Contact Tracing -- Old Models and New Challenges
aa r X i v : . [ q - b i o . P E ] A ug Contact Tracing – Old Models and New Challenges
Johannes M¨uller , , Mirjam Kretzschmar Mathematical Institute, Technische University M¨unchen, Boltzmannstr. 3, 85748 Garching, Germany Institute for Computational Biology, Helmholtz Center Munich, 85764 Neuherberg, Germany Julius Center for Health Sciences and Primary Care, University Medical Center Utrecht, UtrechtUniversity, Heidelberglaan 100, 3584CX Utrecht, The Netherlands
Abstract
Contact tracing is an effective method to control emerging diseases. Since the 1980’s,mathematical modelers are developing a consistent theory for contact tracing, with the aimto find effective and efficient implementations of contact tracing, and to assess the effects ofcontact tracing on the spread of an infectious disease. Despite the progress made in the area,there remain important open questions. In addition, technological developments, especiallyin the field of molecular biology (genetic sequencing of pathogens) and modern communi-cation (digital contact tracing), have posed new challenges for the modeling community. Inthe present paper, we discuss modeling approaches for contact tracing and identify some ofthe current challenges for the field.
Emerging and re-emerging infectious diseases like SARS, Ebola, Lassa fever, Tuberculosis, andmost recently SARS-CoV-2, require rapid responses and targeted control measures. It is best ifan immunization of the population is possible - however, in case of emerging diseases, often thenecessary vaccines are not available yet, or not available in sufficient quantities. An alternativeapproach is to stop infection chains by non-pharmaceutical control measures, such as reducinginfectious contacts by social distancing, and testing and isolating infectious individuals. Thiscan be done by screening if sufficient testing facilities are available, and in addition tracing andquarantining contact persons of infected index cases [58].Mass screening as a stand-alone control measure is effective if prevalence is high, and cheap, rapid,and reliable diagnostic procedures are available. Otherwise, most tested persons are uninfectedsuch that even a small probability for a false positive test result leads to a large number of falsealarms, whereas only few infected persons can be identified.Contact tracing (CT) is a more focused method: Once an infected individual is diagnosed andisolated, contact persons are identified who had potentially infectious interactions with that indexcase. In general, the prevalence within this group will be much higher than that in the generalpopulation, such that it may be effective to screen these persons, and to quarantine or isolatethem. CT acts on several levels of the transmission process:
Individual level:
Infected persons arediagnosed early and are isolated or receive medical help.
Population level:
Transmission chainscan be detected and stopped, which reduces the effective reproduction number.
Medical/scientificlevel:
It is possible to study infector–infectee pairs, and to learn about who infected whom in theoutbreak. This gives information on risk factors, transmission modes, infectivity, and generation1ntervals of the infectious disease. This information can be central in the design of further controlmeasures.The main challenge in mathematical modeling of CT is the individual-based character ofthe process. Information about the health status of single individuals and the time course ofthe contacts between those individuals is necessary. An appropriate formulation of the processis only possible at the microscopic level. However, the main interest is at the mesoscopic andmacroscopic level: Not single individuals but the spread of the infection is in the focus of theinterest. Model approaches need to span several scales, from the individual to the population.Similar to CT, also the transmission of infection is a process occurring between individuals. Thatprocess can be readily approximated by mean field models as the Kermack-McKendrick model.For the epidemic process, it is well established how to bridge the scales. CT, in contrast, followsthe interaction given by transmission events, so can be viewed as a kind of superinfection: Onecould consider CT as an infection that follows the paths of the primary infection, and removesin that way infected individuals. This process superimposed on the transmission process createsa high degree of dependency between individuals. Therefore, lifting a model description for CTfrom the individual level to the population level involves more technical difficulties than for thecase of a pure infection process.Intuitively, the CT and transmission processes are racing each other. Starting at an indexcase, the infection spreads to contacts, while with CT we aim to catch up. This picture alreadyindicates many of the properties that make a disease controllable by CT [30]: (a) a sufficientlylarge fraction of cases develops symptoms and is tested and diagnosed; (b) contacts need to bewell identifiable and tracable; (c) the disease spreads slowly enough to allow CT to catch up,even if identification, testing, and quarantining of contacts comes with a certain delay, and (d)diagnostic tests are able to readily identify symptomatic and asymptomatic persons.Here, we review published literature on mathematical models for CT and their applications. Wethen identify and discuss open problems and current challenges for the theory of CT.
As a rigorous mathematical analysis of detailed models for CT is rather challenging, a multitudeof simplifying approaches have been established. Among the first papers is a study by Heth-cote and Yorke concerning interventions for gonorrhoea [39]. They include CT in their modeland suggest that the effect of CT is a reduction of the transmissibility of the infection. In thesame spirit, [45] proposes that CT is based on the identification of infected persons by theirinfected contacts. In that, the “superinfection” is taken verbally, and infecteds are removed witha term that resembles the incidence term (product of discovered infecteds and prevalence). Inthose models, the term representing CT is not based on first principles, but rather defined ad hoc .For models, that are based on first principles, three main directions can be identified:1. Simulation models that directly simulate individuals and large populations;2. Pair approximation models that are extensions of mean field models incorporating infor-mation about correlations of pairs of individuals;3. Stochastic and deterministic models that are based on a rigorous analysis of a simplebranching process modeling CT. 2e discuss these in the following sections, and also the phenomenological approaches, which arerelevant for practical applications. phenomenological approaches, which are relevant for practicalapplications.
