During the Covid-19 epidemic, one of the limiting factors that affected the capability to handle the spread of the disease was the small number of available tests. This lack of information has created major issues in several countries and promoted the idea that testing is essential in the control of an epidemic (Salath et al. (2020) ).
Recent research works support the importance of testing to effectively control an epidemic, see Brotherhood et al. (2020) ; Wang (2020) ; Eichenbaum et al. (2020) . In this regard, the selection of the individuals to be tested has become a major concern in many countries. However, to the best of the authors' knowledge, research on how to define these testing policies is still at a very early stage (Nowzari et al. (2016) ). This observation is testified by the de facto policies applied by decision makers during the Covid-19 epidemic. Among the various policies we can mention the use of contact tracing of individuals exposed to positive cases (Cereda et al. (2020) ), contact tracing combined with additional random testing (Shim et al. (2020) ), the use of exhaustive control of new arrivals in isolated communities ), and the testing of people with high number of human interaction such as health care personnel (Padula (2020) ). It is worth to note that most of these strategies rely on the appearance of symptomatic cases and required the use of hard lockdown policies to be effective.
Interestingly enough, the selection of individuals to test has important similarities with the problem of sensor selection for state estimation in the context of Wireless Sensor Networks. In both cases only a limited amount of information on a partially unknown process can be retrieved due to a limited amount of resources, the number of available tests or the channel bandwidth. The objective is then to optimize where to collect the measurements based on the available information and on the model of the process. Sensor selection has been an active field in the last two decades: a method based on convex optimization is proposed by Joshi and Boyd (2008) , a stochastic policy is studied by Gupta et al. (2006) and the optimal periodic policy for two sensors is given by Shi et al. (2011) . In the case of a general number of sensors the problem has been explored by Vitus et al. (2012) over a finite horizon, by Mo et al. (2014) over the infinite horizon, and for a general number of independent dynamical systems by Han et al. (2017) . However most of available works on sensor selection focus on real-valued dynamical systems, while the case where the process state assumes values from a finite set is at the best of our knowledge still largely unexplored.
The first step to propose an effective smart testing is the selection of an adequate model to monitor the epidemic. Compartmental epidemic models proved to provide accurate estimations of the dynamics of an epidemic (Brauer (2008) ). These models can be divided in deterministic models, governed by differential equations (McCluskey (2010) ), or stochastic models, where the heterogeneity of small communities can be better represented (Bjørnstad et al. (2002) ; Lopez-Herrero and Amador (2017)). New models tailored for the Covid-19 case have been developed seeking for more suitable approaches for the design of control strategies, e.g. Giordano et al. (2020) ; Casella (2020) ; Franco (2020) . However, the nature of compartmental models implies an homogeneously distributed population with random mixing between individuals, which does not inform about the granular distribution of the disease. To model the granularity of the spread of a disease, network diffusion models provide a better insight of the population's distribution and allow to identify the critical clusters of the spreading.
The most common network diffusion models are based on Stochastic Cellular Automata (SCA), where the spread of the disease depends on the interaction between neighboring cells (Mikler et al. (2005) ; White et al. (2007) ). This idea has been lately extended to more complex network topologies (Li et al. (2014) ; Keeling and Eames (2005) ). In these complex network models the interactions between individuals are modelled as the edges of a graph. This representation makes it possible to also model time-varying interactions, as well as selective quarantine policies (e.g. by removing the connections of certain individual with the rest of the population). From the theoretical viewpoint it is possible to prove that any SCA model is equivalent to a Markov chain (Ruhi and Hassibi (2015)). As we will discuss later on in this paper, this fact, although important from the theoretical viewpoint, is however not very useful in practice as the resulting Markov chain has a number of states that is exponential in the number of the states of the SCA.
It is important to mention that while the use of network models has been often overlooked due to the difficulty to monitor and define the interactions in real communities, in the authors' opinion the conception of more advanced tracking systems during the last pandemic leads naturally to this kind of approaches.
The problem of estimating the state of partially observable dynamic networks has been object of only a few studies in the last few years. One of the most studied problems is the estimation of the source of an information spread in networks using only limited observations. Zhu and Ying (2016) ; Zhu and Ying (2014) propose a sample path algorithm to estimate the location of a source of information or a disease. Alexandru and Dragotti (2019) extend this idea to the case where multiple rumours are spread and the time of the origin of the information is unknown. These works provide interesting idea that can be possibly adapted for the estimation of the evolution of an epidemic over a network.
