COVID-19 pandemic has been already modeled using optimal control techniques. For instance, in [22] , optimal control theory is used to explore the best strategy for implementation while waiting for the development of a vaccine. More specifically, they seek a solution minimizing deaths and costs due to the implementation of the control strategy itself.
The same objective is studied in [45] by the means of an age-structured optimal control model. Specifically, a model with a double continuous structure by host age and time since infection is proposed. In [9] , the authors also propose an optimal control model with infection age. The main difference from the previous model is the consideration of the peak value in the objective function that leads to a problem with state constraints. This work has generated an open access code, based on the optimal control toolbox BOCOP. An SIR model is studied in [4] to derive a simple but mathematically rigorous criterion for designing optimal transitory NPIs. In particular, the authors found that reducing the reproduction number below one is sufficient but not necessary. This condition may be prescribed according to the maximum health services' capacity.
In all of the cited articles, (optimal) control strategies are obtained numerically or analytically in closed-loop form (as in [4] ) requiring the assumption that the state variable is perfectly observed.
Moreover, these works consider that the effects of NPIs are perfectly controllable by the decision maker without taking into account that a decision regarding the application or release of an NPI may require time for the population to adapt. These considerations are important when the proposed trajectories are composed by bang-bangs, singular arcs or saturation of constraints, because in practice, these controls may be very difficult to implement.
Additionally, since NPIs activation/release decisions based on the observation of indicators are made at specific given moments (for instance, lockdowns may be activated on Mondays for the whole week, or always at a given time of the day, etc.), they would not be faithfully represented through the aforementioned approaches. This is a key aspect when one of the objectives of the study is to provide recommendations to decision makers.
Thus, our first aim is to propose a framework for representing the decision-making process related to the application of NPIs based on the observation of epidemiological indicators. The proposed framework uses event-triggered controls in discrete time. To the best of our knowledge, the notion of event-triggered control was first introduced in [6 ] and allows to model update instants (in the decision, or control) by the violation of a condition depending on the state of the system, instead of assuming a constant or a periodic updating. As mentioned above, the event-triggered decisions is precisely the framework such as the NPIs that are being applied in several places, because depending on whether or not some indicators are in violation of their respective thresholds, the NPIs are applied or lifted. In recent years, event-triggered control techniques have attracted increasing interest with studies focusing on the stabilization of event-triggered implementations and the existence of a minimum interevent time (see for instance [7, 10, 23, 28, 38, 47, 49, 56] ). In our work, these topics are not addressed because it is assumed that the stabilization is not an issue, in the sense that we are not concerned with the final state of the system but rather we seek to be able to determine what type of indicator-based strategy is more suitable for use. Ensuring the minimum interevent time is not an issue in our framework, because we impose this minimum interevent time to avoid the application and lifting (or vice versa) of an NPI in a short period of time, which is the practice we have observed in cited examples. Moreover, we consider the possibility that decisions can be taken (apply or release an NPI) only at some prescribed times, for instance, only on Mondays, making the modeling of the decision-making process more realistic.
In addition to realistically representing the decision-making process based on the observation of indicators, our approach allows the assessment and comparison of the use of indicators from a costbenefit perspective. Indeed, based on the discussion given in this introduction section, we assume that a decision maker aims to obtain as few as possible of some outcomes that are consequences of the NPIs. The peak of ICU beds demand and the total number of days in lockdown are two examples of such undesired outcomes. Therefore, if the policy for activating or releasing NPIs is triggered by some observation or, more specifically, when a given indicator (computed in terms of these observations) is above or below a certain threshold, we can link all of these thresholds with their respective observations. Thus, considering the trigger threshold as a parameter, one can construct the curve of outcomes associated with a given indicator. This curve lies in the space of outcomes and it is called the trade-off curve. This curve allows the identification of the trade-offs between different outcomes.
Moreover, after computing these curves for several indicators, they can be compared to choose an indicator that will determine the policy. Indeed, for instance, if one of the curves dominates the others in the Pareto sense (that is, the curve is below the others), this immediately suggests that the corresponding indicator is the most suitable for the outcomes considered in the analysis.
On the other hand, when no domination occurs between the curves, a decision can be taken by setting all but one outcomes as objectives (viewed as upper bounds). This allows the determination of the best indicator as that with the lowest value for the outcome not considered as the objective. Thus, the proposed methodology allows the comparison of the use of indicators to trigger either the application or release of NPIs.
The rest of this paper is organized as follows. In section 2, we introduce the proposed framework that we use to model the making decision process, that is, discrete-time control systems and eventtriggered controls. In section 3, we define the trade-off curve associated with a trigger indicator and show how to compare these curves in order to choose the best indicator to activate/release NPIs. To illustrate the proposed modeling framework and methodology for the assessment of the indicators, in section 4 we discuss two case-studies, corresponding to Chile and China, where different indicators are evaluated. Finally, some concluding remarks are stated in section 5.
