SIR epidemics and vaccination on random graphs with clustering
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Abstract
In this paper we consider Susceptible \(\rightarrow \) Infectious \(\rightarrow \) Recovered (SIR) epidemics on random graphs with clustering. To incorporate group structure of the underlying social network, we use a generalized version of the configuration model in which each node is a member of a specified number of triangles. SIR epidemics on this type of graph have earlier been investigated under the assumption of homogeneous infectivity and also under the assumption of Poisson transmission and recovery rates. We extend known results from literature by relaxing the assumption of homogeneous infectivity both in individual infectivity and between different kinds of neighbours. An important special case of the epidemic model analysed in this paper is epidemics in continuous time with arbitrary infectious period distribution. We use branching process approximations of the spread of the disease to provide expressions for the basic reproduction number \(R_0\), the probability of a major outbreak and the expected final size. In addition, the impact of random vaccination with a perfect vaccine on the final outcome of the epidemic is investigated. We find that, for this particular model, \(R_0\) equals the perfect vaccineassociated reproduction number. Generalizations to groups larger than three are discussed briefly.
Keywords
SIR epidemics Configuration model Clustering Branching processes Vaccination1 Introduction
One of the most important factors that determine the fate of an outbreak of an infectious disease is the contact pattern of individuals in the population. The frequency and duration of the contacts between individuals typically depend on the nature of their relationship. For this reason, recent interest has focused on the impact of the underlying social network on the spread of the disease. The social network is typically represented by a random graph (Newman et al. 2002), in which the nodes or vertices represent individuals and the edges represent social contacts between the individuals. Two nodes that share an edge are called “neighbours”.
A popular choice when generating random graphs with a specified degree distribution is the configuration model (CM). It was introduced by Bollobás (1980) for the special case where the degree distribution is degenerate (i.e. every node of the graph has the same degree) and extended to more general degree distributions by Molloy and Reed (1995, 1998). There is a vast literature on epidemics on configuration model graphs [see e.g. Andersson (1999), Britton et al. (2007), Janson et al. (2014), Barbour and Reinert (2013), Bhamidi et al. (2014)].
An important feature of the configuration model is that, under mild regularity conditions on the degrees, this type of graph is asymptotically unclustered. That is to say, it contains virtually no groups and short circuits. Real world networks do, however, typically exhibit clustering (Newman 2003), and there are a number of graph models that do allow for group structure (Bollobás et al. 2011; Karoński et al. 1999; Newman 2002). Epidemics on graphs with group structure were studied by Trapman (2007); Ball et al. (2009, 2010, 2014); Coupechoux and Lelarge (2015); Britton et al. (2008).
In this paper, we use a generalized version of the configuration model to incorporate clustering of the social network in the analysis of the spread of an infectious disease. The configuration model with clustering (CMC) was independently introduced by Miller (2009) and Newman (2009). It is an extension of the CM in the sense that, for each node u, in addition to the degree of u one also specifies the number of pairs of neighbours of u that are in turn neighbours of each others. In other words, one specifies the number of triangles (with nonoverlapping edges) of which u is a member [see Sect. 2.1 for a precise definition of the graph model]. This allows for graphs with nonnegligible clustering and a specified degree distribution. That is to say, the CMC deviates from the classical Erdős–Rényi graph model (Erdős and Rényi 1959) in two fundamental ways: it allows for for a nonPoissonian degree distributions and is asymptotically clustered. Epidemics on this type of graph have previously been studied by Miller (2009) and Volz (2011). Miller (2009) investigated the impact of clustering on the epidemic threshold, formulated as a bond percolation problem. This means that the infectivity of infected individuals is assumed to be homogeneous; an infected individual transmits the disease to each of its neighbours independently with some fixed probability T. Volz (2011) investigated the time evolution and final size of epidemics on CMC graphs under the assumption of exponentially distributed infectious periods during which individuals contact neighbours at a constant rate.
