Stochastic effects in population dynamics : simulations and analytical models

Departamento de Físi a da Fa uldade de Ciên ias

Sto hasti ee ts in population

dynami s: simulations and

analyti al models

Ganna Rozhnova

Doutoramento em Físi a

Departamento de Físi a da Fa uldade de Ciên ias

Sto hasti ee ts in population

dynami s: simulations and

analyti al models

Ganna Rozhnova

Tese orientada pela Professora Doutora Ana Nunes

Doutoramento em Físi a

as the Theoreti al Physi ist, is working: one introdu es a model

be- auseone anworkwithit. IfNaturedisagreeswiththemodel,Nature

will be denied a degree in Theoreti al Physi s. After some time,

Na-ture usuallyreapplies forthe examinationsin ea realsystem isfound

orresponding roughly to the Theoreti al Physi ist's model.

First of all, I would like to say the words of gratitude to my

par-ents who have always believed in me and supported me under any

ir umstan es.

Tothe sameextent,I amgratefultomy's ienti mother', myPh.D.

supervisor Prof. Ana Nunes. The door of her o e has been always

open to me both for s ienti dis ussions and for solving personal

problems I have en ountered during the last years. Looking ba k I

realize I annot wish tohave had abetter supervisor than her.

I am thankful to Prof. Alan J. M Kane for the dis ussions we had

during hisvisits toLisbon.

Dis ussions with my olleague Dr. Andrea Parisihave been very

im-portant atseveral stages of my Ph.D. studies. I espe iallyappre iate

his help during the lastproje t in solving a big part of problems

re-lated to the arbitrary pre ision al ulations. In onne tion with the

same proje t I thank Carles Simó who kindly provided me the ode

in Fortranwith the Runge-Kutta7-8pro edure.

Intera tions with my friend Dr. S.N. Dorogovtsev over a number of

years have leadmetothe appre iationand abetterunderstanding of

omplex networks.

I am grateful to my old friend Dr. I. Safonov for introdu ing me to

new software tools,toHelena Cruz,PamelaPa ianiand CarlosReis

for a pleasant working atmosphere in our o e and to the members

ofthe Physi sofBiologi alSystemsResear hGroupofthe Center for

This work ould not have been realizedwithoutthe nan ialsupport

from the Portuguese Foundation for S ien e and Te hnology under

grant SFRH/BD/32164/2006 and from Calouste Gulbenkian F

In this thesis, we address the problem of the modeling of y ling

behavior observed in many systems of populationdynami s. This is

done by studying the role of intrinsi sto hasti ee ts ausedby the

dis rete andrandom hara teroftheintera tionsbetweenindividuals

in anite population(demographi sto hasti ity).

Wefo us on simplesto hasti models whi hin lude anunfor ed and

a periodi allyfor ed epidemi modelfor awell-mixed population, an

epidemi andapredator-preymodelforapopulationwithanimpli it

representation of the network of onta ts between individuals, and

study their long term behavior. We show that, rst, depending on

the parameters of the model, demographi sto hasti ity an generate

oherent u tuations whi h behave as sustained os illations.

More-over, in systems whose sizes represent real populations the role of

sto hasti ee ts be omes fundamental for the interpretation of the

histori aldata. Se ond, thepowerspe trumof theseu tuations an

be al ulated analyti ally using the van Kampen's expansion, both

for the well-mixed unfor ed sto hasti model and for the sto hasti

models with spatial orrelations and periodi for ing.

The appli ation ofthis methodtothe sto hasti epidemi models for

realisti parameter values of hildhoodinfe tious diseases shows how

the powerspe trum al ulated forea hmodelis hangedinthe

pres-en e oftheseadditionalingredients. Namely,westudysystemati ally

how the amplitude and the oheren e of the sto hasti u tuations

hanges, fordierent populationsizes and dierentparameter values.

Wealso showthat, ingeneral, neitherthe fullstru ture of the power

The on lusionsofthisthesis on urwiththoseofseveralre ent

stud-iesto support the view thatthe role of the intrinsi sto hasti ee ts

is fundamental in the explanation of y ling behavior in the systems

of population dynami s.

Keywords: Population dynami s, re urrent epidemi s, hildhood

Nesta tese aborda-se o problema da modelação do omportamento

í li oobservadoemmuitossistemasdadinâmi adepopulações. Esta

abordagem entra-senoestudodaimportân iadosefeitosesto ásti os

intrínse os ausados pelo ará ter dis retoealeatóriodas intera ções

entre osindivíduos dapopulação(esto asti idade demográ a).

Consideram-semodelosesto ásti ossimples,entreosquaisummodelo

epidémi o não forçado ou om forçamentoperiódi o parauma

popu-lação bem misturada, um modelo epidémi o e um modelo

predador-presa in luindo a representação implí ita da rede de onta tos da

populaçao, e estudam-se os seus omportamentos no longo termo.

Mostra-seque,porumlado,dependendodosparâmetrosdadinâmi a,

a esto asti idade demográ a pode gerar utuações oerentes que se

manifestam omo os ilações sustentadas, de modo que, em sistemas

ujos tamanhos representem populações reais, o papeldos efeitos

es-to ásti os passa a ser entral para a interpretação da fenomenologia

eque, poroutrolado, oespe tro destasutuaçõespode ser al ulado

analiti amente ombase naexpansãode vanKampen, tantopara

sis-temas om a oplamentoglobal esem forçamento omo parasistemas

om orrelações e om forçamentoperiódi o.

A apli ação deste método aos modelos esto ásti os de infe ção om

parâmetrosrealistasparaasdoençasinfe iosasinfantismostra omo

oespe tro al ulado para adamodelo semodi anapresença destes

ingredientes adi ionais. Nomeadamente, faz-se o estudo sistemáti o

da dependên ia da amplitude e da oerên ia das utuações

esto ás-ti as para tamanhos de população diferentes e valores dos

As on lusões desta tese e de todoum onjunto de trabalhos

analíti- os e simula ionaisre entes favore em avisãoqueo papel dos efeitos

esto ásti os intrínse os é fundamental para a expli ação do

ompor-tamento í li o emsistemasde dinâmi ade populações.

Palavras have: Dinâmi a de populações, epidemias re orrentes,

List of Figures xiv

List of Tables xx

1 Introdu tion 1

1.1 Review of the literature . . . 1

1.2 The problemaddressed inthe thesis . . . 4

1.3 The stru ture of the thesis . . . 5

2 Deterministi and sto hasti frameworks inthe mean eld ap-proximation 10 2.1 Introdu tion . . . 10

2.2 The epidemi model . . . 10

3 Deterministi and sto hasti frameworks in the pair approxi-mation 20 3.1 Introdu tion . . . 20

3.2 The epidemi model . . . 21

3.3 The mean eld approximation versus the pair approximation . . . 30

3.4 Dis ussion and on lusions . . . 33

4 Sto hasti dynami s on random graphs versus the determinis-ti pair approximation 36 4.1 Introdu tion . . . 36

4.2 The epidemi model . . . 37

4.3 The predator-prey model . . . 43

5 Sto hasti dynami s on random graphs versus the sto hasti

pair approximation 51

5.1 Introdu tion . . . 51

5.2 The epidemi model . . . 52

5.3 The mean eld approximation versus the pair approximation  revisited . . . 64

5.4 The pair approximationurn model . . . 69

5.5 Dis ussion and on lusions . . . 73

6 Sto hasti dynami s on random graphs versus higher order approximations 74 6.1 Introdu tion . . . 74

6.2 The epidemi model . . . 74

6.3 Dis ussion and on lusions . . . 85

7 Sto hasti ee ts in a seasonally for ed epidemi model 87 7.1 Introdu tion . . . 87

7.2 The seasonally for ed deterministi epidemi model . . . 88

7.3 The seasonally for ed sto hasti epidemi model . . . 94

7.4 Results . . . 100

7.5 Dis ussion and on lusions . . . 114

8 Con lusions 117 9 Appendix 120 9.1 Appendix A . . . 120 9.2 Appendix B . . . 126 Referen es 128 Resumo em português 141

