Stochastic epidemics on randomnetworks Abid Ali Lashari

Abid Ali Lashari    Stochastic epidem

ics on ran

dom n

etworks

Doctoral Thesis in Mathematical Statistics at Stockholm University, Sweden 2019

Department of Mathematics

ISBN 978-91-7797-661-5

Stochastic epidemics on random networksAbid Ali Lashari

Academic dissertation for the Degree of Doctor of Philosophy in Mathematical Statistics atStockholm University to be publicly defended on Thursday 16 May 2019 at 10.00 in sal 14, hus5, Kräftriket, Roslagsvägen 101.

AbstractThis thesis considers stochastic epidemic models for the spread of epidemics in structured populations. The asymptoticbehaviour of the models is analysed by using branching process approximations. The thesis contains four manuscripts.

Paper I is concerned with the study of the spread of sexually transmitted infections, or any other infectious diseaseson a dynamic network. The model we investigate is about the spread of an SI (Susceptible → Infectious) type infectiousdisease in a population where partnerships are dynamic. We derive explicit formulas for the probability of extinction andthe threshold parameter R0 using two branching process approximations for the model. In the first approximation somedependencies between infected individuals are ignored while the second branching process approximation is asymptoticallyexact and only defined if every individual in the population can have at most one partner at a time. By comparing the twoapproximations, we show that ignoring subtle dependencies in the dynamic epidemic model leads to wrong prediction ofthe probability of a large outbreak.

In paper II, we study a stochastic SIR (Susceptible → Infectious → Removed) epidemic model for the spread of anepidemic in populations structured through configuration model random graphs. We study the asymptotic (properly scaled)time until the end of an epidemic. This paper heavily relies on the theory of branching processes in continuous time.

In paper III, the effect of vaccination strategies on the duration of an epidemic in a large population is investigated. Weconsider three vaccination strategies: uniform vaccination, leaky vaccination and acquaintance vaccination.

In paper IV, we present a stochastic model for two successive SIR epidemics in the same network structured population.Individuals infected during the first epidemic might have (partial) immunity for the second one. The first epidemic isanalysed through a bond percolation model, while the second epidemic is approximated by a three-type branching processin which the types of individuals depend on their status in the percolation clusters used for the analysis of the first epidemic.This branching process approximation enables us to calculate a threshold parameter and the probability of a large outbreakfor the second epidemic. We use two special cases of acquired immunity for further evaluation.

Keywords: Branching process, Configuration model, Random graph, Epidemic process, Final size, Thresholdbehaviour, Duration of an epidemic, Vaccination.

Stockholm 2019http://urn.kb.se/resolve?urn=urn:nbn:se:su:diva-167373

ISBN 978-91-7797-661-5ISBN 978-91-7797-662-2

Department of Mathematics

Stockholm University, 106 91 Stockholm

STOCHASTIC EPIDEMICS ON RANDOM NETWORKS 

Abid Ali Lashari

Stochastic epidemics onrandom networks 

Abid Ali Lashari

©Abid Ali Lashari, Stockholm University 2019 ISBN print 978-91-7797-661-5ISBN PDF 978-91-7797-662-2 Printed in Sweden by Universitetsservice US-AB, Stockholm 2019

To my Parents, Saba andMusa.

List of Papers

The following papers, referred to in the text by their Roman numerals, areincluded in this thesis.

PAPER I: Abid Ali Lashari, Pieter Trapman. (2018). Branching processapproach for epidemics in dynamic partnership network. Jour-nal of Mathematical Biology, 76 (1): 265-294.DOI: 10.1007/s00285-017-1147-0

PAPER II: Abid Ali Lashari, Ana Serafimovic, Pieter Trapman, The dura-tion of an SIR epidemic on a configuration model.(Under review)

PAPER III: Abid Ali Lashari, Pieter Trapman, Effect of vaccination on theduration of an SIR epidemic in homogeneously mixing and struc-tured populations.(Manuscript)

PAPER IV: Abid Ali Lashari, Frank Ball, David Sirl, Pieter Trapman, Mod-eling the spread of two successive SIR epidemics on a configu-ration model network.(Manuscript)

Paper I is included with permission from the publisher.

Author’s contribution

PAPER I: “Branching process approach for epidemics in dynamic partner-ship network.”

This paper is based on an idea of Pieter Trapman. Abid AliLashari did the modelling together with Pieter Trapman andwrote most of the manuscript.

PAPER II: “The duration of an SIR epidemic on a configuration model.”

This paper is based on the master thesis of Ana Serafimovic ona problem proposed by Pieter Trapman. Ana Serafimovic de-veloped the model and analysed the model together with PieterTrapman. Abid Ali Lashari and Pieter Trapman extended theresults and wrote the manuscript.

PAPER III: “Effect of vaccination on the duration of an SIR epidemic inhomogeneously mixing and structured populations.”

In this paper, Abid Ali Lashari derived the analytical results andwrote the manuscript with input from Pieter Trapman.

PAPER IV: “Modeling the spread of two successive SIR epidemics on a con-figuration model network.”

