model hierarchies in edge-based compartmental modeling for ...model hierarchies in edge-based...

J. Math. Biol. (2013) 67:869–899DOI 10.1007/s00285-012-0572-3 Mathematical Biology

Model hierarchies in edge-based compartmentalmodeling for infectious disease spread

Joel C. Miller · Erik M. Volz

Received: 11 October 2011 / Revised: 10 July 2012 / Published online: 22 August 2012© Springer-Verlag 2012

Abstract We consider the family of edge-based compartmental models for epidemicspread developed in Miller et al. (J R Soc Interface 9(70):890–906, 2012). Thesemodels allow for a range of complex behaviors, and in particular allow us to explicitlyincorporate duration of a contact into our mathematical models. Our focus here is toidentify conditions under which simpler models may be substituted for more detailedmodels, and in so doing we define a hierarchy of epidemic models. In particular weprovide conditions under which it is appropriate to use the standard mass action SIRmodel, and we show what happens when these conditions fail. Using our hierarchy,we provide a procedure leading to the choice of the appropriate model for a givenpopulation. Our result about the convergence of models to the mass action modelgives clear, rigorous conditions under which the mass action model is accurate.

Mathematics Subject Classification 92D30

Electronic supplementary material The online version of this article(doi:10.1007/s00285-012-0572-3) contains supplementary material, which is available to authorized users.

J. C. Miller (B)Departments of Mathematics and Biology, Penn State University, University Park, USAe-mail: [email protected]

J. C. MillerFogarty International Center, NIH, Bethesda, USA

J. C. MillerDepartment of Epidemiology, Center for Communicable Disease Dynamics,Harvard School of Public Health, Boston, USA

E. M. VolzDepartment of Epidemiology, University of Michigan, Ann Arbor, USA

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http://dx.doi.org/10.1007/s00285-012-0572-3

870 J. C. Miller, E. M. Volz

1 Introduction

The spread of epidemics through populations is affected by many factors such as theinfectiousness of the disease, the duration of infection, the distribution of contactsthrough the population, and the typical duration of contacts. Typically for predictingepidemic spread or intervention effectiveness, we want an accurate, but simple, modelthat captures the relevant effects.

In our earlier work Miller et al. (2012), we introduced the edge-based compart-mental modeling approach for the spread of SIR diseases in populations with differentcontact dynamics. Table 1 summarizes the models and their underlying assumptions.In particular we showed that edge-based compartmental models can capture contactduration and social heterogeneity (variation in contact levels) simultaneously in math-ematically and conceptually simple terms. The models we studied all assumed thatthe population was made up of individuals who were identical except for their contactlevels. We modeled the population as a network, with nodes representing individualsjoined by edges representing potentially transmitting contacts. We also assumed asimple disease, with transmission occurring at rate β per edge and recovery occurringat rate γ . In this paper we investigate the relationships between models and how tochoose the simplest model appropriate for a given population.

The contact duration falls into three possibilities: it can be permanent, finite, orfleeting. In the permanent case, a contact that exists at any time has always existedand will always exist. In the finite case contacts may change over time. In the fleetingcase, contacts are so brief that over any macroscopic time scale an individual samplesa very large number of neighbors, and so it is safe to assume that the total contact timewith infected individuals matches its expected value.

The distribution of contact levels can be split into two types. In the first class ofmodels we discuss (expected degree models), we assign an expected degree κ to a node,with different nodes having different values of κ . The probability that an edge existsbetween two nodes is proportional to the expected degrees of each node. Edges arecreated independently of one another, so the existence of an edge between u and v doesnot alter whether an edge can exist between u andw. Note that the expected degree can

Table 1 The basic models and their underlying assumptions

Model Contact duration Heterogeneity type

Configuration model (CM) Permanent Actual degree k

Mixed Poisson (MP) Permanent Expected degree κ

Dynamic variable-degree (DVD) Finite Expected degree κ

Dynamic fixed-degree (DFD) Finite Actual degree k

Mean field social heterogeneity (MFSH) Fleeting Contact rate k or κ

Dormant contact (DC) Finite active and dormant Maximum degree km

Mass action (MA) Fleeting None

The models are distinguished based on how contacts are formed and broken. Contacts are assumed toform and break at some rate which can vary from zero (permanent) to infinite (fleeting). Depending on theprocess governing contact formation, we may know the actual degree k of an individual or we may knowits expected degree

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Hierarchy of epidemic models 871

take any non-negative real value, and we assume that the distribution of κ is given bythe probability density function ρ(κ). If the contacts are permanent, then we arrive at a“Mixed Poisson” network: a node with a given expected degree κ has its actual degreechosen from a Poisson distribution. These are also called Chung–Lu Networks afterChung and Lu (2002); these are a type of inhomogeneous random graphs Bollobáset al. (2007), van der Hofstad (2012), and are almost identical to network classesintroduced sin Britton et al. (2006), Norros and Reittu (2006). If existing contactsbreak at rate η and (independently) an individual with expected degree κ forms newcontacts at rate ηκ , then at any time the networks will be Mixed Poisson networks,but the structure will vary in time. The degree of any individual varies in time, but hasaverage equal to its expected degree. These are the dynamic variable degree networks.If the duration is so short that at every time the contacts are newly selected, then thisis the mean field social heterogeneity model (in the expected degree formulation).

In the second class of models (actual degree models) we assign an actual integernumber of contacts k, the degree, to each individual though in the case of the dormantcontact model not all of these contacts must be active at all times. We think of anindividual as having k stubs (or half-edges) which pair randomly with stubs of othernodes to form edges. In this case the existence of an edge between u and v removesan available stub from u, and so it affects the probability of an edge between u andw. The distribution of degrees is given by the probability mass function P(k). Ifthe contacts are permanent, this is a configuration model network Newman (2003).If individuals break contacts in such a way that they immediately form a new contactwith other individuals who are simultaneously breaking a contact, then degrees do notchange: this is the dynamic fixed degree model. If contacts break so quickly that at eachmoment the contacts are newly selected, then this is the mean field social heterogeneitymodel (in the fixed degree formulation). In a final model, the dormant contact model,individuals have a given number km (assigned randomly to each individual) of possiblecontacts, referred to as “stubs”. A given stub is either active (involved in a contact)or inactive (not in a contact). Inactive stubs join with other stubs at rate η1 and activestubs dissolve their contact at rate η2. This dormant contact model reduces to any ofthe other expected or actual degree models in appropriate limits.

The distinction between the expected degree models and the actual degree modelsbecomes apparent when we calculate the probability that an individual is susceptible.For the expected degree models, there is a continuum of risk levels and so we will haveto calculate the per-expected degree probability of not having been infected. In contrastfor the actual degree models the risk is discretized. The calculation is slightly differentin each case, but the underlying concepts are the same. The expected degree modelstend to be marginally more difficult conceptually, but they are simpler mathematically.

The equations produced by edge-based compartmental models are surprisinglysimple. Nevertheless the cases with simpler assumptions lead to simpler equations,and so it is worth knowing what conditions allow the use of a simpler model.In general when contact duration is short compared to the infection and recoverytime scales it is safe to use the fleeting contact models, while if contact duration islong compared to the duration of the epidemic it is safe to use the permanent contactmodels. However, there are some less obvious limits. If the average degree 〈K 〉 islarge while the rate of transmission per edge scales such that β is of order 1/ 〈K 〉 and

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the recovery rate γ is fixed, then the probability that any given edge transmits evenonce is small. The probability it transmits twice is negligible: the disease “sees” anedge at most once, so whether it is permanent, fleeting, or finite has no impact ondisease spread. Thus we may treat the model as if the contacts are fleeting, which sim-plifies the equations. If further the contact distribution is such that the contact levelsare generally close to the mean, then we can neglect variation in contact levels, andturn to the simple mass action model of Kermack and McKendrick (1927), Andersonand May (1991). The precise condition required for this is somewhat technical: wemust have | ⟨K 4

⟩ − 〈K 〉4 |/ 〈K 〉4 � 1 and 〈K 〉 � 1 with β = β 〈K 〉 fixed. Whenthis does not hold, there can be a significant deviation away from the mass actionmodel.