An alternative approach to the surveillance of epidemics within networks can be found on the use of a sentinel system to estimate the evolution of the disease as done by Braeye et al. (2019). A sentinel system involves a limited network of selected reporting sites monitoring the disease in small portions of the population. The obtained data is used to estimate the behaviour of the entire network. Souty and Boëlle (2016) estimate the total number of cases of influenza based on the population density associated to each reporting site. Although this approach uses the density of population to improve the estimation of the state of the epidemic, the total population is still divided into small clusters with homogeneous distribution and interactions.
At the time of the writing of this paper, some early work presenting attempts to define smart testing and quarantine policies have been just published. In particular Berger et al. propose a policy based on conditional quarantine and random testing. However, the model based on partial observations assumes that tested negatives are "tagged" and they remain observable after a single test. In another recent paper on the subject by Kasy and Teytelboym (2020) , the trade-off between quarantine and testing is regarded by defining a certain threshold based on the infection probability and related to the cost of testing or quarantining an individual. In this case, the partial information is inferred based on the social group of the individual rather than its interactions within the network.
The main contribution of this paper is to propose a smart testing strategy to select the individuals to be tested based on the estimated probability of infection of each individual. As a first step we propose an approximated estimation of the current state of the epidemic which is computationally inexpensive. On the basis of this estimation, the testing policy is defined as a constrained optimization problem. This testing policy is coupled with a selective confinement policy which allows to only confine few individuals of the population based on the outcome of the tests. Numerical simulations show the advantage of this approach with respect to policies based on random testing both in terms of number of infected individuals and in terms of number of individuals put in quarantine at each time. In particular, on a population of 10'000 individuals, the total number of infected is reduced to one third and the total amount of days spent in quarantine is reduced to one half. The proposed algorithms can be used in a centralized way (e.g. by a decision maker) but they are also suitable to work in a distributed privacy-aware fashion and to integrate with tracing devices.
The remainder of the paper is organized as follows. In Section II the proposed model of the epidemic is presented. Sections III and IV introduce the exact and the approximated estimations to the evolution of the epidemic. Section V defines the testing strategy and Section VI the quarantine actions. In Section VII several simulations demonstrate the performance of the proposed strategies. Section VIII provides conclusions and future works.
Consider a population of N individuals where a disease is spreading. Each individual can be susceptible, infected, or removed. The spreading of the epidemic is modelled according to the following assumption.
The exposure to an infected individual is a necessary but not-sufficient condition for a susceptible individual to become infected. Indeed, the contagion actually takes place if some events (e.g exchange of body fluids for flu-like illnesses) have occurred and thus it is intrinsically stochastic. Motivated by these considerations, we model the transmission of the disease through random variables. Similarly, also recovery is modelled as a random variable to capture the uncertainty of the recovery process.
Mathematically, each individual i has at fixed time instants, say every day, t a state ξ i (t) ∈ {S, I, R} that can take three logical states:
• S -susceptible, the individual is healthy and was never infected before, so it is susceptible of being infected;
• I -infected, the individual is infected and can infect others;
• R -removed, the individual has been infected in the past and cannot be infected anymore (because immune or dead).
We denote with u i (t) ∈ {0, 1} the binary stochastic input representing the stochastic contagion event at time t. The variable takes value u i (t) = 1 if the ith individual has been infected between time t and time t + 1, and u i (t) = 0 otherwise. To characterize u i we introduce the transmission variables T ji (t) ∈ {0, 1} which takes value T ji (t) = 1 if the infection is transmitted from j to i between time t and time t + 1 given that the individual j was infected and the individual i was susceptible. In the same way r i (t) ∈ {0, 1} denotes the binary stochastic variable representing the stochastic recovery event at time t. In particular, r i (t) = 1 if the ith individual becomes removed between time t and time t + 1, and r i (t) = 0 otherwise. The state of each individual evolves according to the following equation
The state evolution of each individual is depicted by Fig 1. From the last equation it is clear that an individual i can be infected by individual j if individual j was infected, i.e. ξ j (t) = I, and if the transmission occurred, i.e. T ji (t) = 1. The modelling of variable T ji (t) is summarized by the following assumption.