In this paper, Abid Ali Lashari and Pieter Trapman developedthe model and wrote the manuscript while David Sirl and FrankBall contributed with valuable comments and discussions.

Acknowledgements

First, I would like to thank my supervisor Pieter Trapman for his guidance,encouragement and dedicating time to me whenever it was needed. He mademe push past my limits to accomplish my goals. Thanks to Tom Britton fordiscussions and guidance on mathematical modeling of infectious diseases.

I would also like to thank all colleagues who have made the Division ofMathematical Statistics a great work place. I am grateful for the support ofall my PhD fellows, especially Theresa for being a wonderful officemate, Se-bastian for being nice and translating the summary of papers in Swedish andStanislas for being my lunch buddy. I am eternally thankful to my family fortheir prayers and support throughout my studies. I am grateful for the supportand Duas of all my friends, especially Waqas, Habib, Asif and Nabeel, to Ma-teen for being my Jigar.

I appreciate the financial support from the Swedish Research Council andthe Department of Mathematics, Stockholm University.

Most importantly, I am extremely grateful to Almighty Allah for providingme with the ability, understanding and insight to carry out this work and whohas been with me from the start till the end of this work (Alhamdulillah).

Last but not least, I would like to express my deep appreciation to my wifeSaba who filled a giant hole in my life, for her efforts to make me a betterperson, for her loving and caring attitude and for giving birth to an adorableand wonderful boy Musa.

Contents

List of Papers i

Author’s contribution iii

Acknowledgements v

1 Introduction 1

2 Epidemic models 32.1 The beginning of epidemic modeling . . . . . . . . . . . . . . 32.2 Branching processes . . . . . . . . . . . . . . . . . . . . . . 5

2.2.1 Generating functions and the extinction probability . . 52.2.2 Approximating epidemics by branching processes . . . 6

2.3 Vaccination strategies . . . . . . . . . . . . . . . . . . . . . . 7

3 Spread of epidemics on networks 93.1 Random networks and epidemics . . . . . . . . . . . . . . . . 93.2 The configuration model network . . . . . . . . . . . . . . . . 10

3.2.1 The SIR epidemic on networks . . . . . . . . . . . . . 10

4 Overview of the papers 13

5 Sammanfattning 17

References 21

1. Introduction

Infectious diseases such as smallpox, the Black Death and the Spanish In-fluenza which killed millions have shaped the history of mankind (Morenset al., 2008). In recent years, infectious diseases such as swine flu (influenzatype A virus) and the Ebola virus have shown considerable impact and emergedas a serious public health problem to the society (Camacho et al., 2014; NovelSwine-Origin Influenza A (H1N1) Virus Investigation Team, 2009; WHO EbolaResponse Team, 2014). Despite of substantial advancements in medical andhealthcare technology, infectious diseases such as measles, malaria, and HIVremain major threats to public health in many parts of the world (Lopez andMurray, 1998; Murray et al., 2014). Outbreaks of such epidemics remind usthat the battle against infectious disease is not over and the epidemics are anactual problem for health authorities.

Indeed the spread of infectious diseases has declined in most parts of theworld, but continuous appearance of outbreaks of new infectious diseases isstill a significant problem in the modern world. In infectious disease epidemi-ology, the three major tasks for health authorities are to first understand thepossible causes of the disease, then make predictions of the course of dis-ease, and finally to develop methods for its prevention and control. The usualprocedure for this is obtaining and analysing observed data because humanexperimentation is often not possible or ethical. In such cases, understanding,assessing and predicting the dynamics of the spread of infectious diseases isessential and for this mathematical modeling plays an important role (Heester-beek et al., 2015).

There has been a significant interest in understanding the infectious dis-ease epidemiology and use of mathematical models in the spread of epidemics(Chubb and Jacobsen, 2010). Mathematical models have played a valuable rolein providing insight into most effective control measures such as vaccinationstrategies or the required infrastructure for the public health decision-making(see, Bacaër, 2011; Hollingsworth, 2009, for an overview of the history ofmathematical modeling). However, numerous challenges are involved in de-veloping a complete understanding of epidemics and for an overview of someof the recent important challenges in epidemic modeling we refer the readerto the special issue of epidemics by Lloyd-Smith et al., 2015 and in particularsee, e.g. Britton et al., 2015; Eames et al., 2015; Pellis et al., 2015 (all papers

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in that issue).The specific goal of this thesis is to study the spread of epidemics on ran-

dom networks by developing new stochastic epidemic models. We identifythe problems that arise in the approaches used for computing important epi-demiological quantities in some existing studies and provide new techniquesfor the computation of those important quantities. In particular, the main theo-retical results provide answers to questions: what is the probability of a largeoutbreak (the epidemic becomes established and infects a nonnegligible pro-portion of the population)? What fraction of the population is infected or willbe infected if a large outbreak is possible and occurs? What is the duration ofa large outbreak? What is the effect of vaccination on the duration of an entireepidemic.

