Weakening connections in heterogeneous mean-field models
WWeakening connections in heterogeneous mean-field models
C. Dias and M. O. Hase Escola de Artes, Ciˆencias e Humanidades, Universidade de S˜ao Paulo,Av. Arlindo B´ettio 1000, 03828-000 S˜ao Paulo, Brazil ∗ Two versions of the susceptible-infected-susceptible epidemic model, which have different trans-mission rules, are analysed. Both models are considered on a weighted network to simulate amitigation in the connection between the individuals. The analysis is performed through a hetero-geneous mean-field approach on a scale-free network. For a suitable choice of the parameters, bothmodels exhibit a positive infection threshold, when they share the same critical exponents associatedwith the behaviour of the prevalence against the infection rate. Nevertheless, when the infectionthreshold vanishes, the prevalence of these models display different algebraic decays to zero for lowvalues of the infection rate. ∗ [email protected] a r X i v : . [ c ond - m a t . s t a t - m ec h ] F e b I. INTRODUCTION
Recently, the COVID-19 outbreak imposed a new lifestyle to face this disease. It also reminded us that new infectiousdiseases can emerge and the difficulties to make predictions on the course of its diffusion based on incomplete data (forinstance, due to underreporting) have emphasized the importance of mathematical modelling of infection spreading[1, 2].A simple and standard scheme that is popular in mathematical modelling in epidemiology is the division of thepopulation into compartments, which characterises the possible states of individuals with respect to the disease:infected, susceptible, recovered, et cætera . The main strategy is to write a set of differential equations that describesthe flow of the population between the compartments. Traditionally, these equations have adopted the hypothesisof “homogeneous mixing”, in which an individual has the same probability of being in contact with any member ofthe compartment [3]. Although this supposition has been popular for decades, it leads to an oversimplification thatcontrasts with the heterogeneous organization in the network of human contacts [4–6]. This observation has pointedout the importance of combining both epidemiology and network theory in order to study the spreading of infectiousagents based on a realistic interconnection between individuals [7].Apart from the complex structure of the human contact network that is vital to monitor the spreading of diseases,the recent COVID-19 pandemic has reinforced the importance of keeping social distance, especially when no otherefficient alternative to contain the propagation of the infection is known. As a consequence, the physical ties betweenpeople are effectively weakened, either by deliberate seclusion or by forced isolations (like hospitalisations). In thiswork, we examine the effects of the epidemic spreading in a scenario where the contact between individuals is mitigated(especially, the contact to infected ones). The main strategy to accomplish this goal is to introduce a tunable parameter ω that controls the probability of linking between the members of the system. This modification is equivalent to analysea class of weighted networks [8], and the definition of epidemiological models on such graphs has also been consideredby several authors [9, 10].We test our ideas on the susceptible-infected-susceptible (SIS) model, where each member of the system (representedby the vertices of a network) can be in one of two states: susceptible or infected. The former can be infected bydirect contact with an infected individual, while the latter can recover spontaneously to integrate the set of susceptiblemembers. We perform the analysis of the SIS dynamics, with weakening effects in the connections between individuals,on two similar - but different - systems. In the investigation of infection spreading, many transmission mechanisms canbe proposed [11]. We highlighted two of them, which we call, following [12], the standard SIS (s-SIS), and threshold
SIS(t-SIS) models. In the former, an infected node infects each susceptible neighbour with a rate λ . On the other hand, int-SIS model, a susceptible node is infected, with rate λ , if at least one of its neighbours is infected, thus characterisinga threshold process [13, 14] (see also [15, 16], where a similar idea is present in the “first epidemic model”). Ourinvestigation for both s-SIS and t-SIS models is based on the heterogeneous mean-field (HMF) approximation [17, 18],which allows the inclusion of heterogeneities (present in real networks) at the level of degree distributions. Despitethis appealing feature of the HMF, it allows a finite infection threshold for a SIS dynamics on a scale-free network[18]. In other words, it implies the existence of a critical point that separates a disease-free absorbing state fromthe active phase, where the infection persists. This is in