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Optimal control of the COVID-19 pandemic - Nature

1, we show that the SAIRP model (as described above and in detail in “Methods”) fits well the con- firmed active infected cases in Portugal from March 2, 2020 until  

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| (2021) 11:3451 | www.nature.com/scientificreports w w w

—Ó—oe—"'

x x z

SARS-COV2virusishigher1

t,while

Department of Mathematics, Center for Research and Development in Mathematics and Applications (CIDMA),

Departamento

Departamento de Física Aplicada,

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the beginning of the epidemic till time t). Regarding the active cases, 6,035,791 (99%) suer mild condition of

the disease and 65,488 (1%) are in serious or critical health situation 2 . In Portugal, the rst conrmed 2 infected

cases were reported on March 2, 2020, and the Government ordered public services to draw up a contingency

plan in line with the guidelines set by the Portuguese Public Health Authorities. On March 12, 2020, it was

declared State of Emergency. In the following week, additional measures were adopted, such as: prohibition of

events, meetings or gathering of people, regardless of reason or nature, with 100 or more people; prohibition of

drinking alcoholic beverages in public open-air spaces, except for outdoor areas catering and beverage estab

lishments, duly licensed for the purpose; documentary control of people in borders; the suspension of all and

any activity of stomatology and dentistry, with the exception of proven urgent situations and non-postponable.

Teaching as well as non-teaching and classroom training activities were suspended from 16th March 2020
3 ; the air

trac to and from Portugal was banned for all ights to and from countries that do not belong to the European

Union, with certain exceptions. Actually, the Portuguese were advised to stay at home, avoiding social contacts,

since 14th March 2020, inclusive, restricting to the maximum their exits from home. From March 20 on, it was

mandatory to adopt the teleworking regime, regardless of the employment relationship, whenever the functions

in question allow. On May 2 the emergency status was canceled (duration of 45 days). Aer the 45 days of state

of emergency, the Government progressively established measures for the reopening of the economy but with

rules for the control of the spread of the virus. Portugal is still insituation of alert, and the situation of calamity

and contingency can be declared, depending on the region and the number of active cases. According to the

Portuguese Health Authorities, as of the writing, there has not been an overload of intensive care services; since

the beginning of the Portuguese outbreak the intensive medicine capacity increased from 629 to 819 beds (+23%)

(data from June 14, 2020); the health authorities objective is to reach, by the end of 2020, a ratio of 9.4 beds per

100 thousand inhabitants. Moreover, Portugal did not enter a rupture situation; at the peak of the epidemic (in the

end of April, beginning of May), there were 1026 intensive care beds; the levels of intensive medicine occupancy,

by June 14, 2020, were of 61% at national level and 65% in the Lisbon and Vale do Tejo region 4

e way we manage today the pandemic is related to the ability to produce quality data, which in turn will

allow us to use the same data for mathematical modeling tasks, that are the best framework to deal with upcoming

scenarios 5 . Many eorts have been done in this eld 6-9 . e adjustment of the model parameters in a dynamic

way, through the imposition of limits on the system in order to optimize a given function, can be implemented

through the theory of optimal control 10

e usefulness of optimal control in epidemiology is well-known: while mathematical modeling of infectious

diseases has shown that combinations of isolation, quarantine, vaccination and/or treatment are oen necessary

in order to eliminate an infectious disease, optimal control theory tell us how they should be administered, by

providing the right times for intervention and the right amounts 11,12 . is optimization strategy has also been

used in some works within the scope of COVID-19. Optimal control of an adapted Susceptible-Exposure-Infec

tion-Recovery (SEIR) model has been done with the aim to investigate the ecacy of two potential lockdown

release strategies on the UK population 13 . Other COVID-19 case studies include the use of optimal control in USA 14 . Optimal administration of an hypothetical vaccine for COVID-19 has been also investigated 15 ; and an expression for the basic reproduction number in terms of the control variables obtained 16 . According to the

most recent pandemic spreading data, until a large immunization rate is achieved (ideally by a vaccine), the

application of so-called nonpharmaceutical interventions (NPIs) is the key to control the number of active

infected individuals 17

Here we are interested in using optimal control theory has a tool to understand ways to curtail the spread of

COVID-19 in Portugal by devising optimal disease intervention strategies. Moreover, we take into account several

important issues that have not yet been fully considered in the literature. Our model allows the application of

the theory of optimal control, to test containment scenarios in which the response capacity of health services is

maintained. Because the pandemic has shown that the public health concern is not only a medical problem, but

also aects society as a whole 18 , the dynamics of monitoring the containment measures, that allow each indi vidual to remain in the protected P class, is here obtained through models of analysis of social networks, which

dierentiates this study getting closer to the real behavior of individuals and also predicting the adherence of

the population to possible government policies.