In the following chapters of the thesis, we provide an intuitive introductionto stochastic models for epidemics on random networks and the applicationof branching process approximations in analysing the dynamics of the infec-tious disease spread on those random networks. We give a brief introductionof some basic deterministic and stochastic epidemic models, basic epidemicquantities and the theory of branching processes. Moreover, a short introduc-tion of the random graphs constructed through a configuration model networkand its application to the modeling of epidemics on random networks is alsoprovided. In the end, we give an overview/summary of the papers included inthis thesis.

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2. Epidemic models

In this chapter, first we very briefly give a review of early work in epidemicmodeling and define the SIR (Susceptible-Infective-Removed) epidemic model.Next, we discuss the basic theory of branching processes and its application toepidemics and provide a brief introduction of some vaccination strategies inepidemic modeling.

2.1 The beginning of epidemic modeling

One of the earliest publications addressing the mathematical modeling of epi-demics dates back to 1760 and is written by Daniel Bernoulli (see Bacaër,2011; Hethcote, 2000). In this paper Daniel Bernoulli created a mathemat-ical model to analyse the impact of vaccination on the spread of smallpox.The mathematical modeling of epidemics has developed systematically andgained popularity with the benchmark paper of Ross (1911) on discovering themosquito based transmission of malaria and the contribution of Kermack andMcKendrick (1927) (see Diekmann et al., 2012, 1995, and references therein).The pioneering work of Kermack and McKendrick established the determinis-tic compartmental epidemic modeling.

In the widely studied SIR epidemic models (which is a special case of theKermack and McKendrick model), the population is split into three epidemio-logical classes:

• Susceptible individuals who have not yet been infected by the diseasebut are at risk of becoming infected.

• Infectious individuals who have been infected and can transmit the in-fection to other (susceptible) individuals during their infectious period.Infectious individuals stays infectious for some time (random/fixed) af-ter which they eventually recover from the infection.

• Recovered/Removed individuals who have had the disease but now areno longer spreading the disease. Recovered individuals are unable toacquire or transmit the infection because they are no longer infectiousand are fully immune (or dead).

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At any given time (depending upon the disease stage) an individual belongsto one of the above epidemiological classes. The individual can change classby moving either from S to I or from I to R. Eventually the epidemic endswhen there are no more infectious individuals in the I-class. The SIR modelis more realistic for diseases leading to lifelong immunity after infection (e.g.chickenpox, measles).

The evolution of the number of susceptible, infectious and recovered in-dividuals over time in Kermack and McKendrick’s special case SIR model isusually studied through three differential equations (see, e.g. Diekmann et al.,2012). In this deterministic model the demography (births, deaths and migra-tion) is ignored and the assumption is that the population size is effectivelylarge and fixed. Another simplifying assumption is that individuals mix homo-geneously. With this assumption the infectious contact pattern is ignored and itis assumed that an individual is equally likely to interact with any other individ-ual in the population (regardless of geographic location, age, etc.). Althoughthose simplifying assumptions seem unrealistic, the mathematical models withthose assumptions have long been providing important insight and predictivecapability. However, the deterministic model is not useful in predicting quan-tities such as the probability of a large epidemic outbreak. Furthermore, thedeterministic models are not useful for studying the start and the end of anepidemic because those models do not handle small numbers of infecteds well(see, e.g. Andersson and Britton, 2012; Diekmann et al., 2012).

As a prime example of the stochastic version of the SIR epidemic model,we consider the Markov SIR epidemic model. In this model, the duration of theinfectious period is stochastic and it is assumed to be exponentially distributed,and during the infectious period the infected individuals make close contactsat a constant rate. If the individuals who comes into contact with infectiousindividuals are susceptible they becomes infectious immediately.

For a more extensive survey of stochastic epidemics and literature reviews,the reader can consult Andersson and Britton, 2012 and Britton, 2010.

The fundamental parameter in both stochastic and deterministic epidemicsthat governs the epidemic dynamic is the basic reproduction number, usuallydenoted by R0. It is usually defined as the expected number of secondary infec-tions that a typical newly infected individual produce in an entirely susceptiblepopulation during the early phase of an epidemic. The critical value of R0 = 1is known as the epidemic threshold. This R0 will also play an important role inthis thesis.

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2.2 Branching processes

In this section, we briefly give an overview of the basic theory of the branch-ing processes that is used in the analysis of the infection spread on randomnetworks in all the papers of this thesis. In the following we define the so-called single-type Galton-Watson process and the probability of extinction ofthis branching process.

Consider a population in which the number of individuals in the nth gen-eration is Zn. The difference between male and female is ignored. Each indi-vidual produces a (random) number of individuals independently of the otherindividuals and the numbers of children are identically distributed. The num-ber of individuals in (n+1)th generation of the branching process is

Zn+1 =Zn

∑i=1

ζi,

where the ζi are independent and identically distributed random variables withdistribution

P(ζi = k) = pk, k = 0,1,2 · · · .

So ζi is the number of offspring of the ith individual and pk is the probabilitythat a single individual produces k offspring.

