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The authors have declared that no competing interests exist.

Current address: Centre for the Mathematical Modelling of Infectious Diseases, London School of Hygiene and Tropical Medicine, London, UK is current address for W.Waites

Existing compartmental mathematical modelling methods for epidemics, such as SEIR models, cannot accurately represent effects of contact tracing. This makes them inappropriate for evaluating testing and contact tracing strategies to contain an outbreak. An alternative used in practice is the application of agent- or individual-based models (ABM). However ABMs are complex, less well-understood and much more computationally expensive. This paper presents a new method for accurately including the effects of Testing, contact-Tracing and Isolation (TTI) strategies in standard compartmental models. We derive our method using a careful probabilistic argument to show how contact tracing at the individual level is reflected in aggregate on the population level. We show that the resultant SEIR-TTI model accurately approximates the behaviour of a mechanistic agent-based model at far less computational cost. The computational efficiency is such that it can be easily and cheaply used for exploratory modelling to quantify the required levels of testing and tracing, alone and with other interventions, to assist adaptive planning for managing disease outbreaks.

The importance of modeling to inform and support decision making is widely acknowledged. Understanding how to enhance contact tracing as part of the Testing-Tracing-Isolation (TTI) strategy for mitigation of COVID is a key public policy questions. Our work develops the SEIR-TTI model as an extension of the classic Susceptible, Exposed, Infected and Recovered (SEIR) model to include tracing of contacts of people exposed to and infectious with COVID-19. We use probabilistic argument to derive contact tracing rates within a compartmental model as aggregates of contact tracing at an individual level. Our adaptation is applicable across compartmental models for infectious diseases spread. We show that our novel SEIR-TTI model can accurately approximate the behaviour of mechanistic agent-based models at far less computational cost. The SEIR-TTI model represents an important addition to the theoretical methodology of modelling infectious disease spread and we anticipate that it will be immediately applicable to the management of the COVID-19 pandemic.

This is a

Since the beginning of 2020, the W orld has been in the midst of a COVID-19 pandemic, caused by the novel coronavirus SARS-C

The World Health Organisation has recently updated their guidance on this, recommending a six point strategy that requires firstly assuring that the pandemic spread has been suppressed, and is followed by detecting, testing, isolating and contact-tracing of infected individuals [

Mathematical modelling has figured prominently in decision making around control and containment of

Mathematical models have a long history of being used to describe the spread of infectious diseases from plague outbreaks more than a century ago [

While classic compartmental models can easily be used to simulate some interventions analogous to parameter changes, they cannot readily include contact tracing of infected individuals unless vast assumptions are made. This is because modelling contact-tracing is intrinsically reliant on individual behaviour within a network structure. Previous work on Ebola [

In this paper we develop an extension to the classic Susceptible-Exposed-Infectious-Removed (SEIR) model [

We note that slightly different nomenclature for SEIR has ben used by different authors. Exposed means infected but not yet infectious and is sometimes called Latent. Infectious is sometimes called Infective and represents individuals capable of transmitting the disease. Removed is often called Recovered, though we opt for the former as it indicates that those individuals are no longer causing infection but we make no statement about whether they are removed through recovery or death.

Due to its relative simplicity, SEIR-TTI is applicable across a spectrum of diseases. With appropriate parametrisation, it can be used anywhere a standard SEIR model can be used with the same caveats and limitations.

Though we are clearly motivated by the current COVID-19 pandemic and wish to understand how interventions like TTI can be used to contain it, we do not claim that we are modelling it in particular. Our contribution is a mathematical tool and software implementation that can be used for understanding TTI, not a model of COVID-19.

The method that we present is general and can also be applied to other compartmental models, with the standard caveat that with more compartments comes more work to determine the appropriate rates that need to be informed by data. We validate our SEIR-TTI ODE model against a mechanistic agent-based model where testing, tracing and isolation of individuals is explicitly represented and show that we can achieve good agreement at far less computational cost. We also provide a flexible software package at

We design a compartmentalised model describing the populations of susceptible (

These models are widely used to describe the spread of various infectious diseases with disease progression captured by movement of individuals sequentially between compartments accounting for progression from susceptible individuals (

SEIR is a compartmentalised model describing susceptible (_{I} represents contact with an infectious individual. Non-infectious individuals having been isolated through contact tracing have, in effect, been misdiagnosed. Individuals transition between compartments _{X→Y} which we derive in the text.

