key: cord-0719240-n3lzrnjm authors: Scarabel, F.; Pellis, L.; Ogden, N. H.; Wu, J. title: A renewal equation model to assess roles andlimitations of contact tracing for diseaseoutbreak control date: 2021-01-02 journal: nan DOI: 10.1101/2020.12.27.20232934 sha: 0d2b3c3c475789f8c5b0cab5a09d671e79f420b6 doc_id: 719240 cord_uid: n3lzrnjm We propose a deterministic model capturing essential features of contact tracing as part of public health non-pharmaceutical interventions to mitigate an outbreak of an infectious disease. By incorporating a mechanistic formulation of the processes at the individual level, we obtain an integral equation (delayed in calendar time and advanced in time since infection) for the probability that an infected individual is detected and isolated at any point in time. This is then coupled with a renewal equation for the total incidence to form a closed system describing the transmission dynamics involving contact tracing. We define and calculate basic and effective reproduction numbers in terms of pathogen characteristics and contact tracing implementation constraints. When applied to the case of SARS-CoV-2, our results show that only combinations of diagnosis of symptomatic infections and contact tracing that are almost perfect in terms of speed or coverage can attain control, unless additional measures to reduce overall community transmission are in place. Under constraints on the testing or tracing capacity, the interruption of contact tracing may be irreversible and, depending on the overall growth rate and prevalence of the disease, may lead to outbreaks even in cases when the epidemic was initially under control. Used in combination with other public health efforts for diagnosis of symptomatic individuals or preventive screening, contact tracing has the potential to detect asymptomatic or pre-symptomatic individuals rapidly and efficiently, thus preventing further transmission chains. Moreover, by targeting specifically individuals who have been exposed to the infection, it allows resources to concentrate on the population at risk, in contrast with other untargeted interventions like mass testing, physical distancing or mass quarantining. For these reasons, contact tracing is one of the most cost-effective and widely adopted non-pharmaceutical interventions to counteract the spread of infectious diseases in the absence of effective treatments and vaccines. It was implemented successfully for the control of the spread of coronaviruses including Severe Acute Respiratory Syndrome (SARS) and Middle East Respiratory Syndrome (MERS) [1] , and Ebola virus [2, 3, 4] . In the ongoing battle against the severe respiratory disease caused by the coronavirus SARS-CoV-2, contact tracing has been used to mitigate the global spread of COVID-19 almost universally but with various degrees of success and failure [5] . Many studies, based on different mathematical models, investigate mechanisms of contact tracing and its efficacy. Modelling contact tracing is challenging: a satisfactory mathematical description should account at least for concurrent screening/diagnosis programs that can initiate contact tracing, and the contacts between individuals that occurred prior to that moment. For these reasons, many mathematical models have been formulated as stochastic branching processes or agent-based models, which allow to keep track of the epidemiological status of each individual in the population together with all their infectious contacts [6, 7, 8, 9, 10, 11] . As such, these models require large computations to simulate the epidemics and, although they can provide evidence of the simulation results, extrapolating general insights can be challenging. The few deterministic 2 . CC-BY 4.0 International license It is made available under a perpetuity. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint models available in the literature [6, 12, 13, 14] are typically obtained as approximations of a corresponding stochastic model under simplifying assumptions. Here, we present what, to our knowledge, is the first deterministic model that overcomes these limitations by employing both delayed (in time) and advanced (in time since infection) temporal variables, as will be explained below. We develop a deterministic mixed-type renewal equation model for transmission dynamics that incorporates diagnosis of individuals from symptoms and contact tracing. We focus on forward contact tracing [6, 15] : when an infected individual is detected, only its suspected secondary cases are traced. In contrast, backward contact tracing aims at finding the infector of the index case [16] . The modelling framework is motivated by a classical renewal equation approach [17, 18] where rates are described as a function of time, or age, since infection (i.e., the time passed since the infection event), thus allowing to incorporate realistic distributions of infectiousness and incubation period. The contact tracing process is modelled by describing the probability that an infected index case is detected by contact tracing. This event happens only if the infector of the index case was detected, by diagnosis or contact tracing, hence triggering the search of the secondary cases. This formulation naturally leads to a mixed-type integral equation, delayed in time and advanced in the age since infection, since it depends both on the infection event in the past and on the likelihood that the infector is later tested and traced: in terms of age since infection, the infector is always "older" than the infectee. By describing the relation between infectee and infector, the model enables us to link the probability of being traced with the overall epidemic in the population and to capture the intrinsic correlation between diagnosis of individuals and contact tracing, which is typically one of the main bottlenecks of deterministic approaches [14] . For an emerging outbreak, we provide explicit formulas for the basic reproduction number R 0 , describing the average number of secondary cases generated 3 . CC-BY 4.0 International license It is made available under a perpetuity. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint by one infectious individual in an otherwise susceptible population, and the effective reproduction numbers in the presence of isolation of individuals upon diagnosis and contact tracing. We show that the outbreak can be brought under control only if the proportion of transmission that occurs before isolation is less than 1/R 0 , regardless of how isolation is achieved. This is in agreement with the observation by Fraser et al. [14] , that relate the controllability of an epidemic with R 0 and the fraction of pre-symptomatic transmission. This also shows that, for a disease that is highly infectious or with large amount of pre-symptomatic transmission, a contact tracing program is unlikely to be effective unless other social distancing measures are in place. The public health response to an emerging outbreak in the absence of vaccine involves a coordination and combination of measures including diagnostic testing, contact tracing, and reduction of community transmission via personal protective equipment, physical distancing or, in the case of COVID-19, mass quarantine. Some measures are more disruptive than others to private life and society, and each one impacts the infection dynamics differently. Estimating the feasibility and cost-effectiveness of each measure is key for a successful containment, and requires careful evaluation of several epidemiological indicators. We here ignore the effect of delays in contact tracing and focus on two different characteristics: the tracing window, defined as the length of the period of time preceding detection during which contacts are traced and isolated, and the tracing coverage, defined as the fraction of contacts that are effectively traced. We use parameters from COVID-19 to illustrate several mechanisms of success and failure of contact tracing to prevent the outbreak, in relation to diagnosis of symptomatic individuals (in terms of speed and efficacy of diagnosis) and additional public health measures that aim at reducing overall transmission. Finally, we address a question of public health relevance in the presence of limited testing or tracing capacity, namely the short-term interruption of contact tracing. We show 4 . CC-BY 4.0 International license It is made available under a perpetuity. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint that, with limitations on capacity, a short-term interruption of contact tracing may not be reversible, in the sense that reintroducing contact tracing may not recover the negative growth rate pre-interruption, if capacity is breached. This example underscores how, in the presence of resource constraints, even measures that may seem of minor impact assume an important role in maintaining control [1, 19] . The mathematical model is introduced in Section 2 in the fully nonlinear context, whereas important concepts in the case of an emerging outbreak, including the effective reproduction numbers in the presence of diagnosis and contact tracing, are the focus of Section 3. Finally in Section 4 we consider some specific scenarios and focus on some epidemiological and public health insights from our model-based simulations and analyses. We assume that individuals can either be detected because they develop symptoms (or from any other screening program that does not depend on the overall epidemic) or from contact tracing. We talk about diagnosis of individuals if they are detected from symptoms and tracing if they are detected via a contact tracing process, whereas we talk about detection of individuals either from symptoms or contact tracing. We only consider forward contact tracing: when one individual is detected (index case) and contact tracing is initiated, only secondary contacts of the index case are isolated, rather than their infector. We assume that, upon detection (whether it is by diagnosis or contact tracing), individuals are immediately isolated and do not contribute to the force of infection, and contact tracing of their secondary cases is immediately initiated. This also implies that contact tracing is iterative, in the sense that contacts of contacts are indefinitely (and instantaneously) traced. Throughout this study, we assume that only infected individuals can trigger contact tracing. In reality, this would rely on a monitoring 5 . CC-BY 4.0 International license It is made available under a perpetuity. