Resul s in Physics 40 (2022) 105855
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Resul s in Physics
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Modi ied SIR model o COVID-19 ansmission dynamics: Simula ion wi h
case s udy o UK, US and India
P ana i Rakshi a, Soumen Kuma b, Samad Noeiaghdam c,d, Unai Fe nandez-Gamiz e,∗,
Mohamed Al anji , Shyam Sunda San a g,h
aDepa men o Compu e Science and Enginee ing, JIS College o Enginee ing, Kalyani, Wes Bengal, India
bAssocia e Consul an and Da a Scien is in Ta a Consul ancy Se ices L d., Kolka a, Wes Bengal, India
cIndus ial Ma hema ics Labo a o y, Baikal School o BRICS, I ku sk Na ional Resea ch Technical Uni e si y, I ku sk, 664074, Russia
dDepa men o Applied Ma hema ics and P og amming, Sou h U al S a e Uni e si y, Lenin p ospec 76, Chelyabinsk, 454080, Russia
eNuclea Enginee ing and Fluid Mechanics Depa men , Uni e si y o he Basque Coun y UPV/EHU, Nie es Cano 12, 01006 Vi o ia-Gas eiz, Spain
Depa men o Ma hema ics, College o Science, King Khalid Uni e si y, Abha 61413, Saudi A abia
gDepa men o Ma hema ics, JIS College o Enginee ing, Kalyani, Wes Bengal 741235, India
hDepa men o Ma hema ics, Applied Science Clus e , Uni e si y o Pe oleum and Ene gy S udies (UPES), Deh adun, U a akhand 248007, India
ARTICLE INFO
Keywo ds:
COVID-19
SIR model
P edic ion
Asymp oma ic
R-Squa e sco e
ABSTRACT
Co ona i us disease 2019 (COVID-19) is an in ec ious disease and has sp ead o e mo e han 200 coun ies
since i s ou b eak in Decembe 2019. This pandemic has posed he g ea es h ea o global public heal h
and seems o ha e changing cha ac e is ics wi h al e ing a ian s, hence a ious epidemiological and s a-
is ical models a e ge ing de eloped o p edic he in ec ion sp ead, mo ali y a e and calib a ing a ious
impac ing ac o s. Bu he aysmp oma ic pa ien coun s and demog aphical ac o s needs o be conside ed
in model e alua ion. He e we ha e p oposed a new se en compa men al model, Suscep ible- Exposed-
In ec ed–Asymp oma ic–Qua an ined–Fa al–Reco e ed (SEIAQFR) which is based on classical Suscep ible-
In ec ed-Reco e ed (SIR) model dynamic o in ec ious disease, and conside ed ac o s like asymp oma ic
ansmission and qua an ine o pa ien s. We ha e aken UK, US and India as a case s udy o model e alua ion
pu pose. In ou analysis, i is ound ha he Rep oduc i e Ra e (𝑅0) o he disease is dynamic o e a long
pe iod and p o ides be e esul s in model pe o mance (>0.98 R-squa e sco e) when model is i ed ac oss
smalle ime pe iod. On an a e age 40%−50% cases a e asymp oma ic and ha e con ibu ed o model accu acy.
The model is employed o show accu acy in co espondence wi h di e en geog aphic da a in bo h wa e o
disease sp ead. Di e en disease sp eading ac o s like in ec ion a e, eco e y a e and mo ali y a e a e well
analyzed wi h bes i o eal wo ld da a. Pe o mance e alua ion o his model has achie ed good R-Squa e
sco e which is 0.95 − 0.99 o in ec ion p edic ion and 0.90 − 0.99 o dea h p edic ion and an a e age 1% − 5%
MAPE in di e en wa e o he disease in UK, US and India.
In oduc ion
Vi us causes in ec ions which la e u ns in o disease in human &
animal body and a e he e o e ha e an impac on medical, social, and
economical li e. No el Co ona i us (COVID-19) caused an ou b eak
o a ypical pneumonia i s in Wuhan, China in Decembe 2019 [1]
and hen ex ended i s deadly cha ac e is ics h oughou he whole
wo ld in ec ing mo e han 210 million popula ion wo ldwide aking
a dea h oll o 4.47 million [2] ill da e as o 27 h Aug 2021. A e
assessing i is amazing sp eading powe and signi ican ha m owa ds
human ci iliza ion, he Wo ld Heal h O ganiza ion (WHO) announced
∗Co esponding au ho .
E-mail add esses: [email p o ec ed] (P. Rakshi ), [email p o ec ed] (S. Kuma ), [email p o ec ed],[email p o ec ed] (S. Noeiaghdam),
[email p o ec ed] (U. Fe nandez-Gamiz), [email p o ec ed] (M. Al anji), [email p o ec ed],[email p o ec ed] (S.S. San a).
his new pneumonia ou b eak a ‘‘global pandemic’’ [3]. Though e e
wi h coughing and espi a o y p oblems a e he main symp oms o
COVID-19 which esemble he p ima y indica ions o simple lu, bu
la e complexi y a ises esul ing e y low oxygen sa u a ion in body.
P ima ily i was app ehended ha Co ona i uses a e ansmi ed in
wo modes: d ople s du ing coughing o sneezing and con ac ans-
mission (con amina ed hands). Disin ec an s like hyd ogen pe oxide
and sodium hypochlo i e, e c. [4] can des oy his i us. I is di icul
o ind ou he sp eading mechanism o COVID-19 because i can be
silen ly ca ied by and ansmi ed om people wi hou showing any
h ps://doi.o g/10.1016/j. inp.2022.105855
Recei ed 1 Ap il 2022; Recei ed in e ised o m 22 July 2022; Accep ed 22 July 2022
Resul s in Physics 40 (2022) 105855
2
P. Rakshi e al.
symp oms [5], bu his is c i ical o p e en he pandemic ou b eak.
The sp ead o he disease is igh ening and a h ea o human ci iliza-
ion. Though he adminis a ion o accina ion s a ed ac oss se e al
coun ies, WHO s ill ecommends a guideline like social dis ancing,
sani iza ion o hands, usage o ace masks o a oid i is successi e
wa es. Go e nmen o each coun y has also aken a ious s a egies
like comple e lockdown, banning in e na ional physical anspo , and
qua an ining he suspec ed pe sons o con ol he epidemic p og ession
a i s ea ly s ages. As an example, coun ies such as Sou h Ko ea [6,7]
quickly s a ed o ace con ac s o in ec ed pe sons, hence iden i ying
he epicen e o he ou b eak, and p e en ing huge impac .