The novelty of our model is that we have within each compartment included subgroups of people diagnosed and undiagnosed with the virus, attributable to reported and unreported diagnosis. Individuals in our model are defined to be diagnosed either through testing or putatively through tracing. Diagnosed individuals are then isolated.

Before introducing contact tracing, we examine the standard SEIR model with testing. These results, and those in the following section, use the system of differential equations as described in detail in the Methods. We choose a relatively large initial number of infectious individuals merely for illustrative purposes as it renders the dynamics clearer—the more aggressive testing regimes would result in immediate containment of a small outbreak which would be difficult to see whereas a large outbreak nevertheless takes some time to contain. The parameters have the usual meaning, with values fixed for the purposes of this section: ^{7} individuals is the total population, ^{5} is the initial number of infected individuals, ^{−1} is the incubation rate, the rate of leaving the exposed state and becoming infectious; and ^{−1} days^{−1} is the rate of recovery, or leaving the infectious state. These values result in a basic reproduction number of _{0} = 3. In the simplest case, testing is conducted at random at some rate

Representative trajectories from this system for various values of _{e}(_{e}(

The dynamics represented here are for a scenario with normal contact, _{e}(

The red line is is given by the equation

The above shows that, whilst testing and isolating alone can be sufficient to control an outbreak, it would take a herculean effort on its own. Without any form of distancing (

The central mathematical result is the expression for the rate at which individuals are isolated due to contact tracing,
_{U}, and isolated, _{D}, sub-compartments, the rate of moving between them is proportional to the probability of having had contact with an infectious individual conditional on being in _{U}.

The effects of contact tracing is shown in ^{−1}days^{−1} and the tracing rate is fixed at ^{−1}days^{−1}. The tracing success rate, ^{−1}days^{−1}. One would expect that testing and isolating individuals, on average, after they have recovered and it is too late would be insufficient to contain an outbreak. Indeed it is not sufficient, but it does reduce the maximum number of infected individuals somewhat. However, since tracing happens as a consequence of testing, it amplifies its effectiveness. This can be seen in the figure where even a modest tracing success rate of 30-40% results in a substantial reduction of more than half the peak infections.

The dynamics presented here are the same as those of ^{−1}days^{−1}, meaning testing of infectious individuals on average once per two weeksonce per week. The rate of contact tracing is set at ^{−1}days^{−1}, meaning that it takes on average two days to trace a contact. A variety of values of tracing success rate,

The relationship between testing rate and tracing rate can be seen from

The central result of this paper is not specific observations about how testing and contact tracing affect the propagation of epidemics, though those are valuable, but a technique to compute these effects efficiently. This technique allows consideration of larger populations than would be possible with agent- or individual-based models allowing for the exploration of many different scenarios. Figs

It could be argued that it is sufficient to capture these dynamics in an agent-based model for modest populations and simply rescale the output for large populations. That approach is not sound for two reasons that are easily seen. First, small outbreaks. Imagine a hypothetical country of 70 million people with 100,000 infections. Proportionally, that is 14.3 infections in a population of 10,000. There is a non-negligible probability that an outbreak of size 14 will die out on its own. This will be accounted for by the ABM but is not a realistic possibility for an outbreak of 100 thousand. Scaling therefore suggests fundamentally different results. Second, without intervention, the number of infectious individuals will reach a maximum as the available pool of susceptible individuals becomes depleted. This takes longer in a large population simply because the pool is larger. If timing of the peak of an outbreak is a quantity of interest, a scaled ABM will give the wrong result.