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted January 2, 2021. ; protocol for which all contacts are immediately tested, so that it is immediately (and with certainty) known which individuals are actually infected and trigger further tracing only for those. The diagnosis process. We denote by h d (τ) the rate of an individual with age since infection (hereafter, ASI) τ being diagnosed for symptoms. We will assume that this is a given known model ingredient, determined both by the disease and by the surveillance strategy, and, for simplicity, we assume it is independent of time (although, in general, diagnosis and testing protocols may vary during the course of an epidemic). Note that, in prescribing h d , we are implicitly assuming that distinct individuals are independent of each other in terms of diagnosis. Typically, h d has compact support (i.e., h d (τ) = 0 if τ is larger than a fixed threshold), since infected individuals can be diagnosed only within a certain time after infection (for example, because after a while the diagnostic test gives a negative result). Let F d (τ) denote the probability that an infected individual has not yet been diagnosed by ASI τ. This is given by where F d (τ) denotes the cumulative distribution function of the time to diagnosis, i.e., the probability that an individual has been diagnosed before ASI τ. From (2.1) it is also clear that h d can be computed from the prescribed cumulative distribution by is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted January 2, 2021. ; https://doi.org/10.1101/2020.12.27.20232934 doi: medRxiv preprint being diagnosed at time τ after infection can be related to the distribution of the incubation time by specifying a suitable density function for the delay between symptom onset and diagnosis, as suggested in [14] . The contact tracing process. Let h c (t, τ) denote the rate at which an individual of ASI τ at time t is detected by contact tracing. The probability that an individual is traced in [t, t + dt] is given by h c (t, τ) dt, and the probability that an individual of ASI τ at time t has not yet been detected by contact tracing is Note that we explicitly include the dependence of h c on time t, since the contact tracing process is intrinsically connected with the underlying epidemic. Moreover, since the contact tracing process alters, but in turn is also affected by, the transmission process, h c (t, τ) is an unknown dynamical quantity in this model. the incidence at time t, we can write where N 0 is the total population (assumed to be constant), and S(t) and Λ(t) are, respectively, the total number of susceptibles and the total force of infection in the population at time t. The susceptible population satisfies S (t) = −y(t), Let β(τ) describe the infectiousness, that is, the per capita contribution to the force of infection, of an infected individual with ASI τ. As for h d , in prescribing β we are implicitly assuming that distinct individuals are independent of each other in terms of transmission. We can also split describes the probability that an infected individual has not yet recovered at ASI τ, and φ represents the infectivity of an individual that has not yet recovered. With this notation, the density of infective (not yet recovered) individuals at time t with ASI τ is Assuming that detected individuals are instantly isolated and do not contribute further to the force of infection, the density of active infective individuals with ASI τ, i.e., individuals who are infective and not isolated, From (2.5) and (2.8) we obtain 8 . CC-BY 4.0 International license It is made available under a perpetuity. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted January 2, 2021. ; which, for given F , is a renewal equation for the incidence y(t). To describe the evolution of the contact tracing rate h c in time, we look at an index case that has ASI τ at time t (i.e., was infected at time t − τ), and compute the probability that the index case is detected by contact tracing in [t, t + dt], which itself depends on the probability that contact tracing is initiated for the infector of the index case, see also Figure 1 . We first study the relation between the ASI of infectee and infector. The probability ψ(t, s) ds that a random infector at time t has ASI in [s, s + ds] is given by Let ε c denote the tracing coverage (i.e., the fraction of contacts that are effectively traced). We can compute explicitly the probability h c (t, τ) dt that an individual that was infected at time t − τ is traced in [t, t + dt] (see also Figure 1 is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted January 2, 2021. ; by Note that the integral at the right-hand side is computed with respect to s. Simplifying dt from both sides and using (2.10), we obtain where F is defined by (2.3) and (2.4), and S is given by (2.6). Equation (2.11) is delayed in t and advanced in τ. In general we assume that only the contacts that took place in a certain time window before detection of an individual are traced. Under this assumption, h c (t, τ) = 0 if τ is larger than the length of the tracing window. Assuming that h d is prescribed, we obtain a closed system by coupling (2.9) with (2.11). Assuming that the integration bounds are finite, which can be obtained by setting β to zero after a certain threshold, the model can be is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted January 2, 2021. ; https://doi.org/10.1101/2020.12.27.20232934 doi: medRxiv preprint (2.7). Moreover, regarding the diagnosed and traced individuals we have Constraints on the capacity of contact tracing resources can be modelled by imposing that the total number of contacts traced in a certain time interval [t 1 , t 2 ] is bounded: where K is a fixed constant describing the maximal capacity. Similarly, constraints on the total diagnosis capacity can be described by To study the effect of limits to the tracing or testing capacity, we considered a state-dependent probability of tracing or diagnosis. In the case of limited tracing capacity, we modified ε c bŷ where K denotes a maximal instantaneous capacity and the denominator is computed from (2.13). In the case of maximal diagnosis capacity it was necessary to consider a time dependent rate of diagnosisĥ d (t, τ) defined aŝ . CC-BY 4.0 International license It is made available under a perpetuity. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted January 2, 2021. ; https://doi.org/10.1101/2020.12.27.20232934 doi: medRxiv preprint During an emerging epidemic, in the approximation S(t) ≈ N 0 , the force of infection is approximately exponential of the form where r represents the Malthusian parameter, or real-time growth rate, of the epidemic. Under this assumption, the solutions of (2.11) are stationary in time, 3) determines r and h c . It is interesting to note that, even in the setting of an emerging epidemic, the contact tracing process is described by a nonlinear equation (3.2). We can define three reproduction numbers: the basic reproduction number is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted January 2, 2021. ; https://doi.org/10.1101/2020.12.27.20232934 doi: medRxiv preprint tracing are in place. From (2.5) it is easy to verify that The controllability of an infectious disease outbreak depends not only on the basic reproduction number R 0 , but also on the proportion of transmission occurring before symptom onset, defined as where F s (τ) is the probability of not having developed symptoms by ASI τ [14] . In the presence of diagnosis only, or with additional contact tracing, the amount of transmission occurring before isolation is given by θ d = R d /R 0 and θ d,c = R d,c /R 0 , respectively. It is therefore evident that, in the absence of vaccine or alternative intervention measures like physical distancing, an emerging disease which is by itself able to spread (R 0 > 1) can be brought under control only if the proportion of transmission occurring before isolation (either by diagnosis or contact tracing) is smaller than 1/R 0 . A related important concept used to measure the effectiveness of contact tracing is the amount of onward transmission prevented by isolation [8, 9] : this amounts to 1 − θ d for diagnosis only, and to 1 − θ d,c for diagnosis and contact tracing combined. In particular, the transmission prevented by contact tracing only (in addition to diagnosis) is θ d − θ d,c . . CC-BY 4.0 International license It is made available under a perpetuity. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted January 2, 2021. ; Table 1 : Parameter values for COVID-19. Note that the infectiousness profile and the incubation period are independent of each other. With the specified parameters, approximately 45% of the transmission occurs prior to symptoms, consistently with [20, 21] . The probability density of the time to diagnosis is obtained by shifting the density of the incubation period to the right by a fixed amount, corresponding to the diagnosis delay, with a 2-day delay in diagnosis corresponding to approximately 75% of the transmission occurring prior to diagnosis. Figure 2 : Curves R d = 1 and R d,c = 1 varying the proportion of individuals who are effectively diagnosed (ε d ), the delay from symptom onset to diagnosis, and the proportion of contacts who are effectively traced (ε c , specified in the legend). For the underlying epidemic we fixed R 0 = 2.5 (left) and R 0 = 1.5 (right). In the former case, diagnosis alone is never sufficient to prevent the spread of the epidemic. For the numerical simulations in this section we used parameters specific to the COVID-19 pandemic, summarised in Table 1 14 . CC-BY 4.0 International license It is made available under a perpetuity. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted January 2, 2021. ; Figure 3 : Effect of the tracing coverage (ε c ) and the length of the tracing window on R d,c (left) and on the fraction of prevented onward transmission in addition to diagnosis (right). We fixed R 0 = 2.5 and assumed no delay in diagnosis. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted January 2, 2021. ; is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted January 2, 2021. ; https://doi.org/10.1101/2020.12.27.20232934 doi: medRxiv preprint tacts are considered for tracing). The influence of these aspects strongly depends on the amount of pre-symptomatic transmission and the infectiousness relative to symptoms [20, 25] . For fixed infectiousness and incubation period distributions as in Table 1 , Figure 3 shows the effect on R d,c and on the onward transmission is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted January 2, 2021. On the left, capacity is on tracing (max 10 −7 total tracing rate); on the right, capacity is on diagnosis (max 10 −7 total diagnosis rate). First row: 50% of infected individuals are diagnosed (ε d = 0.5), the epidemic grows in the presence of contact tracing; the interruption causes a temporary increase in the real-time growth rate and the breaching of capacity (which would have occurred eventually anyway), which leads to increasing growth rate. Second row: 60% of infected individuals are diagnosed (ε d = 0.6), the epidemic is controlled in the presence of contact tracing; interruption of contact tracing increases the value of the growth rate, but reintroducing contact tracing after T = 30 days brings the growth rate to the original level; this is delayed in the case of capacity on diagnosis (right), as the capacity is breached during the contact tracing interruption. Third row: 60% of infected individuals are diagnosed (ε d = 0.6), the epidemic is controlled in the presence of contact tracing; the interruption of contact tracing for T = 50 days causes the breaching of capacity, so that when contact tracing is resumed it is not sufficient to control the epidemic, and an outbreak is observed. The initial condition for the infected individuals was taken as 10 −7 e r d t , where r d is the growth rate due to diagnosis only. Oscillations in the observed growth rate are an effect of the numerical method. 