Compa men al models which di ide he o al human popula ion
in o classes acco ding o he s a e o he disease a e he mos popula
ma hema ical models [8]. Among hem, he classic SEIR model is he
mos widely adop ed o pandemic disease and also cha ac e izes he
COVID-19 ou b eak [9]. In simple SEIR model, he numbe o people in
di e en ca ego y like in ec ed o eco e ed o dead a e measu ed on
he basis o di e en pa ame e s o disease dynamics like he numbe
o con ac s pe uni ime, he ansmission a e o he disease, he ime
pe iod o disease incuba ion, a e o eco e y o he sick pe son o be
cu ed and he a e o dea h.
We know ha due o he lack o p ope de ec ion echnique, he
asymp oma ic pa ien s who ha e mild o no symp om, a e no consid-
e ed in he con i med case coun , and his is he un ouched a ea in basic
SEIR model. To es ima e he un aceable con ac s some me hods ha e
been discussed in [10]. In his wo k, we aim o conside he "Asymp-
oma ic" class as a i al ac o in epidemic olume de e mina ion and
has in oduced new compa men in exis ing model. I he in ec ed
iden i ied case is conside ed he ip o an icebe g, he e is a la ge
hidden popula ion ca ying he i us wi hou any symp oms due o lack
o public heal h su eillance and e ec i e es ing. We ha e measu ed
he dynamic na u e o he di e en key epidemic pa ame e s which
con ibu e o he sp ead o COVID-19, in a ela i ely e ec i e way
in di e en pe iods o disease ou b eak ac oss mul iple geog aphical
a eas. The e ec i eness o la ency pe iod, in ec i i y a e, eco e y a e
and se e al ac o s like human immuni y, social awa eness had been
conside ed while calib a ing he model in di e en phases o ou b eak
in India, UK and US.
In Sec ion "Backg ound" we ha e discussed he backg ound o his
p esen wo k, Sec ion ‘‘Ma e ial and me hods’’ desc ibes abou he
ma e ial and me hod in which he p oposed model, model pa ame e
e c. a e discussed, he esul s, ou come analysis ha e been discussed
in Sec ion "Resul s and Discussion" while Sec ion "Fu u e Wo k" deals
wi h he u u e scope, conclusion.
Backg ound
In di e en coun ies a ious ma hema ical modeling has been p o-
posed and de eloped o unde s and he dynamics o Co id 19 sp ead.
The epidemiological beha io o his i us can be e alua ed as a
complex pa e ns/sys em h ough ma hema ical models. This s a is ical
da a ega ding he disease sp ead can be analyzed and in e p e ed o a
model e olu ion which will help o p edic he disease cha ac e is ics,
o impose con ainmen , o con ol and mi iga e p obable ou b eak. The
al eady known pa ame e s abou he i us sp ead like he numbe o
in ec ed coun , he numbe o eco e ed people and he dea h coun
can be ed in o he model o e ine and calib a e i mo e accu a ely.
The dynamic equa ions o ma hema ical models can p o ide powe ul
insigh s in o he dynamics. These me hods hold good o he disease
whe e a la ge numbe o unknown, unce ain ac o s con ibu e o he
disease g ow h.
Rela ed wo ks
Ba is a e al. [11] p oposed suscep ible in ec ious eco e ed (SIR)
model which ied o o ecas he numbe o Co ona i us pandemic
cases and inal size, which seemed o be e y accu a e conside ing a
he mon h o Feb ua y 2020 da a, bu he cu en end has p o ed
he p edic ion no so e ec i e. Huang Y e al. [12] u ilized he SIR
model equa ions and es ima ed he u u e in ec ed coun due o Co ona
i us in Japan, Sou h Ko ea, I aly, and I an. Incuba ion pe iod has no
been conside ed while cons uc ing his model. Fo ios e al. [13] has
adop ed echniques which ollow simple ime se ies o ecas ing. They
p edic s he u u e wi h non con e gence g ow h. Leona do e al. [14],
conside ed he la en pe iod in a compa men al model o SEIR whe e
he in ec ion sp eads wi h changeable con ainmen s. The exponen-
ial cu e da a ep esen s a good p edic ion in dea hs and eco e ed
coun . Ca cione e al. [15] has used he modi ied SEIR model and
a ied di e en ini ial pa ame e s alues o calib a e he model. While
analyzing he esul s, hei model is e ined wi h numbe o dead
indi iduals. They ound a ange o incuba ion pe iod and Rep oduc ion
numbe o i he eal-wo ld da a and calcula e he in ec ion a ali y
a e (IFR). FaïçalNdaï ou e al. [16] has p oposed a new compa men al
model conside ing supe –sp eade s. Mo eo e , hey conside a dea h
compa men . They ha e men ioned he "Asymp oma ic" class in hei
model. They ha e done a good sensi i i y analysis o hei model and
nume ical simula ion wi h he case s udy o Wuhan. Eugene B. Pos -
niko e al. [17] p opose a simple SIR model wi h p edic i e es ima ion
conside ing co ela ion o epidemic dynamics. Peng e al. [18] has
in oduced qua an ined s a e o p opose a new SEIR model and also
done a sensi i i y analysis. Mukesh Jakha e al. [19] has made a s a e-
wise analysis o India. They ha e made a s udy and o ecas based on
he demog aphical di e si y. Also, he phase wise Lockdown popula ion
densi y is conside ed while p edic ing he g ow h o he epidemic. A
de ailed analysis and compa a i e s udy o he di e en o ecas ing
me hodology has been ound in [20]. Subhas Khajanchi e al. [21] has
in oduced ma hema ical modeling conside ing di e en Indian s a es
and hey calcula ed R0 alue and o he sensi i e pa ame e s using he
pa ial ank co ela ion coe icien s echniques.
