Salgın Hastalıkların Yayılımı icin MatematikselModeller ve Uygulamalar

Ayse Humeyra BilgeKadir Has Universitesi

25 Kasım 2017,“KADIN VE BILIM V”TKMD ve DUKAM Etkinligi

Bu seminerde sunulacak sonuclar F. Samanlioglu, O.Pekcan, Y.Ozdemir, S.Ogrenci, O. Ergonul, M.V. Gurol ile ortak calısmalaradayalıdır.Calısmalarda kullanılan denysel ve gozlemsel veri O. Ergonul, G.Evingur, S. Kara, D. Kaya tarafından temin edilmistir.

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Ana hatlar I

I (1) Salgın hastalıkların yayılımı icin SIR/SEIR ve SIS modelleri

I (2) SIR modellerinin 2009 H1N1 salgınına uygulamaları

I (3) SIR modelinin polimerizasyon (kimyasal jellesme) surecineuygulamaları

I (4) SIS modelinin fiziksel jellesme surecine uygulamaları

I (5) Sigmoidal egriler ve “kritik noktalar.”.

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Matematiksel modeller I

I Salgın hastalıkların yayılımı icin matematiksel modeller1927’de Kermack ve McKendrick tarafından onerilmistir.

I Bu modellerde toplumdaki bireyler hastalıga iliskindurumlarına gore belli gruplara ayrılıp bunlar arasındakigecislerin matematiksel modeller aracılıgı ile belirlenir(compartmental models).

I Bu cercevede gelistirien modellerin en temelleri,“Susceptible-Infected-Recovered (SIR)” ve“Susceptible-Infected-Susceptible (SIS)” modelleridir.

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Matematiksel modeller II

I SIR modellerinde hastalık bitiminde bagısıklık kazanılır (virutikhastalıklar)

I SIS modellerinde hastalik bitiminde bagısıklık kazanılmaz(bakteriyel hastalıklar)

S → I → R

S → I → S

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Susceptible-Infected-Removed: SIR modeli (Bagısıklık var)I

I SIR modelinde, bagısıklıgı olmayan kisler (susceptible) enfekteolmakta, belli bir sure icin enfeksiyonu cevrelerine yaymakta,bu surenin sonunda ise (SIR) modellinde kalıcı olarak bagısıklıkkazanmaktadır. Virutik hastalıklar, kızamık vb. cocukhastalıkları ve enfluenza tipi hastalıklar bu modele ornektir.

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Susceptible-Infected-Removed: SIR modeli (Bagısıklık var)II

dS

dt= −βSI , dI

dt= βSI − ηI , dR

dt= ηI

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Susceptible-Infected-Removed: SIR modeli (Bagısıklık var)III

IdS

dt= −βSI , dI

dt= βI (S − η

β),

dR

dt= ηI

Yukarıdaki SIR modelinde toplumdaki birey sayısı sabittir. Bumodel kısa sureli salgınlar icin gecerlidir. S + I + R = 1alınabilir. I = 0 denge noktasını olusturmaktadır.

I ηβ oranı R0 ile gosterilmekte ve “Basic Reproduction Number”olarak adlandırılmaktadır.

I S − ηβ > 0 ise I ’nın turevi pozitiftir ve enfekte olanların sayısı

artar. S − ηβ = 0 oldugu anda enfeksiyonun yayılma hızı

maksimuma erismistir, bundan sonra duser.

I S ≤ 1 oldugu icin, R0 = βη < 1 ise salgın yayılamaz,

sonumlenir.

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Susceptible-Infected-Removed: SIR modeli (Bagısıklık var)IV

I Salgının ne kadar siddetli olacagını R0 belirler, ne zamanortaya cıkacagını ise baslangıc kosulları belirler.

I SIR sisteminde dogum ve olumler gozonune alınırsa ilgincdinamikler ortaya cıkar. Parametrelerin zamana gore degiskenolması soz konusu olabilir.

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Susceptible-Infected-Removed: SIR modeli (Bagısıklık var)V

I Hastalıgın bir kulucka suresi olabilir. Bu donemde, kisi enfekteolmustur ancak cevresini enfekte edememektedir. Bu model“Susceptible-Exposed-Infected-Removed” SEIR olarakadlandırılır. Bu tur modeller SIR modeline benzer, amahastalıgın yayılımı daha yavastır.

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Susceptible-Infected-Removed: SIR modeli (Bagısıklık var)VI

I Asılama yapılırsa, bireyler, enfekte olmadan S ’den dogrudan Rgrubuna gecerler. Hastalık oldurucu degilse, saglık acısındangereklilik yoksa, sagını onlemek icin tum bireyleri asılamakgerekmez, S grubundaki kisilerin oranının η

β esik degerininaltına dusmesi yeterlidir.