contrast with a rigorous result that states the location ofthis critical point to be at zero for an uncorrelated scale-free random network [19]. The vanishment of the criticalpoint is, however, predicted by the SIS model in the quenched mean-field (QMF) [20, 21] approach for networks thatare governed by a power-law degree distribution [22, 23]. The QMF captures the structure of the network throughits adjacency matrix, and the discrepancy coming from the quenched or annealed (of which HMF is an example)treatment in many statistical models is not surprising [24, 25]. All the models mentioned above are usually treatedat mean-field level, whose analysis does not mandatorily demand metric features of the problem. Nevertheless, theincorporation of the information based on spatial structure of the environment, like in metapopulation models [26–28],is beneficial to epidemiology.The layout of this paper is as follows. We briefly review the s-SIS and t-SIS models in Section 2 to introduce somenotations. In Section 3, we show the modification necessary in these models to introduce weakening effects on thelinks between individuals. Then, we examine the s-SIS and t-SIS models with weakening in connections in Sections 4and 5, respectively. The results from both models are compared in Section 6 and the conclusions are delivered in thelast Section. II. HETEROGENEOUS MEAN-FIELD MODELS
In this work, we examine the SIS model, where an infected individual infects, by contact, a susceptible one witha rate λ , and an infected recovers spontaneously with a rate µ . In an uncorrelated network, the assumption ofthe “homogeneous mixing”hypothesis leads to an infection-free absorbent phase for a sufficiently low infection rate.Nevertheless, increasing this rate over a critical threshold λ c , which is inversely proportional to the mean degree ofthe graph, implies the persistence of the disease [29].Henceforth, we consider the cases where the network of contacts does not have a homogeneous structure. Anepidemiological model defined on a heterogeneous (in the sense of degree distribution) network was first investigatedin the framework of a HMF theory [17, 18]. In this approach, the time evolution of ρ k , which is the probability of avertex with degree k being infected, is given by ∂ t ρ k = − ρ k + λk (1 − ρ k ) Θ k , (1)where ∂ t ≡ ∂∂t stands for the (partial) derivative with respect to time. This is a s-SIS model, where an infected vertexinfects each susceptible neighbour with a rate λ . We treat vertices that share the same degree on equal footing, butwe will see that the probability of being infected differs if they have different connectivity. In (1), the recovering rate µ is taken to be µ = 1 without loss of generality. This procedure is always possible by choosing a suitable time scale,and then the infection rate λ becomes numerically equivalent to the spreading rate λ/µ . In the last term, Θ k standsfor the probability that a link originated from a vertex of degree k connects to an infected node.In this work, we consider an uncorrelated network, where the probability Θ k is independent of k , and we replacethis symbol by Θ. As a consequence of this assumption, the probability q ( k | k (cid:48) ) of a vertex of degree k (cid:48) linking to anode of degree k can be cast as [5, 30] q ( k | k (cid:48) ) = q ( k ) = kP ( k ) (cid:104) k (cid:105) , (2)where P is the degree distribution and (cid:104) k α (cid:105) := (cid:80) k k α P ( k ) stands for the α -th moment. For simplicity, we write q ( k )instead of q ( k | k (cid:48) ), since q does not depend on the degree of the source vertex in an uncorrelated network. Therefore,the probability Θ can be written as Θ = (cid:88) k q ( k ) ρ k = 1 (cid:104) k (cid:105) (cid:88) k kP ( k ) ρ k , (3)which is the sum of all disjoint events of linking to an infected node of degree k .Contrary to the s-SIS model described above, we may conceive a different infection mechanism, where a susceptiblevertex changes its state if at least one of its neighbours is infected [11]. This is the t-SIS model, since the infection isa threshold process. The HMF master equation in this scenario is given by ∂ t ρ k = − ρ k + λ (1 − ρ k ) (cid:104) − (1 − Θ) k (cid:105) . (4)Again, a suitable choice in time scaling leads to a unitary recovery rate. As in the previous model, we consider anuncorrelated network, and the factor (1 − Θ) k is the probability that none of the k neighbours of a vertex is infected,which follows from the assumption that it constitutes k independent events. As a consequence, 1 − (1 − Θ) k is theprobability that at least one of the k neighbours is infected. Despite some similarities (shown later), this is a differentmodel from the s-SIS, although some confusion may arise [11].In this work, we mainly investigate the stationary state, where ∂ t ρ k = 0. The quantification of the infection isexamined through the prevalence ρ = (cid:88) k P ( k ) ρ k , (5)which is the stationary infection probability of the system. III. SIS MODELS WITH WEAKENED CONNECTIONS