Mo i a ion o he esea ch
Though all he abo e wo ks h ow ligh on he espec i e a eas
du ing analysis, bu in mos o he cases he pa ame e s es ima ion
and p edic ion is la gely based on he s a is ics o con i med cases,
whe eas we el ha in a de eloping coun y like India wi h i s a ie y
o popula ion, he e a e la ge numbe o unde ec ed cases which will
con ibu e o u u e sp ead signi ican ly. Also, we ha e ep esen ed
he model wi h UK and US cases o asce ain he accep abili y ac oss
a ious geog aphical da a. In his s udy, we ha e analyzed he numbe
o asymp oma ic cases and un epo ed in ec ious cases as impac ing
ac o s, whe eas he numbe o epo ed in ec ious cases and dea h
coun s a e conside ed o i he model pa ame e wi h he eal-wo ld
s a is ical da a. While all he o he p e ious esea che s ha e ocused
on a pa icula Geog aphical a ea wi h a speci ic ime span o hei
model calib a ion and e alua ion, we ha e simula ed he model beha -
io wi h espec o di e en demog aphic da a like UK, US and India
in bo h i s wa e and second wa e o he disease peak wi h a ying
pa ame e alues.
Ma e ial and me hods
A i s , we will b ie ly discuss he p ope ies o basic Suscep ible-
Exposed-In ec ed-Remo ed (SEIR) sys em wi h ou s a es o he dis-
ease. Suscep ible a e heal hy indi iduals bu ha e a chance and ul-
ne able o ge in ec ed. Exposed a e indi iduals who ha e had con ac
wi h an in ec ed pe son bu no ye conside ed as in ec ed. In ec ed a e
Resul s in Physics 40 (2022) 105855
3
P. Rakshi e al.
Fig. 1. P oposed SEIAQFR model wi h se en compa men s/classes.
indi iduals who es ed posi i e o in ec ion and eco e ed is indi id-
uals who a e decla ed cu ed. We ha e employed model analysis wi h
exis ing da a o di e en coun ies by a ying he pa ame e s and ini ial
assump ions and conside ed he epo ed a ali y numbe and in ec ed
coun as e e ence while es ablishing he model dynamics. He e we
ha e emphasized on cu en and p edic i e end o he ou b eak
wi h eal wo ld da a along and hence he model calib a ion becomes
complex and lexible o accommoda e impac ing ea u es.
P oposed model (SEIAQFR)
Now, le us desc ibe ou p oposed model which conside s 7-
compa men /classes o he disease and he elabo a ion o s a e ans-
o ma ion is depic ed in Fig. 7. Suscep ible (S): They a e he indi iduals
o he o al popula ions (N) who a e exposed o he disease sp ead.
Exposed (E): The indi iduals ha ha e been exposed o came in o
con ac o In ec i e pe sons. A his s age pe son is in ec ed bu no
ye de eloped any symp oms, hence no iden i ied as in ec ious o
sp ead he disease. Disease is in a la en s age. People goes om S o E
depending on he numbe o con ac s wi h In ec ed (I) indi iduals and
he a e o in ec ion sp eading.
In ec ed (I): The indi iduals hose a e e ealing symp oms o Co id-
19, iden i ied and symp oma ically in ec ed. They a e also in ec i e, so
hey can sp ead he disease(see Fig. 1).
Asymp oma ic (A): These g oups a e in ec ed and in ec i e, hey
may ac as a hidden ca ie o he disease, bu no iden i ied as hey
a e no showing any disease symp oms and due o lack o apid es s
conduc ed in mass, hey a e no es ed posi i e.
Qua an ined (Q): When in ec ed people a e con i med wi h Co id-
19 a e p ope medical es and ei he hospi alized o home qua an-
ined.
Reco e ed(R): Pe sons eco e ed om epidemic disease, so asymp-
oma ic indi iduals may ge au oma ically eco e ed wi hou hei
disease de ec ion & ea men , whe eas Qua an ined pe sons ge well
a e p ope ea men .
Fa al (F): This g oup o people ailed o su i e a e COVID-19
diagnosis.
Model pa ame e s and s a e ansi ions
Di e en model pa ame e s con ibu ing o his model is desc ibed
below:
𝛽: This deno es a e age con ac equency i.e., Numbe o people an
in ec ed pe son can in ec /day. This con ols he a e o sp ead, which
ep esen s he p obabili y o ansmi ing disease be ween a suscep ible
and in ec ious indi idual. In o he wo ds, gi en a popula ion N, among
hem ini ially suscep ible is S and I is in ec ed people, hen in ec ed
people s a impac ing suscep ible popula ion a a a e o 𝛽 o make
hem exposed COVID-19 i us.
𝛼: Exposed (E) becomes in ec ious a he a e o 𝛼i.e. (1
𝛼)is he a e -
age ime equi ed o an exposed indi idual be ween his ini ial con ac
o an in ec ious pe son and showing he symp oms o COVID-19. The e-
o e (1
𝛼)o 𝛼−is he incuba ion pe iod o he disease. Assuming ha
among Exposed (E) pe sons, ac ion p de elops symp oms and become
symp oma ically in ec ed (I) whe eas (1-p) emains Asymp oma ic (A)
a e he incuba ion pe iod.
𝛾: The symp oma ic(I) a e mo ed o Qua an ined(Q) s a e ei he in
hospi al o a home a a a e o 𝛾so (1
𝛾o 𝛾−) is he in ec ious pe iod
du ing which a symp oma ic in ec ed pe son can sp ead in ec ion. And
ob iously, a e being qua an ined he is isola ed and canno sp ead
in ec ion. We assume ha all iden i ied symp oma ic pe sons will be
qua an ined, hence p obabili y is 1.
𝜑: The asymp oma ic people(A) also con ibu e o Symp oma ic
In ec ed (I) g oup by de eloping symp oms a e (1/𝜑) days i.e., a a
a e o 𝜑, o he wise hey ge au oma ically eco e ed om in ec ion
wi hou showing any symp oms and hence wi hou unde going any
ea men . The asymp oma ic (A) a e emo ed a a a e 𝜑, a ac ion
𝑞o hem going in au oma ic emission and ac ion (1 − 𝑞)becomes
symp oma ic (I).
𝜃: This deno es Reco e y Ra e a e ea men i.e., he Qua an ined
people(Q) ge well a a a e o 𝜃. i.e. (1/𝜃o 𝜃−) is he a e age
qua an ined pe iod in hospi al/home o he eco e ed pe son du ing
ea men .