In this paper we begin by providing a flow chart leading to selection of the appropri-ate model for a given population. Following the flow chart, we introduce the hierarchyof the models which underlies the flow chart. We describe the precise assumptions ofthe models and sketch their derivation. Throughout we assume that the epidemic isstarted by a very small initial proportion infected. We demonstrate some of the sim-pler parts of the hierarchy. After that, we consider some of the more difficult aspectsof the hierarchy. We finally discuss some of the implications and limitations of ourapproach. In particular, we note that we give a simple heuristic for when the massaction equations are appropriate. In the Supplementary Information (SI) we providemore rigorous justifications for the claims in the hierarchy section. Because of itsimportance, we include the rigorous proof of conditions under which the mass actionequations hold in the main text.

2 Model selection

In Fig. 1 we present a flow chart that can be used to select the appropriate modelfor a given population. The conditions depend on the degree distribution and therate at which edges change. These are relatively straightforward to measure for apopulation Mossong et al. (2008), Wallinga et al. (2006), Salathé et al. (2010). Fromthe observations of population and disease parameters, we can choose the simplestmodel to accurately represent disease spread in a given population. The equations foreach model are developed in Sect. 3.

Most existing modeling of infectious disease spread are based on the mass actionmodel Kermack and McKendrick (1927). This flow chart gives appropriate conditionsunder which this model is reasonable. In general, we must have most degrees closeto the average degree, but at the same time either contacts must have high turnover orthe probability of transmission per contact is very low. If these conditions hold, it isappropriate to use the MA model.

Before using this flow chart it is always prudent to be sure that the population doesnot violate other assumptions of the models. For example, different contact structurebetween age groups may require more consideration. If there are important features notcaptured by these models it may be possible to develop a custom model that capturesthe relevant detail (see Miller and Volz 2011; Volz et al. 2011). Otherwise we may notbe able to rely on edge-based compartmental models and may have to use simulation.

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Fig. 1 Flow chart for model choice. Depending on the how the edge turnover rates compare to the infectionand recovery rates, as well as the degree distribution, we arrive at different models. The model abbreviationsare as in Table 1. This flow chart assumes that all structure in the population is due to heterogeneity in contactlevels

Some features, such as non-constant rates of infection or recovery or a latent periodcan be captured by straightforward generalizations of the models.

The questions asked in the flow chat are somewhat vague in the sense that whethera number is large or not is somewhat a matter of opinion. In Sect. 3 we address this in

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more detail. We show how the results of the models converge as the parameter valueschange.

Although most of these models are new, some of these (or closely related models)have been applied previously. An early version of the CM model was introducedin Volz (2008), and an earlier, closely related approach which only gives final sizeinformation has been widely used (e.g., Newman 2002; Meyers 2007; Miller 2007).The DFD model has been used by Volz and Meyers (2007), Volz et al. (2010) to studysyphilis and HIV infections. The MFSH models have been used widely, primarily tounderstand HIV spread Anderson and May (1991), May and Anderson (1988), Mayand Lloyd (2001), Moreno et al. (2002), Pastor-Satorras and Vespignani (2001).

3 The model hierarchy

In this section we investigate the hierarchy of Fig. 2 underlying the flow chart in Fig. 1.We first give a brief overview of the standard simple model, the mass action model. Wethen consider the hierarchy of models. We find it convenient to consider the expecteddegree models before the actual degree models. We consider the models roughly inorder of increasing complexity, explaining the underlying assumptions and sketchingthe derivation of the equations from Miller et al. (2012). If we can reduce the modelto another model, we explain why this should happen and give some details of themathematical explanation. More complete derivations are in the SI.

The mean field social heterogeneity models require further attention. We have twoformulations, one in terms of expected degree κ and the other in terms of actual degreek. We will not address this initially while we discuss the main structure of the hierarchy,but at the end of this section we show that the two formulations are mathematicallyequivalent in the sense that each can be derived from the other. So we do not distinguishbetween the two models in the hierarchy of Fig. 2. We will also show that all othermodels reduce to mean field social heterogeneity in the large 〈K 〉 limit if β 〈K 〉 andγ are both constant. In turn we have conditions under which the mean field socialheterogeneity model reduces to the mass action model. Given any model with β 〈K 〉and γ fixed, 〈K 〉 → ∞ and

⟨K 4

⟩/〈K 〉4 → 1, we arrive at the mass action model.1

3.1 The mass action model

The mass action (MA) model assumes that all individuals have the same rate of contactformation k (or κ) and edges (contacts) are sufficiently short-lived that we may neglectcontact duration. The transmission rate per contact is β, and so the combined trans-mission rate is β = βk. If we take S, I , and R to be the proportion of the populationthat is infected, then Fig. 3 leads to

S = −β I S, I = β I S − γ I, R = γ I

1 It is worth noting that in the expected degree formulations the average fourth power of the degree and theaverage fourth power of the expected degrees are not equal, but if the ratio with 〈K 〉4 tends to 1 for either,it does for the other as well.

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Fig. 2 The hierarchy structure. The edges summarize the relations between the models. A solid arrow fromone model to another represents that the target model can be derived as a limiting case of the base model.We give heuristic explanations in the text for these and more rigorous derivations in the SI. A dashed arrowdenotes that the target model can be derived as a special case of the base model. The dashed arrows arestraightforward and the justification is given in the main text

Fig. 3 The flow diagram for the MA model. S is the proportion susceptible, I the proportion infected, andR the proportion recovered. The flux from I to R is γ I , and the flux from S to I is β I S

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However, this may be simplified somewhat by noting that S + I + R = 1. We replacethe equation for I with I = 1 − S − R

S = −β I S, I = 1 − S − R, R = γ I

3.2 Expected degree models

We now study the expected degree models for which each node has an expecteddegree κ assigned using the probability distribution function ρ(κ). We allow κ to bea continuous variable. At any given time, the probability that two nodes u and v sharean edge is proportional to κuκv , and each edge is assigned independently of all others.

We briefly sketch the approach used to derive equations in these networks. Fulldetails are in Miller et al. (2012). We define Θ as a function of time such that 1 −Θ

represents the per-unit κ probability of having been infected. To be precise let u and vbe nodes such that v has a slightly larger κ than u : κv = κu +�κ with �κ � 1. Bycalculating the increased risk to v, we can arrive at a formula for s(κ, t) the probabilityan individual with a given κ is susceptible. In the small �κ limit, (1 − Θ)�κ is theprobability that the small amount of extra κ v has has ever contributed an edge that hastransmitted to v. The probability that u is susceptible is s(κu, t) and the probabilitythat v is still susceptible is s(κu +�κ, t) = [1 − (1 −Θ)�κ]s(κu, t)+ O(�κ2).

Taking �κ → 0, we have ∂s/∂κ = (1 −Θ)s. So s(κ, t) = exp[−κ(1 −Θ)] andthe probability a random node is susceptible is S(t) = Ψ (Θ(t)) where

Ψ (x) =∞∫

0

e−κ(1−x)ρ(κ) dκ

The difference in the various expected degree models is in how long edges last: theymay be permanent, fleeting or finite.

For each system, we sketch the derivation of the equations. A full derivation andcomparison with simulations for each model appear in Miller et al. (2012).Our focusin this paper is on understanding how the systems relate to one another rather thandetails of the derivation. We begin by considering permanent edges.