2 Mathematical framework based on event-triggered controls 2.1 Discrete-time control systems Given an initial time t 0 ∈ N, a horizon time T ∈ N (with T > t 0 ), an initial state x 0 ∈ X and a finite sequence of controls u = (u(t)) T t=t0 = (u(t 0 ), . . . ,u(T )), we consider the discrete-time control system:
In this setting, f : [[0 : N ]] × X × U −→ X is the dynamics, X is a vector space, called the state space, and U is a compact set of a given vector space satisfying 0 ∈ U (representing no action).
The latter is called the control space. Here, when p ≤ q, [[p : q]] denotes the collection of all integers between p and q (inclusive).
Several epidemiological models can be represented as system (1) . For instance, the state x ∈ X can model the number of individuals in different stages of a disease (e.g., SEIR compartmental models; See, for instance, [11] and references therein) in different populations (e.g., counties, cities, age-groups, etc.), and the control u ∈ U can model the application of one or more NPIs to some groups of the population under study.
We denote by U the collection of all possible controls; that is,
A solution of the control system (1) associated with a control u ∈ U is an element of the space
that satisfies the initial time condition x(t 0 ) = x 0 .
For ∆ ∈ N, with 0 ≤ ∆ < T − t 0 , we consider feedback controls based on the recent history of the state, that is, the state in the current time t and in the previous
Therefore, for t ∈ [[t 0 : T ]], we introduce the notation
If t − ∆ < t 0 , in the definition of x ∆ (t), we repeat the initial state x 0 in the first t 0 − t + ∆ components.
Motivated by the current practice, we consider x ∆ (t) instead of x(t) for defining our policies.
Indeed, as mentioned in the introduction, the use of epidemiological indicators for triggering either the application, or the release, of NPIs considers the state of the disease in a window of time and not instantaneously.
To reduce the control variability, we turn to the event-triggered state feedback control based on the recent history of the state. For this purpose, we introduce an event-triggered set S ⊆ X ∆+1 in order to determine the variations in the control law which in turn are produced by a transition of the recent history x ∆ (t), either from S to its complement S c , or vice versa. To compute these transitions, we define the indicator (XOR) function ϕ S :
The event-triggered set S will then define the application (x ∆ / ∈ S) or release (x ∆ ∈ S) of an NPI. Now, we define the following event-triggering mechanism
and
where the trigger times t k are updated according to rule (5) given below, and the controller u : [[0 :
withû − ,û + : [[0 : T − t 0 ]] × U −→ U reference time-varying controls associated with the NPI to be applied. Functionsû − (·,u) andû + (·,u) satisfyû − (0,u) =û + (0,u) = u for all u ∈ U. The controller u(τ,x ∆ ,u) is then defined (in (4)) through two types of policies depending on whether or not x ∆ belongs to the event-triggered set S. Thus, this models the application of a given NPI or the release of this measure. This policy, consisting of applying or not some NPIs, is eventually time-varying on the intervals [[t k : t k+1 ]], and can model a variable intensity or an adoption degree that changes with time. For instance, in a population where an NPI has been applied for a long period of time, when the intervention is lifted, some time is required to return to the activity levels (of contact rates, mobility, etc.) identical to those prior to the application of the NPI. In addition, the controller takes into account the last control u applied in the previous interval [[t k−1 : t k ]] with the aim of allowing (but not limiting to) smooth variations of the applied policies. The trigger times t k in (3) are given by
In the above definition, we have considered a minimum fixed period of time t min ∈ [[∆ : T − t 0 ]] during which the event-triggered control is applied, thus imposing a minimum interevent time. The interevent time is assumed to be greater or equal to ∆ which is the time window where the state variable is considered in order to have enough time to observe the effects of the application of the NPI, or of its release. On the other hand, the element
in (5) is the sequence of states obtained from
with initial condition x(t k ).
More generally, trigger times in (5) can be defined as
where the function ξ day :
] can be introduced in order to impose a change in the control policy in the prescribed periods. For instance, if the time unit is day and the application (or release) of an NPI must start on a Monday, when t 0 is a Monday, we take t min a multiple of seven and the function ξ day (t) = 7 t 7 , where ⌈α⌉ is the smallest integer that is also greater or equal than α.
To summarize, the event-triggered feedbacks considered in this paper are defined by: 4. Controller u: the function defining the control through mechanism described in (2), (3), and (4) . The definition of the controller u depends on functionsû − andû + in (4) and on ∆ and S.
An event-triggered feedback will be denoted by u = m(x 0 ,∆,t min ,u,S) ∈ U in order to highlight that the associated control u is completely defined by the event-triggered mechanism (the four elements listed above) and the initial condition x 0 at time t 0 .
Hence, given the mathematical formalism described above, the main objective of this paper is to compare different even triggered controls assuming that all of them have common controllers u with same functionsû − andû + in (4) (that is, all of the policies apply the same NPI, and in the same manner). In addition, since the initial state x 0 is fixed for all of our analyses, we simplify the notation of event-triggered feedbacks by u = m(∆,t min ,S).