We propose a deterministic SAIRP mathematical

model for the transmission dynamics of SARS-CoV-2 in a homogeneous population, which is subdivided into

ve compartments depending on the state of infection and disease of the individuals (see Supplementary Fig.1):

S , susceptible (uninfected and not immune); A , infected but asymptomatic (undetected); I , active infected (symptomatic and detected/conrmed); R , removed (recovered and deaths by COVID-19); P , protected/pre vented (not infected, not immune, but that are under protective measures). e class P represents all individuals that practice, with daily ecacy, the so-called non-pharmaceutical

interventions (NPIs), e.g., physical distancing, use of face masks, and eye protection to prevent person-to-person

transmission of SARS-CoV-2 and COVID-19. Based on recent literature 19 ,20 , we assume that the individuals in the class P are free from infection, but are not immune and, if they stop taking these measures, they become susceptible again, at a rate ?1β , where w represents the transition rate from protected P to susceptible S and

m represents the fraction of protected individuals that is transferred from P to S class (see Supplementary Fig.1

for the diagram of the model; for the equations and a description of the parameters, see the “Methods" section).

In Fig.1, we show that the SAIRP model (as described above and in detail in “Methods") ts well the con-

rmed active infected cases in Portugal from March 2, 2020 until July 29, 2020 (a total of 150 days), using the

data from e Portuguese Public Health

Authorities

21
. More precisely, based on daily reports from the Portuguese

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Public Health Authorities, that provide information about the conrmed infected cases, recovered, and deaths,

the active cases are therefore the result of subtracting to the cumulative conrmed cases the sum of the recovered

and deaths by COVID-19. See section “Methods" for the parameter values and initial conditions used, as well

as their justication.

Most of the parameter values of the

SAIRP model are xed for the 150 days considered. However, we ana

lyzed the model in three dierent time intervals from the rst conrmed case, on March 2, until July 29, and the

parameters ? , p and m take dierent values in these three time intervals. At rst, we consider the time interval

going from the rst conrmed infected individual (March 2) until May 17, that is, 15 days aer the end of the

three Emergency States in Portugal. Here, despite the fraction of susceptible individuals

S that are transferred to

class P being 1

2 (see Table3 in “Methods"), meaning that approximately β of the population was

protected due to the COVID-19 connement policies during the three emergency states (suspension of activities

in schools and universities, high risk groups protection and teleworking regime adoption) 21
,22 , the number of

infected individuals increased exponentially (red curve in Fig.1). e second time interval goes from May 17

until June 9, the period when the number of new infected individuals grows slower comparing with the begin-

ning of the outbreak. In this time period, and aer the end of the three emergency states (during 45 days), the

fraction of susceptible individuals that could stay protected decreased ( 3 m

1 ), which, together with a low

rate of m 2 m , explains the progressive decrease of I (yellow curve in Fig.1). Finally, the model was applied

to the period going from June 9 until July 29, 2020. In that case, with the gradual opening of the society and

economy, the value for 3 becomes smaller and t increases as the number of active infected individuals started to rise again (green curve in Fig.1). For these parameter values 0 t , with 1t?[][ , we estimated the parameter values 0 (see “Methods" for details on the estimation of the parameters). e pandemic evolutions along past months, in dierent regions

worldwide, demonstrated that the behavior of the population is of crucial inuence. Same control policies,

implemented in dierent regions, resulted in dierent outcomes. Even more, the same policies, implemented at

dierent times, may produce dierent outcomes as the social state of opinion also changes with time.

We aim to incorporate the state of people"s opinion into the SAIRP model in order to analyze its inuence.

e process is divided into three steps. First, we calculate, from empirical data, the social network describing the

social interactions for Portugal at two dierent moments of time (April and July 2020). With this information,

we consider a simple opinion model that provides a probability distribution function that we interpret as the

distribution of opinions to follow government policies (distributed from zero to one, zero meaning no intention

to accept the policies and one total acceptance). As a nal step, we introduce this probability distribution func-

tion into the SAIRP model by modulating the access to class P. e details on the construction of the network, describing the social inter-

actions, are explained in the “Methods" section. Just note that in both cases analyzed (April and July 2020) the

network topology is quite dierent, reecting a dierent social state. Each network is composed by a set of

nodes (corresponding to dierent users or persons) and the connections with other nodes in the network. Both

networks built, as described, constitute some kind of ngerprint of the social situation in Portugal at the specic

periods of time considered.

We use this network topology in order to incorporate a model of opinion. For that, we consider now that each

node in our network is endowed with some dynamical equations, which allow to determine its state of opinion,

combined with the information that it is coming through the network. e opinion dynamical equations are

based on the logistic equations and they are fully described in the “Methods" section. e combined eect of

051001050

Time (days)

00.511.522.5

Fraction of active infected

10 -3

Model: March 2 - May 17

Model: May 17 - June

9

Model: June 9 - July 29

Real data

June

9May17May 24

Figure11.

