RESEARCH Re is a Mexicana de F´
ısica 67 (1) 123–136 JANUARY-FEBRUARY 2021
Ma hema ical modeling and o ecas ing o COVID-19:
expe ience in San iago de Cuba p o ince
E. E. Rami ez-To es
Depa amen o de Biom´
edica, Facul ad de Ingenie ´
ıa en Telecomunicaciones, In o m´
a ica y Biom´
edica,
Uni e sidad de O ien e, San iago de Cuba, Cuba.
A. R. Sel a Cas a˜
neda
Depa amen o de Telecomunicaciones, Facul ad de Ingenie ´
ıa en Telecomunicaciones, In o m´
a ica y Biom´
edica,
Uni e sidad de O ien e, San iago de Cuba, Cuba
Y. Rod ´
ıguez-Aldana
Cen o de Es udios de Neu ociencias y P ocesamien o de Se˜
nales,
Uni e sidad de O ien e, San iago de Cuba, Cuba
S. S´
anchez Dom´
ınguez
Depa amen o de Ma em´
a ica, Facul ad de Ciencias Na u ales y Exac as,
Uni e sidad de O ien e, San iago de Cuba, Cuba
L. E. Vald´
es Ga c´
ıa
Cen o P o incial de Higiene, Epidemiolog´
ıa y Mic obiolog´
ıa, San iago de Cuba, Cuba
A. Pal´
u-O ozco
Cen o P o incial de Higiene, Epidemiolog´
ıa y Mic obiolog´
ıa, San iago de Cuba, Cuba
E. R. Oli e os-Dom´
ınguez
Di ecci´
on de In o ma izaci´
on, Uni e sidad de O ien e, San iago de Cuba, Cuba
L. Zamo a-Ma amo os
Depa amen o de Ma em´
a ica, Facul ad de Ciencias Na u ales y Exac as,
Uni e sidad de O ien e, San iago de Cuba, Cuba
R. Lab ada-Cla o
Depa amen o de Ma em´
a ica, Facul ad de Ciencias Na u ales y Exac as,
Uni e sidad de O ien e, San iago de Cuba, Cuba
M. Cobas-Ba is a
Depa amen o de Ma em´
a ica, Facul ad de Ciencias Na u ales y Exac as,
Uni e sidad de O ien e, San iago de Cuba, Cuba
D. Sedal-Yanes
Facul ad de Ciencias Sociales, Uni e sidad de O ien e, San iago de Cuba, Cuba
O. Sole -Na i˜
no
Facul ad de Ciencias Sociales, Uni e sidad de O ien e, San iago de Cuba, Cuba
P. A. Vald´
es-Sosa
Join China-Cuba Lab o F on ie Resea ch in T ansla ional Neu o echnology, Sichuan, China
J. I. Mon ijano
Ins i u o Uni e si a io de In es igaci´
on de Ma em´
a icas y Aplicaciones,
Uni e sidad de Za agoza, Za agoza, Spain
L. E. Be gues Cab ales
Cen o Nacional de Elec omagne ismo Aplicado,
Uni e sidad de O ien e, San iago de Cuba, Cuba.
e-mail: be [email p o ec ed]
Recei ed 19 July 2020; accep ed 28 Sep embe 2020
124 E. E. RAMIREZ-TORRES e al.,
In he p o ince o San iago de Cuba, Cuba, he COVID-19 epidemic has a limi ed p og ession ha shows an ea ly small-numbe peak o
in ec ions. Mos published ma hema ical models i da a wi h high numbe s o con i med cases. In con as , small numbe s o cases make
i di icul o p edic he cou se o he epidemic. We p esen wo known models adap ed o cap u e he noisy dynamics o COVID-19 in
he San iago de Cuba p o ince. Pa ame e s o bo h models we e es ima ed using he app oxima e-Bayesian-compu a ion amewo k wi h
dedica ed e o laws. One pa ame e o each model was upda ed on key da es o a el es ic ions. Bo h models app oxima ely p edic ed he
in ec ion peak and he end o he COVID-19 epidemic in San iago de Cuba. The i s model p edic ed 57 epo ed cases and 16 un epo ed
cases. Addi ionally, i es ima ed six ini ially exposed pe sons. The second model o ecas ed 51 con i med cases a he end o he epidemic.
