Effectiveness of isolation measures with app support to contain COVID-19 epidemics: a parametric approach.

In this study, we analyze the effectiveness of measures aimed at finding and isolating infected individuals to contain epidemics like COVID-19, as the suppression induced over the effective reproduction number. We develop a mathematical model to compute the relative suppression of the effective reproduction number of an epidemic that such measures produce. This outcome is expressed as a function of a small set of parameters that describe the main features of the epidemic and summarize the effectiveness of the isolation measures. In particular, we focus on the impact when a fraction of the population uses a mobile application for epidemic control. Finally, we apply the model to COVID-19, providing several computations as examples, and a link to a public repository to run custom calculations. These computations display in a quantitative manner the importance of recognizing infected individuals from symptoms and contact-tracing information, and isolating them as early as possible. The computations also assess the impact of each variable on the mitigation of the epidemic.

Introduction

Main concepts and goals

This study aims to develop a probabilistic model to predict the effectiveness of containing an epidemic such as COVID-19 with measures aimed at finding and isolating infected individuals. More precisely, we are interested in modeling such “isolation measures,” by which we mean finding and isolating infected people via their symptoms and contact tracing, to predict the impact of these measures on the effective reproduction number of the epidemic. Special attention is dedicated to the case in which contact tracing is achieved, for a part of the population, through a mobile application.

Studies such as Ferretti et al. (2020) have underlined the role of asymptomatic and presymptomatic transmission in the COVID-19 outbreak, and the consequent importance of using a mobile application for efficient contact tracing. This insight has also led to the development of models to quantitatively assess the impact of a contact tracing app on the epidemic, primarily through agent-based approaches like in Pathogen Dynamics Group (2020).

In this paper, we propose an analytical approach to answer the following questions: How is the effective number of an epidemic impacted when isolation measures are in place versus when they are not, and what are the main factors contributing to the reduction in ? We take the effective reproduction number in the absence of isolation measures, denoted by , as an input of our model, which is thus independent of any underlying epidemic model. Moreover, our approach is parametric in that we concentrate the quantitative description of the isolation measures into relatively few, comprehensible parameters that comprise the input of the model. These parameters include the share of the population using an app, the share of people who self-isolate upon testing positive, and more.

Previous studies concerning the impact on the epidemic of isolating infected individuals include (Müller et al. 2000), which proposes a generative stochastic model of SIR-type, and Fraser et al. (2004), which uses an analytical method more similar to our own. The subject has also been addressed recently in Scarabel et al. (2021) using a deterministic dynamical model.

The starting point of our analysis is the effective reproduction number in the absence of isolation measures,1 that we consider as given. When discussing modeling “isolation measures,” we refer to policies focused on selectively isolating infected individuals after these individuals have been found through contact tracing or because they have displayed symptoms. We do not refer to generalized actions like imposing a lockdown, whose impact on the epidemic is considered already known and encompassed in .

is defined, for any absolute time t, as the expected number of cases generated by a random individual who was infected at time t during their lifetime. This quantity can be written as an integralwhere is the infectiousness (also called effective contact rate): is a function describing the expected number of cases generated by an individual infected at time t, per unit of infectious age, that is the period of time (measured in days) elapsed from the time of infection of the individual. So, for example, the numberis the expected number of people infected between 24 and 72 h from the infector’s moment of infection. Note that the normalization is the PDF of the generation time, the time taken by an individual infected at t to infect a different individual.2

In this study, we set up a methodology and a model to analyze changes in the reproduction number when the population is subject to isolation measures, including the support of an app for individuals who have tested positive, and depending on some parameters of simple interpretation. We denote bythe effective reproduction number in presence of isolation measures, and we compute as a function of ,3 other epidemiological data such as the symptom onset distribution, and some parameters describing the isolation measures, such as the probability that an infected, symptomatic individual gets a test, or the probability that a recipient of the infection gets notified when their infector receives a positive test. We only model how isolation measures work and how they affect the epidemic,4 without assuming anything about how the epidemic itself develops. In particular, our model is agnostic of any particular form for and .