3.2.1 Mixed Poisson

In the mixed Poisson (MP) model, the population is assumed to be static, so that ifa contact ever exists, then it has always existed and will always exist. The relevantflow diagram is in Fig. 4. We set ΦS, ΦI , and ΦR to be the per-unit κ probabilities ofhaving a susceptible, infected, or recovered neighbor that has not transmitted. For theMP model,Θ = ΦS +ΦI +ΦR . We can solve forΦS = Ψ ′(Θ)/Ψ ′(1), and show thatΦR = β(1−Θ)/γ from which we can findΦI in terms ofΘ . So Θ = −βΦI becomesan equation just in terms of Θ . Putting this all together, the governing equationsare

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Fig. 4 The MP flow diagram. (Left) Edge quantities:ΦS , ΦI , ΦR , and 1−Θ are the per-unit κ probabilitiesof having a neighbor which is susceptible, infected and has not transmitted, recovered and did not transmit,or which has transmitted respectively. (Right) Individual quantities: S the probability of being susceptible,I the probability of being infected, and R the probability of being recovered

Fig. 5 The MFSH flow diagram (for the expected degree formulation). (Left) The variables ΦS , ΦI , andΦR are the per-unit κ probabilities for an individual to be connected to a susceptible, infected, or recoverednode. However, because the edges are fleeting, these change over quickly, and are thus always equal to�S ,�I , and �R . The variable 1 − Θ remains the per-unit κ probability of having received infection.(Center) The variables �S ,�I , and �R are the proportion of contacts at any given time which are madeby susceptible, infected, or recovered individuals respectively. (Right) The variables S, I , and R remain theproportion of individuals who are susceptible, infected, or recovered

Θ = −βΘ + βΨ ′(Θ)Ψ ′(1)

+ γ (1 −Θ)

R = γ I, S = Ψ (Θ), I = 1 − S − R

3.2.2 Mean field social heterogeneity [expected degree formulation]

We now consider the opposite limit in which edges are fleeting. The mean field socialheterogeneity (MFSH) model Anderson and May (1991), May and Anderson (1988),May and Lloyd (2001), Moreno et al. (2002), Pastor-Satorras and Vespignani (2001)generalizes the MA model by allowing for variations in contact rate among the peo-ple. At any given time the node is expected to have κ edges, but they change overrapidly. The relevant flow diagram is shown in Fig. 5. This introduces some newvariables, �S,�I , and �R which are the probabilities a new contact is with a sus-ceptible, infected or recovered individual. Unlike the MP case, Θ = ΦS +ΦI +ΦR .Through some simplifications similar to the MP case, we find �R = γ (1 − Θ)/β,so ΦI = �I = 1 − Ψ ′(Θ)/ψ ′(1) − γ (1 − Θ)/β. This leads to a similar system ofgoverning equations

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Fig. 6 DVD flow diagram. The variables have the same meanings as in the MFSH case, but because edgeshave nonzero duration, we must include the possibility that a partner changes status while an edge exists.Note that ΦS = �S = Ψ ′(Θ)/Ψ ′(1). The fluxes in and out of ΦI lead to a differential equation whichcan be integrated to give an expression for ΦI in terms of Θ and �R . This leads to a differential equationfor Θ in terms of Θ and �R

Θ = −β + βΨ ′(Θ)Ψ ′(1)

+ γ (1 −Θ)

R = γ I, S = Ψ (Θ), I = 1 − S − R

These equations differ in only one term from the MP equations.The only difference in the assumptions of the MFSH model and the MA model is that

the MA model assumes all contact rates are the same. Indeed, if all expected degrees arethe same κ , thenΨ (Θ) = exp[−κ(1−Θ)]. We set R = γ (1−Θ)/β, I = 1− S − R,and S = Ψ (Θ). We first note that with thisΨ (Θ), we find Θ = −β I . Setting β = βκ

and taking the time derivatives of S and R, we see that S = −β I S and R = γ I . Thuswe have arrived at the MA equations.

More generally we expect that if the variation in contact rate is sufficiently small, theMFSH model should behave like the MA model. In Sect. 3.4.3 we discuss this further.

We note that in Miller et al. (2012), the final size derivation for this model has asign error. Setting Θ = 0, and solving, we find Θ = β

γ(−1 + Ψ ′(Θ)

Ψ ′(1) ) + 1. Solvingthis equation and setting R(∞) = 1 − Ψ (Θ) gives the final size.

3.2.3 Dynamic variable-degree

In the dynamic variable-degree (DVD) model, an individual may create or terminateedges at any time. A node with expected degree κ creates edges at rate κη. Any exist-ing edge breaks at rate η, so a node with expected degree κ will on average have κedges, though the value fluctuates. The flow diagram in Fig. 6 is similar to the previousdiagrams. We can use the flow diagram to find a differential equation for ΦI , whichcan be solved in terms of�R . However, we cannot analytically solve for�R in termsof Θ , so we require additional equations. The governing equations are

Θ = −βΘ + βΨ ′(Θ)Ψ ′(1)

+ γ (1 −Θ)+ η

(1 −Θ − β

γ�R

),

�R = γ�I , �S = Ψ ′(Θ)/Ψ ′(1), �I = 1 −�S −�R,

R = γ I, S = Ψ (Θ), I = 1 − S − R.

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The new variables �S,�I , and �R give the probabilities that a newly formed edgewill connect to a susceptible, infected, or recovered node respectively.

If the changeover rate is sufficiently fast we anticipate that the model should reduceto the MFSH model. This statement can be made precise by considering a susceptiblenode u. The node u is constantly creating new edges and breaking existing edges. Itsrisk of infection depends on how many infected neighbors it has. If edges are long-lasting, then knowing that u has not been infected suggests that its neighbors may stillbe susceptible. However, if an edge to an infected neighbor is likely to break beforethe neighbor transmits, then knowing that u is still susceptible says little about currentneighbors. In mathematical terms this means if η/β is large, we anticipate that theDVD model reproduces the MFSH model. This is made more rigorous in the SI.

In the opposite limit, we would expect that having small η leads to an effectively sta-tic network, so the MP model should result. In fact this is true, but the precise conditionis somewhat more subtle than might be anticipated. It is not enough that η/β and η/γbe very small because the epidemic can last for many generations. In practice the staticmodel will work well at early times, but may fail at later times as the contact structureaccumulates changes. We set t0 to be a time early enough that the number of infectionsby time t0 is very small, but large enough that shortly thereafter the total number ofinfections is no longer negligible. At some later time t the MP model will be reason-able if η(t − t0) is small. A more precise condition that the MP model is reasonable ifη(�I +�R)/r � 1 where r is the early exponential growth rate is also described inthe SI.

We show convergence of the DVD model to the MP and MFSH heterogeneitymodel as η → 0 or η → ∞ in Fig. 7. The technical mathematical details showingthis convergence are in the SI.

3.3 Actual degree models

For our second class of models, we assume that each node has an actual degree kassigned using the probability mass function P(k). The degree must be a non-negativeinteger. Each individual is given k stubs, and at any time those stubs may join in pairswith stubs of other nodes to form edges. In most models we assume that the stubs arealways in pairs (though the partner may change), so the degree is k. In the dormantcontact model, we allow stubs to be active or dormant, and thus take km [distributedaccording to P(km)] to be the maximum degree of a node, with ka and kd the activeand dormant degrees respectively ka + kd = km . In all of these models, we use θ(t)to denote the probability that a stub has not transmitted infection from a neighborto its node by time t , and φS, φI , and φR to denote the probabilities that a stub hasnot transmitted infection and currently connects to a node of the given status. Theprobability a node with a given k is susceptible is θk , and taking a weighted averageover all k, we find S(t) = ψ(θ(t)) where

ψ(x) =∑

k

P(k)xk

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Fig. 7 Convergence of DVD to MP and MFSH models. We consider the DVD model with varying valuesof η. The probability density function for the expected degree is ρ(κ) = eκ/[exp(10)− 1] for κ ∈ (0, 10)and 0 otherwise. This yields Ψ (x) = [1 − exp(−10(1 − x))]/[10(1 − x)]. We take β = 0.2 and γ = 1. Asη decreases (top), the MP model results, while as η increases (bottom) the MFSH model results. Note thedifference in axes from top to bottom

As above, we sketch the derivations of the equations. Full details of the deriva-tions and comparison with simulation are in Miller et al. (2012). We begin again withpermanent contacts.

