Compu e Me hods and P og ams in Biomedicine 211 (2021) 106399
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Compu e Me hods and P og ams in Biomedicine
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COVID-19: Es ima ion o he ansmission dynamics in Spain using a
s ochas ic simula o and black-box op imiza ion echniques
Ma cos Ma abuena
a , ∗, Pablo Rod íguez-Mie
b
, Ca los Ga cía-Meixide
c
, Vic o Lebo án
a
a
CiTIUS (Cen o Singula de In es igación en Tecnoloxías In elixen es), Uni e sidade de San iago o Compos ela, San iago de Compos ela, Spain
b
Toxalim (Resea ch Cen e in Food Toxicology), Uni e si é de Toulouse, INRAE, ENVT, INP-Pu pan, UPS, Toulouse 31300, F ance
c
Uni e sidade de San iago de Compos ela, San iago de Compos ela, Spain
a i c l e i n o
A icle his o y:
Recei ed 19 Ap il 2021
Accep ed 31 Augus 2021
Keywo ds:
Epidemic models
COVID-19
Compu ing science
S ochas ic p ocesses
E olu iona y compu a ions
a b s a c
Backg ound and objec i es: Epidemiological models o epidemic sp ead a e an essen ial ool o op imizing
decision-making. The cu en li e a u e is e y ex ensi e and co e s a wide a ie y o de e minis ic and
s ochas ic models. Howe e , wi h he inc ease in compu ing esou ces, new, mo e gene al, and flexible
p ocedu es based on simula ion models can assess he e ec i eness o measu es and quan i y he cu -
en s a e o he epidemic. This pape illus a es he po en ial o his app oach o build a new dynamic
p obabilis ic model o es ima e he p e alence o SARS-CoV-2 in ec ions in di e en compa men s.
Me hods: We p opose a new p obabilis ic model in which, o he fi s ime in he epidemic li e a u e,
pa ame e lea ning is ca ied ou using g adien - ee s ochas ic black-box op imiza ion echniques simu-
la ing mul iple ajec o ies o he in ec ion dynamics in a gene al way, sol ing an in e se p oblem ha is
defined employing he daily in o ma ion om mo ali y eco ds.
Resul s : A e he applica ion o he new p oposal in Spain in he fi s and successi e wa es, he esul o
he model confi ms he accu acy o es ima e he se op e alence and allows us o know he eal dynamics
o he pandemic a pos e io i o assess he impac o epidemiological measu es by he Spanish go e nmen
and o plan mo e efficien ly he subsequen decisions wi h he p io knowledge ob ained.
Conclusions: The model esul s allow us o es ima e he daily pa e ns o COVID-19 in ec ions in Spain
e ospec i ely and examine he popula ion’s exposu e o he i us dynamically in con as o se op e a-
lence su eys. Fu he mo e, gi en he flexibili y o ou simula ion amewo k, we can model si ua ions
—e en using non-pa ame ic dis ibu ions be ween he di e en compa men s in he model— ha o he
models in he exis ing li e a u e canno . Ou gene al op imiza ion s a egy emains alid in hese cases,
and we can easily c ea e o he non-s anda d simula ion epidemic models ha inco po a e mo e complex
and dynamic s uc u es.
©2021 The Au ho (s). Published by Else ie B.V.
This is an open access a icle unde he CC BY license ( h p://c ea i ecommons.o g/licenses/by/4.0/ )
1. In oduc ion
The sp ead o SARS-CoV-2 is gene a ing unp eceden ed heal h
and socio-economical c isis wo ldwide, being one o he mos sig-
nifican challenges in Eu ope since Wo ld Wa II. In he ligh o
his eme gency, he go e nmen s ough o o ganize an app op i-
a e schedule and op imize poli ical decisions based on scien ific
e idence o a oid he collapse o he heal hca e sys em, educe
i us- ela ed mo ali y and minimize he po en ial e ec s o an
economic ecession [39,44,50,51] .
∗Co esponding au ho .
E-mail add ess: [email p o ec ed] (M. Ma abuena).
Gi en he i al capaci y o he i us o sp ead and he lack o
e ec i eness o p e en i e measu es, many coun ies ha e been
sys ema ically o ced o lock down he popula ion empo a ily. Al-
hough hese policies may help con ol he sp ead o he i us,
hey a e economically unsus ainable o e ime. In his ega d, o e-
cas ing he e olu ion and consequences o he pandemic based
on he exposu e o he popula ion becomes a c i ical ac o in
decision-making [23,40] . Howe e , i is fi s necessa y o assess he
cu en sp ead o he epidemic o igo ously p edic hese e ec s,
which is o en unknown due o he limi ed acking o new in ec-
ions and ac i e cases.
A he beginning o he 20 h cen u y, he fi s ma hema ical
models o s udy he dynamics o an epidemic we e in oduced.
P obably he bes -known me hod is he suscep ible-in ec ed-
h ps://doi.o g/10.1016/j.cmpb.2021.106399
0169-2607/© 2021 The Au ho (s). Published by Else ie B.V. This is an open access a icle unde he CC BY license ( h p://c ea i ecommons.o g/licenses/by/4.0/ )
M. Ma abuena, P. Rod íguez-Mie , C. Ga cía-Meixide e al. Compu e Me hods and P og ams in Biomedicine 211 (2021) 106399
eco e ed model (SIR). SIR model and i s a ia ions [33,38] di ide
he popula ion in o compa men s, and using di e en ial (de e -
minis ic) equa ions, he numbe o indi iduals in each o he com-
pa men s o e ime a e es ima ed. Since hen, many new a ia-
ions o hese models ha also in ol ed s ochas ic e sions ha e
been in oduced in he li e a u e (see o e iew [3,5,49,70] o
o he con empo a y examples [14,59] ).
Despi e he eno mous p og ess wi h hese models in ecen
decades, hei di ec applica ions can be limi ed in se e al se -
ings. Fi s , mos models explain he dynamics o he epidemic a
he popula ion le el [28,37,41] excluding ele an indi idual in e -
ac ions. Second, model-specific assump ions can be es ic i e and
abs ac ed om p ac ice. Fo example, p ac i ione s use Poisson’s
homogeneous p ocess o handle he mechanism o new in ec ions
o pa ame ic dis ibu ions ha de e mine ime ansi ions [29,42] .
Thi d, in oducing model e o mula ions in p ac ice can be chal-
lenging and ime-consuming wi h he cu en op imiza ion s a e-
gies o he li e a u e-based p ima y on designed specific p oce-
du es wi h likelihood equa ions [10] . We belie e his is a c i ical
ac o limi ing he pe o mance o ini ial expe imen s and he use
o no el and non-s anda d o mula ion o epidemic models in a
ou ine and s aigh o wa d manne .