2.1 Phase diagram in the

(λ, γ)

plane a ording to the MFA-SIRS deterministi model. Region I represents sus eptible-absorbing

states. InregionIIa tivestates anbeasymptoti allystablenodes

orasymptoti allystablefo i. The riti allineseparating regionsI

and II is given by the equation

λ

MFA

= 1/4

where we hose time

units su h that

δ = 1

and dened

λ = kλ

¯

,

k = 4

. The insets (a) and (b) demonstrate disease extin tion and disease persisten e in

the populationfor parametervaluesin regionsI and II,respe tively. 16

2.2 Plotsofthesus eptibleandinfe tivePSNFs(greenandbla klines,

respe tively) as fun tions of angularfrequen y for the MFA-SIRS

epidemi model. Parameters:

γ = 0.02

,

δ = 1

,

λ = 10

¯

. . . 18 3.1 Phase diagramin the

(λ, γ)

plane for the MFA-SIRS and the P

A-SIRS deterministi models and parameter values for measles (

△

), hi ken pox (

◦

), rubella (



) and pertussis (

⋄

) from data sour es forthe pre-va inationperiod. The stars arethe parameter values

used in Figure3.4. . . 22

3.2 (a) Analyti al and numeri al PSNFs of the infe tives for model

(2.1)-(2.3)with

γ = 0.1

and

λ = 2.5

;(b) thesamefor model (3.9)-(3.17); ( ) a similar plot for model (3.9)-(3.17) with

γ = 0.034

and

λ = 2.5

, noti ethe lin-logs ale; (d) lo ationof the parameter values hosen for (a) and (b) ( ir le)and for( ) (square). . . 31

3.3 (a) and (a) Plots of the peak amplitude

A

of the PSNF of the PA model(3.9)-(3.17)as afun tionof

N

forthe parameter values hosen for (b) and ( ) inFigure 3.2. . . 32

3.4 The steady-state infe tive density given by the PA deterministi

model (dashed blue lines), and by simulations of the PA and the

MFA sto hasti models (solid bla k and green lines, respe tively)

in regions III and II for

N = 10

7

. Parameters are (a)

λ = 7.5

,

γ = 0.01

and (b)

λ = 9

,

γ = 0.01

. . . 34

4.1 Phase diagraminthe

(λ, γ)

plane ofthe PA-SIRSmodelfor

δ = 1

and

k = 2.1, 3, 4, 5

(darker shades of gray for smaller values of

k

). Region I (regions II and III) represents sus eptible-absorbing states (a tive states with nonzero infe tive and sus eptible

densi-ties). The dashed lines orrespond tothe trans riti al bifur ation

urves, see Eq. (4.4); the dottedlines are asso iated witha

super- riti al Andronov-Hopf bifur ation. In region II the xed points

are asymptoti ally stable nodes or asymptoti ally stable fo i, in

region III asymptoti ally stable solutions are limit y les. The

markers depi ting dierent phases (I,II, III) are shown for

k = 2.1

. 39 4.2 For

δ = 1

,

k = 4

, omparisonof the solutions ofthe PA

determin-isti model(dashedlines)withtheresultsofsto hasti simulations

(solidlines) onaRRG-

4

with

N = 10

6

nodes forparameter values

in region II. Sus eptible (infe tive) densities are plotted in gray

(bla k). Parameters: (a)

γ = 2.5

,

λ = 2.5

; (b)

γ = 0.1

,

λ = 2.5

. . . 41 4.3 (a) and (b) Comparison of the solutions of the PA

determinis-ti model(dashed lines) with the results of sto hasti simulations

(solidlines) onaRRG-

4

with

N = 10

6

nodes forparameter values

in region III. Sus eptible (infe tive) densities are plotted in gray

(bla k); ( ) and (d) are plots of the results of simulations (bla k

line)andthe orrespondingsolutionsofthePA-SIRSdeterministi

model(blue line) inthe

(P

S

, P

I

)

plane. Parameters:

δ = 1

,

k = 4

,

γ = 0.025

,

λ = 2.5

. Initial onditionsin (a)and ( ) are equal, and the same is true for (b) and (d). . . 42

4.4 Phase diagram for the predator-prey model for

k = 4

. (a)Region I representsprey-absorbing states andregions II and III represent

a tivestates withnonzero densitiesof predatorsand prey. Several

riti al lines of trans riti al bifur ations between prey-absorbing

and a tive phases are shown for

k = 4

and dierent values of

p

2

. (b) The a tive states an be asymptoti ally stable nodes or asymptoti allystablefo iasinregionIIorstablelimit y lesasin

region III. The phase with stable os illatory behavior is bounded

bysuper riti alAndronov-Hopfbifur ation urveplottedfor

k = 4

and dierentvalues of

p

2

for omparison. . . 46 4.5 For

k = 4

,

p = 0

,

d = 0.017

,

p

2

= 0

, omparison of the solutions

of the PA deterministi modelgiven by Eqs. (4.9)-(4.13) (dashed

lines) with the results of sto hasti simulations (solid lines) on a

RRG-

4

with

N = 10

6

nodes for parameter values in region III.

Predator (prey) densities are plotted in gray (bla k). The three

panels show time interval of equal length in the beginning, in the

middle and inthe end of the same time series. . . 47

5.1 S hemati representationofanodeinstate

α

anditsnearest neigh-bors in states

β

i

, where

i = 1, . . . , 4

and

α, β

i

∈ {S, I, R}

, in a RRG-

4

. . . 54 5.2 Exampleofa ongurationbeforeandafterre overyof the entral

node. The total numberof re overy ongurations is15. . . 56

5.3 Example of a onguration before and after infe tion. The total

number of infe tion ongurations is10. . . 57

5.4 Densities ofthe infe tivesand ofthe sus eptibles inthePA(green

lines) and averaged numeri al time series (bla k lines) obtained

from Monte Carlo simulations of the SIRS sto hasti model on a

RRG-

4

with

N = 10

6

nodes. All plots were obtained for

δ = 1

,

λ = 2.5

and

k = 4

. Parameters: (a)and (b)

γ = 0.09

; ( ) and (d)

5.5 Analyti al PSNFs of the infe tives and of the sus eptibles in the

PA(greenlines)andaveragednumeri alPSNFs(bla klines)

al u-lated from Monte Carlosimulationsof the SIRSepidemi pro ess

on a RRG-

4

with

N = 10

6

nodes. All plots were obtained for

δ = 1

,

λ = 2.5

and

k = 4

. Parameters: (a) and (b)

γ = 0.09

, lin-linplot; ( ) and (d)

γ = 0.04

, lin-log plot. . . 62 5.6 Analyti al(bla k lines)and numeri al(purplelines)PSNFsof the

infe tives for the MFA model and the PA models A and B.

