\({ SI}\) infection on a dynamic partnership network: characterization of \(R_0\)
Abstract
We model the spread of an \({ SI}\) (Susceptible \(\rightarrow \) Infectious) sexually transmitted infection on a dynamic homosexual network. The network consists of individuals with a dynamically varying number of partners. There is demographic turnover due to individuals entering the population at a constant rate and leaving the population after an exponentially distributed time. Infection is transmitted in partnerships between susceptible and infected individuals. We assume that the state of an individual in this structured population is specified by its disease status and its numbers of susceptible and infected partners. Therefore the state of an individual changes through partnership dynamics and transmission of infection. We assume that an individual has precisely \(n\) ‘sites’ at which a partner can be bound, all of which behave independently from one another as far as forming and dissolving partnerships are concerned. The population level dynamics of partnerships and disease transmission can be described by a set of \((n+1)(n+2)\) differential equations. We characterize the basic reproduction ratio \(R_0\) using the nextgenerationmatrix method. Using the interpretation of \(R_0\) we show that we can reduce the number of statesatinfection \(n\) to only considering three statesatinfection. This means that the stability analysis of the diseasefree steady state of an \((n+1)(n+2)\)dimensional system is reduced to determining the dominant eigenvalue of a \(3\times 3\) matrix. We then show that a further reduction to a \(2\times 2\) matrix is possible where all matrix entries are in explicit form. This implies that an explicit expression for \(R_0\) can be found for every value of \(n\).
Keywords
\({ SI}\)infection Mean field at distance one Dynamic network Concurrency \(R_0\)Mathematics Subject Classification
34D20 92D301 Introduction
The role that concurrent partnerships might play in the spread of HIV in subSaharan Africa is the subject of an ongoing debate. While simulation studies have shown the large impact that concurrency potentially has on the epidemic growth rate and the endemic prevalence of HIV (Kretzschmar and Morris 1996; Morris and Kretzschmar 1997, 2000; Eaton et al. 2011; Goodreau 2011), the empirical evidence for such a relationship is inconclusive (Lurie and Rosenthal 2010; Reniers and Watkins 2010; Tanser et al. 2011; Kenyon and Colebunders 2012).
Mathematical modelling results have played a key role in fuelling the debate (Watts and May 1992; Kretzschmar and Morris 1996; Morris and Kretzschmar 1997, 2000; Eaton et al. 2011; Goodreau 2011). However, a mathematical framework suitable to derive analytical results is still lacking. At present, simulation studies prevail, and general theory is mainly focused on static networks (Diekmann et al. 1998; Ball and Neal 2008; House and Keeling 2011; Lindquist et al. 2011; Miller et al. 2012; Miller and Volz 2013). This motivated us to develop and analyse a mathematical model for the spread of an \({ SI}\) (Susceptible–Infectious) infection along a dynamic network.
In a previous paper (Leung et al. 2012) a model for a dynamic sexual network of a homosexual population is presented that incorporates demographic turnover and allows for individuals to have multiple partners at the same time, with the number of partners varying over time. This network model can be seen as a generalization of the pair formation models (that describe sequentially monogamous populations) to situations where individuals are allowed more than one partner at a time. Pair formation models were first introduced into epidemiology by Dietz and Hadeler (1988) and extended in various ways (Kretzschmar et al. 1994; Inaba 1997; Kretzschmar and Dietz 1998; Xiridou et al. 2003; Heijne et al. 2011; Powers et al. 2011). In the present generalization, individuals have at most \(n\) partners at a time. We call \(n\) the partnership capacity. In the partnership network individuals are, essentially, collections of \(n\) ‘binding sites’ where binding sites can be either ‘free’ or ‘occupied’ (by a partner). In the case that \(n=1\) we recover the pair formation model of a monogamous population.
Consider an individual in the sexual network. Since individuals may have several partners simultaneously, the risk of acquiring infection depends on that individual’s partners, but also on their partners, and so on. We would need to keep track of the entire network to fully characterize the risk of infection to an individual. Here we introduce an approximation rather than taking full network information into account: we assume that properties concerning partners of partners can be obtained by averaging over the population. This approximation is termed the ‘mean field at distance one’ assumption (‘mean field at distance one’ should be read as one term; from here on we write this without quotation marks). This assumption relates to what is called ‘effective degree’ in Lindquist et al. (2011), where transmission of infection along a static network is studied (we are, apart from Britton and Lindholm 2010; Britton et al. 2011), not aware of any analytical work so far, on disease transmission across dynamic networks with demography (see e.g. Altmann 1995, 1998; Ferguson and Garnett 2000; Bansal et al. 2010; Kiss et al. 2012; Miller and Volz 2013) and references therein for models incorporating dynamic partnerships in a demographically closed population).