The final goal of the model we propose is to understand the most important leverages that may facilitate optimization to better direct efforts of decision-makers, scientists, and developers. Such factors include app efficiencies, timeliness of notifications, app adoption in the population, and others.

The assumptions of the model and outline of the paper

The model developed in Sect. 2 is the translation into mathematical terms of the following assumptions, that describe an idealized schema in which infected individuals acknowledge their illness and take measures to avoid infecting others.The equations derived from these hypotheses produce an algorithm that computes the time evolution of the key quantities. This is summarized in Sect. 2.5.

An infected individual who shows symptoms is immediately5 notified that they should take a test (which does not discount the possibility that they acknowledge this necessity independent of an external input). This process does not always necessarily occur, but does so with a probability .

Given a infector–infectee pair, when the infector tests positive after the contagion, the infectee is immediately notified to take a test, with probability .

In either scenario, after an infected individual is notified to take a test, they take a test which will return a positive result after a time from the notification, which is distributed according to a given distribution (possibly reaching to account for the case in which the individual is never tested or never receives the positive outcome).

Immediately upon receiving the positive outcome of the test, an average infected individual will self-isolate with probability . Put differently, the number of individuals they infect from this moment is reduced by a factor compared to the scenario in which they do not take any isolation measures.

Note that in our model we are only considering forward contact tracing, i.e., infectees are notified of the positive result of their infectors, but not vice-versa. Doing otherwise would significantly complicate the discussion. This is probably the main limitation of the model, which may thus underestimate the effectiveness of the isolation measures: While backward contact tracing is in general less effective when timeliness in isolating infected individuals is key, it must be noted that its effect may be significant for epidemics for which super-spreaders, i.e. individuals that infect a large number of people, have a major impact on the contagion. Such individuals may be identified more easily thanks to backward tracing. A treatment of backward tracing in the context of a generative model is covered in Müller et al. (2000, §3.1).

Subsequently, in Sect. 3 we consider a more complex model. Instead, we assume that the population is split into two groups, depending on whether or not they use a mobile application for epidemic control. The parameters and are different, depending on whether they refer to individuals who use the app.

Finally, in Sect. 4, we apply these models by computing the suppression of for specific choices of the input parameters, particularly to assess the importance of such parameters. As for the input parameters that describe the epidemic, we use data relative to COVID-19. All these data are taken for a single source (Ferretti et al. 2020). It should be noted that these quantities are still preliminary, have quite large uncertainties, and are not necessarily the most up-to-date. However, we stress that these data are only used as inputs in all our computations, which can be easily reproduced and extended by using the code available in the open repository (Maiorana and Meneghelli 2021). It would be immediate to redo the computations with different inputs, to reflect any new understandings the scientific community should gain on COVID-19. In addition to this, in Sect. 4.2.3 we briefly check the robustness of our results with respect to changes in some epidemic data, namely the share of infected individuals that are asymptomatic, the contribution of those individuals to the reproduction number, and the generation time distribution.

The paper includes an Appendix where the main steps of the mathematical model are proven rigorously, in a framework where the hypotheses can be formulated precisely using the language of probability theory.

Discussion of the results

Summing up, this paper introduces a model of targeted isolation measures—with special attention paid to those based on contact tracing—in the context of an epidemic with given dynamics. It studies the impact of these, measured as the change in the key indicators of the epidemics (first of all, the reproduction number) with respect to the situation without measures. It presents a methodology to turn the assumptions defining the model into mathematical equations, without assuming an underlying model explaining the time evolution of the epidemic. In particular, the formalism developed in the Appendix allows a careful and exact development of the theory, in which all the interdependencies of the involved quantities are clarified. We end up with with a set of equations that express the relevant quantities in terms of those relative to previous times, giving a deterministic time evolution.

These equations (summarized in Sect. 2.5 for the “homogeneous” setting) are quite complex, reflecting the non-triviality of the assumptions about how isolation measures work. This makes it hard to analyze them analytically, for example, to study the asymptotic behaviour of the solutions, as was done in Fraser et al. (2004). On the other hand, our treatment allows us to refrain from making strong and unrealistic independence assumptions about the involved quantities, and leaves us greater freedom in setting up the hypotheses of how contact tracing works (for example, the isolation of contact-traced individuals is not assumed to be certain, nor immediate). And, notably, it allows us to numerically compute, with arbitrary precision, the time evolution of the reproduction number (and, hence, of the epidemic size) starting from the “default” reproduction number , other epidemiological data, and the parameters introduced in Sect. 1.2 describing the isolation measures.