𝛿: This deno es Mo ali y/Fa ali y Ra e. Un o una ely, qua an ined
pe sons (Q) ge deceased a 𝛿 a e wi h p obabili y 𝑥, hen (1/𝛿o 𝛿−)
is he a e age days spen by any pa ien be o e dea h since ea men
has been s a ed and he is qua an ined. So, qua an ined pe sons (Q)
ge eco e ed a a p obabili y o (1 − 𝑥). Basic Rep oduc ion Numbe
(𝑅) = 𝛽
𝛾de e mines he se e i y o sp ead a any ime and a key ac o
in his model. We ha e p oposed he i us ansmission by he ollowing
nonlinea o dina y di e en ial equa ions (ODE) as SEIAQFR model:
𝑑𝑆
𝑑𝑡 = − 𝛽𝑆(𝑡)𝐼(𝑡)
𝑁,
𝑑𝐸
𝑑𝑡 =𝛽𝑆(𝑡)𝐼(𝑡)
𝑁−𝛼𝐸(𝑡),
𝑑𝐴
𝑑𝑡 =𝛼(1 − 𝑝)𝐸(𝑡) − 𝛷𝐴(𝑡),
𝑑𝐼
𝑑𝑡 =𝛼𝑝𝐸(𝑡) + 𝛷(1 − 𝑞)𝐴(𝑡) − 𝛾𝐼(𝑡),
𝑑𝑄
𝑑𝑡 =𝛾𝐼(𝑡) − 𝜃(1 − 𝑥)𝑄(𝑡) − 𝛿𝑥𝑄(𝑡),
𝑑𝑅
𝑑𝑡 =𝛷𝑞𝐴(𝑡) + 𝜃(1 − 𝑥)𝑄(𝑡),
𝑑𝐹
𝑑𝑡 =𝛿𝑥𝑄(𝑡).
In his s udy, ou model is used wi h he ollowing assump ions:
Resul s in Physics 40 (2022) 105855
4
P. Rakshi e al.
•Cons an popula ion numbe s (an equal numbe o dea hs bal-
ances he numbe o bi hs)
•Homogeneous popula ion which means each indi idual has he
same oppo uni y o make con ac wi h o he indi iduals.
•The sp ead o he co ona i us only occu s be ween human o
human.
•Indi iduals a ec ed by COVID-19 can ei he eco e o dies.
•Indi iduals who ha e eco e ed canno be in ec ed again.
•In ec ion can only sp ead when he pe son is in in ec ious s age,
all o he possibili ies like Qua an ined pa ien in Hospi al o home
canno sp ead in ec ion, so hey can ei he eco e o die.
In his s udy, we will analyze he numbe o asymp oma ic cases
and un epo ed in ec ious cases, as well as he numbe o epo ed
in ec ious cases o i he model pa ame e wi h he eal-wo ld da a.
The model building and simula ion has been de eloped in py hon.
Pa ame e es ima ion
Acco ding o p esen scena io, he numbe o Dea hs (F) and he
numbe o Reco e ed (R) om Qua an ined ( om hospi alized o
home) and he es s posi i e (In ec ed-I) cases a e a ailable in Go e n-
men published s a is ics. Since he alues o he abo e pa ame e s a y
on se e al ime dependen ac o s, we ha e ca e ully es ima ed hem
acco ding o he demog aphic ea u es, pe iods o in ec ion sp ead,
socio-economic s uc u e and public heal h. The ini ial alues o In-
ec ed, Reco e ed and Dead (𝐼0, 𝑅0, 𝐹0) a e a ailable om he da ase o
espec i e popula ion, whe eas he (𝐸0, 𝑄0, 𝐴0) a e calcula ed acco d-
ing o he ini ial sp ead o disease in ha egion and also dependen
on he ime e e ence o he model. In his op imiza ion p oblem o
de e mining he co ec coe icien alues o all pa ame e s a e e y
c i ical, ei he o e i ing o unde es ima ion happen due e y small
a ia ion o ime dependen model dynamics. To ix his, we i s ly
de e mine some pa ame e alues om he p e ious analysis o disease
dynamics and hen i o he s o achie e he bes possible esul s. Du ing
he analysis o di e en da a o he li e a u es [15,19,21] we decided
o simula e ou model wi h da a o India du ing i s i s and second
su ge, as well as Coun y wi h less huge popula ion like US and UK
whe e he e was an e iden su ge o Co id-19 wi h high in ec ion
a e . Though he e a e a lo o o he ac o s like Lockdown pe iod,
social dis ancing awa eness, p e en i e measu es o ace mask and
hand washing, and he UnLock-1 ha e impac s on he ime se ies
da a o i he model, bu he analysis om medical bulle in [22] and
discussion o heal h p o essionals pu he ac ual pic u e o he disease
sp ead. Incuba ion Pe iod (𝛼−) and In ec ious pe iod (𝛾−) a e wo
impo an pa ame e s o de e mine he disease sp eading and hey a y
in di e en coun y in di e en ime phase o hei pandemic ou b eak.
No mally he e is a linea p opo ional ela ionship i.e., 𝛼−∝𝛼−. Again,
when he se e i y o disease sp eading is as e i.e., he 𝑅0is inc easing
o highe alue, and hen 𝛼−and 𝛾− end o he lowe side which helps
he disease o sp ead mo e sha ply. The 𝛼−has gained a ange o 5–20
days depending upon he locali y and in ec ion peak, so on a e age,
i akes 10–15 days o an in ec ious indi idual o ge symp oms and
become (I) om (E) s a e.
Now acco ding o he da a om medical esea ch all o e he
wo ld, he COVID-19 i us has a ypical con agious pe iod o a ound
5–7 days a e in ec ing a pe son, so 𝛾−is conside ed. This 𝛾− alue
holds good o US, whe eas UK ha e a ange o 11–20 and India
ha e 5–16 days depending upon he in ec ing s ain o he disease and
he con amina ion eloci y. These pa ame e s (𝛼−and 𝛾−) a e ge ing
changed wi h p opo ional o e e y changing i us cha ac e is ic and
impo an se e i y de e minan . The o he pa ame e s like 𝛿−( he a e -
age day spen be o e dea h), 𝜑−( he a e age day o ge eco e ed om
asymp oma ic condi ion) and 𝜃−( he a e age day o ge eco e ed om
a e p ope ea men ) a e dependen on he pa ien ’s immuni y powe
and medical acili y o ha geog aphy. So, in ou model op imiza ion,
Table 1
S a is ics o SEIAQFR model pa ame e alues o bes i o cu en ac ual da a o US,
UK and India in di e en phases o disease sp ead.