3.3.1 Configuration model

The configuration model (CM) networks are similar to the MP networks. In a CM net-work, the exact degree of an individual is assigned. A node is given k stubs, assignedusing the probability mass function P(k). Once all stubs are assigned to nodes, stubsare randomly paired into edges. The resulting network is static. We define θ to be theprobability that the neighbor along a random stub from u has not transmitted infectionto u (so 1 − θ is the probability the neighbor has transmitted). Using the flow diagramin Fig. 8, we find that

θ = −βθ + βψ ′(θ)ψ ′(1)

+ γ (1 − θ)

R = γ I, S = ψ(θ), I = 1 − S − R

This is similar to the MP model. In fact, it is possible to show that a MP networkis a special case of the CM networks. In MP networks, a node u with expecteddegree κu has its actual degree chosen from a Poisson distribution of mean κu

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Fig. 8 The flow diagram for the CM model. (Left) Edge quantities: the variables φS , φI , and φR are theprobabilities that a neighbor of a given node has not yet transmitted to that node and has the given status.The variable θ = φS + φI + φR is the probability that the neighbor has not yet transmitted. We can solvefor φI in terms of θ , which leads to a simple differential equation for θ in terms of θ . (Right) Individualquantities S, I , and R are the proportions in each state

Fig. 9 The MFSH flow diagram (for the actual degree formulation). (Left) The variables φS , φI , andφR denote the probabilities that a stub has not yet transmitted infection to an individual and is currentlyconnected to a partner of a given type. These sum to θ , the probability the stub has not yet transmittedinfection. Each of these is θ times the corresponding πS , πI or πR . (Center) The proportion of stubs thatbelong to susceptible, infected, or recovered individuals or equivalently, the probability a new edge is withan individual of each type. (Right) The proportion of the population in each state

(in the limit of a large network). Thus in an MP network the probability a nodehas degree k is P(k) = ∫ ∞

0 [e−κκk/k!]ρ(κ) dκ . Thus ψ(x) = ∑k P(k)xk =∫ ∞

0 e−κ (∑κk xk/k!)ρ(κ) dκ = Ψ (x), and so the MP model emerges as a special

case of the CM model.2

3.3.2 Mean field social heterogeneity [actual degree formulation]

In the actual degree version of mean field social heterogeneity (MFSH), each individ-ual has some number of stubs k assigned independently of other individuals. Stubschange edges quickly so that the neighbor at any given time has no bearing on who theneighbor is later. We use the flow diagram in Fig. 9. The new variables πS, πI , and πR

represent the proportion of all stubs that belong to susceptible, infected, or recoverednodes. A newly formed edge connects a node to a susceptible, infected, or recovered

2 In finite networks, there are some correlations between degrees that are different between the models, butthis disappears in the large network limit.

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Fig. 10 The DFD flow diagram. (Left) The arrows at the top of the diagram represent breaking edges. Thetotal rate at which stubs that have not yet transmitted infection break is ηθ . The proportions of these thatthen join up with individuals of a given type is proportion to the corresponding π variable. We cannot finda simple relation giving φI , so our equations cannot simplify as much as in previous models

node with probabilities πS, πI , and πR respectively. We can find that πR = −γ θ/βθfrom which we can find πR in terms of θ . This gives πI in terms of θ which allowsus to find a differential equation for θ in terms of θ . The governing equations are

θ = −βθ + βθθψ ′(θ)ψ ′(1)

− γ θ ln θ

R = γ I, S = ψ(θ), I = 1 − S − R

Using techniques similar to those for the expected degree formulation of the MFSHmodel, we can show that if all degrees are the same, then this reduces to the MA model.Similarly, if the degrees are sufficiently close to the mean degree in the sense that⟨K 4

⟩/ 〈K 〉4 → 1 and both γ and β 〈K 〉 are fixed, then the solution again converges

to that of the MA model. In Sect. 3.4.1 we show that in fact this model is equivalentto the expected degree formulation of the MFSH model.

3.3.3 Dynamic fixed-degree

In the dynamic fixed-degree (DFD) model a node is given k stubs, which are pairedwith stubs of other nodes into edges. As time progresses, an edge may break, and thefreed stubs immediately form edges with stubs from other edges that break at the sametime, a process we refer to as “edge swapping”. The rate any edge breaks is η. FromFig. 10, the resulting equations are

θ = −βφI ,

φS = −βφIφSψ ′′(θ)ψ ′(θ)

+ ηθπS − ηφS,

φI = βφIφSψ ′′(θ)ψ ′(θ)

+ ηθπI − (β + γ + η)φI ,

πR = γπI , πS = θψ ′(θ)ψ ′(1)

, πI = 1 − πR − πS,

R = γ I, S(t) = ψ(θ), I (t) = 1 − S − R.

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Fig. 11 Convergence of DFD to CM and MFSH models. We consider the DFD model with varyingvalues of η. The degrees are k = 6, k = 8, and k = 10, with probability 1/3 each. This yields ψ(x) =(x6 + x8 + x10)/3.We take β = 0.2 and γ = 1. As η decreases (top), the CM model results, while as ηincreases (bottom) the MFSH model results. Note the difference in axes from top to bottom

Unlike previous models, these equations cannot be simplified into a single equationfor θ .

The DFD model plays the same role in the actual degree case that DVD modelplayed in the expected degree case. It experiences similar limiting behavior. If η/β islarge, we recover the MFSH model. Alternately the CM is an accurate approximationso long as η(t − t0) is small where t0 is a time around when the epidemic begins toinfect significant numbers. Again, a more precise condition that η(πI + πR)/r � 1where r is the early exponential growth rate is described in the SI.

Figure 11 shows the convergence of this model to the CM and MFSH models asη → 0 or η → ∞.

3.3.4 Dormant contacts

We finally move to the dormant contact (DC) model which captures all the previousexpected and actual degree models as limiting cases. In the DC model, each node isgiven km stubs [with km chosen using P(km)]. However, only a fraction of them areactive. At any given time, the node will have ka active stubs and kd dormant stubs, sokm = ka +kd is the maximum number of active stubs. Active stubs become dormant atrate η2 and dormant stubs become active at rate η1. We defineψ(x) = ∑

kmP(km)xkm .

Using Fig. 12, the governing equations are

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Fig. 12 The flow diagram for the DC model. Now when a stub ceases to be in an edge, it enters a dormantphase until it finds a new partner. The new variable φD represents the probability a stub has not yettransmitted infection to an individual and is dormant. The new ξ variables represent the proportion of allstubs which belong to nodes of a given type and are part of edges (i.e., are active). We set ξ = ξS + ξI + ξRto be the proportion of stubs that are active. The π variables are now the proportion of stubs which belongto nodes of a given type and are available to form edges (i.e., are dormant). We set π = πS + πI + πR tobe the proportion of stubs that are dormant. The probabilities a new edge is with an individual of each typeare πS/π, πI /π and πR/π . Again, we are not able to significantly simplify the resulting equations

θ = −βφI

φS = −βφIφSψ ′′(θ)ψ ′(θ)

+ η1πS

πφD − η2φS

φI = βφIφSψ ′′(θ)ψ ′(θ)

+ η1πI

πφD − (η2 + β + γ )φI

φD = η2(θ − φD)− η1φD

ξR = −η2ξR + η1πR + γ ξI , ξS = (θ − φD)ψ ′(θ)ψ ′(1)

, ξI = ξ − ξS − ξR

πR = η2ξR − η1πR + γπI , πS = φDψ ′(θ)ψ ′(1)

, πI = π − πS − πR

ξ = η1

η1 + η2, π = η2

η1 + η2

R = γ I, S = ψ(θ), I = 1 − S − R

The new variable φD represents the probability that a stub has not transmitted infectionto its node and is currently dormant. The variables πS, πI , and πR now measure theproportion of all stubs which are both dormant and belong to a susceptible, infected,or recovered node, with π = πS + πI + πR = η2/(η1 + η2) the probability a stub isdormant. The ratios πS/π, πI /π , and πR/π give the probabilities that a newly formededge connects to a susceptible, infected, or recovered node. The variables ξS, ξI , andξR give the probabilities that a stub is active and belongs to a susceptible, infected, orrecovered node, with ξ = ξS +ξI +ξR = η1/(η1 +η2) the probability a stub is active.

