Lockdown Measu es and hei Impac on Single- and Two-age-s uc u ed Epidemic Model o he
COVID-19 Ou b eak in Mexico
J. Cue as-Ma a e
G upo de F´
ısica No Lineal, Depa amen o de F´
ısica Aplicada I,
Uni e sidad de Se illa. Escuela Poli ´
ecnica Supe io , C/ Vi gen de ´
A ica, 7, 41011-Se illa, Spain
Ins i u o de Ma em´
a icas de la Uni e sidad de Se illa (IMUS). Edi icio Celes ino Mu is. A da. Reina Me cedes s/n, 41012-Se illa, Spain
P. G. Ke ekidis, Q. Y. Chen, and G. A. Ke ekidis
Depa men o Ma hema ics and S a is ics, Uni e si y o Massachuse s, Amhe s , MA 01003-4515, USA
V´
ıc o Villalobos-Daniel
Na ional Cen e o Disease P e en ion and Con ol P og ams - CENAPRECE,
A enida Benjam´
ın F anklin, 132, 11800-Ciudad de M´
exico, CDMX
Z. Rap i
Depa men o Ma hema ics and Ca l R.Woese Ins i u e o Genomic Biology, Uni e si y o Illinois a U bana-Champaign
Y. D ossinos
Eu opean Commission, Join Resea ch Cen e, I-21027 Isp a (VA), I aly
The ole o lockdown measu es in mi iga ing COVID-19 in Mexico is in es iga ed using a comp ehensi e
nonlinea ODE model. The model includes bo h asymp oma ic and p esymp oma ic popula ions wi h he la e
leading o sickness (wi h eco e y, hospi aliza ion and dea h possibili ies). We conside he si ua ion in ol ing
he imposed applica ion o pa ial social dis ancing measu es in he ime se ies o in e es and ind op imal pa a-
me ic i s o he ime se ies o dea hs (only), as well as o ha o dea hs and cumula i e in ec ions. We discuss
he me i s and disad an ages o each app oach, we in e p e he pa ame e s o he model and assess he ealis ic
na u e o he pa ame e s esul ing om he op imiza ion p ocedu e. Impo an ly, we explo e a model in ol ing
wo sub-popula ions (younge and olde han a speci ic age), o mo e accu a ely e lec he obse ed impac as
conce ns symp oms and beha io in di e en age g oups. Fo de ini i eness and o sepa a e people ha a e ( yp-
ically) in he ac i e wo k o ce, ou pa i ion o popula ion is wi h espec o membe s younge s. olde han he
age o 65. The basic ep oduc i e numbe o he model is compu ed o bo h he single- and he wo-popula ion
a ian . Finally, we conside wha would be he impac on he numbe o dea hs and cumula i e in ec ions upon
imposi ion o pa ial lockdown (in ol ing only he olde popula ion) and ull lockdown (in ol ing he en i e
popula ion).
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I. INTRODUCTION
COVID-19, he disease caused by he no el co ona i us SARS-CoV-2 has, as o Augus 2020, a ec ed 216 coun ies and
changed he daily li es o billions o people [1]. I has a he same ime been he ocus o nume ous s udies, bo h clinical and
ma hema ical in na u e. The s udy o compa men al models ha add ess he sp eading o such epidemics has a ime-hono ed
his o y since he seminal con ibu ion o [2], which by now has been summa ized in a ious e iews [3] and books [4–6]. In
ecen yea s, a ia ions o such models ocusing on he pa icula i ies o co ona i uses ha e been inco po a ed, such as he ole
o asymp oma ic ca ie s o he i us, bo h as ega ds ea lie CoV examples, such as MERS (see o a ela ed example he wo k
o [7]), and la ely in he case o COVID-19 (see o a ela ed example he wo k o [8]).
Following on some o hese mo e ecen de elopmen s, he p esen s udy ocuses on a comp ehensi e compa men al epi-
demiological model ha akes in o accoun some o he in icacies o COVID-19, while conside ing i s applicabili y o an u gen
and impo an case example, namely he coun y o Mexico. Mo e speci ically, he model is an ex ension o he s anda d SEIR
(Suscep ible, Exposed, In ec ious, Reco e ed) model ha includes a p esymp oma ic s age, du ing which a pe son expe iences
no symp oms, bu is ne e heless in ec ious [9]. The p oposed ma hema ical se up also accoun s o bo h he asymp oma ic
in ec ious and symp oma ic in ec ious indi iduals. Asymp oma ic in ec ious cases ha e been ound in nume ous s udies and
epo s a gue ha hey may be signi ican ly unde - epo ed [8, 10], which may complica e mi iga ion e o s such as con ac
acing and sel -isola ion. Those wi h se e e disease symp oms may equi e leng hy hospi aliza ion, which has s ained he
heal h sys em o many coun ies [11]. In ligh o ha , he model also includes a compa men desc ibing he hospi aliza ions.
Ano he dis inc i e ea u e o he disease is he nonhomogenei y wi h which i mani es s in di e en age g oups, especially as
i pe ains o symp om se e i y and mo ali y isk [12, 13]. O he ac o s, such as p eexis ing condi ions and in e gene a ional
con ac s may also play a ole [14]. While popula ion age-s uc u e may be o en a e aged ou and deemphasized in nume ous
modeling a emp s [15], in ou model we choose o conside bo h a single age-g oup and a wo age-g oup e sion o he model.
The a ionale behind his choice is he mul i old inhomogenei y in he popula ion o a ious coun ies (including ou example
o in e es ). Fi s ly, as men ioned abo e, he se e i y in younge people (especially child en [16]) is smalle han ha in adul s.
Secondly, olde and mo e ulne able people may shed mo e i al pa icles, hus being mo e in ec ious [17]. Thi dly, con ac s
pe day [18] and he con ac ne wo k i sel o olde people is di e en om hose o younge people. The pa i ion especially
be ween p o essionally ac i e (i.e., non- e i ed) indi iduals and e i ees is impo an in connec ion o he abo e wo poin s, bo h
as ega ds he di e en ial in a e age numbe o con ac s o hese wo g oups, and as ega ds he po en ial ulne abili ies he eo .