However, doing this requires some approximations and it is important to understand where and how well these approximations hold. To do this, we compare with two different agent based models as described in the methods, and show that our method agrees well for a large range of physically interesting and realistic parameter values. The first ABM reproduces the same uncorrelated processes as the ODEs, with agents moving between compartments at constant rates, without any correlation with their time of arrival in them. This results in an exponential distribution of the times that each agent spends in a given state. However, in reality, the distribution of these times is rarely exponential; more realistic choices are distributions with a maximum at

A comparison of the ODE and the first type of ABMtwo systems for reasonable parameter values is shown in

Here the population size is 10000 individuals with 100 initially infected. The testing rate is ^{−1}days^{−1}, and tracing rate and success probability are _{U}, _{U}, _{U} and _{U}. The bottom row are those representing isolated—diagnosed or distanced—individuals, _{D}, _{D}, _{D}, _{D}. The heavy orange curves are the output of the ODE-based simulation. The teal curves are the average output of the agent-based simulation, with envelopes for one and two standard deviations.

There exist extreme scenarios where the ODE performs poorly at reproducing the mean trajectory of the ABM system. An example is shown in ^{−1}days^{−1}. This circumstance violates the assumption underlying

This plot shows the effect of very low levels of testing, ^{−1}days^{−1}. In this circumstance, the number of traceable susceptible individuals takes on unphysically high values, shown by the red line in the top left panel. This results in an overestimation of the maximum number of unconfined exposed and infectious individuals and a corresponding underestimation of the effect of contact tracing in preventing infection in this scenario.

^{−1}days^{−1} and _{0} = 2

Here the population size is 10000 individuals with 100 initially infected. The testing rate is ^{−1}days^{−1}, the base testing rate is _{0} = 2_{U}, _{U}, _{U} and _{U}. The bottom row are those representing isolated—diagnosed or distanced—individuals, _{D}, _{D}, _{D}, _{D}. The heavy orange curves are the output of the ODE-based simulation. The violet curves are the average output of the agent-based simulation with correlated transitions, with envelopes for one and two standard deviations.

As in ^{−1}days^{−1}, the base testing rate is _{0} = _{U}_{U}_{U}_{U}_{D}_{D}_{D}_{D}

We consider the problem of determining the effect of testing and contact tracing in a population,

to keep track of whether people have been isolated from the rest (either due to testing positive, or having been traced as a contact of someone who tested positive)

to keep track of whether people have been in contact with an infectious individual recently enough to be potential targets for tracing.

Ordinary compartment models like SEIR are designed to separate individuals into distinct, non-overlapping groups. This is not a problem for the first feature, as people who are isolated and people who are not constitute entirely distinct sets. We therefore can represent unconfined and isolated individuals simply by doubling the number of states, labeling _{U}, _{U}, _{U} and _{U} the Undiagnosed people who are respectively Susceptible, Exposed, Infectious, or Removed, and similarly, _{D}, _{D}, _{D} and _{D} the ones who have been Diagnosed or otherwise Distanced from the rest of the population, by means of home isolation, quarantine, hospitalisation and such.

However, dealing with contact tracing is harder, as it can not be achieved with separate compartments. Here we take two approaches. First, we describe an agent-based model that simulates contact tracing with an approximation of how it could take place in real life. This agent-based model serves as our reference. Then we describe fully our compartment model, and, relying on a system of second order Ordinary Differential Equations (ODEs), we introduce the concept of overlapping compartments. Overlapping compartments represent model states that are not mutually exclusive, so that it is possible for an individual to belong in more than one of them e.g. be infected and contact-traced, or exposed and tested. We define equations for this model in order to represent the processes that happen in the agent-based model, providing the comparisons seen above in the Results section.

Among the possible measures to suppress an epidemic, contact tracing is defined as “an extreme form of targeted control, where the potential next-generation cases are the primary focus” [

We start by defining our modified SEIR model in agent-based form. The model features _{U}, _{D}, _{U}, etc. respectively the numbers of individuals in each combination of those states, and _{T} the total number of such traceable individuals. This contact matrix encapsulates a history of interactions in a way that is realistic but is not possible to represent directly in ODE form. It is specifically the functioning of this individual contact matrix that we claim to reproduce at the population level with our ODE formulation below.