In conclusion, an oppo une epidemiological in es iga ion, along wi h he low numbe o ini ially exposed indi iduals, migh pa ly explain
he a o able e olu ion o he COVID-19 epidemic in San iago de Cuba. Wi h he a ailable da a, he simples model p edic ed he epidemic
e olu ion wi h g ea e p ecision, and he mo e complex model helped o explain he epidemic phenomenology.
Keywo ds: COVID-19 epidemic; ma hema ical modelling; app oxima e Bayesian compu a ion; Poisson noise.
PACS: 02.70.Uu; 02.30.Zz; 87.10.Ed; 87.15.Aa; 87.19.Xx
DOI: h ps://doi.o g/10.31349/Re MexFis.67.123
1. In oduc ion
The ongoing COVID-19 epidemic ep esen s a global chal-
lenge o heal h-ca e sys ems. The SARS-COV-2 co on-
a i us causing he COVID-19 is highly ansmissible [1,2].
Ne e heless, he epidemic p opaga ion in a egion/coun y is
in luenced by se e al ac o s, including go e nmen al mea-
su es [3]. The Cuban go e nmen has implemen ed se e al
s a egies o social isola ion o con ol his epidemic. As a
esul , a small numbe o cases is epo ed in he p o ince
o San iago de Cuba. Ma hema ical models a e use ul o
unde s anding he COVID-19 ansmission and he in luence
o go e nmen in e en ions [4–7]. Cu ie e al. [4] sugges
ha sys em dynamics (di e en ial-equa ion-based modeling)
is sui able o model go e nmen decisions ha cause social
dis ancing and isola ion.
The mos common sys em dynamics in epidemic mod-
eling a e SIR (suscep ible - in ec ious - emo ed) and SEIR
(suscep ible-exposed-in ec ious- emo ed) ype. The classic
SIR and SEIR models assume ha indi iduals change classes
(e.g., om suscep ible o in ec ed) a cons an a es. This as-
sump ion is no ealis ic in mos si ua ions [8]. Addi ionally,
i does no explain why he e ec i e ep oduc ion numbe o
COVID-19 changes o e ime [9]. Some au ho s in oduce
ime-dependen pa ame e s o o mula e ealis ic models o
he cu en pandemic [5,10–12].
A challenge o COVID-19 modele s is o desc ibe he
di e ence be ween o al cases and diagnosed cases. One ap-
p oach is o exclude his di e ence and i he model o e-
po ed da a [16–18]. Ano he op ion is o conside unde-
ec ed cases as a compa men o he di e en ial equa ions
sys em [6,7,12,19,20]. Finally, some au ho s a oid he com-
plica ion o adding a new compa men , and hey model he
e o wi h di e en ad hoc me hods [5,10,21,22].
Ma hema ical models ha e been used o o ecas he dy-
namics o COVID-19 in delimi ed geog aphical a eas. These
models usually i big-numbe da a a e se e al days o epi-
demic e olu ion [7, 10, 23–26]. The o icial da a epo ed
om San iago de Cuba show a maximum alue o 36 ac-
i e cases eached 31 days a e he i s con i med case.
This peak is ollowed by a dec easing end. These small
and noisy da a make di icul he model i ing. A common
s a egy o desc ibe COVID-19 dynamics is o upda e one o
mo e model pa ame e s based on key days o go e nmen ac-
ions [7,10,12].
O dina y di e en ial equa ion (ODE) based models a e
widely used o cap u e he kine ic o biological sys ems and
o p edic he beha iou o ime-dependen a iables. The
minimiza ion o a cos unc ion (as a s a egy o i ing a
gi en dynamic model) does no ake in o accoun he unce -
ain y o pa ame e s. Addi ionally, his app oach can lead o
un ealis ic i ings [27].
Bayesian in e ence o e s a pla o m ha deals na u ally
wi h bo h p oblems, bu i equi es he calcula ion o like-
lihood unc ions [13–15, 28]. In many p ac ical p oblems,
likelihood unc ions become analy ically in ac able; ne e -
heless, da a can be simula ed om a pa ame e ec o by
applying some algo i hm ha depends on he model. Thus,
se e al echniques ha e been de eloped o in e pa ame e s
wi hou using likelihood unc ions. These echniques a e usu-
ally called app oxima e Bayesian compu a ion (ABC) o ee
likelihood in e ence [28,29].