In this section, we specify the event-triggered set S in the definition of event-triggered controls introduced in section 2.2. The particular forms considered of the set S are provided by an epidemiological indicator (observation). Thus, considering the space of the recent history of states X ∆+1 and an indicator I : X ∆+1 −→ R, we will work with event-triggered sets S in the form
where θ ∈ R is a threshold for the indicator I. 
b) the mean of differences of instantaneous observations:
c) variation rate of instantaneous observations:
d) variation rate of differences of instantaneous observations:
The specific instances of observations O can be the number of infected people [1, 19, 18, 46, 51] , number of deaths associated with COVID-19 [51] , excess mortality due to pneumonia [51] , number of tests [46] , positivity rate [18, 51] , COVID-19 total hospitalizations [24, 18, 46, 51] and the number of people in ICU beds [18, 46, 51] , among others. Additionally, the formalism introduced above permits to consider more elaborated indicators I computed from x ∆ (t), such as the effective reproductive number [18, 51] .
We update the notation of an event-triggered feedback by u = m(∆,t min ,I,θ), noticing that the event-triggered set S now depends on indicator I and threshold θ.
For a fixed initial state x 0 ∈ X and a given control sequence u ∈ U, we assume that m ∈ N (m ≥ 2) outcome measures are observed. These outcomes are represented by a function P : X × U −→ R m and, in our convention, we want these to be as low as possible. Examples of these outcomes are the peak demand of ICU beds, the total number of deaths (directly attributed to the COVID-19 outbreak) and the total number of days of lockdown (i.e., shelter-in-place), or of any other NPI.
The first two outcomes are related to the health cost of to the pandemic, while the third-one is a proxy for the economic and societal cost of the policy. Intuitively, there should be a trade-off between the different outcomes, in the sense that a lower outcome of one kind (e.g., peak demand of ICU beds) will lead to a greater outcome of the other kind (e.g., total time of NPI), as it is proven in [4] . Of course, one can take a group of outcomes where the mentioned trade-off does not exist (e.g., peak demand of ICU beds and the total number of deaths), but even in these cases it is interesting to quantify the links between these outcomes. Therefore, we propose to compute the outcomes given by an event-triggered feedback u = m(∆,t min ,I,θ) when the threshold θ varies.
With this information, decision makers can observe the links between the outcomes provided by the policy based on the indicator I and, for instance, to evaluate the marginal variation of one outcome when other outcome decreases/increases. For a given event-triggered feedback u = m(∆,t min ,I,θ), we define the trade-off curve associated with indicator I, and parametrized by thresholds θ, as follows
where D I ⊆ R is the domain for the thresholds associated with indicator I.
In fig. 1 , we illustrate the trade-off curve C(x 0 ,∆,t min ,I) when the observed outcomes are m = 2, that is, P(x 0 ,u) = (P 1 (x 0 ,u),P 2 (x 0 ,u)) with u = m(∆,t min ,I,θ). In this figure, we use the simplified notation P(x 0 ,u) = (P 1 (θ),P 2 (θ)) ∈ R 2 , for highlighting the dependence of the outcomes on thresholds. In this curve, we represent two outcomes exhibiting a trade-off (i.e., greater value of one outcome leads to a lower values of the other outcome). 
To compare the efficiency of two event-triggered feedbacks based on two different epidemiological indicators I a and I b , we propose to compute the associated trade-off curves described in the previous section. If the policies based on indicators I a and I b consider observation time windows ∆ a and ∆ b , respectively, and minimum interevent times t a min and t b min , respectively, then we compare the trade-off curves C a = C(x 0 ,∆ a ,t a min ,I a ) and (6) . Recall that the outcomes are represented by the function P : X × U −→ R m , with m ≥ 2, and the aim is to have the m outcomes P(x 0 ,u) = (P 1 (x 0 ,u), . . . ,P m (x 0 ,u)) to be as low as possible.
Then, the calculation of the trade-off curves C a and C b , consists of the evaluation of function P for event-triggered controls u a (θ a ) = m(∆ a ,t a min ,I a ,θ a ) and
If a decision maker considers as objective (maximal bounds) the first m − 1 outcomesP 1:m−1 = (P 1 , . . . ,P m−1 ) (e.g., peak demand of ICU beds), the corresponding m-th outcomes P a m and P b m (e.g., total time of NPI) obtained using event-triggered feedbacks based on indicators I a and I b can be computed such that
Thus, the decision maker should consider the policy based on I a or I b comparing P a m and P b m decision depending on the given objective outcomesP 1:m−1 . In fig. 2 , we show two trade-off curves for m = 2. In this figure, for small objective values of outcomeP 1 , it is better to use a policy based on indicator I b . Otherwise, for larger values ofP 1 , it is recommended to use a policy based on I a .