Fraction of conrmed active cases per day in Portugal. Red line: from March 2 to May 17, 2020.

Yellow line: from May 17 to June 9, 2020. Green line: from June 9 to July 29, 2020. e drastic jump down in

the real data (black points) corresponds to the day when the Portuguese authorities announced 9844 recovered

individuals on May 24.

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the opinion model for each node, together with the inuence of the information coming through the network,

results in an opinion distribution function. e results are presented in Fig.2. To each opinion in the x-axis it

corresponds a probability to occur. In the two cases considered (April and July 2020) the opinion distribution

appears very polarized, but in July we can detect a clear decrease in the intention to follow government imposed

policies. is reects the experience of the situation as it happened, during the worst of the pandemic (April)

people were eager to follow any policy that helped reducing the impact of the disease, while in July more people

changed the opinion and decide to oppose the restriction policies. Our aim now is to couple the previous SAIRP model with

opinion distributions. For this purpose, instead of using a deterministic approach, we nd more feasible a multi-

agent based approach with stochastic dynamics, where a large number of individuals conform a mobility net

work and infected nodes can spread the disease through its connections with susceptible individuals 8 . e con sidered synthetic population is built according to the Watts-Strogatz model 23
, so it has small-world properties and high clustering. In particular, we considered a synthetic network with an average connectivity

1β and

a probability of long range connections of

2 . Following the main idea of the SAIRP model, each node can be

in one of the dierent compartments. Susceptible nodes can become asymptomatic by interactions with either

asymptomatic or infected nodes, or become protected with probability

β3 , at each time step. At the same time,

asymptomatic individuals are detected with probability m and conwrmed infected individuals can recover with probability

1 . Finally, protected individuals become susceptible again with probability m . e network is initial-

ized with a discrete number of infected individuals and then these processes are evaluated until the dynamics of

the disease become stationary. We now introduce the opinion distributions through the protected P compartment. Considering the opinion probability distributions, P(u) (Fig.2) for each node of the synthetic population we assign an opinion value drawn

from P(u). Next, instead of having a xed value for p and m, we consider that each node has its own probabilities

of becoming protected and susceptible again, 2 and m , and that these probabilities are given by the opinion value of the particular node. While we can directly identify 3 with , t has to be related to the complementary of : 0 t?0 . Note that the meaning of the extreme values of the opinions are either to follow the directives and stay at home (if 1 0 t?[] ) or not (if 0 β1.1 ). In this way, the opinion distributions overlap smoothly with

the transition to the protected compartment. Finally, following the infection rate of the deterministic model,

=m1= , we consider that the infection process occurs along the connection of an infected node i with a

susceptible node j with probability p1=0t . In this way, the infection process is also weighted by the opinion value of the susceptible node.

Remark

Although the values of

7 are directly related to the β values, their index j belong to completely dierent networks. On one hand, from the social network we extract the opinion distribution

P(u), from which we build

a new distribution P(p) with identical probabilities but applied to the epidemiological network (the one where we simulate the infective stochastic dynamics), assigning each node a value 2 0 e results of the SAIRP model with the opinion distributions included are presented in Fig.3a. e red

crosses mark the experimental observations until May 17 and the blue line is the wt to the SAIRP model with the

opinion distribution. e model simulation was repeated 12000 times in order to gain statistical signiwcance,

i.e., the evolution of the number of infected individuals shown in Fig.3 is consistent and does not depend on a

limited number of realizations, but is rather generic as the average over a signiwcantly large number of simula-

tions. e parameters used for these simulations are in Table4, in the “Methods" section.

In Fig.3b, the results of the SAIRP model, coupled with the opinion distributions, are shown for the two situ-

ations considered. e blue line corresponds to the situation in April 2020. e yellow line shows a possible line

of evolution of the pandemic in case the distribution of opinion is such as in July 2020 (the rest of the parameters

Figure=2.

Probability distribution ( m20 ) for each opinion ( = ). e opinion ranges from zero to one, zero

meaning no intention to follow the government policies while one means complete adhesion to this policy. e

blue values correspond to the Portuguese situation in April 2020 while the yellow ones are for the situation in

July 2020.

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were kept as in the blue curve). Note that the yellow line shows a much worse scenario and it is a direct conclu

sion of a change in the distribution of opinions. We obtain optimal control strategies that respect the following important constraints. (i)

One needs to ensure that the number of hospitalized individuals with COVID-19 is such that the health system

can respond to the other diseases in the population, in order that the mortality associated with other causes does

not increase. (ii) It is important that the number of active infected individuals is always below a critical level.