Individual (or agent) based models (IBMs) [111] are perhaps the first choice to formulate aprocess as complex as CT. Individual-based models describe the fate of every single individual ina population and their interactions. IBMs explicitly incorporate a contact graph, which can be assimple as a complete graph where every individual has contact with every other individual (fullyconnected population), a random graph as described by the configuration model, or a small worldgraph [79, 52, 88] that reflects local and long distance contacts. Conceptual, parsimonious modelsare used to address the influence of the contact graph structure on transmission dynamics [55, 57].The most detailed IBMs describe graphs that aim to represent existing societies with cities, workplaces, and schools [78]. It depends on the aim of the model how detailed the contact graph, thestate of an individual, and its behavior is formulated. As such a model is algorithmic, there isalmost no limit for the degree of detail that can be included. However, very detailed models areoften faced with the problem of lack of data for an appropriate parametrization.In any case, the advantages of individual-based models for CT are three-fold: First, they aresimple to formulate and to communicate; second, they allow to directly represent CT in a realisticway and in that foster and guide the development of more abstract, analytical models; and third,they provide sufficient detail, such that results can be used in public health decision making.IBMs are frequently used to investigate specific infectious diseases, and to extract relevantinformation about the effectiveness of intervention strategies. Contact tracing is commonlyperformed to improve case finding for sexually transmitted diseases as gonorrhoea and chlamydia;for these diseases, contacts can be clearly defined - recent sex partners - and a large fraction ofinfections are asymptomatic, such that contact tracing greatly enhances the possiblity of findingand treating infected persons. Also, the time scale of transmission of these infections is sufficientlyslow to enable finding and treating contacts on a faster time scale than the generation interval.Finally, as sexual partnerships are often long lasting, treating contacts may prevent re-infectionsof the index case [63, 33, 109, 61, 60, 2]. Several IBM models investigate Tuberculosis [106, 107,49, 76] (see also the review article [10] and references therein), Smallpox [95], also in connectionwith bioterrorism [27], measles [69], and Ebola [101]. Also, the effect of CT on control ofSARS has been simulated by various IBMs, see the e.g. [58, 70], and particularly the reviewarticle [67] and references therein. Peak et al. [93] compare effectiveness of CT for severalinfections, including Ebola, influenza, and SARS. In [4, 5], CT is analyzed from a public healtheconomics point of view. The effect of CT on the SARS-CoV-2 pandemic was investigated basedon IBMs in [13, 15, 37, 38, 53, 66, 72, 104, 94].
Stochastic processes are hard to analyze, while many tools for deterministic models are available.Mean field equations are a well established, heuristic approach to reformulate individual-basedmodels as described above in terms of ordinary differential equations (ODE’s). In the limit, ifthe population size tends to infinity, the approximation becomes exact in case of a homogeneouspopulation; that precisely is the way to derive a deterministic model as the Kermack-McKendrick(or SIR) model from a stochastic process for an infection. Instead of counting the numberof individuals of different types, the expected relative frequencies are addressed by the ODEmodel. Some transitions, as recovery of an individual, are independent of the states of all other3ndividuals. In that, the expectations exactly satisfy an ODE. Other transitions, as transmissionof infection, are based on interaction of individuals. At that point, only an approximation ispossible that neglects all correlations in the contact graph. Formally, the expectation of theproduct of random variables is replaced by the product of expectations [85, chapter 3.1.10].Under appropriate conditions, particularly if the contact graph is a complete graph (contactshappen between any of the individuals), this procedure becomes rigorous if the population sizebecomes large or infinite. The success of ODE models to describe the time course of real-worldepidemics is an a posteriori justification for simple deterministic models. In many cases, theyallow for a deeper understanding of the underlying mechanism, and for powerful predictions [51].All information about correlations are lost in the transition from an IBM to a mean fieldmodel. In that, a mean field model is insufficient to appropriately describe the infectious processon an inhomogeneous contact graph, particularly if the contact graph is strongly locally clustered:Is this the case, the neighbor of an infected individual often already is infected, and the spread ofinfection slows down. In the 1990’s, mainly driven by Japanese [99] and British [50, 51] groups,an improved mean field approximation was developed, the pair approximation. Here, not onlyexpectations but also the correlations are formulated in an ODE model. That is, not only theexpected number of individuals in state S , I , or R , say, but also the expected number of edgesconnecting e.g. I with I or S with I individuals are followed in the system of ODE’s. As themodel incorporates information about correlation, it can – up to a certain degree – mimic theslow down of the spread caused by spatial correlations. The disadvantage of this approach is theexploding number of equations needed, as not only one equation per state is required, but alsoone for each pairwise combinations of two states. Therefore, most authors do not perform ananalysis based on the theory of dynamical systems, but the resulting ODEs are solved numerically.Furthermore, it turns out that – for strongly localizing graphs – the conclusions are rather ofqualitative than quantitative nature. If the contact graph is well mixing (high degree of thenodes, and no clustering), the pair approximation becomes better (which is also the case for themean field approximation). Recent developments allow even an exact analysis by the “messagepassing method”, a subtle further development of the pair approximation [48, 102, 110] for trees.Concerning the modeling of CT, pair approximation keeps a central piece of information thatis dismissed by the mean field approximation: We know how likely it is that a neighbor of aninfected individual is infected. In that, it is possible to remove infected neighbors of an indexcase [25, 44]. This idea has been discussed in a series of