As in s a is ical lea ning heo y, we can say ha he e is no uni-
e sal model o all scena ios. Ins ead, we p obably ha e o design
specific models ollowing he exis ing epidemiological e idence o
each si ua ion and in oduce he p io knowledge ob ained in o
models.
Simula ion echniques a e a p ominen al e na i e me hod o
build complex and mo e ealis ic epidemiological models a a high
compu a ional cos . Howe e , hei use is no new, and se e al
agen models ha e appea ed in he li e a u e [75,79] , which al-
low modeling he possible impac o di e en in e en ions on he
e olu ion o a pandemic. Fo ins ance, we can s udy he impac o
accina ion, social dis ance, o lockdown policies in he educ ion
o in ec ions o mo ali y [32] . Mo e specifically, some o he spe-
cific ad an ages a e summa ized below:
• We can in oduce a wide a ie y o dis ibu ions in he compo-
nen s o he model ha can be specified wi h in ac able com-
plex likelihood equa ions [17] o e en non-pa ame ic assump-
ions.
• Simula ion models allow he in oduc ion o pe sonal in o ma-
ion o indi iduals, such as age and o he co a ia es ele an
o disease mani es a ion, wi hou in oducing challenges in he
model implemen a ion, unlike classical epidemic models.
• Adding some cons ain s in o he model, such as he social
in e ac ions be ween indi iduals, is no complica ed om a
model design pe spec i e and only inc eases compu a ion de-
mands.
A co ne s one in expanding his a ea o esea ch is he abili y
o ob ain eliable solu ions o he unde lying op imiza ion p ob-
lem wi hou eso ing o p oblem-specific op imiza ion s a egies.
Ad ances in compu a ional powe and he field o Black-Box op-
imiza ion [66] can be an essen ial miles one in achie ing such a
goal and being able o examine di e en models wi hou consum-
ing much ime using gene al p ocedu es. Howe e , some imes, his
s a egy equi es high-compu ing en i onmen s. Only by e alua ing
an objec i e unc ion can hese algo i hms lea n easonable solu-
ions pe o ming mul iple simula ions.
In his pape , we explo e his idea. Using a flexible ye s aigh -
o wa d dynamic p obabilis ic model ha we designed based on
he biological e idence o he onse o he pandemic, we es i-
ma e he se op e alence in di e en egions o Spain along di -
e en wa es. We also econs uc he dynamics o in ec ions and
eco e ies in di e en compa men s o answe specific epidemi-
ological ques ions, such as when he amous in ec ion peaks hap-
pened. To do his, we sol e an in e se p oblem wi h he mo al-
i y eco ds o es ima e some specific model pa ame e s, such as
he daily a e o in ec ions. In his ask, we use, o he fi s ime
in his a ea, he CMAES algo i hm [30] , one o he s a e-o - he-a
Black-Box s ochas ic op imiza ion me hods ha ha e been in ou
p e ious es s mo e compe i i e han o he exis ing algo i hms.
We mus no e ha ou p ima y pu pose in ma hema ical mod-
eling is no o make o ecas s abou he dynamic e olu ion o he
pandemic. Ins ead, he aim o ou p oposal is o pe o m back-
cas ing: o e ospec i ely econs uc he dynamics o in ec ions
while es ima ing se op e alence in he di e en compa men s o
he model. By es ima ing his in o ma ion, we can be e cha ac-
e ize he conc e e mechanisms o i us ansmission in he e -
i o ies analyzed. Thus, o example, we can guide poli ical deci-
sions in a mo e efined sense by es ablishing mo e ad anced and
pe sonalized epidemiological h esholds o de e mine lock-down
policies, acco ding o each e i o y’s specific socio-economic and
heal hca e ac o s and he dynamic e olu ion o he numbe o in-
ec ions d awn by ou model in he di e en compa men s.
1.1. Ou line
The a icle s uc u e is as ollows: Fi s , we in oduce ou new
ma hema ical model o es ima e he sp ead o COVID-19 in egions
and coun ies oge he wi h he model op imiza ion s a egy used.
Then, we in oduce some his o ical backg ound on he e olu ion
o he COVID-19 pandemic in Spain. Also, some demog aphic and
economic cha ac e is ics o he Spanish popula ion a e p esen ed.
Nex , we e alua e he beha io o he model and we illus a e i s
use ulness, pe o ming di e en analyses ac oss se e al Spanish e-
gions, epo ing he day- o-day e olu ion o suscep ible, in ec ed,
and eco e ed pa ien s. Finally, we discuss he esul s, he model
limi a ions, and he powe and alue o he new me hodology p e-
sen ed in he exis ing li e a u e.
1.2. Aims o he analysis
In o de o show he use ulness and b oad po en ial o ou p o-
posal o p ac i ione s, we pe o m di e en analyses ha allow
answe ing he ollowing epidemiological ques ions:
1. Wha was he sp ead o he i us in he fi s wa e in di e en
egions o Spain like?. Fo example, when did he peak o in-
ec ions occu ?. How many in ec ed people we e he e in Spain
a he end o he lockdown policies?
2. Using a longe ime ame, un il Ma ch 1, 2021, how we e he
o e all dynamics o SARS-CoV-2 in he Spanish popula ion as
a whole?. Fo example, he heal hca e si ua ion was c i ical in
Oc obe o 2020, and he e we e discussions abou applying a
na ional lockdown; Wha could be he eal epidemiological si -
ua ion a ha ime?
3. Gi en ha , om a heo e ical poin o iew, we can econs uc
he dynamics o in ec ions wi h ou model, how was he ac ual
day- o-day capaci y o de ec new cases in Spain?
2. Ma hema ical model and op imiza ion s a egy
2.1. Model elemen s
Suppose ha D = { 0 , 1 , . . . , n } is he se o days unde s udy.
Conside he ollowing andom p ocesses whose domain is defined
on D.
• S( ) : Numbe o people suscep ible o become in ec ed on day
.
• I
1
( ) : Numbe o in ec ed indi iduals who a e incuba ing he
i us on day .
2
M. Ma abuena, P. Rod íguez-Mie , C. Ga cía-Meixide e al. Compu e Me hods and P og ams in Biomedicine 211 (2021) 106399
Fig. 1. Diag am o s a e changes in ou model.
• I
2
( ) : Numbe o in ec ed people who ha e passed he heo e -
ical incuba ion pe iod and who: (i) don’ show symp oms o (ii)
symp oms a e mild on day .
• I
3
( ) : Numbe o in ec ed people who ha e passed he incuba-
ion pe iod and do show mode a e o se e e symp oms on he
day .