ABC can be used o es ima e pa ame e s when he a ail-
able da a o an in ec ious disease is coa se [30]. Besides,
ABC accomplishes pa ame e s es ima ion o models ha in-
clude dedica ed laws o desc ibe he e o be ween o al cases
and epo ed cases.
The aim o his pape is o adap wo models epo ed in
he li e a u e o cap u e he dynamics o he noisy and small-
numbe da a o COVID-19 in San iago de Cuba. Fo his,
we modi y an ex ended SEIR model [10] by in oducing a
dynamic-mode gamma dis ibu ion o desc ibe he e o be-
ween o al and epo ed cases. Addi ionally, we adap he
classic SIR model epo ed in [31] by in oducing Poisson
noise in he obse a ion o emo ed cases. Fo bo h mod-
els, we upda e a pa ame e on c i ical days o go e nmen
ac ions and es ima e he pa ame e alues wi h ABC. We a e
no awa e ha hese modi ica ions ha e been epo ed in he
li e a u e.
Re . Mex. F´
ıs. 67 (1) 123–136
MATHEMATICAL MODELING AND FORECASTING OF COVID-19: EXPERIENCE IN SANTIAGO DE CUBA PROVINCE 125
2. Me hods
2.1. Da a
Because no pe sonal da a o pa ien s we e used, he e hical
app o al and indi idual consen we e no applicable.
We used o icial da a o cumula i e, in ec ed, and e-
mo ed cases o COVID-19 in San iago de Cuba p o ince.
These da a we e epo ed om Ma ch 20 h o May 4 h,
2020. Daily da a was p o ided by he P o incial Di-
ec ion o Public Heal h o San iago de Cuba and i-
sualized in an in o ma ion boa d om a web pla o m
(h p://www.co id19scu.uo.edu.cu). Fo y-nine pa ien s wi h
SARS-COV-2 we e con i med using eal- ime polyme ase
chain eac ion (PCR) es s. PCR es s we e made in he Lab-
o a o y o Molecula Biology, P o incial Cen e o Hygiene,
Epidemiology and Mic obiology o San iago de Cuba.
2.2. Epidemiological e idence
The i s day o he COVID-19 in San iago de Cuba p o ince
coun ed when he i s case was diagnosed (Ma ch 17 h,
2020). The i s case (day one), he second case (day ou ),
and he hi d case (day se en) we e con ac s o di e en
a ele s om a ious coun ies wi h COVID-19 ansmis-
sion epo ed. The numbe o diagnosed cases inc eased
om day se en and hey we e con ac s o h ee i s con-
i med cases. The epidemiological in es iga ion e ealed ha
he p ima y in ec ious ocus was small in San iago de Cuba
p o ince due o he small numbe o a ele s ha a i ed in
his p o ince. All con i med cases wi h COVID-19 we e a -
ended and ea ed in he D . Jos´
e Joaqu´
ın Cas illo Duany
hospi al o San iago de Cuba. The con i med pa ien s e-
mained a leas 14 days admi ed o he hospi al. When wo
PCR es s we e nega i e, pa ien s we e sen home unde med-
ical supe ision. Addi ionally, all con ac s o hese con i med
pa ien s we e quickly isola ed du ing 14 days o de ec symp-
oma ic and non-symp oma ic cases. A e 14 days o isola-
ion, uncon i med cases o COVID-19 we e sen home and
daily e alua ed by physicians.
The go e nmen o San iago de Cuba ook di e en ac-
ions ha es ained COVID-19 ansmission, such as he u -
ban and in e p o incial a el es ic ions; wo ke s we e en-
cou aged o wo k om hei homes; sa e anspo a ion was
p o ided o essen ial people in he di e en p io i ized ac i -
i ies; school ac i i y was empo a ily suspended a all educa-
ional le els; social mo emen was es ained; and he modi-
ied qua an ine was es ablished on he case clus e s o impo -
ance o disease ansmission.
2.3. Me hodology
In o de o unde s and and p edic COVID-19 in San iago
de Cuba, we ha e chosen wo de e minis ic models: one
mo e explana o y and he o he mo e pa simonious. The i s
model allowed o elabo a e hypo heses abou some possible
causes o he ew epo ed cases. The second model was used
o make as e and accu a e p edic ions o he epidemic e o-
lu ion. Fo bo h models, we upda ed a pa ame e on key days
o go e nmen ac ion. Also, we included e o models o ex-
plain he a iabili y o he obse ed da a. De ails o he cho-
sen models a e shown in he ollowing sec ion.