papers [108, 26, 52, 40, 41]. Thecomparison with Monte Carlo simulations of the stochastic process indicates that the results arevalid especially for large, homogeneous graphs. Most of these papers investigate the effectivenessof CT in different contact graph structures. In [40] it is found that CT is more effective inclustered than in homogeneous populations. In [24] recursive and one-step tracing is compared,and “targeted CT”, that is, CT focusing on a risk group, is analyzed. Recursive and targetedCT were found to be particularly effective. The models for CT based on pair approximationare rather conceptual models that allow addressing fundamental questions than models used toquantitatively predict the effect of CT on the spread of real-world infections. One of the fewexception is [18], where pair approximation is applied to predict the impact of CT on chlamydiaprevalence. A simpler approach, in which only pairs are taken into account, which may formand dissolve, is used in [36] to investigate the impact of screening and CT, again addressing theprevalence of chlamydia. At the onset of an outbreak spreading in a homogeneous population, it is possible to replacethe epidemic process by a birth-death process of independent individuals, where “birth” means4 new infection and “death” recovery. Particularly the interaction of two infectious individualsis unlikely and negligible in a large, homogeneous population [6]. The statistics of this process,e.g. the size and structure of connected components is known [86]. On top of this linear birth-death process, a process of CT can be formulated. It is possible to rigorously analyze thisstochastic process [84, 7]: CT mainly affects the removal rate of infected individuals. That is,in order to address the stochastic process, the probability to be infectious at a given time afterinfection is determined. This probability is the central function that allows to readily determinethe effective reproduction number, or the doubling time of an infection. Various aspects of CTcan be investigated in this context, as the recovery and infectivity depending on the time sinceinfection [84, 7], estimation of the tracing probability from data [82, 12], or the effect of a tracingdelay [8, 83]. Strictly spoken, the analysis and the results are valid only for the onset (or duringthe decay before exinction) of the outbreak, if direct contacts between infecteds are unlikely tohappen. However, using heuristic arguments, the removal rate can be approximated also in thecase of high prevalence, and a modified mean field equation has been proposed [84].Similar to pair approximation, the approach is not suited for a complex contact graph struc-ture with small homogeneous clusters that only weakly interact. It is interesting that the centralidea for the analysis of CT on the one hand, and the message passing methods used in recentversions of pair approximation on the other hand, bear a remarkable similarity. In [89], thebranching process analysis is generalized from homogeneous populations to populations with aprescribed contact graph, and ideas are discussed how to merge the branching process analysisand the pair approximation. Also [59] investigates CT on a random graph, this time via anapproximation using generating functions for he degree distribution of neighbors of randomlychose infected individuals. They find, that the degree of an individual detected by backwardtracing roughly behaves as the expectation of the squared degree, indicating the high efficiencyfor CT to detect the persons that are best connected.Brown et al. [14] developed a sophisticated ODE-approximation of the branching-process struc-ture and applied that to Ebola. Becker et al. [9] also proposed a simplified version of the modeland investigated the SARS epidemic in a deterministic model with household-structure. Kret-zschmar et al. [62] used a branching process to model ring vaccination for smallpox. This modelwas recently developed further to investigate the effectiveness of CT for SARS-CoV-2 [65, 64].Based on a branching-process formulation for CT, Tanaka [104] analysed data for SARS-CoV-2to estimate the fraction of asymptomatic cases.
Phenomenological approaches are not rooted in an analysis of stochastic processes that obviouslymodel CT in an adequate way. In that, it is difficult to understand if the terms chosen to modelCT are adequate. The advantage of these models is their simplicity – they are ODE modelswith rather simple structure (concerning CT) and can be readily analyzed or simulated. In that,these models are suited to address real world epidemics. The difficult part is the interpretationof the results (again concerning CT), as the clear connection with first principles is not obvious.Often, the tracing part is either formulated as a linear removal term, or as a mass action term.Many of these models are applied to the HIV infection, where mostly a mass action termis used to represent CT [21, 45, 45, 20, 87, 1], but sometimes also simply a linear term [43].In [20, 42] several ways to model CT – linear, mass action, and a saturation function – arecompared with data. Interestingly enough, results in [42] indicate that a mass action term forCT is inferior to a linear term or a saturation function. In a similar spirit, Clarke et al. [19] adjusta power low term for CT based on simulations from an IBM. Also models for Chlamydia [35],Tuberculosis [3], Smallpox [47], Ebola [11], and models for SARS-CoV-2 [34, 73] are based on5his phenomenological modeling approach.Fraser [30] proposed an idea with a slightly more profound connection to CT. The modelis based on age-since-infection. In that, it is similar to the branching-process analysis, but CTis formulated as a linear effect. In that article, characteristics in the timing between onset ofsymptoms and infectivity are identified that make an infection controllable by CT. Chen etal. [16] took up that approach to analyze a model for the SARS infection and Ferretti et al. [29]applied the model to the SARS-CoV-2 epidemic in Italy.
The efforts of the last 40 years to develop a toolbox for CT have delivered a considerable amountof modelling approaches and results. There is a general agreement about a fundamental modelstructure, that can be readily realized in IBMs, and there are various ways to analyze thestochastic process either approximately or rigorously – at least, if we stick to simple models.Many simple ad-hoc models are published. Nevertheless, some of the central questions are stillopen.
Some questions and problems have been debated for quite a while. We pick a few of these“classical questions” which we consider as interesting and/or of practical need, and discuss them.
Modeling CT.