• R
1
( ) : Numbe o eco e ed cases which a e s ill able o in ec
on he day .
• R
2
( ) . Numbe o eco e ed cases ha a e no able o in ec
anymo e on he day .
• M ( ) : Numbe o dea hs on day .
Hence o h, we will deno e by I( ) = I
1
( ) + I
2
( ) + I
3
( ) he
numbe o in ec ed people a ime ∈ Dand R ( ) = R
1
( ) + R
2
( )
he numbe o eco e ed people.
The abo e andom p ocesses desc ibe he dynamic o popula-
ion indi iduals in sepa a e compa men s. We di ided he in ec ed
and eco e ed indi iduals in a b oade and specific axonomy o
he pa icula case o he COVID-19 han he classical epidemiolog-
ical models [5,38] . The e a e wo main easons o his. Fi s , he
pa ien s es ed by heal hca e a e usually hose ound in I
3
. In his
case, he e is an essen ial co pus o p io knowledge abou how
hey e ol e, and in case o dea h, hei su i al ime. Second, he e
is e idence ha he e a e eco e ed pa ien s who can s ill in ec
o he s.
2.2. Basic model defini ion
The causal mechanism o newly in ec ed indi iduals is in o-
duced below. Fo each day ∈ D, we assume ha he new in ec-
ions I
new
1
( ) a e gene a ed by he indi idual in e ac ion o he sus-
cep ible people wi h in ec ed pa ien s and he eco e ed cases ha
hey can s ill con amina e (pa ien s ha belong o s a es I
1
, I
2
, I
3
,
R
1
).
Fo mally, we assume ha i an indi idual can con amina e, i
does so acco ding o a andom a iable X ∼Poisson (R
i
( )) , being
R
i
( ) he a e age numbe o new in ec ions ha can cause each
pe son in he day . I is na u al o assume ha he unc ion R
i
( )
ollows a dec easing end in he fi s mon hs o he epidemic, ba-
sically due o wo easons: (i) qua an ine policies ha e been sys-
ema ically in oduced along wi h di e en coun ies and egions.
(ii) he numbe o suscep ible people dec eases o e ime, while
he numbe o in ec ed people can inc ease. These ac s indica e
ha in ou pa icula se ing, i is mo e complica ed o in e ac
wi h non-in ec ed people.
Once a new in ec ed pe son a i es o he model (see Fig. 1 ),
we assume ha he ansi ions be ween he di e en g aph
s a es a e modeled by a p obabili y law ha e ifies he ol-
lowing condi ions: (i) he ansi ion p obabili ies a e indepen-
den o he absolu e ins an when such ansi ion akes place
Table 1
Random a iables o he ime o each ansi ion.
T ansi ion Random a iable Used e e ences
I
1
→ I
2 Gamma
(5 . 807 , 0 . 948) Abdel-Salam and Mollazehi [1] , Laue
e al. [43]
I
1
→ I
3 Gamma
(5 . 807 , 0 . 948) Abdel-Salam and Mollazehi [1] , Laue
e al. [43]
I
2
→ R
1 Uni o m
(5 , 10)
I
3
→ R
1 Uni o m
(9 , 14) Abdel-Salam and Mollazehi [1]
I
3
→ M Gamma
(6 . 67 , 2 . 55) Abdel-Salam and Mollazehi [1] , No el
e al. [57] , Salje e al. [68] , Ve i y
e al. [81]
R
1
→ R
2 Uni o m
(7 , 14) Bi e al. [7] , Ehmann e al. [22]
Table 2
P obabili y o each ansi ion.
Coefficien Value Used e e ences
α0.8 Day [19] , Mizumo o e al. [53] , Nishiu a e al.
[56] , Taba a e al. [77]
β0.06 Dudel e al. [20] , Fauci e al. [24] , Mahase [48] ,
Rajgo e al. [65] , Ve i y e al. [80] , Wu e al. [83]
(ii) he p obabili ies depend only on he cu en s a e o he
pa ien ega dless o he p e ious pa h in he g aph. In pa ic-
ula , gi en he S a es = {I
1
, I
2
, I
3
, R
1
, R
2
, M} , and α, β∈ [0 , 1] ,
we ha e: P (I
2
|I
1
) = α, P (I
3
|I
1
) = 1 −α, P (M|I
3
) = β, P (R
1
| I
3
) =
1 −β, P (I
3
|I
2
) = 1 , P (R
2
|I
1
) = 1 ; all o he ansi ions ake a alue
equal o ze o in p obabili y. Mo e schema ically, he P , p obabili y
ansi ion ma ix, be ween e en s is shown in he Eq. (1) .
P =
⎡
⎢
⎢
⎢
⎢
⎢
⎢
⎣
I1 I2 I3 R 1 M R 2
I1 0 α1 −α0 0 0
I2 0 0 0 1 0 0
I3 0 0 0 1 −ββ 0
R 1 0 0 0 0 0 1
M 0 0 0 0 1 0
R 2 0 0 0 0 0 1
⎤
⎥
⎥
⎥
⎥
⎥
⎥
⎦
. (1)
Addi ionally, Table 1 shows he andom a iables ha model
he ime be ween ansi ions oge he wi h he e e ences used in
ou elec ions o eal examples o COVID-19 in Spain.
Table 2 shows he alues used o model he ansi ion p oba-
bili ies. The Supplemen a y Ma e ial p o ides specific de ails abou
how he men ioned pa ame e s and unc ions we e selec ed.
Finally, as we defined he model abo e, we ha e a con inuous-
ime p obabilis ic model. Howe e , he su oga e a iables o fi
and compa e he model esul s a e eco ded daily in eal-wo ld
si ua ions. Consequen ly, in ou implemen a ion we pe o m he
simula ion be ween ansi ions and new in ec ions on a daily basis
and we unca e he co esponding con inuous ime in days.
2.3. S ochas ic model implemen a ion
Ou model does no ha e a closed- o m solu ion. The e o e, in
a eal-wo ld se ing, i is necessa y o use s a is ical simula ion
me hods o app oxima e specific popula ion cha ac e is ics o he
s ochas ic p ocess as quan ile unc ions. Also, we mus fi some
pa ame e s o he model o cha ac e ize he beha io o he s udy
popula ion. Fo his pu pose, we use a sample o he deceased pa-
ien s {M
1
, M
2
, . . . , M
s
} along he se o days O = { 1 , . . . , s } .