I Asılama kampanyasının etkili olabilmesi icin erken baslamasıcok onemlidir.

I SIR ve SEIR sistemleri ancak numerik olarakcozulebilmektedir.

I Stokastik SIR ve SEIR modelleri konusunda pek cok calısmabulunmaktadır.

I Salgın hastalıklara iliskin gozlemler genellikle R(t)degiskeninine iliskindir. Sadece bu gozlemlerden hareketle,denklem sisteminin parametrelerin bulunması gerekmektedir.

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Susceptible-Infected-Removed: SIR modeli (Bagısıklık var)VII

I EN ONEMLISI: SIR ve SEIR modelleri baslangıcta, ayrıkdenklemler olarak tanımlanmıstır. Limit durumunda yukarıdaverilen denklmeleri degil, integro-diferansiyel denklemlervermektedir. Halen yaygın olarak kullanılan denklemleryaklasık denklemlerdir ve hem R(t) hem de I (t)’nin gozlenmisoldugu 2009 H1N1 salgını verisi diferansiyel denklem sistemiile uyusmamaktadır.

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Susceptible-Infected-Susceptible: SIS modeli (Bagısıklıkyok) I

I SIS modelinde ise enfeksiyon suresinin sonunda bagısıklıkkazanılamamaktadır. Genel olarak bakteriyel hastalıklar bugruba girer.

dS

dt= −βSI + ηI ,

dI

dt= βSI − ηI .

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Susceptible-Infected-Susceptible: SIS modeli (Bagısıklıkyok) II

IdS

dt= −βSI + ηI ,

dI

dt= βSI − ηI .

SIS modelinde 2 denge noktası vardır, I = 0 ve S = ηβ .

I Bu modelde de S + I = 1 alınabilir. Buradan I icin “lojistikdenklemi” elde edilir.

dI

dt= βI

[(1− η

β

)− I

]I Bu denklemin analitik cozumu kolayca bulunabilir

(genestirilmis lojistik fonksiyonu ileride verecegiz).

I Gerek SIS modelindeki I (t) egrisi, gerekse SIR modelindekiR(t) egrisi iki yatay asimptot arasında monoton artan“sigmoid” olarak adlandırılan egrilerdir.

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Uygulamalar I

I Salgın hastalıkların yayılımının gozlenmesi sorunlu bir surectir.Grip benzeri hastalıklarda, vakaların sadece bir bolumu kayıtaltına alınabilmektedir. Genel olarak hastalıga baglı olumlereait kayıtların dogru oldugu dusunulebilir.

I Olum kayıtlarından hareket ederek, hastalanıp da iyilesenlerinoranını belirlemek baslı basına bir problemdir. Bunu“normalize R(t) egrisinden parametre bulunması” olaraktanımlıyoruz.

I 2009 H1N1 salgını sırasında Prof. Dr. Onder Ergonultarafından Istanbul’daki belli baslı hastanelerde H1N1 teshisiile yatan hasta, taburcu olma ve olum vakaları derlenmistir.Bu veri, sadece R(t) degil I (t) bilgisini de icermesi acısındanonemlidir.

I 2009 H1N1 salgını sırasında cesitli Avrupa ulkelerinindekihaftalık olum vakaları sayısı web ortamında yayımlanmıstır.

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Uygulamalar II

I Calısmalarımızda, SIR modeli 2009 H1N1 salgını verisineuygulanarak Istanbul ve cesitli Avrupa ulkelerine ait veriincelenmistir.

I Kimyasal jel olusumu surecinin gozlendigi deneysel verinin,kimyasal aktivasyonun derecesine baglı olarak SIR veya SEIRsistemlerinin R(t) egrisini takip ettigi gozlenmis, sistemparametreleri bulunmustur.

I Fiziksel jel olusumu, zaman yerine sıcaklıga baglı olan tersinir(reversible) bir surectir. Bu surecin SIS modelinin I (t) egrisineuydugu gozlenmis ve sistem parametreleri bulunmustur.

I Jellesme olaylarında, sıvı halden jel olusumuna gecis zamanı“jel noktası” olarak adlandırılmaktadır. Bu nokta deneyselolarak gozlemlenebilmekte, ancak sigmoidal egrinin hanginoktasında olması gerektigi teorik olarak acıklanamamaktadır.

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Uygulamalar III

I Sigmoidal egrilerle ilgili calısmalarda, genel varsayımlar altındabu egrilerin yuksek mertebeden turevlerinin mutlakdegerlerinin maksi,muma eristigi ti noktalarının yakınsak birdizi olusturdugu kanıtlanmıstır. Bu nokta, sigmoidal egrinin 1.turevinin Fourier donusumu ile belirlenmistir.