The branching process is said to extinct by generation n if Zn = 0. Triviallyif Zn = 0, then Zn′ = 0 for all n′ ≥ n. Let ζi ∼ ζ and µ := E[ζ ] denote theexpected number of offspring of a single individual. The branching processis said to be subcritical, critical or supercritical according to whether µ < 1,µ = 1 or µ > 1, respectively. The subcritical and critical branching processesalmost surely becomes extinct while the supercritical branching process has apositive probability of avoiding extinction (see Harris, 2002; Jagers, 1975). Toshow this we use generating functions as introduced below.

2.2.1 Generating functions and the extinction probability

We now define the probability generating functions (PGFs) describing the off-spring of individuals in the branching process. For s ∈ [0,1], the probabilitygenerating function f (s) for the branching process with offspring distributiongiven through pk for k ∈ N is defined by

f (s) = E[sζ ] =∞

∑k=0

P(ζ = k)sk =∞

∑k=0

pksk.

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The generating function of the number of individuals in the nth generation ofthe branching process conditioned on a single ancestor is

fn(s) =∞

∑k=0

P(Zn = k)sk.

Since f1 = f , and using the independence property of offspring distribution wehave that

fn(s) = E[sZn ] = E[E[sZn | Z1]

]= E

[E[sZn−1 ]Z1

]= f ( fn−1(s)) .

We now consider the probability of extinction of the branching process.Let q denote the extinction probability of the above mentioned Galton-Watsonbranching process initiated by one single ancestor. In order for the offspring ofan individual to go extinct, all the offspring of its children should go extinct.Therefore,

q =∞

∑k=0

P(Z1 = k)qk = f (q)

and we know from standard branching process theory that q is the smallestsolution in [0,1] of s = f (s). Furthermore, for s ∈ (0,1)

limn→∞

fn(s) = q.

The above mentioned Galton-Watson process describes a family tree from ageneration perspective in which a population only contains individuals of asingle generation at a time. Generalizations of the above setup to multitypebranching processes and branching processes in real time is well studied. Fordetailed descriptions of possible generalizations of branching processes withbiological applications, we refer the reader to e.g. Haccou et al., 2005; Harris,2002; Jagers, 1975.

2.2.2 Approximating epidemics by branching processes

It is well known in epidemic literature that the beginning of an SIR epidemic(when the fraction infected is still small) in a homogeneously mixing popu-lation can be analysed by a suitable branching process (see Andersson andBritton, 2012; Ball and Donnelly, 1995; Britton, 2010, and references therein).In those branching process approximations, infecting someone corresponds togiving birth while death of individuals corresponds to actual death or recoveryfrom infection. Such branching processes have shown to be good approxi-mations to the stochastic epidemic processes during the initial stages of theepidemics, when the initial number of infectives is small and the number of

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susceptibles is still large (see e.g. Ball and Donnelly, 1995), and also duringthe final stages of the epidemics (Ball, 1983; Lashari et al., 2018). The initialstage of the epidemic is the most crucial one because during that stage it can bedetermined if there is a possibility for the epidemic to take off or the epidemicwill die out. The basic reproduction number R0 corresponds to the offspringmean of the particular branching process. In particular, R0 determines whetherthe infection will die out with probability 1, or whether the epidemic is capableof causing a large epidemic outbreak.

In many scenarios one may be interested in the real-time development ofthe number of infected individuals in a population. For example, in epidemi-ology, the real-time growth rate during the initial phase of the epidemic isan important parameter with a clear biological implications (Trapman et al.,2016) as we will show in the second and third paper of this thesis. The epi-demic/branching process growth/decay rate α corresponds to the so-calledMalthusian parameter for the population growth/decay rate and is the solu-tion of

1 =∫

∞

0e−αt

µ(t)dt,

where µ(t) represent the expected rate at which infected individual make in-fectious contacts with other individuals t time units after it was infected. Notethat

R0 :=∫

∞

0µ(t)dt.

If R0 > 1, the epidemic/branching process is said to be supercritical and α > 0exists while if R0 = 1, the epidemic/branching process is critical and in thiscase α = 0. If R0 < 1, the epidemic/branching process is said to be subcriticaland α might exist and if it does then α < 0.

2.3 Vaccination strategies

An important application of modeling the infectious disease spread is to informpublic health decision-makers to how to reduce the impact of the diseases. Inorder to prevent or reduce the impact of a disease, it is sometimes possible tovaccinate individuals prior to the introduction of the disease. In paper III, weconsider the effect of three vaccination strategies (defined below) to controlthe spread of a disease.

• Uniform vaccination: in this strategy individuals are chosen uniformlyat random from the population and then they are vaccinated.

• Leaky vaccination: a vaccine that reduces the susceptibility and infec-tivity of every vaccinated individual but does not eliminate the potential

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for infection is called a leaky vaccine.

• Acquaintance vaccination: in this strategy some “acquaintances” of ran-domly selected individuals are vaccinated.