As a case s udy, we ocus on he COVID-19 ou b eak in Mexico. While s udies o Mexico based on ma hema ical models
exis , hey di e om ou s in se e al signi ican ways. Some igno e social-dis ancing and o he mi iga ion measu es [19], o he s
ocus on he es ima ion o R0and in ec ions, using a Bayesian hie a chical model [20], and ye o he s ha e since become
ou da ed [21]. Mexico aces a unique challenge, due o he p e alence o COVID-19 isk ac o s, such as obesi y, diabe es and
hype ension, among i s popula ion [22]. This is being e lec ed in he epo ed da a and ou p edic i e esul s, which show
almos as many a ali ies in he <65 yea s old g oup as in he >65 yea s old g oup. We eel ha hese pa icula ea u es o
he Mexican popula ion in conjunc ion wi h he la ge numbe o in ec ions and especially o dea hs in he coun y wa an an
examina ion h ough he p ism o di e en age-s uc u ed models (e.g., single-age s. wo-age models; in u u e s udies, possibly
u he pa i ioning may be o in e es ) and an assessmen o he po en ial impac o lockdown measu es in he immedia e u u e.
Following he o mula ion o he single-popula ion model and he esul s ob ained h ough i in sec ion II, we con inue wi h
he wo age-g oup model in sec ion III. In each case, we ob ain he op imal pa ame e s o he model in ma ching he a ailable
da a ega ding dea hs, which a e conside ed o be he single mos eliable piece o a ailable in o ma ion. We do discuss he
ad an ages and disad an ages o po en ially ma ching he numbe o cumula i e in ec ions (and he numbe o dea hs). Once
he op imal i ing pa ame e s a e ob ained we assess he impac in bo h dea hs (bu also cumula i e in ec ions) o immedia e
lockdown measu es in ei he he case o he en i e popula ion o in ha o jus he olde age g oup. The p edic ion o he model is
ha housands o li es may be sa ed in jus he ollowing mon h alone, should such measu es be imposed e ec i e immedia ely.
In sec ion IV we summa ize ou conclusions and p esen ideas o u u e in es iga ion.
II. SINGLE-POPULATION MODEL
A. Equa ions
In he model p esen ed he ein, we modi y somewha he se up o he ea lie wo k o a subse o he p esen au ho s [23],
by inco po a ing he e ec o p esymp oma ic indi iduals. Mo e conc e ely, we s a wi h a suscep ible (S) popula ion ha can
become exposed (E) o he SARS-CoV-2 i us upon in e ac ion wi h h ee ca ego ies o al eady in ec ed indi iduals: (a) he
p esymp oma ic (P), indi iduals who a e in ec ed, in ec ious, and e en ually will de elop symp oms; (b) he asymp oma ic (A),
indi iduals who a e in ec ed, in ec ious, and will no de elop (clinical) symp oms; and (c) he symp oma ically in ec ed/sick
(I) popula ion membe s ca ying he i us (in ec ious). Upon such in e ac ion, he suscep ible become exposed o he i us.
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No ice ha in he suscep ible popula ion, he e is he po en ial o “ emo al” (i.e., dea h) by o he causes, ia he las e m o he
co esponding ODE p opo ional o µ, howe e his e m is p ac ically i ele an o ou pu poses. Fo his eason, we se µ= 0.
Once a membe o he popula ion becomes exposed (E), a la en pe iod (τl= 1/σ1) o he i us ollows (expec ed o be in he
icini y o 3 days [24]), du ing which he exposed popula ion is in ec ed bu no in ec ious. A e his pe iod, we assume ha he
hos can na u ally be pa i ioned o ei he asymp oma ic (A) o p esymp oma ic (P). The ac ion o he o me is φ, while o he
la e 1−φ. While bo h A and P play a ole (along wi h he in ec ed I) in u he ansmi ing he i us, and indeed A ha e been
a gued o play a c ucial ole [9, 10], i is only P ha will p esen symp oms a e an addi ional ime scale, he p eclinical pe iod
τp= 1/σ2. The incuba ion pe iod, i.e., he pe iod om in ec ion o he de elopmen o (clinical) symp oms, o his pa i ion
τinc =τl+τp= 1/σ1+ 1/σ2is o he o de o 5 days [24], and ep esen s he ime ill he onse o symp oms.
F om he e on, he asymp oma ics A con inue as i no hing happened, gi en ha hey ha e minimal o no symp oms. Thei
pa h is only owa ds eco e y wi h a cha ac e is ic a e MAR, o wi h a ime scale ep esen ing he asymp oma ic in ec ious
pe iod τA
in = 1/MAR ha is ypically expec ed o be on he o de o 7days. This is why in mos coun ies qua an ine is
expec ed o las a ound 14 days (7 days o in ec iousness ill eco e y and ano he 7 ill eco e y o any o he pe son ha may be
in ec ed among he ones in close con ac wi h he pe son o in e es ). In o de o dis inguish hose eco e ed om asymp oma ics
(which canno be di ec ly moni o ed, unless ex ensi e es ing is pe o med in a communi y) om hose coming om a pa h o
symp oms/sickness (which can be –a leas pa ially– moni o ed), we deno e hose eco e ed om A as AR.
The pa h o he p esymp oma ics P is mo e complica ed. Indeed, hese may s ill eco e (R) wi hou he need o hospi aliza-
ion and wi hou se e e mani es a ion o symp oms (going h ough class I). Howe e , in a ac ion γo he cases hospi aliza ion
is needed. Reco e y o hospi aliza ion in he model is associa ed wi h a ime scale 1/M, he symp oma ic in ec ious pe iod
τI
in = 1/M. Subsequen s eps in ol e a ac ion ωo he hospi alized ha die, o e a ime scale 1/ψ and a ac ion 1−ω
ha eco e o e a ime scale 1/χ. The abo e o e s an, in p inciple, comple e desc ip ion o he modeled quan i ies wi hin ou
sys em. We should no e ha β e e o he coe icien s o in e ac ion be ween A (o P) and S, as well as I and S, leading o new
in ec ions; hese a e, espec i ely, βSA and βSI . We no e he e, ha we somewha abuse no a ion as a as β’s a e conce ned.