Coun y Span 𝛾−𝛼−𝑅 𝛿−𝛷−𝜃−𝑝 𝑞 𝑥
UK Sep2020–Feb2021 20 12.5 2.7 18 16 12 0.6 0.6 0.009
UK Sep2020–Dec2020 22 21 2.9 18 16 12 0.6 0.55 0.0095
UK Dec2020–Feb2021 11.6 14 3 18 14 12 0.75 0.74 0.007
US Ma ch2020–May2020 7 6 2.8 14 10 14 0.68 0.65 0.032
US Ma ch20–Ap il2020 5 5 3.45 12 10 10 0.77 0.65 0.045
US Ap il2020–May2020 7 7 3.45 12 10 10 0.77 0.65 0.016
India Ma ch 2021 12 11.5 2.59 14 9.35 14 0.56 0.61 0.0003
India Ap il 2021 11.5 12 3.65 12 9 14 0.56 0.6 0.0045
India May 2021 16 10 3.4 10 9.2 14 0.56 0.61 0.0034
India May2020–Sep2020 4.5 5 1.76 18 15 10 0.75 0.8 0.004
India Ma ch2021–May2021 11.5 13.5 2.6 14 14 12 0.52 0.57 0.0015
hese h ee pa ame e s alues a e kep almos cons an h ough he
analysis pe iod o Peak o O -Peak pe iod o a speci ic egion, e -
e enced in Table 1. While i ing he model, obse a ion pe iod o
UK, US and India is conside ed a he peak ou b eak o he disease
a ha Geog aphy: Sep embe 2020–Feb ua y 2021 i.e. 180 days o
UK, Ma ch 2020–May 2020 i.e., 90 days o US and o India we ha e
conside ed wo pe iods May 2020–Sep 2020(Co ona i s wa e), Ma ch
2021–May 2021(Co ona second wa e). Also, hese spans a e di ided
in o sho e ime span o i he model wi h mo e accu acy and a ying
pa ame e alues. Fo Ini ial alues o ODE equa ions: 𝑆0is he o al
popula ion, whe eas 𝐼0, 𝑅0and 𝐹0a e aken om he ac ual da a [23]
and 𝐸0, 𝐴0and 𝑄0is assumed o be in ini ial alues o ha egion
and aken as a ixed pe cen age o o al popula ion acco ding o he
popula ion densi y. While i ing he cu e in smalle span o du ing he
Co id-19 s wa e, he ini ial pa ame e s a e conside ed as co esponding
da ase synch oniza ion wi h ou come o he p e ious ime span in ou
Model e.g., he May mon h Cu e i ing s a s wi h he da a alues
gene a ed as a esul o Ap il mon h cu e i .
While i ing he model, se e al simula ions esul he alue o 𝑝, 𝑞
and R, hence 𝛽. The 𝑥is a ying o di e en coun ies which esembles
o he ac ual da a o his coun y. The ep oduc ion a e (R) plays a
i al ole in eaching he peak, while he la ge alue e lec s a sha p
ise and pandemic ou b eak, bu he smalle alue ep esen s he slow
sp ead, bu o a longe du a ion. Fo India, ou analysis pe iod co e s
he Lock down pe iod in a con olled en i onmen and as well as he
Unlock o Phase 1 na ionwide. Du ing he ini ial pe iod o Lockdown
and he Vi us sp ead, as he medical diagnosis and es s esul s we e
doub ul in India whe eas o US and UK, we we e con iden abou he
epo ed ou come, so pe iods we e chosen acco dingly. Also he second
wa e is conside ed while p edic ing he India da a om Ma ch 2021
o May 2021. Also, we ha e aken a longe pe iod o i ou model
and compa e he esul wi h ac ual da a as compa ed o o he a ailable
model unde esea ch. We ha e plo ed he cu es o i ou model wi h
he ac ual dea h and in ec ed numbe as pe he s a is ics [22] and he
o he cu es a e de i ed.
Resul s and discussion
Conside ing he exis ing da a o he In ec ed and Dea h [23], he
ollowing Table 1 and co esponding cu es (Figs. 2–7) a e ob ained o
US, UK and India. As men ioned abo e, he model has been calib a ed
on he basis o he o al numbe o in ec ed and casual ies o a ce ain
ime pe iod. In his model, Rep oduc ion Ra e (R) ac s as h eshold o
de e mine whe he he in ec ion disease sp ead is coming ou as an
epidemic o going o die down. Theo e ically, a ini ial he in ec ed
coun inc eases exponen ially and a a peak he a e o change o
in ec ed popula ion is ze o i.e., 𝑑𝐼(𝑡)
𝑑𝑡 = 0 and he cu e s a declining,
hus when 𝑡→∞, S and I →∞; hence he p edic ion o 𝑡is impo an .
Bu he hidden disease sp ead by asymp oma ic ca ie (A) and sudden
su ge o in ec ion cause he inc ease in R alue and hus impac ing he
cu e o disease ise and die down. Mo eo e , he Vaccina ion p ocess
has also accele a ed he disease con olling imeline.
Resul s in Physics 40 (2022) 105855
5
P. Rakshi e al.
Fig. 2. UK: Ac ual and Model p edic ed Cu e alue o In ec ed and Dea h coun in di e en ime phases o disease (a) In ec ed Coun : pe iod: Sep 2020–Feb 2021 (b) Dea h
Coun : pe iod: Sep 2020–Feb 2021 (c) In ec ed Coun : pe iod: Sep 2020–Dec 2020 (d) Dea h Coun : pe iod: Sep 2020–Dec 2020 (e) In ec ed Coun : pe iod: Dec 2020–Feb 2021
( ) Dea h Coun : pe iod: Dec 2020–Feb 2021. Red line: P edic id alue, Blue line: Ac ual alue. (Fo in e p e a ion o he e e ences o colo in his igu e legend, he eade is
e e ed o he web e sion o his a icle.)
Fig. 3. UK: (a) Model SEIAQFR p edic ed pa ame e s ajec o ies du ing peak pe iod Sep 2020 - Feb 2021 (b) Model SEIAQFR P edic ed Pa ame e s alue o pandemic’s p edic ed
li espan.
