as f increases, but there are still non-negligible number of undetected infections even in an ideal scenario of f = 1. The undetected infections continue to spread the infection leading to a large overall infection count. On the other hand, in case of 2-hop contact tracing, even a small increase in f can significantly reduce the number of undetected infections to almost zero. Thus infections are being detected and consequently isolated (quarantined). Thus not many nodes can spread the infection, which consequently significantly reduce the number of infections with even a small increase in f. It is clear that 2-hop contact tracing can effectively contain the epidemic even in challenging environments with a small f, primarily because it can detect and therefore isolate most of the infected individuals.

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Fig 8. The expected number of infections (I), and the number of detected infections (D) and undetected infections (U) as a function of activation probability f.

We use the data-driven network for this figure.

https://doi.org/10.1371/journal.pone.0288394.g008

Next, we analyze the connectivity pattern of those cases infected and detected for different contact tracing strategies through computing 1) i(k), the probability that a node of degree k has been infected by time τ, 2) d(k), the probability that a node of degree k has been detected by time τ, and 3) u(k) = i(k) − d(k), the probability that a node of degree k has been infected but not detected by time τ. In Fig 9a, we plot i(k), d(k), and u(k) when no contact tracing is performed. We set f = 0.7 and the attack rate to 0.8. This shows that i(k) increases with increase in the degree k, i.e., the greater number of contacts an individual has, the more likely he is to get infected. We observe that d(k) also increases with increase in degree k; this is because nodes with higher number of contacts are more likely to be infected and consequently more likely to be detected after showing symptoms. However, u(k) is as large as d(k). Therefore, a large number of superspreaders (i.e., infected nodes with large degrees) are evading detection when no contact tracing is performed.

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Fig 9. Infections, detected infections, and undetected infections as a function of degree k for different contact tracing strategies.

This figure shows the probability that a node of degree k has been infected i(k), detected d(k), and infected but not detected u(k)≔ i(k) − d(k). We use the data-driven network for this figure.

https://doi.org/10.1371/journal.pone.0288394.g009

An outbreak can be effectively controlled only if the superspreaders are detected early on. When 1-hop contact tracing is performed, u(k) decreases significantly across the values of degree k as compared to the case of no contact tracing (Fig 9b). In particular, u(k) displays a maximum at some degree k, and beyond this maximum value, u(k) decreases with increase in k. Thus contact tracing is detecting superspreaders of very high degrees. Moreover, as shown in Fig 9c, 2-hop contact tracing demonstrates maximum efficiency. u(k) approaches zero for all values of degree k meaning that infected nodes have almost zero probability of evading detection.

3.2.2 Cost-effectiveness of contact tracing.

We have shown that detecting and isolating the infected individuals through 2-hop contact tracing can stop greater number of infected individuals from further spreading the disease as compared to 1-hop contact tracing. But 2-hop contact tracing can also involve an increase in the number of people tested and quarantined as compared to 1-hop and no contact tracing schemes. We show how our framework can determine if this is indeed the case by allowing the evaluation of the above quantities which represent the costs associated with contact tracing strategies. This in turn helps public health authorities decide the level of contact tracing a public health system ought to opt for based on the magnitude of the costs it can afford. Using the results from our framework, we argue that 2-hop provides a more favarable cost-benefit tradeoff compared to 0-hop and 1-hop.

We first compare the quarantining cost, i.e., the total number of people that need to be quarantined (equivalently, the total number of people detected), over the course of 1 year (i.e., τ = 365) for different types of contact tracing strategies. We compute the ratio RD_k-hop of the total number of individuals detected by time τ under k-hop contact tracing to that under 0-hop.