In this section, a modification in the models presented in the previous section is proposed, where the connectionsare weakened. We mean by this that the probability Θ of linking to an infected vertex is decreased. This situationmay simulate a scenario where infected individuals become harder to be reached (by isolating themselves, beinghospitalised, et cætera ). The modification is introduced by replacing the linking probability (2) by q ω ( k ) = k ω P ( k ) (cid:104) k ω (cid:105) , (6)where ω is a parameter that we assume to belong in the interval ( −∞ , ω = 1 recovers the usual case onan uncorrelated network. Therefore, the probability of a link achieving an infected link isΘ ω = (cid:88) k q ω ( k ) ρ k = 1 (cid:104) k ω (cid:105) (cid:88) k k ω P ( k ) ρ k . (7)The index ω in Θ ω is written to remember that we are dealing with the modified version of the model.It is important to stress that the probabilities q and q ω are quantities associated to graph properties and are notrelated to the epidemiological context. Nevertheless, they appear, respectively, in the probabilities Θ and Θ ω only.This is the reason we can treat the replacement of Θ by Θ ω as a weakening of the links to infected vertices, becausecontacts between susceptible ones do not play any role in SIS dynamics. The model defined in (7) is equivalent todefine the SIS model on weighted networks [8]. IV. STANDARD SIS MODEL WITH WEAKENED CONNECTIONSA. General setting
We examine the stationary state of the s-SIS model, which consists of the master equation (1) and the probability(7). Some of the procedures are well-known, but we present them for completeness. In steady state, ∂ t ρ k = 0, equation(1) implies ρ k = λk Θ ω λk Θ ω , (8)which shows the inhomogeneous structure of the network, because the infection probability depends on the degree ofthe vertex. Inserting (8) into (7) leads to Θ ω = g (Θ ω ) , (9)where g (Θ ω ) = 1 (cid:104) k ω (cid:105) (cid:88) k k ω P ( k ) λk Θ ω λk Θ ω . (10)Since Θ ω is a probability, one has 0 ≤ Θ ω = g (Θ ω ) ≤
1. It is also immediate that g (0) = 0, g (cid:48) (Θ ω ) > g (cid:48)(cid:48) (Θ) < g (cid:48) stands for the derivative of g ). Hence, g is an increasing and concave function in [0 ,
1] such that g (0) = 0 and g (1) ≤
1. This last upper bound follows also from g (Θ ω = 1) = 1 (cid:104) k ω (cid:105) (cid:88) k k ω P ( k ) λk Θ ω λk Θ ω ≤ (cid:104) k ω (cid:105) (cid:88) k k ω P ( k ) 1 + λk Θ ω λk Θ ω = 1 . (11)The equation (9) can be solved by searching for the intersection points of the graphs y = Θ ω and y = g (Θ ω ) in theplane y × Θ ω , as one can see in figure 1. The critical point is obtained from the condition g (cid:48) (0) = 1, which implies λ c = (cid:104) k ω (cid:105)(cid:104) k ω +1 (cid:105) . (12)This result could have been obtained through a simpler approach [11]. However, by the present method we can besure that there are at most two solutions in the model.The analysis of the system above is based on the equations (9) and (10) with the constraint Θ ω (cid:54) = 0. Concretely,it implies the investigation of the equation1 (cid:104) k ω (cid:105) (cid:90) ∞ d k k ω P ( k ) λk λk Θ ω = 1 , (13)where the continuum approximation was invoked. FIG. 1. Sketch of the graphical solution of (9). Left: Trivial solution at Θ ω = 0. Right: Trivial solution at Θ ω = 0 and anon-trivial positive solution. B. Scale-free network
We examine the s-SIS model on a scale-free network, where the degree distribution scales as P ( k ) ∼ k − γ , withweakened connections gauged by the parameter ω . Let us first establish some notations and results. Assuming thatthe minimum degree of each vertex in the network is m , one has P ( k ) = ( γ − m γ − k γ ( k ≥ m ) . (14)We assume that γ > (cid:104) k ω (cid:105) = ( γ − m ω γ − ω − ≤ ω ≤ , (15)where the average is taken with respect to the distribution (14). From (12), the critical point is λ c = 1 m (cid:18) γ − ω − γ − ω − (cid:19) , (16)and we have to admit the condition γ − ω − > γ − ω − λm(cid:15) γ − ω − (cid:90) ∞ (cid:15) d uu γ − ω − (1 + u ) = 1 , (17)where (cid:15) := λm Θ ω . We are especially interested in the behaviour of the system close to the infection threshold. Inthis region, the integral in (17) can be examined analytically, as shown in Appendix A. Here, we will just state theresults.The evaluation of the prevalence (5) in s-SIS model close to the critical point benefits also from the results ofAppendix A. When γ >
2, we have ρ (cid:39) (cid:18) γ − γ − (cid:19) ( λm Θ ω ) , (18)where λm Θ ω (cid:39) (cid:20) m ( γ − ω − π sin (( γ − ω − π ) (cid:21) − γ + ω λ − γ + ω , < γ − ω < (cid:20) sin ( − ( γ − ω − π ) λπ ( γ − ω − (cid:21) γ − ω − ( λ − λ c ) γ − ω − , < γ − ω < λ (cid:18) γ − ω − γ − ω − (cid:19) ( λ − λ c ) , γ − ω > . (19) FIG. 2. Graph ρ × λ for the s-SIS model with weakened connections; here, γ = 2 .