It is relatively straightforward to see that if η1 is much larger than η2, then theproportion of time a stub is dormant is tiny, π � 1. Consequently at any moment anode is expected to have ka = (1 − π)km ≈ km active contacts. As η2/η1 shrinks,this approximation improves. So in this limit, the DC model should reduce to the DFDmodel with edges breaking at rate η2. Indeed, in the equations above, if we take η = η2

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and assume η1 � η, then φD is negligibly small, so the φD terms drop out of the φS

and φI equations. The values of πS and πI both go to zero as η1 grows, but a littlemore care shows that πS/π approaches ξS and πI /π approaches ξI and the ξ variablessolve the same equations in this limit as the π variables in the DFD equations.Thusthe DC equations reduce to the DFD equations in this limit. More details are in the SI.

In the opposite limit, if η2 is large compared to η1, then stubs spend most timedormant, ξ � 1. If the number of stubs is sufficiently large, the expected number ofactive stubs κ = E[ka] = ξkm will not be negligible. The variation in the number ofactive stubs will be significant relative to its expected value κ . However, κ � km sothe number of dormant stubs kd = km − ka can be approximated as kmπ . The rate thatdormant stubs become active is η1, so the rate at which the node forms new edges isapproximately η1kd = η1kmπ which itself is η2κ , while each existing edge breaks atrate η2. This is the assumption underlying the DVD model. A careful analysis of theequations show that indeed we can reduce them to the DVD model if η2 � η1. Moredetails are in the SI.

Alternately, we can have the DC model converge to the MFSH model if 〈K 〉 → ∞,as long as β 〈K 〉 remains fixed. In this case we find that the MFSH model is a goodapproximation, using β = βξ as the transmission rate. The underlying argument ofthis is that as 〈K 〉 increases, the probability of transmission per edge decreases. Conse-quently, it becomes unimportant whether the edge has long or short duration becauseit is incredibly rare for the disease to try to transmit along the same edge twice. Thetotal infectiousness of an individual is then given by the number of active edges timesthe per-edge infection rate. We can assume that only a proportion ξ of the stubs areactive at any time, and we can absorb this into β. So an individual with k stubs causestransmission at rate βk where β = ξβ. This is independent of how (or if) the edgesare changing in time.

Figure 13 shows how the DC model converges in the various limits.

3.4 Further analysis of mean field social heterogeneity models

In this section we show that the two MFSH models are equivalent. We also show thatall the non-mass action models described here reduce to the MFSH model if the aver-age number of contacts is large, and the probability of infection per edge scales like1/ 〈K 〉. Technically, our results show that there is no difference between the results ofany of the (non-MA) models in this limit, so any of these models would be appropriate,but generally the equations of the MFSH model are simplest so we use it.

3.4.1 Equivalence of mean field social heterogeneity formulations

There are two formulations of the MFSH model, one with expected degrees and theother with actual degrees.

In the expected degree formulation, each individual has a given expected degree κ(which varies by individual). On average, at any given moment of time the individualhas κ contacts. The exact number of contacts may vary from moment to moment,but how many contacts exist at one moment and who those contacts are with are

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Fig. 13 Convergence of DC to DFD, DVD, and MFSH models. We take β = γ = 1. (Top) We considerthe convergence of the DC model to the DFD model as η1 increases. We take ψ(x) = (x2 + x8)/2.For the DC model we take η2 = 1 and vary η1. As η1 increases, the DC model converges to the DFDmodel result for η = 1. (Middle) We consider the convergence of the DC model to the DVD model as η1decreases and the degrees increase to compensate. For the DC model, we have ψ(x) = (x1/ξ + x4/ξ )/2where ξ = η1/(η1 + η2). We use η2 = 1 and vary η1. For the DVD model, we take η = 1 and Ψ (x) =(exp[−(1− x)]+exp[−4(1− x)])/2. (Bottom) We consider the convergence of the DC model to the MFSHmodel with η1 = η2 and η1, η2 → ∞. We take ψ(x) = (x2 + x8)/2. The MFSH model has the same ψ ,but uses βξ = β/2 as the transmission rate. Note that for all three calculations, the η1 = 1 curves are thesame

independent from any other moment. Thus over a short period of time, we may safelyassume that the total number of contacts with infected individuals matches the expectednumber.

In the actual degree formulation, each individual is assigned an actual degree kand given k stubs. At any time each of those k stubs forms an edge with a stub fromanother node, but who the stub connects to changes rapidly. Again, over a short periodof time, we may assume that the number of contacts with infected individuals matchesthe expected number.

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Actual degree formulation as a special case of expected degree formulation Becausecontacts are very short, over any time interval, the total contact time is well-approximated based on the expected amount of contact at any given moment. Thuswhether the individuals have the same number of contacts at each moment, or whetherthe amount varies from moment to moment, the effect over any macroscopic time inter-val is the same. Thus the actual degree model should be a special case of the expecteddegree model. In the expected degree formulation, the probability that a node with κexpected contacts is susceptible is exp[−κ(1 −Θ)] and in the actual degree formula-tion the probability that a node with k contacts is susceptible is θk . If κ = k, we expectthese to equal, and so we anticipate θ = exp(Θ − 1), or Θ = 1 + ln θ . Plugging thisinto the expected degree equations, we arrive at the actual degree equations. So thetwo methods are equivalent subject to a change in variables.

Expected degree formulation as a limiting case of the actual degree formulation Thefact that the actual degree formulation leads to the expected degree formulation isbased on similar reasoning. The underlying additional idea is that any continuous dis-tribution can be approximated by a sufficiently well-refined discrete distribution. Wethen need a way to take a well-refined discrete distribution and rescale it so that allthe probability is massed at integer values. To do this we make the observation thatfor the MFSH models, we can multiply every individual’s (expected or actual) degreeby L with no impact on the epidemic so long as we also divide the transmission rateby L . The resulting equations remain unchanged.

To make this more precise, consider a given β and ρ(κ) and assume L is large. Weapproximate the continuous distribution of κ using a discrete distribution where theprobability of k/L is

∫ (k+1)/Lk/L ρ(κ) dκ . Increasing L gives a finer scale approxima-

tion. We then multiply every k/L by L , to get P(k) = ∫ (k+1)/Lk/L ρ(κ) dκ where k is an

integer. So long as we divide β by L , the spread of the epidemic on a population withthe given P(k) and the rescaled β will closely approximate the epidemic spread in theoriginal expected degree population, with the approximation improving as L → ∞.We find that the actual degree equations converge to the expected degree equations asL → ∞. Full details are in the SI.

As an example we take the uniform distribution from 0 to 10 for κ . So Ψ (x) =(1−exp[−10(1−Θ)])/[2(1−Θ)]. We initially start with a discrete distribution on theintegers 0 through 9 with weight 1/10 on each. So our initial ψ(x) is

∑9k=0 xk/10 =

(x10 − 1)/10(x − 1) with β = β. We then refine the distribution. For given L wehave ψ(x) = ∑10L−1

k=0 xk/10L = (x10L − 1)/[10L(x − 1)], and we take β = β/L .Convergence is shown in Fig. 14.