If the vaccine is perfect then the critical vaccination coverage required to pre-vent a large outbreak of an epidemic in a randomly mixing population is givenby 1−1/R0.

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3. Spread of epidemics onnetworks

In this chapter we define the random graph models based on the configurationmodel and the spread of SIR epidemics on the network. Moreover, we brieflydescribe branching process approximations of an epidemic on random graphs.

3.1 Random networks and epidemics

In recent years, there has been a lot of research on modeling stochastic epi-demics spreading through networks of individuals (Davis, 2017; Miller andKiss, 2014). Diseases like HIV are clearly dependent on the contact structureof the infected individuals and indeed for all sexually transmitted disease thesimplifying random mixing assumption is not valid.

The network is typically represented by a random graph (Newman, 2002).The graph can be viewed as a collection of vertices/nodes that represent indi-viduals and edges represent connections/relationships of individuals (for ex-ample, social contacts with family members, work colleagues, and friendsetc.). The degree of a node is the number of edges attached to that node. Twonodes are neighbors if they share a common edge.

The network used to analyse the spread of an infectious disease can eitherbe static or dynamic. In paper I (Lashari and Trapman, 2018) the network weconsider is dynamic while in all other papers of this thesis static network isconsidered. The epidemic is modeled in populations having a “ConfigurationModel” structure (defined in the next section). This special model structure ischosen since an epidemic on a general network is too difficult to analyse. Thisconstruction allows to compute important quantities of interest such as thereproduction number, the probability of a major epidemic outbreak, the finalsize of an epidemic and the duration of an epidemic on the random network.Below we formally introduce the random graph and the epidemic model. Itis important to note that all the results are asymptotic as the population sizen→ ∞.

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3.2 The configuration model network

The models we investigate consider the spread of an infectious disease througha population of n individuals. The population is assigned a network struc-ture according to the configuration model (see, e.g. Durrett, 2007, chap. 7 orVan der Hofstad, 2016, chap. 9). In order to create the graph describing pos-sible relations/contacts, we assign a random number of so called half-edges(edges with just one endpoint attached to a vertex) to each individual/vertex inthe population according to independent draws from a given distribution. De-note this distribution by D, having mass function pk =P(D= k), k = 1,2,3 · · · .We assume that E(D) < ∞. The number of half-edges are independent andidentically distributed according to D. The half-edges are then paired uni-formly at random which in turn form the edges of the random graph. In thisnetwork construction, every vertex has the right degree, although some prob-lems such as self-loops or parallel edges might arise. The degree of a uni-formly chosen individual is distributed according to D. By the constructionof the graph, the degrees of neighbors of the uniformly chosen individual, orneighbors of any given individual do not have degree distribution D but havethe size biased degree distribution D given by

pk = P(D = k) =kpk

E(D), k = 1,2,3 · · · .

Usually we want graphs to be simple, which means that there are no par-allel edges (several edges connecting the same two vertices) and self-loops(edge connecting a vertex to itself). The assumption E[D]< ∞ (or equivalentlyE[D2] < ∞) ensures that the probability of obtaining a configuration modelwith no parallel edges and self-loops is bounded away from 0 as n→ ∞ (see,e.g. Van der Hofstad, 2016, p. 219). Furthermore, the graph generated througha configuration model is locally tree-like (as n→ ∞ with probability tendingto 1, the “neighborhood” of a given vertex contain no short circuits). Thetree-like structure allows branching process approximations for the spread ofepidemics on graphs (see, e.g. Andersson, 1998; Britton et al., 2007).

3.2.1 The SIR epidemic on networks

We consider the evolution of SIR epidemics on graphs. A vertex is said besusceptible, infectious, or recovered if the individual it represents is in thisspecific “infection state.” Infectious vertex stay so for a random period of time(the length of its infectious period) during which it makes infectious contactswith any given neighbor according to independent homogeneous Poisson pro-cesses. A contact made by an infectious individual with any other individual is

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called an infectious contact. When the infectious period of an infectious indi-vidual ends it becomes removed and then plays no further role in the spread ofan epidemic. A susceptible individual becomes immediately infectious (birthin the branching process) when it is contacted by an infected individual. Theepidemic is assumed to starts at time t = 0 when all individuals are susceptibleexcept for a single infectious individual chosen uniformly at random from thepopulation. The infectious periods and the Poisson processes describing thecontact processes are all independent. The epidemic stops when there is noinfected individual left in the population.

The early stages of an epidemic spreading across the network is analysedby a (forward) branching process. This branching process approximates thenumber of infected individuals by taking into account who an individual willinfect forward in time. The initial generation in this branching process cor-responds to the single infectious individual chosen uniformly at random fromthe population. In general, the generation k consists of individuals who werepreviously uninfected but becomes infected by an infectious contact with in-dividuals in generation k− 1. In the early stages of the epidemic as n→ ∞

it is more likely that each of the contact made by an infective individual willbe with uninfected individuals apart from contacts with the “infector.” Thatis why the early spread of an epidemic is well approximated by a (forward)branching process (see, e.g. Ball and Sirl, 2013; Ball et al., 2009, for rigoroustreatments of the forward branching process). The threshold parameter thatdetermines whether a major epidemic outbreak is possible is determined bywhether or not the (forward) branching process avoid extinction. For conver-gence of the probability of a major outbreak to the survival probability of theforward branching process see, Ball and Sirl, 2013; Ball et al., 2009.