We simulate the model using Gillespie’s algorithm [

contact between a random individual and one belonging to _{U}, with rate _{U}. The contact is stored in the contact matrix. If the individual happens to belong in _{U}, with likelihood _{U} individual becomes _{U};

progression of the disease for an

recovery from the disease, or removal due to hospitalisation or death, for an

diagnosis by regular testing of an _{U} individual, with rate _{D}; all its past contacts, retrieved from the contact matrix, are marked as traceable with likelihood _{D} was marked as traceable, it is unmarked (as they’re already in isolation and there is no need to trace them any more);

release from isolation of an _{D} individual, making them _{U}, with rate _{D};

release from isolation of an _{D} individual, making them _{U}, with rate _{D};

contact tracing of a traceable individual with rate _{T}. The individual is moved from _{U} to _{D}, where

The transitions described above can be intuitively seen as corresponding to the ones that would happen in an idealised real-life version of epidemic spread with testing and contact tracing. The biggest deviation from reality is the perfect mixing of the population implied by the first process. The testing and tracing processes are parametrised by

We also define a second ABM, for the purpose of investigating how time correlation between events affects the results. In regular ODE models, and in the ABM that was described above, transitions between states happen at fixed rates but are completely uncorrelated; this results in an exponential distribution of times each agent spends in a given state. In real life, this is obviously an unrealistic scenario: in particular, the time necessary for someone to recover from a disease is generally better described by a peaked distribution [^{−1}. Similarly, contacts between individuals happen regularly at intervals of ^{−1}.

The main difference between this model and the previous one is in the mechanism used to describe testing. In the regular ABM and in the ODE model, a single parameter ^{−1} = 14 days might mean that everyone who is infected gets tested 14 days after infection, or that 50% get infected 7 days after, and the rest not at all. These can lead to very different outcomes in this model; in particular, if ^{−1} > ^{−1}, no one will be tested before recovering, and thus, testing is as good as non-existent. For this reason, for this specific model, we further split the parameter in two:
_{0} being the ‘base’ testing rate and 0 ≤ _{θ} ≤ 1 the fraction of individuals in the _{0} are defined as input parameters and _{θ} is derived from their ratio. The waiting time for a test will then be _{θ} of agents, infinite for everyone else.

We begin by introducing the ODE form of the standard SEIR model [_{X→Y} as the rate at which members of the population move from compartment _{S→E} is the rate at which Susceptible members of the population are Exposed to the virus. In addition, for convenience when discussing movements that can happen due to multiple phenomena, we might add a superscript, such as

With this notation, the differential equations that describe the standard SEIR model have the following form,

The terms in the above differential equations are defined in the usual way as,

While this formulation treats the populations as continuous analytical functions, in general these equations describe the mean trajectory of what is fundamentally a stochastic system. This stochastic system can be simulated with Gillespie’s algorithm and, up to this point, is equivalent in the continuous limit to an agent-based model featuring the same compartments and transition rates.

Now we add diagnosis to our description. Four more compartments, _{D}, _{D}, _{D} and _{D}, are created to keep track of population cohorts who have been identified as potentially infected, and thus isolated from the rest of the population as a measure to limit the spread of the disease. Disease progression is not affected by this process; therefore,

Including isolation will change the infection rate, as unlike population _{U}, the isolated population _{D} does not contribute to further infection. Hence we do not include an infection term here. This is an idealisation. In reality isolation will not be perfect, and we can imagine a reduced ‘cross-infection’ rate in which some people belonging to _{U} are infected by people in _{D}. This could happen with medical professionals treating infectious patients or care workers who maintain a quarantine facility. We could even consider infection of people in _{D} due to those in _{D}, such as a patient in home isolation infecting their family. However, for present purposes, we will work in an ideal situation where isolation is perfect.