2.4. Models
Lin e al. [10] epo ed an SEIR model wi h h ee addi ional
classes: he o al popula ion size (N), he public pe cep ion
o isk (D), and he cumula i e cases numbe (C). Also, hey
modeled he zoono ic ansmission du ing he i s mon h and
hen only pe son- o-pe son ansmission, aking in o accoun
he emig a ion a e inside he coun y.
Unlike [10], we did no conside he zoono ic ansmis-
sion because in ec ion in Cuba was no o igina ed by animals.
Also, we neglec ed he emig a ion a e o wo easons: i s ,
he ew days elapsed be o e he anspo es ic ion by ai ,
land, and sea; second, he social isola ion ac ions aken, such
as emo e wo k, s opping school ac i i ies, among o he s. As
a esul o hese wo assump ions, we ob ained he ollowing
model, named model-I:























































dS
d =−β( )
NSI
dE
d =β( )
NSI−σE
dI
d =σE−γI
dR
d =γI
dD
d =dγI−λD
dC
d =σE,
(1)
wi h
β( ) = β0(1 −α)µ1−D
N¶k
,(2)
whe e σ−1,γ−1,d,λ−1,β( ),β0,αand ka e he mean la-
en pe iod, he mean in ec ious pe iod, he p opo ion o se-
e e cases, he mean du a ion o public eac ion, he dynamic
ansmission a e, he ini ial ansmission a e, he go e n-
men al measu e s eng h and he in ensi y o indi idual e-
sponse, espec i ely. The las exp ession o Eq. (1) ollows
om C=I+R. As αwas conside ed a s epwise unc ion
in [10], we used wo miles one da es: he days when in e -
p o incial anspo (Ap il 10 h, 2020) and u ban anspo
(Ap il 25 h, 2020) we e es ic ed.
To desc ibe he e o in epo ed da a, we assumed ha
he di e ence CT−CRbe ween o al cases (CT) and e-
po ed cases (CR) is a gamma p obabili y dis ibu ion. The
assump ion o a disc e e andom a iable as a con inuous one
Re . Mex. F´
ıs. 67 (1) 123–136
126 E. E. RAMIREZ-TORRES e al.,
p o ided some ad an ages in he assessmen o pa ame e es-
ima ion. Fo ins ance, i was possible o pe o m a nonpa a-
me ic con inuous hypo hesis es o e esidues. The shape
pa ame e o he gamma dis ibu ion was es ima ed, and he
scale pa ame e was changed dynamically. This app oach al-
lowed us o place he mode close o ze o in he mos op-
imis ic scena ios. We used an inc emen al e o in he e-
po ed a e du ing he ea ly days o he epidemic. A dedi-
ca ed law was o mula ed o he numbe o un epo ed cases
pe each epo ed one (U( )), gi en by
U( ) = (p +δ, i < dT
pdT+δ, i ≥dT,(3)
whe e pand δa e uning pa ame e s and dTis he ime om
which COVID-19 ansmission showed highe egula i y in
he case coun . We empi ically se he alues in p= 0.01
day-1,δ= 0.04 and dT= 8 days. We assumed ha U( )
inc eased du ing he days when almos no new cases a ose,
and a e ha , i emained oughly cons an . This assump ion
is consis en wi h a p e ious s udy [32] ha epo ed g ow h
in undocumen ed cases du ing he ea ly days o he epidemic.
The exp ession o he dynamic mode o e o (M( )) is
M( ) = CT
U( )+ 1.(4)
The di e ence be ween CTand M( )was conside ed a ime-
dependen app oxima ion o unde ec ed cases.
Model-II consis ed o a s anda d SIR model [31], adap ed
wi h he in oduc ion o Poisson noise in he emo ed cases.
Fo he pa ame e calib a ion o model-II, epo ed cumula-
i e and emo ed cases we e conside ed. We desc ibed β( )
in he model-II as a s epwise unc ion ha included changes
in key da es. The e o was assumed as a s anda d no mal
dis ibu ion o model-II.