We stress that, despite our extensive use of the language of probability theory, our model of the isolation measures is deterministic: It works as if the full history of the epidemic, with or without isolation measures, is given, and uses some parameters describing the mean efficacy of the isolation measures on the population. It then expresses in terms of and these parameters.

Note also that, in this paper, we always refer to as the case reproduction number. Sometimes, the instantaneous reproduction number is instead used in the literature when monitoring the evolution of an epidemic.6 Our choice is also connected to the way in which we formulate our mitigation hypotheses in Sect. 2 in terms of parameters , , , which we consider depending on absolute infection times rather than notification and isolation times. An alternative formulation following the latter option would add some slightly more cumbersome formulae but otherwise no essential complications of note to the treatment.

A limitation to the model comes from our homogeneous-mixing hypotheses regarding contact tracing and isolation policies: The only heterogeneity taken into account is the separation between individuals who do or do not use an app in Sect. 3. For example, the fact that, in reality, individuals belonging to the same household are more easily traced (in addition to being more easily infected by each other) is not taken into account. Besides the absence of backward contact tracing, mentioned in Sect. 1.2, other limitations may be attributed to the specific form of the hypotheses. However, many changes to the assumptions could be taken into account within the same mathematical framework: Features such as a different delay in testing for symptomatic or contact-traced individuals, or the existence of a targeted quarantine for potential infected individuals (even before they get tested) could be modeled without adding conceptual complications.

By using the model in Sect. 4 to compute the reduction in , we can recognize how isolation measures, particularly app-mediated isolation measures, can play an important role in mitigating epidemics like COVID-19. However, our results show how the impact of such measures is strongly sensitive to parameters describing their efficiency and timeliness: For example, the reduction in quickly becomes insignificant as the time taken to get a positive test result (and then to start isolating) grows past a few days (see Fig. 2).

Fig. 2

as a function of the time from notification to positive testing

The computations relative to the case in which an app is used show the importance of having an app which is effective at spotting infections, maximizing the fraction of true-positives.7 Past studies like Bendavid et al. (2021) and Li et al. (2020) suggest that “standard” contact tracing measures used by healthcare systems may be less efficient (fewer truly infected individuals are recognized) and slower when compared to an app (usually, several days elapse between symptom onset, the first medical visit, and the test outcome). In the computations, we model this fact by setting different parameters for people using an app and people who don’t, with the latter parameters left to reasonably low values. We analyze how the impact on the epidemics depends on these parameters and the app adoption rate (Fig. 9), showing how these are all key factors in reaching satisfactory epidemic mitigation levels.

Fig. 9

as a function of app adoption

The mathematical model in the homogeneous population setting

In this section, we develop the core mathematical model of the paper. We do so with a simplified scenario in which the same isolation measures apply to the entire population, thus eliminating the need to distinguish between those who do and who do not use an app. Some mathematical derivations require extra care, and their complete proofs have been moved to the Appendix to prevent this section from being loaded with many formulae and a heavier formalism.

Notations and conventions

We consider random variables on the sample space of all infected individuals, describing (absolute) times at which certain events happen: (time of infection), (time of symptom onset), (time of infection notification), (time of positive test). These variables can take as a value to express the cases in which an event never takes place (this is useful when writing relations between them).

As we want to relate these variables to the reproduction number , which measures the average number of people infected by an individual infected at a given time t, it is logical that all these variables refer to the infectious age (that is, the time from the infection) of the average individual infected at t: so we have, for example, the relative time of symptom onset, which is the -valued random variableWe can assume that this variable is independent of the contagion time t. Hence, we denote it by . Analogously, we have the random variables (time of notification for an individual infected at t, measured since t), (time of positive test for an individual infected at t, measured since t).