3.4.2 Reduction of all models to mean field social heterogeneity modelat large average degree

One particularly important limit corresponds to people having many contacts, but alow probability of transmitting per contact before recovering. That is, 〈K 〉 is large, andβ/γ is comparably small. In this limit, we expect that the duration of contact becomesunimportant, because the infection is unlikely to cross an edge more than once so thedisease has no way to know how long the edge lasts. Heterogeneity in contact levels

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Fig. 14 Convergence of actual degree formulation of MFSH to expected degree formulation. The actualdegree formulation of the MFSH model for the appropriate discrete distribution (described in text) convergesto the expected degree formulation of the MFSH model with contact rates chosen uniformly from 0 to 10

Fig. 15 Convergence to MFSH. (Top) Convergence of the expected degree models assuming a uniformdistribution from 0 to 2 〈K 〉. (Bottom) Convergence of the actual degree models assuming a third each with〈K 〉 /2, 〈K 〉, and 3 〈K 〉 /2 stubs. The solid curves correspond to 〈K 〉 = 1 (top) and 〈K 〉 = 2 (bottom). Thedotted curves correspond to various larger values of 〈K 〉. Because β decreases to balance the increase in〈K 〉, the MFSH curves are the same for all 〈K 〉

may still play an important role. Figure 15 shows the convergence of our models tothe MFSH model as 〈K 〉 increases for different distributions. If the heterogeneity issufficiently small, it is reasonable to expect that the MA SIR model is appropriate.

In Sect. 4.2 we rigorously derive the MFSH model from the DC model assuming〈K 〉 → ∞ with fixed β 〈K 〉 and fixed γ . A similar proof will apply for the othermodels, or we can simply argue that the DC model reduces to all of the others, andcareful attention to detail shows that they inherit this limiting case.

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3.4.3 Reduction of MFSH to MA model if⟨K 4

⟩/ 〈K 〉4 → 1 and β 〈K 〉 fixed

It is straightforward to show that the MFSH model becomes the MA model if everynode has the same contact rate. However more generally, we would expect that if thecontact rates are sufficiently close together, the model should behave like the MAmodel. In fact this holds if

⟨K 4

⟩/ 〈K 〉4 → 1 with β 〈K 〉 fixed. Intuitively, if the

number of nodes with higher or lower contact rates is very small, their contributionto the spread of the disease is drowned out by the average nodes, and so the diseaseshould spread as if the contact rate were homogeneous. We do not have a good intuitiveexplanation for why the precise condition relies on

⟨K 4

⟩/ 〈K 〉4. However, we prove it

rigorously in Sect. 4.3.We expect this case to be particularly relevant if the average degree is large and

infectiousness is low. Combining this with the previous result, we conclude that allmodels converge to the MA model under appropriate conditions as 〈K 〉 → ∞.

4 Rigorous proof of convergence to mass action equations

In this section, we give a rigorous proof of one of the most significant results. Namely,if 〈K 〉 is large, but

⟨K 4

⟩/〈K 〉4 is approximately 1 and β 〈K 〉 and γ are small compared

to 〈K 〉, then regardless of which model we use, the result is well-approximated by theMA equations.

We first show some examples, and then provide the details needed to prove the resultfor the DC model. The same result for simpler models can be derived by the samemanner (though in some cases the proof will be simpler). For our proof, we assumewe have a sequence of populations and diseases indexed by n such that 〈K 〉 →∞,

⟨K 4

⟩/ 〈K 〉4 → 1 with ξβ 〈K 〉 fixed. We show that this converges to the mass

action equations using β = ξβ 〈K 〉.We do the proof in two steps. First, we show that in the limit of large 〈K 〉 and constant

βξ 〈K 〉, the DC equations converge to the MFSH equations. We then show that theMFSH equations converge to the MA equations if additionally

⟨K 4

⟩/ 〈K 〉4 → 1.

4.1 Examples

As examples, we consider several different population structures using the MP modelin Fig. 16. In the first, we take an exponential distribution of expected degrees. To varythe average degree, we change the decay rate of the exponential.

ρ(κ) = e−κ/〈K 〉

〈K 〉

This does not satisfy the conditions that⟨K 4

⟩/ 〈K 〉4 → 1. In the other exam-

ples, we take a uniform distribution, with expected degrees chosen uniformly from(〈K 〉 − 〈K 〉α , 〈K 〉 + 〈K 〉α). That is,

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Fig. 16 The expected degree distributions for our examples of convergence as 〈K 〉 → ∞, with and without⟨K 4

⟩/ 〈K 〉4 → 1

Fig. 17 Sample convergence as 〈K 〉 → ∞. We use the exponential distribution described in Fig. 16. The

distribution of expected degrees decays exponentially and⟨K 4

⟩/ 〈K 〉4 does not approach 1 as 〈K 〉 grows.

The solutions appear to converge, but they do not converge to the MA SIR model

ρ(κ) =

⎧⎪⎨

⎪⎩

0 κ < 〈K 〉 − 〈K 〉α1

2〈K 〉α 〈K 〉 − 〈K 〉α < κ < 〈K 〉 + 〈K 〉α0 κ > 〈K 〉 + √〈K 〉

This satisfies the condition for any α < 1. Using⟨K 4

⟩to denote the average of the

4th power of the expected degree,3 we have⟨K 4

⟩= 〈K 〉4 + 2 〈K 〉2+2α +〈K 〉4α /5. If

α = 1, then⟨K 4

⟩/ 〈K 〉4 does not approach 1. The distribution would not make sense

for α > 1 since some nodes would have negative expected degree.For all distributions, we take β = 2/ 〈K 〉 and γ = 1. The corresponding MA

model has β = 2. In Fig. 17 we plot the results for the exponential distribution. Thisdistribution does not satisfy the conditions, and we see that the solutions do converge,but not to the MA model. In Fig. 18, we take the uniform distribution from 〈K 〉−〈K 〉αto 〈K 〉 + 〈K 〉α . We again see that when the conditions are not satisfied (α = 1), thesolutions may still converge, but not to the corresponding MA model. However, whenthe conditions are satisfied, the solutions converge to the MA model.

3 Our condition that⟨K 4

⟩/ 〈K 〉4 → 1 applies equally for the fourth power of actual or expected degree

since at leading order they are the same.

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Fig. 18 Sample convergence as 〈K 〉 → ∞. We use the uniform distribution described in Fig. 16. In thefirst, second, and third rows, we use α = 1, 0.85, 0.5 respectively. The condition fails for α = 1 and thesolutions do not converge to the MA model, but they converge to the MA model for smaller α

4.2 DC to MFSH

We now begin our proof that the DC model converges to the MA model. We beginby showing that it converges to the MFSH model. We will use the following resultseveral times in the proof: Assume g(t) is a function of time and

g = −c(t)g + f (t)

If c(t) ≥ C > 0 for all time and | f | ≤ F , and further |g(0)| ≤ F/C , then for alltime |g(t)| ≤ F/C . To see this, assume that f and g are positive. If g is increasingbeyond F/C , then −cg + f must be negative, violating the assumption that g is

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increasing. Similar results hold for negative values of f and g. Our proofs skip someof the intermediate steps. Complete details of all the algebra is in the SI.

We want to prove that if degrees increase while the infection rate simultaneouslydecreases, with γ constant, then we must arrive at the mass action equations. Weintroduce the notation o(1) to denote a function whose value goes to zero as 〈K 〉 → ∞.Our result above implies that if f = o(1) and c is bounded away from 0 as 〈K 〉 changes,then g = o(1). We will show that neglecting small terms in the large degree limit theDC model becomes

θ = −βθζR = γ I, S = ψ(θ), I = 1 − S − R

where ζ = 1 − θψ ′(θ)/ψ ′(1)+ (γ /β) ln θ .We define ζ = πI + ξI This is the proportion of all stubs which belong to infected

nodes. We prove a series of results:

1. θ = 1 + o(1).2. φD = π + o(1).3. η1πR = η2ξR + o(1).4. φS = ξ

ψ ′(θ)ψ ′(1) + o(1).