Assumption 2 The transmission of the disease T ji (t) form an infected individual j to a susceptible individual i is a Bernoulli random variable with mean w ij (t), such as T ij ∼ B(w ij ). Moreover T ji (t) is independent of T mn (k) ∀m, n, k = i, j, t and of the initial state ξ n (0) ∀n. The mean values are symmetric, i.e. w ij (t) = w ji (t). For any pair i, j of individuals that have no contacts w ij (t) = 0.
The variable r i (t) is modelled according to the following assumption,
In general the system is partially observable as symptoms only appear in a small percentage of the population. The appearance of symptoms is modeled by the random variable e i (t) ∈ {0, 1} taking value e i (t) = 1 if i-th individual is infected and shows symptoms between time t and time t + 1, and 0 otherwise. We model it according to the following assumption.
Assumption 4 The appearance of symptoms e i (t) is a Bernoulli random variable with mean θ i constant over time. Moreover e i (t) is independent of e j (k) and r j (k) ∀j, k = i, t, of T mn (k) ∀m, n, k, and of the initial state ξ n (0) ∀n.
In this paper we consider the case in which only a limited amount N T of tests are available at each time t. We will assume that the tests are perfectly reliable, i.e. when a test is performed on State Estimator
Quarantine Selection
No other information is provided by the test, so it is not possible to distinguish if an individual is susceptible or recovered. Additional information is provided by symptomatic individuals. Formally, let S(t) = {s 1 (t), s 2 (t) , . . . , s M(t) (t)} be the set containing the indices of the individuals that are tested at the time instant t and of the individuals who show first symptoms at time t. Note that the cardinality M (t) of the set is time-dependent since the number of symptomatic individuals is not constant. For ease of notation we introduce the binary variable x i (t) taking value x i (t) = 1 if ξ i (t) = I, and x i (t) = 0 otherwise. Then, the observed output at time t can be then expressed as
while the set of the observed outputs up to time instant t is Y 0:t = y(0), y(1), . . . , y(t) and it represents the information available at time instant t.
The goal of this paper is to define a policy for the choice of the individuals to be tested that, in conjunction with a selective quarantine policy, is able reduce the spread of the disease while at the same time keeping the number of individuals in quarantine limited. To do so, we tackle the problem by proposing the closed-loop structure reported in Fig.2 consisting of three stages: 1) estimation of the states x i (t) using the information available Y 0:t from the feedback of the outputs; 2) selection of the N T individuals to test by optimizing a reward function R(·); and 3) based on the output y(t), execution of control actions through selective quarantine.
The following sections focus on the derivation of a proper state estimation given the information available Y 0:t and the definition of a suitable cost function R(·).
The state of the whole population is defined by the vector
Under Assumptions 1,2,3, at any time t, the next state of the population ξ(t + 1) depends only on the current state of the population ξ(t). Accordingly, in line of principle, the stochastic process describing the evolution of the epidemic satisfies the Markov property and can be represented by a Markov chain.
To model the dynamics of the Markov chain, we have to derive the transition matrix A ∈ 3 N ×3 N whose entries are
where z, v ∈ {S, I, R} N represent two possible states of the network, and z, v represent the indices of the transition matrix associated to them. Under Assumptions 1,2,3, the next states of two individuals are independent given the previous state of the population. It follows that
allowing to compute the transition probabilities of the network as a derivation of the individual transition probabilities between the states of each single node. Since only the transition from susceptible to infected depends on the state of other individuals, the following simplification holds
The state transition probability of any node at time t can be computed as
while the probability of remaining in a given state is
All other transitions have probability 0. A major difficulty in our setting is that the evolution of the system can be only observed by symptomatic individuals and selective tests on the population. Since M (t) < N , the Markov model is hidden and can be only partially observed through the output. The complete characterization of the state given the available information is provided by the joint distribution p(ξ(t), Y 0:t ). For a given Y 0:t , the joint distribution can be represented by a vector of dimension 3 N with entries p(z, Y 0:t ) =p t (z) ∀z ∈ {S, I, R} N . In the case of Hidden Markov Models, this joint distribution can be easily computed by means of the forward algorithm (see Blunsom (2004) ; L. R. Rabiner (1989) ), providing the following expression
The computation of P (y(t)|ξ(t) = z) is then easy: P (y(t)|ξ(t) = z) = 1 if the state ξ(t) = z gives the output y(t), and 0 otherwise, namely if y(t) is not a possible output for the state z.