We first plot RD_1-hop with respect to f and attack rate. Fig 10a shows that as f decreases and attack rate increases, RD_1-hop increases; RD_1-hop is greater than 1 in higher region of attack rate and lower region of f, i.e., 1-hop contact tracing quarantines a greater number of individuals compared to the case of no contact tracing. In particular, when the attack rate is very high, 1-hop contact tracing quarantines a greater number of individuals regardless of the value of f. On the other hand, Fig 10b shows that RD_2-hop is less than 1, and that RD_2-hop is significantly lower than RD_1-hop, across almost all values of f and attack rate we consider. Thus, 2-hop contact tracing substantially decreases the overall number of quarantines (i.e. quarantine cost) compared to traditional 1-hop contact tracing regardless of f and attack rate, despite the fact that 2-hop controls the outbreak more effectively. Moreover, 2-hop contact tracing significantly reduces the number of quarantines required even compared to the case of no contact tracing in all but extremely challenging environments; see bottom right region surrounded by black line in Fig 10b which corresponds to very low f (therefore most contacts can not be traced) and very high attack rate.

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Fig 10. The ratios (a) RD_1-hop and (b) RD_2-hop are expressed in color as a function of attack rate (x-axis) and activation probability f (y-axis).

We choose different color codes to represent different ranges of values of the ratios (i.e., RD_1-hop and RD_2-hop) corresponding to values of f and attack rate. We have highlighted the case RD_k-hop = 1 (black line) signaling that 0-hop and k-hop quarantine the same number of individuals. We use the data-driven network for this figure.

https://doi.org/10.1371/journal.pone.0288394.g010

We now explain the apparently counter-intuitive result, that is, as to why 2-hop contact tracing quarantines substantially fewer number of individuals as compared to 1-hop or 0-hop contact tracing. Recall that , where is the ratio of the number of individuals detected via k-hop contact tracing by time τ to the number of individuals detected under 0-hop and is the ratio of the number of individuals detected because they tested as a result of showing symptoms by time τ to the number of individuals detected under 0-hop. We plot RD_k-hop, , and for different contact tracing strategies, as a function of f when attack rate is 0.8 (Fig 11). For 1-hop contact tracing (left plot in Fig 11), the number of quarantines required slowly decreases (from 1.13 to 1) as f increases. In particular, increases with f while decreases. Nevertheless, both detection via contact tracing and symptom-based detection contribute significantly to the number of quarantines for all values of f. On the other hand, in case of 2-hop contact tracing (right plot in Fig 11), even a small increase in f reduces to a value close to 0. This suggests that the number of individuals with symptoms (and therefore the number of individuals infected as a certain fixed fraction of infected individuals develop symptoms) reduce to near 0 with even a modest increase in f (the probability that a contact can be traced). Thus, the overall infection count becomes very small with a modest increase in f. Once the overall infection count becomes small, the number of individuals detected would have to become small, this in turn reduces the number of primary contacts and therefore the number of secondary contacts traced. Thus, also rapidly decreases with increase in f. Naturally then RD_2-hop, which equals , also significantly reduces as f increases.

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Fig 11. RD_k-hop, , with respect to the activation probability f.

We use the data-driven network for this figure.

https://doi.org/10.1371/journal.pone.0288394.g011

It however turns out that 2-hop contact tracing needs to test a greater number of individuals compared to 1-hop contact tracing. Fig 12 shows that the ratio of total tests via 2-hop contact tracing to total tests via 1-hop (i.e., T_2-hop/T_1-hop) is greater than 1 across the values of f and attack rate we consider. In addition, the ratio decreases with f and increases with attack rate. Thus, 2-hop contact tracing incurs greater testing cost as compared to 1-hop contact tracing.

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Fig 12. The ratio of total tests via 2-hop contact tracing to total tests via 1-hop (i.e., T_2-hop/T_1-hop) are expressed in color as a function of attack rate (x-axis) and activation probability f (y-axis).