5. The curves were obtained from numericalresolution of equation (17). Although the prevalence does not depend on λ and m independently (actually, it is a function of λm , as one can easily see from (17)), we fixed the value of m to 1 to show the dependence on the infection rate only. This result recovers the ones obtained by [18].The following conclusions can be established. Firstly, the infection threshold vanishes if γ − ω ≤
2. However, theway the infection probability ρ (and also Θ ω ) approaches zero should be examined carefully. For γ − ω <
2, we observean algebraic decay ρ ∼ λ − γ + ω , while a non-algebraic decay, ρ ∼ e − λm , is predicted if γ − ω = 2. This last case isrealized [17] in the usual s-SIS model defined on a Barab´asi-Albert network [31].For γ − ω >
2, a positive critical point emerges. In the interval 2 < γ − ω <
3, we see an algebraic decay of theorder parameter ρ ∼ ( λ − λ c ) γ − ω − . On the other hand, if γ − ω = 3, the relation Θ ω ln Θ ω ∼ ( λ − λ c ) holds closeto the critical point, discarding an algebraic relation between Θ ω (and, consequently, ρ ) and the difference λ − λ c .Finally, if γ − ω >
3, the criticality of the system is described by ρ ∼ ( λ − λ c ).From (19), the probability Θ ω depends on the parameters γ and ω through its difference γ − ω , and this property isobserved in both critical exponents and the prefactors. However, this type of dependence is broken when we considerthe prevalence ρ due to the factor γ − γ − in (18). V. THRESHOLD SIS MODEL WITH WEAKENED CONNECTIONS
In this section, we investigate the stationary state of the t-SIS model defined by the master equation (4). In thisregime, one has ρ k = λ (cid:104) − (1 − Θ) k (cid:105) λ (cid:104) − (1 − Θ) k (cid:105) = 1 −
11 + λ (cid:104) − (1 − Θ) k (cid:105) . (20)Let us now include the weakening factor in the model by inserting (20) into (7), which leads toΘ ω = 1 − (cid:104) k ω (cid:105) (cid:88) k k ω P ( k ) 11 + λ (cid:104) − (1 − Θ ω ) k (cid:105) , (21)where the degree distribution is given by (14) and the mean value (cid:104)·(cid:105) is taken with respect to this scale-free distribution.We assume, again, that the minimum degree of the graph is m . Define, now, the probability ϕ ω := 1 − Θ ω , (22)which is just the complementary one to Θ ω . From this notation, our principal equation, (21), can be cast as ϕ ω = h ( ϕ ω ) , (23)where h ( ϕ ω ) := 1 (cid:104) k ω (cid:105) (cid:88) k k ω P ( k ) 11 + λ (1 − ϕ kω ) . (24)Noting from (24) that h (0) ≤ h (1) = 1, h (cid:48) ( ϕ ω ) > h (cid:48)(cid:48) ( ϕ ω ) >
0, and following a similar reasoning of section IVthat led to (12), we can determine the critical point λ c through the relation h (cid:48) (1) = 1, which implies λ c = (cid:104) k ω (cid:105)(cid:104) k ω +1 (cid:105) , (25)and is exactly the same value obtained in the previous s-SIS model [11].The system (23)-(24) has a trivial solution ϕ ω = 1, which corresponds to Θ ω = 0. We will now search for thenontrivial solutions of this system, and keep in mind that ϕ (cid:54) = 1. In continuum approximation, we have ϕ ω = 1 (cid:104) k ω (cid:105) (cid:90) ∞ d k k ω P ( k ) 11 + λ (1 − ϕ kω )= ( γ − ω − m γ − ω − (cid:90) ∞ m d kk γ − ω
11 + λ (1 − ϕ kω ) . (26)The asymptotic behaviour for ϕ ∼ − of the integral in (26) is examined with detail in Appendix B. From this result,we can characterise the prevalence (5) in this model, which is ρ (cid:39) (cid:18) γ − γ − (cid:19) ( λm Θ ω ) , (27)where λm Θ ω (cid:39) [ m Γ(2 − γ + ω )] − γ + ω λ − γ + ω − γ + ω , < γ − ω < (cid:20) λ γ − ω − ( γ − ω − A (1) (cid:21) γ − ω − ( λ − λ c ) γ − ω − , < γ − ω < λ (cid:104) m (1 + 2 λ ) (cid:16) γ − ω − γ − ω − (cid:17) − (cid:105) ( λ − λ c ) , γ − ω > , (28)where A (1) := (cid:82) ∞ yy γ − ω − e − y ( λ − +1+ e − y ) ( λ − +1 − e − y ) .We discuss this result comparing to the ones obtained in the s-SIS model in the next section. VI. COMPARISON OF S-SIS AND T-SIS MODELS WITH WEAKENED CONNECTIONS