Nex , we suppose ha ou model ( M) is dependen on a
ec o o pa ame e s θ= (θ1
, θ2
) ∈ R
p
1
×R
p
2 (wi h p
1
+ p
2
= p),
whe e θ1 is a ec o o dimension p
1
, defined in be o ehand,
and θ2 mus be es ima ed om he sample. Fu he mo e, le us
assume ha he ini ial s a e o he sys em is cha ac e ized by
S = (S(0) , I
1
(0) , I
2
(0) , I
3
(0) , R
1
(0) , R
2
(0) , M (0)) ∈ N
7 and T =
(T
1
(0) , T
2
(0) , T
3
(0) , T
4
(0) , T
5
(0) , T
6
(0)) ∈ N
m
×. . . ×N
m
. Shas he
3
M. Ma abuena, P. Rod íguez-Mie , C. Ga cía-Meixide e al. Compu e Me hods and P og ams in Biomedicine 211 (2021) 106399
numbe o elemen s o each compa men o he model on day
0. T also con ains he amoun o emaining days o comple e he
ansi ion hey a e in o each indi idual in he ini ial s a e, be-
ing m a na u al numbe ha ep esen s he maximum numbe o
egis e ed days.
To simpli y he no a ion, o each day ∈ D, we deno e he a -
e age dead ajec o y by he unc ion Mean (θ1
, θ2
, S, T )( ) .
The nex s ep is o es ima e
ˆ
θ2
. To do his, we p opose o sol e
he ollowing op imiza ion p oblem:
ˆ
θ2
= a g min
θ2
∈ S⊂R
p
2
s

i =1
ω
i
(M
i
−Mean (θ1
, θ2
, S, T )(i ))
2
, (2)
whe e ω = (ω
1
, . . . , ω
s
) is a weigh ed ec o ha can help o im-
p o e model es ima ion. Examples o hese weigh s may be:
ω
i
= M
i
/

s
i =1
M
i
o ω
i
= (1 / M
i
) / (

s
i =1
1 / M
i
) (i = 1 , . . . , s ) .
A his poin , i is ele an o no e ha he abo e op imiza ion
p oblem (2) , as o mula ed, includes he possibili y o in oducing
cons ain s in he pa ame e s’ space. The p e ious ac is essen ial
because we o en ha e p io knowledge o he ange o pa ame-
e s, o we can e en in oke he biological in e p e a ion o pa am-
e e s o answe his ques ion. Thus, by in oducing his knowledge,
we can sp ead up he speed o op imiza ion Black-Block echniques
significan ly.
In Eq. (2) , we ha e used he eal mean ajec o y. Howe e , in
p ac ice, his is unknown, and we mus app oxima e i using sim-
ula ion. Nex , we un B di e en simula ions, and we deno e by
M (θ1
, θ2
, S, T ) =
1
B

B
i =1
M
i
(θ1
, θ2
, S, T ) , he es ima ed mean a-
jec o y. M
i
(θ, θ2
, S, T ) (i = 1 , . . . , B ) deno es he esul o simula-
ion numbe i .
So he op imiza ion p oblem o be sol ed is:
ˆ
θ2
= a g min
θ2
∈S⊂R
p
2
s

i =1
ω
i
(M
i
−M (θ1
, θ2
, S, T )(i ))
2
. (3)
Schema ically, he o e all op imiza ion p ocess is desc ibed be-
low.
1. Define an ini ial θ0
2
and un B imes M(θ1
, θ2
, S, T ) . We de-
no e by M
1
(θ1
, θ0
2
, S, T ) , . . . , M
B
(θ1
, θ0
2
, S, T ) , di e en esul s
a e ob ained.
2. Es ima e he mean ajec o y M (θ1
, θ0
2
, S, T ) =
1
B

B
i =1
M
i
(θ1
, θ0
2
, S, T )
3. Es ima e he mean squa e e o ˆ
RSS
0
=

s
i =1
ω
i
( M (θ1
, θ0
2
, S, T )(i ) −M
i
)
2
.
4. To cons uc a succession o ec o s { θj
2
}
R +1
j=1
so ha ˆ
RSS
0
> ˆ
RSS
1
> ˆ
RSS
2
> . . . > ˆ
RSS
R +1
. Fo example wi h a s ochas ic op imiza-
ion sol e .
5. S op a e R + 1 i e a ions and e u n θR +1
2
as he op imal pa-
ame e o he p oblem.
In ou pa icula se ing, θ2
con ains he pa ame e o he R
i
( )
unc ion ha a e defined in Sec ion 2.2 . In ou p elimina y expe -
imen s, we assumed ha hei unc ional o m is equal o R
i
( ) =
min { C, ae
−(b + c
2
+ d
3
+ e
4
+
5
)
} whe e a ∈ [0 , 3] , b ∈ [ −1 , 1] , c ∈ [0 , 1] ,
d ∈ [0 , 1] , e ∈ [0 , 1] , ∈ [0 , 1] wi h θ2
= (a, b, c, d, e, ) ∈ [0 , 3] ×
[ −1 , 1] ×[0 , 1] ×[0 , 1] ×[0 , 1] ×[0 , 1] and Cis a posi i e cons an
fixed 0.005. Howe e , ou final elec ion a e se e al expe imen s
and sensi i i y analysis wi h di e en amily o unc ions ha in-
clude he p e iously exponen ial o in e se sigmoid/logi unc ions
is, R
i
( ) = d +
a
1+ b
−( −c)
, whe e a ∈ [0 , 5] , b ∈ [0 , 1] , c ∈ [ −30 , 30] ,
d ∈ [0 , 0 . 1] .
2.4. S uc u al limi a ions o he model in COVID-19 pandemic
The beha io o ou model is p ima ily de e mined by he pa-
ame e s α, β, and he unc ion R
i
( ) , while small a ia ions in
he dis ibu ions unc ion o he ansi ions imes should no ha e
a significan impac on he se op e alence es ima ions. R
i
( ) is es-
ima ed om obse ed mo ali y eco ds. Howe e , he pa ame e s
α, βa e fixed, wi h s a ics alues o e ime, and pe haps, in p ac-
ice, hei alue should a y in successi e wa es. In addi ion, α,
β, de e mine he in ec ion a ali y a e ( IF R ), he quo ien be ween
a ali ies, and he numbe o in ec ions. In pa icula , in ou model
IF R , is gi en, IF R % = ((1 −α) ∗β) ∗100 .
Some s udies ha e shown ha in he cu en pandemic, IF R
is he gold s anda d epidemiological indica o o moni o ing he
se e i y wi h which he i us has a ec ed di e en coun ies
[58,60] . Howe e , dynamic es ima ion o IF R is challenging o pe -
o m since a p ecise app oxima ion o he eal numbe o in ec ed
people is gene ally only possible in a single ime poin , hanks o
se op e alence s udies.