Namely, i is ele an conside he ansmission a e β=˜
β
N, whe e Nis he o al popula ion. Wha we epo in he ables
ha ollow is ac ually ˜
β. In e ms o he ele an ime scales, he in ec ious pe iod associa ed wi h symp oma ically in ec ed
is τI
in = 1/M, wi h asymp oma ic indi iduals τA
in = 1/MAR, whe eas he in ec ious pe iod associa ed wi h p esymp oma ic
indi iduals is τP
in =τp+τI
in = 1/σ2+ 1/M.
The ansc ip ion o he abo e s eps in equa ions leads o he ollowing ODEs:
dS
d =−βSA( )S(A+P)−βSI ( )SI −µS
dE
d =βSA( )S(A+P) + βSI ( )SI −σ1E
dP
d = (1 −φ)σ1E−σ2P
dA
d =φσ1E−MARA
dAR
d =MARA
dI
d =σ2P−MI
dH
d =γMI −(1 −ω)χH −ωψH
dR
d = (1 −γ)MI + (1 −ω)χH
dD
d =ωψH
A schema ic diag am o he model is shown in igu e 1.
In he p ac ical aspec s o wha ollows, we will conside he da a o Mexico, wi h a o al popula ion o 127,575,528 people
(in 2019) and mo e han 430,000 con i med cases and 47,000 dea hs by he beginning o Augus 2020. Da a we e aken om
“Di ecci´
on Gene al de Epidemiolog´
ıa (DGE)” o “Gobie no de M´
exico” [28]. These da a ha e he pa icula i y o including
each clinical case, om which we e ie e h ee basic pieces o in o ma ion: he da e when symp oms s a , he dea h da e (i
applicable) and he age. The da a a e upda ed on a daily basis. As he way o measu ing always implies an unde es ima ion o
he numbe o cases and dea hs in he days close he epo ’s (especially because o he delay in dea h communica ions o he
DGE), we ha e pe o med i s up o da es abou 20 days om he epo da e (i.e. he epo da e is July 29 and i s a e pe o med
un il July 10).
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SEP
A AR
I
H
R
D
𝜔𝜔𝜓𝜓
MAR
𝝋𝝋𝝈𝝈1
(1-𝝋𝝋)𝝈𝝈1𝝈𝝈2(1-𝝲𝝲)M
𝝲𝝲M
(1-𝜔𝜔)𝟀𝟀
𝞫𝞫SII
𝞫𝞫SAA
𝞫𝞫SPP
EAAR
La en pe iod 1/𝝈𝝈1
In ec ious pe iod 1/MAR
EPIR
In ec ious pe iod 1/𝝈𝝈2+1/M
Incuba ion pe iod 1/𝝈𝝈1+ 1/𝝈𝝈2
P eclinical pe iod 1/𝝈𝝈2
FIG. 1. Schema ic diag am o he single-popula ion model (le panel) and a diag am o he wo disease p og ession pa hways: he asymp-
oma ic and he symp oma ic one ( igh panel, based on a ele an a ia ion o [8] adap ed o he speci ic compa men s and ime scales o he
p esen model).
Ou p incipal diagnos ic quan i ies in o de o ob ain he op imal pa ame e s o he model will be he ime se ies o dea hs
D( ), bu we will also moni o he cumula i e in ec ions. The la e in he ealm o he p esen model amoun s o C( ) =
I( ) + H( ) + R( ) + D( ), i.e., he sum o he indi iduals going h ough he pa o he ne wo k in ol ing he symp oma ically
in ec ed. We ake = 0 as Ma ch 22, and i da a o July 10. This e lec s ou e o o be (in e ms o he o al numbe s o bo h
diagnos ics) well wi hin he “well mixed” egime om he beginning o he in ec ion whe e he ODEs o in e es a e expec ed
o be ele an .
We a e pa icula ly in e es ed in he e ec o non-pha maceu ical in e en ion s a egies on he sp eading and de elopmen o
he disease. Such s a egies ende he ansmission a es ime dependen . On Ap il 21 ( q= 30) a ligh o m o social dis ancing
was en o ced in Mexico. In addi ion, we will conside a numbe o scena ios acco ding o which a mo e se e e lockdown may
be imposed on Augus 10 ( L= 141).
We a e pa icula ly in e es ed in how non-pha maceu ical in e en ion s a egies modi y he sp eading and de elopmen o he
disease. Such s a egies ende he ansmission a es ime dependen . On Ap il 21 ( q= 30) a ligh o m o social dis ancing
was en o ced in Mexico. In addi ion, we will conside a numbe o scena ios acco ding o which a mo e se e e lockdown may
ha e been imposed on Augus 10 ( L= 141).
The e ec o in e en ions on he o e all ansmission a e βmay be es ima ed by conside ing biological and physical p ope -
ies o expelled espi a o y d ople s, which a e he ca ie s o he pa hogens and speci ically o SARS-CoV-2. The ansmission
a e is usually w i en as β=cp, wi h c he numbe o con ac s pe day a suscep ible indi idual has and p he ansmission p ob-
abili y. Fo pa hogen ansmission ia in ec ious espi a o y d ople s (o diame e d), be hey ai bo ne (ai bo ne ansmission) o
se led (con ac ansmission), he ansmission a e has been exp essed as, c . Re s. [25] and [26] ( o ai bo ne ansmission)
β=βd
κd
αd
,(1)
whe e κdis he espi a o y d ople emission a e ( i al shedding) by e.g. b ea hing, speaking, coughing, sneezing, αdis he
d ople e ec i e emo al a e, by e.g. g a i a ional se ling, ambien ai low, pa hogen inac i a ion, and βd he ansmission a e
pe deposi ed espi a o y d ople . The subsc ip “d” e e s o a speci ic d ople size. As we will use i Eq. (1) o es ima e how he
o e all ansmission a e changes we will neglec i s complex dependence on d. Las ly, he d ople ansmission a e βddepends
on he numbe o e ec i e con ac s a suscep ible has wi h o he indi iduals, he numbe o pa hogens con ained in an in ec ious
d ople (i s pa hogen load) , and he i us ansmission p obabili y pe inhaled/deposi ed d ople .