Obviously, a multitude of models are in use. Some models, as the stochasticIBMs, directly simulate CT. In an IBM, it is easy to incorporate CT appropriately. Stochas-tic simulation models have the advantage that their outcomes are easy to interpret, but thedisadvantage that they cannot be analyzed analytically. To obtain insight into the parameterdependence of CT and to develop general rules for CT is not straightforward or may be impos-sible if we exclusively rely on IBMs.Other approaches, as the pair approximation models or models based on branching processes, usefirst principles to develop the model structure. Therefore, also these models are able to reflectCT in an appropriate manner. Even if approximations are used to derive simplified mathemat-ical structures, it is straightforward to check their accuracy (compare the analytical results tosimulations of the original models). In that, these models are appealing. However, in some cases,the derivation is rather technical and not straightforward to communicate. This class of modelsare well suited for theoretical considerations. For practical purposes, simpler model structuresseem to be desirable.At that point, the phenomenological approach comes in. Here, the models are mostly compart-mental (ODE) models, as often used in ecology and epidemiology. These models can be readilyanalyzed and simulated. In that, practical applications of such simple models are easily possible,which is the strength of the phenomenological approach. The drawback is the fact that theyare not rooted in first principles. It is hard to assess whether the model structure appropriatelyreflects reality. E.g., if CT is formulated as a linear term, this formulation is contradicting theobservation that CT is based on correlations and dependencies between index cases and theirinfectees/infector (as a rule, dependencies are expressed by nonlinear terms). Another aspect isthe parametrization of the models: Some central parameters can be obtained by observations onthe micro scale. E.g., the probability to detect a contact can be estimated using the number ofdetected cases per index case [82, 12]. As phenomenological approaches dismiss the micro-scaleand directly jump to the macro-scale, information that is available on the micro-scale is almostimpossible to incorporate. 6asy to use models that are well accepted and approved by the modeling community are neces-sary but not yet in sight.
Backward/Forward tracing.
Even in one of the very first papers about CT, the seminal workby Hethcote and Yorke [39], the distinction between backward and forward tracing is mentioned.Backward tracing means that the infector is detected by an infectee who becomes an index case,while in forward tracing the infector is the index case, while the infectee is detected. Since thisfirst article, the relative importance of backward- and forward tracing is under discussion. Anindividual only has one infector, but in general several infectees. This fact might indicate thatforward tracing is more important. On the other hand, if we randomly select an individual in anatural contact graph, the neighbor of this individual will on average have more contacts thanan average individual. This finding, also called the “friendship paradox” in the context of socialnetworks, has been proposed to be used in an early warning system for influenza [17]: Studentswere asked to name two friends. The authors of the study monitored the occurrence of influenzaamong those named persons. It turned out that the incidence of influenze increased two weeksearlier than it did among the average students. This observation indicates that backward tracingalso is of importance, as most likely we will find persons who have many contacts and already hadthe chance to infect many of them. This is especially important for CT in sexually transmittedinfections, which are often circulating in highly connected core groups.The question of whether there are super-spreaders also falls within this context [71, 32, 77].The frequency and the importance of super-spreaders for the dynamics of infection, and also forthe impact of CT, has not yet been conclusively clarified; [58, 59] argue verbally that backwardtracing is readily able to identify super-spreaders, s.t. a combination of backward- and forwardtracing is efficient, also in the presence of super-spreader events. In [89], quantitative comparisons(based on analytic results for a branching-process model for CT on random tees) indicates thatthe effect of CT decreases with the variance of the degree distribution. This, in turn, is a hintthat CT performs better without super-spreader events. Also simulation studies [56] point inthat direction (at least if we consider the expected final size). The timing (latency and incubationperiod) of the infection itself will also have some influence. There is a need to investigate theeffect of super-spreaders on CT more in depth.
Endemic equilibrium.
CT is also performed if the infectious process is in its endemic equi-librium. In this case, in average each infected individual only has one infector and one infectee( R eff = 1). Why does CT pay in that situation? For sexually transmitted diseases (STD), it isclear that the partner is of high risk, and a couple should rather be considered as a single entity.Partner notification in faithful pairs is a simple case of CT. It is more interesting to note that forsome STDs asymptomatic persons can be infectious for a long time (several months). In that,particularly the tracing of asymptomatic persons may be decisive. CT is a method to find thesehighly infectious persons with possibly many contacts, who are difficult to localize otherwise. Tracing probability.
In order to monitor the effect of CT, it is desirable to estimate thetracing probability, that is, the fraction of identifyable contacts among all infectious contacts.While data for the number of detected cases are available, the number of missed cases is usuallyunknown. In [82, 12], some statistical methods are developed to estimate the tracing probability.A related problem is the estimation of the abundance of asymptomatic cases from tracing data.[23] proposes a method based on household models for Yaws, a disabling bacterial infection, whileTanaka [104] aims to estimate the percentage of asymptomatic cases for SARS-CoV-2 using thebranching process approach for CT. However, these questions are rather neglected by the recentliterature and deserve a deeper investigation. 7 .2 CT and genetic sequence data
In recent years, genetic sequencing of pathogen DNA became rather cheap, and genetic sequencedata are readily available. Methods of population genetics are used to, e.g., estimate the preva-lence of an infection [31]. Genetic sequence data is also used to identify clusters of infections,and even to identify and refine transmission trees (infector/infectee relations) within a clus-ter [97, 92, 22, 91]. However, the combination of data from CT and from sequencing techniquesis hardly exploited by now. It is natural to ask what epidemiology could gain from that combi-nation.Clearly, from samples of infector-infectee pairs, the mutation rate of the pathogen can be esti-mated. Moreover, as an infection event forms a bottle neck for the population of the pathogen,the time since infection can be estimated, and in that, it is possible to narrow down the time ofthe infectious contact. A comparison of different infector/infectee pairs might help to sharpenthe estimations for the prevalence.The usage of genetic sequence data together with methods from population genetics is ratheryoung, and many powerful methods – as the SMC [75] – are rather recent developments. Weexpect that useful tools become available in the near future.