Se e al s udies ha e in es iga ed which ac o s influence he
alue o he IF R . The p ima y sou ces o a ia ions a e age and sex
[76] . I he dis ibu ion o new in ec ions is uni o m along ime e-
ga ding hese a iables, we can assume cons an s and s a ics al-
ues o αand βin di e en ime-spans.
Tes ing ha assump ion is necessa y o de e mine i we mus
a y he coefficien s α, and β, o e di e en wa es. Howe e , since
we do no know abou he ue in ec ions in successi e wa es, i
is no i ial o alida e his hypo hesis empi ically, and some s a-
is ical es ima ions a e needed.
In Spain, hea h ins i u ions pe o med an ambi ious and unique
longi udinal epidemiological s udy o know he pa e ns o i us
expansion in se e al ime poin s ha allow us o d aw es ima ion
abou IF R . In pa icula , we ha e in o ma ion on in ec ions a he
beginning o June and he end o No embe 2020. Using his in o -
ma ion, we can es ima e he di e ences in IF R be ween he fi s
and successi e wa es. We mus no e ha he Spanish si ua ion in
he fi s wa e was c i ical, wi h many p oblems in he elde ly pop-
ula ion, pa icula ly in nu sing homes, and and so changes in he
IF R along ime a e expec ed.
2.5. Model implemen a ion o handle mul i-wa es
Analogous o an in e en ion analysis in he con ex o ime se-
ies, we need o upda e he daily in ec ion unc ion o be able o
model well he eali y in a leas he ollowing wo si ua ions: a
he end o each lockdown, and a new e u n o no mal, o sho ly
be o e an explosi e g ow h in he numbe o new cases o dea hs
occu s, and no lockdown policies a e applied. In hese si ua ions,
he e a e ab up changes in daily in ec ion ends, and he e o e
he unc ional o m o he unc ion R
i
( ) needs o be modified.
Conside
0
= 0 <
1
<
2
< . . . <
m
, m empo al poin s, defined
wi h expe knowledge, and, in which, we hypo hesize ha he
end o he daily in ec ion unc ion is modifiable be ween di e -
en {
s
}
m
s =0
, o example be ween wo wa es. Then, we p opose o
define he unc ion R
i
( ) , as a piecewise unc ion, dependen on
he local unc ions R
1
i
( ) , ... , R
m
i
( ) , ha is,
R
i
( ) =
⎧
⎪
⎪
⎨
⎪
⎪
⎩
R
1
i
( ) ∈ [0 ,
1
)
R
2
i
( ) ∈ [
1
,
2
)
.
.
.
R
m
i
( ) ∈ [
m −1
,
m
) .
In ou fi s in successi e wa es, we assume ha he unc ional
o m o each R
s
i
( ) (s = 1 , . . . , m ) is iden ical, and equal o p io
unc ion R
s
i
( ) = d
s
+
a
s
1+ b
−( −c
s
)
s
( o all ∈ [
s −1
,
s
) , and any s ∈
{ 1 , . . . , m } ), whe e, he sub-index s , deno es he dependence o pa-
ame e s, o ime-pe iod [
s −1
,
s
) . Then, he numbe o ini ial ee-
model pa ame e s is mul iplied by he numbe o pe iods, m , con-
side ed.
4
M. Ma abuena, P. Rod íguez-Mie , C. Ga cía-Meixide e al. Compu e Me hods and P og ams in Biomedicine 211 (2021) 106399
Rega ding pa ame e s αand β, ou implemen a ion allows
a ying hese pa ame e s be ween {
s
}
m
s =0
. Le be αs and βs
, he
men ioned pa ame e o he in e al [
s −1
,
s
) . In his pape α=
α1
= . . . = αm
. Howe e , in he global Spanish analysis along di -
e en wa es, βwill be a y dynamically.
2.6. Con o mal simula ion bands o quan i y model unce ain y
Quan i y model unce ain y is a c i ical poin in o de o in-
e p e a e he esul s ob ained oge he wi h hei na u al limi s.
In epidemic modeling, acco ding o [21] , we can decompose he
model unce ain y in h ee di e en sou ces o e o :
• Da a unce ain y : Unce ain y in specified model pa ame e s,
es ima ed ex e nally om da a, o in da a o which models a e
fi ed.
• S ochas ic unce ain y : Unce ain y de i ed om he me hod o
simula ion.
• S uc u al unce ain y : Unce ain y in he op imal model s uc-
u e, o de i ed om he use o mo e han one model s uc u e
o a gi en ques ion.
In ou se ings, we use he dynamic e olu ion o mo ali y as a
sou ce o in o ma ion. Then, di ec ly epo ing he “S ochas ic Un-
ce ain y” d awn o he s ochas ic simula o as a measu e o un-
ce ain y is un ealis ic, since we can es ic he numbe o possi-
ble scena ios ha happen in p ac ice, acco ding o mo ali y esid-
uals. Mo e o mally, ou sou ce o in o ma ion is a co ela ed ime
se ies ha de e mines he possible eali y ha occu ed, and o
a gi en configu a ion o he ec o pa ame e θ, we should be e-
mo ed a significan ac ion o un ealis ic simula ion scena ios.
“Da a unce ain y” is no i ial o inco po a e in his ype
o model, and he non-pa ame ic boo s ap solu ion p oposed in
D’Agos ino McGowan e al. [21] ha e impo an limi a ions, since
i does no conside he dynamic and co ela ion s uc u e o daily
epo s. Mo e gene al, boo s ap s a egies ha can handle he spe-
cific dependence s uc u e o daily epo s can be adap ed as ou
se ing, such as block o wild boo s ap, bu i is some hing ou o
he scope o his wo k.
Tes ing “S uc u al unce ain y” is also challenging. The e may
be no uni e sal model in an epidemic modeling con ex , and pe -
haps he bes model o each si ua ion is a combina ion o b oad
dic iona y o simple models ha change dynamically as he pan-
demic e ol es.
In his pape , we p opose o use a solu ion simila o he
one p oposed in Shen e al. [73] , which consis s o selec ing a
small ac ion o simula ions, acco ding o e o c i e ia, e.g., mean
squa ed e o , ha shape he possible eal in ec ion ajec o ies.
Wi h he emaining ajec o ies, we apply specific con o mal in e -
ence echniques ha , o he bes o ou knowledge, ha e no been
applied p e iously o he con ex o s ochas ic simula ion models.
Con o mal in e ence me hods a e a gene al me hodology o quan-
i y model unce ain y [72] , wi h well-es ablished heo e ical oun-
da ions [45] , and we e used in an ex ensi e lis o machine lea n-
ing and s a is ics p oblems (see o example a con empo any ap-
plica ion [46] ).