From the joint distribution, the conditional distribution is
The optimal estimate of the i-individualx i (t|t) is then
This procedure allows to obtain the probability of each individual to be infected at time t given the complete vector of observations Y 0:t . However, in spite of allowing to compute the exact probability, this approach requires the computation of all the transition probabilities of the matrix A and the use of a vector variable of size 3 N , which is not computationally feasible even for small populations.
Due to the prohibitive burden of an exact probability computation, in this paper we propose an approximated low-computational algorithm to estimate such probability. The proposed approximated estimation is based on the idea of temporal and spatial truncation of the updates and is also suitable for decentralized implementations More precisely, we propose to propagate the information from testing only to individuals that are the most correlated to the tested individuals, namely the ones that have direct contact with tested people, while for the remaining part of the population the update is performed based on previous estimates and the topology of the network representing the population. In the same way, only a limited amount of past state estimates are assumed to be affected by the new information. This approximation allows to retrieve the information regarding the individuals that are most affected by the result of each test while keeping a limited computational time.
We define the estimate of the state of the individual i aŝ
where the local information set for the i-th individual I 0:t i is defined as
In order to keep the computations associated to the i-th individual limited, we define a local approximated estimation which can be retrieved based only a partial knowledge of the network. In particular, we will focus only on individuals with whom the i-th individual was in contact: in the case of untested individuals we will use only the previous local estimatesx k (t−1|t−1) while in the case of tested individuals we will use the current state and the updated estimationx k (τ |t), τ ≤ t−1 of past states.
Denote by X 0:t i the (random) vector collecting all the random variables x i (τ ) τ ≤ t, namely X 0:t i = x i (0), . . . , x i (t) ′ . With a little misuse of notation X 0:t i = 0 denotes the case where all the past states x i (τ ), τ ≤ t, are equal to 0. We assume that the random variables T ji (τ )x j (τ ), j = i, are conditional independent given X 0:τ i = 0. Similar assumptions are made by Valdano et al. (2015) and Boguná et al. (2013) . The rationale is that, if the individual i has always been healthy, the coupling between two of his neighbours m and n is negligible, as in the case when individual i is the only connection between m and n or, even if w mn (τ ) = 0, contacts of n are enough different from contacts of m. It follows that
To further simplify the estimation algorithm we will simplify the stochastic recovery with a deterministic one based on the average recovery time D. Then we havê 
We compute P (X 0:t i = 0 | I 0:t i ) and P (X 0:t−D i = 0 | I 0:t i ) as
where the last equality holds since T ji (τ )x j (τ ) j = i are conditional independent given X 0:τ−1 i = 0. To obtain the numerical value of P (X 0:τ i = 0 | I 0:t i ) would require to compute P (x j (τ − 1) = 1 | X 0:τ−1 i = 0, I 0:t i ) that in turn would require P (x k (τ − 2) = 1 |X 0:τ−1 j = 0, X 0:τ−1 i = 0, I 0:t i ) and so on. Since this propagation is very computationally expensive we make the approximation that
The underlying assumption is that the state of an individual and those of its neighbors are independent. The accuracy of this assumption has been explored by Gleeson et al. (2012) where it has been shown that the dynamics are well approximated if the degrees of closest neighbours are high.
To keep a limited number of computations, we also make the following approximation
with initialization P (x j (τ ) = 1 | I 0:τ i ) =x j (τ |τ ). Roughly speaking, if individual j has direct contact with a tested individual k and individual i has direct contact with j but not with k, the state estimates of j will be corrected based on the outcome while the state estimates of i will use the old estimation of j, as derived without the knowledge of the outcome. This means that we use the information regarding the outcome from the tests to only update the direct neighbours of a tested individual.
Since the update of each individual uses only knowledge from local connections, new information can be used differently for tested individuals, individuals with a direct contact with them, and the remaining of the population. for τ ≤ t − D as no additional information on past states is given by a negative outcome. As x i (t) = 0, x i (τ ) may be equal to 1 only if a contagion occurs in the interval (τ − D, t − D), therefore the infection probability is updated aŝ
for τ = t − D + 1, . . . , t and
For the case of a positive result, x i (t) = 1, we have 
for τ = t − D + 1, . . . , t and
Let the neighbours of a tested individual be defined by the set Q(t) = {i | ∃w ik (τ ) = 0, k ∈ S(t), τ < t} which represents the set of individuals that has been in contact at least once with at least a tested individual. According to the definition of local information set, the update of the estimation exploits also the updated estimate of the past states of tested individual.