In this section we need to understand how to describe the random variables , , , and their relation to the reproduction numberbased on the assumptions of Sect. 1.2. The finite parts of these random variables are described using improper CDFs, denoted by , , and respectively, whose limit for (representing the probability that each time is less than infinite) may be less than 1. So, for example, denotes the probability that an individual infected at t tests positive within a time from the time of infection. is the probability that the same individual eventually tests positive.

Further auxiliary variables are introduced later on.

The suppression model for

Recall from Sect. 1.2 how we assume that self-isolation works: If an infected individual tests positive, then they immediately self-isolate, resulting in a reduction, on average, of the number of people they subsequently infect by a multiplicative factor , which we assume given, and possibly depending on the time t at which the individual was infected.8

We can then determine a relation between the “default” reproduction number density , its correction as a result of the isolation measures, and the distribution of the relative time at which individuals infected at t receive a positive test result. This relation holds for any t greater or equal to the time at which the isolation measures are enacted.

For simplicity, let’s assume for a moment that receiving a positive test and infecting someone (assuming no isolation measures) at a given infectious age are independent events. By , an individual who was infected at t has already received a test with probability . In such a case, the number of people they infect per unit time is . Alternatively, if the individual has not received a test by (which happens with probability ), they do not self-isolate, and the average number of people they infect per unit time is just . In summary, we have, for any ,This is analogous to Eq. 6 in Fraser et al. (2004). To illustrate further, suppose that all infected individuals test positive at the same infectious age , i.e. is a Heaviside function with step at : then we have for and for .

However, the above result relies on the assumption of independence between testing positive and the number of people the individual would infect without isolation. In practice, this is not an adequate reflection of what occurs. For example, with COVID-19, it is known that a significant proportion of the infected population is asymptomatic, and less contagious—see e.g. Mizumoto et al. (2020) and Ferretti et al. (2020). Given the lack of symptoms, this population has a lower probability of self-isolating. To overcome this factor, we introduce a new random variable G, which has a finite range that describes the severity of symptoms of an infected individual. It is assumed to be independent of the time of symptom onset, but it is related to the number of infected people and the probability of the individual recognizing their own symptoms. Then, to write a relation between and , we restrict the relevant random variables to each possible value of G: for any we denote bythe probability that an individual infected at time t and with severity g has tested positive by . Similarly, we denote bythe average number of people infected by an individual infected at t and with severity g, and by the analogous quantity in absence of isolation measures. Assuming now that for a given g the number of people infected (without isolation) and the event of being tested are independent, we write our “suppression formula” asIn Sect. A.3 we include a careful derivation of this formula. Note that the relations with the aggregate variables arewhere is the probability that an infected individual has symptoms with severity g.9

Also, in Sect. 4 we always take G to assume the values 0 and 1 only, to describe asymptomatic versus symptomatic infected individuals. However, this formalism allows for a greater diversification of , according to the severity of the illness.

We end this subsection with an example of an application of (2) in a simplified scenario. Suppose that G only takes the values 0 and 1, describing asymptomatic and symptomatic infected individuals, and that each constitutes half of the population. Suppose also that , and that asymptomatic individuals are never tested, so that , while symptomatic individuals are tested immediately after infection, so that , where is the Heaviside function. Then, we have and , so that . Had we used Eq. (1) instead, we would have ended up with , which does not take into account the fact that isolating symptomatic individuals has a greater impact on the reduction of than isolating the same proportion of randomly chosen individuals.

First considerations on the variables , , and

The distribution of the time of symptom onset is independent of the isolation policy and is considered as given throughout the paper, although its specific shape is irrelevant in this section.10

The description of is addressed in the next subsection. Here, we only consider its relation with : Having assumed that the time between notification and testing positive is described by a given random variable , which is independent from and for simplicity constant in absolute time, we haveThe relation still holds if we restrict it to individuals with a given severity g, and hencewhere is the improper CDF of .