5. πI = πζ + o(1) ξI = ξζ + o(1).6. φI = ξθζ + o(1)7. θ = −βθζ + βo(1).8. ζ = 1 − θ

ψ ′(θ)ψ ′(1) + (γ /β) ln θ + to(1).

If we drop the error terms in the final two results, then we have the equations governingthe mean field social heterogeneity model in the actual degree case, with β = βξ

playing the role of the transmission rate. The existence of the error terms means thatat earlier time, the approximation is better, and it can deviate more as time increases.The solution converges uniformly for any t less than any arbitrary chosen value T as〈K 〉 grows. So for large enough 〈K 〉, the region over which the MFSH model givesan accurate approximation will include the entire period of the epidemic.

1. θ = 1 + o(1).To estimate the probability a stub receives at least one transmission, we first esti-mate the expected number of transmissions a stub receives, which gives a boundon the probability of at least one transmission. Each time a stub receives infec-tion, this corresponds to a transmission from the neighboring node through oneof its stubs. So the expected number of inward transmissions equals the expectednumber of outward transmissions. This is an upper bound on the probability a stubreceives an inward transmission, 1 − θ . After some straightforward calculations,we can show that

β

β + η2 + γ

1

1 − η1η2(η2+γ )(η1+γ )

is a bound on 1 − θ . So long as γ = 0, this is o(1) since β = o(1). Thusθ = 1 + o(1).

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If γ = 0, this argument can be salvaged by a cruder bound. At time t after theinitial time, the expected number of transmissions is bounded by βt = to(1). Sochoosing T as described above, we will still be able to prove uniform convergencefor t ≤ T since θ = 1 + T o(1) = 1 + o(1) if T is held fixed. If T is chosen to belarge enough initially, we have uniform convergence during the dynamic phase ofthe epidemic.

2. φD = π + o(1).We show φD − π = o(1):

d

dt(φD − π) = η2(θ − φD)− η1φD

= η2[1 + o(1)] − (η1 + η2)φD

= (η1 + η2)π − (η1 + η2)φD + o(1)

= −(η1 + η2)(φD − π)+ o(1)

In the next to last step we used the fact that π = η2/(η1 +η2) so η2 = (η1 +η2)π .So φD − π is at most o(1)/(η1 + η2) = o(1).

3. η1πR = η2ξR + o(1).We set xS = η1πS − η2ξS, xI = η1πI − η2ξI , and xR = η1πR − η2ξR . Using theprevious results we can show that xS = o(1). Since the sum xS + xI + xR equalsη1π − η2ξ which is zero, we conclude xI = −xR + o(1). Some algebra showsxR = −(η1 + η2)xR + γ xI = −(η1 + η2 + γ )xR + o(1), from which the resultfollows quickly.

4. φS = ξψ ′(θ)ψ ′(1) + o(1).

We have

d

dt

[φS − ξ

ψ ′(θ)ψ ′(1)

]= φS − ξ θ

ψ ′′(θ)ψ ′(1)

= −η2φS + η1πS

πφD − βφI

ψ ′′(θ)ψ ′(θ)

(φS − ξ

ψ ′(−θ)ψ ′(1)

)

=−η2φS + η1πψ ′(θ)ψ ′(1)

−βφIψ ′′(θ)ψ ′(θ)

(φS −ξ ψ

′(θ)ψ ′(1)

)+ o(1)

where we used the fact that πS = φDψ′(θ)/ψ ′(1) and φD = π + o(1) and that

ψ ′(θ)/ψ ′(1) ≤ 1. Since η1π = η2ξ , we have

d

dt

[φS −ξ ψ

′(θ)ψ ′(1)

]= −η2

(φS −ξ ψ

′(θ)ψ ′(1)

)−βφI

ψ ′′(θ)ψ ′(θ)

(φS −ξ ψ

′(θ)ψ ′(1)

)+ o(1)

= −(η2 + βφI

ψ ′′(θ)ψ ′(θ)

) (φS − ξ

ψ ′(θ)ψ ′(1)

)+ o(1)

Since ψ ′′(θ)/ψ ′(θ) is nonnegative, the coefficient of φS − ξψ ′(θ)/ψ ′(1) in thefinal expression is at least η2, so we conclude that φS = ξψ ′(θ)/ψ ′(1)+ o(1).

5. πI = πζ + o(1) ξI = ξζ + o(1).

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We first define q = (−φD + πθ)ψ ′(θ)/ψ ′(1). Since θ = 1 + o(1) and φD =π + o(1), we have −φD + πθ = o(1). Since 0 ≤ ψ ′(θ) ≤ ψ ′(1), we haveq = o(1). We will show that πI − πζ − q = o(1) and so πI = πζ + o(1).

d

dt(πI − πζ − q) = −πS − πR − πζ − q

= − d

dt

(φDψ ′(θ)ψ ′(1)

)− γπI − η2ξR + η1πR

−π(

− d

dt

(θψ ′(θ)ψ ′(1)

)− γ ζ

)

− d

dt

((−φD + πθ)

ψ ′(θ)ψ ′(1)

)

= −γ (πI + πζ − q)+ o(1)

So πI − πζ − q = o(1), and since q = o(1), πI = πζ + o(1). A similar proofshows that ξI = ξζ + o(1).

6. φI = ξθζ + o(1)We will actually show that φI + φS = ξθζ + ξθ2ψ ′(θ)/ψ ′(1)+ o(1).Once this result is shown, we use the fact that φS = ξψ ′(θ)/ψ ′(1) + o(1) =ξθ2ψ ′(θ)/ψ ′(1) + o(1) to get our final result. We begin by taking the derivativeof φI + φS − ξ [θζ + θ2ψ ′(θ)/ψ ′(1)]

d

dt

(φI + φS − ξ

[θζ + θ2ψ

′(θ)ψ ′(1)

])

= φI + φS − ξ

[θ ζ + θ ζ + θ θ2ψ

′′(θ)ψ ′(1)

+ 2θ θψ ′(θ)ψ ′(1)

]

After some manipulations (see SI) we have

d

dt

(φI + φS − ξ

[θζ + θ2ψ

′(θ)ψ ′(1)

])

= −(η2 + γ )

(φI + φS − ξ

[θζ + θ2ψ

′(θ)ψ ′(1)

])+ o(1)

So φI + φS − ξ [θζ + θ2ψ(θ)/ψ ′(1)] = o(1). Because θ = 1 + o(1) and φS =ξψ ′(θ)/ψ ′(1)+ o(1) it follows then that φI = ξθζ + o(1).

7. θ = −βθζ + βo(1).This step is trivial. Since φI = ξθζ + o(1), we have

θ = −βφI

= −βθζ + βo(1)

8. ζ = 1 − θψ ′(θ)ψ ′(1) + (γ /β) ln θ + to(1).

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We make the observation that ζ = − ddt [θψ ′(θ)/ψ ′(1)] − γ ζ . We have

ζ− d

dt

[1−θ ψ

′(θ)ψ ′(1)

+(γ /β) ln θ

]=− d

dt

[θψ ′(θ)ψ ′(1)

]−γ ζ+ d

dt

[θψ ′(θ)ψ ′(1)

]− γ θβθ

= −γ ζ − γ θ

βθ

From our previous result we have ζ = [−θ + βo(1)]/βθ . Substituting this in wehave

ζ − d

dt

[1 − θ

ψ ′(θ)ψ ′(1)

+ (γ /β) ln θ

]= γ θ

βθ− γβo(1)

βθ− γ θ

βθ= o(1)

Our result follows immediately.

So the DC model reduces to the actual degree formulation of the MFSH model. Becausethe expected degree formulation contains the actual degree formulation as a specialcase, it suffices now to prove that the expected degree formulation converges to theMA model in the appropriate limit.

4.3 MFSH to MA

Our proof that the MFSH model converges to the MA model does not require that〈K 〉 → ∞, but it does require

⟨K 4

⟩/ 〈K 〉4 → 1. However, in order to conclude that

the other models approach the MA model, we need the conditions of the previoussection to hold as well.