The probability relative to the initial time instant is not changed
By using the information from the neighbours that have been tested at time instant t we can update the probabilities as
using (21), where
Note that the previous update takes advantages from the knowledge of the update estimate of the past state of tested individuals. The last equality holds only if individual i has not been tested before, otherwise P (X 0:τ−1 i = 0 | I 0:t−1 i ) would be different according to the update relative to a tested individual. In that case, the correction procedure starts from the instant where the individual was tested. The correction procedure works if more than one neighbours have been tested even in different time instants. Note that c 2,i (τ, t) > 1 if x j (t) = 0 and c 2,i (τ, t) < 1 if x j (t) = 1. Finally, the infection probability at time t is computed aŝ
For each individual not having direct contact with any tested individual, the open-loop estimate is computed as P (X 0:t i = 0 | I 0:t−1 i ) based on the previous estimates P (x j (t − 1) = 1 | I 0:t−1 i ) provided by its neighbours as
Other required values are updated according to
The state estimation scheme proposed above performs a hierarchical update of the infection probability. This update is structured around individuals that are tested at time t, the neighbors of the tested individuals and the remaining of the population. At each time instant, the estimation is thus divided into 3 levels of update based on the derivations obtained in the previous subsection:
• First level: Tested individuals, using the output y(t) from the performed tests In line of principle, buffers of increasing length are needed to store past probabilities. In the spirit of a temporal truncation of the updates, since the current test outcomes bring little information on the oldest states except for positive tested individuals, we assume that for untested individuals past probabilities older than D w are not affected by the new outcomes, i.e. 
Under this approximation, in terms of information storage, the local update of the current state estimate requires the storage of the following two buffers of information for each individual: i) the Susceptibility buffer
. . , P (X 0:t i = 0 | I 0:t i )} and, ii) the Infection probability buffer
Remark 1 An interesting feature of the proposed approach is that it is not only computationally efficient and thus can be used to compute probabilities in a centralized way for a given community, but that it can be also implemented in a decentralized manner. This is the case if each individual is equipped with a smart device (e.g. a smartphone) provided with small computational capabilities and able to communicate with other devices and with a central testing unit, see Fig. 5 . The introduction of contact tracing mechanism has been indeed taken into account by many countries during the last months and software are already available in the market. With respect to them, our algorithm can be implement based on the same hardware and with a larger amount of transmitted data. In particular when individuals get in contact during the day, their previous estimatex i (t − 1|t − 1) has to be exchanged. Remote communications of the outcome of the tests y(t) and the updated estimates of the previous statesx j (τ |t), j ∈ S(t) of the tested individuals than has to be performed once per day from the main server to the individuals. An explanatory representation is given in Fig. 4 . No information on interactions is communicated neither to the central unit or other individuals so that privacy is preserved and vulnerability of a central data collector is avoided. Then, each individual transmits the updated estimatesx i (t|t) to the server which decides who to test the next day. 
Similarly to the literature on sensor selection, it is possible to formulate the test selection problem as a constrained optimization problem based on the state estimate. Formally, we introduce the binary control variable γ i (t) taking value γ i (t) = 1 if i-th individual is selected at time instant t to be tested at the next time instant t+1, and γ i (t) = 0 otherwise, while we denote γ(t) = (γ i (t), . . . , γ N (t)) ′ . Then the test selection problem can be formulated as
where R(·) is a suitable reward function. Several possibilities exist for the choice of the cost function. Differently from most of the works on sensor selection for remote estimation, we avoid to adopt the error covariance matrix because it is computationally infeasible for large N . In this paper we propose to maximize the expected value of the number of detected positive individuals, that is
This policy is equivalent to select the N T individuals whose probability of being infectedx i (t|t) is the highest.
The outcomes of the tests are exploited to act on the population through a selective quarantine. Formally, we introduce the control variable q k (t) such that q k (t) = 1 if k-th individual is selected to be quarantined at time instant t, and q k (t) = 0 otherwise. In this paper for any positive i we propose to quarantine the L closest neighbours, i.e. the L individuals with the highest weight w ij (t). The parameter L can be properly tuned to trade-off between the total number of quarantined for positive and the expected number of infected (but not detected) that are quarantined because they have a direct contact with a positive. We consider that individuals will leave quarantine after D Q days.