Finally, we need to incorporate mechanisms to move individuals between the _{U} who, each day, are diagnosed with the disease. We note that this parameter does not refer to any specific testing procedure; it just represents the total of people who are recognised as having the disease. It can represent, for example, actual testing for a specific pathogen as well as clinical diagnosis. We only focus on the category of _{U} as these are the patients who are most likely to realise they are sick and seek medical help. This generic testing process is described by the equation,

In addition, people will be released from isolation after a finite time without symptoms. For this reason, we don’t include a mechanism for people in _{D} to return to the

With this model adaptation, a single infected individual can now take two paths:

_{U} → _{U} → _{U} → _{U}, in which they are exposed to the disease, become infectious, and finally recover, without being isolated or diagnosed, as in the normal SEIR model, or,

_{U} → _{U} → _{U} → _{D} → _{D} → _{U}, in which, after becoming infectious, they are identified, isolated, removed from the pool of those who can infect other susceptible people, and after recovering, released from isolation.

Having these two paths allows attainment of some degree of control of the epidemic; however, it must be noted that while we have introduced them, the states _{D} and _{D} are here left unused. This is because at this stage we associate testing with symptomaticity; there is yet no mechanism other than by diagnosis to identify someone who could be infected. This is especially problematic in terms of the impossibility of isolating exposed people. These are individuals with a latent infection who will soon become infectious. Isolating them pre-emptively would contribute a great deal towards suppressing the epidemic. For this reason, we move on to include contact tracing as a means of preventive isolation.

We’ve seen previously that it is intuitive how contact tracing can be represented in an agent-based model, in which individuals are simulated and each has an history of contacts with other members of the population. It is not as obvious how to treat contact tracing in a compartment model, where there is no memory of the histories of contacts of specific individuals, but only average quantities. We outline here a probabilistic method for doing this.

Let us define Pr(_{U}) = _{U}/_{I}) the probability of an individual of having had contact with an infectious individual in the past where that infectious individual is still infectious. The latter detail is important because here we consider only “next-generation” tracing; in other words, we only try to trace the direct contacts of those infectious individuals who were found to test positive. This is a conservative assumption. It could be possible to make contact tracing more effective by also tracing one generation further (the contacts of the contacts), but because the process requires exponentially more resources with each generation with decreasing likelihood of correctly identifying exposed or infectious individuals, we simply opt to neglect that possibility. Therefore, in this model the only people who can be traced are those whose most recent infectious contact is _{T}) the probability of an individual of being traced. All these probabilities are functions of time, and quantities that evolve with the model itself.

First, we rewrite the probability of being traced is
_{T}|_{I}) is the conditional probability of being traced given that one has had an infectious contact in the past, and Pr(_{T}|¬_{I}) the probability of being traced given that one has not. Clearly, Pr(¬_{I}) = 1 − Pr(_{I}). If we ignore the possibility of false positives, then Pr(_{T}|¬_{I}) = 0, namely, a person can only be traced if they did have an infectious contact in the past. If we then set an ‘efficiency’ parameter

To derive transition rates among compartments, we consider that individuals will be traced proportionally to how quickly the infectious individuals who originally infected them are, themselves, identified. We add a factor

The difficulty is then computing the exact probabilities. These are functions that, in general, vary in time and require a certain degree of information about the past. We need to define useful assumptions and approximations in order to work with these probabilities in a model that inherently lacks any memory about the individual histories of the elements of its population.

One simple assumption for Exposed and Infectious individuals is

Another limit of this assumption is that we have defined Pr(_{I}) as the probability of having had an infectious contact

Estimating Pr(_{I}|_{U}) and Pr(_{I}|_{U}) is more complicated. One possible approximation is to work as if _{U} were constant on the time-scales of interest; in that case we would have
_{U} state. Putting together recovery, regular testing, and contact tracing, we find _{U} might still be infected, and thus only has a probability 1 − _{U}, as it may often reflect reality very poorly.