I Bu limit noktası, “sigmoidal egrinin kritik noktası” olarakadlandırılmıs, ve deneysel jelesme egrilerindeki jel noktası ileortustugu gosterilmistir.

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SIR ve SEIR modelleri, R0 I

SIR Modeli : S ′ = −βSI , I ′ = βSI − ηI , R ′ = ηI .

t0 5 10 15 20 25 30 35 40 45 50

0

0.1

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1SIR Model, R0=3

SIR

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SIR ve SEIR modelleri, R0 III Basic Reproduction Number, R0 = β/η, R(t)’nin

baslangıctak4 artıs hızının bir olcutudur (uygun yaklasımaltında R(t) baslangıcta ustel olarak artar, R0 da bu ustelartısın katsayısıdır.

I I (t) nin maksimuma etistigi tm zamanın, S(t)’nin 1/R0’adustugu zamandır.

I Ancak en onemlisi, R(t)’nin t sonsuza giderken limiti, Rf

degeri, R0 tarafından belirlenir.

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SIR ve SEIR modelleri, R0 III

SEIR Modeli : S ′ = −βSI , E ′ = βSI−εE , I ′ = εE−ηI , R ′ = ηI .

t0 50 100

0

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1SIR Model, R0=3

t0 50 100

0

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1SEIR Model, R0=3

Her iki modelde de enfeksiyon suresi 3 gundur; SEIR modelindeayrıca 3 gunluk bir kulucka suresi vardır. Her iki modelde deR(t)’nin limit degeri sadece R0’a baglıdır. SEIR modelindekikulucka suresinin etkisi, R(t)denki artısın daha yavas olmasıdır.

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SIR ve SEIR modelleri, R0 IV

I Her iki modelde de, Si = S(0) ve Ri = R(0) olmak uzere, Rve S

S(R) = Sie−βη

(R−Ri ), R = − ηβ

ln(S/Si ) + Ri ,

seklinde ifade edilebilir.

I SIR modelinde, I , I = 1− R − S esitliginden bulunur ve SIRmodeli R cinsinden cozulmus olur.

I SEIR modelinde ise, sadece I + E = 1− R − S diyebiliriz, yanitam cozum yoktur.

I Bu sistemlerde denge konumu I = 0 ve I = E = 0 ile ifadeedilir. limt→−∞ R(t) = 0 ve limt→−∞ S(t) = 1 ise, S0 ve R0

keyfi degildir:

S = e−βηR, R = − η

βln(S)

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SIR ve SEIR modelleri, R0 V

I Her iki modelde de Rf = limt→∞ R(t) ve R0 = β/η arasında

Rf + e−βηRf = 1 bagıntısı vardır, yani, R0, Rf tarafından

belirlenir: βη = − ln(1−Rf )

Rf

R01 2 3 4 5 6 7

Rf

0

0.1

0.2

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0.9

1SIR,SEIR Models

80% Rf forbeta/eta=2

flu typeepidemics

smollpox, rubellaSARS

Rf=0.999 forR0=bet/eta=7

measles:bet/eta 12-15

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SIR ve SEIR modelleri, R0 VII R0 hem hastalıgı yapan etmenin hem de hastalıgın yayıldıgı

toplumun ozelliklerine baglıdır, dolayısıyla aynı hastalık icin

toplumdam topluma degisebilir.

I R0 = βη sadece Rf belirlenebilir ama Rf ’yi dogru olarak kestirmek

cok zordur. Ornegin hastalıga baglı olumlerin, D(t), R(t)’yi

yansıttıgını dusunebiliriz ama bu oran da, saglık sisteminin

ozelliklerine baglı olarak, degiskendir.

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SIR ve SEIR modelleri, R0 VII

I 2009 H1N1 salgını icin cesitli Avrupa ulkelerindeki hastalıgabaglı olumler, ulke nufusları ile normalize edilerek verilmistir.

Health Index0.82 0.84 0.86 0.88 0.9 0.92 0.94 0.96 0.98

Rel

ativ

e F

atal

ities

2

4

6

8

10

12

14

16

Norway

Estonia

Hungary

Lithuania

Romania

Czech Rep.