Note that in line of principle other quarantine strategies can be designed based on probabilities of infection of the neighbours of a positive tested, as well as preventive quarantine based only on the state estimate, and they will be the subject of future investigations.
This section shows, through numerical simulations, the effectiveness of the proposed solution by comparing it to current approaches.
The simulation setup considers a closed population of 10'000 individuals with the following parameters regarding the spread of the disease • R 0 = 2. This value is equivalent to a virus with high spreading, e.g. the Covid-19, when no social distance measures are adopted.
• 0.05% of the population is initially infected.
• 20% of new infected present symptoms of the disease before the recovery.
• 0.5% of the population can be tested at each sampling time, corresponding to N T = 50.
• The L = 5 closest individuals of each individual with positive test are put in quarantine for D Q = 14 days. When in quarantine, all the arcs weights w ij are reduced to 1/100 of their normal value.
The underlying graph representing the population distribution is such that each individual belongs to one or more clusters (namely a subset of individuals where each one is connected to each other) to mimic families, offices, habitual relations and activities etc. Dimensions and average weights are set in a realistic way (e.g. families consists of up to 6 individuals, average weights in a family are twice the ones in a small office). A number of random links are also added to the network. For the sake of simplicity the topology is assumed to be fixed.
Initial conditions ξ(0), i.e. which individuals are initially infected, are stochastically generated based on the initial probability of each node to be infected. To test the robustness of the proposed strategy, we assume that the probability distribution of the initial conditions is perturbed up to the 10%. It is also assumed that 10% of the arcs of the graph are unknown.
The presented simulations compare three different scenarios:
• No measures. In this case, no measures are applied to the population in terms of distancing nor quarantine.
• Random testing. This scenario presents a policy where only the control action is performed in a smart manner, namely the quarantine based on the graph representation. The testing, on the other hand, is performed based on randomly selected individuals.
• Smart testing. This scenario follows the proposed control scheme where the testing policy is based on the probability of an individual to be infected, Eq. (34). Given the stochastic nature of the model, 100 simulations have been generated for each scenario. The spread of the infectious disease is monitored for a time span of 300 days.
As we can see in Fig. 6 , the proposed control mechanism (testing and conditional quarantine) is effective in reducing the total number of infected people in a given temporal window.
The proposed strategy is effective also in mitigating the epidemic outbreak by reducing the peak of active cases, as shown in Fig. 7 . By looking to the zoom in Fig. 8 we can appreciate that the peak is lower in the case of the smart testing approach than in the case with random testing. This is an important result since it is fundamental to have a low number of active cases to avoid the health-care system maximum capacity to be reached.
The number of people in quarantine at each time instant is depicted in Fig. 9 . Although not intuitive, this plot shows lower numbers of people in quarantine for the smart testing policy, indicating that the improvement in performance does not require a greater number of people in quarantine but that actually can be achieved with less but better focused quarantines. In these numerical simulations it can be seen that the maximum number of people simultaneously in quarantine never exceeds the 0.5% of the total population. Table I shows that the averaged total number of days lost through quarantine is more than the double in the case of random testing, 5756.2 days, with respect to the smart testing strategy, 2775.2 days. This is a very promising result especially from an economic point of view since it would limit the social and economical impact of the measures. A synoptic overview of the numerical simulations is reported in Table I . The results show the clear improvement on the containment of the epidemic, in terms of both active cases and people in quarantine, by using a testing and quarantine policy based on the presented probability estimation algorithm.
In this paper we presented a novel testing strategy to smartly select the individuals to be tested during an epidemic. This policy is based on a decentralized state estimation of the status of the epidemic obtained from the outcome of the tests.
The testing policy is defined as an optimization problem based on the state estimation. The proposed estimation algorithm is computationally inexpensive and can even be implemented in a distributed fashion. The numerical results based on Montecarlo simulations demonstrate that the use of the proposed scheme, testing and selective quarantine, significantly reduces the total number of infected people as well as the peak of active case and the number of people put in quarantine.
Future works will focus on the link between the test selection objective functions and the quarantine policies.