The idea to use data from mobile phones to trace contacts is rather recent. First practical at-tempts to use digital sensors (RFID chips) to observe contacts and to investigate an empiricallyvalidated contact network in relatively small communities (school, hospital, conference) reachback to 2010 [98, 46, 103]. Soon it became clear that risk evaluation based on mobile phone datais possible and useful [68, 74]. Perhaps the first paper that considered digital CT (DCT) was anIBM-based simulation study by Farrahi et al. in 2014 [28]. Several manuscripts in the context ofSARS-CoV-2 also address DCT [13, 15, 73, 29, 38, 64].The ability to localize persons using data generated by smartphones already detects trafficjams [90], and is a source of major concern as these data could also be misused [105]. DCTcould help to improve some crucial shortcomings of classical CT. The advantages of DCT are therapid identification of infectious contacts, and the possibility to also rapidly inform contactees.In comparison with conventional CT, DCT has the potential to strongly reduce the tracing delay,and to increase the tracing probability. In that, infections may become controllable that were notcontrollable before. However, these potential benefits come with a number of technical, social,medical, and practical challenges. Contacts identified by mobile devices need to correlate withinfectious contacts. A rapid and cheap test for the infection is necessary, as the number of per-sons to test will be much higher than in conventional CT. Aspects of privacy and data protectionare crucial elements of DCT. Apps need to be accepted in the society. The concept of DCT, itsstrengths and its weaknesses have to be communicated clearly, in a way that citizens understandand accept DCT. The system can only work if a large part of the population participates inDCT.In the present note, we rather focus on new modeling challenges posed by DCT, and less on thesocial and technical aspects, though all of these points are intertwined.
Correlation between individuals.
Often, the mathematical analysis of CT models focus on theonset of the outbreak. In that, correlations (particularly between infected individuals) that donot come from an infector/infectee relation can be neglected. At present, conventional CT canafford to trace contacts only if rather few index cases are diagnosed. Once the number of diag-nosed infected persons exceeds several hundred in a community, the effort to identify manuallymany contacts per person is not feasible anymore.8CTS promises to overcome some of the logistic problems of CT and to allow for the identifi-cation of a huge number of contacts, even if a high number of infected individuals are present.The number of reported contacts per day and person in Europe is in the magnitude of 10 [81],with a large standard deviation (which is in the same range as the average number of contacts).If we have a tracing window of one week, we easily estimate 70 contact persons per index case(with a high variance). Depending on the nature of the infection, we can (and should!) imme-diately inform not only the direct contactees but also the contactees of the contactees (secondlevel tracing). Note that second level tracing is dissimilar to the classical two-step tracing: Intwo-step tracing a contactee is first tested, and if the test is positive, he/she becomes an indexcase and the snowballing continues, while in second level tracing the contactees of contactees areimmediately informed about their possible exposure, without waiting for a test result of contactpersons. In that, we easily arrive at 500 direct and indirect contactees per index case. That is,the fraction of the population affected by DCT is around 500 times larger than the number ofindex cases, however, the number of actually exposed persons might be small. From a practicalaspect, it is of utmost importance to identify features of an infection that imply the necessity offirst- second- or even higher-level tracing.The difference to the conventional procedure is the possibility for rapid and immediate informa-tion of direct and indirect contactees, without waiting for medical tests and diagnoses. Even ina moderate outbreak, it is likely that the groups of contactees related to different index caseswill overlap. These effects tremendously complicate the mathematical analysis of the models. Aclear challenge is the identification of techniques that allow for the mathematical analysis of thissituation.
High number of contactees / practical protocol.
As discussed above, DCT can identify ahigh number of contractees and inform them about direct or indirect infectious contacts. Frompractical considerations, a reduction of that number is desirable. Technical devices may give ascore to each contact, and estimate the probability for infection. That might help, but it alsomight be the case that scores are not reliable.Most likely, we are faced with a serious practical problem: If all casual contacts are reported,many persons might receive a warning rather often. In that case, fatigue sets in and warningsare not taken seriously anymore. The DCT becomes useless. A possibility to escape this riskis the choice of a smaller subgroup from the set of contactees. One possibility could be simplya random sub-set. More effective is a classification of contactees in different risk groups, e.g.according to the number of recent contacts. Particularly asymptomatic persons that spread theinfection for a longer time will appear among repeatedly identified contactees.Modeling approaches need to clarify the situation and to estimate the number of warnings aperson receives under realistic conditions. In consequence, different strategies of reducing thenumber of contactees to be informed have to be defined. Models allow to estimate the impactof these strategies, and to filter out reasonable policies for DCT. Many parameters will influencethe outcome, as the availability of tests, the social/contact structure, and the abundance of atracing app on mobile phones.
DCTS induce an inhomogeneity in the population.
The population is divided in a subpopu-lation with, and a subpopulation without a tracking device. In that, contacts within the DCTS-subpopulation are readily identified, while contacts between the two subpopulations or amongthe non-DCTS-subpopulation can only be identified by conventional CT. That situation is com-pletely different from an imperfect test, which leads to a certain probability of undetected infec-tious contacts. It is likely that the acceptance of a tracing device correlates with social factorsas education, political, or religious orientation. In that, connected subgroups appear that escapethe detection by DCTS. These subgroups might form a reservoir for the infection, from which9mportation to the general population may occur. The fraction of the population equipped withDCTS-equipment alone is not decisive, but also the distribution in a heterogeneous population.However, even for a homogeneous population, the division into two subgroups might lead tounforeseen effects. This aspect is a new one that deserves deeper investigation.