Greece

Slovenia

Ireland

Netherlands

GermanySweden

France

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SIR ve SEIR modelleri, R0 VIII2009 H1N1 salgınında Istanbul icin gunluk hastane basvuruları ve

kumulatif olumler asagıda verilmistir. Hastaneye basvuru oranının

degisken oldugu gozlenmektedir. Olum vakalarında, 15 gunden uzun

yogun bakım sureleri, veriye 15 gun olarak aktarılmıstır. Hastaneye

basvuruların maksimumu ile kumulatif olumlerin buklum noktası arasında

8 gun kadar bir kayma gozlenmektedir. Bu da SIR diferansiyel denklemi

ile uyumsuuz ancak orjinal, ayrık SIR modeli ile (integral denklemini

veren sistem ile) uyumludur. (Data: O. Ergonul)

0 20 40 60 80 100 120 140 160 1800

5

10

15

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25

30

35

40

45

50H1N1 2009 Epidemic, Istanbul

Days

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SIR ve SEIR modelleri, R0 IX

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10

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50Fatality Data, 2009 H1N1 Epidemic, Istanbul

Days (Starting from September 1, 2009)

Dur

atio

ns o

f sym

ptom

s an

d ho

spita

lizat

ion

Yogun bakımda gecen sureler dikkate alındıgında, olum zamanlarını,

yogun bakıma yatıstan en fazla 15 gun sonrası olarak almak daha

dogrudur. Yukarıdaki sekilde, acık renk ile, semptomların baslangıcı, koyu

renk ile yoggun bakımda kalma suresi belirtilmistir. Buradan, salgının

baslangıcındaki olum zamanlarının, R(t)yi daha daogru yansıttıgı

soylenebilir.

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Inverse problem: Finding system parameters fromnormalized data I

I We aim to find the parameters of the SIR or SEIR model fromfatality data. This would be easy if the final proportion ofR(t) were known, but since the proportionality constantbetween R(t) and D(t) is unknown, fatality data is anormalized data.

I In [BSE], we proved that there is a unique SIR model fitting anormalized curve of removed individuals R(t)

I This is not valid for the SEIR model: In an analysis for fittingthe SEIR model to fatality data we observed that theparameters of “best fitting models” (within %1− 3) followedcertain patterns. We proved there were infinitely many SEIRmodels fitting a normalized curve of R(t) (fatalities). (sum ofthe incubation and infection periods is an INVARIANTQUANTITY)

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Inverse problem: Finding system parameters fromnormalized data II

I

I Let {S , I , R} and {S ,E , I ,R} be solutions of the SIR andSEIR models with parameters (β, η) and (β, ε, η). Letlimt→∞R = Rf and limt→∞R = Rf , and let tm and tm be thetimes corresponding to the inflection points of R(t) and R(t).Assume that Rf = Rf , Rm = R(tm) = R(tm),

R ′m = dRdt (tm) = dR

dt (tm). Then,

β

η=β

η,

1

η=

1

η+

1

ε.

I The SIR model is a good approximation to the SEIR modelfor low values of R0

I Although the model fitting R(t) is unique for the SIR model,we have seen that, there are too many “best fitting” models(NEARLY INVARIANT QUANTITIES)

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Inverse problem: Finding system parameters fromnormalized data III

0 50 100 150 2000

0.1

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0.7

0.8Rf=0.75

SEIRSIR

0 50 1000

0.1

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0.9

1Rf=0.99

SEIRSIR

(a) R(t) for the SIR and SEIR models for Rf = 0.75; (b) R(t)for the SIR and SEIR models for Rf = 0.99.

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2009 H1N1 Epidemic in Europe I

I The model is fitted to data by solving the ODE’e for a wide range

of parameters and finding the ones that fit best with respect to

certain norms.(Czech Republic. β/η = 1.3)

0 100 200 300−0.2

0

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0.8

1

1.2

Days

Epi

dem

ic D

ata

Czech Republic

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1

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2009 H1N1 Epidemic in Europe III Normalized curves look the same but final proportion of R(t)

is different (Czech Republic. β/η = 1.5)

0 100 200 300−0.2

0

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1

1.2

Days

Epi

dem

ic D

ata

Czech Republic

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1

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2009 H1N1 Epidemic in Europe IIII The range of “best fitting” models is not narrow. Czech Republic:

Various parameters that fit the data within 2.5% in the L2 norm and

in less than 0.02 in the sup norm.

0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1x 10

−3

Normalized Error

Curvatures

0.2 0.4 0.6 0.8 11

1.5

2

2.5

Normalized Error

beta/eta

0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

Normalized Error

d(Rm/Rf)dt

0.2 0.4 0.6 0.8 10

0.2

0.4

0.6

0.8

Normalized Error

Rm/Rf

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2009 H1N1 Epidemic in Europe IVI Although the SIR model can be recovered uniquely from

fatality data, the inverse problem is ill posed/there arequantities that are nearly invariant

30 40 50 60 700

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1

Week

Nor

mal

ized

Fat

ality

Czech Republic: Best Models in L2 Norm

1 1.5 2 2.52

3

4

5

6

7

8

9

10

R0

1/et

a, D

ays

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2009 H1N1 Epidemic in Europe V