Similarly, the expected final size of a major outbreak (proportion of indi-viduals who ultimately become recovered) corresponds to the survival prob-ability of an approximating (backward) branching process. This branchingprocess is a good approximation of a generation perspective on an individual’ssusceptibility set which is a key tool for finding the fraction of the populationthat ultimately becomes infected by a major outbreak.

Let V = {v1,v2, · · · ,vn} denote the set of susceptible vertices of the undi-rected graph generated through the configuration model. We construct the di-rected graph on V, in which for any vi and v j a directed edge from vi to v j ispresent if and only if vi, if infected during the epidemic, would make contactwith v j during its infectious period. Then, the susceptibility set of v j is thecollection of all those vertices in the directed graph (including v j itself) thatcan be reached from v j by following a path backwards. Note that any givenvertex vi becomes ultimately infected in the epidemic if and only if its suscep-tibility set contains the initial infective (see, e.g. Paper II, Section 3 or Ball and

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Neal, 2002 for details of susceptibility sets). The schematic illustration of thesusceptibility set is shown in Figure 3.1.

u

Figure 3.1: The solid red nodes represent the susceptibility set of node u in thedirected random graph representation of an epidemic.

The susceptibility set is approximated by a (backward) branching process,which we now briefly describe. Let vi denote a uniformly at random choseninitial individual. In the (backward) branching process, we consider how manyneighbors make infectious contact with that individual. The 0-th generation inthis branching process consist of individual vi. In general, the generation kcomprises individuals who enter the susceptibility set by making infectiouscontact with an individual in generation k− 1 (for details, see Ball and Sirl,2013; Ball et al., 2009).

For an overview of the asymptotic results that relate expected final size ofan epidemic outbreak to the survival probability of the (backward) branchingprocess see, e.g. Ball and Neal, 2002, 2008; Ball et al., 2009.

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4. Overview of the papers

The understanding of the structure of a network in the absence of infection isvery important for modeling the spread of the infectious diseases on networks.There has been a considerable interest in epidemics on networks as a way torepresent the complex structure of contacts that are capable of spreading infec-tion in a population (see Ball and Neal, 2008; Durrett, 2007; Newman, 2002,and references therein). Leung et al. (2015) consider a (deterministic) modelfor the spread of an SI (Susceptible→ Infectious) type epidemic on a networkof dynamic partnerships. The network is dynamic in the sense that individualsenter the population (by birth) and leave the population (by death) and the part-nerships (edges in the network) are forming and dissolving while the epidemicspreads through the population. A key ingredient in the model of Leung et al.(2015) is the “mean field at distance one” assumption which is a (non-exact)approximation of the distribution of the number of partners of partners of anewly infected individual. In dynamic network model with demography, thedegrees of partners are not independent (Leung et al., 2017) however Leunget al. (2015) ignored the degree dependence between partners. This approxi-mation allowed them to write down a system of differential equations to obtainanalytic expressions for the basic reproduction number R0.

In paper I, we extend the model of Leung et al. (2015) by considering epi-demic and population dynamics as stochastic processes. We discuss the com-plications that come by ignoring the degree dependencies between the part-ners while modeling the spread of infectious disease on a dynamic network.We propose a new scheme of modeling demography with the assumption thatthe new individuals have a stationary distribution (degree distribution) for thenumber of partners. That is to say that the new individuals immediately forma (random) number of partnerships upon their arrival in the population withindividuals already in the population. Under this assumption the dependenciesbetween the number of partners of an individual are absent and the ‘mean-fieldat distance one’ of (Leung et al., 2015) is no longer required.

The early stages of a stochastic SI type epidemic is approximated by asuitable branching process as the network size tends to infinity. In particular,we derive the expressions for two of the most fundamental quantities in thetheory of epidemiology: the reproduction number R0 that determines whetheror not an epidemic with few initial infectives can become established and the

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probability of a large outbreak of a disease that becomes established. Thesequantities have importance for health officials in understanding and controllingthe spread of epidemics. We explored different strategies to derive explicitexpressions for these two quantities for an SI type epidemic that spread acrossa dynamic network.

Another area of research is to investigate the duration of the epidemic.This is not studied very often (but see Barbour and Reinert, 2013; Barbour,1975). In case of a large outbreak, the duration of an epidemic is important,for instance, during the outbreak travel/trade bans are imposed on the coun-tries within which the disease is spreading. The longer durations of epidemicsmight lead to severe economic costs for those countries. Therefore, in the sec-ond paper we focus on the duration of an SIR (Susceptible, Infectious, Recov-ered) epidemic in a population structured through a random graph, generatedby the configuration model. We focus especially on the final stages of the largeepidemic outbreak and analyse the large graph limit of an SIR epidemic. Theaim is to derive the asymptotic form of the time needed for an infection tocompletely disappear from the population as the size of the graph tends to in-finity. In order to achieve this we investigate the so-called epidemic-generatedgraphs. To obtain the proposed results, we couple both the start and the end ofthe epidemic with suitable branching processes where we use a supercriticalbranching process for the start of an epidemic and a subcritical one for the endof an epidemic.