We first make the observation that if⟨K 4

⟩/ 〈K 〉4 → 1, then

⟨K 2

⟩/ 〈K 〉2 → 1

as well. By Jensen’s inequality, we have 〈K 〉2 ≤ ⟨K 2

⟩and

⟨K 2

⟩ ≤√⟨

K 4⟩. So 1 =

〈K 〉2 / 〈K 〉2 ≤ ⟨K 2

⟩/ 〈K 〉2 ≤

√⟨K 4

⟩/ 〈K 〉2 =

√⟨K 4

⟩/ 〈K 〉4 → 1. So

⟨K 2

⟩/ 〈K 〉2 is

bounded from below by 1 and from above by something converging to 1.We first prove two lemmas. We define s1 and s2 by

s1 = 1

〈K 〉Ψ ′′(Θ)Ψ ′(1)

− S

s2 = 1

〈K 〉Ψ′(Θ)− S

We show s1 and s2 go to zero. We have

limn→∞ |s1| = lim

n→∞

∣∣∣∣Ψ ′′(Θ)〈K 〉2 − S

∣∣∣∣

= limn→∞

∣∣∣∣∣

∫ ∞0 (e−κ(1−Θ) − S)(κ2 − ⟨

K 2⟩)ρ(κ) dκ

〈K 〉2

∣∣∣∣∣

(see SI for details).

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We now use the Cauchy–Schwarz inequality to bound this. We have

∣∣∣∣∣

∫ ∞0 (e−κ(1−Θ) − S)(κ2 − ⟨

K 2⟩)ρ(κ) dκ

〈K 〉2

∣∣∣∣∣

≤ | ∫ ∞0 (e−κ(1−Θ) − S)2ρ(κ) dκ|1/2| ∫ ∞

0 (k2 − ⟨K 2

⟩)2ρ(κ) dκ|1/2

〈K 〉2

≤ | ∫ ∞0 (κ2 − ⟨

K 2⟩)2ρ(κ) dκ|1/2

〈K 〉2

where we use the fact that | exp[−κ(1 − Θ)] − S| ≤ 1 to show that the first termin the numerator of the first equation is at most 1. All that remains is to expand thenumerator and bound it.

| ∫ ∞0 (κ2 − ⟨

K 2⟩)2ρ(κ) dκ|1/2

〈K 〉2 = | ∫ ∞0 (κ4 − 2κ2

⟨K 2

⟩ + ⟨K 2

⟩2)ρ(κ) dκ|1/2

〈K 〉2

=∣∣∣∣∣

⟨K 4

⟩ − ⟨K 2

⟩2

〈K 〉4

∣∣∣∣∣

1/2

But our assumption is that as n → ∞, both⟨K 4

⟩/ 〈K 〉4 and

⟨K 2

⟩/ 〈K 〉2 go to 1. Thus

this goes to zero as n → ∞. This completes the proof that s1 → 0.To show that s2 → 0 is similar.

|s2| =∣∣∣∣

1

〈K 〉Ψ′(Θ)− S

∣∣∣∣

=∣∣∣∣∣

∫ ∞0 (e−κ(1−Θ) − S)(κ − 〈K 〉)ρ(κ) dκ

〈K 〉

∣∣∣∣∣

≤ | ∫ ∞0 (e−κ(1−Θ) − S)ρ(κ) dκ|1/2| ∫ ∞

0 (κ − 〈K 〉)ρ(κ) dκ|1/2〈K 〉

≤ | ⟨K 2⟩ − 〈K 〉2 |1/2

〈K 〉and as n → ∞, this tends to 0 as well.

We have our bounds on s1 and s2, so we are now ready to complete the proof that themean field social heterogeneity model converges to the mass action model as n → ∞.

We define �I = −Θ/β = 1 − Ψ ′(Θ)/Ψ ′(1)− γ (1 −Θ)/β. This represents theprobability that a new neighbor is infected. We expect �I to be very close to I andset y = �I − I . Then

y = �I − I

= −Θ Ψ′′(Θ)Ψ ′(1)

+ γ Θ/β − [−S − R]

123


= −Θ Ψ′′(Θ)Ψ ′(1)

+ γ Θ/β − [−ΘΨ ′(Θ)− γ I ]

= β�IΨ ′′(Θ)Ψ ′(1)

− γ�I − β�IΨ′(Θ)+ γ I

= β(I + y)Ψ ′′(Θ)Ψ ′(1)

− γ y − β(I + y)Ψ ′(Θ)

= β 〈K 〉 (I + y)[(S + s1)− (S + s2)] − γ y

= β(I + y)(s1 − s2)− γ y

We know that I + y is at most 1, and s1 and s2 both tend to zero as n increases, so wehave y approaches −γ y as n → ∞. Note also that at early time �I and I are bothvery close to 0 so y begins as a very small number. It is straightforward to concludethat y can never grow larger than the maximum value of s1 which tends to zero asn → ∞. Consequently, y → 0 as n → ∞.

We finally have

S = −β�IΨ′(Θ)

= −β�I (S + s2)

= −β I S − β yS − β ys2

Since y and s2 both tend to 0 as n → ∞, we have S = −β I S. This means that wehave the mass action model in the limit.

5 Discussion

We have shown a number of relations between the various edge-based compartmentalmodels for infectious disease spread originally derived in Miller et al. (2012). We usedthis to develop a flow chart (Fig. 1) which can guide the choice of appropriate modelfor a given population.

We note that while the edge-based compartmental models allow us to capture effectsthat were previously inaccessible through analytic techniques, there are still manyeffects that are not captured by the models considered here. Our flow chart does notaddress these. It is always prudent to consider the disease and population to ensurethat the assumptions of our models are not strongly violated. A number of adaptationsof the mass action SIR models exist for populations in which the disease has multiplestages, or the population has important substructures. In an upcoming paper Miller andVolz (2011) we show that the general edge-based compartmental modeling approachcan be used to accommodate many of these effects.

There are two interesting open (and related) questions which we call attention to.Firstly, we showed that if 〈K 〉 → ∞ and

⟨K 4

⟩/〈K 〉4 → 1 with β 〈K 〉 fixed, the models

converge to the mass action model. However, there are many cases where⟨K 4

⟩/〈K 〉4

approaches some other value, and our calculations appear to converge and to somelimit. It would be interesting to identify what the relevant reduced equations are in thislimit. Secondly, it is perhaps surprising that the condition for convergence to the mass

123


action model depends on⟨K 4

⟩/ 〈K 〉4 → 1 rather than

⟨K 2

⟩/ 〈K 〉2 → 1. It can be

shown analytically that if⟨K 2

⟩/ 〈K 〉2 → 1 but

⟨K 4

⟩/ 〈K 〉4 → 1 then the early expo-

nential growth rate is higher than the mass action model. Our numerical calculationssuggest that there is an early phase that deviates from mass action, but after that phasethe solutions are indistinguishable. The early phase becomes shorter as 〈K 〉 grows.So we believe that there is pointwise convergence although not uniform convergence.The underlying explanation for this is that in this limit there is a vanishingly smallproportion of nodes with very high degree. They are quickly infected in an epidemicand then removed from the active population, leaving behind a population that followsthe mass action dynamics. We have not investigated either of these questions in anydetail.

Acknowledgments JCM was supported by (1) the RAPIDD program of the Science and TechnologyDirectorate, Department of Homeland Security and the Fogarty International Center, National Institutes ofHealth and (2) the Center for Communicable Disease Dynamics, Department of Epidemiology, HarvardSchool of Public Health under Award Number U54GM088558 from the National Institute Of GeneralMedical Sciences. EMV was supported by NIH K01 AI091440. The content is solely the responsibilityof the authors and does not necessarily represent the official views of the National Institute Of GeneralMedical Sciences or the National Institutes of Health.

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