30 40 50 60 700

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1

Week

Nor

mal

ized

Fat

ality

Sweden: Best Models in L2 Norm

1.8 2 2.2 2.4 2.69

10

11

12

13

14

15

16

R01/

eta,

Day

s

Data for Sweden

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2009 H1N1 Epidemic in Europe VI

We have shown that the “pulse width” or the “duration of theepidemic” is nearly invariant. [Samanlioglu, Bilge]

0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

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0.8

1

1.2

1.4

1.6

1.8

2Rf versus eta for Fixed Pulse Width

Rf

eta

1020304050607080

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Remarks I

I The early and late phases of an epidemic may be different dueto health care practices

0.5 0.6 0.7 0.8 0.92

4

6

8

10

Rf

Per

cent

age

Err

or

France

0.35 0.4 0.45 0.5 0.552

4

6

8

10

Rm/Rf

Per

cent

age

Err

or

France

0 0.5 1 1.5 22

4

6

8

10

eta

Per

cent

age

Err

or

France

0.11 0.12 0.13 0.14 0.152

4

6

8

10

d(Rm/Rf)/dt

Per

cent

age

Err

or

France

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Remarks III Vaccination has a huge effect on total fatalities but it cannot

be observed on normalized data (we can omit it inmathematical models)

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Days

R(t) with Vaccination

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1

1.2

Days

Normalized R(t) with Vaccination

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Remarks IIII Late vaccination has very little effect. This explains the

difference between fatalities in Norway and Sweden.

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1

Time (Days)

SIR Model: R0=1.3; Vaccination Rate 30%

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Time (Days)

SIR Model: R0=1.3; Vaccination Rate 30%

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Remarks IVI Long vaccination campaigns are not that useful

−80 −60 −40 −20 0 20 40 60 800

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1

Days

Rf/R

f0: 1

0%−

50%

Tot

al V

acci

natio

n

Relative Improvment in Rf: 14 Day Pulse Vaccination

−80 −60 −40 −20 0 20 40 60 800

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1

Days

Rf/R

f0: 1

0%−

50%

Tot

al V

acci

natio

n

Relative Improvment in Rf: 28 Day Pulse Vaccination

−80 −60 −40 −20 0 20 400

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1

Days

Rf/R

f0: 1

0%−

50%

Tot

al V

acci

natio

n

Relative Improvment in Rf: 70 Day Pulse Vaccination

−80 −75 −70 −65 −60 −55 −50 −450

0.05

0.1

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0.25

0.3

0.35

0.4

Days

Rf/R

f0: 1

0%−

50%

Tot

al V

acci

natio

n

Relative Improvment in Rf: 140 Day Pulse Vaccination

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Remarks VI The ordinary differential equation (ODE) model is just an

approximation for the spread of an epidemic.

I In the study of 2009 H1N1 Istanbul data, we observed that I (t) andR ′(t) become correlated only after a shift of about 8− 9 days.

I We have shown that a discrete model on daily subdivisions agreeswith the data.

I Increasing the number of subdivisions, we arrive to an integralequation, which is the original model proposed by Kermack andMcKendrik in 1927.

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Remarks VI

I Spatial spread can be modeled in terms of diffusion (spread of pestin medieval ages) or by hopping through nodes on a network (spreadof a pandemic through airports) or a combination of both (diseasesspread by flies, by diffusion in 100-300 meters, by hopping via citybuses).

I Compartment models are also studied in terms of networks. A

remarkable result, by Dr. Duygu Balcan, based on the properties of

networks, states that a pandemic cannot be stopped completely by

precautions at the airports.

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Sol-gel transition for (irreversible) chemical gels I

I The sol-gel phase transition of polyacrilamide-sodium alginatecomposite with low and high Sodium Alginate concentrations isstudied in [Evingur,Tezcan,Erim, Pekcan, Phase Transitions (2012)].The steep, low amplitude curves correspond to low SAconcentrations obeying percolation model (SIR model) The slowerrising high amplitude curves obey classical model (SEIR model).The gel points are determined by a dilatometric technique.

0 20 40 60 80 100 1200

50

100

150

200

250

300

Time (Minutes)

Flu

ores

cenc

e In

tens

ity

Classical/Percolation Type Phase Transition

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Sol-gel transition for (irreversible) chemical gels II

I We modeled the gelation curves by the SIR system and obtainedgood agreement. [Bilge,Pekcan, Gurol, (2012)]

0 20 40 60 80 100 1200

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

Time (Minutes)

Rel

ativ

e S

tren

gth

SIR Model Solutions/Percolation Type Transition Data

SEIR

I In experimental work and in the SIR model, the critical point islocated in between the zeros of the third and second derivatives; itis closer to the zero of the third derivative for high activation (lowSA int experimental work, high k/η in the SIR model).

