key: cord-0913918-2n5hlwz8
authors: Du, Marvin
title: Contact Tracing as a Measure to Combat COVID-19 and Other Infectious DiseasesHighlights
date: 2021-12-08
journal: Am J Infect Control
DOI: 10.1016/j.ajic.2021.11.031
sha: 7d987102361b78cdf170480b261aa31594896e6e
doc_id: 913918
cord_uid: 2n5hlwz8

Background: Most of the mathematical modeling studies on COVID-19 transmission are based on continuous deterministic models that do not consider the characteristics of social network. Methods: The effect of contact tracing on mitigating COVID-19, and other infectious diseases in general, is studied in a small-world network. This network has its advantages over the commonly used continuous deterministic mathematical models in that the characteristics of social networks can be properly incorporated. Results: Simulation results show that for the original strain of SARS-CoV-2, contact tracing can play an important role to reduce and delay the peak daily new cases. New cases can be reduced by using symptom onset to isolate tracked individuals, but the benefit can be greatly enhanced by testing asymptomatic and pre-symptomatic individuals on the 6th to 8th day of infection. For the delta variant, or other variants of much higher infectivity, contact tracing alone cannot significantly lower the number of daily new cases but is able to delay the peaks greatly, thus affording more time to explore and implement pharmaceutical interventions. Conclusions: Contact tracing can be a very powerful tool to combat COVID-19 caused by the original strain or any variant of SARS-CoV-2. In order to make contact tracing effective, every effort is needed to expand the pool of contact tracing and provide all necessary support to the self-quarantined.

Nonpharmaceutical Interventions (NPIs), such as facemask wearing, business and school lockdowns, and contact tracing, are effective tools to combat infectious diseases such as COVID-19 that have devastated people"s lives around the globe 1 . When a vaccine is not available, NPIs can help "flatten the curve" which helps gain valuable time to develop, manufacture, and distribute the vaccine. Even after a significant fraction, or even most, of the general population has been vaccinated, breakthrough infection Mathematical modeling plays an essential role to combat infectious diseases 3 . Modeling results provide insight to how various NPIs slow down the spread of infections and rationalize when to initialize and terminate the NPIs. Up to now, most of the epidemiological modeling works are based on continuous deterministic compartment models 3, 4 . An example is the popular continuous SIR model in which the general population is distributed in three compartments: susceptible, infectious, and recovered. Of course, different variants of SIR models allow different or additional states such as reinfected, latent, etc. An intrinsic and vital shortcoming with the continuous deterministic models is that the infection transmission is assumed to be homogeneous 5 , i.e., all susceptible people have equal chances to get infected by a given infectious person regardless of whether they have direct contact with the infected. More fundamentally, continuous deterministic models fail to take into consideration the interactions among people in society since no contact network can be homogeneous. This deficiency inevitably diminishes the validity of the modeling results.

The homogeneous assumption associated with the continuous deterministic compartment models can be avoided if the spread of infectious diseases is studied in complex networks 6 . For infectious disease transmissions, several important metrics of the networks must be taken into consideration. The first is the shortest path length that characterizes the distance among vertices in the network. Another is the clustering coefficient that represents the relative propensity of two vertices, given that they share a common neighbor (in the layman"s words, friend"s friend is more likely a friend). The third is the degree distribution which reflects that some vertices have more connected vertices than others. There are two types of networks that possess two (out of three) ideal characteristics. Scale-free networks have the right short distances among vertices and a scale-free degree distribution which is supposed to be the right feature for social networks, but their clustering coefficients are normally too small 7 . On the other hand, small-world networks have the correct short distances among vertices and high clustering coefficient, but they have an exponential degree distribution 8 . It is true that certain scale-free networks can be designed to have the right clustering coefficient, but in the meantime, doing so will introduce unwanted features into the networks. For example, a scale-free network can be developed to have the correct short distances among vertices and a high clustering coefficient, but some vertices in the network are connected to all other vertices 9 , a feature that does not resemble social networks. Additionally, a recent study has shown that social networks are not necessarily scale-free, and many of them are at best weakly scale-free 10 . This implies that a scale-free network may not be the best choice as the paradigm on which infection transmission is studied.