The main contribution of our research is that we extend the results of Miller (2009) and Volz (2011) by allowing for heterogeneous infectivity, i.e. by allowing for some infected individuals to be more contagious than others or for individuals to exhibit different contact behaviors for different types of neighbours. Such heterogeneity may, for instance, reflect variability in the infectious period or contact preferences on the part of individuals. We provide expressions for the probability of a major outbreak and the final size of a major outbreak. A key tool in our analysis is the approximation of the epidemic seen from a “generation of infection” or “rank” perspective by a multitype Galton Watson branching process. This approximation, which is interesting in its own right, gives rise to the rankbased reproduction number \(R_0\) [see e.g. Pellis et al. (2008, 2012)]. We note that especially allowing for heterogeneity in the infectivity of individuals requires a more intricate branching process approximation than a model with homogeneous infectivity [as analysed by Miller (2009)]. To see this, consider an individual v which is infected through a triangle \(\varDelta \). The “local epidemics” in \(\varDelta \) and in other triangles v is part of all depend on the infectivity of v and are therefore in general not independent.
The second contribution of this paper concerns vaccination. We investigate the impact of uniform vaccination (i.e. vaccinated individuals are selected uniformly at random) with a perfect vaccine (i.e. a vaccine that provides full and permanent immunity to the disease). We find that it is necessary to vaccinate a fraction \(11/R_0\) of the population in order to prevent a major outbreak of the disease, as in the case of homogeneous mixing. We illustrate our findings with numerical examples.
This paper is structured as follows. In Sect. 2 we provide the preliminaries for the model. In Sect. 2.1 we give a more detailed description of how graphs are generated in the CMC and investigate the asymptotic clustering of such graphs and in Sect. 2.2 the epidemic model is specified. Sections 2.3 and 2.4 contains an overview of the concept of reproduction numbers and the necessary branching process background. In Sect. 3, we derive expressions for the probability of a major outbreak and the expected final size under the assumption of an unvaccinated and fully susceptible population, and in Sect. 4 the analysis is repeated under the assumption of uniform vaccination with a perfect vaccine. We illustrate our findings with numerical examples presented in Sect. 5 and discuss possible extensions in Sect. 6.
2 Preliminaries
2.1 The configuration model with clustering
For this reason, selfloops and multiple edges are negligible in the limit as \(N\rightarrow \infty \). In the remainder of this paper, we ignore the small differences in the topology of the graph that arise from erasing multiple edges or selfloops. In addition, we ignore the small differences in effective degree distribution that arise from erasing halfedges so that the number of single and triangle halfedges are multiples of two and three, respectively.
 (A1)
\(E(\varDelta ^2)<\infty \text { and } E(S^2)<\infty . \)
 (A2)
\(P(\max (\varDelta ,S)\ge 2)>0\) and \(E( \varDelta S)>0\).
2.1.1 Clustering coefficient of \(G_N\)
As stated in the following proposition, CMC graphs have asymptotically nonzero clustering as \(N\rightarrow \infty \). An analogous result for fixed degree sequences was presented in Newman (2009). Let \(\overset{P}{\longrightarrow }\) denote convergence in probability.
Proposition 1
The proof is presented in the Appendix.
2.1.2 Downshifted sizebiased degrees
2.2 The epidemic model
We use an SIR model to investigate the dynamics of the spread of the disease. At any given time point, the population is divided into three groups, depending on health status. The groups are susceptible (S), infectious (I) and recovered (R) [see e.g. Britton (2010), Diekmann et al. (2013)]. Individuals of the population make contact with other individuals at (possibly random) points in time. If, at some time point, an infectious individual contacts a susceptible individual then the susceptible individual instantaneously becomes infectious. An infectious individual will cease to be contagious after a period of time, which we call the infectious period of the individual in question, and is then transferred to the recovered group. Recovered individuals are those that are immune to the disease. Individuals belonging to this group play no further role in the spread of the disease. Because of this last observation, we can treat individuals that die because of the disease as “recovered”. In summary, we allow only the transitions \(S\rightarrow I\) and \(I\rightarrow R\). Note that the population is assumed to be closed; we ignore births, deaths and migration.