Pa-rameters: (a),(b)and ( )

γ = 0.1

,

λ = 2.5

[ ir lein Figure3.2(d)℄; (d),(e) and (f)

γ = 0.034

,

λ = 2.5

[square inFigure3.2(d)℄. . . 66 5.7 Realization of the PA-SIRS urn model. The 1st urn (above) and

the 2nd urn (below) are shown. The numbers of balls satisfy the

ratios

m

1

: m

2

: (N − m

1

− m2) = 2 : 1 : 7

and

m

3

: m

4

:

(km

1

− m3 − m

4

) = 5 : 15 : 80

for the 1st urn and for the 2nd urn, respe tively. Parameters:

N = 1200

,

k = 4

,

δ = 1

,

γ = 0.15

,

λ = 7.5

. . . 71

6.1 (a) Linear and (b) "T-like" quadruplets both belong to the lass

of open quadruplet ongurations, i.e. lusters of four nodes that

donot ontain any loop. . . 79

6.2 Timeevolutionofsus eptible(leftpanels)andinfe tive(right

pan-els) densities for parameter values in the endemi region of the

phasediagramofthe PAmodelaspredi tedbytheTAmodel(red

lines), thePA model(blue lines)and the resultsof sto hasti

sim-ulations (bla k lines) on a RRG-

3

with

N = 10

6

nodes. All plots

were obtained for

δ = 1

,

λ = 15

and

k = 3

. Parameters: (a) and (b)

γ = 2

;( ) and (d)

γ = 0.2

; (e) and (f)

γ = 0.05

. . . 82 6.3 The same data as in Figure6.2with

δ = 1

,

λ = 2.5

for

k = 4

and

RRG-

4

with

N = 10

6

nodes. Parameters: (a)and (b)

γ = 2.5

; ( ) and (d)

γ = 0.1

;(e) and (f)

γ = 0.025

. . . 84 7.1 The fra tionof infe tive individualsasa fun tionof time for

peri-odi solutions of Eqs. (7.2)-(7.4). Parameters: (a)

β

1

= 0.05

, (b)

7.2 Theupperpanelsshowtheauto orrelationfun tionsofthe

sto has-ti u tuations of infe tives. The theoreti al urves are plotted in

blue, and the urved omputed from the simulations are plotted

inbla k. The lowerpanels showthe orresponding powerspe tra.

The verti al helper lines mark the frequen ies predi ted by Eq.

(7.42). The dashes(dotted) linesare al ulatedby takingthe plus

(minus) in the equation. The system size used for simulationis

10

8

.103

7.3 Power spe tra of the numberof infe tives al ulated from

simula-tionsforseveral systemsizes

N

. Thesimulationsof500years were run for

N = 10

8

_{, 10}

7

_{, 5 × 10}

6

_{, 10}

6

andof 100years for

N = 5 × 10

5

.

Averagesover

10

3

realizationsweremadetoobtainea h urve. The

for ingamplitudeislowand orrespondstotheannuallimit y le,

β

1

= 0.02

. Theverti alhelperlinesmarkthefrequen iespredi ted by Eq. (7.42). . . 105

7.4 Power spe tra of the numberof infe tives al ulated from

simula-tionsforseveral systemsizes

N

. Thesimulationsof500years were run for

N = 10

8

_{, 10}

7

_{, 5 × 10}

6

_{, 10}

6

andof 100years for

N = 5 × 10

5

. Averagesover

10

4

(

10

3

)realizationsweremadetoobtainthepower

spe tra for

N = 10

8

(for the remaining system sizes). The

for -ing amplitude is

2.5

higher than inFig. 7.3 but the deterministi system is still in the annual limit y le regime,

β

1

= 0.05

. The verti alhelper lines markthe frequen ies predi ted by Eq. (7.42). 107

7.5 Power spe tra of the numberof infe tives al ulated from

simula-tionsforseveral systemsizes

N

. Thesimulationsof500years were run for

N = 10

8

_{, 10}

7

_{, 5 × 10}

6

and of 100 years for

N = 10

6

. A ver-ages over

5 × 10

3

(

10

3

)realizationswere madetoobtainthe power

spe tra for

N = 10

8

(for the remainingsystem sizes). The for ing

amplitude orresponds to the biennial limit y le,

β

1

= 0.2

. The verti al helper lines markthe frequen ies predi ted by Eq. (7.42). 108

7.6 Power spe tra of the numberof infe tives al ulated from

simula-tions for several system sizes

N

. The blue(bla k) urveswere ob-tained for

β

1

= 0.1

(

β

1

= 0.12

, respe tively). In the deterministi model the period doubling o urs at

β

′

1

≈ 11.479

. The frequen- iesoftheverti alhelperlines orrespondtotheannuallimit y le.

Thesimulationsof500yearswereusedfor

N = 10

8

_{, 10}

7

_{, 5×10}

6

_{, 10}

6

and of 100 years for

N = 5 × 10

5

. Averages over

10

3

realizations

were madeto obtainall urves. . . 110

7.7 Power spe tra of the numberof infe tives al ulated from

simula-tions for twosystem sizes

N

and two sets of initial onditions(for the blue urve simulations started from random initial onditions

and for the bla k urves the initial onditions were hosen lose

to the deterministi triennial y le). In the left (right) panel the

verti al helper lines mark the predi ted peak frequen ies for the

triennial (annual) limit y le. The amplitude of seasonal for ing

orresponds to oexisting stable annual and triennial limit y les

in the deterministi model,

β

1

= 0.1

. . . 111 7.8 The infe tive density re orded from a typi al realization of the

sto hasti modelstartingonthedeterministi trienniallimit y le.

The seasonal for ing amplitude is

β

1

= 0.1

. . . 112 7.9 The infe tive density re orded from a typi al realization of the

sto hasti modelstarting on the deterministi annual limit y le.

5.1 The values of the peak frequen y,

ω

peak

, amplitude of the peak,

A

peak

, overall ampli ation,

M

, and oheren e,

C

, a ording to the analyti alpredi tionsofthe threemodels: the MFAsto hasti

model, the PA oarsegrainedsto hasti model(modelA) and the

PA detailed sto hasti model (model B). Parameters orrespond

tothe (a),(b)and ( ) plots of Figure5.6. . . 68

5.2 DataasinTable5.1. Parameters orrespondtothe (d),(e)and (f)

plots of Figure5.6. . . 68

6.1 Analyti al and numeri al values for the mean,

N

L

, and the vari-an e,

V ar(N

L

)

,of the numberof loopsof length

L = 3, 4

inRRG-

4

. 76 7.1 Parameter values for measles that will be used in this Chapter.

A ording to Eq. (7.8) this set orresponds to

R

0

= 15.74

. The amplitude of seasonal for ing will be varied so that the solutions

ofEqs. (7.2)-(7.4) exhibitstablelimit y lesof period

T

indi ated in the parenthesis next tothe

β

1

value. . . 91 7.2 Floquet multipliers,

λ

1

, λ

2

, λ

3

, and Floquet exponents,

ρ

1

, ρ

2

, ρ

3

,

forlimits y lesofdierentperiodsandseveralvaluesofthefor ing

Introdu tion

1.1 Review of the literature

The models of population dynami s are among the rst examples of the

intro-du tion of quantitative methods in biologi ally inspired problems [14℄. In the

ontextofe ologyandepidemiology,many ontributions,rst,frommathemati s

[5, 6℄ and, more re ently, fromphysi s [7℄ have inthe ourse of the last entury

helped to explain the fundamental aspe ts inthese phenomena and to onstru t

models apableofreprodu ingthe phenomenologyandofoutliningsound

strate-gies of intervention.

Avery ommonapproa hinpopulationdynami sstudies,and oneofthe rst

to have been proposed, is toassume that populationis not spatially distributed

so that individuals mix perfe tly and onta t ea h other with equal probability

[5℄. Thus, in the limit of innite population the time evolution of the system

is des ribed in terms of the densities of dierent subpopulations and governed

by a set of ordinary dierential equationswhi h an be dedu ed from the law of

mass a tion [5,8℄. The Kerma k-M Kendri k modelof epidemi s [9℄, the

Lotka-Volterra model of predator-prey intera tions [10, 11℄ and their extensions [5, 8℄

are lassi al examples of this simple deterministi approa h. In spite of the fa t

that these traditional analyti models assume trivial intera tion stru tures [7℄

and are unable to modelu tuations [1214℄ or any type of orrelations [15, 16℄

in a population they were a starting point for many further elaborations of the

Manyofthemorere ent ontributions totheeldofpopulationdynami sare

relatedwiththemodi ationofthetraditionalmodelssoastoin ludethedis rete

andrandom hara teroftheintera tionsbetweenindividuals[19,20℄andalsothe

stru ture of onta ts in a population[7, 21℄  two main aspe ts whose impa t

on the behaviorof amodelhas been shown tobevery important. Some of these

works (e.g. [22, 23℄) had themselves a high impa t on the s ienti ommunity,

and there exists today a widespread a knowledgment of the fundamental role

these twoingredientsplay inthe phenomenologyofthe dynami sof populations.