The mean field at distance one assumption is a moment closure approximation obtained by ignoring certain correlations between the states of two individuals that are in a partnership and, as a consequence, this assumption is inconsistent with the assumptions that underlie the partnership network (see e.g. Ferguson and Garnett 2000; Kamp 2010; House and Keeling 2011; Taylor et al. 2012) and references therein for different moment closure approximations on networks). However, this assumption allows us to write down a closed system of ODEs to describe an approximation of the \({ SI}\) infection on the partnership network. If a partnership capacity \(n\) is given, then we have an \((n+1)(n+2)\) dimensional system of ODEs.
A large part of the paper is devoted to characterizing the basic reproduction number \(R_0\) and proving its threshold character for the nonlinear system of ODEs. This system is quite large already for small \(n\). However, by considering only statesatinfection and using the nextgeneration matrix approach, \(R_0\) can be characterized as the dominant eigenvalue of an \(n\times n\) matrix. Using the interpretation we can further reduce this and \(R_0\) can ultimately be characterized as the dominant eigenvalue of a \(2\times 2\) matrix where the entries of this matrix are explicit, and therefore also \(R_0\) has an explicit expression. In fact, we are able to interpret \(R_0\) in terms of individuals (which are considered in the model specification) and in terms of binding sites.
The structure of the paper is as follows. First, in Sect. 2, we consider the partnership network of Leung et al. (2012) and summarize the main results needed for this paper. Next, in Sect. 3 we superimpose an \({ SI}\)infection on the network and specify the model assumptions. Particular attention is given to the mean field at distance one assumption. The rest of the paper is devoted to characterizing the basic reproduction number \(R_0\). For this, in Sect. 4, we first consider the linearisation of the system.
In Sect. 5, which constitutes the core of the paper, we characterize \(R_0\) in terms of newly infected binding sites that produce newly infected binding sites. We introduce a transition matrix \(\varSigma \) and a transmission matrix \(T\) and define \(R_0\) as the dominant eigenvalue of the next generation matrix \(T\varSigma ^{1}\) (Diekmann et al. 2013, Section 7.2). The building blocks for an explicit expression for \(R_0\) are presented in Appendix C. We also show that \(R_0\) thus defined can be interpreted as the basic reproduction ratio for individuals, since individuals can be considered to be collections of \(n\) binding sites. Section 5 can be read independently of the rest of the paper.

\({{\mathrm{sign}}}(R_0 1) = {{\mathrm{sign}}}(r)\) where \(r\) is the Malthusian parameter (i.e. the dominant eigenvalue) of the matrix \(T+ \varSigma \)

the linearised system derived in Sect. 4 can be mapped in a natural way to the bindingsite system defined by the matrices \(\varSigma \) and \(T\), while preserving positivity.
2 The partnership network
In this section we will give a summary of the specification of the partnership network and of the main results presented in Leung et al. (2012).
Consider a population of homosexual individuals—all with partnership capacity \(n\). The partnership capacity is the maximum number of simultaneous partners an individual may have. One may think of an individual as having \(n\) binding sites. Binding sites are either ‘occupied’ (by a partner) or ‘free’. We assume that binding sites of an individual behave independently from one another as far as forming and dissolving partnerships are concerned. Furthermore, individuals enter (‘birth’) and leave (‘death’) the sexually active population.
Lemma 1
Lemma 2
Note that Lemma 2 does not imply that the states of the two individuals in this partnership are independent of one another. Indeed, they are not. Information about the number of partners of one of the individuals provides some information about the duration of the partnership and thus influences the probability that the other individual has \(k\) partners (or, in other words, there exists degree correlation in this network); see Appendix B for explicit calculations for \(n=2\). (We have, so far, not calculated degree correlations for general \(n\).)
Note that the model specification is deterministic in the sense that it concerns expected values for a population of infinite size. Partnership formation is at random between two free binding sites. As a consequence of mass action and infinite population size, partnership formation with oneself or multiple partnerships with the same individual occur with probability zero. For the same reason clustering does not occur in the network. It should be possible to formulate a stochastic version for a population of size \(N\) and derive the present description by considering the limit \(N\rightarrow \infty \). We conjecture that all the previous statements hold in the limit. In particular clustering disappears in the limit, i.e. the probability that a path of a fixed finite length contains a loop goes to zero in the limit.