More specifically, we consider an SIR epidemic in a generation framework on the clustered static graph \(G_N\) and assume possible heterogeneity in infectivity, both between different individuals (individual heterogeneity) and between different kinds of edges (edge heterogeneity). Individual heterogeneity means that some infected individuals are more contagious than others. Such heterogeneity may, for instance, arise from variability in the infectious period. Edge heterogeneity reflects that individuals may exhibit different contact behaviors for different types of neighbours ; an individual may for instance prefer to spend more time with its triangle neighbours at the expense of spending less time with its single neighbours.
To construct a model that captures such heterogeneities, let \(T=(T_{s}, T_{\varDelta })\) be a random variable with support in \( [0,1]^2\), and let \(\{T_i\}_{i=1}^N=\{(T_{s}^{(i)}, T_{\varDelta }^{(i)})\}_i\) be a sequence of independent copies of T. We allow for any dependence structure between \(T_{s}\) and \(T_{\varDelta }\). Each node \(v_i\) of \(G_N\) is equipped with a twodimensional transmission weight\(T_i\). If \(v_i\) gets infected, then each susceptible single neighbour (neighbour by virtue of a single edge) of \(v_i\) gets infected by \(v_i\) independently in the next generation with probability \(T_{s}^{(i)}\), and each susceptible triangle neighbour (neighbour by virtue of a triangle edge) of \(v_i\) gets infected by \(v_i\) independently in the next generation with probability \(T_{\varDelta }^{(i)}\) (conditioned on \(\{T_i\}_i\)). Node \(v_i\) thereafter becomes recovered, playing no further role in the epidemic. An infected node transmits the disease independently of the transmissions from other infected nodes. An infected node does not, however, transmit the disease to its neighbours independently, unless the distribution of T is degenerate. Conditioned on the transmission weights \(\{T_i\}_i\) and the structure of \(G_N\), the number of single and triangle neighbours that an infected node \(v_i\) makes (infectious) contact with while infectious has a binomial distribution with parameters \((S_i, T_{s}^{(i)})\) and \((\varDelta _i, T_{\varDelta }^{(i)})\), respectively.
The spread of this epidemic can be fully captured by a directed graph [see e.g. Pellis et al. (2012), Kenah and Miller (2011)]. To construct such directed graph from an undirected CMC graph \(G_N\), we replace each undirected edge of \(G_N\) by two parallel directed edges, pointing in the opposite direction. The weight of an edge \((v_i,v_j)\), which represents the (potential) transmission time from \(v_i\) to \(v_j\), is taken to be 1 if \(v_i\) would make infectious contact with \(v_j\) if infected, and \(\infty \) otherwise. The individuals ultimately infected are then the individuals that can be reached from an initial case by following a path consisting of directed edges with finite edge weights.
2.3 Reproduction numbers
A key quantity in the study of epidemics is the basic reproduction number, often denoted by \(R_0\). It is usually defined as the expected number of infected cases caused by a “typical” infected individual in an otherwise susceptible population (Diekmann et al. 1990). For most stochastic epidemic models [including SIR epidemics in homogeneous mixing populations (Britton 2010), populations with households (Ball et al. 2016) and epidemics on networks (Britton et al. 2007)] it has the threshold property that a major outbreak is possible if and only if \(R_0>1\).
For models where a suitable generation based branching process approximation is available, \(R_0\) is usually defined as the Perron root (the dominant eigenvalue, which exists and is realvalued by assumptions A1 and A2, see for instance Varga (2009, Chapter 2) of the mean matrix of the approximating Galton Watson branching process. This is the definition used in this article. By standard branching process theory, the interpretation of \(R_0\) as the expected number of cases caused by the typical individual in the early phase of the epidemic and its threshold properties are retained by this definition. The threshold property of \(R_0\) is made precise in Theorem 1 below.