Describing

In this subsection, we consider the random variable and study the relations with it that formalize the assumptions of Sect. 1.2, namely:We introduce two new random variables relative to individuals infected at a given time t, describing the receiving of a notification for either cause:The relation between these new variables and isIn terms of improper CDFs, and assuming independence of the two notification times, this givesDescribing requires the introduction of an additional random variable , that gives, for any individual infected at t, the time elapsed between the the infection time of their infector and t. In particular, we need the joint distribution of and the severity G, that can be described in terms of improper CDFs : Letdenote the probability that, given an individual infected at t, their infector has severity g and was infected at a time . Note that these improper CDFs satisfy a normalization conditionand they are completely determined by quantities relative to times preceding t, namely the number of infected people and the infectiousness (more details on how they are computed are deferred to Sect. A.5).

When an infected individual shows symptoms, they receive an immediate notification to get tested, with probability depending on the severity g of symptoms, and possibly on the infection time t.

Immediately after an infector tests positive, each infectee is notified of the risk, with probability . If the contagion takes place after the positive test, then the infectee is never notified.

We denote by the time from infection at which an individual infected at t and with severity g is notified because of symptoms. We assume that this happens with probability at the time of the symptom onset, so its improper CDF is simply11

We denote by the time from infection at which an individual infected at t receives a notification resulting from the positive test of their infector. Below, we see how to describe this.

Now, the notification time of an individual infected at t is by hypothesis equal to the testing time of the infector minus the generation time , but only if the notification actually occurs, which happens with probability provided that the contagion took place before . Hence, to get the improper CDF we should first average , translated to the left by , over all possible values of , each weighted by the probability of the generation time being . In doing this we should also treat separately the different severity levels that the infector may have, as these impact the testing time distribution. So should look like a sumThis formula doesn’t take into account that by assumption the notification can only occur after the contagion time, meaning that must be supported on positive numbers. This is considered by replacing the integrand with the probabilityAlso, in averaging the CDFs we should take into account the fact that the testing time of the infector is not distributed like the testing time of an arbitrary individual: Having infected someone at the infectious age , the infector is more likely than average to be tested after , or to never receive a test. As we will show carefully in the Appendix, to take this into account we need to divide the integrand by the same suppression factor that appears in Eq. (2), evaluated at . We conclude that, for any , we haveThis result is proven rigorously in Sect. A.6.

Summary and discrete-time algorithm

In this section, we have translated the hypotheses made in Sect. 1.2 into mathematical equations describing a dynamical system. In doing this, we added a few natural assumptions of independence between the variables under considerations, namely:Putting all the equations together, we see that we can compute, at any time t, the suppressed infectiousness in terms of the parameters , , , of the model, the default infectiousness and the other known epidemiological quantities, and the distributions relative to previous times .

the assumption in Sect. 2.2 that the testing time of an individual with given severity is independent from their default infectiousness

the assumption of independence between notification times and and the testing delay

To add an initial condition to the dynamical system, we assume that the isolation measures start at a given absolute time , so that for .12 Hence, all individuals infected at will never take a test (even after ) and never self-isolate. As a consequence, the effective reproduction number is for , while it gets reduced according to Eq. (2) for . In particular, individuals infected at can only be notified of the need to take a test through symptoms, so that .

Our set of equations can be approximated with arbitrary precision to a discrete-time algorithm that computes how the epidemic evolves, given the above data.13 This is the algorithm used in the calculations of Sect. 4.

Summing up, for each time , the algorithm works as follows:

Compute the number of individuals infected at t and the improper CDFs , from and for (as detailed in Sect. A.5).

Compute the distribution of as in Eq. (4):

Compute the distribution of from and the distribution of , for , using Eq. (6). If , just take .

Compute the distribution of using Eq. (5), that is and then the distribution of from via Eq. (3).

Compute using the distribution of , via Eq. (2):

The extended model including the use of an app for epidemic suppression

So far, we have operated under the hypothesis that the ability to inform infected people that their source has been infected can be described by a single (possibly time-dependent) parameter . Now, let’s suppose that the population is divided into people who use an app for epidemic control and people who do not. This forces us to complicate the model of Sect. 2 because, when we analyze the distribution of the notification time for people with the app, we need to apply different weights to the cases in which the source of the contagion has the app or does not. We also leave open the possibility that people using the app may have a different probability of requiring a test because of their symptoms.