If P b m < P a m for all objective outcomesP 1:m−1 , it means that the curve C b dominates curve C a and therefore, the policy based on I b is always better. This situation is illustrated in fig. 3 for Figure 2 : Illustration of the trade-off curves C a and C b corresponding to event-triggered feedbacks based on indicators I a and I b , when the number of outcomes m = 2 and there is no domination. Figure 3 : Illustration of trade-off curves C a and C b corresponding to event-triggered feedbacks based on indicators I a and I b , when the number of outcomes is m = 2 and one curve dominates the other.
Finding a control u ∈ U that minimizes the m outcomes given by the vector P(x 0 ,u) = (P 1 (x 0 ,u), . . . ,P m (x 0 ,u)) is evidently a multicriteria optimization problem. Therefore, we only propose to compare single outcomes obtained from two policies when the other (m − 1) outcomes are fixed and treated as objectives. This avoids the need to consider outcomes of different nature and expressed in different units in a single objective function. On the other hand, to compare two policies when several outcomes are simultaneously considered (i.e., not fixed) is technically possibly, although it is not straightforward.
being infected. In this stage, they do not have symptoms but can infect other people with a lower probability than those in the infectious group described below. Mild infected or subclinical (denoted by I m ) is an infected population that can also infect other people. however, in our posterior analysis, we will neglect this source of infection because we assume that hospitalized individuals are highly isolated. After leaving the ICU bed, they either come back to the basic facilities in the hospital or die. The term dead (denoted by D) accounts for people who did not survive the disease.
The evolution of the previous state variables is described by the following discrete-time system,
where natural births and deaths are not considered because their effects are negligible.
The above system is a type of discrete-time SEIR (or SEIRHD) model. It aims to better describe an outbreak where part of the population has been infected by a virus, with many people with no symptoms or only mild symptoms. It was found that this is the case for SARS-CoV-2, as has been reported in several papers in the literature [26, 32, 35] . The model we use can be extended to consider age ranges and evaluate other indicators or policies such as the schools reopening or the progressive lifting of quarantines according to age groups as explored in [54, 53] . However, to illustrate the proposed methodology, we have only considered one age class.
The structure of this mathematical model, along with the transitions among the different stages, is shown in fig. 4 .
XxEXx We proceed now to describe the main components of system (7) . First, we consider the time interval [[t 0 : T ]], with days as the unit of time, where t 0 is the initial time from which we start the assessment of different policies (September 21, 2020) and T is the time horizon. In this example, the time horizon is the simulation time and must be large enough for all of the considered policies (NPI triggered by indicators) to be able to reduce the number of infected people to zero (September 21, 2025), without considering other interventions.
The contagion rate (see the first and second equation in (7)) is
where the specific rates β E , β I m , β I , β H , β H c can be written as the product of the contagion probabilities (p E , p I m , p I , p H , and p H c ) and the reference values of contact rates (c ref
, each value associated with a contagious stage of the disease (indicated in the subindex). Therefore, one has
Our goal is to influence the contagion rate Λ(x(t)) with a control policy u(t), representing the application of NPI, in this example, a lockdown, having as effect a reduction in the contact rates.
We denote byΛ(x(t),u(t)) the controlled contagion rate and we define controlled specific contagion ratesβ E ,β I m ,β I ,β H ,β H c as follows
where u = (u E ,u I m ,u I ,u H ,u H c ) ∈ [0,1] 5 is the initial vector of control variables that later will be reduced.
We shall assume that c ref H = c ref H c = 0 because we suppose that the hospitalized patients are highly isolated. Therefore, u H and u H c are no longer considered as control variables. This approach is also used in [33] . Additionally, we assume that the policy for the exposed and mild-infected individuals is the same and thus u E = u I m , which will be denoted u for simplicity. This means that NPI have the same effects (in terms of the contact rates reduction) for the people in the incubation stage and for the infected population with mild symptoms because these populations are not detected, so that NPI such as lockdowns are assumed to affect them similarly. Finally, we also assume that NPI do not have additional effects in terms of the contact rates reduction, in the infected population with symptoms (and then detected and reported by authorities) because we assume that these individuals are already isolated. Thus, we set u I = 1 −δ withδ ∈ (0,1) representing the fraction of nonisolated infected (detected) population. Our vector of control variables is then reduced to a single input u ∈ [0,1] and the (controlled) contagion rate becomeŝ
Furthermore, we set 1 −δ as the upper bound for the control variable, thus representing that NPI cannot isolate people to a greater extent than the isolation already imposed on the infected and detected population. Therefore, the single control variable u belongs to the control space
In the following, we setδ = 0.2. Now we can write the (autonomous) dynamics f : X × U −→ X, where the state space is X = R 8 + , defining the discrete-time control systems of this example as in (1) as follows
where the controlled contagion rateΛ(x,u) is given by (9) . The rest of parameters defining the above dynamics are described in appendix A where we describe the calibration procedure for this example.