2.5. Pa ame e es ima ion
Pa ame e es ima ion was pe o med wi h he ABC app oach
[28,33,34]. We p e e ed he sequen ial Mon e Ca lo (ABC
SMC) me hod desc ibed in he Algo i hm 4.8 o [34] because
o i s well-documen ed esul s. Ne e heless, he ABC SMC
me hod can pe o m slowly compu ing adap i e ole ances.
This sample (Algo i hm 1 in Appendix A) ou sma s i s
p edecesso s in au oma ic compu ing o ole ances based on
he e ec i e sample size o he p e ious popula ion. We used
N= 104,h0= 108,T= 5 and a Gaussian accep ance
ke nel unc ion Kh. We e e eade s o [34] o a deepe
unde s anding o ABC sample s and hei adjus able pa am-
e e s.
The eweigh s ep o he chosen ABC SMC sample led
o an op imiza ion p oblem in he calcula ion o hm. Thus,
i may in oduce delays in he compu a ion p ocess, espe-
cially i he pos e io dis ibu ion is dis an om he p io
dis ibu ion. The e o e, we in oduced he hyb id e sion
o ABC echniques, named ABC H algo i hm (Algo i hm 2
in Appendix B) as a hyb id o ejec ion sample (ABC RS)
and Ma ko chain Mon e Ca lo (ABC MCMC) app oaches.
Al hough we ha e no demons a ed he e godici y o he
ABC H algo i hm, we empi ically asce ained ha i ound
a maximum-a-pos e io i (MAP) loca ed wi hin he pa ame-
e egion de e mined by he ABC SMC algo i hm, consum-
ing less compu a ional ime. Hence, we p e e ed o use he
MAPs ob ained by he ABC H sample o quick p edic-
ion and he ABC SMC p ocess o mo e in-dep h pos e io
Bayesian analysis. Speci ically, we used he ABC SMC e-
sul s o compu e he c edible in e als o pos e io dis ibu-
ions. De ails o he algo i hms ABC SMC and ABC H a e
shown in A and B, espec i ely.
As a as we know, he ABC H sample is a new idea, and
i s main disad an age may be he inaccu acy o he sampling
om he pos e io dis ibu ion. This issue was no c i ical be-
cause we aimed o gi e a quick p edic ion based on he MAP.
Ne e heless, we hypo hesize ha o a easonably high alue
assigned o he ou h sampling choice (w3, co esponding o
a con en ional ABC MCMC), i is possible o ob ain pos e-
io dis ibu ions simila o ABC SMC esul s. Fu he e-
sea ch is equi ed o e i y his hypo hesis. We used he
scheme w={0.2,0.2,0.3,0.3} o sampling choice. O he
alues used in ABC H sample we e N= 103,h= 800, and
a Gaussian accep ance ke nel unc ion Kh.
Because ac ual da a can be di ec ly compa ed o simu-
la ed da a o ODE models, i was no necessa y o use sum-
ma y s a is ics as c i e ia o di e gence. We used he sum o
squa ed e o s as he dis ance c i e ion, which is closely e-
la ed o he likelihood [33,35,36], and sui able o in ec ious-
disease modeling [30]. We used a me hodology o he p io -
dis ibu ion choice ha consis ed o se e al ials mimicking
di e en a o able and un a o able scena ios, ollowed by
eedback om epidemiologis s.
Values o d= 0.2and σ= 1/3day-1 in model-I we e e-
po ed in [10]. The emaining pa ame e alues and he ini ial
condi ions E(0) and D(0) we e es ima ed wi h ABC s a ing
om locally uni o m p io dis ibu ions.
In ou app oach, wo key da es o a el es ic ions di-
ided he e olu ion o he epidemic in o h ee pe iods. Th ee
go e nmen al ac ion s eng hs we e es ima ed o hese pe-
iods: be o e a el es ic ions (α1), a e in e p o incial
a el es ic ion (α2), and a e he u ban public anspo
es ic ion in San iago de Cuba (α3). The h ee alues β1,β2,
and β3we e conside ed in model-II aking in o accoun he
same c i e ia o key da es men ioned abo e.
We calcula ed he coe icien o de e mina ion R2 o
he eg ession analysis o MAP i o assess he quali y o
pa ame e -es ima ion esul s o models I and II. We pe -
o med he equen is and nonpa ame ic Mann-Whi ney U
es [37,38], aking as null hypo hesis ha he ac ual esidues
ob ained a ose om he assumed gamma dis ibu ion. Fu -
he mo e, an app oxima ion o maximum likelihood (ML) a-
io be ween models (ML o model-I o ML o model-II) was
compu ed wi h ABC ou pu .