\(P=(P_k)\) for a random individual,

\(q=(q_k)\) for an individual who just acquired a partner (but is otherwise randomly chosen),

\(Q=(Q_k)\) for an individual in a randomly chosen partnership.
3 Superimposing transmission of an infectious disease
We consider an \({ SI}\) infection spreading on the dynamic sexual network described in Sect. 2. We assume that individuals become infectious at the very instant that they become infected and stay infectious (with the same infectiousness) for the rest of their life.
3.1 istates and idynamics
The model specification begins at the ilevel (i for individual). We classify individuals as either susceptible (indicated by the symbol \(\)) or infectious (indicated by \(+\)). We assume that the \(\pm \) classification has no influence whatsoever on partnership formation and separation nor on the probability per unit of time of dying.
The state of an individual is now a triple \((x, k_,k_+)\), where \(x\) is either \(+\) or \(\) and \(k_\) and \(k_+\) are nonnegative integers with \(0\le k_ +k_+\le n\). The \(x\) specifies whether the individual itself is susceptible or infectious, \(k_{}\) specifies the number of its susceptible partners, and \(k_+\) specifies the number of its infectious partners.
3.1.1 Demographic change of istates
Consider an individual and suppose it does not die in the period under consideration. There are two types of state transitions: those that contribute to demography and those that involve transmission of infection.
We let \(F_\) denote the fraction of the total pool of binding sites that is free and belongs to a susceptible individual and let \(F_+\) denote the fraction that is free and belongs to an infectious individual so \(F_+F_+=\bar{F}\). We shall say that a binding site is susceptible or infectious if the ‘owner’ is so.
3.1.2 Transmission (mean field at distance one)
It is also possible that a partner \(v\) of \(u\) (with \(u\) either susceptible or infectious) becomes infected by one of \(v\)’s infectious partners (which includes \(u\) if \(u\) is infectious). Of course the probability that this happens depends on the actual configuration in terms of number of partners of \(v\) and their infection status. That information is, however, not incorporated in our description.
3.1.3 ilevel dynamics
3.2 Bookkeeping on the plevel and feedback
We have now specified the ilevel dynamics. In this section we consider the plevel (p for population) and the feedback to the ilevel via the variables \(F_\pm \) and \(\varLambda _\pm \).
3.2.1 Bookkeeping
3.2.2 Feedback
Consider a transition of an individual \(u\), with \(u\) in state \((\pm ,k_,k_+)\rightarrow (\pm ,k_{}1,k_++1)\). This transition occurs when a susceptible partner \(v\) of the focus individual \(u\) in state \((\pm ,k_,k_+)\) gets infected. The rate at which \(v\) gets infected depends on the number of infectious partners \(v\) has. However, we only know that \(v\) is a susceptible partner of \(u\).
Note that, from an individualbased perspective, (16) and (17) are the only formulas consistent with our assumption that \(u\)’s susceptible partners are subject to a force of infection \(\beta \varLambda _\pm \) depending only on \(t\) and \(u\)’s infection status \(\pm \) (and not on the number of susceptible and infectious partners of \(u\) cf. Appendix B). Hence our choice of the term ‘mean field at distance one’ for the latter assumption.
3.3 The plevel differential equations
3.3.1 Consistency relations
4 Linearisation and the map \(L\)
In this section we linearise system (18) around the diseasefree equilibrium. Next we show that we can reduce the dimension of the linearised system and consider only the variables \(P_{(,k_,1)}\) and \(P_{(+,k_,k_+)}\). In Sect. 6 we will use this reduced linearised system to prove that the basic reproduction number \(R_0\), that we characterize in Sect. 5, indeed provides a threshold value of 1 for the disease free steady state of system (18) to become unstable. To this end we define a map \(L\) in Sect. 4.2, which allows us to relate, in the linearisation, populationlevel fractions of individuals (that we consider in the present section) to fractions of binding sites (that we consider in Sect. 5).
4.1 Linearisation
Next, note that we can use relationship (21) in order to replace \(F_\) by \(\bar{F}F_+\) (note that this last expression does not involve any variable of the form \(P_{(,k,0)}\)). Next, we can reduce the dimension of the system by \(n+1\) by eliminating the \(P_{(,k,0)}\), \(k=0,\ldots ,n\), from the system using relation (20).