It is the author"s choice to use a small-world network to study the infection transmission and in particular, the effects of contact tracing as one of the NPIs to mitigate infectious diseases such as COVID-19. As mentioned previously, two desirable features possessed by small-world networks are the short distances among vertices and large clustering coefficient. It is not clear how the unappealing characteristic of non-scale-free degree distribution affects the modeling results. As such, the primary hypothesis of this study is that small-world networks can be used to study how infection spreads.

It is helpful to give a brief description of the objectives of the present study. First of all, it exposes a potentially important flaw embedded in the most widely used continuous deterministic mathematical models of infection transmission and offers a more rigorous and physically sound alternative, the smallworld network model. Secondly, it demonstrates that the small-world network model can meaningfully redress the shortfall of underestimating the effectiveness of contact tracing. Thirdly, it provides insights for public health officials and the means to enhance the efficacy of contact tracing. By expanding the pool for contact tracing, testing effectively, and making every effort to ensure that all quarantined people are provided reasonable accommodations so that they can remain in quarantine until they are cleared to go back to normal lives.

The original Watts-Strogatz small-world network 8 connects a total of N vertices along a circle by both short-range regular edges and long-range random edges. Each vertex is connected to its k neighbors on each side by regular edges, and some of edges are re-wired randomly to connect remote vertices. The number of random edges is specified by a parameter p, which is the percentage of edges that are rewired. For example, p = 0.01 means that one percent of the regular edges are rewired, and those rewiring edges are called random edges. In this study, N, k, and p are set to be 100,000, 100 and 0.01. A constraint that needs to be satisfied is that so that the network is connected by random edges.

A variant 11 of the Watts-Strogatz model 8 is used in the present study. Instead of rewiring regular edges to get random edges, random edges are randomly added to connect vertices to remote ones. In this variant, p signifies the ratio of random edges to total edges. Fig 1 shows that, over a wide range of p, the network model has the correct average short distance and large clustering coefficient.

The average shortest distance between vertices and clustering coefficient with N = 100,000 and k = 100.

In the network, each vertex represents a person in society. The state of each vertex can be susceptible, infected, recovered, or dead. A susceptible vertex gets infected by one of the connected infectious vertices. The newly infected goes through the recovery process in a given period of time, and during the process, it can be reclassified as dead with a probability specified as the fatality rate of the disease. In the meantime, the vertex can infect other susceptible vertices. The recovered vertex gains immunity and normally will not get infected again; however, for COVID-19, evidence exists that the immunity can be lost 12 . Another parameter is the fraction of asymptomatically infected among all infected people, R asymp . It is reported that among all the infected, 27% to 57% could be asymptomatic 18, 19 , so in this work, I set R asymp = 0.3. The fatality rate, R fatality , is also highly uncertain, as in certain regions it was high but substantially lower in other regions. Without loss of generality, I assume it to be 0.05 since it accounts for only a very small fraction of the infected, so it barely changes the dynamics of the spread of disease on the network.

Finally, based on epidemiological data 12 it is assumed that a recovered individual has a probability of 20% to get reinfected within 180 days.

Wells et al. 20 

Once a decision is made to start contact tracing, all subsequently infected vertices are tagged and traced.

This corresponds to the real-world situation that once some individuals are confirmed to be infected, all persons that have had meaningful close contacts with these infected individuals are contacted and instructed to take appropriate steps to reduce the chances of passing on the infection to others. Here, "meaningful" means the contact is long enough and close enough so that the virus can be transmitted from one person to another. The U.S. Centers for Disease Control and Prevention (CDC) issued guidance that requires the contact to be within 6 feet for at least 30 minutes. It might be argued that the guidance is too restrictive and may fail to tag and track many potentially infected people.