Re . Mex. F´
ıs. 67 (1) 123–136
MATHEMATICAL MODELING AND FORECASTING OF COVID-19: EXPERIENCE IN SANTIAGO DE CUBA PROVINCE 127
TABLE I. Summa y able o model-I pa ame e s and esul s.
Pa ame e No a ion P io o MAP MAP
ixed alue (ABC SMC) (ABC H)
T ansmission a e β0U(0.1,1.5) 0.4811 days-1 0.4340 days-1
CI(0.3680,0.5468)
α10.5245 0.6179
CI(0.3717,0.7534)
Fi ing pa ame e α2U(0.1,1) 0.2855 0.1801
CI(0.1775,0.3692)
α30.9665 0.9417
CI(0.8527,1.0000)
In ensi y o esponds k U(600,4000) 3430 3405
CI(3335,3598)
Mean la en pe iod σ-1 3days - -
Mean in ec ious pe iod γ-1 U(1,40) 21 days 25 days
CI(1,33)
P opo ion o d0.2- -
se e e cases
Mean du a ion o λ-1 U(10,30) 20 days 50 days
public eac ion CI(1,39)
Ini ial suscep ible S(0) 9.41 ·105- -
popula ion
Ini ial exposed E(0) U(1,10) 6 6
popula ion CI(4,7)
Ini ial in ec ious I(0) 1 - -
popula ion
Ini ial emo ed R(0) 0 - -
popula ion
Ini ial isk pe cep ion D(0) U(1,103) 163 148
CI(142,188)
Ini ial cumula i e cases C(0) 1 - -
All simula ions we e made in a 256-co e high-
pe o mance-compu ing (HPC) p ocesso wi h 256 GB RAM
using Py hon 3.6 [39]. The HPC machine was acqui ed by
he Flemish De elopmen Coope a ion h ough VLIR-UOS
(Flemish In e uni e si y Council-Uni e si y Coope a ion o
De elopmen o Belgium) in he con ex o he Ins i u ional
Uni e si y Coope a ion p og am wi h Uni e sidad de O i-
en e, San iago de Cuba.
3. Resul s
Model-I simula ion (Fig. 1) was pe o med wi h MAPs calcu-
la ed wi h ABC SMC. All pa ame e s and esul s o model-I
i ing a e summa ized in Table I. The Mann-Whi ney U es
pe o med on he esidues o model-I i did no ejec he null
hypo hesis based on a 5% signi icance le el.
Figu e 2 was ob ained using he MAPs o pos e io dis-
ibu ions o model-II pa ame e s ha we e compu ed wi h
ABC SMC. These MAPs and o he alues om model-II a e
summa ized in Table II.
In all ials made, he ABC H sample consumed less ime
and eached simila pa ame e egions o he ones ob ained by
ABC SMC (Figs. 3-6) o he same p io dis ibu ions.
Model-I p edic ed a o al o 73 in ec ed a he end o he
COVID-19 epidemic in San iago de Cuba, wi h 21% o un-
documen ed cases. The inal alue o M( )was 57, being he
model-I p edic ion o con i med cases. Model-II o ecas ed
51 con i med cases a he end o he epidemic. Addi ion-
ally, he ML a io calcula ed was 0.941; and he alues o de-
e mina ion coe icien s we e R2= 0.9450 o model-I and
R2= 0.9781 o model-II.
Re . Mex. F´
ıs. 67 (1) 123–136

128 E. E. RAMIREZ-TORRES e al.,
FIGURE 1. Es ima ed e olu ion o COVID-19 epidemic in San i-
ago de Cuba wi h model-I. The selec ed pa ame e alues o sim-
ula ion co espond o he MAP o he app oxima e pos e io dis i-
bu ion. The model-p edic ed and epo ed in ec ed indi iduals a),
and he cumula i e b), bo h p edic ed in simula ion (CTand CR
p edic ed) and obse ed (CR) cases a e plo ed. The shaded a ea
is limi ed by he ex emes o 95% c edible in e als o pos e io
dis ibu ions ob ained wi h he ABC SMC sample .