Remark 1
Intuitively, one would expect that, in the linearisation, for \(k_+\ge 2\), \(P_{(,k_k_+)}(t)=0\) for all \(t\) if \(P_{(,k_k_+)}(0)=0\). Indeed, in the beginning of an epidemic very few individuals in the population are infectious. It is already very unlikely for a susceptible individual to have an infectious partner, so the probability that a susceptible individual has more than one infectious partner should be negligible. That this is indeed the case, is established in the following lemma.
Lemma 3
Proof
 Step 1.

Observe first that the differential equations for \(P_{(,k_,k_+)}\), \(k_+\ge 2\), form a closed system, i.e. they do not depend on the remaining variables (see (25)).
 Step 2.
 Observe that this closed system has a certain hierarchical structure, viz. the subsystem for the variables\(0\le j\le k\), depends on the variables of the subsystems with a lower value of \(k\), but not on the variables of any subsystem with a higher value of \(k\) (the reason is that both \(F_+\) and \(\varLambda _\) were put equal to zero to derive the equations that we consider; recall that we focus on \(nk\ge 2\)).$$\begin{aligned} P_{(,j,nk)}, \end{aligned}$$
 Step 3.
 For \(k=0\) we haveso, if \(P_{(,0,n)}(0)=0\), then \(P_{(,0,n)}\equiv 0\).$$\begin{aligned} \frac{dP_{(,0,n)}}{dt}=\left( (\sigma +\mu )n+\mu +\beta n\right) P_{(,0,n)} \end{aligned}$$
 Step 4.

Consider \(k=1\). The diagram in Fig. 1 shows at once that the zero state is globally stable, i.e. if \(P_{(,j,n1)}(0)=0\), then \(P_{(,j,n1)}\equiv 0\), \(j=0,1\). For \(k=2\), we have the diagram in Fig. 2, which shows that if \(P_{(,j,n2)}(0)=0\), then \(P_{(,j,n2)}\equiv 0\), \(j=0,1,2\).
4.2 The map \(L\)
5 Dynamics of the binding sites of an infectious individual: characterization of \(R_0\)
By exploiting that an individual can be considered as a collection of \(n\) binding sites that behave independently from one another as far as separation or acquiring a new partner is concerned and by using our mean field at distance one assumption, we are able to characterize \(R_0\) in terms of binding sites. In this section we only use the interpretation of the model and we do not use the system (18) or its reduced linearisation (26). We characterize \(R_0\) as the dominant eigenvalue of a nextgeneration matrix (NGM) that we construct using the interpretation of the model.
The entries in the NGM can be viewed as expected offspring values for a multitype branching process (Jagers 1975; Haccou et al. 2005), with the two matrixindices specifying the type at birth of, respectively, offspring and parent. Several slightly different branching processes may yield the same NGM and for the deterministic theory (which is what we deal with here) there is no need to choose one of these as ‘the’ underlying process. A branching process corresponding to the NGM is subcritical when \(R_0<1\) and supercritical when \(R_0>1\). But does such a branching process indeed correspond to the linearisation of (18) in the disease free steady state? Especially for \(n>1\) this is a nontrivial question. In Sect. 6 we will therefore prove that \(R_0\), as computed from the NGM, is indeed a threshold parameter with threshold value one for (18).
First, in Sect. 5.1, we consider the case \(n=1\). In Sect. 5.2 we generalize the transition and transmission scheme to \(n>1\), and in Sect. 5.3 we characterize \(R_0\) on the level of binding sites. We conclude this section by showing in Sect. 5.4 that \(R_0\) also has an interpretation in terms of individuals. The explicit expression for \(R_0\) and the remainder of its derivation is left for Appendix C.
Consider the usual setting for determining \(R_0\), i.e. suppose that we have a population in which only a few individuals are infectious and all others are susceptible. We are interested in the expected number of secondary cases caused by one ‘typical’ infectious case.
5.1 The case \(n=1\)

\(A\)—free

\(B\)—occupied by a susceptible partner

\(C\)—occupied by an infectious partner
We can characterize \(R_0\) by constructing an NGM \(K_1\) that involves a transmission part \(T_1\) and a transition part \(\varSigma _1\).
Recall that we use the convention that, for a transition matrix \(M=(m_{ij})\), \(m_{ij}\) denotes the probability per unit of time at which a transition from \(j\) to i occurs (instead of the transition from i to \(j\), as it is common in the stochastic community).