The insight that spa e (or onta t) mediated intera tion exists and plays an

important role in the e ology and epidemiology of populations is not new [5,6℄.

Atthedeterministi level,thespa eingredient anbein orporatedexpli itlyinto

the model by the introdu tion of a new ontinuous variable representing spa e

[5℄. One then investigates the dynami s using rea tion-diusion based models.

Mathemati ally,thismeansthattheequationsunderstudyarepartialdierential

equations [24℄. Rea tion-diusion models have been largely analyzed and shown

to give reasonable predi tions for problems where the intera tion depends on

the distan e between individuals,namely, inplant e ology, in problems of inse t

and animal dispersion, et . [25℄. Another su essful example of modeling based

on this approa his the Fisher-Kolmogorov like equation[5, 26℄for the spread of

buboni plagueinEuropebetween 1348and1350,alsoknownastheBla kDeath

[27℄. The rea tion-diusionapproa hfails, however, in the des ription of human

intera tions that an be viewed as omplex networks [2834℄. Another obvious

limitation of these models is that by their very nature they do not in orporate

dis reteness of individuals and therefore ignore u tuations that are present in

real populations [35, 36℄.

Anotherin reasinglypopularapproa h tothe study of epidemi or

predator-prey dynami s is by means of simulations of sto hasti models onlatti es or on

more general graphs, destined for implementation on a omputer [3740℄. In

this approa h,the intera tions between individuals propagate through a onta t

network [4144℄ where the dis rete variables asso iated to ea h node represent

the presen e and attributes of anindividual, e.g. sus eptible, infe tive or

literature, thesimulationstudiesongraphsrangingfromlatti esto omplex

net-works have been usually ombinedwith analyti alwork[48,49℄. The onne tion

between expli it sto hasti simulations and deterministi mean eld

approxima-tion models valid for innite population [49℄ lies in rather heuristi analyti al

approa hes to the derivation of dierential equations for the time evolution of

the densities of nodes  and higher order spatial ongurations  in the

dif-ferent states dened by the underlyingsto hasti model. In these equations,the

stru ture of onta tsinthepopulationiseithernotpresentatall[5053℄,oronly

impli itly present through the pair [5460℄ or more elaborated approximations

[6168℄, or, yet, in networks in whi h the degree heterogeneity is signi ant, it

intervenes through the degree distribution of nodes only [6973℄. Thus, in this

line of resear h, apart fromthe analysis and interpretation of the resultsof

sim-ulations for the purpose of the model's on eptual dis ussion, the study of the

regimesof validityof dierent meaneld approximationsisof interestbothfrom

the theoreti al pointof view and fromthe pointof view of appli ations[7480℄.

Finally, the most re ent ontributions onsist in the analyti des ription of

the intrinsi sto hasti ee ts ausedbytheo urren eofrandomintera tionsin

a nite population (dubbed demographi sto hasti ity in [22, 39℄), in luding or

notin ludingarepresentationofthenetworkof onta ts [8187℄. Thefoundation

of this new line of resear h has been laid down in [81℄ where a general

me ha-nism of resonant ampli ation of demographi u tuations has been proposed

to des ribe the y ling behavior of predator-prey systems. It was noti ed that

whileitisdi ulttoexplainthe y lingbehaviorofpredator-preysystems using

simple deterministi models valid for innite population (e.g. without re urring

to spe i nonlinear terms in the deterministi equations), large u tuations or

y les are present in the asso iated sto hasti models. The explanation of this

phenomenon wasgivenanalyti allyfromthestatisti alphysi s'perspe tive. A

-nitesto hasti system anbeimaginedasperturbedbyaninternal(demographi )

noisewhoseamplitudeisdeterminedbytheparametersandthesizeofthesystem,

and automati allya hievesresonan e atafrequen y endogenoustoea h system.

The analyti al des ription of the sto hasti u tuations was given by expanding

the master equation of the sto hasti modelaroundthe equilibrium pointof the

The equations for the u tuations resulting from the expansion are linear and

allow for the al ulation of the u tuations power spe trum. This phenomenon

willbefurtherdis ussedbelow, in onne tionwith there urren eandperiodi ity

of epidemiologi aldata re ords.

1.2 The problem addressed in the thesis

A key test forthe assumptionsunderlyingthe modelsand the newideas brought

totheeldofpopulationdynami shasalwaysbeentheirabilitytoexplain

spatio-temporalpatternsobservedinrealdata. Childhoodinfe tiousdiseaseshaveoften

been taken as a ase study and model testing ground, be ause de ades long of

fairly well time resolved data re ords are available,on one hand, and be ause of

their dierent phenomenology despite the similarities in ontagion me hanisms

and in infe tious, latent and immunity waning typi al times [89℄. Cy les are a

very striking behavior of epidemiologi al systems [90, 91℄ also seen in

predator-prey systems [35, 36℄  a ase in point is the pattern of re urrent epidemi s of

many hildhood infe tions.

The ontroversy in the literature over the driving me hanisms of the

perva-sive noisy os illationsobserved in these systems has been going onfor long [92℄,

be ausethe simplestdeterministi models predi tdamped, instead of sustained,

os illations [93℄. One of the aspe ts of this ontroversy is whether these

me h-anisms are mainly external or intrinsi , and the ee ts of more realisti latent

and infe tious period distributions [9496℄, of seasonal for ing terms [97102℄

and of higher order non-linear intera tion terms [103, 104℄ have been explored

in the framework of a purely deterministi des ription of well-mixed, innite

populations. These more elaborate models exhibit os illatory steady states in

ertain parameterranges, and have ledtosu essful modelingwhen external

pe-riodi for ing is of paramount importan e [105, 106℄, but they fail to explain

the widespread non-seasonalre urrent outbreaks of hildhood infe tions[93℄. In

a deterministi framework of this kind, the role of sto hasti ity is merely that

of ausing a lo al disease extin tion [12, 107℄ or making the system to swit h

between oexisting attra tors of the deterministi model[98, 99, 108℄. Another

realdataaretheeviden eofsto hasti ee tsthatwouldshowupinthereal

pop-ulations asnoisyos illationswiththe same frequen y asthe damped os illations

of the deterministi system [109, 110℄.

This thesis argues for the fundamental role of sto hasti ity in the long term

behavior of disease spread. The on urrent on lusions of a whole set of re ent

analyti al and simulation studies [81, 83, 84, 86℄, in luding those developed in

this thesis, are the following. First, depending onthe parametersof a given

dis-ease, demographi sto hasti ity an generate oherent u tuationswhi hbehave

assustained os illations. Moreover, insystems whosesizesrepresentreal

popula-tions, the role ofsto hasti ee ts be omesfundamentalfor theinterpretationof

the histori aldata re ords. Se ond, thepowerspe trumoftheseu tuations an

and should be al ulated analyti ally using the van Kampen's expansion, both

for well-mixed unfor ed systems and for systems with spatial orrelations and

periodi for ing. The appli ation of this method tomodels of infe tion for

real-isti parameter values of hildhood infe tious diseases will show how the power

spe trum al ulatedforea hmodelis hangedinthepresen e oftheseadditional

ingredients. Namely,asystemati study ofhowthe amplitudeand the oheren e

of the sto hasti u tuations hanges, for dierent population sizes and

dier-ent parameter values, will be presented. It will also be shown that, in general,

neither the full stru ture of the powerspe trumnor the main frequen y (or

fre-quen ies) orresponding to the sto hasti u tuations an be fully predi ted by

the onventional deterministi theory.

Ihopethatboththeideasandtheresultsofthisthesiswillbeofinteresttothe

s ienti ommunity,andwill ontributetotheunderstandingofthephenomenon

of y les invarious systems of populationdynami s.