4. Discussion
COVID-19 dynamics in San iago de Cuba is di icul o cap-
u e empi ically wi hou he in oduc ion o s epwise unc-
ions o pa ame e αin model-I and pa ame e βin model-II.
Resul s show ha model-I (Fig. 1) and model-II (Fig. 2), i -
ed wi h ABC algo i hms, can p edic an ea ly small-numbe
peak o in ec ed cases and he app oxima e ending o he
COVID-19 in San iago de Cuba. The non- ejec ion o he
null hypo hesis on he esiduals allows us o con inue consid-
e ing he alidi y o he e o law o model-I.
All MAPs calcula ed wi h ABC H a e wi hin c edible
in e als ob ained wi h ABC SMC, excep pa ame e λo
model-I. E en o his pa ame e , he his og am gene a ed by
ABC H pa ially o e lapped he one gene a ed by ABC SMC.
We deduce ha bo h algo i hms ind he same pa ame e e-
gion o he chosen p io . A isual compa ison o Fig. 3 wi h
Fig. 4, and Fig. 5 wi h Fig. 6, e eals ha ABC H does no
sample om he same dis ibu ion as ABC SMC. Al hough
his does no a ec he use o ABC H in his wo k, a modi i-
ca ion o he ABC H sample is equi ed o make i a ully
FIGURE 2. Es ima ed e olu ion o COVID-19 epidemic in San i-
ago de Cuba wi h model-II. P edic ed in ec ed indi iduals ollow
he dynamics o he epo ed ones a). A long- e m scena io o bo h
epo ed and p edic ed cumula i e cases, and he a ea limi ed by
he ex emes o he 95% c edible in e als o pos e io dis ibu ions
ob ained wi h he ABC SMC sample b) a e plo ed.
unc ional ABC algo i hm. The scheme w={1,0,0,0} o
sampling choice co esponds o a ABC RS p ocess [33,40].
The scheme w={0,0,0,1} esembles an ABC MCMC
me hod [41]. ABC RS and ABC MCMC a e algo i hms ha
sample om adequa e pos e io dis ibu ions bu in an ine i-
cien ime, so o he in e media e schemes may be e alua ed
in u he wo ks.
The compu ed pe cen age o undocumen ed cases is in
con as wi h he 86% o un epo ed in ec ed cases es ima ed
o China [43]. This esul sugges s ha he limi ed p og es-
sion o he COVID-19 epidemic in San iago de Cuba may
be due o oppo une epidemiological in es iga ions, e ec i e
con ol measu es in each sou ce o in ec ion, and a low num-
be o ini ially exposed indi iduals (E(0) = 6). Addi ionally,
case clus e s p e ail o e ansmission clus e s o COVID-
19 in San iago de Cuba p o ince. The policymake s used
hese es ima es o complemen an in es iga ion in o he ini-
ial in ec ious load o SARS-COV-2 in San iago de Cuba.
The MAPs o αpos e io dis ibu ions we e es ima ed a
α1= 0.5245,α2= 0.2855 and α3= 0.9665 (Table I). Value
o αis expec ed o g ow as go e nmen measu es inc ease
bu , su p isingly, he alue a e he closu e o anspo a ion
Re . Mex. F´
ıs. 67 (1) 123–136
MATHEMATICAL MODELING AND FORECASTING OF COVID-19: EXPERIENCE IN SANTIAGO DE CUBA PROVINCE 129
TABLE II. Summa y able o model-II pa ame e s.
Pa ame e No a ion P io o MAP MAP
ixed alue (ABC SMC) (ABC H)
β10.5674 days-1 0.5149 days-1
CI(0.3067,0.7534)
T ansmission a e β2U(0.1,0.9) 0.5855 days-1 0.6056 days-1
CI(0.4794,0.7088)
β30.3555 days-1 0.3548 days-1
CI(0.2443,0.4525)
Mean in ec ious pe iod γ-1 U(1,10) 2 days 2days
CI(1,3)
In e al expec a ion λ U(1,20) 18 days 18 days
o emo ed CI(12,23)
Ini ial suscep ible S(0) 9.41x105- -
popula ion
Ini ial in ec ious I(0) 1 - -
popula ion
Ini ial emo ed R(0) 0 - -
popula ion
be ween p o inces is lowe han be o e (α2< α1). We con-
side wo possible explana ions o his esul :
Fi s , he in e p e a ion o αas go e nmen ac ion is
somewha o ced. In a p e ious s udy [42], αwas in e p e ed
as seasonali y o ansmission associa ed wi h he school cal-
enda o a simila o mula ion o β( ). Consequen ly, he
in e p e a ion o α emains open.