Social dis ancing, o o he lockdown measu es ha es ic human mobili y, dec eases he a e age numbe o daily con ac s,
and possibly he a e age du a ion o con ac , he eby dec easing βd(and consequen ly β). In wha ollows we deno e his
ac ional dec ease as ηSA and ηSI . Ano he common in e en ion measu e is he use o su gical ace masks. Mil on e al.
(2013) [27] a gued ha hei use p oduced a 3.4- old educ ion in i al ae osol shedding. Face masks also ende physical
con ac o in ec ed hands wi h suscep ible a eas on an indi idual’s ace (mou h, eyes, nose) mo e di icul . They, also, modi y
he expelled ai low, wi h consequen ial e ec s on d ople anspo and dispe sion in he en i onmen and hei ai bo ne li e ime
( hus hei emo al a e). We sugges ha he combined e ec o wea ing su gical ace masks is p ima ily condi ioned by he
dec ease in i al shedding. We, hus, es ima e ha hei use (i he whole popula ion used hem con inuously and co ec ly i ed)
would dec ease he o e all ansmission a e o ∼0.2β.
The e ec o lockdown measu es is modelled he ein by he pa ame e ζ, Eq. (2): we se he a e lockdown ansmission a e
o be ζβ. This pa ame e inco po a es he e ec o all lockdown measu es, including he equi emen ha ace mask be wo n.
As he p e ious es ima e leads o a conside able dec ease in he ansmission a e, we decided o be conse a i e and we a ied
ζ om 1.0 o 0.5.
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These ea u es should be kep in mind, as we aim no only o cap u e he cu en end o he pandemic, bu also o sugges
mi iga ion s a egies ha may educe he cumula i e in ec ions, as well as he a ali ies as a esul o COVID-19 in he ime
se ies o in e es , namely in he case o Mexico. In wha ollows, we sugges possible scena ios o social dis ancing, and we
acco dingly e alua e hei impac as ega ds he po en ial educ ion induced in he numbe o dea hs and cumula i e in ec ions
o he ime se ies o in e es ha is ob ained om he DGE da a.
To explo e he impac o u he lockdown e ec s o mi iga e he sp ead o he in ec ion, we will conside he ollowing ime
dependence o he β’s:
βSI ( ) =βSI ηSI + (1 −ηSI )1− anh[2( − q)]
2
βSA( ) =βSA ηSA + (1 −ηSA)1− anh[2( − q)]
2+ηSA(ζ−1)1 + anh[2( − L)]
2(2)
wi h 0< ζ ≤1, as a gued, and q, Lp e iously speci ied. Fo ζ= 1, Eqs. (2) educe o he equa ions modelling he dec ease
in he numbe o pe sonal con ac s when a ligh o m o social dis ancing is imposed (no lockdown, no equi emen o wea
ace masks). Figu e 2 shows he ime-dependence o he β’s. The i s jump in he alue o βSI and βSA (i.e., he ansmission
a es o in ec ed and p esymp oma ic-asymp oma ic i us ca ie s) occu s a q. The nex jump occu s a Land is assumed o
ake place only o he p esymp oma ic o asymp oma ic ca ie s o he i us h ough he imposi ion o lockdown es ic ions a
= L.
As ou da a will show below, he g ow h o a ali ies as well as in ec ions in he coun y is conside able and hus ou model
sugges s he ele ance o he applica ion o signi ican lockdown es ic ions so ha he p og ess o he in ec ion be subs an ially
cu bed.
In he pa ame e s o he single-popula ion model, we assume he ollowing cons ain s:
•ηSI ≤1
•ηSA ≤1
•5≤1/σ1+ 1/σ2≤6
The i s wo a e a he i ial (and wi hou loss o gene ali y) implying ha we go om a alue o he ansmission a e β, o a
lowe alue β×ηin bo h he in e ac ions o S wi h A (o P) and in hose o S wi h I. The hi d cons ain is based on obse a ions
associa ed wi h SARS-CoV-2 [24] posi ioning he incuba ion pe iod associa ed wi h his i us be ween oughly 5and 6days
om he s a o exposu e.
FIG. 2. Single-popula ion model. T ansmission- a e ime dependence (βSI and βSA). Fo he asymp oma ic ca ie s o he i us he ans-
mission a e, which ini ially dec eases due o social dis ancing ( e lec ed in ηSA) is u he assumed o dec ease by a ac o ζ(shown o
di e en cases), e lec ing he e ec o obliga o y wea ing o (e ec i e) ace masks o o he lockdown-based dec ease o con ac s. Fo he
in ec ed popula ion ins ead, we assume ha βSI dec eases only once due o social dis ancing, (ηSI ), bu i does no u he dec ease, gi en he
sel -isola ion o such indi iduals due o he p esence o symp oms.
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B. Resul s
We now p esen he esul s de e mined by an op imiza ion p ocedu e ha iden i ies unde he abo e minimal cons ain s he
op imal pa ame e s o he model in compa ison o he ime se ies o da a ob ained om he Go e nmen o Mexico [28]. Table I
shows he op imal pa ame e s ound om minimizing he no m
N=X
iα1|log(Cnum( i)) −log(Cobs( i))|+α2|log(Dnum( i)) −log(Dobs( i))|(3)
wi h α1= 0,α2= 1 as displayed in Fig. 3. We will compa e hese esul s sho ly wi h he case o α1=α2= 0.5shown in
Fig. 4. The a ionaliza ion o hese wo choices is as ollows. In he o me case, we ake he iew ha he only “g ound u h”
da a is ha o he dea hs; in ac , e en hose can be unde -es ima ed (dea hs wi h “suspicion” o COVID-19, bu no de ini i e
es ) o –pe haps less likely– o e -es ima ed (dea hs a ibu ed o COVID-19 wi hou explici es ing), bu he e we will assume
ha his is he mos well-de ined piece o da a, as is gene ally expec ed o be he case. On he o he hand, in ec ions a e b oadly
expec ed o be unde - epo ed. This is because many o he cases wi h symp oms do no ge o be se ious enough o lead o
hospi aliza ion o o be epo ed. In ha ligh , i is expec ed ha assuming he dea hs as g ound u h, we should expec o
ind a signi ican o e -es ima ion o he numbe o in ec ions (we e u n o his poin below). I , on he o he hand, we ” o ce”
he model o ma ch he cu en epo ing o in ec ions, hen we will end up wi h a be e app oxima ion ”on a e age” o bo h
cu es bu wi h a po en ial ad e se by-p oduc in he esul ing numbe o dea hs ha we will discuss below. This will be due, in
addi ion, o issues o measu emen e o s in he se ing o cumula i e incidences, as discussed, e.g., in [29].