1.3 The stru ture of the thesis

The followingparagraphs giveabriefdes riptionof the ontentsand theoriginal

ontributions of ea h of the following Chapters of the thesis. Chapter 2

orre-sponds to the presentation of the basi epidemiologi al model for a well-mixed

populationthe deterministi andthe sto hasti versionsofthe ompartmental

to this model of the method of the van Kampen's expansion for the al ulation

of the power spe trumof sto hasti u tuations.

InChapter 3asto hasti versionoftheSIRSmodelinthepairapproximation

for anetworkof homogeneousdegree

k

isintrodu ed. This model orrespondsto animpli itand minimalrepresentation ofthe spatial orrelationsinany network

model with this property. For

k = 4

, the bifur ation diagram of the pair ap-proximation modelin the limitof innite populationis des ribed. This analysis

ompletes the results in the literature revealing the existen e of an os illatory

phaseinasmallparameterregion orrespondingto hildhoodinfe tiousdiseases.

Apart fromthis phase and an absorbing phase where atrivial equilibriumisthe

onlyattra torthereexistsaphasewheretheonlyattra torisanendemi

equilib-rium. Theu tuationsintheendemi phaseare studiedusingthevanKampen's

expansion applying the method des ribed in Chapter 2 to the sto hasti pair

approximation model. The onditions of onvergen e of the expansion for the

u tuations about the endemi equilibrium break down onthe boundary of the

endemi phase with the os illatory phasewhi h orresponds to the emergen e of

os illationsthat s ale with the system size. The analysis of the power spe trum

intheendemi phasedemonstratestwointerestingee ts. Onone hand,the

u -tuations are more oherent and of mu h higher amplitude than those observed

in the model without spatial orrelations. This ee t had been des ribed inthe

literature on the basis of systemati simulations in 'small-world' networks, and

here itistreatedanalyti allyforthe rst time. On theotherhand, thedominant

frequen y inthe u tuations powerspe trumisshifted signi antlywith respe t

tothefrequen y ofthe damped os illationsinthe vi inityoftheendemi

equilib-rium of the model for a well-mixed innite population. That is, the presen e of

spatial orrelations modies not only the amplitude and oheren e but alsothe

frequen y of sto hasti u tuations.

Chapter 4 dis usses the relevan e of the os illatoryphase des ribed in

Chap-ter 3 and in general of the sto hasti SIRS model in the pair approximation by

omparing the results of this modelwith the results of sto hasti simulationson

regular random networks of degree

k

. This omparison is extended to a lass of predator-prey models whi h also exhibit an os illatory phase in a small

observed inthesimulationsonnetworks,and thatthe qualityofthe pair

approx-imation deterioratesin the parameter region lose to the boundary of this phase

forboth lassesofmodels. Takingintoa ountthatregularrandomnetworksare

thebest andidatesforagoodapproximationatthe levelofpairsthis omparison

shows that the os illatory phase annot explain sustained os illations observed

in real systems. On the other hand, the possibility of another type of

nonlin-earities in the deterministi equations orresponding to innite population has

beenlargely explored inthe literature ofthe lastde ades withoutithavingbeen

possibletoobtainrobustos illationsinbiologi allyplausiblemodels. Hen e, this

negative result reinfor es the importan eof sto hasti ee ts in the explanation

of the sustained os illations observed in the time series of various

epidemiolog-i al and e ologi al systems su h as, for example, re urrent epidemi s typi al of

hildhood infe tiousdiseases.

Chapters 5 and 6 address the problem of the failure of the pair

approxima-tion found in Chapter 4. In Chapter 5 a pair approximation sto hasti model

for a homogeneousnetwork of degree

k

is onstru ted by modifyingthe pair ap-proximation sto hasti modeldeveloped in Chapter 3. This is done through the

in lusion of a mi ros opi des ription of all possible transitions in the modied

model. It is found that the power spe trum of the sto hasti u tuations

mea-suredfromsimulationsonregularrandomnetworksisapproximated mu hbetter

by the analyti al power spe trum omputed from the van Kampen's expansion

of themodiedsto hasti model. However, the behaviorofboth pair

approxima-tion models is qualitativelythe same and, in parti ular, the modied sto hasti

model also predi ts an os illatory phase whi h is suppressed in the simulations.

In Chapter 6, it is shown that to approximate the behavior of simulations on

networks for the parametervaluesof hildhoodinfe tious diseasesitis ne essary

to go beyond the pair approximation and onsider models losed at the level of

triplets.

Chapter 7 deals with a model with periodi for ing representing a seasonal

for ing whi h is widely a epted as being an important fa tor in the dynami s

of hildhood infe tious diseases. The method of al ulation of the power

spe -trum of the sto hasti u tuations based on the van Kampen's expansion has

autonomous or afor ed (non autonomous)system that represents the modelfor

innite population. Applying this method to the

sus eptible-exposed-infe tive-re overed (SEIR) modelwith a periodi for ing in the infe tion rate, the power

spe trum of u tuations around the dominant attra tor is omputed for

dier-ent values of the for ing amplitude. The analyti al results agree perfe tly with

the power spe trum measured from sto hasti simulations and allow to predi t

the positions of frequen ies of the seasonal or non-seasonal peaks for hildhood

infe tions. The on lusions of this study reinfor e the view that it is the

oher-entu tuations aroundthedominantattra tor andnot the ompetitionbetween

dierent attra tors of the for ed deterministi system that shape the time series

observedinthemajorityofthein iden ere ordsfor hildhoodinfe tiousdiseases.

Finally,inChapter8wesummarizethemain on lusionsofthepresentstudy.

Theoriginalresultspresented inthisthesis have beenpublished orsubmitted

as follows.

Chapter 2and Chapter 3have been published as:

•

G. Rozhnova and A. Nunes. Flu tuations and os illationsin a simple epi-demi model. Phys. Rev. E 79, 041922 (2009) and Virtual Journal of

Biologi al Physi s Resear h 17(9), May 1 (2009).

Chapter 4has been published as:

•

G. Rozhnova and A. Nunes. SIRS dynami s on random networks: simu-lations and analyti al models. In Complex S ien es: Complex 2009, Part

I, LNICST 4, edited by J. Zhou (Springer Berlin Heidelberg, 2009), pp.

792797

•

G. Rozhnova and A. Nunes. Population dynami s on random networks: simulations and analyti al models. Eur. Phys. J. B 74, pp. 235242

(2010).

Chapter 5and Chapter 6have been published as:

•

G.Rozhnova andA.Nunes. Clusterapproximationsforinfe tiondynami s onrandomnetworks. Phys. Rev. E 80, 051915(2009)andVirtual Journal

Chapter 7has been submitted as:

•

G. Rozhnova and A. Nunes. Sto hasti ee ts in a seasonally for ed epi-demi model. June, (2010).

Deterministi and sto hasti

frameworks in the mean eld

approximation

2.1 Introdu tion

In this Chapter, we introdu e the theoreti al framework of the mathemati al

modeling of epidemi s in the homogeneously mixed population. In both the

de-terministi and in the sto hasti des riptions, we make use of the on ept of

ompartments whi h is at the basis of modern epidemiology. The

ompartmen-tal approa h assumes that a population an be divided intoseveral lasses with

respe t to the individuals' disease status, su h as, for example, sus eptibles,

in-fe tivesand re overed. The individualswithinea h ompartment are onsidered

to be homogeneously mixed, and their dynami s between the ompartments is

spe ied by the rate onstantsdepending onthe etiologyof a given disease.

2.2 The epidemi model

In order to x notation in the simplest setting, let us start by onsidering the

sto hasti and deterministi des riptions of the

is when the populationis assumed to be homogeneouslymixed.