The generalization of the homogeneous scenario to this case is quite straightforward. In any case, some more mathematical detail has been added in Sect. A.7.

Parameters and random variables in the two-component model

A share of the infected population, perhaps depending on the absolute time t, uses an app that may do the following:We then distinguish into and , describing the probability that an individual infected at t, respectively with or without the app, is notified of the need to be tested given that they have symptoms with severity g. Note thatso that this distinction does not complicate the model, and is made only for adding clarity in the computations.

It gives the users clear instructions on how to behave when they have symptoms indicative of the disease, assuming that this can increase the probability that an infected individual asks the health authorities to be tested because of their symptoms.

It notifies the users when they have had contact with an infected individual who also uses the app, assuming that this can increase the probability that an infected individual asks the health authorities to be tested because of contact with an infected person.

The increased complexity of this situation lies in the fact that now has to be replaced by two parameters and , describing the probabilities that, given an infector–infectee pair, the positive testing of the infector occurred after the infection caused a notification to be sent to the infectee, respectively in the cases that both the infector and the infectee have the app, and that at least one of them does not have the app. Note that there is no relation between and and the general as simple as Eq. (7).

We also distinguish each random variable between people with the app and people without it. For example, the time of notification due to contact now reads for people with the app and for people without it. The relation between their improper CDFs isWe have analogous formulae for and , while there is no need to make a distinction for .

Likewise, we have to separate into two components and , namely, the average number of people infected by someone infected at t who has or does not have the app, respectively:Analogous relations hold when restricted to individuals whose illness has a given severity g.

It is reasonable to assume that having or not having the app is independent of symptom severity, so that, for example, the fraction of individuals infected at time t using the app and with severity g is . Also, while of course having an app does impact the testing time distribution and the infectiousness, we can safely suppose that it is independent of the default infectiousness, i.e. the number of people an individual would have infected in the absence of measures. This is why in this scenario the suppression formula (2) simply becomesfor .

The mathematical relations between the random variables

Now, we can write the new relations between the random variables. Eq. (4) is replaced byThe relations (3), (5) immediately extend to each component.

The distributions of and can be computed similarly to as we did in Sect. 2.4 for the homogeneous case. But now, for each of them Eq. (6) needs to be split into two parts, accounting for the cases in which the source of the infection has or doesn’t have the app:For we get a similar equation with replaced by : In this case, it doesn’t matter whether or not the infector has the app. The equation simplifies to a form analogous to Eq. (6), namelyAgain, we refer to the Appendix for a greater mathematical rigor: The last two equations are derived in greater detail in Sect. A.7.

Scenarios and calculations

In this section, we use the models introduced in Sects. 2 and 3 to numerically compute the suppression of due to isolation measures in certain scenarios.

The results reported here, as well as new custom calculations, can be obtained by cloning the public Python repository (Maiorana and Meneghelli 2021).

General considerations

Some inputs of the algorithm developed are parameters or distributions describing the features of the epidemic under consideration. In this section, we focus on COVID-19, and we make the following assumptions, taking all the epidemic data from Ferretti et al. (2020) (Table 1, in particular) for convenience:All these assumptions hold throughout the whole section except for Sect. 4.2.3, where we check how the results change using different epidemic data. The other parameters of the model, describing the isolation measures, are selected later.

The incubation period is distributed according to a log-normal distribution: where denotes the CDF of the standard normal distribution. The parameters and used here imply that the mean incubation period is days.

The default infectiousness distribution is assumed to depend on the absolute time t only via a global factor, so that where (which also represents the default generation time distribution) integrates to 1. It is described by a Weibull distribution with mean 5.00 and variance 3.61: with , .