We choose different color codes to represent different ranges of values of the ratio (i.e., T_2-hop/T_1-hop) corresponding to values of f and attack rate. We use the data-driven network for this figure.

https://doi.org/10.1371/journal.pone.0288394.g012

We now argue that for any contact tracing scheme, the number of individuals quarantined should be considered a more significant component of the overall cost than the number of tests. Note that various countries, at least the developed ones, could quickly scale up testing capacity within a few months of start of Covid-19. In general, an individual finds it more disruptive to be quarantined than to be tested because the former usually involves several days of not being able to discharge regular professional and social responsibilities. The following have been documented in 2 years of Covid. Long-term quarantines significantly increase non-COVID-19 fatalities [17]. During lockdowns, which represents long-term quarantines, there has been an increase in drug overdose deaths (eg, in Ontario and British Columbia), possibly as a result of isolation [18]. Lockdowns have been associated with an increase in domestic violence (e.g., 30% more in France and 25% more in Argentina) [19]. Lockdowns have wreaked havoc on economies around the world, especially in developing nations [20, 21]. Quantifying the negative effects of all of the aforementioned are challenging. Overall, nevertheless, the substantially greater quarantine cost may be more important than other costs.

Thus, since 2-hop contact tracing significantly lowers 1) the most significant component of the costs, and 2) the overall infection count compared to 1-hop and 0-hop, the cost-benefit tradeoff for 2-hop may be considered more favorable compared to 1-hop and 0-hop.

3.3 Computation time

We finally compare the computational time required in generating the output of the MMCA model formulation with that of running 500 runs of MC simulations. We observe the change in computation time required for both with the size of Erdős Rényi random network, from 1, 000 to 20, 000 nodes and fixed mean degree 〈k〉 = 8. We set β = 0.001 and f = 0.5. Fig 13 shows that computation times for both MC simulations and MMCA formulation increase linearly with the network size (i.e., the numbers of nodes), but the computation time for MC simulations increases much more rapidly. In addition, MC simulations usually requires a significantly larger number of iterative runs for accurate statistics based on samples (simulations run). As a result, the computation time for MC simulation increases even more steeply with network size if a greater number of runs are made, and the difference in computation time between MC simulation and MMCA may become greater as the former requires greater accuracy. In practice, contact networks can be large, e.g., population in large cities in US are of the order of millions (e.g., New York city had more than 8 million residents in 2021) while small towns in US have thousands of residents. Thus, MMCA can scale to contact networks that arise in practice in a computationally efficient manner, which MC simulations may not be able to accommodate in reasonable computation times. We used Amazon Web Services (AWS) m5a.large instances to compare the computation time.

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Fig 13. Computation time as a function of population size for Erdős Rényi random network.

https://doi.org/10.1371/journal.pone.0288394.g013

4.1 Combination of contact tracing and random testing

We consider a combination of contact tracing and random testing in which a node can be tested with probability δ for each time step even if he is not in a k-hop neighborhood of a detected node or even if he does not show symptoms. We consider k = 1, 2 in this section, but the framework directly generalizes to larger values of k through an extension of the k-hop contact tracing framework provided in the S1 Text. Specifically, earlier, pre-symptomatic and asypmtomatic nodes could be detected only through contact testing; now such nodes can be detected either through random testing or through contact tracing. Earlier symptomatic nodes could be detected through contact tracing, or testing with probability ω because of showing symptoms; now they may also be detected through random testing.

Under this approach, only the value of the probability that a node i in the state X is detected through testing at time t, , need to be adapted. For k = 1, (16)

When k = 2, the term in bracket […] need to be replaced by in Eq 4. The rest of the framework presented in Section 2.1 holds as is once the modified value of is used therein.