In the sto hasti version of the MFA-SIRS model a losed populationof size

N

, at a given time

t

, onsists of

m

1

individuals of type

S

(sus eptible),

m

2

individuals of type

I

(infe ted and infe tious), and

(N − m

1

− m

2

)

individuals of type

R

(re overed). The individuals are grouped simplyby theirtype and are identi alwithinagiven lass. Nowwepostulatethatthe dynami softhe system

an beessentiallydes ribed by the following three pro esses:

1. Re overy

I −→ R

with arate

δ

; 2. Immunity loss

R −→ S

with arate

γ

; 3. Infe tion

S −→ I

with a rate

λm

¯

2

/N

.

Therst andthe se ondpro esses involveonlyone individualofagiven lass

and are independent ofthe density ofindividualsinother lasses. Thethird

pro- ess involves two individualssimultaneously be ause an infe tion an take pla e

only when there is a onta t between a sus eptible and an infe ted individual.

In the MFA formulation of sto hasti models, the ommon way to dene the

transition rates is a ording to the law of mass a tion. Namely, we onsider a

homogeneousapproximationin whi hthe probability forea h individualof lass

S

to intera t with an individual of lass

I

is simply proportional to the density of infe ted individuals,

m

2

/N

. The rate dening onstant

λ

¯

depends both on the infe tion probability and on the onta t rate in the population. In the ase

when the sto hasti MFA is onsidered as an approximation of a sto hasti

lat-ti e(network)modelthis onstantin ludesaparameter hara terizingthelatti e

(network)topology. Forexample,forgraphsof xed oordinationnumber

k

(the number of nearest neighbors of a node) the onstant

¯

λ

is equal to the infe tion probability

λ

multiplied by

k

.

In the sto hasti frameworkthe way to pro eed then isto hoose individuals

randomlyandtoseewhetherthey hangestatea ordingtothepro essesdened

above. The state of the system

m = (m

1

, m

2

)

is dened by the number of sus eptible and infe ted individuals and hanges dis retely whenever one of the

during a time interval

∆t

depends on the urrent number of individualsin ea h lass and therate onstants. Forexample,the state

m = (m

1

, m

2

)

will hangeto the state

m

′

_{= (m}

1

, m

2

− 1)

if a re overy event is exe uted. The probability of this o urren e in

∆t

is given by

Q

_(m

′

_{, t + ∆t|m, t) = δm}

₂

_{∆t + o(∆t),}

where

o(∆t)

denotes termsthat are negligibleforsmall

∆t

. As explainedbefore, in the ase of an infe tion event the probability that infe ted and sus eptible

individuals meet is taken to be proportional to their urrent numbers in the

population.

In this way, the transition rates asso iated to the three pro esses postulated

above an be written asfollows:

T

m

1

−1,m

2

+1

m

1

,m

2

= ¯

λm

1

m

2

N

,

(2.1)

T

m

1

,m

2

−1

m

1

,m

2

= δm

2

,

(2.2)

T

m

1

+1,m

2

m

1

,m

2

= γ(N − m

1

− m

2

).

(2.3) Here

T

m

1

+k

1

,m

2

+k

2

m

1

,m

2

denotes the transitionrate fromstate

(m

1

, m

2

)

tostate

(m

1

+

k

1

, m

2

+ k

2

)

,

k

i

∈ {−1, 0, 1}

,where

i = 1, 2

.

Notethatbe ausethe transitionratesdependonlyonthe urrentstateofthe

system and not on the previous states the underlying pro ess is Markov. The

standard analyti way to deal with this kind of pro esses is to write down a

master equation whi h gives a omplete des ription of the time evolution of the

postulatedsto hasti model[88, 111℄. Giventhe initialandboundary onditions,

its solutionisthe probability of havingthe system ina given stateat time

t

and is equivalent to the full sto hasti simulation of the model. If the range of the

sto hasti variable is a dis reteset of states with labels

m

this equation has the followinggeneral form:

dP(m, t)

dt

=

X

m

′

_6=m

h

T

m

m

′

P

(m

′

, t) − T

m

′

m

P

(m, t)

i

,

(2.4)

where

T

m

m

′

denotes transitionratesfromotherstates

m

′

tostate

m

and vi eversa for

T

m

′

m

.

Takingintoa ount Eqs. (2.1)-(2.3) the master equation(2.4) for the MF

A-SIRS sto hasti pro ess be omes:

dP(m

1

, m

2

, t)

dt

= T

m

1

,m

2

m

1

−1,m

2

P

(m

1

− 1, m

2

, t) + T

m

1

,m

2

m

1

,m

2

+1

P

(m

1

, m

2

+ 1, t)

− T

m

1

−1,m

2

+1

m

1

,m

2

+ T

m

1

,m

2

−1

m

1

,m

2

+ T

m

1

+1,m

2

m

1

,m

2

P(m

1

, m

2

, t)

+ T

m

1

,m

2

m

1

+1,m

2

−1

P

(m

1

+ 1, m

2

− 1, t).

(2.5)

The omplete solution of this dierential-dieren e equation

P

(m

1

, m

2

, t)

gives the probability of nding the system in state

(m

1

, m

2

)

for all allowed sets of integers

m

1

,

m

2

attime

t ≥ 0

subje tto the initial,normalizationand boundary onditions. In general, it is not easy to solve this equation analyti ally but it

is quite straightforward to analyze it for large but nite

N

using van Kampen's system size expansion[88℄. In that spirit, we set

m

1

(t) = NP

S

(t) +

√

Nx

1

(t),

(2.6)

m

2

(t) = NP

I

(t) +

√

Nx

2

(t).

(2.7)

In both equations,therst ma ros opi termss ale withthe system size

N

. The fun tions

P

S

(t) = lim

N →∞

m

1

(t)/N

and

P

I

(t) = lim

N →∞

m

2

(t)/N

are densities of

sus- eptible and infe ted populations whi h have to be adjusted so as to satisfy the

deterministi equations of motion inthe MFA.

x

1

(t)

and

x

2

(t)

are the new vari-ables whi hdenote sto hasti u tuationsaround the orrespondingsolutions of

the MFA deterministi equationsand repla e

m

1

(t)

and

m

2

(t)

, respe tively. The time-dependent transformations (2.6)-(2.7) from

m

1

(t)

,

m

2

(t)

to

x

1

(t)

,

x

2

(t)

in-volvingfun tions

P

S

(t)

,

P

I

(t)

omefromthefa tthatoneexpe tstheprobability distribution fun tion

P

(m

1

, m

2

, t)

tohave a sharp peak around the ma ros opi values

m

1

(t) = NP

S

(t)

,

m

2

(t) = NP

I

(t)

withawidthoforderof

√

fun tions

P

S

(t)

,

P

I

(t)

followthe motionof the peakintime. Under the transfor-mations (2.6)-(2.7) the fun tion

P

(m

1

, m

2

, t)

of

m

1

(t)

,

m

2

(t)

transforms intothe fun tion

Π(x

1

, x

2

, t)

of

x

1

(t)

,

x

2

(t)

a ording to

P

_(m

₁

_{, m}

₂

_{, t) = P(NP}

_S

₊

√

_{N x}

₁

_{, NP}

_I

₊

√

_Nx

₂

_{, t) ≡ Π(x}

₁

_{, x}

₂

_{, t).}

(2.8)

Substituting Eqs. (2.1)-(2.3) and Eqs. (2.6)-(2.8) into Eq. (2.5), the

large-N

expansion of Eq. (2.5) an be ee tively arried out (see Appendix A for te hni al details). The leading-order terms of the expansion give rise to the set

of the MFA-SIRS deterministi equations of motion for the average sus eptible

and infe tivedensities 1 :

dP

S

dt

= γ (1 − P

S

− P

I

) − ¯λP

S

P

I

,

(2.9)

dP

I

dt

= ¯

λP

S

P

I

− δP

I

.

(2.10)

The above set of dierential equations is well known and is usually written as

a phenomenologi al des ription of the SIRS epidemi at the population level.