Now we describe the elements for defining the event-triggered controls introduced in section 2.2. The considered observation time window and the minimum time of implementation (or minimum interevent time) will be ∆ = t min = 14 days. The event-triggering mechanism, as described in (2) and (4), is defined by u ref = 1 −δ (associated with u(t 0 ) in (2)) and as controllersû − ,û + : [[0 : (4)), we consider max-linear functions of the form
These controllers permit to monotonically decrease/increase the strength of the NPI in t min days until saturating it. We have decided to consider that our control saturates in t min days, coinciding with the minimum interevent time assuming that at the end of this period of time, the effect of NPI (or their release) is fully accomplished, producing the maximal isolation (u = 1 −δ) or reaching the normal contact rates equal to those before the pandemic (u = 0).
To define trigger indicators I : X ∆+1 −→ R and then the event-triggered sets S, as in section 3, we consider the following instantaneous observations of the state O : X −→ R: Thus, the indicators considered in this example are computed from the observations defined in (O1) and (O2) as follows:
(a) the mean of instantaneous observations:
(b) the mean of differences of instantaneous observations:
Thus, we will assess four indicators associated with two instantaneous observations (O1) and (O2) and with two different ways to consider them (a) and (b). Finally, to compare the performance of different event-triggered controls based on the indicators introduced previously (as introduced in section 3.1 and section 3.2), we consider two outcomes, that is, the function of outcomes P : X × U −→ R 2 is given by P(x 0 ,u) = (P 1 (x 0 ,u),P 2 (x 0 ,u)), where the event-triggered control u is defined by u = m(∆,t min ,I,θ), for a given trigger threshold θ, from an initial state x 0 at initial time t 0 . The outcomes considered are:
(P 1) Peak of ICU demand:
H c (t);
(P 2) Total number of days in lockdown. This is expressed as a percentage (over the total simulation time) and is computed differently depending on whether our NPI is applied at the initial time t 0 or not. If the NPI is applied at t 0 , it is given by
We note that in the above expressions, k u is the total number of the switches triggered by the policy u. cases (based in observation (O2), depicted by continuous curves) in practice can be less reliable that the indicator corresponding to the mean of the difference of the hospitalized patients in ICU beds (based in observation (O1) considered as (b)) that is shown by the orange dashed curve.
In table 1, we present the associated threshold for each indicator, having as objective a peak of ICU demand of P 1 (x 0 ,u) = 1,200 beds (see (P 1)). This value corresponds to an approximation of the total ICU beds available during October 2020 for the Metropolitan Region according to Chilean official data [20] . In this table, we also show the percentage of days in lockdown P 2 (x 0 ,u) (see (P 2)) associated with each indicators for the objective mentioned above.
In this example, we consider the compartmental model published in [32] for the spread of COVID-19
in China. The model is composed by nine state variables corresponding to different stages of the Table 1 : Thresholds and percentages of days in lockdown P 2 (x 0 ,u) (see (P 2)) associated with four assessed indicators, considering a peak of ICU demand objective of P 1 (x 0 ,u) = 1,200 beds (see (P 1)), in Metropolitan Region (Chile).
disease: susceptible (S), exposed (E), infectious (I), infectious but undetected (I u ), hospitalized that will recover (H R ), hospitalized that will die (H D ), recovered after being detected (R d ), and recovered after being infectious but undetected (R u ), and dead by COVID-19 (D). Thus, the vector of the state variables is
Since the model in [32] is established in continuous-time, we consider a very small time step (one hour). Then, we use the standard Euler method to discretize the dynamics in [32] with a step ∆t = 1/24 to obtain f : X × U −→ X as in (1) . Thus, for this case-study, the (autonomous) dynamics that define the discrete-time control systems are described by the following:
Here, the controlled contagion rateΛ(x,u) is given bŷ
where a single control variable u ∈ U = [0,1 −δ] has been considered. This control is associated with the implementation of NPIs such as lockdowns. It multiplies all of the contagion rates (β E , β I , β Iu , β HR and β HD ), because its main effect is to reduce the contact rates among all individuals.
Parameterδ ∈ (0,1) represents the fraction of the population that reduces their contact rates during a lockdown (or more generally the application of a given NPI). The latter is considered to be different from zero because, even during lockdowns, some basic services must still operate. In the following, we setδ = 0.25.
The parameters defining the dynamics f : X × U −→ X in (11) are given by the vector (β,γ,φ) ∈ R 5 + ×R 5 + ×[0,1] 2 . Vector β = (β E , β I , β Iu ,β HR , β HD ) ∈ R 5 + contains all of the specific rates associated with the contagious stages of the disease. Parameters γ = (γ E , γ I , γ Iu ,γ HR ,γ HD ) ∈ R 5 + are the mean rates of the transition from these respective stages to the subsequent stage. In other words, for a disease stage X ∈ {E,I,I u ,H R ,H D } parameter γ −1 X days represents the mean duration of stage X. Finally, the vector φ = (φ IHR ,φ IHD ) ∈ [0,1] 2 allows us to describe the distribution from infected people (I) to the next three stages: infected but undetected (I u ), hospitalized that will recover (H R ), and hospitalized that will die (H D ). Indeed, the fraction of infected people that are not detected is given by 1 − φ IHR − φ IHD , while φ IHR and φ IHD represent the fraction passing from infected to the hospitalized stages H R and H D , respectively. We note that in [32] , both fractions are equivalently written in terms of the case fatality ratio and of the fraction of infected people that are detected. The values of (β,γ,φ) used in our simulations correspond to those used in [32] for experiment EXP 29M ; see Table 3 in [32] .