The s-SIS and t-SIS models, as seen in the previous sections, share some common features like the infection thresholdand some critical exponents. In the region 2 < γ − ω <
3, both models have the same nontrivial exponent γ − ω − , andif γ − ω >
3, they displayed the mean-field exponent 1. One can see in figure 4 that although the prevalence curve ishigher in s-SIS than in the t-SIS model (as pointed out by [11]), the critical exponents are the same.The main difference between these two models, however, is observed when the infection threshold vanishes. As onecan see from (19) and (28), the exponent with which the prevalence decays to zero differs in both cases, and this factis supported by numerical calculations, as seen in figure 5.The above results can be summarized by the diagram of figure 6. We noticed that the weakening effect in connections,as defined in (6), can shift the critical behaviour of the prevalence. Furthermore, the weakening parameter ω and thescale-free exponent γ are closely tied in the probability Θ ω , which depends on the difference γ − ω . This observationsextends also to the prefactor of the leading term of Θ ω in both s-SIS and t-SIS models, as one can see in (19) and(28). However, the same property is not shared with the prevalence ρ , where γ and ω become two independentparameters, as (18) and (27) indicate: although all the critical exponents are functions of the difference γ − ω also inthe prevalence, the same can not be said to the prefactors of both s-SIS and t-SIS. FIG. 3. Graph ρ × λ for the t-SIS model with weakened connections; here, m = 1 and γ = 2 .
5. The curves were obtainedfrom numerical resolution of equation (26).FIG. 4. Graph ρ × ( λ − λ c ) in log-log scale, with γ = 3 .
25 and m = 1; in the left figure, ω = 0 .
5, while ω = 0 . λ − λ c ) γ − ω − (left figure) and ( λ − λ c ) (right figure).FIG. 5. Graph ρ × λ in log-log scale, with γ = 2 . ω = 0 . m = 1. The upper dotted line is proportional to λ − γ + ω ,while the lower one is proportional to λ − γ + ω − γ + ω ; they were drawn just for visual aid. FIG. 6. Phase diagram γ × ω . As usual, the exponent β is defined through ρ ∼ ( λ − λ c ) β . The shaded region corresponds to λ c = 0; otherwise, λ c >
0. Here, β = − γ + ω and β = − γ + ω − γ + ω for the s-SIS and t-SIS HMF models, respectively. No algebraicdecay of the order parameter is observed along the thick lines. VII. DISCUSSION AND CONCLUSIONS
In this work, we have investigated two versions of the SIS epidemic model by a heterogeneous mean-field approach.These two models, s-SIS and t-SIS, have different transmission mechanisms, but share some properties like the locationof the infection threshold and similar critical behaviour in some cases.The main contributions of this work can be summarized as follows. Firstly, we have included and examined theinfluence of a weakening factor in the connections between vertices. This modification, characterised by a weakeningfactor ω in the linking probabilities of the underlying network of connections, may simulate a mitigation in the contactbetween people. We could observe that the role played by this parameter was, effectively, a redefinition of the scale-free exponent γ to γ − ω for the probability Θ ω of a link reaching an