The joint effort of modeling and theoretical investigation of CT has borne many fruits, as theestimation of the impact of CT on the spread of infections, or methods to assess the influenceof the tracing delay. Nevertheless, questions of practical importance still deserve our attention.New challenges, e.g. due to developments in genetic sequencing of pathogens and DCTS, havearisen. Simulation models promise first answers on a rapid time scale, but it will take some timeuntil these new aspects are fully understood analytically, and this understanding is translatedinto practice.
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Literature overview
The table orders the papers primarily according to the infection they aim to investigate resp. forthe rather theoretical papers, by the method used. author infection method term for CT CTde-lay remarks outcome(w.r.t. CT)Becker et al.2005 [9] SARS phenomen. /next genera-tion operator reduction ofincidence - article exclusivelyaims at Reff, nodynamics. formula for Reff; socialdistancing together withCT can control SARSChen et al.2006 [16] Influenza,Measles,Chicken-pox,SARS phenomen. /PDE fixed frac-tion of newlyinfecteds areeventuallytraced - based onFraser [30]; onlyReff, no dynamics;airborne infection Probability to control anoutbreak is estimatedKwok et al.2019 [67] SARS review articleFraser et al.2004 [30] Theory &Influenza,SARS,Small-pox,HIV phenomen. /PDE fixed frac-tion of newlyinfecteds areeventuallytraced - basic model, usedin applications [16,29] timing of incubation pe-riod and latency period iscentral for CTLloyd-Smith et al.2003 [70] SARS phenomen. /time-discretestoch. simon pop.-level increasedtransitionrate toquarantine + CT not explicitlyformulated crucial that CT is imple-mented at the beginningof the outbreakBradshaw etal. 2020 [13] SARS-CoV-2 IBM, homo-geneous pop-ulation direct formu-lation + preprint; DCTconsidered, focuson onset, Reff andprob. for majoroutbreak backward tracing and highabundance of DCT de-vices necessaryBulchandaniet al.2020 [15] SARS-CoV-2 IBM, homo-geneous pop-ulation direct formu-lation - preprint; DCT con-sidered, focus ononset, heuristic for-mula for Reff High DCT-device cover-age necessaryFerretti et al.2020 [29] SARS-CoV-2 phenomen. /PDE fixed frac-tion of newlyinfecteds areeventuallytraced + based onFraser [30]; DCTconsidered, heuris-tic formula forReff SARS-CoV-2 controllableby DCTGiordano etal. 2020 [34] SARS-CoV-2 ODE/SIR increasedrate forremoval - detailed ODEmodel CT is a central element incontrolling the infectionHellewell etal. 2020 [37] SARS-CoV-2 IBM, homo-geneous pop-ulation direct formu-lation - onset of epidemicconsidered high tracing probabilitynecessary to control theinfectionHernandez-Orallo et al.2020 [38] SARS-CoV-2 IBM, inho-mogeneouspopulation+ ODE direct formu-lation (IBM)/ fixed frac-tion of newlyinfectedsare eventu-ally traced(ODE) + simulations basedon empirical con-tact network CT needs to be precise inorder ot avoid many per-sons in quarantine uthor infection method term for CT CTde-lay remarks outcome(w.r.t. CT)Keeling et al.2020 [53] SARS-CoV-2 IBM, homo-geneous pop-ulation direct formu-lation - preprint; timeand intensity ofcontacts vary,simulation-basedestimation of Reff high tracing probabilitynecessary to control theinfectionKim et al.2020 [54] SARS-CoV-2 Stochasticbranchingprocess verbally - preprint; DCT con-sidered; Heuristiccalculations of effi-ciency heuristic formula for effi-ciencyKretzschmaret al. [65] SARS-CoV-2 IBM, homo-geneous pop-ulation direct formu-lation + preprint; Modelbased on Kret-zschmar(2004) [62] middle range tracingprobability necessary tocontrol diseasesKretzschmaret al. [64] SARS-CoV-2 IBM, homo-geneous pop-ulation direct formu-lation + preprint; Modelbased on Kret-zschmar [62];DCT considered,combined withconventional CT tracing delays need tobe minimized for effectiveCT; DCT might be a wayto speed up the processKucharski etal. [66] SARS-CoV-2 IBM, inho-mogeneouspopulation direct formu-lation - preprint; DCT con-sidered, combinedwith conventionalCT CT more efficient thanmass testingLorch etal. [72] SARS-CoV-2 IBM withdiscrete spa-tial structure one-steptracing, if atsimilar timesin the samelocation - DCT considered,combined withconventional CT DCT efficient particularlyin case of a low fraction ofquarantined personsLunz etal. [73] SARS-CoV-2 phenomen. /ODE SEIR removal rate,mass action - preprint; thetracing rate is con-nected to contactheterogeneity butnot based on firstprinciples optimal CT defined asminimizing the total num-ber of individuals that gointo quarantine during heoutbreakTanaka etal. [104] SARS-CoV-2 IBM, homo-geneous pop-ulation direct formu-lation - simulation of clus-ters detected by CTas input for stats(see also Blum [12]) Bayesian parameter esti-mation based on CTBerge et al.2018 [11] Ebola phenomen. /ODE, SEIR fixed fractionof newly in-fecteds go toquarantine - CT not explicitlyformulated stationary states and theirstability analyzedBrowne et al.2015 [14] Ebola branchingpro-cess/ODE,SEIR fixed fractionof newly in-fecteds go toquarantine + fraction of detectedcases is computedbased on thebranching-process paper aims at a theoreticalframework that is feasiblefor practical applicationsShahtori etal. 2018 [101] Ebola IBM, homo-geneous pop-ulation direct formu-lation + onset of infectiononly crucial that CT is imple-mented at the beginningof the outbreakRivers et al.2014 [96] Ebola phenomen.,stoch. simon pop.