We use this framework to compare 2-hop contact tracing with the combination of 1-hop contact tracing and random testing. For the latter, i.e., ‘1-hop + random testing’ approach, we choose δ so that the total number of tests for ‘2-hop’ and ‘1-hop + random testing’ are almost the same (i.e., ) (Note that T_{1-hop+ random} can be obtained from Section 2.1 by using computed above for k = 1 in place of computed for k = 1 in Section 2.1.). We choose τ as 365 days as before. We then compare the total number of infections for ‘2-hop’ and ‘1-hop + random testing’. Fig 14 shows that, despite both 2-hop’ and ‘1-hop + random testing’ incurring the same number of tests, 2-hop contact tracing provides significant advantage over ‘1-hop + random testing’ strategy in terms of outbreak size for all attack rates. This demonstrates that the cost-benefit tradeoff can be enhanced through an implementation of the multi-hop contact tracing.

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Fig 14.

(a) The ratio of number of tests via 2-hop contact tracing to that under the combination of 1-hop contact tracing and random testing, . (b) The ratio of cases infected via 2-hop contact tracing to the cases infected via ‘1-hop + random testing’, . We use the data-driven network for this figure.

https://doi.org/10.1371/journal.pone.0288394.g014

4.2 Contact tracing under testing fatigue

We here study a realistic generalization of our strategies which reduces the average daily number of tests of contact tracing strategies and thereby satisfy budgetary constraints on this average through appropriate selection of parameters. In Section 2, for the sake of model simplicity, we had assumed that individuals can be repeatedly tested if traced from contacts who test positive regardless of the outcome of their previous tests. But in reality, individuals develop a testing fatigue if they are frequently subjected to repeated tests, and the tests are negative. Due to this fatigue, individuals may therefore refuse to be tested shortly after a negative test, unless they develop symptoms, even if traced from contacts who test positive. Public health bodies may also avoid recommending tests for an individual shortly after he tests negative unless he develops symptoms, to avoid testing fatigue, elicit his cooperation in future, and satisfy constraints on the overall number of tests. We therefore consider that an individual is not tested for a random duration after he is tested negative unless he develops symptoms, even if he is traced from a 1-hop or 2-hop contact; if he tests positive he is quarantined and not tested again as we don’t consider reinfection. The random duration is geometrically distributed with mean 1/ξ. We show that ξ can be tuned to have both 1-hop and 2-hop satisfy budgetary constraints on daily average number of tests, reduce the difference between the daily average number of tests of 2-hop and 1-hop, and still reduce the outbreak size through 2-hop compared to 1-hop.

Modeling the above requires a generalization of our state diagram and modeling equations (Fig 15). We consider that right after an individual tests negative he becomes “not-ready-to-test” for a random time. In this condition he does not test unless he develops symptoms. We split the states up to symptomatic (namely, susceptible, latent, presymptomatic, asymptomatic) into two distinct states representing whether 1) individuals are ready to test or 2) otherwise. Thus, susceptible state is split into ready-to-test susceptible (S1) and not-ready-to-test susceptible (S2), latent state is split into ready-to-test latent (L1) and not-ready-to-test latent (L2), etc.

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Fig 15. A pictorial illustration of state transitions when there is test fatigue.

https://doi.org/10.1371/journal.pone.0288394.g015

Once an S1 (S2, respectively) individual is infected, he transitions to L1 (L2, respectively). An S1 (L1, respectively) individual can be tested as a result of contact tracing, if he is tested, he tests negative and transitions to S2 (L2, respectively). An S2 (L2, respectively) individual is not tested, even if he is traced from a 1-hop or 2-hop contact, but he transitions to S1 (L1, respectively) at any given time with probability ξ. Basically S1’s possible next states are S2, L1, S2’s possible next states are S1, L2.