In e p e a ion o exis ing da a
Wi h he ollowing pa ame e s, we ha e implemen ed he p oposed
model using py hon. He e we ha e analyzed on he da a o UK, US
and India wi h hei esponse o COVID 19. As men ioned ea lie ,
he ac ual in ec ed and dea h numbe a e compa ed wi h he model
p edic ed numbe wi h a ying pa ame e alues. Di e en coun ies
ac ual Dea h and In ec ed coun is compa ed wi h he model p edic ion
o phases o he disease ou b eak.
UK: The o al Popula ion o his is 68 000 000 and he ac ual
da a conside ed o i wi h ou model is he peak span o he disease
sp ead Sep embe 2020–Feb ua y 2021 (160 days). Fig. 2.(a) and (b)
Resul s in Physics 40 (2022) 105855
6
P. Rakshi e al.
Fig. 4. US Ac ual and Model p edic ed Cu e alue o In ec ed and Dea h coun in di e en ime phases o disease (a) In ec ed Coun : pe iod: Ma 2020–May 2020 (b) Dea h
Coun : pe iod: Ma 2020–May 2020 (c) In ec ed Coun : pe iod: Ma 2020–Ap 2020 (d) Dea h Coun : pe iod: Ma 2020–Ap 2020 (e) In ec ed Coun : pe iod: Ap 2020–May 2020
( ) Dea h Coun : pe iod: Ap 2020–May 2020.
has ma ched he model p edic ion wi h ac ual da a o he 160 days o
he dea h and in ec ed coun , whe eas he (c), (d) & (e), ( ) analyzes
he pa e n wi h small span o in e al by di iding he 160 days in o
smalle span o 100 and 60 days. When he disease was in i s ea ly s age
o sp eading, he incuba ion and in ec ious pe iod was longe (20–22
days), bu in 2nd hal hey ge s sho ened wi h an inc eased R alue, 3
compa ed o p e ious 2.7.
The alue o 𝛿−, 𝜑−and 𝜃−a e almos kep cons an o e lec
ha in as uc u e and human immuni y a e esponding uni o mly
h ough he whole pe iod. As he model is calib a ed wi h he ini ial
assump ions o 1 000 000 as exposed, 80000 asymp oma ic and 70 000
qua an ined, and he in ec ed, eco e ed and dea h coun a e 340546,
1847 and 41 697 espec i ely om he a ailable da a. While he 1s
ime pe iod (Sep 20 o Dec 20) uses he abo e said ini ial assump ion,
he 2-nd pe iod is plo ed agains he ou come o he 1s pe iod . In
Fig. 3.(a) and (b) depic s he cu es ela ed o he all ypes o a ec ed
coun s o he pe iod o 160 days and he o al span(app ox 350 days) o
pandemic end espec i ely. Fig. 3. p o es ha due o he bes medical
sys em in he wo ld in UK, hough he exposed and in ec ed cu e is
highe han he asymp oma ic and qua an ine cu e, bu he sp ead is
con olled.
US: The same analysis o ou model i wi h espec o p edic ion
o dea h and in ec ed is done du ing Peak in ec ious pe iod o US.
He e he o al span is aken om Ma 2020–May 2020 o 80 days
and hen smalle ime pe iod o Ma ch 2020–Ap il 2020 and Ap il
2020–May 2020 is conside ed. The esul s a e cap u ed in Fig. 4.(a)–
( ). The popula ion is 33 000 000 and ini ial alues o 𝐸0, 𝐴0and 𝑄0a e
900 000, 700 000 and 300 000... The sho e pe iod cu es o Fig. 4.(c),
(d) and Fig. 4.(e), ( ) shows be e esul s compa ed o he la ge
pe iod cu e o Fig. 4.(a) and (b). Analy ically, li e sa ing ea men
signi ican ly con ols he dea h a e compa ed o p edic ion in Fig. 4.( )
while he in ec ion cu e Fig. 4.(e) is plo ed wi h same pa ame e
calib a ion. Fig. 5(a) and (b) p o ides a p obable indica ion o end o
his pandemic, he highe R and smalle 𝛾−and 𝛼−in US has sho ened
he pandemic li espan o 250 days compa ed o UK.
India: India being a de eloping coun y and ha ing a popula ion o
1 300 000000 showed a good con olling o e i s phase, bu second
wa e le a bad impac . In bo h he cases, asymp oma ic cases play a
i al ole as o simula e he model pe o mance. Fig. 6(a), (b) and (c)
ha e simula ed he pandemic i s wa e ime pe iod o 120 days (May
2020 o Sep 2020). Ini ial alues o 𝐸0, 𝐴0and 𝑄0a e 100000. 60 000
and 30000 and he end indica e a la ge numbe o qua an ined a e
he end o i s wa e.
The second phase ou b eak in India in Fig. 7(a)–(h) has a o al
pe iod o 72 days (10 h Ma ch 2021 o 20 h May 2021) and also hei
mon h wise indi idual cu e.
Resul s in Physics 40 (2022) 105855
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P. Rakshi e al.
Fig. 5. US (a) Model SEIAQFR p edic ed pa ame e s ajec o ies du ing peak pe iod Ma 2020–May 2020 (b) Model SEIAQFR P edic ed Pa ame e s alue o pandemic’s p edic ed
li espan.
Fig. 6. India Ac ual and Model p edic ed Cu e alue o In ec ed and Dea h coun in di e en ime phases o disease du ing i s ou b eak in i s wa e (a) In ec ed Coun : pe iod:
May 2020–Sep 2020 (b) Dea h Coun : pe iod: May 2020–Sep 2020 (c) Model SEIAQFR P edic ed Pa ame e s alue o pandemic’s p edic ed li espan du ing i s wa e in India.
Fig. 6(a), (b) shows an imp o emen o medical ea men in he las
span o Sep embe 2020, so ha he expec ed dea h a e supe sedes
he ac ual alue. In Fig. 6(c), he qua an ined coun is g ea es and
ob iously a la ge numbe o asymp oma ic has a con ibu ion in ha ,
Fig. 7(a)–(h) shows ha he indi idual cu e o indi idual mon h
pe o ms be e han he whole pe iod o second wa e (Ma ch 21 o
May 21). As an example, In May i s week, we ha e seen ha he
Hospi al s ay becomes longe due o in ec ion(inc easing 𝛾) and he
demise o people in sho e pe iod o ime( educing 𝛿) due o del a
a ian a ack, so adjus ing he pa ame e s he R is ound o be smalle
han p e ious mon h, bu wi h a inc eased mo ali y a e. The o e all R
sco e is impac ed due o he heal h sys em be e men and he in ec ion
& dea h a e dec eases.