The individuals being tagged and tracked are to be instructed to watch for symptoms and/or take a test on a pre-determined date. If symptoms show up or if the test result is positive, the individual will be put in self-quarantine for a certain period. Many jurisdictions including the CDC set the period to be 14 days.

There are two possible scenarios of quarantine. The first one does not involve testing and is solely based on the onset of symptoms, i.e., the tagged and tracked individuals put themselves in quarantine once they exhibit certain symptoms. The second scenario involves testing which allows both the symptomatically and asymptomatically infected to be quarantined. According to Wells et al. 20 , the optimal testing time is day six after being exposed to the virus.

The infection profile implies that, on average, all infected individuals recover in a period of 21 days 20,21 , but in reality, this is hardly the case. Therefore, testing results should be used to determine if an individual is fully recovered and allowed to exit quarantine.

The In each simulation, 100 realizations are carried out so that stable statistics can be obtained for each simulation scenario.

Numerical First of all, the effect of t isolated , the time when the newly infected and tagged individuals are isolated to prevent the spread of the virus, is examined, as shown in Fig 2. For comparison, the base case, i.e., the case without implementing vertex tagging and isolation, is also shown.

Clearly, contact tracing plays an important role to slow down the spread of infection. When half of the symptomatic infections can be isolated upon symptom onset, the peak daily cases can be cut by more than a half. The effect can be further augmented if testing is conducted on the tracked individuals. If test results become available so that the infected can be isolated on the 8 th day of infection, the daily new cases can be reduced significantly since both pre-symptomatic and asymptomatic cases can be partly isolated. If the infected can be isolated on the 6 th day of infection, the effect is almost equivalent to "crush the virus", i.e., the infection transmission can be stopped. One important implication of this finding is, for the original strain of SARS-CoV-2 (its basis reproduction number has been widely shown to be around 2.5), the 50% tracking rate is not reachable since "crush the virus" is rarely seen as the outcome of implementing all NPIs. Thus, it is impossible that contact tracing alone can be as effective as implementing all NPI measures. It also should be noted that, according to Wells et al. 20 , the optimal test day is day six since the average latent period is about three days. This means that isolating the tracked individuals in 6-8 days is

probably the best that can be done if only one test is to be conducted to decide entry to quarantine. For all quarantined people to exit quarantine, another test is to be conducted so that exiting quarantine is allowed only for the fully recovered.

The rebound of the fraction of susceptible population after reaching the lowest reflects the effect of reinfection. As mentioned previously, it is assumed that 20% of the recovered will lose their immunity within 180 days. infections. In the calculation, it is assumed that 50% of the newly infected can be tracked and will be isolated either when symptoms show up or six days after being infected, whichever comes first.

Obviously, contact tracing should be started as early as practically possible, which is confirmed by the plots . Fig 3(a) shows that starting the process early, and in particular, starting contact tracing at the very beginning of the outbreak can reduce the daily number of new cases substantially.

Comparing the curves in Fig 3 and A comparison between Figs. 2 and 4 indicates that 50% isolation upon symptom onset alone is equivalent to 30% isolation when both symptoms and testing based 6-day isolation are used. This is not surprising because the average incubation period is taken to be 5.8 days, i.e., a half of the symptomatically infected will not show any symptoms in the first six days of infection, so the testing results will drive both the asymptomatically infected and pre-symptomatically infected into selfquarantine.

As the percentage of tracking and isolation increases, the effect of "flatten the curve" becomes "crush the virus". As shown in Fig 4(b) , although "crush the virus" is good for controlling the current outbreak, it leaves essentially the entire population susceptible for infection.