Second, he inc ease in PCR es s coincides wi h he pe-
iod co esponding o α2. Nei he he de e minis ic model
no he e o law explici ly akes in o accoun he a iabili y
o PCR es s. This a iabili y in luences he numbe o de-
ec ed cases, and he e o e he es ima ion o all ansmission-
a e pa ame e s.
Thus, in ou esul s, we do no in e p e αas go e n-
men ac ion. We conside αas a mul i ac o ial pa ame e
ha wo ks as a calib a o o β( )dynamics. The only dy-
namic a iable he explici ly in luences β( )in Eq. (2) is
D[10,42]. This in luence is in ag eemen wi h he epidemio-
logical and sociological esea ches ca ied ou in San iago de
Cuba, which con i m he educ ion o he ansmission a e by
he high- isk pe cep ion o bo h decision-make s and popula-
ion. Ne e heless, Ddepends explici ly on I, which in u n
depends on E; so he a iable Dis a ec ed by changes in
hese s a e a iables; and he e o e, hey indi ec ly in luence
he dynamics o β( ).
The es ima ed alue k= 3430 (k= 1117 in [10]) may in-
dica e a good esponse om indi iduals in San iago de Cuba
o COVID-19 epidemic.
The mean in ec ion pe iod o model-I may be explained
by he inclusion o un epo ed cases and he delay in he e-
po s o hose eco e ed. The alue o his pa ame e o
model-II is plausible because i is he numbe o days ex-
pec ed o a diagnosed pa ien o eco e , and he diagnosis
is usually con i med se e al days a e he exposi ion. The
in e al expec a ion o emo ed, λP oisson = 18 days wi h
CI [12,23], seems consis en wi h he pa ien -discha ge pol-
icy implemen ed by he Minis y o Public Heal h o he Re-
public o Cuba.
One possible explana ion o he ML a io is ha he e -
o law o model-I (Eq. (3)) can only pa ially desc ibe he
complex a iabili y o he epo ing a e. As model-II is mo e
pa simonious han model-I, we did no pe o m some u he
analysis like Akaike’s in o ma ion c i e ion (AIC) o Bayes
ac o , which should a ou model-II o e model-I. On he
o he hand, he model-I allowed us o es ima e he numbe
o indi iduals ini ially exposed, and o compu e he pe cen -
age o un epo ed cases. We a gue ha simple models like
model-II should be p e e ed o accu a e p edic ion, whe eas
mo e explana o y models like he model-I a e o be used o
phenomenological analysis.
The cha on he le in Fig. 2 shows a ema kable acking
o he da a a ia ion by he i ed cu e. A con en ional SIR
model canno achie e his. Model-II accomplishes his ack-
ing by di iding he adjus able pa ame e s o he SIR model
in o h ee ime pe iods. Fu he mo e, he cases eco e ed on
he day es ima ed by he SIR model a e no sub ac ed, bu on
he day expec ed by he Poisson e o . In o he wo ds, model-
II is ad an ageous o e a con en ional SIR as i is a piecewise
SIR wi h a Poisson noise in he numbe o emo ed cases.
Re . Mex. F´
ıs. 67 (1) 123–136
130 E. E. RAMIREZ-TORRES e al.,
FIGURE 3. Pos e io densi y es ima ion o model-I pa ame e s using ABC SMC sample and COVID-19 da a om San iago de Cuba. The
es ima ed pa ame e s a e β0,k,λ,α1,α2,α3,γ,E(0), and he scale pa ame e o he gamma dis ibu ion (αGAM ).
Re . Mex. F´
ıs. 67 (1) 123–136
MATHEMATICAL MODELING AND FORECASTING OF COVID-19: EXPERIENCE IN SANTIAGO DE CUBA PROVINCE 131
FIGURE 4. Pos e io densi y es ima ion o model-I pa ame e s using ABC H sample and COVID-19 da a om San iago de Cuba. The
es ima ed pa ame e s a e β0,k,λ,α1,α2,α3,γ,E(0), and he scale pa ame e o he gamma dis ibu ion αGAM .
Re . Mex. F´
ıs. 67 (1) 123–136