Additionally, the initial conditions x(t 0 ) = (S 0 ,E 0 ,I 0 , I u0 ,H R0 , H D0 , R d0 ,R u0 ,D 0 ) ∈ X = R 9 + were obtained from the calibration realized with the complete set of data available in [32] , that is, our initial condition vector x(t 0 ) corresponds to the last day output (March 29, 2020) of the fitted model in [32] for experiment EXP 29M . These values are given in table 2.
It is important to note that a very strict lockdown was considered in the experiment EXP 29M in [32] . This explains the very low values obtained for the infected stages in table 2. Recall that our initial time (t 0 = March 29, 2020) corresponds to the final time in that experiment after the application of the lockdowns.
Regarding the other elements necessary to pursue our analysis, the event-triggered controls introduced in section 2.2 are defined similarly to those in section 4. observation time window and the minimal time of implementation (or minimal interevent time)
will be ∆ = t min = 14 days. The event-triggering mechanism described in (2) and (4) is defined by (2)) and by controllersû − ,û + : (4) ) that are considered as max-linear functions of the form
As in the previous case-study analyzed in section 4.1, our control saturates in t min days, coinciding with the minimum interevent time. Thus, we assume that at the end of this period of time, the effect of NPI (or their release) is fully accomplished, producing maximal isolation (u = 1 −δ) or reaching the normal contact rates equal to those prior to the pandemic (u = 0). To define trigger indicators I : X ∆+1 −→ R and then the event-triggered sets S (cf. section 3), we also consider the following instantaneous observations of the state:
(Õ1) Number of hospitalized people:
(Õ2) Number of detected infectious people:
O(x(τ )) = I(τ ).
As established in section 2.2, given an observation time window of 14 days (i.e., ∆ = 14), the indicators to be used are defined for the recent history of the state
Thus, the indicators considered in this case-study are computed from observations defined in (Õ1) and (Õ2) as follows:
(a) the mean of instantaneous observations:
(b) the mean of differences of instantaneous observations:
Finally, for the assessment of the different event-triggered controls policies based on the indicators above, we apply the methodology described in section 3.1 and section 3.2. Thus, we consider the function of outcomes P : X × U −→ R 2 given by P(x 0 ,u) = (P 1 (x 0 ,u),P 2 (x 0 ,u)), where the event-triggered control u is defined by u = m(∆,t min ,I,θ) (for a given trigger threshold θ, from an initial state x 0 at initial time t 0 ), and the outcomesP 1 (x 0 ,u) andP 2 (x 0 ,u) are given by the following:
(P 1) Peak of hospitalization demand:
where H u R (t) + H u D (t) corresponds to the total number of hospitalized patients at period t.
(P 2) Total number of days of lockdown: This outcome was already defined in section 4.1, outcome (P 2). For the sake of completeness, we recall its computation here:
If the NPI is applied at t 0 , it reads as follows
We note that in the previous expressions, k u is the total number of the switches triggered by the policy u.
Hence, the trade-off curves (see section 3.1)
for the four indicators introduced above (two instantaneous observations (Õ1) and (Õ2) and to the two different ways to consider them (a) and (b)) are depicted in fig. 6 . Figure 6 : Trade-off curve for our case-study based on [32] for the spread of COVID-19 in China. Four indicators are considered: nmber of hospitalized people (observation (Õ1)) considering the mean (indicator (a), blue dashed curve) and the mean of difference (indicator (b), orange dashed curve) and the number of active cases (observation (Õ2)) considering the mean (indicator (a), blue continuous curve) and the mean of difference (indicator (b), orange continuous curve). fig. 6 shows that the 14-day rolling average of infected and detected people (observation (Õ2) considered as (a)) is clearly the best indicator among the four analyzed indicators. The second best indicator is the mean of total hospitalizations due to COVID-19 (observation (Õ1) considered as (a)), but its performance is considerably weaker. For instance, when we consider an objective peak hospitalization demand of 7 million of beds (that is,P 1 (x 0 ,u) = 7,000,000; see outcome (P 1)), the percentage of time in lockdowns (outcome (P 2)) for the policies associated with each indicator is given byP 2 (x 0 ,u) = 31% and 57%, respectively, over the total analyzed time. This demonstrates the strong advantage of using a policy associated with the rolling average of infected people with respect to the other analyzed indicators in this case. Therefore, our analysis suggest the use of this indicator as the basis for an event-triggered policy for this case study.
In Table table 3 we show the percentage of the days in lockdownP 2 (x 0 ,u) (see (P 2)) associated with each indicator when the objective peak hospitalization demand isP 1 (x 0 ,u) = 7,000,000 beds (see (P 1)). We also present the associated thresholds θ for these four indicators. Table 3 : Thresholds and percentages of days in lockdownP 2 (x 0 ,u) (see (P 2)) associated with the four assessed indicators considering the peak hospitalization demand ofP 1 (x 0 ,u) = 7,000,000 beds (see (P 1)) in China as the objective.