For many models, including epidemics on graphs generated by the CM (Britton et al. 2007) and the standard stochastic SIR epidemic model [i.e. individuals mix homogeneously, see for instance Britton (2010)], \(R_V=R_0\). That is, vaccinating a fraction \(11/R_0\) of the population with a perfect vaccine is sufficient to surely prevent a major outbreak. On the other hand, for the households and householdsworkplaces model with uniform vaccination, \(R_V\ge R_0\) (Ball et al. 2016) with strict inequality possible. In Sect. 4.1 we show that for the model analysed in this report, \(R_V=R_0.\)
2.3.1 Epidemics in continuous time: the rankbased approach
As mentioned above, heterogeneity in infectivity might arise from heterogeneity in the infectious period; an important special case of the above described model is epidemics in continuous time with random infectious periods where contacts between individuals take place according to point processes on \(\mathbb {R}_{\ge 0}\). Ignoring the real timedynamics of an epidemic does not impact results that concern the final outcome of the epidemic. This result was first presented by Ludwig (1975), see also Pellis et al. (2008) or Kiss et al. (2017, section 6.2.3) for a more recent discussion. This leads us to the often more tractable rankbased approach.
In order to define the rank of a vertex, denote the initial case by \(v_*\). The rank of a node v in \(G_N\) is the distance from \(v_ *\) to v, if every edge along which the disease would be transmitted is assigned the edge weight 1, and every other edge is assigned the edge weight \(\infty \). That is, the rank of v is the smallest number of directed edges that have to be traversed in order to follow a path of (potential) transmission from \(v_*\) to v. We may then analyse the spread of the disease by letting generation n of the epidemic process consist of the individuals of rank n. If, for instance, \(v_1\) is the first node in a triangle consisting of the nodes \(v_1,v_2,v_3\) to be infected, and \(v_1\) infects \(v_2\) and thereafter attempts to infect \(v_3\), then \(v_3\) is attributed to \(v_1\) regardless of whether \(v_1\) or \(v_2\) infected \(v_3\). This is illustrated in Fig. 2.
2.4 Branching process approximations
To analyse the spread of the disease in the early stages of the epidemic, we employ a multitype branching process approximation. The graph \(G_N\) may be constructed by joining the halfedges in any (possibly random) order, provided that the uniform matching is not violated. In particular, the graph \(G_N\) may be constructed (or explored) as the epidemic progresses; starting with the initial infected case \(u^{*}\) we sequentially match the halfedges along which the disease is transmitted. In the early phase of the epidemic, short cycles (except for the triangles formed by triangle edges) are unlikely to appear. For these reasons, the early spread of the disease is well approximated by a suitably chosen branching process.
Similarly, a branching process approximation can be used to approximate the expected final size of the epidemic (Ball et al. 2009, 2010, 2014). In the graph representation of an epidemic, an individual contracts the disease if and only if there is a path of directed edges with finite edge weights from the initial case to the node representing the individual in question.
By reversing the direction of the edges of the graph representation of an epidemic, but keeping the weights, the expected final fraction of the population infected in a major outbreak and the probability of a major outbreak are interchanged (Miller 2008), provided that the initial case is chosen uniformly at random. The process so obtained is called the backward epidemic process of the node v. If the underlying epidemic model is such that the backward epidemic process can be well approximated by a branching process, then we can use this branching process to compute the asymptotic distribution of the proportion of the population that ultimately escapes infection. This is made precise in the following theorem, due to Ball et al. (2014, Theorem 3.5), who proved the theorem for the related model of random intersection graphs. The statement of Theorem 1 carries over to the forward and backward branching processes considered in this paper. We omit the proof, which is analogous to the proof presented in Ball et al. (2014), see also Ball et al. (2009).
Theorem 1
In other words, in the limit of large population sizes, the epidemic “takes off” with probability \( 1q\), and if this happens a fraction \( 1q_b\) of the population is ultimately infected (with probability converging to 1 as \(N \rightarrow \infty \)). Note that since \(R_0\) is defined as the Perron root of the mean matrix of the forward branching process, \(q<1\) if and only if \(R_0>1\).
3 An epidemic in a fully susceptible population
We now have the tools to analyse the spread of an infectious disease on a graph generated by the CMC. In the present section, the population is assumed to be fully susceptible to the disease, apart from the initially infectious individual.