-level.+ODE,SEIR increased di-agnosis rate - CT not explicitlyformulated reduction of Reff byaround 30% possible uthor infection method term for CT CTde-lay remarks outcome(w.r.t. CT)de Ara-zoza et al.2002 [21] HIV phenomen. /ODE SI mass action - three classes ofinfecteds modeled:infecteds do /donot know theirinfection, AIDS stability analysis of sta-tionary points, compari-son with dataCl´emen¸conet al.2008 [20] HIV phenomen./ stoch.process(populationlevel), SDE,ODE severalterms: lin-ear, massaction, sat-urationfunction - extension of deAzoza [21] development of statisti-cal tools (maximum likeli-hood estimators)Blum et al.2010 [12] HIV phenomen./ stochasticmodel onpopulationlevel linear andsaturation + see also [82, 104] Bayes interference (ABCand Metropolis Hastings)for estimating ct probabil-ityHsieh et al.2005 [42] HIV phenomen. /ODE SI severalterms: lin-ear, massaction, sat-urationfunction - extension of deAzoza [21] mass action inappropri-ate, linear or saturationterm for CT betterHsieh et al.2010 [43] HIV phenomen. /ODE SI saturationfunction;two-steptracing - extension of deAzoza [21] stability analysis of sta-tionary points, Reff; two-step tracing superior overone-step tracingHyman et al.2003 [45] HIV phenomen. /ODE SI mass action - two models: coregroup, differentstages of HIV Reff, sensitivity analysisMellor et al.2001 [76] HIV and Tu-berculosis IBM withhouseholdstructure screening thehousehold - casual contacts arenot traced; HIVand Tuberculosis atthe same time cross-tracing of HIV andTuberculosis is effectiveNaresh et al.2006 [87] HIV phenomen. /ODE SI fixed frac-tion of newlyinfectedsknow theirinfection - CT is not triggeredby diagnosis, butinfections of knowninfecteds stability analysis of sta-tionary pointsClark et al.2012 [18] Chlamydia pair approx. removal rateon infected-diagnosedpairs - based on [25, 40] CT is efficient particularlybelow a certain prevalenceClark et al.2013 [19] Chlamydia IBM +phenomen.ODE SI IBM: direct;ODE: powerlaw in I - power law adaptedto IBM simula-tions; optimalresource allocations(CT/screening) CT is efficient and uses re-sources efficientHeffernan etal. 2009 [35] Chlamydia phenomen. /ODE SI mass actionlaw - model includesrandom screening(yield index cases)and CT model results in line withdataHethcote etal 1984 [39] Gonorrhea phenomen.ODE decreased in-cidence - introduce backwardand forward tracing pioneering work about CTKretzschmaret al. 1996 [ ? ] Gonorrhea,Chlamydia IBM withhouseholdstructure identificationof a fractionp of partners - one-step tracing prevalence for differentcontrol scenarioshline uthor infection method term for CT CTde-lay remarks outcome(w.r.t. CT)Kretzschmaret al.2009 [61] Chlamydia IBM direct formu-lation three different IBMmodels previouslypublished by dif-ferent authors arecompared the results of the mod-els are somewhat different,due to their complexityTurner et al.2006 [109] Chlamydia IBM direct formu-lation + model with pair for-mation, CT withinpairs Model fits data, and yieldscomparable results com-parable studiesAparicio etal. 2006 [3] Tuberculosis phenomen. /ODE SEIR fixed frac-tion of newlyinfecteds areidentified - CT not explicitlyformulated simulation of prevalenceBegun et al.2013 [10] Tuberculosis review articleKasaie et al.2014 [49] Tuberculosis IBM withhouseholdstructure screening thehousehold - contacts outsidethe household arenot traced household tracing reducesthe incidence by 2%-3%Tian et al.2011 [106] Tuberculosis IBM, inho-mogeneouspopulation direct formu-lation + different scenarios,sensitivity analysis simulation of prevalenceTian et al.2013 [107] Tuberculosis IBM, inho-mogeneouspopulation direct formu-lation + Follow-up ofTien [106] simulation of prevalenceAgarwal etal. 2012 [1] Influenza phenomen. /ODE SIR fraction ofnewly in-fecteds go toquarantine - two risk classesin susceptibles areconsidered Dynamical systems analy-sis, ReffEichner2003 [27] Smallpox phenomen./ Stochasticmodel onpopulationlevel all closecontactsand fractionof casualcontacts aretraced - age of infection in-cluded in the model critical tracing probabilityestimatedKaplan et al.2002 [47] Smallpox phenomen. /ODE saturationfunction - contactees who aretraced are vacci-nated mass vaccination superiorto vaccination triggeredby CTKretzschmaret al. [62] Smallpox Stochasticbranchingprocess direct formu-lation + ring vaccinationtriggered by con-tact tracing delay in CT is crucialPorco et al.2004 [95] Smallpox IBM withhouseholdstructure direct formu-lation + one step and two-step tracing com-pared massive CT and ring vac-cination can control theoutbreak.Liu et al.2015 [69] Measles IBM, inho-mogeneouspopulation direct formu-lation + complex/realisticcontact structure CT can significantlycontribute to control ameasles outbreakBall et al.2011 [7] Theory Stochasticbranchingprocess direct formu-lation - SIR, focus onfixed and exponen-tially distributedinfectious period analytic