Ou obse a ion in Fig. 3 is ha he model can p o ide an excellen i o he o al numbe o dea hs, bu in ha case, he e
is a conside able o e -p edic ion o he numbe o in ec ions, p esumably because nume ous o he incu ed in ec ions a e no
epo ed in he o icial da a. The esul s can be compa ed wi h Fig. 4 whe e bo h dea hs and cumula i e in ec ions a e a emp ed
o be i ed, i.e., α1=α2in he minimiza ion p ocedu e abo e. He e, we see ha while he model is capable o doing a e y
adequa e job in ollowing C( ), i also does a easonable job o cap u ing D( ).Ne e heless, he e is a ca ea o he la e . A
close obse a ion o he semiloga i hmic scale o he g aph will lead he as u e eade o obse e ha as he model is ying o
juggle he op imiza ion o bo h ime-se ies, i sligh ly o e -p edic s dea hs ea ly on, sligh ly unde -p edic s hem a he middle
o he ime se ies and e en ually sligh ly o e p edic s again a longe imes, likely p edic ing a much mo e ca as ophic scena io
(wi h mul iple hund eds o housands o dea hs a he end o he examined e olu ion) han is wa an ed by he da a ends. Fo
his eason, we will s ick o he conside a ion o he o me case o Fig. 3 he ea e .
Be o e we discuss he implica ions o mi iga ion s a egies, le us b ie ly commen on he op imal pa ame e alues iden i ied
by he model, as illus a ed in Table I. The la en pe iod 1/σ1is indeed ound o be in he icini y o 3 days (2.8765), while
he o al incuba ion pe iod is com o ably wi hin he p esc ibed in e al o 5-6 days (1/σ1+ 1/σ2≈5.3836). The ime scale
o eco e y o asymp oma ics is a li le unde 7 days as expec ed (1/MAR = 6.0983), while he ime scale o going om
symp oma ic in ec ed o hospi aliza ion is close o 4 days which is also ai ly easonable (1/M = 3.6221). The model p edic s
a la ge ac ion o asymp oma ics (φ= 0.8134) which is in line wi h discussions such as ha o [10]. Howe e , such esul s
should be aken wi h a g ain o sal . This is because o issues associa ed wi h he no ion o iden i iabili y [30]. In pa icula , a
sys ema ic analysis o he model in connec ion o iden i iabili y ( he o mal ma hema ical de ails o which a e ou side he scope
o he p esen wo k) sugges ha φi sel will no end up being an iden i iable pa ame e , bu a he he p oduc o φwi h β’s
will be on such. As a esul , we do no signi ican ly ocus on he la ge alue ob ained o φ, bu we do no e i . In e es ingly,
he model p edic s ha oughly 30% o hose symp oma ically in ec ed need hospi aliza ion, while he es eco e . O hose
hospi alized, nea ly 30% esul in a ali ies, leading o a dea h pe cen age o abou 10% among hose ha p esen symp oms.
Indeed, his is a a he signi ican a ali y pe cen age accoun ing o he la ge numbe o dea hs in he popula ion. The a e age
ime scale o eco e y upon hospi aliza ion is abou 10.7days (1/χ), while ha o dea h (1/ψ) is ound o op imally be nea 11.5
days. Howe e , he e oo we should highligh ha ωand ψa e no independen ly iden i iable (only hei p oduc is) and nei he is
(1 −ω)and χ(again only hei p oduc is). Hence, he ele an op imiza ion pa ame e alues should be conside ed as coming
oge he wi h he co esponding ca ea s (and he ele an p oduc s as being he genuine model de i ed quan i ies), despi e hei
ealis ic indi idual alues. I is ele an o no e ha a he ini ial ime o he model, i is no known how many membe s o he
popula ion a e exposed, asymp oma ic o p esymp oma ic. We hus op imize hose pa ame e s oo, ob aining he las 3 en ies
o he Table.
Ou aim is now o explo e he impac o mi iga ing measu es educing he ob ained op imal case by a ac o o ζ < 1(ζ= 1 is
he cu en op imiza ion esul wi hou addi ional measu es). Table II summa izes he esul s o Fig. 5, in which he p edic ions
o ζ= 0.9,0.7,0.5a e displayed. I can be seen ha a dec ease o βby a ac o be ween 0.9and 0.5could ha e a ca aly ic
esul as conce ns he p edic ions o he model bo h o he dea hs and as ega ds he cumula i e in ec ions. In pa icula , a
dec ease o ζby a ac o o e en as li le as 0.9will educe he numbe o dea hs by mo e han 800, only wi hin he ime ame
be ween Augus 10 and Sep embe 10, while a dec ease by a ac o o 1/2is p edic ed wi hin his single popula ion model o
sa e close o 4000 li es in his in e al alone. The co esponding e ec o he numbe o in ec ions is pe haps e en mo e s iking
(also accoun ing o he ac ha many o hese in ec ions could lead o a ali ies a a la e s age). In pa icula , a dec ease o ζ
o 0.9leads o abou 24000 less in ec ions, while o a ac o o 1/2would lead o abou 100000 less in ec ions again jus wi hin
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TABLE I. Op imal pa ame e s o he single popula ion model. Fo a discussion he eo , see he ex .