We consider for example the total number of members of _{U} who also have had recent infectious contacts, _{I}|_{U}) = Pr(_{I}|_{U})_{U}. We can describe these in first approximation as
_{X}(_{I}, meaning the survival function of the total number of infectious individuals, _{U} + _{D}, because here we focus on overall infectiousness, not the fact that one might have been isolated before recovery. Note, however, that only _{U} individuals participate in contacts. The reason that this is an approximation is that we’re not excluding the _{I}|_{U}) from the pool of _{U} that can be contacted, and thus there is a risk of double counting. That risk will remain negligible as long as _{I}|_{U})/_{U} is small; therefore, this model will perform better in a regime in which there are few infectious individuals, and thus, few contacts. This is in fact the regime in which contact tracing is most likely to be feasible in practice, to control small outbreaks rather than in presence of an uncontrolled epidemic. Regardless, we show in the Results section that even when this approximation does not hold, while it results in oscillatory behaviour early on, it still generally adequately describes the overall trends and long term equilibrium. _{I} =

Given the similarities between these equations and the ones describing the compartment models, it is natural to think of creating a specific compartment for _{I}|_{U}). This is in fact what we do. There is, however, an important difference from regular compartments, because this compartment does not include individuals that exclusively belong to it; rather, it overlaps with _{U}. It is more of a device used for book-keeping purposes, to compute the integral in _{I}|_{U}), _{I}|_{U}) and _{I}|_{U}), which leads, using

There is a lot going on in Eqs

elements are ‘created’ for each state proportionally to the rate of contact with individuals belonging to _{U}, adjusted with 1 − _{U} to account for the likelihood that the contact is infective. These terms are ‘sources’ and can be recognised by having an arrow with nothing on its left in the subscripts;

elements ‘decay’ at a rate that amounts to _{U} → _{D}. These terms are ‘sinks’ and can be recognised by having an arrow with nothing on its right in the subscripts;

elements move between compartments following the usual transitions that control the dynamics of the SEIR model (infection, progression of the disease, recovery). These terms are analogous to the corresponding ones connecting _{U} states, and contribute the remainder of the hazard function for each _{U} to

It must also be noted that, in practice, considering _{I}|_{U}) = _{U} and _{I}|_{U}) = _{U}, which removes the need for two of the four compartments above and simplifies the equations to

A few words are necessary on the hazard function for the _{U} → _{D} transitions. This is approximated as _{U} and _{U} even though that is not precisely correct; the correct hazard function would be _{I}|_{U})/_{U}, but that introduces a risk of instability for small values of _{U}. We justify this choice by the following reasoning. In a weak testing regime (_{I}|_{U})/_{U} might be high due to a great number of infected individuals, but in principle should never be greater than 1 (modulo the point above about double counting). Therefore, the hazard function is dominated by _{I}|_{U})/_{U}, will be very small, and this assumption will at most end up underestimating the effect of contact tracing (by causing a faster decay in _{I}|_{U}) than otherwise would happen). The examples shown in the Results section illustrate how this affects the simulations—in general, leading to good predictions for the behaviour of the _{U} and _{U} compartments.

Eqs

Parameter | Description |
---|---|

Population size | |

Average contacts per day | |

Transmission rate per contact | |

^{−1} |
Incubation period (time from exposed to infectious) |

^{−1} |
Recovery period (time from infection to recovery) |

Testing rate of infectious individuals | |

Efficiency or success rate of contact tracing | |

Contact tracing rate |

We implement the above ordinary differential equations and agent-based model in our PTTI Python package (

The PTTI package provides a declarative language for specifying simulations of models implemented as Python objects. It supports setting of model parameters, simulation hyper-parameters as well as interventions that modify parameters at particular times to conduct piece-wise simulations reflecting changing conditions in a convenient and user-friendly way. We hope that this software formulation will be useful for easy and rapid exploration of the effects of different intervention scenarios for disease outbreak control.

Our work outlines a method for extending the classic SEIR model to include Testing, contact-Tracing and Isolation (TTI) strategies. We show that our novel SEIR-TTI model can accurately approximate the behaviour of agent-based models at far less computational cost. Our adaptation is applicable across compartmental models (e.g. SIR, SIS etc) and across infectious diseases. We suggest that the SEIR-TTI model can be applied to the COVID-19 pandemic to understand the impact of possible TTI strategy to control this outbreak.