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Normalized higher derivatives and time domain plots forthe SIR model

0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 1−1

−0.5

0

0.5

1SIR Model, k=3,eta=1, S0=0.9

S

Der

ivat

ives

of R

0 2 4 6 8 100

0.2

0.4

0.6

0.8

1

Time

S,I,

R

SIR Model, beta=3,eta=1, S0=0.9

SIR

(a) The first 24 derivatives of R(t) normalized to 1 for k = 3, η = 1 and S0 = 0.9. The curves are plotted againstS(t). The phase transition point, indicated by (∗) is expected to be located at Sc = 0.632.

(b) The time domain plots of the solution curves S , I , R. S(t) is monotone decreasing while R(t) is monotone

increasing. The sequence of points converging to the phase transition point tc are shown on each curve by (.). The

phase transition point tc indicated by (o), is located between the maximum of zero I , tm denoted by (∗) and the

inflection point of I , ta denoted by (+). For k = 3, tc is located at 78% left of tm in the interval (ta, tm)43 / 67

Normalized higher derivatives and time domain plots forthe SIR model

0 0.2 0.4 0.6 0.8 1−1

−0.5

0

0.5

1SIR Model, k=10,eta=1, S0=0.9

S

Der

ivat

ives

of R

0 1 2 3 4 50

0.2

0.4

0.6

0.8

1

Time

S,I,

R

SIR Model, k=10,eta=1, S0=0.9

SIR

(a) The first 24 derivatives of R(t) normalized to 1 for k = 10, η = 1 and S0 = 0.9. The curves are plottedagainst S(t). The phase transition point, indicated by (∗), is expected to be located at Sc = 0.534.(b) The time domain plots of the solution curves S , I , R. S(t) is monotone decreasing while R(t) is monotoneincreasing. The sequence of points converging to the phase transition point tc are shown on each curve by (.). Thephase transition point tc indicated by (o), is located between the maximum of zero I , tm denoted by (∗) and theinflection point of I , ta denoted by (+). For k = 10, tc is located at 94% left of tm in the interval (ta, tm)

The critical point moves towards left as R0 increases.

0 0.5 1 1.50

0.2

0.4

0 0.5 1 1.50

0.5

0 0.5 1 1.50

0.5

1

0 0.5 1 1.50

0.5

1

k=5

k=4

k=3

k=10

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Mathematical Definition of the Critical Point IMotivation:

I The physical laws governing the system at the initial phasewhere x < xc and at the final phase where x > xc may bedifferent.

I The generally accepted functional form is piecewise smooth,with characteristic power laws at both sides of the criticalpoint. Junction conditions ensure that y(x) is continuous atxc .

I If the system is approximated by a dynamical systemy ′ = F (y) where F is smooth; the non-analyticity at thejunction will be reflected to the derivatives at the criticalpoint.

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Mathematical Definition of the Critical Point III y(x) is a smooth sigmoidal curve, i.e, monotone increasing

with horizontal asymptotes as t → ±∞.

I {xm,i} and {xa,i}, i = 1, 2, . . . , are the set of points where thederivatives of odd and even order reach their extreme values.

I If the sequences {xm,i} and {xa,i} are both convergent andthey have a common limit xc , this limit is called “the criticalpoint of the phase transition”.

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Mathematical Definition of the Critical Point III

I The ith zero of y (n) is x in. The inflection point of y(x) is theunique zero of its second derivative y (2), denoted by x1

2 .

I x12 lies in between the two zeros of the third derivative x1

3 andx2

3 , and we have the order relations x13 < x1

2 < x23 .

−4 −3 −2 −1 0 1 2 3 4−5

−4

−3

−2

−1

0

1

2

3

4

5y=atan(t), First 4 derivatives

I The fourth derivative has three zeros, satisfying the orderrelations

x14 < x1

3 < x24 , x2

4 < x23 < x3

4 .

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Mathematical Definition of the Critical Point IV

−4 −3 −2 −1 0 1 2 3 4−150

−100

−50

0

50

100

150y=atan(t), First 6 derivatives

I Although the zeros of the third and fourth derivativesalternate, we can’t say anything about the relative positions ofx1

2 and x24 .

I Thus, the observed regular behavior of the zeros near the“critical point” is not a straightforward consequence of thealternation of zeros of successive derivatives.

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Mathematical Definition of the Critical Point V

I The extrema of odd derivative correspond to the zeros of evenderivatives and vice versa. Thus, if the critical point is torepresent some type of break-point, the best that we canexpect is that the sequences, {xm,i} and {xa,i}, converge to acommon limit point xc .