Pa ame e Symbol Op imal alue Pa ame e Symbol Op imal alue
T ansmission a e [pe day] βSI 1.4789 In ec i i y pe iod (A) [days] τA
in 1/MAR 6.0983
T ansmission a e [pe day] βSA 0.1982 Con e sion ac ion (I o H, R) γ0.2991
Social dis ancing e ec ηSI 0.5806 Con e sion ac ion (H o R, D) ω0.3095
Social dis ancing e ec ηSA 0.4585 Reco e y pe iod(H o R) [days] 1/χ 10.6927
La en pe iod [days] τl1/σ12.8765 H o D pe iod [days] 1/ψ 11.5331
P eclinical pe iod [days] τp1/σ22.5071 Ini ial popula ion ac ion (E) E(0)/I(0) 2.0851
A/P pa i ioning φ0.8134 Ini ial popula ion ac ion (A) A(0)/I(0) 2.0532
In ec i i y pe iod (I) [days] τI
in 1/M 3.6221 Ini ial popula ion ac ion (P) P(0)/I(0) 0.5462
FIG. 3. Single-popula ion model. Numbe o cases C( )(le ) and o dea hs D( )( igh ) ound by minimizing no m (3) wi h α1= 0 and
α2= 1. To p oduce he i we ha e used he da a un il July 10. The p edic ion o he op imized model is shown by he solid line, while he
o icial ime se ies [28] is gi en by do s.
FIG. 4. As in he p e ious igu e, bu o α1=α2. The numbe o cases C( )is cap u ed signi ican ly be e , bu he lowe accu acy in
cap u ing D( )and i s implica ions a e u he discussed in he ex .
his 31-day pe iod. One can clea ly see he signi ican po en ial impac o u he lockdown measu es, a ea u e ha may be
wo hwhile o ac o in o u he public heal h conside a ions.
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TABLE II. Single-popula ion model. P edic ions o Cand Da Sep embe 10 i lockdown measu es had been applied on Augus 10.
ζ= 1 ζ= 0.9ζ= 0.7ζ= 0.5
C( o al) 1090578 1066436 1024199 988931
C( om Aug. 10) 292666 268524 226289 191022
D( o al) 83593 82744 81203 79848
D( om Aug. 10) 24362 23514 21972 20617
FIG. 5. Single-popula ion model. E olu ion o he numbe o cases (le ) and o dea hs ( igh ) o di e en ζi lockdown had been applied
on Augus 10 ( = 141). No ice he signi ican cu bing o he pandemic as a esul o such in e en ion measu es, especially so a he le el o
long e m e ec s in he case o ζ= 0.7and ζ= 0.5.
As ou inal commen abou he single popula ion a ian o he model, we no e ha an impo an quan i y in epidemiological
models o his gene al ype is he no ion o he basic ep oduc i e numbe R0; see, e.g., [3]. The numbe e ec i ely ep esen s
he expec ed new in ec ions (so-called seconda y in ec ions) om a single in ec ion in a popula ion whe e all subjec s a e
suscep ible. Using he so-called nex -gene a ion app oach [31], we can de e mine he basic ep oduc i e numbe on he basis o
he pa ame e s o he model and he suscep ible popula ion [see he Appendix o de ails], acco ding o he exp ession:
R0=βSA(1 −φ)
σ2
+βSAφ
MAR
+βSI (1 −φ)
MS∗,(4)
wi h S∗ he ini ial suscep ible popula ion, namely S∗= 1 as pa ame e s βa e no malized by N, as discussed abo e. Fo he
pa ame e s in Tab. I, one ob ains a alue o R0= 2.0755. The e ec i e ep oduc i e numbe a he beginning o social dis ancing
measu es ( = q) is Re= 1.5745 which is unable o mi iga e he pandemic e ec s. In o de ha Re<1, i is needed ha ζis
smalle han 0.8506. Tha is why among he case examples ha we conside ed, hose wi h ζ= 0.7and 0.5p esen a signi ican
dec ease in he numbe o dea hs no only imminen ly (i.e., wi hin he in e al o Augus 10 o Sep embe 10) bu also o e he
longe scale p edic ion o Fig. 3.
III. TWO-POPULATION MODEL
A. Equa ions
We now u n o he wo-popula ion a ian o he model. Recall ha due o he di e en s uc u al cha ac e is ics o he wo
popula ions as conside ed he ein, namely below and abo e 65 yea s, we expec ha his model will be mo e adequa e in cap u ing
bo h he dea hs and he cumula i e in ec ions o he ull popula ion. This is because on he one hand, he younge popula ion in
ou conside a ions is mo e ac i e (belonging ypically in he wo k o ce), hence ca ies a di e en numbe o con ac s. On he
o he hand, he olde popula ion has i s own ulne abili ies o he impac o he i us SARS-CoV-2 and he associa ed disease,
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namely COVID-19. On he o he hand, as explained in he In oduc ion, he p e alence o a ious isk ac o s wi hin he Mexican
popula ion [22] ende his pa i ion e en mo e ele an o conside owa ds cap u ing he de ailed da a ends.
Fi s ly, we discuss he ma hema ical s uc u e o he model. He e, we basically assume ha each o he popula ions has i s
own se o pa ame e s. Thus, he supe sc ip ywill deno e he pa ame e s associa ed wi h he younge popula ion, while he
supe sc ip owill be connec ed o he olde popula ion. Na u ally, he numbe o a iables (S, E, A, P, I, H, R, D)now doubles
wi h each pa ha ing a younge and an olde componen . I is wo hwhile ha o some quan i ies ha a e associa ed wi h
he i us, such as σ1and σ2, cha ac e izing he la en and incuba ion ime he eo , we assume hese o be independen o age.