5.2 Generalization of the transition and transmission matrix: \(n>1\)
Now consider the case \(n>1\). In this case, an individual is a collection of \(n\) binding sites. These binding sites may be free, occupied by a susceptible or occupied by an infectious individual, i.e. in states \(A\), \(B\), or \(C\), respectively. An infectious individual can infect a susceptible individual in the population if it has a binding site that is occupied by a susceptible individual. In that case, that binding site becomes occupied by an infectious individual. Similar to the \(n=1\) situation we observe that if a binding site makes a transition from ‘occupied by a susceptible individual’ to \(C\), it creates a new infectious individual in the population. However, we need to know in which states the \(n\) binding sites of this new infectious individual are. Obviously, one new infectious binding site is in state \(C\), viz. the binding site still occupied by its epidemiological parent. In order to know the states of the other \(n1\) binding sites, we need to know the number of (susceptible) partners of this individual at epidemiological birth.
Naively, motivated by Lemma 2, one would think (as we did at first) that the number of partners of a newly infected individual is \(k\) (i.e. 1 binding site in state \(C\), \(k1\) binding sites in state \(B\) and \(nk\) binding sites in state \(A\)) with probability \(Q_k\). The computation of the corresponding \(R_0\) is rather straightforward (using the method explained in Appendix A for \(n=1\)). However, one can check numerically that the stability switch of the disease free steady state of (18) does not coincide with \(R_0=1\) when \(R_0\) is defined in this manner. We conclude that the premise is wrong. In retrospect this makes sense. First of all, we know that \(q\) differs from \(Q\), where \(q\) and \(Q\) are defined by (7) and (8), respectively. In our model description we keep track of the number of partners of an individual. We use mean field at distance one for the partners of partners of this individual (and this shows up in the \(\varLambda _\pm \) in the transmission events). So we need to do the same when characterizing \(R_0\) and also take into account the partners of susceptible partners. Therefore, we need to extend the information that is tracked in the scheme.

\(A\)—free

\(B_j\)—occupied by a susceptible partner that has \(j\) partners in total, \(j=1,\ldots , n\)

\(C\)—occupied by an infectious partner.
The other elements of \(\varSigma \) have the following interpretation. Note that, in the beginning of an epidemic, a binding site in state \(A\) acquires a susceptible partner at rate \(\rho \bar{F}\). The probability that, just after the moment of acquisition, this susceptible partner has in total \(j\) partners is \(q_j\) in accordance with (7). Therefore, the rate at which a binding site in state \(A\) transits to state \(B_j\) is \((\varSigma )_{B_j,A}=\rho \bar{F} q_j\). In a similar way one can use the interpretation (and the flowchart in Fig. 4) to find the other entries for the matrix \(\varSigma \).
The probability that a partner \(w\) of \(v\) has \(k\) partners depends on the state of \(v\), where \(v\) is in state \((+,j1,1)\) immediately after infection by \(u\). However, as another manifestation of the mean field at distance one assumption, we approximate this probability by only taking into account that the susceptible individual \(w\) has at least one partner \(v\). Therefore, we assume that \(w\) has \(k\) partners with probability \(Q_k\) (cf. Lemma 2). In other words, we assume that a binding site of \(v\) occupied by a susceptible partner, i.e. a binding site in the set \({\varvec{B}}\), is in state \(B_k\) with probability \(Q_k\).
In Sect. 5.4 we shall show that we can identify the \(\phi _j\) with an individual in state \((+,j1,1)\), which allows us to interpret \(R_0\) in terms of individuals. But first, in Sect. 5.3, we focus on the interpretation in terms of binding sites.
5.3 \(R_0\) in terms of binding sites
Theorem 1
\(R_0\), defined as the dominant eigenvalue of \(T(\varSigma )^{1}\), is a threshold parameter with threshold value one for the zero state of (35).
Note that \(T(\varSigma )^{1}\) is an \((n+2)\times (n+2)\) matrix. Also, elements \((T(\varSigma )^{1})_{xy}\) can be interpreted as the expected number of binding sites in \(x\) created by one binding site in \(y\), where \(x,y\in \{A,B_1,\ldots ,B_n,C\}\). This gives us an interpretation of \(R_0\) in terms of binding sites \(A,B_1,\ldots ,B_n,C\). However, we can reduce the characterization of \(R_0\) to a problem involving a \(3\times 3\) matrix by averaging the \(B_j\) in the right way (and this allows us to consider binding sites in \(A\), \({\varvec{B}}\), \(C\) only). We show this in the remainder of this subsection.