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Mathematical Properties I

I The whole real line consists of the “final set” (defined by Polya); i.e,the zeros of the derivatives fill the real line. There is no “gap”.

I This does not guarantee the convergence of these sequences.Furthermore, even if they converge, there is a priori no reason forthe equality of their limits.

I In fact, the convergence rates of the sequences {xm,i} and {xa,i} aredifferent and there seems to be a gap in the set of zeros.

I For the logistic growth we can see that the gap closes when we plotthe zeros of about 200 derivatives.

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Mathematical Properties II

−1 −0.5 0 0.5 1−1

−0.5

0

0.5

1Logistic Growth: Higher Derivatives

y

Hig

her

Der

ivat

ives

0 50 100 150 2000

0.2

0.4

0.6

0.8Logistic Growth: Convergence of the Absolute Extrema to x=0

Order of the DerivativeLoca

tion

of th

e A

bsol

ute

Ext

rem

um

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Physical gels I

I Physical gels don’t fit to the solutions of the SIR/SEIRmodels. They are well approximated by the generalizedlogistic growth curve.

I The standard logistic growth curve is the solution of theSusceptible-Infected-Susceptible (SIS) epidemic model.

I We generalized this model to obtain the generalized logisticgrowth as solutions (Ogrenci, Pekcan, Bilge, Kara).

I We studied the reversible sol-gel/gel-sol transitions ofCarrageenan and High Melting Point (HMP) and Low MeltingPoint (LMP) agarose gels at different concentrations follow ahysteresis loop under heating and cooling.

I This effect also can be modeled by the SIS system.

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Physical gels IIThe generalized logistic growth curve is

y = y0 + a

1 + e−x − x0

b

−c

Fitting of the model to the data is done easily by the SigmaPlotsoftware. The parameters are

I y0 is the vertical shift

I x0 is the horizontal shift. In [BO] we showed that the criticalpoint of the generalized logistic growth curve id x = 0.Therefore x0 is the critical point of the sigmoidal curve, hencethe gel point of the transition.

I a and b give the vertical and horizontal scalings.

I c is the parameter that determines skewness.

I For a symmetrical curve, c = 1 and x0 = 0.

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Physical gels III

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The Mathematical Description of the Critical point of aSigmoidal Curve I

I Let y(t) be a monotone increasing function with horizontalasymptotes y1 and y2 as t → ±∞ and with lim

t→±∞y (n)(t) = 0 for all

n ≥ 1, i.e, a sigmoidal curve. Let tn be the point where the nthderivative of y(t) reaches its global extremum. If the sequence {tn}converges, its limit is called the “critical point” of the sigmoidalcurve y(t).

I We proved that for sigmoidal curves satisfying fairly generalassumptions on their Fourier transform, the sequence {tn} isconvergent and we call it “the critical point of the sigmoidal curve”.

I We gave the proof in the case where the nth derivative has exactlyn − 1 zeros. But the result is valid without this assumption.

I For example, if y(t) be a finite sum of Gaussians, it can be shownthat the critical point is the location of the Gaussian with thesmallest spread (standard deviation).

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The Mathematical Description of the Critical point of aSigmoidal Curve II

I For odd sigmoidal curves, we observed that the critical point wasalways located at t = 0, i.e, at the zero of the first derivative, asexpected, but we were unable to prove this even for the simplestfunctions.

I We outline the steps leading to the proof of the existence of thecritical point

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Preliminaries I

I Fourier transform: Let f (t) be in L1. Then its Fouriertransform F(f ) = F and the inverse transform F−1(F ) = fare defined by

F (ω) =1√2π

∫ ∞−∞

f (t) e−iωt dt,

f (t) =1√2π

∫ ∞−∞

F (ω) e iωt dω.

I The effect of differentiation in the time domain ismultiplication by iω in the frequency domain, i.e,F(f (n)(t)) = (iω)nF (ω).

I Multiplication and convolution in the time and frequencydomains are related by F(f (t) g(t)) = 1√

2πF (ω)G (ω).

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Preliminaries II

I The “modulation” of low frequency signal in the time domainis the multiplication of this signal by a sinusoidal function offixed (usually high) angular frequency ω0. In the frequencydomain, the Fourier transform of the low frequency function isconvolved with the Fourier transform of the sinusoid.

I The Fourier transform of a pure sinusoid is represented byDirac δ functions occurring at ±ω0 and convolution carries thespectrum of the low frequency signal to the frequencies ±ω0.

I It follows that multiplication in the time domain by a complexexponential results in a shift in the frequency domain.Similarly, multiplication by a linear phase factor in thefrequency domain leads to a shift in the time domain, as givenbelow:

F(f (t)e iω0t) = F (ω − ω0), F−1(e−iαωF (ω)) = f (t − α).