Las ly, we explo e a mildly aniso opic a ian o he model whe e in e ms o he in e ac ions βo≡βoy =βyo =βoo and
βy≡βyy. Tha is o say, we assume ha he olde popula ion has a di e en in e ac ion wi hin i sel and wi h he younge
indi iduals, han he younge membe s o he popula ion be ween hemsel es [18]. This is a easonable assump ion unde he
p esen condi ions whe e he mo e sensi i e olde membe s o he popula ion a e ad ised o educe hei in e ac ions. While, in
p inciple, we could ha e used a ully aniso opic a ian o he model, we ind i ele an o a emp o educe he o e all numbe
o model pa ame e s, hence he abo e choice.
dSy
d =−βyy
SI ( )SyIy−βyy
SA( )Sy(Ay+Py)−βyo
SI ( )SyIo−βyo
SA( )Sy(Ao+Po)
dEy
d =−σ1Ey+βyy
SI ( )SyIy+βyy
SA( )Sy(Ay+Py) + βyo
SI ( )SyIo+βyo
SA( )Sy(Ao+Po)
dPy
d = (1 −φy)σ1Ey−σ2Py
dAy
d =φyσ1Ey−My
ARAy
dAy
R
d =My
ARAy
dIy
d =σ2Py−MyIy
dHy
d =γyMyIy−(1 −ωy)χyHy−ωyψyHy
dRy
d = (1 −γy)MyIy+ (1 −ωy)χyHy
dDy
d =ωyψyHy
dSo
d =−βoo
SI ( )SoIo−βoo
SA( )So(Ao+Po)−βoy
SI ( )SoIy−βoy
SA( )So(Ay+Py)
dEo
d =−σ1Eo+βoo
SI ( )SoIo+βoo
SA( )So(Ao+Po) + βoy
SI ( )SoIy+βoy
SA( )So(Ay+Py)
dPo
d = (1 −φo)σ1Eo−σ2Po
dAo
d =φoσ1Eo−Mo
ARAo
dAo
R
d =Mo
ARAo
dIo
d =σ2Po−MoIo
dHo
d =γoMoIo−(1 −ωo)χoHo−ωoψoHo
dRo
d = (1 −γo)MoIo+ (1 −ωo)χoHo
dDo
d =ωoψoHo
(5)
In a na u al ex ension o wha we assumed o he i ing o he single popula ion model, we assume ha as a esul o he
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chance o be hospi alized. In o he se ings whe e he e is hea y es ing in ol ed, asymp oma ics may be coun ed in he epo ed
in ec ions: while his is no an explici assump ion o he model, we did no coun any ac ion o asymp oma ics in he cumula-
i e in ec ions when p esen ing he ele an compa ison. Addi ionally, also, some o he hospi alized indi iduals may ansmi
he i us (e.g. o medical pe sonnel) despi e he much mo e subs an ial heal h and sa e y p o ocols applicable wi hin hospi als.
In any e en , we conside hese ea u es o be he excep ion a he han he ule and hence ha e excluded hem om ou mo e
mains eam conside a ions.
Wi hin he ealm o he model, we ha e exposed he meaning o he ele an pa ame e s (e.g., ansmission a es, la en and
incuba ion imes, ac ions o asymp oma ics s. p esymp oma ics, o hospi alized s. di ec ly eco e ed, and o eco e ed
s. dying indi iduals a he hospi al; also he ime scales o he la e pa i ions we e conside ed). I.e., we ha e a emp ed o
assign an epidemiological meaning o ou di e en pa ame e s and o examine he associa ed esul s o he op imal i o hese
pa ame e s o he da a om a speci ic ime se ies o illus a e he “ easonable” na u e o he indings. No ice ha in addi ion
o explaining he i ing p ocess ( o ei he dea hs o dea hs and cumula i e in ec ions), we ha e aken he app oach o using a
minimal numbe o assump ions o a oid cons aining he sys em o he deg ee possible. We ha e also illus a ed how he model
can be pa i ioned o di e en age g oups, based on he da a ha may be a ailable o he coun y o egion o in e es . He e, we
ha e op ed o conside he simples pa i ion o 2-age models, ye while edious, i is s uc u ally s aigh o wa d (and o some
in e es in i s own igh ) o gene alize conside a ions o many age g oup models.
As ou p o o ypical illus a ion o choice, we ha e used he case o da a om Mexico which ha e been a ailable h ough [28].
This is a case whe e a signi ican numbe o cases has a isen and he conside a ion o po en ial u he lockdown measu es is an
impo an opic o ongoing deba e. Indeed, ou indings sugges ha he p esen measu es appea no o be su icien o mi iga e
he ca as ophic consequences o he pandemic, since he cu en alue o he basic ep oduc i e numbe o he epidemic is
R0>1and hence he si ua ion appea s o need u he mi iga ion measu es and s a egies o a oid signi ican loss o li e. In ha
ein, we ha e discussed in he ealm o he model wha consequences di e en measu es would ha e a he le el o dea hs and
o cumula i e in ec ions. I was ound, e.g., ha a educ ion o ansmission a es by a ac o o a ound 1/2 ia social dis ancing
o ela ed measu es would lead o a non i ial cu bing o he ampan g ow h o bo h C( )and D( ). The ele an educ ions
could o he o de o 100000 in e ms o in ec ions and o mo e han 2000 in e ms o dea hs in he in e al o he nex 30 days
alone. While ou esul s a e only sugges i e (and ele an wi h he con ex o he model), we hope ha hey may be o some
alue owa ds policy conside a ions in he nea u u e.
Na u ally, he e a e nume ous di ec ions ha a e wo hwhile o conside owa ds ex ensions o he p esen wo k. A na u al
ea u e o many o he models (e.g., associa ed wi h he US [15], bu also elsewhe e) is he inco po a ion o unce ain y. We
a e cu en ly in he p ocess o buidling in o he o mula ion an unce ain y quan i ica ion amewo k on he basis o polynomial
chaos conside a ions [34] and he use o sui able dis ibu ions o quan i ies such as he ansmission a es. Cons uc ing a
obus such amewo k would be o conside able alue o models such as he one p oposed he ein. In addi ion, as indica ed
in nume ous cases, he e a e da a b oken down by age g oups (e.g., no only o Mexico, bu also o o he coun ies such as
Po ugal [35], e c.). Clea ly, a gene aliza ion o he model ha conside s he da a by decade would o e a mo e comple e and
sys ema ic pic u e o he impac o COVID-19 o di e en sub-popula ions and hence hei po en ial isk. This would o e , in
u n, a clea e pic u e o which age g oups o a emp o p o ec and would be wo hwhile (e en i somewha cumbe some).