The main result of paper II (Theorem 2.2) states that the (properly scaled)duration of the epidemic is of the order of logn, where the involved constantsare expressed as functions of the Malthusian parameters of suitable continuoustime branching processes. This result then generalizes the results of Barbourand Reinert, 2013, who do not study the end of the epidemic, to a wider rangeof degree distributions and infectious time distributions. The result is provenessentially in two steps: first showing that the first time the number of infec-tious individuals is of order n is of order logn (Lemma 2.6), and then inves-tigating the process between the latter time and the extinction time is also oforder logn (Lemma 2.7). The proof of the first result heavily relies on existingresults on branching processes (Jagers, 1975), and on a generalization of theresults in (Barbour and Reinert, 2013) to degree distributions and infectioustime with unbounded supports.

In paper III we use the model presented in paper II, by including vacci-nation in the model, and explore the effect of various vaccination strategies onthe duration of the end of an SIR epidemic. We consider the effect of a uniformand a leaky vaccine in both homogeneous mixing and structured populations.We show that the uniform vaccine in homogeneously mixing populations andstructured populations can have different effects on the duration of the end of

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an SIR epidemic. In uniformly/homogeneous mixing population, we find thatuniform vaccination with a perfect vaccine either prevent the large outbreak orit increase the duration of the epidemics whereas uniform vaccination in struc-tured populations may lead to both an increase and decrease in the durationof the epidemic. Furthermore, the leaky and uniform vaccine have the sameeffect on the duration of the epidemic in homogeneously mixing populationswhereas we conjecture that a leaky vaccine always increases the duration of theepidemic in structured populations. For acquaintance vaccination, we providea recipe for calculating the asymptotic duration of the epidemic. We showthat acquaintance vaccination can both increase and decrease the asymptoticduration of an epidemic.

In the last paper, we study a model in which two SIR type epidemicsspreads successively through a single population modeled by a configurationmodel network. The infection with the second epidemic might depend on priorinfection with the other epidemic. We assume that the first disease spreadsthrough the network and causes a major epidemic outbreak infecting a non-negligible fraction of the population. Individuals infected during the first epi-demic might becomes partially or fully immune for the second one. The spreadof the first epidemic is analysed through a bond percolation model while theearly stages of the second epidemic is approximated by a multitype branchingprocess. The results are illustrated through two examples. The first exampleis for the case in which individuals infected in the first epidemic are indepen-dently either completely susceptible or completely immune to the second epi-demic. The second example is for the case in which the individuals who haverecovered from the first epidemic have reduced susceptibility and infectivityfor the second epidemic.

A three type branching process is used to calculate a threshold parame-ter, R(2)

0 , that determines whether a major outbreak of the second epidemiccan occur or not. A recipe of computing the probability of a large epidemicoutbreak of the second epidemic is also provided. Furthermore, our paper gen-eralizes some epidemic models discussed in previous literature (Bansal andMeyers, 2012; Newman, 2005; Newman and Ferrario, 2013)). That is to saythat the models in (Newman, 2005; Newman and Ferrario, 2013) and (Bansaland Meyers, 2012) are special cases of our model.

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5. Sammanfattning

Det är viktigt att förstå strukturen av ett nätverk för att kunna modellera sjuk-domsspridning på nätverket. Det har hittills funnits ett stort intresse för epi-demier på nätverk, vilket har visat sig som ett lämpligt sätt att representerakomplexiteten i hur en sjukdom sprids i en population (se Ball och Neal, 2008;Durrett, 2007; Newman, 2002). Leung et al. (2015) betraktar en deterministiskmodell för en SI (Susceptible → Infectious) epidemi på ett nätverk med dy-namiska förhållanden. Nätverket är dynamiskt i den mening att individer födsin i populationen och dör, och under sin livstid formar och avslutar partner-skap med andra individer. Ett viktigt antagande i modellen är mean field atdistance one, vilket är en approximation av fördelningen av antal partners avpartners av en nyligen smittad individ. I ett dynamiskt nätverk med demografi,gradtalet av partners är inte oberoende mellan individer (Leung et al., 2017).Dock ignorerade Leung et al. (2015) detta. Denna approximation tillät dem attbeskriva dynamiken i nätverket med ett system av differentialekvationer. Urdessa härleds sedan ett analytiskt uttryck för reproduktionstalet R0.