Below, we in oduce he specific ma hema ical de ail o he
used con o mal simula ion bands ha can handle he e ocedas ic
noise. Fi s , suppose ha θ= (θ1
,
ˆ
θ2
) is he op imal pa ame e
configu a ion, whe e
ˆ
θ2 was es ima ed acco ding o me hodology
p oposed in he Sec ion 2.3 . Then,
1. Pe o m B = 10 , 0 0 0 simula ion o he model M(θ1
,
ˆ
θ2
, S, T )
and e alua e he mean squa e e o me ic (RSS). ˆ
RSS
s and
M
s
(θ1
,
ˆ
θ2
, S, T ) ( s = 1 , . . . , B ), deno e he esul s o i e a ion s
o mean squa e e o and he es ima ion o mo aly eco ds in
he simula ion model espec i ely.
2. Le Sel = { i ∈ { 1 , . . . , B } : ˆ
RSS
i
≤ˆ
RSS
(10 0 0)
} , he se o index o
simula ion wi h he lesse o equal 10 0 0 alue o RSS es ima-
ions. ˆ
RSS
(10 0 0)
deno e he elemen 10 0 0, conside ing he o de
sample o { ˆ
RSS
s
}
B
s =1
.
3. Using he subsample o dea h simula ion ajec o ies
{ M
i
(θ1
,
ˆ
θ2
, S, T ) }
i ∈ Sel
, es ima e poin wise he s anda de-
ia ion ˆ σ( ) , ∀ ∈ O.
4. Define he con o mal sco e Sco e
i
= max
∈O
| M
i
(θ1
,
ˆ
θ2
, S, T )( ) −M
|
ˆ σ( )
i ˆ σ( ) > 0 , ∀ i ∈ Sel. O he wise Sco e
i
is equal o 0.
5. Calcula e he quan ile q
α= a g min
∈ R
+
{

10 0 0
s =1
1 { Sco e
s
≤ }
10 0 0
≥α} ,
wi h α= 0 . 95 , o gua an ee dis ibu ional in e als ha co e
a confidence le el o 90% .
6. Define o each ∈ O, he confidence in e al p edic ion
as [ M (θ1
,
ˆ
θ2
, S, T )( ) −ˆ σ( ) q
α, M (θ1
,
ˆ
θ2
, S, T )( ) + ˆ σ( ) q
α] ,
whe e M (θ1
,
ˆ
θ2
, S, T ) deno e he simula ion mean ha co e-
spond wi h ou gi en mo ali y es ima ions.
7. Finally, o build confidence bands o he es o he s ochas-
ic p ocess ha makes up ou epidemic model, we mus se-
lec simula ion ajec o ies ha lead o he mo ali y ou come
alling wi hin he mo ali y band calcula ed in s ep 6).
A key elemen o con o mal in e ence is he selec ion o con-
o mal sco es. In his pape , he con o mal sco e has been es imed
using he geome y o he sup eme no m || ·||
∞
. || ·||
∞ is o en
used in he analysis o s ochas ic p ocesses in he field o unc-
ional da a analysis o es ima e confidence bands (see o example
[27] ).
2.7. Ou p obabilis ic model p oposal in he li e a u e
The li e a u e on epidemic modeling is e y b oad and includes
bo h mechanis ic models buil om causal epidemiological knowl-
edge and mo e s a is ical app oaches ha exploi in o ma ion om
his o ical da a wi h pu ely p edic i e models [9,41] . Nowadays, i is
no easy o es ablish a bounda y be ween bo h app oaches since,
on many occasions, bo h me hodologies a e used om he same
poin o iew o e en join ly.
Ou model defini ion is no e y complica ed om a ma he-
ma ical poin o iew. Howe e , i in oduces new challenges om
he compu a ional and modeling poin o iew: he simula ion o
he ajec o y o each indi idual along he popula ion, he in o-
duc ion o p obabili y dis ibu ions ha go beyond he exponen-
ial law as adi ional models do [5] , and he use o la en in ec-
ions models h ough a non-homogeneous Poisson p ocess, ex end-
ing in his sense he Ma ko ian p ope y [5] . In [25] , he au ho s
define a model simila o ou s and p opose a esolu ion amewo k
wi h Bayesian es ima ion me hods. Howe e , we assume ha spe-
cific pa ame e s a e known om he epidemiological scien ific e -
idence. Ou app oach using mo ali y eco ds [61] allow us o ob-
ain eliable se op e alence es ima es, bu he difficul y o he es-
ima ion inc eases. We also do no in oduce a Bayesian app oach
bu a equen is app oach wi h i s ad an ages and disad an ages
as he need in he Bayesian pa adigm o selec ing p io unc ions.
Finally, wi h he philosophy o ou simula ion model, we can con-
side complex ex ensions wi hou making oo many changes in he
implemen a ion.
2.8. Model op imiza ion wi h a CMA-ES black-box sol e
In ou se ing, we mus find op imal pa ame e s aking in o
accoun andomness in app oxima ing he mean. To do his, we
should eso o s ochas ic op imiza ion algo i hms.
Many s ochas ic algo i hms a e a ailable in he li e a u e. S ill,
based on he excellen p elimina y esul s, we ha e decided o use
5

M. Ma abuena, P. Rod íguez-Mie , C. Ga cía-Meixide e al. Compu e Me hods and P og ams in Biomedicine 211 (2021) 106399
a s a e-o - he-a e olu iona y algo i hm: he CMA-ES [30] . CMA-
ES is an e olu iona y-based de i a i e- ee op imiza ion echnique
ha can op imize a wide a ie y o unc ions, including noisy unc-
ions, like he one we use in ou me hod. One su ey o Black-Box
op imiza ion s a egies ound ha CMA-ES ou anked 31 o he op-
imiza ion algo i hms, pe o ming excep ionally well on “difficul ”
unc ions o la ge dimensional sea ch spaces [31] . F om a heo e -
ical poin o iew, CMA-ES can be seen as a pa icula case o he
Expec a ion-Maximiza ion algo i hm (EM) [8] .
We p o ide specific algo i hm s eps in Supplemen a y Ma e ial.
2.9. In e se p oblem beha io
The model fi wi h he mo ali y eco ds in okes new chal-
lenges in model iden ifica ion so ha he in e se p oblem is well-
defined. We fixed some model pa ame e s acco ding o exis ing
scien ific e idence o add ess his issue o make he p oblem mo e
egula ized. A po en ial al e na i e o fi ing mo e pa ame e s wi h
he da a is o ans o m he objec i e unc ion in o a mul iobjec i e
op imiza ion p oblem, conside ing he daily cases o o he ICU in-
dica o s as addi ional sou ces o in o ma ion. Howe e , in he ea ly
s ages o he pandemic, and e en nowadays, he e we e essen ial
doub s abou he eal capaci y o de ec ion o new in ec ions.