Analysis and discussion
Analyzing Figs. 2–7and Table 1, we ha e he ollowing obse a ion
while i ing he in ec ed and dea h coun wi h he ac ual alue:
•When he Rep oduc ion Fac o (R) is changed om smalle o
highe alue, he ising o in ec ion cu e is sha pe and he
cu es ge la ened wi h he lowe alues o R. The alue o he
ep oduc ion ac o (R) and he incuba ion pe iod (𝛼−)de e mines
he sp ead o epidemics, R a ec s he in ensi y o sp ead and 𝛼−
a ec s he speed o ansmission di ec ly. We can obse e om
Table 1 and Figs. 2–7 ha high R and less 𝛼− alues makes an
highe in ec ed a e in US in he i s phase o COVID and UK
emained in bes posi ion wi h high 𝛼− alue and less R alue,
while India s ands in be ween ha means as ea ly as he exposed
pe sons becomes in ec ious and hei con ac wi h o he pe sons
a e mo e, hen he g ow h o disease is also highe . The smalle
in ec ious pe iod (𝛾−)also slows down he sp ead.
•Longe 𝛿− alue la ens he dea h cu e whe eas he 𝜑and 𝜃
also con ibu e o ise o he cu e alue. UK and India pe o ms
be e o comba he disease wi h espec o mo ali y, so longe
hospi al s ay, bu be e su i al a e. om q alue, The immuni y
o all coun ies UK, US and India a e esponding almos in simila
p opo ion.
•𝑝, 𝑞 moni o s he in ec ed coun cu e, while changing he 𝑥 alue
has a di ec impac on he dea h cu e. F om he 𝑝 alue a ia ion,
he asymp oma ic cases a e ollowing he pa e n o India > UK >
US which e lec s he same ela ion wi h he coun ies popula ion
Resul s in Physics 40 (2022) 105855
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P. Rakshi e al.
Fig. 7. India Ac ual and Model p edic ed Cu e alue o In ec ed and Dea h coun in di e en ime phases o disease du ing i s ou b eak in second wa e in 2021 (a) In ec ed
Coun : pe iod: Ma 2021–May 2021 (b) Dea h Coun : pe iod: Ma 2021–May 2021 (c) In ec ed Coun : pe iod: Ma 2021 (d) Dea h Coun : pe iod: Ma 2021 (e) In ec ed Coun :
pe iod: Ap 2021 ( ) Dea h Coun : pe iod: Ap 2021 (g) In ec ed Coun : pe iod: May 2021 (h) Dea h Coun : pe iod: May 2021.
densi y. So Asymp oma ic case ∝Popula ion Densi y, which is
e y ue o such con agious disease.
•In ou model he asymp oma ic cases (𝑝 alue de i a ion) a e
ge ing e alua ed o be 30%–40%, which is inline wi h di e en
s a is ical analysis in India [24].
•As in his model, qua an ined means he pa ien is ei he kep
in home isola ion o hospi alized o ea men , so om all he
epidemic o ecas ing, i is e iden ha he numbe o qua an-
ined pe sons a e g ea e han he in ec ed pe sons a any ime,
because o he slow a e o ge ing eco e ed o dea h compa ed
o he in ec ion a e. The obse ed cases in di e en s a es whe e
he numbe o home qua an ined a e ge ing inc eased due o
sho age o hospi al bed, as a esul he se ious pe sons a e dying
wi hou p ope a angemen o en ila ion & ICCU bed. In sho ,
Resul s in Physics 40 (2022) 105855
9
P. Rakshi e al.
he e is a posi i e co ela ion be ween economic de elopmen and
mo ali y a e in India as compa ed o US and UK. The hospi al
beds and in ec ed co ela ion is es ablished in [25].
•All he cu es in Fig. 5, clea ly depic he ac o e y high eco -
e y numbe as compa ed o he da a ha ha e been published [23,
24]. This is because o he eason ha he e is a la ge numbe o
asymp oma ic cases who a e ge ing eco e ed au oma ically in
sho e ime compa ed o a pe son who a e ge ing qua an ined
and ea ed.
I we deno e om Exposed s a e, asymp oma ic cases ime o eco e
as a and in ec ed pe son ime o eco e du ing p ope ea men &
qua an ine as i, hen
𝑡𝑎=𝑖𝑛𝑐𝑢𝑏𝑎𝑡𝑖𝑜𝑛 𝑝𝑒𝑟𝑖𝑜𝑑+𝑖𝑛𝑓 𝑒𝑐𝑡𝑖𝑜𝑢𝑠 𝑝𝑒𝑟𝑖𝑜𝑑+𝑞𝑢𝑎𝑟𝑎𝑛𝑡𝑖𝑛𝑒𝑑 𝑝𝑒𝑟𝑖𝑜𝑑 =𝛼−+𝛾−+𝜃−
and
𝑡𝑖=𝑖𝑛𝑐𝑢𝑏𝑎𝑡𝑖𝑜𝑛 𝑝𝑒𝑟𝑖𝑜𝑑 +𝑡𝑖𝑚𝑒 𝑓 𝑜𝑟 𝑎𝑢𝑡𝑜𝑚𝑎𝑡𝑖𝑐 𝑟𝑒𝑐𝑜𝑣𝑒𝑟𝑦 𝑜𝑓 𝑝𝑎𝑡𝑖𝑒𝑛𝑡 =𝛼−+𝜑−
F om bo h he s udy o all coun ies we ind ha 𝑡𝑎> 𝑡𝑖. Fo his
ime di e ence, a any poin o ime (𝑡)in ou de i ed model, almos
80% asymp oma ic cases ge s eco e ed in sho e pe iod o ime.
So, we can say ha he ac ually in ec ed pe sons a e much highe
han he con i med cases, because o he p esence o many asymp o ic
indi iduals o he less numbe o es ings.
•Lockdown has a posi i e impac on he g ow h o Co ona. F om
all igu e, he ini ial a ali y a e is a pa o ou model p edic ion
whe eas he e ec i e social awa eness and Lockdown es ic ion
along wi h expe ise o handle Co id pa ien s ha e enabled a 10%
di e ence be ween p edic ed and ac ual dea h coun o each s a e.