It is seen that contact tracing can be a very effective means to reduce the spread of the original strain of SARS-CoV-2. Tracking and isolating even only 10% of infected people can make a meaningful difference. Therefore, every effort should be made to track as many potentially infected people as possible, including relaxing the definition of close contact so that more infected people can be tracked and isolated. For example, many jurisdictions define close contact as within 6 feet for more than 15 It is worth noting that the virus spread reduction due to contact tracing in the present small-world network model is much more significant than that in the continuous deterministic models. For example, results 25, 26 from continuous deterministic models with comparable input parameters show a much less profound reduction of infection cases. A specific example 26 is with a 40% testing coverage, 1 day turnaround time of test results, and 60% contact tracing ratio, the peak infection number is reduced by only 50%, a level much lower than Figs 2 and 4 imply.

This subsection explores the effect of contact tracing on the behavior of infection transmission for a variant virus that has a much larger basic reproduction number than the original strain of SARS-CoV-2.

It is now known that the delta variant, which was raging in the summer of 2021 in the United State, has a basic reproduction number around 6 16, 17 . This subsection presents modeling results obtained with R 0 = 6. In the modeling, all other epidemiological parameters are assumed to be the same as in the previous section, and the infection profile is scaled up so that the sum of the daily infectivity rate is R 0 = 6. As expected, implementing contact tracing earlier in the process will bring better outcomes in terms of lowering and delaying the peak daily cases. Compared to the results obtained with R 0 = 2.5, the effectiveness of contact tracing in bringing down the number of infections is not so great. Nonetheless, contact tracing is still quite a useful mitigation measure. Compare the curve for the non-mitigation base case in Fig 5(a) with those in Fig 7(a) . It is seen that if 30% of the new cases can be tracked and isolated, the peak daily new cases can still be lowered from 2,171 to 1,715 (out of 100,000 vertices). Even if the isolation is solely based on symptom onset, the peak number is still reduced to 1,920 with a 30% tracking and isolation rate. 

Among all NPIs to combat COVID-19, contact tracing is the least intrusive measure to the general public. As such, it should be utilized as the first choice for mitigation. It also works for other infectious diseases because it can reduce the spread of the disease in the broad public.

A small-world network model is used to study how contact tracing affects the infection transmission for COVID-19 and infectious diseases in general. It is better based than the conventional continuous deterministic mathematical models since it accounts for the essential characteristics of social networks.

The continuous deterministic models tend to underestimate the efficacy of contact tracing, which may negatively affect public health officials allocating resources to this effort.

For the original strain of the SARS-CoV-2, contact tracing alone can play a significant role to lower and delay the peak of daily new cases. Using symptom onset to decide entry to quarantine can help, but the benefit can be augmented greatly if test results are used to isolate asymptomatic and presymptomatic individuals. Another finding is that the sooner contact tracing is implemented, the better the outcome.

For the delta variant or other variants with high infectivity, contact tracing alone may not be a winning strategy but it can certainly make a difference. Unlike the original strain of virus with a moderate basic reproduction number, starting contact tracing sooner seems to reduce the peak daily new cases only slightly. However, it can delay the peaks significantly, affording more time for medical professionals and public health authorities to develop pharmaceutical intervention measures.

To deal with outbreaks caused by variants of SARS-CoV-2, preexisting immunity (gained either from vaccination or previous infection) can play a vitally important role to reduce the number of daily new cases. Therefore, vaccinating the general public not only helps combat the current outbreak but also makes future outbreaks easier to deal with.

For both low and high infectivity diseases, increasing the percentage of tracked and isolated individuals among all the infected always leads to lowered numbers of new cases. This means that every effort should be made to enable more infected people to be tracked and isolated. This may include defining "close contact" more reasonably, making use of digital equipment such as smart phones to find individuals who may have close contact with those known to be infected, using test results to determine entry and exit to quarantine, providing support to the self-quarantined so they do not need to make trips to grocery stores and restaurants, maintaining daily contact with the self-quarantined, and providing support wherever and whenever necessary.

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