The importance of modelling to support decision making is widely acknowledged, but models are far more useful when they can accurately represent the classes of interventions that are being considered [

Our work is novel as it is to date, and to the best of our knowledge, the first deterministic model to explicitly incorporate contact tracing. Previously, an attempt to model contract tracing was made by Fraser et al. [

Unlike this model, we have explicitly incorporated in our framework tracing level of both exposed and infectious people—hence allowing the pool of traced people to be increased and specifically accounting for the two groups. Furthermore, we also consider that those traced will be isolated with certain probability and hence we view isolation as follow-on process from tracing and dependent on it. The main purpose of the model in [_{e}(0)) contact tracing needs to be added to the set of control measures. But the issue with an emerging pandemic, such as

Contact-tracing has been until now typically modelled successfully with agent-based models. We are aware that agent-based models allow more realistic infectiousness profiles to be incorporated, and we have done so in our other work [

An important aspect of our approach is that our ODE formulation

Namely, agent-based models are formulated in terms of local interactions among individuals and exhibit emergent behaviour at the population level. For interesting agent-based models, it is usually difficult to obtain any explicit connection between the local interactions and the population-level dynamics except through simulation and inspection of the results. We argue that our work here shows such an explicit connection: we have been able to capture the dynamics that arise at a population level from testing and contact tracing. We show that this is correct by demonstrating good agreement with the population-level dynamics that emerge from the agent-based formulation where only local interactions are specified.

The SEIR-TTI model here considers disease propagation in the classical well-mixed setting. This is appropriate especially in circumstances where data are sparse and gives qualitatively similar results to those from fine-grained models that might otherwise provide more quantitatively accurate results if only more detailed data were available. In particular, well-mixed models do not include any notion of the network of contacts across which a contagion spreads in the real world. In reality, individuals in a large population are not equally likely to have contact with one another and it has long been known [

Another extension is investigating the extent to which individual decisions about compliance with measures to reduce disease propagation (voluntary distancing, wearing of masks, etc.) affect the success of containment. A game-theoretical approach such as that considered by Zhao et al. [

An important next step in this work is the real-time policy driven application of SEIR-TTI. As our next piece of work we are planning to explore how SEIR-TTI model can be combined with economic analysis to guide decisions around optimal design of a TTI strategy that can suppress the

This paper shows how to extend compartmental models to incorporate testing, contact tracing and isolation. The resulting SEIR-TTI model is a key development in the widely used SEIR models, and an important step if these are to be useful in policy decision making during outbreaks. The long and successful history of testing, contact tracing and isolation in slowing and stopping the spread of infectious diseases is well known [

The design of policies that include a variety of infectious disease control tools, and understanding and applying them in ways that are effective for society at large, is critical. Tools and models that allow policymakers to better understand the policies and the dynamics of a disease are therefore critical. If making policy decisions without evidence is flying blindly, making decisions without understanding the consequences of the various control measures is flying without flight controls. Models like SEIR-TTI can inform policymakers of the role that testing and tracing can play in preventing the spread of disease. Combined with economic and policy analysis, this can enable far better decision making both in the immediate future, and in the longer term. The next step in our work is indeed this: the application of the SEIR-TTI model combined with economic models to investigate the effect of different TTI strategies to conquer the

Dear Dr Panovska-Griffiths,

Thank you very much for submitting your manuscript "Testing, tracing and isolation in compartmental models" for consideration at PLOS Computational Biology. As with all papers reviewed by the journal, your manuscript was reviewed by members of the editorial board and by several independent reviewers. The reviewers appreciated the attention to an important topic. Based on the reviews, we are likely to accept this manuscript for publication, providing that you modify the manuscript according to the review recommendations.

We apologize for the delay in our decision. It was extremely challenging to find experts who had the capacity or the willingness to review your work. Unfortunately, we ended up with only one scientific review report. This report is, in our view, insightful and generally positive. Because we, the editors are also positive about your work we decided to take a decision with fewer reports than usual.