Theorem 2
\(R_0\), defined as the dominant eigenvalue of \(K\), where \(K\) is defined by (36), is a threshold parameter with threshold value one for the zero state of (35).
Proof
We have defined \(R_0\) as the dominant eigenvalue of \(T(\varSigma ^{1})\) and this \(R_0\) is a threshold parameter of the linear system corresponding to the matrix \(T+\varSigma \) according to Theorem 1. We will show that \(T(\varSigma ^{1})\) and \(K\) have the same dominant eigenvalue.
Consider the definition of \(K\) given by (36). This definition allows for an interpretation of the elements \(k_{x,y}\). Indeed, \(k_{x,y}\) can be interpreted as the expected number of binding sites in \(x\) created by one binding site in \(y\), with \(x,y \in \{ A,{{\varvec{B}}},C\}\). Therefore, we call \(K\) the NGM on the level of binding sites, and \(R_0\) can be interpreted as the expected number of secondary cases caused by a typical newly infected binding site in the beginning of an epidemic. Note that when \(x\) or \(y\) equals \({\varvec{B}}\) we specify a probability distribution rather than a specific state.
The relation (36) completely characterizes the matrix \(K\). However, using the interpretation, we can give explicit expressions for the entries of \(K\); see Appendix C. In this appendix it is also shown that, in order to find \(R_0\), we can reduce \(K\) to a \(2\times 2\) matrix and calculate the dominant eigenvalue of this smaller matrix. By combining (55)–(57), (59), and (61)–(63) we then find \(R_0\) given as an explicit function of the model parameters.
We have characterized \(R_0\) in terms of binding sites, both by considering all possible states \(\{A,B_1,\ldots , B_n, C\}\) and by considering \(\{A,{\varvec{B}}, C\}\). This allows for an interpretation of \(R_0\) in terms of binding sites. As we next show, we may also interpret \(R_0\) in terms of individuals.
5.4 \(R_0\) in terms of individuals
The model description is on the level of individuals, so it is only sensible that, in this section, we concern ourselves with the interpretation of \(R_0\) in terms of individuals, i.e. the interpretation of \(R_0\) as the expected number of secondary cases caused by a typical newly infected individual (rather than binding site) in the beginning of an epidemic.
Individuals can be considered as collections of \(n\) binding sites. We find the relation between the binding site level and the individual level as follows. Recall (33), where we see in the second equality that the \(\phi _j\) are a linear combination of the \(\psi _A\), \(\psi _{{\varvec{B}}}\), and \(\psi _C\). Note that \(\phi _j\) is a collection of \(n\) infectious binding sites, \(nj\) in state \(A\), 1 in state \(C\), and \(j1\) in states \(B_l\), \(l=0,\ldots ,n\) (and where the infectious binding site is in state \(B_l\) with probability \(Q_l\)). We can identify \(\phi _j\) with an individual in state \((+,j1,1)\). Note that the \((+,j1,1)\) are the possible states of an individual at epidemiological birth. For the case \(n=1\), we have \(\phi _1=\psi _C\) only (which corresponds to the only stateatepibirth \((+,0,1)\) since an infectious individual at epibirth is in a partnership with its epidemiological partner).
The matrix \(K^\mathrm{ind}\) is completely characterized by the identity (37). But, as in the case of \(K\), we can use the interpretation to give a more explicit expression for the entries of \(K^\mathrm{ind}\); see Appendix F.
5.5 \(R_0\): equivalence of different interpretations
6 Proof that \(R_0\) is a threshold parameter
Recall that, using the mean field at distance one assumption, we have written down a system of differential equations to describe the transmission of the infectious disease across the dynamic network. We will refer to the system (18) of differential equations for the fractions of the population of individuals in states \(\ell \), \(\ell =(\pm ,k_,k_+)\) as the \(P\)system. In Sect. 4 we have linearised this system around the diseasefree steady state and we were able to restrict this linearised system to the fractions \(P_{(,k,1)}\) and \(P_{(+,k_,k_+)}\). In Sect. 5 we considered binding sites of an infectious individual (in the linearisation!) and these binding sites could be in \(A\), \({\varvec{B}}\), and \(C\). This led to the \(ABC\)system (35). \(R_0\), defined as the dominant eigenvalue of \(K\), is a threshold for the stability of the zero state of (35); this was formulated in Theorem 2. In this section we will prove that \(R_0\) is also a threshold for the stability of the diseasefree steady state of system (18). We do so by relating the reduced linearisation (26) of the \(P\)system to the \(ABC\)system (35).