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The existence and non-existence of the critical point I

I The generalized logistic growth family of curves and its limitingfunction the Gompertz curve to illustrate the existence and thenonexistence of the critical point and its location.

I The generalized logistic growth curve with horizontal asymptotes at−1 and 1 is given by

y(t) = −1 + 2[1 + ke−βt

]−1/ν,

where k > 0, β > 0 and ν > 0. The parameter k can be adjustedby a time shift, β corresponds to a scaling of time and ν is the keyparameter that determines the shape of the growth. For ν = 1,k = 1 and β = 2 we obtain the standard logistic growthy(t) = tanh(t).

I The Gompertz curve obtained as the limit of the generalized logisticfamily for k = 1/n, ν = 1/n, as n→∞ is the function

y(t) = −1 + 2e−e−βt

.

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The existence and non-existence of the critical point II

I The critical point of the standard logistic curve is located at t = 0;

the choice k = 1 ensures that the critical point of the generalized

logistic curve is also is located at the same point. The Gompertz

curve has no critical point, because the points tn move to negative

infinity.

I

-3 -2 -1 1 2 3

-1.0

-0.5

0.5

1.0

Normalized Derivatives of the Standard

-2 -1 1 2 3 4

-1.0

-0.5

0.5

1.0

Normalized Derivatives of the Generalized Logistic Growth

-4 -2 2 4

-1.0

-0.5

0.5

1.0

Normalized Derivatives of Gompertz Curve

(a) (b) (c)

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Why all derivative agglomerate? I

Higher derivatives of the sigmoidal function behave as localized humps

modulated by sinusoids of increasingly high frequencies. The wave packet

behavior in the time domain corresponds to the band-pass property of its

Fourier transform.

−10 −5 0 5 10−1

−0.5

0

0.5

1

−10 −5 0 5 10−1

−0.5

0

0.5

1

−10 −5 0 5 10−1

−0.5

0

0.5

1

−10 −5 0 5 10−1

−0.5

0

0.5

1

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Why all derivative agglomerate? II

I This is not sufficient for the existence of a critical point (As inthe Gompertz curve), where wave packets move to negativeinfinity.

I (2) These wave packets should be agglomerate as the order ofdifferentiation increases. The property that ensures this is the“asymptotically constant phase” condition, given inProposition 1 below.

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Main Result I

I Claim: Asymptotically constant phase and band-passhypotheses imply the existence of the critical point.

Proposition 1 Let f (t) be the first derivative of a sigmoidalcurve y(t) and f (n)(t) be its nth derivative. If

i the Fourier transform of f (t) has the formF (ω) = |F (ω)|e−iαωe iψ(ω) where α is a constant and ψ(ω)has horizontal asymptotes,

ii for ω > 0, ωn|F (ω)| has a single maximum at ωn and ωn’s areunbounded,

iii the spectrum is localized in the sense that there are constantsωa and ωb (depending on n), such that

limn→∞

∫|ω|<ωa

ωn|F (ω)| dω = limn→∞

∫|ω|>ωb

ωn|F (ω)| dω = 0,

then the sigmoidal curve y(t) has a critical point located att = α.

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Main Result II

I The first condition is the key “asymptotically constant phase”assumption. If F (ω) is as in (i), then an appropriate shift intime eliminates this phase factor and the phase of F (ω)becomes asymptotically constant.

I The requirement of the existence of a single maximum in (ii)is technical; what we need is ensure that, as n goes to infinity,the the Fourier spectrum of the nth derivatives be shifted tothe region where the phase is nearly constant.

I (iii) is again a technical assumption to ensure that thespectrum of the nth derivative is localized. The proof consistsof expressing |f n(t)| using the Fourier inversion formula andproving that for large n, it is less than |f n(0)|. Intermediatesteps include the determination of the location of thespectrum of the nth derivative, from the equalityω/n = F (ω)/F ′(ω). The behavior of the solutions is shown inFigure 3, for the standard logistic growth.

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Main Result III

−10 −5 0 5 10

−1

−0.8

−0.6

−0.4

−0.2

0

0.2

0.4

0.6

0.8

1

Angular Frequency

Location of Maxima of the FT of the Derivatives

Figure: Graphical solution of the equation ω/n = F (ω)/F ′(ω) forthe standard logistic growth.

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Main Result IV

I Main Result: The localization of the Fourier transform andthe asymptotically constant phase assumptions leads to anintrinsic definition of even and odd components of a functionf (t), by choosing the origin of the time axis in such a waythat the Fourier transform has asymptotically constant phase.

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Thank You...

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