2.10. Tuning pa ame e
We pe o med mul iple expe imen s wi h CMA-ES o check
how he op imiza ion sol e beha es. A he same ime, h ough
s a is ical simula ion, we es ima e he a iance o he empi ical
mean by a ying a ali ies in di e en se ings. A e hose ini ial
expe imen s, we decided o es ima e he mean a 300 epe i ions,
ha is, B = 300 . In addi ion, we ha e allowed CMA-ES o un 30 0 0
i e a ions, s a ing he op imiza ion algo i hm om di e en an-
dom poin s. To ob ain he esul s o his pape , CMA-ES, was able
o find he op imal solu ion wi h he unc ion R
i
( ) selec ed in less
han wo hou s. Finally, he loss unc ion used is Mean Squa e E -
o (MSE) wi h w
i
= 1 (i = 1 , . . . , s ) (see Sec ion 2.3 ).
2.11. So wa e de ails and esou ces
Ou p oposal has been implemen ed in se e al p og amming
languages- C++, Py hon and R- al hough he esul s ha a e shown
in his a icle ha e been ob ained wi h Py hon. We op imized he
pa ame e s using lib a y pycma [2] , and numpy has been used o
ma hema ical ope a ions.
In he di e en pe o med s a is ical analyses, we ha e used
R. Plo s ha e been made bo h in R wi h ggplo 2 lib a y and in
Py hon wi h ma plo lib .
Finally, he aining da a used o fi he models in he fi s wa e
can be downloaded a [67] , and [18] , ha ep esen he daily Span-
ish s a is ics o COVID-19 a ali ies. In he mos ex ensi e analysis
o he o e all Spanish popula ion, we use he excess o mo ali y as
a sou ce o in o ma ion. The aw da a o es ima ed excess o mo -
ali y can be ob ained in he public web in e ace ela ed o MoMo-
daily Spanish mo ali y su eillance sys em ( h ps://momo.isciii.es/
public/momo/dashboa d/momo _ dashboa d.h ml ), coo dina ed by
Na ional Ins i u e o Heal h Ca los III (ISCIII).
We elease he code used in his pape o he benefi o he
scien ific communi y a ( h ps://gi hub.com/co id19-modeling ).
3. COVID-19 in Spain
Spain was one o he fi s coun ies wo ldwide o expe ience
he e ec s o COVID-19, a e China and I aly. Howe e , he con-
sequences we e mo e d ama ic despi e he delayed ou b eak s a
wi h espec o hese coun ies. To gi e a be e con ex o he e o-
lu ion o Co ona i us in Spain, in he fi s wa e, and compa e i
wi h o he coun ies, we in oduce some his o ical backg ound:
• Janua y 31s . The fi s posi i e esul was confi med on Spanish
e i o y in La Gome a. A ha ime, he e we e a ound 10,0 0 0
confi med cases wo ldwide.
• Feb ua y 12 h. The Mobile Wo ld Cong ess, one o he mos e-
ma kable echnological cong esses in he wo ld, o be held in
Ba celona, was cancelled.
• Ma ch 8 h. Mul i udinous ma ches we e celeb a ed in Spain.
Also, spo s compe i ions and o he e en s we e held as usual.
• Ma ch 13 h. Mad id epo ed 500 new cases o Co -19 in one
day (64 dea hs o al). Wuhan had gone in o lockdown wi h 400
new cases pe day (17 dea hs o al).
• Ma ch 14 h. Wi h he inc ease in he ou b eak o in ec ions, he
go e nmen decla ed a qua an ine h oughou he coun y.
• Ma ch 21s . Due o an o e loaded heal h sys em, he fi s pa-
ien s s a ed o a i e a new makeshi hospi als.
• Ap il 3 d. Spain accoun s o a o al o 117,710 confi med cases,
su passing I aly o he fi s ime.
• Ap il 6 h. Spain becomes he coun y in he wo ld wi h mo e
dea hs pe million inhabi an s.
• Ap il 9 h. The FMI o ecas s ha 170 coun ies a e going o all
in o ecession his yea in he wo s c isis since he G ea De-
p ession.
• Ap il 18 h. The Spanish go e nmen changes p o ocols o he
daily s a is ics o COVID-19.
Fig. 2 , shows he accumula ed numbe o cases and deceases
espec i ely in he p e ious pe iods in Spain, I aly, China, Uni ed
Kingdom, and he Uni ed S a es acco ding o he da a supplied by
he di e en go e nmen s.
Following he s a is ics o he Popula ion Re e ence Bu eau,
Spain is he 20 h coun y wi h he wo ld’s oldes popula ion [12] .
The coun y demog aphic s uc u e, po e y a es, and epidemio-
logical p ofiles a e essen ial o compa e mo ali y be ween coun-
ies. In he Co ona i us disease, ela i e and absolu e case- a ali y
isk (CFR) [26] inc eases d ama ically wi h age and wi h como -
bidi y, as e idenced by he cu en li e a u e. Rela i e isk can in-
c ease by mo e han 900% in pa ien s o e 60 [85] .
Subsequen ly, we pe o m a desc ip i e analysis in he egions
o Spain ha we analyze in his pape : Galicia, País Vasco, Cas illa
y León, Mad id, Ca aluña. Table 3 con ains he essen ial demo-
g aphic and socioeconomic cha ac e is ics o hese egions. We can
see ha Cas illa y León is he egion wi h he highes p opo ion o
elde ly people. A he same ime, Cas illa y León has he mos de-
localized popula ion cen es, and he País Vasco is he egion wi h
he lowes po e y a e. The na ional po e y a e is highe han
he o he analyzed egions because we do no include he mos
po e y egions.
Spain is a mul icul u al coun y whe e he e a e significan eco-
nomic, geog aphical, social, and demog aphic di e ences h ough-
ou he egions. All hese peculia i ies make Spain an in e es ing
coun y o ex apola e he e ec s o COVID-19 sp ead o o he e-
gions and coun ies.
Finally, in Fig. 3 , we show he e olu ion o in ec ions and a-
ali ies among he egions unde conside a ion in he fi s wa e.
As we can see, Mad id is he mos a ec ed egion, while Galicia is
he leas a ec ed, despi e i s olde popula ion. Howe e , i is es-
sen ial o no e ha he ou b eak began la e , and he con ainmen
was ca ied ou ea lie han in Mad id.
6
M. Ma abuena, P. Rod íguez-Mie , C. Ga cía-Meixide e al. Compu e Me hods and P og ams in Biomedicine 211 (2021) 106399
Fig. 2. Sp ead and numbe o dea hs o Co ona i us in Spain, I aly, China, and he Uni ed S a es. Numbe o accumula ed in ec ed pa ien s (le ) and he numbe o accumu-
la ed dea hs ( igh ) [13] .