It would be great if you would address the points of criticism the reviewer raises, and that we believe are well taken. Especially the point on extending your perspective of ABM by more realistic infectivity profiles and distributions of time spent in each state would be worthwhile to discuss as a limitation. We also encourage you to include a comparison with an ABM that makes more realistic assumptions in this regard, if feasible.

In addition, there is previous work by Fraser et al that you mention in the Introduction (ref 52) that is very similar in scope to your work. It would strengthen your work, in our opinion, if you compared your work to the approach by Fraser et al in more detail in the Discussion.

Lastly, please also note the suggestions in the reproducibility report to improve documentation and reporting of initial conditions.

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Reviewer's Responses to Questions

Reviewer #1: The Reproducibility Report is uploaded as an attachment.

Reviewer #2: In this work, Sturniolo et al. formulate a method for modelling contact tracing in a compartmental model and test this ODE approximation against an ABM with the same structure. Overall, I think this work is valuable and should be published. I particularly liked the clear discussion of when the ODE approximation breaks down.

That said, I do have one major comment relating about the impact of constant infectivity and constant transition rates in ODE models. Ideally, this would require additional comparison between the ODE model and ABM. If this involves a prohibitive amount of work, it could potentially be adequately addressed through additional discussion.

I also have some additional points relating to presentation.

1. Exponential time distributions in ODE and ABM

The advantage of ABMs is discussed in terms of allowing fine-grained modelling of individual variation in susceptibility to disease and contact patterns. The authors point out that we often lack data to parametrise such fine grain models.

However, in the context of contact tracing, ABMs provide additional flexibility for important parameters we do have data about:

i) more realistic infectiousness profiles than the constant infectiousness assumed in ODE models (for covid infectiousness profiles, see for example He et al., 2020 Nature Medicine; Ferratti & Wymant et al, 2020 Science);

ii) more realistic distribution of times spent in each state – for example, fixed time delays for testing and tracing rather than constant rates (which lead to exponential waiting time distributions).

These are of course standard limitations of ODEs, but I think it may be particularly important when the aim is to model contact tracing: the effectiveness of tracing depends on where in the infectiousness profile individuals are isolated. The interplay between assumptions about the shape of the infectivity profile and waiting time distributions may therefore have a significant impact.

This manuscript would be considerable stronger if the authors included a comparison with an ABM with different assumptions of infectivity profile and time distributions (even for just one of the processes – e.g. the time it takes to trace contacts), to give a reader a sense of the magnitude and direction of the error this may introduce.

However, if implementing such a comparison is a prohibitive amount of work, discussing this point might be enough, particularly if the authors are able to provide an intuition for how these approximations might affect the output of the ODE.

2. Framing of testing and interpretation of theta

In the results section, testing and isolation (without tracing) is presented as random testing of the “entire population” (but also, confusingly, “only infectious individuals are tested and isolated”). Theta is therefore interpreted as the average testing rate in the population.

In the methods, the same process is described as infectious individuals experiencing symptoms and seeking diagnosis. Theta is therefore interpreted as the rate of experiencing symptoms and receiving a diagnosis.

Firstly, this inconsistency makes the paper confusing to read.

Secondly, it is not clear to me that that these two framings of theta are equivalent: if the entire population gets tested every 14 days on average, the expected time delay between getting infected and tested would be 7 days.

I think the authors are aware of this, based on the last paragraph on page 6 (or is theta = 1/14 a typo? The value of theta is 1/7 in subsequent figure legends), so it is not clear to me whether I should interpret the result that “testing the entire population every 20 days” as an expected delay from infection to diagnosis of 20 or 10 days.

3. Minor points:

- Calling the D compartment “diagnosed” is confusing when it also applies to susceptible individuals. In particular, this makes figure 1 difficult to interpret.

- Some variables are not defined when introduced (eta in equation 1, C_I in Figure 1).

- R stands for both recovered and reproductive number, which makes figures 2 and 4 a little confusing (from the axes labels, one would assume R(t) is recovered).

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Testing, tracing and isolation in compartmental models

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