3.1 Forward process
 Type 1:
A node infected along a triangle edge whose twin (in the same triangle) is infected at the same time step or earlier
 Type 2:
A node infected along a triangle edge that is not of type 1
 Type 3:
A node infected along a single edge
If \(M_f\) is positively regular (see the last paragraph before Sect. 2.1.1) then \(R_0\) is given by the Perron root of \(M_f\). With little effort, one can use the expected values provided in (6) to show that necessary and sufficient conditions for \(M_f\) to be positively regular are that assumptions A1A2 hold and that \(0<E(T_{s})<1\) and \(0<E(T_{\varDelta })<1\). If some of these conditions are not satisfied, we may analyse the spread of the disease by reducing the number of types of the approximating forward branching process. It is worth pointing out that \(R_0\) only depends on the marginal distributions of \(T_{s}\) and \(T_{\varDelta }\) (via their moments), not on the dependence structure between them.
3.1.1 Probability of a major outbreak
3.2 Backward process
Let w be a given node of \(G_N\), chosen uniformly at random. We use a backward branching process to approximate the probability that w contracts the disease, which by an exchangeability argument equals the expected final size of a major outbreak. The offspring of an individual v in the backward process are the individuals that would potentially have infected v, if they were infected themselves.
 Type 1:
The vertex is included in the susceptibility set by virtue of potential transmission along a single edge
 Type 2:
The vertex is included in the susceptibility set by virtue of potential transmission along a triangle edge
 (\(E_{1}\))

\(v_2\) and \(v_3\) both “infect” \(v_1\)
 (\(E_{2}\))

\(v_2\) infects \(v_1\) and \(v_3\) “infects” \(v_2\)
 (\(E_{3}\))

\(v_3\) infects \(v_1\) and \(v_2\) “infects” \(v_3\)
Standard calculations give that the probability of the union of the events \(E_{1}\)\(E_{3}\) is given by \(p_2=3E(T_{\varDelta })^22E(T_{\varDelta })E(T_{\varDelta }^2)\). Similarly, the probability that neither \(v_1\) nor \(v_2\) will be members of the susceptibility set of v by transmissions within the triangle is given by \(p_0=(1E(T_{\varDelta }))^2\). For later use, denote \(1p_0p_2\) by \(p_1\).
3.2.1 Expected final size of a major outbreak
Let b be the probability generating function of the offspring distribution of the two types of the approximating backward branching process. Furthermore, let \(b_*\) be the probability generating function of the offspring distribution of the ancestor w. Analogously to the forward branching process, the probability that a branching population whose ancestor is of type \(i,\ i=1,2,\) will go extinct is given by \(q^b_i\), where \(\bar{q}_b=(q_1^b,q_2^b)^{\mathsf {T}}\) is the unique solution of \(\bar{q}_b=b(\bar{q}_b)\) in \([0,1)^2\) (recall \(R_0>1\)). The probability of extinction is given by \(b_*(\bar{q}_b)\).
4 Vaccination
4.1 Random vaccination with a perfect vaccine
Assume that a fraction \(f_{\text{ v }}<1\) of the population is vaccinated, and that the vaccinated individuals are chosen uniformly at random (without replacement) from the population. The vaccine is perfect, in the sense that a vaccinated individual gains full and lasting immunity to the disease. If the population size N is large, we may use a slightly different model, where each individual is vaccinated with probability \(f_{\text{ v }}\), independently of the vaccination status of other individuals. By the law of large numbers, for our purposes the models are equivalent in the limit as the population size \(N\rightarrow \infty . \)
 Type 1:
Infected along a triangle edge and has a twin that is known not to be susceptible
 Type 2:
Infected along a triangle edge and has a twin that might be susceptible
 Type 3:
Infected along a single edge
We conclude that, for this particular graph model, equality holds between the basic reproduction number \(R_0\) and the perfect vaccineassociated reproduction number \(R_V\) as defined in (7). Note that \(R_0\) is based on a rankbased perspective of infection and not on “whoinfectedwhom.