approach, boundson Reff, extinction proba-bility uthor infection method term for CT CTde-lay remarks outcome(w.r.t. CT)Ball et al.2015 [8] Theory Stochasticbranchingprocess direct formu-lation + SEIR, follow up ofBall [7] effect of tracing delay,Reff, extinction prob.M¨uller et al.2000 [84] Theory Stochasticbranchingprocess direct formu-lation - focus on age sinceinfection Reff, ODE approximation,critical tracing probabilityM¨uller et al.2007 [82] Theory Stochasticbranchingprocess direct formu-lation - based onM¨uller [84], seealso [12, 104] estimation of tracingprobabilityM¨uller et al.2016 [83], Theory Stochasticbranchingprocess direct formu-lation + based onM¨uller [84] effect of tracing delay andlatency period, ReffOkolie et al.2018 [89] Theory Stochasticbranchingprocess direct formu-lation - connects branchingprocess and pairapprox., based onM¨uller [84] effect of a random contactgraph on CTKlinkenberget al. 2006 [58] Theory &influenza,smallpox,SARS,andfoot-and-mouthdisease Stochasticbranchingprocess direct formu-lation + single-step and re-cursive tracing mostly: single step and re-cousive tracing is equal ef-fectiveShaban et al.2008 [100] Theory Stochasticbranchingprocess direct formu-lation - Vaccination of de-tected individuals Reff, probability for ex-tinction, simulation of fi-nal sizeKojaku et al.2020 [59] Theory Stochasticbranchingprocess direct formu-lation - CT on randomgraph, focus ongenerating func-tions for the degree CT highly efficient asnodes with high degree aretracedKiss et al.2006 [55] Theory IBM, inho-mogeneouspopulation direct formu-lation - isolation of suscep-tible; scale free andPoisson network For scale free networks,tracing effect less sensitiveto the epidemiological pa-rametersKiss et al.2008 [57] Theory IBM, inho-mogeneouspopulation direct formu-lation - assortatively /disassortativelymixing networks;Single-step andrecursive tracing CT more effective in dis-sacociative networks; re-cursive tracing more effi-cientFarrahi etal. [28] Theory IBM, inho-mogeneouspopulation direct formu-lation + first model for digi-tal CT digital CT can be efficienteven if the fraction of app-users is smallEames et al.2002 [25] Theory /STD pair ap-prox.& stochsim. direct formu-lation - this paper intro-duced pair approx-imation for CT modeling CT by pair ap-proximationEames et al.2003 [26] Theory pair ap-prox.& stochsim. direct formu-lation - based onEames [25] critial tracing probabilityEames et al.2005 [52] Theory pair ap-prox.& stochsim. direct formu-lation - based on [25]; fo-cus on different so-cial graphs (smallworld, scale-free) network structure influ-ence efficiency of CT uthor infection method term for CT CTde-lay remarks outcome(w.r.t. CT)Eames2007 [24] Theory pair ap-prox.& stochsim. direct formu-lation - based onEames [25] recursive CT much moreeffective than one-stepCT; “targeted CT”: focuson core groupsHouse et al.2010 [40] Theory pair ap-prox.& stochsim. direct formu-lation - based onEames [25]; focuson different so-cial graphs (smallworld, scale-free) CT has higher efficiency inclustered contact graphsHuerta et al.2002 [44] Theory pair ap-prox.& stochsim. direct formu-lation - develop pair ap-proximation forCT rewiring of contact net-work decreases CTTsimering etal. 2003 [108] Theory pair ap-prox.& stochsim. direct formu-lation - based onHuerta [44] rewiring of contact net-work decreases CTArmbrusteret al. 2007 [4] Theory phenomen. /ODE linear re-moval term - addresses a cost-efficiency analysis CT only cost efficient ifprevalence is smallArmbrusteret al. 2007 [5] Theory phenomen. /ODE linear re-moval term - based on Arm-bruster [4] CT only cost efficient ifprevalence is smallMizumoto etal. 2013 [80] Theory phenomen. /next genera-tion operator reduction ofR0 by a fac-tor - focus on theonset; multitype-branching process,analyzed by gener-ating functions Reff, probability for ex-tinction, duration of a mi-nor outbreakuthor infection method term for CT CTde-lay remarks outcome(w.r.t. CT)Eames2007 [24] Theory pair ap-prox.& stochsim. direct formu-lation - based onEames [25] recursive CT much moreeffective than one-stepCT; “targeted CT”: focuson core groupsHouse et al.2010 [40] Theory pair ap-prox.& stochsim. direct formu-lation - based onEames [25]; focuson different so-cial graphs (smallworld, scale-free) CT has higher efficiency inclustered contact graphsHuerta et al.2002 [44] Theory pair ap-prox.& stochsim. direct formu-lation - develop pair ap-proximation forCT rewiring of contact net-work decreases CTTsimering etal. 2003 [108] Theory pair ap-prox.& stochsim. direct formu-lation - based onHuerta [44] rewiring of contact net-work decreases CTArmbrusteret al. 2007 [4] Theory phenomen. /ODE linear re-moval term - addresses a cost-efficiency analysis CT only cost efficient ifprevalence is smallArmbrusteret al. 2007 [5] Theory phenomen. /ODE linear re-moval term - based on Arm-bruster [4] CT only cost efficient ifprevalence is smallMizumoto etal. 2013 [80] Theory phenomen. /next genera-tion operator reduction ofR0 by a fac-tor - focus on theonset; multitype-branching process,analyzed by gener-ating functions Reff, probability for ex-tinction, duration of a mi-nor outbreak