We simplify the severity of symptoms by considering only two levels of severity: and , respectively for symptomatic and asymptomatic individuals. We take asymptomatic individuals as 40%, and we assume that they account for 5% of .14 In formulae, this means that the input parameters of our model are

As Key Performance Indicators (KPIs) describing the effectiveness of the isolation measures, we look at the reduction of compared to the value it would take in the absence of measures. We call effectiveness of the isolation measures the relative reduction in :Thus, indicates that there is no effect on , while describes a complete suppression of the contagion. We will see in Sect. 4.2.3 that the dependency of on the default reproduction number is very weak.15 As such, attempts to model a realistic profile for have little relevance to our computations, as any choice of leads to almost the same . Thus, in the rest of this section we simply takeAnother useful KPI is the probability that an individual infected at a certain time t is eventually found to be positive, namely the limitIn the remainder of this section, we report the results of some selected calculations, considering both the “homogeneous” scenario of Sect. 2 and, in greater detail, the scenario of Sect. 3, in which an app for epidemic control is used. First, we study how the above KPIs evolve in time for certain input parameter choices. Then, we focus on the limits for of these KPIs, i.e., their “stable” values after a sufficient number of iterations, to study how these vary when we change certain input parameters, leaving the others fixed.

Reduction in with homogeneous isolation measures

First, we perform some calculations in the setting of Sect. 2, where the isolation measures are “homogeneous” within the whole population. We recall the parameters that describe this situation, some of which remain fixed in all the calculations:16

Note that assuming that is a constant random variable means that we are modeling that all individuals notified of the risk test positive, and take the same time to do so. Although unrealistic, this assumption makes little difference to the results. It is made here for simplicity, although it can be easily changed by using a more realistic , when this datum is available.

Time evolution with isolation due to both symptoms and contact-tracing

We now choose the following parameters, describing an optimistic situation, with reasonable efficiencies in spotting infected individuals:

The results are shown in Fig. 1. Note that immediately at drops to around 0.92, as half of the symptomatic individuals are notified as soon as they show some symptoms, and they then infect a reduced number of people. Subsequently, continues to decrease due to contact-tracing, quickly approaching its limit value (i.e., 84% of the value it would have had with isolation due to symptoms only).

Fig. 1

evolution in the homogeneous model, in an optimistic scenario

Dependency on testing timeliness

We now focus on the limit value , investigating its dependency on the time from a notification to the positive result of the test (recall that we are assuming that is a constant random variable).

Like in Sect. 4.2.1, the other parameters are fixed as follows:

The result is plotted in Fig. 2. The effectiveness of the isolation measures improves dramatically with the ability to test (and then isolate) infected individuals as soon as possible after their notification of possible infection.

Dependency on the epidemic data used

In this subsection we briefly explore what happens if we change some of the data describing the epidemic, that were introduced in Sect. 4.1 and used elsewhere in this section. This is done to see how depends on these data. The other parameters, describing the isolation measures, are fixed as usual:

First, we let the fraction of symptomatic individuals vary, along with their contribution to —let us denote it here by —that is elsewhere taken as . Recall thatThe value of for a few choices of and is plotted in Fig. 3, where it is apparent how the result is robust with respect to changes in these input data. Note that if we fix and let vary, then the two components of (and hence those of ) are linearly rescaled, meaning that also changes linearly.

Fig. 3

for some values of and

Second, we fix and as usual, and we modify instead the density of the default generation time, by replacing it withfor . Note that this implies that the expected value of the default generation time (denoted here by ) is multiplied by f:Figure 4 depicts the relation between and , as f varies. As expected, the isolation measures become more effective as the time taken by the infection to be transmitted increases.

Fig. 4

for some rescalings of the distribution of

Finally, Fig. 5 shows how changes slightly as we change the value of (for ).

Fig. 5

for some values of , for

Reduction in in the case of app usage

Now, we focus on applying the model of Sect. 3 to study how is reduced when a fraction of the population uses an app for epidemic control.

In this case, we summarize the input parameters in the following table, fixing some of them to the given values for the rest of the section (unless explicitly mentioned).

Time evolution in an optimistic scenario

We start with an optimistic scenario, where the app is effective at recognizing infected individuals from symptoms and contact-tracing information. The internal predictive models that estimate the probability of an individual being infected have high efficiencies (a situation likely bound to the possibility of training the predictive models on real data, in practice). The app is adopted by a large fraction (60%) of the population, and is trusted, so that most of the people notified take a test and self-isolate. The app also helps a notified individual to get tested more quickly.17

As we start from , we reach a limit value of for an effectiveness of 0.16. Note also that , while . The time evolution of , along with the other main quantities of interest, is shown in Fig. 6.