Table 3
Demog aphic and socioeconomics cha ac e is ics o he Spanish popula ion h oughou some egions: Galicia, País
Vasco, Cas illa y León, Ca alu
´
na, Mad id [34] .
Galicia País Vasco Cas illa León Ca alu
´
na Mad id Spain
Popula ion 2,698,763 2,181,916 2,553,301 7,609,497 6,685,470 47,100,396
A - isk-o -po e y a e 18.8 8.6 16.1 13.6 16.1 21.5
Popula ion densi y 91.28 305.19 25.47 239.01 830.02 93.08
Pe cen age o popula ion by age g oup
0–9 7.50 8.93 7.51 9.78 9.92 9.28
9–18 7.56 8 . 78 7.79 9.74 9.44 9.37
18–30 10.39 10.66 10.67 12.72 12.80 12.42
30–45 21.18 20.28 19.47 22.24 23.24 22.07
45–60 22.77 23.28 23.60 21.77 22.23 22.61
60–80 22.60 21.41 22.30 18.24 17.33 18.70
om 81 on 8.01 6.67 8.65 5.50 5.03 5.56
Fig. 3. E olu ion o accumula ed in ec ed (le ) and dea h pa ien s ( igh ) in Galicia, País Vasco, Cas illa y León, Mad id, Ca aluña.
4. Resul s
4.1. Fi s wa e analysis
In o de o explo e he limi s o he model in a mo e chal-
lenging scena io, we s a he analysis wi h he fi s wa e. In his
pe iod, mos Spanish se op e alence su eys we e pe o med, and
he e o e we ha e a eliable es ima ion o he numbe o in ec ions
acc oss di e en Spanish egions in a single ime poin , allowing us
o e alua e ou model pe o mance. In addi ion, he in o ma ion is
o poo quali y, and epidemiological e idence is sca ce; hus, mak-
ing es ima ions in his scena io is mo e complica ed. Mo e speci -
ically, we es ic he model analysis o Ap il 26s in Galicia, País
Vasco, Cas illa y León, Mad id and Ca aluña. As he e is conside -
able unce ain y abou mo ali y eco ds, we assuming ha hese
wo scena ios hold:
1. We assume ha he numbe o eal dea hs due o Co ona i us
is eflec ed in official eco ds.
7
M. Ma abuena, P. Rod íguez-Mie , C. Ga cía-Meixide e al. Compu e Me hods and P og ams in Biomedicine 211 (2021) 106399
Fig. 4. Resul s in Mad id.
2. We suppose a mo e pessimis ic scena io. We assume ha many
people ha e died o Co ona i us, bu hey ha e no been in-
cluded in he eco ds because a diagnos ic es was no pe -
o med. In pa icula , we shall suppose ha he numbe o
dea hs is wice as high as hose indica ed in he official eco ds
each day.
To display esul s in an easy- o- iew o ma , we g aphically
ep esen he e olu ion o some s a es defined a he beginning o
Sec ion 2.1 in he wo cases conside ed. In addi ion, o gain u -
he insigh s in o he esul s, we show (i) he numbe o people
who may be con amina ed o ha e al eady ansmi ed he i us
as a pe cen age o he popula ion size; and (ii) he a e o new
in ec ions each day (deno ed in he Figu es as λ
).
Finally, we in oduce confidence bands o ou es ima ions using
me hodology desc ibed in Sec ion 2.6 .
He e, we only show he Figu es ha con ain esul s in Galicia
and Mad id. The es o he Figu es a e a ailable in Supplemen a y
Ma e ial.
4.2. Mul i-wa e analysis
To show mo e ecen and in o ma i e esul s on Co ona i us
dynamics in Spain, we adjus ed he model o he o al Spanish
popula ion un il 1 Ma ch 2021. To a oid choosing be ween he wo
scena ios abo e, we use excess mo ali y as a sou ce o in o ma ion
o eed ou model. αhas been selec ed wi h he same c i e ia as
he fi s wa e. Howe e , βis fixed wi h a alue equal o 0.085-in
he fi s pe iod, while he es wi h 0.0425. These alues we e es-
ablished o gua an ee an IF R o 1 . 7% in he fi s wa e and 0 . 85% in
successi e pe iods. Specific de ails abou IF R es ima ions a e ele-
ga ed o he Supplemen a y Ma e ial. Daily in ec ion unc ion R
i
( )
was fi ed as a piecewise unc ion (see Sec ion 2.5 o de ails). In
pa icula , he cu -o poin s selec ed o he jumps a e as specified
below ( Figs. 4–6 ).
1. 1 Ma ch o 30 May.
2. 1 June o 5 July.
3. 6 July o 15 Augus .
8
M. Ma abuena, P. Rod íguez-Mie , C. Ga cía-Meixide e al. Compu e Me hods and P og ams in Biomedicine 211 (2021) 106399
Fig. 5. Resul s in Galicia.
4. 16 Augus o 30 Sep embe .
5. 1 Oc obe o 24 Decembe
6. 25 Decembe o 1 Ma ch.
I is impo an o no e ha hese pe iods co espond o c i ical
e en s in pandemic e olu ion, such as a change in lockdown pol-
i ics, holidays, o o he e en s ha led o ab up changes in he
dynamics o in ec ions.
4.3. Analysis o esul s
The mos ele an esul s in he fi s wa e ( 1 s Ma ch 2020 o
26 h Ap il 2020) a e ou lined below:
• Mad id was he mos a ec ed egion by COVID-19. I we con-
side an ex eme se ing (e.g., he numbe o dea hs is dou-
ble ha epo ed by he Go e nmen ), 22.5% o he popula ion
could ha e been in ec ed o eco e ed om he i us. On Ap il
26s , he e may ha e been almos 1,2 million o pa ien s eco -
e ed.
• Galicia was he egion ha su e ed he mildes e ec s. The pe -
cen age o in ec ed people was less han 2.9%.
• Cas illa y León, Pais Vasco and Ca aluña could ha e su e ed he
e ec s o COVID-19 wi h a p opo ional magni ude. In hose e-
gions, he pe cen age o in ec ions could ha e been be ween 6
and 12% o he popula ion.
• The peak o new in ec ions p obably occu ed a he s a o
qua an ine, while he peak o people who can con amina e ook
place be ween Ma ch 17 and 24.
• The numbe o new in ec ions ha e been d ama ically educed
a e he in oduc ion o con ainmen measu es.
• The mos accu a e scena io is he pessimis ic scena io. The
analysis o he excess mo ali y epo ed in he Supplemen a y
9