The model is dedu ed from the law of mass a tion, on the assumption that the

populationislargeenoughfortheu tuationstobenegligibleandforthenumber

of individuals in ea h lass to be approximated by a real variable that evolves

deterministi allyover time.

TheMFA-SIRSmodelis hara terizedby theexisten eoftwopossiblesteady

states depending on the parameters

λ

¯

,

δ

, and

γ

. We al ulate them by setting the l.h.s. of Eqs. (2.9)-(2.10) equal to zero. Let

P

¯

S

and

P

¯

I

denote the steady state values of the sus eptible and infe tive densities of the MFA-SIRS model.

The rst solution

1

The equations for the average densities

P

S

= hm

1

i/N

and

P

I

= hm

2

i/N

in the limit

N → ∞

an be obtainedby multiplyingEq. (2.5)by

m

1

and

m

2

in turn, andthen summing overallthestatesofthesystem[88℄.

¯

P

I

= 0, ¯

P

S

= 1

(2.11)

orresponds tothe steadystate inwhi hthe wholepopulationissus eptible

(no- oexisten e region). The se ond solution

¯

P

I

=

γ(¯

_{λ − δ)}

¯

λ(γ + δ)

, ¯

P

S

=

δ

¯

λ

(2.12)

is a xed pointwhi h orresponds tononzero sus eptible and infe tive densities

( oexisten e region). This steady state is alled endemi and it an be seen that

it is ameaningfulsolution of the modelonly when

¯

λ > δ

.

The phase diagram obtained from the linear stability analysis of Eqs.

(2.9)-(2.10) is typi al of many simple deterministi models of infe tion dynami s (see

Figure 2.1). The trivialxed point (2.11) whi h orresponds to the

sus eptible-absorbing state is asymptoti ally stable in region I. To the right of the riti al

line dened by

¯

λ

MFA

= δ,

(2.13)

viz in region II, this point be omes unstable giving rise to a nontrivial stable

xed point (2.12) whi h is an a tive state. In region II the xed point an be

an asymptoti ally stable node or an asymptoti ally stable fo us. Although it

is possible to dedu e an analyti al expression

λ = f (γ)

¯

for the urve separating domainsinregionIIa ordingtothewaythenontrivialxedpointisapproa hed,

most ofthetime wewillnot distinguishbetween them. The reasonisthat inthis

study we willbeinterested in omparing the results of analyti models with the

results of sto hasti simulations for the stationary states. At this point,we only

stress that the damped os illationsin the populationdensitiesimplying thatthe

xed point isa stable fo us be ome noti eableonly for small

γ

.

We would also like to draw the reader's attention to the spe i parameters

usedtorepresentthephasediagram. First,wetakethere overyrate

δ

tobeequal to unity whi h sets the time s ale and allows us to deal with a two-dimensional

0

0.25

0.5

0.75

1

1.25

1.5

1.75

2

0

0.1

0.2

0.3

0.4

0.5

0.6

0.7

0.8

0.9

1

0

2.5

5

7.5

10

0

0.2

0.4

0.6

0.8

0

5

10

15

0

0.05

0.1

0.15

PSfrag repla ements

t

t

P

I

P

I

γ

λ

(b) region II, diseasepropagation (a) regionI, disease extin tion

II I

Figure 2.1: Phase diagram in the

(λ, γ)

plane a ording to the MFA-SIRS de-terministi model. Region I represents sus eptible-absorbing states. In region

II a tive states an be asymptoti allystable nodes or asymptoti allystable fo i.

The riti al lineseparating regions I and II is given by the equation

λ

MFA

= 1/4

where we hose time units su h that

δ = 1

and dened

¯

λ = kλ

,

k = 4

. The insets (a) and (b) demonstrate disease extin tion and disease persisten e in the

diagraminsteadof athree-dimensionalone. Se ond,weintrodu eanew

parame-ter

λ

a ordingtotheequation

λ = kλ

¯

sothatEqs. (2.9)-(2.10)be ometheMFA equationsof the sto hasti SIRSpro ess onageneralgraphwhere ea hnodehas

degree

k

. Inthe futuredis ussionofthepairapproximationwewill,inparti ular, refer to the study [75℄ by J. Joo and J. L. Lebowitz who performed sto hasti

simulations of the SIRS pro ess onthe square latti e that motivated our hoi e

of

k = 4

. Inthis notationtheMFA riti alline(2.13) separatingregionsI and II is given by

λ

MFA

= 1/4

.

Turningba ktothesto hasti pro essdenedbyEq. (2.5)anditspowerseries

expansion inthe system-size

N

the next-to-leading order termsof the expansion yieldamultivariatelinearFokker-Plan kequationfortheprobabilitydistribution

fun tion

Π(x, t)

(see Refs. [88, 111℄ and Appendix A):

∂Π(x, t)

∂t

= −

X

i,j

A

ij

∂(x

j

Π(x, t))

∂x

i

+

1

2

X

i,j

B

ij

∂

2

_{Π(x, t)}

∂x

i

∂x

j

.

(2.14)

Here

i, j = 1, 2

and

x = (x

1

, x

2

)

are sto hasti u tuations of the densities of infe ted and sus eptible individuals about their endemi steady state values in

the MFA. A is the Ja obian matrix of Eqs. (2.9)-(2.10) linearized about the

endemi equilibriumsolution(2.12):

A =

1

(γ + δ)

 −γ(γ + ¯λ) −(γ + δ)

2

γ(¯

_{λ − δ)}

0



.

(2.15)

B

is symmetri internal noise ross orrelation matrix derived dire tly fromthe expansion of Eq. (2.5):

B =

δγ

_¯

λ

(¯

_{λ − δ)}

(γ + δ)



2

₋₁

−1

2



.

(2.16)

Sin e the Fokker-Plan k equation(2.14) is linear itssolution,

Π(x, t)

, is a multi-variate Gaussiandistribution ompletelydetermined by the rst and the se ond

moments. We, however, will be interested in studying the stru ture of the

0

0.2

0.4

0.6

0.8

1

0

1

2

3

4

5

6

7

PSfrag repla ements

ω

PSNF s P S

(ω)

P I

(ω)

Figure 2.2: Plots of the sus eptible and infe tive PSNFs (green and bla k lines,

respe tively)asfun tionsofangularfrequen yfortheMFA-SIRSepidemi model.

Parameters:

γ = 0.02

,

δ = 1

,

λ = 10

¯

.

multivariate linear Langevin equationfor the u tuationsto whi h Eq. (2.14) is

mathemati ally equivalent [88, 111℄:

dx

i

(t)

dt

=

X

j

A

ij

x

j

(t) + L

i

(t), i, j = 1, 2,

(2.17)

where

L

i

(t)

are white randomnoise terms with the following properties:

hL

i

(t)i = 0,

(2.18)

hL

i

(t)L

j

(t

′

)i = B

ij

δ(t − t

′

).

(2.19)

The simple Fourier analysis of the linear Langevin equation (2.17) allows to

u tuations (PSNF) in the endemi phase. Denoting the PSNF as the averaged

squaredmodulusoftheFouriertransformofthesto hasti u tuations,weobtain

P S

(ω) ≡

| ˜

x

1

(ω)|

2

₌

B

11

(A

2

12

+ ω

2

)

(D − ω

2

₎

2

_{+ T}

2

_ω

2

,

(2.20) P I

(ω) ≡

| ˜

x

2

(ω)|

2

₌

B

11

(A

2

11

+ A

11

A

21

+ A

2

21

+ ω

2

)

(D − ω

2

₎

2

_{+ T}

2

_ω

2

,

(2.21)

for the sus eptible and the infe tive PSNFs, respe tively. In Eqs. (2.20)-(2.21),

the parameters

D

and

T

are equal to the determinant and to the tra e of the matrix

A

.