6.1 The case \(n=1\)
For \(n=1\) the proof is relatively easy, since there is no distinction between ‘individual’ and ‘binding site’. As the proof provides guiding lines for the general case, we present it first.
6.2 Generalization: \(n>1\)
It remains to prove that the stability switch of the zero state of the \(ABC\)system occurs if and only if the diseasefree state of the \(P\)system (18) switches stability. This will be shown in the remainder of this section.
We will proceed as follows. First we shall prove that \(r_{ABC}\) and \(r_P\) are dominant eigenvalues of the matrices \(T+\varSigma \) and \(M_P\), respectively, in the sense that these eigenvalues are uniquely characterized by the positivity of the eigenvector (up to a multiplicative positive constant).
We show in Lemmas 4 and 5 that \(T+\varSigma \) and \(M_P\) are irreducible matrices. This then allows us to conclude that the dominant eigenvalues of \(M_P\) and \(T+\varSigma \) are real and uniquely characterized by a positive eigenvector (see e.g. Theorem 2.5 of Seneta 1973). In other words, there exists a real eigenvalue \(r_P\) for \(M_P\) for which it holds that \(r_P>\text {Re } \lambda \) for any eigenvalue \(\lambda \ne r_P\) of \(M_P\) and \(r_P\) is uniquely defined by the positivity of the corresponding eigenvector (and similarly with \(r_{ABC}\) replacing \(r_P\) and \(T+\varSigma \) replacing \(M_P\)).
In Lemmas 4 and 5 below we use that a matrix \(M=(m_{xy})\) is irreducible if and only if variable \(x\) communicates with variable \(y\) (\(x\leftrightarrow y\)) for all variables \(x\) and \(y\), i.e. there is a path from \(x\) to \(y\) (\(x\rightarrow y\)), i.e. there are variables \(y_1\), \(y_2\), \(\ldots \), \(y_n\) such that \(m_{y,y_n}\cdots m_{y_2,y_1}m_{y_1,x}>0\), and a path from \(y\) to \(x\) (\(y\rightarrow x\)), i.e. there are variables \(x_1\), \(x_2\), \(\ldots \), \(x_k\) such that \(m_{y,x_k}\cdots m_{x_2,x_1}m_{x_1,y}>0\). Note that the somewhat unusual notation is due to our convention that \(m_{xy}\) denotes the transition from \(y\) to \(x\) (instead of the transition from \(x\) to \(y\), as it is common in the stochastic community).
Lemma 4
\(T+\varSigma \) is an irreducible matrix.
Proof
The flowchart describing the matrix \(\varSigma \) is presented in Fig. 4. We immediately see from this figure that from any state \(x\) there is a path to any other state \(y\), with \(x,y\in \{A,B_1,B_2,\ldots ,B_n,C\}\). It follows that \(\varSigma \) is irreducible. Since \(T\) is nonnegative, also \(T+\varSigma \) is irreducible. \(\square \)
Lemma 5
\(M_{P}\) is an irreducible matrix.
Proof
Finally, consider two variables \(x^=P_{(,k,1)}\), \(x^+=P_{(,l_,l_+)}\) of the matrix \(M_P\). We show that \(x^\leftrightarrow x^+\).
Since \(M_2\) and \(M_3\) are nonnegative and nonzero, there are variables \(y^\), \(y^+\), \(z^\), \(z^+\) such that \(y^\rightarrow y^+\) and \(z^+\rightarrow z^\). Note that, in terms of interpretation, the nonzero elements of \(M_2\) correspond to infection of \(\) individuals by one of their \(+\) partner, i.e. transitions with rate \(\beta \) from fractions \(P_{(,j_,1)}\) to \(P_{(+,j,1)}\). The nonzero elements of \(M_3\) correspond to the feed into the \(P_{(,j_,1)}\) category via the \(F_+\) terms from fractions \(P_{(+,k_,k_+)}\).
Since any two variables \(x^\) and \(x^+\) of \(M_P\) communicate, i.e. \(x^{{\mathrm{\leftrightarrow }}}x^+\), \(M_P\) is irreducible. \(\square \)
We now have all the ingredients to prove that \(R_0\) is a threshold parameter for the disease free state of (18).