Las ly, as di e en coun ies a e del ing in o a e-opening exe cise, he o mula ion o a me a-popula ion model wi h di e en
hubs and a quan i ica ion o he anspo coe icien s be ween hese [36] would be cen al owa ds going beyond he well-mixed
assump ion and ac o ing in a spa ial s uc u e and anspo a ion ea u es wi hin he model. Such di ec ions a e cu en ly unde
ac i e in es iga ion and will be epo ed in u u e wo k.
Appendix A: R0calcula ions
1. One-popula ion model
We use he nex gene a ion ma ix app oach o ind R0[31]. We de ine he ele an ec o s:
F=
















βSAS(A+P) + βSI SI
0
0
0
0
0
0
0
0
















,V=
















σ1E
−(1 −φ)σ1E+σ2P
−φσ1E+MARA
−σ2P+MI
−γMI + (1 −ω)χH +ωψH
βSAS(A+P) + βSI SI +µS
−MARA
−(1 −γ)MI −(1 −ω)χH
−ωψH
















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17
We hen ocus on he 5in ec ious/in ec ed compa men s (E,P,A,I,H) and igno e he es (S,AR,R,D). We ind he
Jacobians o F,Vwi h espec o E, P, A, I, H in he o de in which hey appea . This will yield wo 5×5ma ices:
F=






0βSAS∗βSAS∗βSI S∗0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0
0 0 0 0 0







(A1)
V=






σ10 0 0 0
−(1 −φ)σ1σ20 0 0
−φσ10MAR 0 0
0−σ20M0
0 0 0 −γM (1 −ω)χ+ωψ







(A2)
The basic ep oduc i e numbe is he spec al adius o FV −1which in ou case is
R0= (1 −φ)βSAS∗
σ2
+φβSAS∗
MAR
+ (1 −φ)βSI S∗
M.(A3)
The i s e m is due o he p esymp oma ic hos s P, he second due o he asymp oma ic hos s A, and he las one due o he
symp oma ic in ec ious hos s I. In each e m, he nume a o yields he a e o new in ec ions βSAS∗,βSI S∗and his is hen
mul iplied wi h he a e age du a ion o he s ay in ha in ec ious s age 1
σ2,1
MAR,1
M. Each e m is mul iplied wi h he ac ion
o hos s in ha s age/s a e φ o asymp oma ics and 1−φ o symp oma ics.
2. Two-popula ion model
Using again he nex gene a ion ma ix app oach esul s in he ollowing wo ma ices:
F=


















0βyy
SASyβyy
SASyβyy
SI Sy0 0 βyo
SASyβyo
SASyβyo
SI Sy0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0βoy
SASoβoy
SASoβoy
SI So0 0 βoo
SASoβoo
SASoβoo
SI So0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0
0 0 0 0 0 0 0 0 0 0


















(A4)
V=


















σ10 0 0 0 0 0 0 0 0
−(1 −φy)σ1σ20 0 0 0 0 0 0 0
−φyσ10My
AR 0 0 0 0 0 0 0
0−σ20My0 0 0 0 0 0
0 0 0 −γyMy(1 −ωy)χy+ωyψy0 0 0 0 0
0 0 0 0 0 σ10 0 0 0
0 0 0 0 0 −(1 −φo)σ1σ20 0 0
0 0 0 0 0 −φoσ10Mo
AR 0 0
0 0 0 0 0 0 −σ20Mo0
0 0 0 0 0 0 0 0 −γoMo(1 −ωo)χo+ωoψo


















.
(A5)
Since Vis a block ma ix i holds
V= A0
0D!⇒V−1= A−10
0D−1!,
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(which was no ce i ied by pee e iew) is he au ho / unde , who has g an ed medRxi a license o display he p ep in in pe pe ui y.
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18
om which is eadily ollows ha he spec al adius o FV −1is gi en by
R0=Ryy
0+Roo
0
2+p(Ryy
0−Roo
0)2+ 4Ryo
0Roy
0
2,whe e
Ryy
0= (1 −φy)βyy
SASy
σ2
+φyβyy
SASy
My
AR
+ (1 −φy)βyy
SI Sy
My
Roo
0= (1 −φo)βoo
SASo
σ2
+φoβoo
SASo
Mo
AR
+ (1 −φo)βoo
SI So
Mo
Ryo
0= (1 −φo)βyo
SASy
σ2
+φoβyo
SASy
Mo
AR
+ (1 −φo)βyo
SI Sy
Mo
Roy
0= (1 −φy)βoy
SASo
σ2
+φyβoy
SASo
My
AR
+ (1 −φy)βoy
SI So
My.
Ryy
0and Roo
0a e he basic ep oduc i e numbe s in he young and old age g oups, espec i ely, i hey we e comple ely isola ed
o each o he . Ryo
0is he basic ep oduc i e numbe i suscep ible young hos s come in o con ac wi h only old in ec ious hos s,
and Roy
0is he basic ep oduc i e numbe i old hos s come in o con ac wi h only young in ec ious hos s.
Acknowledgmen s This ma e ial is based upon wo k suppo ed by he US Na ional Science Founda ion unde G an s No.
DMS-1815764 (ZR), PHY-1602994, and DMS-1809074 (PGK). PGK also acknowledges suppo om he Le e hulme T us
ia a Visi ing Fellowship and hanks he Ma hema ical Ins i u e o he Uni e si y o Ox o d o i s hospi ali y du ing his wo k.
Disclaime The iews exp essed in his manusc ip a e pu ely hose o he au ho s and may no , unde any ci cums ances, be
ega ded as an o icial posi ion o he Eu opean Commission.
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(which was no ce i ied by pee e iew) is he au ho / unde , who has g an ed medRxi a license o display he p ep in in pe pe ui y.
The copy igh holde o his p ep in his e sion pos ed Augus 14, 2020. .h ps://doi.o g/10.1101/2020.08.11.20172833doi: medRxi p ep in