In Figure 2.2we plot the PSNFs given by Eqs. (2.20)-(2.21) for a parti ular

set of parameters. Both PSNFs are bell-shaped resonant urves with a peak

situated atawell-dened prin ipalfrequen y, indi atingthe os illatorybehavior

in the sto hasti system. Further examination of Eqs. (2.20)-(2.21) suggests

that the main frequen ies for sus eptibles and infe tivesare not ingeneralequal

(see supplementarymaterialofRef. [84℄). Notealsothatthey donot ne essarily

orrespond tothefrequen y ofthedeterministi dampedos illationsapproa hing

a stable xed point [this frequen y is determined by the absolute value of the

imaginary part of the omplex onjugate eigenvalues of the linearized Ja obian

of the MFA-SIRS deterministi equations (2.9)-(2.10)℄.

The ompletedes riptionoftheapproa hdevelopedinthisChapterasapplied

to anon-spatial predator-preysystem wasgiven ina work[81℄by A.J. M Kane

and T. J. Newman and we refer the reader to this paper for more details. We

also leave the dis ussion of the epidemiologi almeaning and signi an e of the

MFA-SIRSmodelforfuture Chapters. Havingset up the notationour main goal

now is todemonstrate howa similar kindof reasoning as that given here an be

Deterministi and sto hasti

frameworks in the pair

approximation

3.1 Introdu tion

In this Chapter we address the problem of modeling the ombined ee t of

de-mographi sto hasti ity and spatial orrelations in the ontext of the SIRS

epi-demi model. We extend the approa h des ribed in Chapter 2 by relaxing the

homogeneous mixing assumption to in lude an impli it representation of

spa-tial dependen e. We show that the in lusion of orrelations at the level of pairs

leadstodierentquantitativeandqualitativebehaviorsinaregionofparameters

that orresponds to infe tious diseases whi h onfer long lasting immunity. Our

motivation was twofold. On one hand, the homogeneous mixing assumption is

known togive poorresults for latti eornetwork stru tured populations [41, 75℄.

On the other hand, systemati simulations of infe tion on small-world networks

have shown that the resonant ampli ation of sto hasti u tuations is

signif-i antly enhan ed in the presen e of spatial orrelations [86℄. Therefore, apart

from sto hasti ity, the orrelations due tothe onta t stru ture are another key

ingredient to understand the patterns of re urrent epidemi s. One of the main

di ulties in in luding this ingredient is that the relevant network of onta ts

epidemi modelthat leads tothe ordinary pair approximation(PA)equations of

Ref. [75℄inthethermodynami limitasthesimplestrepresentation ofthespatial

orrelations onanarbitrary networkof xed oordinationnumber

k

. The PSNF aroundthesteadystate anbe omputedfollowingtheapproa hofChapter2and

Refs. [81℄and [88℄. The ombined ee t of sto hasti ity and spatial orrelations

has been mu h studied through simulations, but this is an analyti al treatment

of a model that in ludesboth these ingredients.

3.2 The epidemi model

Consider then a losed population of size

N

at a given time

t

, onsisting of

m

1

individualsoftype

S

,

m

2

individualsof type

I

,and

(N − m

1

− m

2

)

individualsof type

R

,modeledasnetworkofxed oordinationnumber

k

. Asbefore,re overed individuals lose immunity at rate

γ

and infe ted individuals re over at rate

δ

. Infe tion of the sus eptible node o urs in a sus eptible-infe ted pair at rate

λ

. Let

m

3

(respe tively,

m

4

and

m

5

) denote the number of pairs between nodes of type

S

and

I

(respe tively,

S

and

R

and

R

and

I

). Intheinnitepopulationlimit, with the assumptions of spatial homogeneity and un orrelated pairs, the SIRS

systemisdes ribedbythedeterministi equationsofthestandardorun orrelated

PA asfollows [75℄:

dP

S

dt

= γ (1 − P

S

− P

I

) − kλP

SI

,

(3.1)

dP

I

dt

= kλP

SI

− δP

I

,

(3.2)

dP

SI

dt

= γP

RI

− (λ + δ)P

SI

+

(k − 1)λP

SI

P

S

(P

S

− P

SR

− 2P

SI

) ,

(3.3)

dP

SR

dt

= δP

SI

+ γ (1 − P

S

− P

I

− P

RI

− 2P

SR

) −

(k − 1)λP

SI

P

SR

P

S

,

(3.4)

dP

RI

dt

= δ (P

I

− P

SI

) − (γ + 2δ)P

RI

+

(k − 1)λP

SI

P

SR

P

S

.

(3.5)

In the above equations the variables stand for the limit values of the node and

0

5

7.5

9 10

15

20

0

0.005

0.01

0.015

0.02

0.025

0.03

PSfragrepla ements

γ

λ

II - stablenodes orstable fo i

III -stable y les

Figure 3.1: Phase diagram in the

(λ, γ)

plane for the MFA-SIRS and the PA-SIRS deterministi models and parameter values for measles (

△

), hi ken pox (

◦

),rubella(



)andpertussis(

⋄

)fromdatasour esforthepre-va inationperiod. The stars are the parameter values used in Figure3.4.

and

P

RI

= m

5

/(kN)

as

N → ∞

. As expe ted, negle ting the pair orrelations and setting

P

SI

= P

S

P

I

in Eqs. (3.1)-(3.2) leads to the lassi equations of the homogeneously mixed or MFA-SIRS deterministi modelstudied inChapter 2.

The steady state solutions of Eqs. (3.1)-(3.5) an be obtained analyti ally

[75℄. Let

P

¯

S

,

P

¯

I

and

P

¯

SI

,

P

¯

SR

,

P

¯

RI

denote the steady state values of the node and pairdensitiesofthePA-SIRSmodel. Then,one steadystateisatrivialxed

point orresponding tozero infe tive density (no- oexisten e region):

¯

P

I

= 0, ¯

P

S

= 1, ¯

P

SI

= 0, ¯

P

SR

= 0, ¯

P

RI

= 0.

(3.6)

infe tive densities( oexisten e region):

0 < ¯

P

I

< 1, 0 < ¯

P

S

< 1.

(3.7) The phase diagram of the PA-SIRS deterministi model is plotted in Figure

3.1. Again, we have set the time s ale so that

δ = 1

and used

k = 4

. The trivialxedpointisstableinregionI(theregiontotheleftfromthesolidorange

line - labelnot shown), and the nontrivial xed point isstable inregion II. The

riti al line separating a sus eptible-absorbing phase (region I) from an a tive

phase(regionII)whereastablesteadystateexists withnonzeroinfe tivedensity

is given by

λ

PA

(γ) =

γ + 1

3γ + 2

(3.8)

for the PA-SIRS model. The dashed line is the riti al line (2.13) separating

region I and region II for the MFA-SIRS model. Note that in the deterministi

frameworkthere isaqualitativedieren einthe behaviorpredi tedbythe MFA

and the PA models. The riti al value of infe tion rate, dened as the smallest

valueof

λ

foragivenrateofimmunityloss

γ

abovewhi htheinfe tionpersists,is independentof

γ

in theMFA.Furthermore, forany

γ

itislowerthanthe riti al infe tion rate in the PA whi h means that it is easier for a disease to spread in

homogeneously mixed than in network stru tured populations.

Inaddition,inthea tivephaseofthePA-SIRSmodelwend forsmallvalues

of

γ

a new phase boundary (solid blue line) that orresponds to an Andronov-Hopf bifur ationandseems tohavebeen missedinpreviousstudiesofthismodel

[75℄. This boundary separates the a tive phase with onstant densities (region

II) from ana tive phase with os illatorybehavior (regionIII). The maximum of

the urve issituatedat

λ ≈ 2.5

,

γ ≈ 0.03

, whi hmeans that the PA-SIRSmodel predi ts sustainedos illationsinthe thermodynami limitwhenlossofimmunity

is mu h slower thanre overy frominfe tion.

Before we onsider a sto hasti model des ribing the behavior of the

u tu-ations superimposed on the deterministi traje tories given by Eqs. (3.1)-(3.5),