18 . CC-BY 4.0 International license It is made available under a perpetuity. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted January 2, 2021. ; and contact tracing maintain control of the epidemic, and interruption of contact tracing is reversible: since tracing capacity is never reached, the reintroduction of contact tracing eventually restores the original growth rate and allows the control of the epidemic. When the constraints are on diagnosis (right panels) the effect is delayed since diagnosis capacity is breached, but contact tracing still brings the growth rate to original levels (and diagnosis again under capacity). The lower panels illustrate a qualitative worst-case scenario in which the epidemic is initially controlled, but interruption of contact tracing is long enough to make the intervention irreversible: even after the reintroduction of contact tracing the growth rate will remain positive, and will increase due to the increasing prevalence. This analysis is particularly relevant for regions with lower prevalence and growth rates (for example, some rural areas), where the disease may be controllable by contact tracing. We have included the computed profiles of h c from the nonlinear model in Figure A. 8, before and after each contact tracing interruption. . CC-BY 4.0 International license It is made available under a perpetuity. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted January 2, 2021. ; Impact of physical distancing versus contact tracing. The effectiveness of a contact tracing policy depends not only on the tracing efforts, but also on the epidemiological properties of the disease. The same contact tracing policy (in terms of tracing coverage and length of the tracing window) can control an epidemic with low transmissibility, but not an epidemic with high transmissibility. Figure 6 shows the relation of R d,c with the contact tracing coverage (ε c ) and the unconstrained reproduction number R 0 . This relation has important public health implications, especially for planning economic investments. Intensifying contact tracing efforts (e.g., by employing larger number of tracers or developing and deploying digital tracing systems), translates into a shift upward in the plane in Figure 6 , whereas reducing overall contact (e.g., by intensifying physical distancing measures), translates into a shift to the left. Whereas contact tracing efforts may be sufficient to prevent the epidemic to take off in some regions, strict physical distancing measures (e.g., mass quarantine) may be eventually unavoidable in regions with higher transmissibility. Considering limitations to the minimum per capita contact rate or to the maximal contact tracing effort (e.g, due to adherence of individuals to isolation or to disclose contacts [26] ) is fundamental for planning interventions. We formulated a deterministic model for disease transmission dynamics including both diagnosis, either from symptom onset and hospitalisation or from any public health screening/monitoring program, and contact tracing. As many im- is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted January 2, 2021. ; https://doi.org/10.1101/2020. 12.27.20232934 doi: medRxiv preprint the model to investigate several mechanisms of public health relevance using parameters characteristic of the COVID-19 pandemic. A first important aspect is the large impact of detection delays on the control of the epidemic. For the COVID-19 pandemic, many researchers noticed early on that, due to the large potential pre-symptomatic transmission, only high contact tracing coverage coupled with very short delays would be effective to stop the spread, thus supporting the implementation of digital contact tracing as a tool to minimise the isolation delays [8, 9, 10, 11, 23] . Our findings show that, even with extremely quick diagnosis and tracing, the coverage level required to reach control in the case of COVID-19 suggests that digital contact tracing is likely insufficient, unless the current low app usage observed in different countries is significantly improved [27] . A significant reduction in the underlying transmission seems anyhow necessary for any realistic contact tracing or diagnosis program to be successful in gaining control, either obtained through physical distancing or by vaccination programs. In the presence of pre-symptomatic transmission, as in the case of COVID-19, the length of the tracing window, i.e., the number of days back from the detection of the index case during which contacts are traced, is fundamental for the effectiveness of contact tracing. For COVID-19, it was suggested that tracing two days before detection may not be sufficient to cover enough pre-symptomatic transmission [20, 21, 25] . We investigated different choices of contact tracing windows from 2 to 5 days and found that, for R 0 = 2.5 and the other parameters assumed here, if less than 80% of contacts are traced, tracing only 2 days backward is not sufficient to control the epidemic. A tracing window of 5 days allows control to be achieved if the tracing coverage is no less than 50%. In the presence of limited testing or tracing capacity, diagnosis or tracing efforts may need to be redistributed among regions with high or low prevalence, with some regions being prioritised over others. We showed how these decisions 21 . CC-BY 4.0 International license It is made available under a perpetuity. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted January 2, 2021. ; https://doi.org/10.1101/2020.12.27.20232934 doi: medRxiv preprint must take into account several aspects related to the underlying transmission and the overall testing/tracing capacity of the system. A short term interruption of contact tracing in a region where the epidemic is apparently under control may have dramatic impact on the outbreak, as, with limited capacity, the intervention may be irreversible: reintroducing contact tracing when the tracing capacity has been exceeded does not recover the original decline. A careful preliminary evaluation is fundamental especially in response to COVID-19 since, due to possibly significant delays between infection and detection, the effects of the interruption of contact tracing will be visible only after several days [28] . To our knowledge, this is the first deterministic model for contact tracing formulated in terms of a mechanistic description of the individual level processes (disease transmission, diagnosis and contact tracing). Most of the current scientific literature addressing contact tracing relies on stochastic agent-based models. Compared to the latter, deterministic models have several advantages: first, they are computationally more efficient and faster to simulate, which is crucial when simulations must be repeated many times, for instance in model fitting and parameter estimation; second, they often allow to obtain analytically tractable and more transparent relations between structural model assumptions and parameter values, and the conclusions. The model has several limitations that could be addressed in future work. First, it is based on the assumption of homogeneous mixing, so clustered contact tracing, for instance focused on households or realistic social networks [11, 29, 30, 31] , is not captured in this framework. Some level of heterogeneity could be incorporated by considering a vector-valued incidence and contact mixing matrices. Second, to simplify the presentation of the model and to obtain preliminary insights, we have here assumed instantaneous contact tracing. More realistic tracing delays could be incorporated. Third, in the current model, the infectivity profile is assumed to be independent of the incubation time. Given the efficacy 22 . CC-BY 4.0 International license It is made available under a perpetuity. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted January 2, 2021. ; of contact tracing is strongly affected by the amount of pre-symptomatic transmission, an important extension would be to incorporate the correlation between individual infectiousness and onset of symptoms. Fourth, the model only tracks infected individuals, ignoring contacts that did not lead to an infection. The advantage is that our construction does not require specifying the contact rate in the overall population, but only the infectivity profile of infected individuals. However, by making assumptions about the overall contact rate, we could fur- is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted January 2, 2021. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted January 2, 2021. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted January 2, 2021. ; https://doi.org/10.1101/2020.12.27.20232934 doi: medRxiv preprint is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted January 2, 2021. ; https://doi.org/10.1101/2020.12.27.20232934 doi: medRxiv preprint is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted January 2, 2021. ; https://doi.org/10.1101/2020.12.27.20232934 doi: medRxiv preprint Figure A.7: Left column: computed h c and cumulative distribution function of the time to isolation for contact tracing, for different lengths of the tracing window. The cumulative distribution function is computed as 1 − F c from (2.3). The right column shows the same quantities (rate and cumulative distribution function) for the overall detection from diagnosis and contact tracing combined. We took R 0 = 1.5, a 2-day diagnosis delay, and ε c = 1. . CC-BY 4.0 International license It is made available under a perpetuity. is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted January 2, 2021. ; https://doi.org/10.1101/2020.12.27.20232934 doi: medRxiv preprint is the author/funder, who has granted medRxiv a license to display the preprint in (which was not certified by peer review) preprint The copyright holder for this this version posted January 2, 2021. ; https://doi.org/10.1101/2020.12.27.20232934 doi: medRxiv preprint Epidemic Models of Contact Tracing: Systematic Review of Transmission Studies of Severe Acute Respiratory Syndrome and Middle East Respiratory Syndrome Implementation and management of contact tracing for Ebola virus disease A model of the 2014 Ebola epidemic in West Africa with contact tracing Role of contact tracing in containing the 2014 Ebola outbreak: a review Contact tracing in the context of COVID-19. 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The authors thank two anonymous reviewers for helpful comments that improved the readability of the manuscript. FS thanks Eugenia Franco for useful discussions on some mathematical aspects of the model.