Presymptomatics (Asymptomatics, respectively) are also split into ready to test and not ready to test versions: I_p1, I_p2 (I_a1, I_a2, respectively). Upon progression of the disease, L1 transitions to I_p1 or I_a1, L2 transitions to I_p2 or I_a2. An individual in I_p1, I_a1 state is tested if he is traced from a 1-hop or 2-hop contact, when he is tested he tests positive and transitions to the detected state. An individual in I_p2 (I_a2, respectively) state is not tested even if he is traced from a 1-hop or 2-hop contact, but at each time t he becomes ready to test, i.e., transitions to I_p1 (I_a1, respectively), with probability ξ. A symptomatic (I_s) is always ready to test as he experiences symptoms. Thus, we don’t split the symptomatic state, and upon progression of the disease, both I_p1, I_p2 become symptomatic. We don’t split the removed (R) state either. Thus, I_p1’s next states are D, I_s, I_p2’s next states are I_p1, I_s, I_a1’s next states are D, R, I_a2’s next states are I_a1, R.

The probability that a susceptible node i is infected (and then transits to Latent state) at time t is the same for S1, S2, as the contact rates and the probability that a contact transmits the contagion do not depend on the readiness to be tested. Hence we still denote it as , i.e., we use a generic S instead of S1, S2. But its value will be different from when the states are not partitioned, i.e., when everyone is willing to be tested all the time. When states are partitioned, its value is: (17) where as before is the probability that node i is in state X at time t, A = (A_jk) is the adjacency matrix of the original graphs G(V, E), in which |V| = N, and A_ij = 1 if there exists an edge between node i and node j, and A_ij = 0 otherwise.

Recall also that and are the probabilities that a node i in the state X is detected and tested at time t, respectively. Note that state names containing the number 2 are neither tested nor detected. To simplify the notation, numbers corresponding to partitions of states (i.e., 1 and 2) are excluded from the notation of these probabilities (e.g., instead of ). The expressions for these probabilities, and , are the same as expressions as in equations in Section 2. The dynamics of spreading of epidemics and different types of contact tracing intervention can be adapted as follows: (18)

We consider an example attack rate, 0.21. We first assume that an individual who tests negative can be tested again after 14 days on average (ξ = 1/14), which is reasonable. This corresponds to a situation where average number of tests per day is less than 27 for 2-hop and 1-hop (the left in Fig 16a). Considering that the population size is 672, the average daily testing load is less than 4% of the total population. However, as shown on the right in Fig 16a, 2-hop can reduce the outbreak size by up to 73% compared to 1-hop and 87% compared to 0-hop.

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Fig 16. The average number of tests per day and fraction of the populace infected via 1-hop and 2-hop as a function of activation probability f.

Figures (a), (b), (c), and (d) show the results for ξ = 1/14, ξ = 1/60, ξ = 1/90 and ξ = 0, respectively. The horizontal dotted lines in the figures on the right represent the number of individuals infected when no contact tracing is performed. We use the data-driven network for this figure.

https://doi.org/10.1371/journal.pone.0288394.g016

We note that as ξ decreases average number of tests per day decreases for both 1-hop and 2-hop, and the average number of tests for both become closer and closer, but throughout 2-hop considerably reduces the outbreak size as compared to 1-hop. When ξ = 1/60, the average number of tests per day is less than 15 for both 1-hop and 2-hop; 2-hop can reduce the outbreak size by up to 55% compared to 1-hop and 69% compared to 0-hop (the right in Fig 16b). When ξ = 1/90, the average number of tests per day is less than 10; 2-hop can reduce the outbreak size by up to 51% compared to 1-hop and 64% compared to 0-hop (the right in Fig 16c). As an example of extreme case, to further constrain the testing budget, we assume that ξ = 0, i.e., individuals who have been tested once will not be tested again. This corresponds to a situation where the average number of tests per day is less than 2, and the difference in the number of tests between 1-hop and 2-hop becomes insignificant (the left in Fig 16d). Nevertheless, the right in Fig 16d shows that 2-hop can reduce the outbreak size by up to 41% compared to 1-hop and 49% compared to 0-hop.

In conclusion, one can satisfy desired constraints on the average number of tests per day by appropriately choosing ξ, and still attain a considerable reduction in outbreak size through 2-hop contact tracing over 1-hop contact tracing.