4.1.1 Probability of a major outbreak
4.1.2 The backward process
 Type 1:
Transmits along triangle edge, no information on vaccination status is available
 Type 2:
Transmits along triangle edge and is known not to be vaccinated since it is successfully infected by its twin
 Type 3:
Transmits along single edge, no information on vaccination status is available
 (\(E_{1}\))
 \(v_2\) attempts to infect \(v_1\) and \(v_3\) attempts to infect \(v_2\), both succeed, and \(v_3\) does not attempt to infect \(v_1\). Or the same thing might happen, with \(v_2\) and \(v_3\) interchanged. This results in one type 1 and one type 2 individual in the approximating branching process. If \(v_1\) is represented by a type 1 or 3 individual this happens with probabilityif \(v_1\) is represented by an individual of type 2 this happens with probability$$\begin{aligned} 2\big (1f_{\text{ v }}\big )^2E(T_{\varDelta })E\big (T_{\varDelta }(1T_{\varDelta })\big ), \end{aligned}$$$$\begin{aligned} 2(1f_{\text{ v }})E(T_{\varDelta })E\big (T_{\varDelta }(1T_{\varDelta })\big ). \end{aligned}$$
 (\(E_{2}\))
 Only one of \(v_2\) and \(v_3\) attempts to infect \(v_1\), and succeeds. The other node does not attempt to infect any node within the triangle. This results in one type 1 offspring. If \(v_1\) is represented by an individual of type 1 or 3 this happens with probabilityif \(v_1\) is represented by an individual of type 2 this happens with probability$$\begin{aligned} 2(1f_{\text{ v }})E(T_{\varDelta })E\big (T_{\varDelta }(1T_{\varDelta })\big ), \end{aligned}$$$$\begin{aligned} 2E(T_{\varDelta })E\big (T_{\varDelta }(1T_{\varDelta })\big ). \end{aligned}$$
 (\(E_{3}\))
 \(v_2\) and \(v_3\) both attempt to infect \(v_1\) and succeeds. This results in two type 1 individuals born in the approximating branching process. If \(v_1\) is represented by an individual of type 1 or 3 this happens with probabilityif \(v_1\) is represented by an individual of type 2 this happens with probability$$\begin{aligned} (1f_{\text{ v }})E(T_{\varDelta }^2), \end{aligned}$$$$\begin{aligned} E(T_{\varDelta }^2). \end{aligned}$$
 (\(E_{4}\))

\(v_2\) attempts to infect \(v_1\) and succeeds. The other node, \(v_3\), attempts to infect \(v_2\), but fails due to \(v_2\) being vaccinated. The individual \(v_3\) does not attempt to infect \(v_1\). In this scenario, \(v_2\) belongs to the susceptibility set of \(v_1\). However, we do not include \(v_2\) is the approximating branching process. This does not have any impact on the result of our analysis, since we are only interested in the probability of extinction of the backward process and \(v_2\) does not produce any offspring in this process.
4.1.3 Expected final size
Let \(b^{(\text{ v })}\) and \(b^{(\text{ v })}_*\) be the probability generating function of the offspring distribution of the three types of the approximating backward branching process and of the ancestor, respectively. Furthermore, let \(\bar{\zeta }_i=({\zeta ^b_{i,1}}, {\zeta ^b_{i,2}}, {\zeta ^b_{i,3}})\) be distributed as the offspring of a type \(i,i=1,2, 3\), individual and denote by \(E_s\) the conditional expectation given that the parent of \(\zeta ^b_{i,1},\zeta ^b_{i,2}, \zeta ^b_{i,3}\) is susceptible. Let further \(\bar{\zeta }_*=({\zeta ^b_{*,1}},{\zeta ^b_{*,2}}, {\zeta ^b_{*,3}})\) be distributed as the offspring of the ancestor. Denote the extinction probability of a process descending from a type i individual by \(q_i^b\), \(i=1,2,3\) and let \(\bar{q}^b=(q_1^b,q_2^b, q_3^b)^{\mathsf {T}}\).
5 Numerical example
Consider for now the special case where \(T_{s}=T_{\varDelta }\). With some abuse of notation we denote \(T_{s}=T_{\varDelta }\) by T, i.e. \(T=T_{s}=T_{\varDelta }\) is onedimensional here. Under very general assumptions, increasing the heterogeneity in infectiousness leads to a decrease in the probability of a major outbreak, the expected final size and \(R_0\) (Kuulasmaa 1982; Meester and Trapman 2011; Miller 2008), see also Ball (1985); Kenah and Robins (2007); Miller (2007). In particular, for a fixed (marginal) transmission probability E(T), the probability of a major outbreak and the expected final size are maximized if \(T=E(T)\) with probability 1 and minimized if \(P(T=1)=E(T)=1P(T=0)\). Similarly, for given E(T), \(R_0\) is maximized if \(T=E(T)\) with probability 1 and minimized if \(P(T=1)=E(T)=1P(T=0)\).