(iii) In order to keep the country “working", there is always a percentage of the population that is susceptible to

get infected. For instance, it is very important to keep schools open, in particular for children under 10/12 years

old; there are always people that do not follow the rules imposed by the government; etc. Roughly speaking,

our goal is to maximize the number of people that go back to “normal life" and minimize the number of active

infected (and, consequently, the number of hospitalized and in ICUs), ensuring that the health system is never

overloaded.

For the hospitalized individuals, the ocial

data for the fraction of hospitalized individuals due to COVID-19, represented by H , with respect to the active

infected individuals I is plotted in Supplementary Fig.2 (a), H/I. We observe that aer a rst period, where all the

active conrmed cases were hospitalized, the so-called containmentphase, the percentage of active infected indi-

viduals that needs hospital treatment is always below 15%. Moreover, aer the end of the emergency states (red

dot in Supplementary Fig.2), the percentage of active infected individuals that needs to be treated at hospitals is

less or equal than 5% (the 15% and 5% are plotted with dotted blue lines in Supplementary Fig.2).

For the percentage of active infected individuals that need to be in intensive care units (ICU), we observe

that (see Supplementary Fig.2 (b)) the proportion of active infected individuals that requires medical assistance

in ICU is always below than ? and, moreover, amer the end of the state of emergency the percentage of active infected individuals in the ICU is always below 1. One of the main challenges, facing countries struck by the

pandemic, is the reopening of the economy while preserving the health of the population without collapsing

the public health system. It is very important to keep the schools open (remember that children under 10/12

years old are not obliged to use a mask in Portugal) and prevent the economy to sink. us, there is a minimum

number of people that need to be susceptible to infection. But we also need to account that the population do

not always follow the rules imposed by governments. We have developed tools to quantify this eect and include

it into the equations. With this idea in mind, we investigate the use of optimal control theory to design strate

Figureβ3.

Evolution of the number of infected individuals (normalized by the total population) with time. (a)

Red crosses correspond to the experimental recordings while the blue line is the t of the SAIRP model with

opinion. e bluish shadow marks the uncertainty of the model. ( b) Blue line is the t of the SAIRP model

coupled with the opinion distribution, corresponding to April 2020, and the yellow line is the evolution of the

model coupled with the state of social opinion as in July 2020.

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| (2021) 11:3451 | www.nature.com/scientificreports/

gies for this phase of the disease. e goal now is to maximize the number of people transferred from class

P to

the class S (that helps keeping the economy alive) and, simultaneously, minimize the number of active infected

individuals and, consequently, the number of hospitalized and people needing ICU (in other words, ensuring

that the health system is never overloaded). We want to impose that the number of active infected cases is always

below 2/3 or 60% of the maximum value observed up to now ( 1 ). is condition warrants that the health system does not collapse. e fraction of protected individuals P that is transferred to susceptible S, is mathematically represented, in

the SAIRP model, by the parameter m. e class of active infected individuals I is very sensitive to the change of

the parameter m (Supplementary Fig.3). Taking into consideration the real ocial data of COVID-19 in

Portugal

21
, let β 2 represent

the maximum fraction of active infected cases observed in Portugal from March 2, 2020 until July 29, 2020. Note

that for

3m1 the constraint m2m3m

is not satiswed for the uncontrolled model (

1). is means that

the need of hospital beds and ICU beds can take vales such that the Health System can not respond, so we take

the maximum value t as a reference point for the state constraints imposed on the optimal control problem,

in order to ensure that in a future second epidemic wave the number of active infected cases remains below a

certain percentage of this observed maximum value. e parameter m in the SAIRP model, is replaced by a control function

0t? . We formulate mathematically

this optimal control problem and solve it (see “Methods"). e control function u takes values between 0 and 1 t , with ? 0,7

β . When the control u takes the value 0

there is no transfer of individuals from

P to the class S; when u takes the value

1 , then m 10.

5 of individuals

in the class

P are transferred to the class S at a rate w (see Table2 in “Methods" for the meaning of parameter w).

We consider a time window of 120 days. In the Supplementary Information, we analyze with more detail the optimal control problem subject to =p1=0t= .67 and ? 7,1

β0 (see Supplementary Figs.4-6 and Sup-

plementary Table1).

Remark

e optimal control problem under the state constraint =m2=0p= .58 is associated with a solution that

implies a substantial and important dierence on the number of hospital beds occupancy and in intensive care

units with respect to the optimal control problem subject to the state constraint

2=tt?2

. e choice of the constraints

1β3=1m1

.9 and 3=pp33 comes from the mathematical numerical simulations carried

out and the number of hospitals beds that the Portuguese Health System has available for COVID-19 assistance.

e controlled solution takes the maximum value in a wrst period of time, followed by a period where

there are no transfer of individuals from the class P to the class S and, at the nal period of time, it takes the

maximum value again (Fig.4a,b). e casequotesdbs_dbs19.pdfusesText_25