Fig. 6

KPIs evolution in the optimistic scenario

Time evolution in a pessimistic scenario

We now run an analogous computation in a “pessimistic” scenario. The app can only recognize infected individuals from contact-tracing information, and not from symptoms ( consequently defaults to the no-app value). In addition, we assume a low efficiency , perhaps due to poor predictive models. Also, only 70% of those testing positive self-isolate.

Even with a high app adoption rate (60% of the population), the effectiveness drops dramatically. We get and . Most notably, the app does not change things much with respect to “standard” isolation measures: and .

Time evolution in the case of gradual adoption of the app

Now, we study the evolution of in a scenario whereby the fraction of people using the app is not constant, but increasing in a linear fashion until it reaches 60% in 30 days:The other parameters are chosen as in the optimistic scenario of Sect. 4.3.1:

As shown in Fig. 7, decreases until stabilizing again to the same value obtained in Sect. 4.3.1, although it takes more time to do so. The limit values of the KPIs are not changed by a gradual adoption of the app, compared with a prompt adoption.

Fig. 7

KPIs evolution in case of gradual adoption of the app

Dependency of effectiveness on the efficiencies and

We now focus on the study of how the limit values of the KPIs change when we vary certain parameters, starting with the app efficiencies and . In Fig. 8, we plot as a function of these two parameters, while the others are fixed to the following values:

Fig. 8

as a function of the efficiencies and

Dependency on the app adoption

In Fig. 9, we can observe the dependency of the effectiveness on the share of the population using the app. The remaining parameters are fixed to these values:

Parameter	Meaning	Value
ssyms	Probability that a symptomatic infected individual is notified of the infection because of their symptoms	Not fixed
sasys	As above, but for the asymptomatic	0
ξ	Probability that someone testing positive self-isolates	Not fixed
Δ^A→T	Time from notification to positive testing	Constant distribution, whose value is not fixed at this moment
t_0	Time at which isolation measures begin	0

Parameter	Value
ssyms	0.5
s^c	0.7
ξ	0.9
Δ^A→T	2

Parameter	Value
ssyms	0.5
s^c	0.7
ξ	0.9

Parameter	Value
ssyms	0.5
s^c	0.7
ξ	0.9
Δ^A→T	2

Parameter	Meaning	Value
ssyms,app	Probability that a symptomatic infected individual using the app is notified of the infection because of their symptoms	Not fixed
s^s,noapp_sym	As above, but for individuals without the app	0.2
sasys,app, s^s,noapp_asy	As with the two parameters above, but for asymptomatic individuals	0, 0
s^c,app	Probability that an infected individual with the app is notified of the infection because of their source having tested positive	Not fixed
s^c,noapp	Probability that an infected individual without the app is notified of the infection because of their source having tested positive	0.2
ξ	Probability that someone testing positive self-isolates	Not fixed
Δ^A→T,app, Δ^A→T,no app	Time from notification to positive testing for people with and without the app, respectively	Constant distributions, whose values are not fixed at this moment
ϵ_t,app	Fraction of the population adopting the app at time t	Not fixed
t_0	Time at which isolation measures begin	0

Parameter	Value
ssyms,app	0.8
s^c,app	0.8
ξ	0.9
ϵ_app	0.6
Δ^A→T,app	2
Δ^A→T,no app	4

Parameter	Value
ssyms,app	0.2
s^c,app	0.5
ξ	0.7
ϵ_app	0.6
Δ^A→T,app	2
Δ^A→T,no app	4

Parameter	Value
ssyms,app	0.8
s^c,app	0.8
ξ	0.9
Δ^A→T,app	2
Δ^A→T,no app	4

Parameter	Value
ξ	0.9
ϵ_app	0.6
Δ^A→T,app	2
Δ^A→T,no app	4

Parameter	Value
ssyms,app	0.5
s^c,app	0.7
ξ	0.9
Δ^A→T,app	2
Δ^A→T,no app	4