I paper I, utökar vi modellen i Leung et al. (2015) genom att betrakta epi-demin och populationen som stokastiska processer. Vi diskuterar hur antagan-det om oberoende gradtal mellan individer (ett felaktigt antagande) påverkarresultaten om modellen. Vi föreslår även ett nytt sätt att modellera demografinunder antagandet att nya individer har en stationär gradfördelning av partners.Alltså, nya individer formar direkt vid ankomst ett slumpmässigt antal part-nerskap med individer redan i populationen, och detta antal beror ej på tidenav ankomst (stationärt). Under detta antagande försvinner beroendet mellanindividers gradtal och antagandet om mean field at distance one behövs intelängre.

Början av en SI-epidemi kan approximeras med en lämplig förgreningspro-cess då storleken på nätverket går mot oändligheten. Speciellt, kan vi användaapproximationen för att härleda uttryck för två av de mest fundamentala stor-heterna i epidemiologi: reproduktionstalet R0 och sannolikheten att ett stortutbrott inträffar. Dessa storheter är viktiga för beslutsfattare inom t ex. Folk-hälsomyndigheten. Vi utforskade olika strategier för att härleda dessa storheterför en SI-epidemi som sprider sig på ett dynamiskt nätverk.

Ett annat forskningsområde är att undersöka hur länge en epidemi pågår.Detta är inte alltid studerat (men se Barbour och Reinert, 2013; Barbour, 1975).

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I fallet av ett stort sjukdomsutbrott är varaktigheten av utbrotten viktig. Tillexempel, rese- och handelsförbud kan vara aktuellt att införa, och därför blirvaraktigheten av epidemin en viktig faktor. En lång varaktighet kan då få sto-ra ekonomiska konsekvenser. I det andra pappret fokuserar vi därför på justvaraktigheten av en SIR (Susceptible, Infectious, Recovered) epidemi på enpopulation representerat av en slumpgraf (konfigurationsmodellen). Särskiltfokus läggs på slutskedet av epidemin. Målet är att härleda en asymptotiskapproximativ av tiden som krävs för att fullständigt försvinna ifrån popula-tionen, där approximationen blir exakt då storleken på populationen (grafen)går mot oändligheten. För att göra detta undersöker vi så kallade epidemiskt-genererade grafer. Strategin är att koppla båda starten och slutet av epideminmed en lämplig förgreningsprocess. Starten på epidemin kan approximerasmed en supercritical förgreningsprocess, och slutet med en subcritical förgre-ningsprocess.

Huvudresultatet i paper II säger att varaktigheten av epidemin (lämpligtskalad) är av ordning logn. Detta resultat generaliserar resultatet av Barbouroch Reinert 2013 (som inte studerar slutet på epidemin), till en bredare klass avgradfördelningar och infektionstider. Resultatet bevisas väsentligen i två steg:först visar vi att tiden då det finns ungefär n smittade individer är av ordninglogn (Lemma 2.6), sedan visar vi att denna tid och tiden till epidemin dör utär också av ordning logn (Lemma 2.7). Beviset för första resultatet bygger påresultat för förgreningsprocesser (Jagers, 1975), och på en generalisering avresultaten ur (Barbour and Reinert, 2013).

I paper III använder vi modellen i paper II, genom att inkludera vacci-nation i modellen, och undersöker effekten av olika vaccinationsstrategier påen SIR epidemi. Speciellt undersöker vi effekten av ett uniform vaccine ochett leaky vaccine i homogena och strukturerade populationer. Vi visar att detförsta vaccinet kan ha olika effekter på epidemin. I en uniform mixing popu-lation, så visar det sig att det första vaccinet antingen stoppar ett stort utbrotteller förlänger tiden på epidemin. I en strukturerad population kan detta vaccinleda till både en förkortning eller en förlängning av epidemin. I den första po-pulationen har båda vaccinen samma effekt på varaktigheten av epidemin. Viundersöker även en vaccinstrategi till acquaintance vaccination, och beskriverhur man beräknar den asymptotiska varaktigheten av epidemin.

I det sista pappret, paper IV, studerar vi en modell där två SIR epidemiersprids genom en population modellerad av konfigurationsmodellen. Vi antar attden första epidemin sprids innan den andra, och spridningen av den andra kansåledes beror på hur den första sprids. Vi antar att den första epidemin orsa-kar ett stort utbrott. Individer smittade under det första utbrotten blir immuna,eller delvis immuna, för den andra epidemin. Hur den första epidemin spridsanalyseras genom bond percolation och det tidiga skedet av utbrotten genom

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en multi-typ förgreningsprocess. Resultaten illustreras genom två exempel. Idet första exemplet antar vi att individer smittade av den första epidemin blir,oberoende av varandra, antigen får fullt eller inget smittoskydd för den andraepidemin. I det andra exemplet får smittade individer delvis smittoskydd förden andra epidemin.

En tre typs förgreningsprocess används för att beräkna R20. Vi visar även

hur man beräknar sannolikheten för ett stort utbrott. Vidare generaliserar papp-ret resultat från (Bansal och Meyers, 2012; Newman, 2005; Newman och Fer-rario, 2013). Alltså modellerna i (Newman, 2005; Newman och Ferrario, 2013)och (Bansal och Meyers, 2012) är specialfall av vår modell.

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