 1.
\(p(2,1)=1\)
 2.
\(p(4,0)=0.95=1p(2,1)\)
 3.
\(p(0,2)=0.95=1p(2,1)\).
As can be seen in Fig. 8, ignoring actual heterogeneity of infectivity in this case leads to an overestimation of the probability of a major outbreak (Fig. 8a, b). This effect is particularly evident in the presence of high clustering; the steeper slope of the curve corresponding to distribution 3 (Fig. 8b) and the relatively low probability of a major outbreak when \(\alpha \) is small can be explained by the fact that the approximating forward branching process is close to being critical when \(\alpha \) is small. Figure 8c, d shows that heterogeneity of infectivity has virtually no impact on the expected final size of a major outbreak and \(R_0\) in the near absence of clustering, which is in line with known results for unclustered networks [see, for instance, section 4 in Miller (2008)]. In the presence of clustering, on the other hand, ignoring heterogeneity of infectivity leads to an underestimation of the expected final size and a substantial overestimation of the critical vaccination coverage \(f_v^{(c)}\). Note that \(R_0\) and \(f_v^{(c)}\) depend on the distribution of T only through the first and second moment of T.
Next, we relax the assumption \(T_{s}=T_{\varDelta }\) and investigate the impact of the correlation \(\rho \) between \(T_{s}\) and \(T_{\varDelta }\) on the spread of the disease. To this end, we consider a model where as before \(T_{s}\) and \(T_{\varDelta }\) both have distribution Beta\((\alpha , \alpha )\) and where the correlation \(\rho =\rho (t)\) may be tuned by varying \(t\in [1,1]\). Here \(\rho (t)\) is increasing in t with \(\rho (1)=1\) and \(\rho (1)=1\). The degree distribution of the underlying graph is given by distribution 1 above.
6 Discussion
In this paper, we have incorporated clustering in the spread of an infectious disease by allowing for groups of size three with nonoverlapping edges. It is, in principle, straightforward to extend the methods used in this paper to larger group sizes. The CMC may, for instance, be generalized to larger group sizes as follows. Let \(K=\{k_1,\ldots , k_r\}\subset \mathbb {N}_{\ge 2}\) be the set of possible group sizes. In the matching procedure, each node is equipped with an rdimensional degree in \(\mathbb {N}_0^r\). The ith component (the \(k_i\)degree) of a degree specifies the number of groups of size \(k_i\) to which the node in question belongs. Analogously to the construction of a CMC graph, groups are then formed by creating one list for each group size; a node with \(k_i\)degree \(d_i\) appears precisely \(d_i\) times in the list corresponding to groups of size \(k_i\). The lists are then shuffled and halfedges of nodes in positions \(k+1,\ldots ,k+k_i\) in the \(k_i\)list are joined. The structure of a graph so obtained would be characterized by fully connected cliques, and similar to that of a random intersection graph (Ball et al. 2014). One possible approach to investigate epidemics on such graphs would be to approximate the spread of the disease by a multitype Galton Watson process where groups (or cliques) are represented by the particles of the branching process. The types of the approximating branching process would then be vectors in \(\mathbb {N}^2\) of the form (m, n), where m represents the size of the clique and n represents the number of members of the clique that the primary case of the clique attempts to infect. Another possible approach would be to use an infinite type branching process in the spirit of Ball et al. (2014). We believe that the result would be analogous to the results obtained in Ball et al. (2014).
Notes
Acknowledgements
This research is supported by the Swedish Research Council (Vetenskapsrådet) Grant 201604566. We wish to thank the members of the journal club on infectious diseases at Stockholm University and Daniel Ahlberg for suggestions that lead to substantial improvements of the paper.
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