•US, UK, India’s i s wa e basic ep oduc ion numbe is on he
lowe side as compa ed o B azil [25] and I aly [26], and his
may be because o he s ic lockdown and immedia e isola ion
o he in ec ed, bu he second wa e in India has shown a ise in
𝑅 alue.
•As hese da a a e dynamic ones a ec ing he cu es in ends,
he e o e e o s in posi i e case de ec ion, epo ing o o he la ge
unce ain ies associa ed wi h hese alues may d ama ically al e
calib a ion pa ame e s and he conclusi e in e ence in e ms o
numbe o in ec ed and associa ed mo ali y.
•The adi ional concep o SIR model wi h a speci ic eloci y o
in ec ion and eco e y a e has been enhanced wi h a a ying
na u e o disease wi h a co ela ion among di e en ac o s. The
a ying alues o 𝛼, 𝛾, 𝑅 es ablish he dynamic cha ac e is ic o
SIR model o e ime pe iod and demog aphic o ien a ion. I e-
lec s he s eng h o medical in as uc u e o he coun y, human
immune sys em o speci ic e hnic, likewise he US, UK and India
has di e en a ying pa ame e alues o ha e a bes i , so hese
pa ame e s a e no cons an s o any epidemic disease e olu ion.
To measu e he pe o mance o ou model we ha e calcula ed some
ele an pa ame e like he MAPE and R-Squa ed me ics and he
esul s a e shown in Table 2.
MAPE (mean absolu e pe cen age e o ) measu es model accu acy
as a pe cen age, and can be calcula ed as he a e age absolu e pe cen
e o o each ime pe iod minus ac ual alues di ided by ac ual alues.
𝑀𝐴𝑃 𝐸 =1
𝑁
𝑁
∑
𝑥=1 |𝑦𝑥−𝑦|
𝑦𝑥
× 100%.
R-squa ed (Coe icien o de e mina ion) ep esen s he coe icien o
how well he alues i compa ed o he o iginal alues. The alue om
0 o 1 in e p e ed as pe cen ages. The highe he alue is, he be e he
model is.
The abo e me ics can be exp essed,
𝑅2= 1 − ∑𝑁
𝑥=1(𝑦𝑥−𝑦)2
∑𝑁
𝑥=1(𝑦𝑥−𝑦)2,
whe e
𝑦𝑥=𝐴𝑐𝑡𝑢𝑎𝑙 𝑣𝑎𝑙𝑢𝑒,
𝑦 =𝑃 𝑟𝑒𝑑𝑖𝑐𝑡𝑒𝑑 𝑣𝑎𝑙𝑢𝑒,
𝑦 =𝑀𝑒𝑎𝑛 𝑣𝑎𝑙𝑢𝑒,
The alue o R-squa e measu e and co esponding MAPE e lec s a
signi ican pe o mance o ou Model p edic ion. Minimizing he Mean
Absolu e E o and Maximizing he R-squa e alue gi es an op imum
model pa ame e o bes i o he ac ual li e da a.
•In case S udying he In ec ed coun o he ac ual and he model
p edic ed alue we ound an accu acy (R-squa e sco e) o > .98
and e o (MAPE) o <5%. o mos o he pe iods. Whe eas
o he Dea h coun o he ac ual and he model p edic ed alue
accu acy (R-squa e sco e) and e o (MAPE) a e a e age 0.95 and
10% espec i ely.
•In India, he model pe o ms bes in 2-nd wa e wi h an 99%
accu acy o in ec ed and dea h (R-Squa e ≈0.99 and MAPE ≈
1%).
•Du ing analysis, i is obse ed ha when he change o 𝑝 alue is
0.11%, i is a ec ing he R-Squa e alue wi h a a ia ion o 60%.
•The R-Squa e and he MAPE alue has a g ea e pe o mance in
he sho e pe iod o ime o model i compa ed o longe ime
span.
•While making he bes i Cu e, o en o achie e op imized MAPE
alue, he accu acy o R-Squa e is ge ing sac i iced.
Fu u e wo k
Model i indica es ha he pandemic end o UK and US a e 350
and 250 days ime pe iod espec i ely, bu he changing beha iou o
i us a ian and endu ance o acqui ed esis ance o i us is a chal-
lenging goal oday and alongside he ecu ence o he in ec ion despi e
accina ion is a ec ing he p edic ion in long e m. The immuniza ion
p ocess and he e icacy o ha agains a pa icula s ain o i us
ha e p oposed a second hough in ou mind o imp o e he model
conside ing di e en uelling ac o s as well as p e en ion echniques.
Unlike adi ional s a is ical analysis, ou model p ope ly ies o assess
and inco po a e impac s o hidden asymp oma ic and in ec ious cases
on he en i e p ocedu e o epidemic. In ou u he s udy, we a e
aiming o inco po a e he ollowing poin s like conside ing a ce ain
quan i y o popula ion wi h s ic con inemen , di e en age g oup,
sex and people wi h co mo bidi y, in oducing he p e en i e measu es
aken by di e en adminis a ion, he i al load o he in ec ions [27],
he s ic ness o lockdown & he mig a ion o people among di e en
places. Though we ha e conside ed he la en pe iod o asymp oma ic
pe son as cons an , i is a ying na u e is ano he majo con ibu ing
ac o o he i us sp ead which should be aken ca e o .
Conclusions
The p esen analysis o he p oposed model indica es a e y high
numbe o peak in ec ions in India by compa ing he exis ing da a o
he model simula ion. Howe e , he pa ame e s chosen o he model
calib a ion a e no unique, as di e en choices o pa ame e alues
wi h he same good i cu e, da a can esul o e y di e en model
p edic ions. Fo example, The epo ed numbe o deceased people
could possibly be unde es ima ed due o undecla ed cases and can also
be o e es ima ed due o he como bidi y ac o s o elde ly whe e he
pa ien s could ha e su i ed om COVID-19, bu o he illness like
exis ing lungs-kidney p oblem, diabe es has accele a ed he a ali y
a e. In a wo d, he age g oup, wea he condi ions o di e en s a es
(as he humid and ainy season in India may pu a b eak in he
ou b eak o he epidemic) can be conside ed as o he impac ing ac o s.