In this work, we addressed two main goals: to propose a modeling framework for representing the decision-making process related to the application of NPI policies based on the observation of epidemiological indicators and to introduce a methodology to compare the effectiveness of these strategies (defined by trigger indicators) with respect to multiple objectives outcomes. The proposed framework is that of event-triggered controls with some specific details such as considering the state in a window of time and imposing a minimum intervention time (or minimum interevent time). Both features were introduced in order to represent the application of the NPIs more realistically.
The two case-studies discussed in section 4 show how our methodology can be used to recommend an indicator-based policy when some given outcomes are treated as objectives in our analysis.
Nevertheless, our approach should be consider only as a guide to suggest a recommendation. In practice, decision makers must also undertake a complementary and more in-depth analysis. For instance, the observability of the indicators under analysis is something that must also be taken into account. Indeed, when an indicator based on active cases has a similar performance to that of another indicator based, for instance, on the number of hospitalized patients (as in the case-study given in section 4.1), a possible recommendation is to consider a policy based on the second type of indicator because the corresponding observations (hospitalized patients) are more reliable and, consequently, the latter is a more robust NPI policy in practice.
As described in this article, our approach permits to represent the decision-making process based on the observation of indicators and compare the use of these indicators to define NPI policies from a cost-benefit perspective. This comparison will depend on the outcomes chosen as objectives and also on the structure of the dynamic model and on the indicators chosen for this comparison. Both of the latter mathematical objects are intrinsically related in the sense that the dynamics considered as a basis in our methodology must allow the computation and simulation of the indicators that we seek to compare in the analysis. Therefore, for assessment of more elaborate indicators such as positivity rates, indicators associated with testing and contact tracing and effective reproduction number, it is necessary to consider dynamical systems with compartmental stages that allows the formulation of such indicators. On the other hand, our methodology was demonstrated for two SEIR-type models that are some of the most common mathematical models in epidemiology. Nevertheless, our methodology can also be adapted to other underlying mathematical models, such as agentbased models. Of course, different dynamics or other underlying mathematical models can lead to different recommendations. However, this issue is inherent to any real-life decision derived from a mathematical analysis. The framework introduced in this paper for assessing indicators-based event-triggered policies raises some interesting problems to study the further theoretical analysis of which can have important repercussions on the decision-making process. We mention three such problems:
• Time consistency: When the analysis introduced in section 3.2 for determining the best indicator to use is carried out at time t 0 , following the policy induced by this indicator, there is no guarantee that this strategy will be the best in the long term. Since the trade-off curves are dynamic objects because they depend on the initial conditions (see (6)), it is possible that with time, another indicator should be chosen for triggering NPIs, because its performance is better considering the future state as the initial condition.
In the terms of trade-off curves and the comparison process proposed in section 3.2, the aforementioned situation occurs when two of these curves (the best chosen at t 0 and other), viewed as curves that evolve in time, intersect in a future time at the level (in the outcomes space) corresponding to the m − 1 outcomes objectives for which the analysis was carried out at time t 0 . To prove that a future intersection between the trade-off curves will not occur will clearly require us to assume certain properties of the dynamics and the compared indicators;
this is an interesting problem. A similar problem in the context of linear system where the performance (i.e., our outcomes) is a quadratic cost, is studied in [5] .
On the other hand, it is natural to consider situations where the outcomes objectives can evolve in time. For instance, at time t 0 the situation is analyzed considering a given ICU maximal capacity but perhaps in some months, this capacity will be increased. In summary, the comparison process of indicators-based event-triggered policies should be implemented continuously across the evolution of the disease in the population, taking into account eventual updating of objective outcomes by the decision makers. Due the latter and the dynamics aspect of trade-off curves, with longer time it is possible that a discarded indicator will in fact become the best indicator.
• Is it better to consider policies based on more indicators? In the analysis proposed in this work, we have considered only policies based on a single indicator. That is, to consider event-triggered sets S, defined in the space of the recent history of states X ∆+1 , in the form
where the indicator I : X ∆+1 −→ R takes values in R and θ ∈ R is a threshold for this indicator. Consideration of more indicators leads to the consideration of a vector-valued function I : X ∆+1 −→ R p , with p ≥ 2, and the inequality I(x ∆ ) ≤ θ in (13) with respect to the componentwise order for a vector of thresholds θ ∈ R p . We wonder if the question of whether more indicators lead to better policies can be formulate as follows: If S(I, θ) ⊆ S(Ĩ, θ)
for all vector of threshold θ (in an appropriate domain) can one conclude that a policy based on I is better than one based onĨ (or vice versa)?