6.3 Characterization of the Malthusian parameter \(r\) (\(=r_{ABC}=r_P\))
7 Looking back and ahead
The overall aim of our research is to formulate and analyse models for the spread of an infectious disease across a network that is dynamic in the double sense that individuals come (by birth) and go (by death) and that links/partnerships are formed and broken. In particular our aim is to investigate the role of concurrency in the spread of sexually transmitted infections.
In Leung et al. (2012) we introduced a class of doubly dynamic network models that are relatively simple to describe, that involve just a few parameters, and for which one can calculate many statistics exactly in explicit detail. The next step, taken here, is superimposing the spread of an infection. In order to retain the simplicity, we again characterize individuals by their dynamic degree (i.e. the current number of their partners), but now include the disease status (\(S\) versus \(I\)) of the individual itself and of its partners. In this bookkeeping scheme we need to account for the infection of a partner by one of its other partners, but the scheme itself does not provide information about partners of partners. Thus we faced a closing problem. The mean field at distance one assumption provided a natural solution.
Originally we thought that this was an assumption because we had not yet found a way to prove it. In a late stage Pieter Trapman pointed the way to the current Appendix B, showing that the assumption is inconsistent with the model itself. We then realised that, in essence, our bookkeeping scheme constitutes a first order description that we close by making the (inconsistent) mean field at distance one assumption. So the deterministic system studied here provides at best an approximation to the large system size limit of a stochastic model.
The great advantage of the deterministic system of dimension \((n+1)(n+2)\) is that it is amenable to analysis. The fact that binding sites operate to some extent independently from each other enables a reduction of the dimension from \((n+1)(n+2)\) to \(2\) in the characterization of \(R_0\). Indeed, we characterized the basic reproduction number \(R_0\) as the dominant eigenvalue of a \(3\times 3\) matrix with elements describing the expected numbers of newly infected binding sites of three different types generated by one infected binding site of either type during its life time. We could then further reduce the \(3\times 3\) matrix to a \(2\times 2\) matrix which lead to an explicit expression for the dominant eigenvalue \(R_0\). We also verified that the basic reproduction number \(R_0\) defined in this way is indeed a threshold parameter for the stability of the disease free steady state of the nonlinear system of model equations. This is done by establishing a relationship between the exponential growth rate \(r\) of the epidemic in the linearised system and the quantity \(R_0\) on the level of binding sites.
The characterization of \(r\) and \(R_0\) opens up the route for investigating the impact of concurrency on the transmission of the \({ SI}\) infection in the dynamic network. We can now study how \(r\) and \(R_0\) depend on the capacity \(n\) when fixing all other parameters at constant values. Furthermore, the relationship between concurrency measures on the one hand and \(R_0\), \(r\), and the endemic steady state on the other, can be analysed. This will be explored in a followup paper. (Concerning the endemic steady state, we will need to derive the equations that characterize it, to investigate the uniqueness and to prove that existence requires \(R_0>1\).)
There are a number of generalisations of the network model that are both useful and feasible. The extension to a heterosexual population requires only the distinction between males and females and some assumptions on the symmetry or asymmetry in rates and partnership capacity between the two sexes. We expect that all results presented here carry, mutatis mutandis, over to that situation. No doubt the model can also be extended to the situation that \(n\) is a random variable with a prescribed distribution.
Other generalisations pertain to the description of infectiousness. An obvious example is a model with two consecutive stages \(I_1\) and \(I_2\), where infectiousness is characterised by \(\beta _i\) in stage \(I_i\). Other compartmental epidemic models could be considered as well, such as \({ SIR}\) and \({ SIS}\). Inclusion of the impact of the disease on mortality is very relevant in the context of HIV. Unfortunately it might turn out to be very hard.
The most stringent limitation of our framework is the assumption that having a partner does not influence an individual’s propensity to enter into a new partnership or its contact rate in other ongoing partnerships. This is clearly at odds with reality (although equally clearly it is an impossible task to disentangle the manifold ways in which dependence ‘works’ in reality). Dependence destroys the basis on which our analytic approach rests.
Be that as it may, we view the work presented here as a first step towards a framework for studying the impact of dynamic network structure on the transmission of an infectious disease.
Notes
Acknowledgments
We thank Pieter Trapman, Martin Bootsma, and Hans Metz for useful ideas and discussions. We thank two anonymous referees for helpful suggestions. KYL is supported by the Netherlands Organisation for Scientific Research (NWO) through research programma Mozaïek, 017.009.082.
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