For triggering NPIs it can be natural to consider various indicators, as in the examples cited in the introductory section, motivated by the reliability (or the lack of) of observations for making decisions. Nevertheless, assuming that all observations are reliables, it is not clear that the addition of indicators for triggering the decisions will induce better policies from the cost-effectiveness viewpoint.
• Stochastic analysis: The framework introduced in this work can be extended considering stochastic dynamics and indicators. In this case, a possible additional outcome that can be considered is the maximum probability of satisfying the other outcomes (seen as objectives), and then the performance of different event-triggered policies can be assessed in terms of that probability, considering the risk-aversion of the decision makers.
To considering stochastic dynamics, the estimated probabilities distributions of involved parameters can be take as one of the output given by the calibration procedure used in section 4.1 and mentioned in appendix A.
Perhaps using simple models (few state variables) it is possible to provide mathematical proofs for the questions posed in the first two points of the previous list, that is, to ensure that the trade-off curves do not intersect and to deduce when an indicator is better than others from the properties of the event-triggered sets S corresponding to those indicators. Eventually the theoretical analysis can be simpler considering feedbacks based only on the instantaneous observations and time-continuous dynamics, a framework have purposefully avoided here because the measurability of the involved functions merits a deeper analysis. The aforementioned problems are beyond the scope of this paper; however, we plan to study them in future works.
A Parameter description and calibration procedure of model in section 4.1
The parameters of system (7) associated with dynamics f : X×U −→ X given by (10) are contained in the vector (β,γ,φ) ∈ R 3 + × R 5 + × [0,1] 5 , where β = (β E , β I m , β I ) ∈ R 3 + is the vector of the specific rates associated with contagious stages of the disease and that are involved in the definition of the controlled contagion rateΛ(x(t),u(t)) given by (9) .
Parameters γ = (γ E , γ I m , γ I ,γ H ,γ H c ) ∈ R 5 + are the mean rates of transition from the respective stages to the next stage. In other words, for a disease stage X ∈ {E,I m ,I,H,H c } parameter γ −1 X days represents the mean duration of stage X.
The vector φ = (φ EI ,φ IR ,φ HR ,φ HD ,φ H c D ) ∈ [0,1] 5 contains different distribution fractions between the different stages. Thus, φ EI is the fraction of exposed people who become infected (with symptoms), φ IR is the fraction of infected people who recover, φ HR is the fraction of hospitalized (in normal services) people who recover, φ HD is the fraction of hospitalized (in normal services) people who die, and φ H c D is the fraction of hospitalized people in ICU beds who die.
The parameters have been estimated using a Hamiltonian Markov chain Monte Carlo estimation method using the Stan library [15] . The method is a Bayesian approach that allows the approximation of the probability densities for each of the parameters in the system when using a sufficient number of iterations. We use the Gelman-Rubin (R) convergence diagnostic and 12 parallel chains to ensure that the separate chains converge to the same values.R compares the variance of each chain to the pooled variance of all chains. Each of the parameters obtained anR ≤ 1.03 where each chain ran 10,000 iterations of which a half were warm-up. For each of the parameters, we set upper and lower limits to restrict the parameter search space to allow realistic combinations only. In addition, we used a weakly informative prior for each of the parameters with mean values obtained from the literature and broad standard deviations.
In table 4, we state for each parameter its limits, prior probability distribution in the form of a mean and standard deviation, and some references for the mean of the used prior distributions.
The model was trained on data from June 20, 2020, until September 20, 2020, in the Metropolitan Region (Chile), considering the total number of COVID-19 cases (symptomatic), the number of patients in ICU, and the number of deceased, according the information collected and published by the Chilean Ministry of Science, Technology, Knowledge and Innovation in [20] . We assume an independent and identically distributed (i.i.d.) error of σ for all data points.
Since there are no data available of the asymptomatic population, we are not able to train the trajectory of asymptomatic cases from E to I m and to R. That is, the parameters φ EI and γ I m are not trained by maximizing the posterior, except for the small influence through β I m that is difficult to distinguish from the other sources of contagion. To mitigate this, we will fix φ EI = 0.6 where we assume 60% of the infections to be symptomatic, following [44] and [ Table 4 : Parameter limits, priors, and mean and standard deviation following MCMC estimation.
The initial conditions x(t 0 ) = (S 0 ,E 0 ,I m 0 ,I 0 ,R 0 ,H 0 ,H c 0 ,D 0 ) ∈ X = R 8 + , have been estimated using weakly informative priors where we take E 0 , I m 0 , I 0 , and H 0 to have scaled Beta priors between 0 and N with α = 1 and β = 4 (parameters of Beta distribution), where N = 7,112,808 is the total population. For the initial number of hospitalized patients in ICU, deceased patients, and total number of infections to date we assume a lognormal distribution with a standard deviation of σ around the data. In this setup, S 0 and R 0 are free variable but we need to restrict at least one of these variables to solve the system. We propose a rough estimate of R 0 to be the total number of infected patients to date plus the percentage that were mildly infected. Here, we assume that the number of people that are currently infected, hospitalized, or deceased are small relative to the total population. S 0 then follows from 