infected node. This property holds for the criticalexponents for the prevalence, but does not extend to its nonuniversal details like the prefactor near the infectionthreshold.Secondly, we investigated both models defined on a scale-free network with degree distribution P ( k ) ∼ k − γ . Whilethe s-SIS model has already been studied, we could characterise the decay to the critical point of the t-SIS dynamics,whose analytical critical properties seem to lack in the literature. We have shown that the prevalence decays as ρ ∼ ( λ − λ c ) β for both s-SIS nd t-SIS models, and the exponent β is the same in some cases: it assumes a nontrivialvalue − γ + ω for 2 < γ − ω <
3, and recovers the mean-field exponent 1 for γ − ω >
3. The novelty is found when thecritical point vanishes: the exponent of the prevalence in the vicinity of zero infection rate is different in s-SIS andt-SIS dynamics: in the former, the stationary infection probability decays as λ − γ + ω , while in the latter, we observed ρ ∼ λ − γ + ω − γ + ω . These results are supported by numerical methods. A. APPENDIX A
In this Appendix, we examine the expansion of the integral I ( (cid:15) ) := (cid:90) ∞ (cid:15) d xx α (1 + x ) (29)for (cid:15) ∼ α >
0. In the main text, we notice that α = γ − ω −
1. We consider the following cases: (i) 0 < α < < α <
2, (iii) α > α = 1 and α = 2.0(i) Case 0 < α <
1. Since I (0) < ∞ , we can write (29) as I ( (cid:15) ) = (cid:90) ∞ d xx α (1 + x ) − (cid:90) (cid:15) d xx α (1 + x )= B (1 − α, α ) − (cid:90) (cid:15) d xx α (cid:0) − x + x − x + · · · (cid:1) , (30)where B (1 − α, α ) = Γ(1 − α )Γ( α )Γ(1) is the Beta function. Performing the integrals, we have I ( (cid:15) ) = Γ(1 − α )Γ( α ) − (cid:15) − α − α + O ( (cid:15) − α ) , (0 < α < . (31)(ii) Case 1 < α <
2. Here, the integral I (0) diverges, but isolating the dominant term (that diverges if (cid:15) ↓ I ( (cid:15) ) = (cid:90) ∞ (cid:15) d xx α − (cid:90) ∞ (cid:15) d xx α − (1 + x ) , (32)where the last integral in (32) converges for (cid:15) = 0 (1 < α < I ( (cid:15) ) = (cid:90) ∞ (cid:15) d xx α − (cid:90) ∞ d xx α − (1 + x ) + (cid:90) (cid:15) d xx α − (1 + x )= (cid:15) − α α − − B (2 − α, α −
1) + (cid:90) (cid:15) d xx α − (1 − x + · · · )= (cid:15) − α α − − Γ(2 − α )Γ( α − O ( (cid:15) − α ) . (33)(iii) Case α >
2. Consider, initially, the case 2 < α <
3. Starting from (32), the last integral in RHS now diverges for (cid:15) →
0. We proceed as in the previous case by isolating the dominant term (that is responsible for the divergence);then, I ( (cid:15) ) = (cid:90) ∞ (cid:15) d xx α − (cid:90) ∞ (cid:15) d xx α − + (cid:90) ∞ (cid:15) d xx α − (1 + x ) . (34)The last integral is well-defined at (cid:15) = 0 for 2 < α <
3; therefore, writing (cid:82) ∞ (cid:15) d xx α − (1+ x ) = (cid:82) ∞ xx α − (1+ x ) − (cid:82) (cid:15) xx α − (1+ x ) leads to I ( (cid:15) ) = (cid:15) − α α − − (cid:15) − α α − B (3 − α, α − − (cid:90) (cid:15) d xx α − (1 + x )= (cid:15) − α α − − (cid:15) − α α − − α )Γ( α − O ( (cid:15) − α ) (35)for 2 < α <
3. Now, even if α ≥
3, the first two dominant terms are the same from (35); therefore, I ( (cid:15) ) = (cid:15) − α α − − (cid:15) − α α − O (1) (36)for α > α = 1 and α = 2. Here, the integrals can be performed by partial fraction decomposition. The results are I ( (cid:15) ) = (cid:90) ∞ (cid:15) d xx (1 + x ) = ln (cid:18) (cid:15) (cid:19) ( α = 1) (37)and I ( (cid:15) ) = (cid:90) ∞ (cid:15) d xx (1 + x ) = 1 (cid:15) − ln (cid:18) (cid:15) (cid:19) ( α = 2) . (38)1Recollecting the results above, we have I ( (cid:15) ) = π sin( απ ) − (cid:15) − α − α + O ( (cid:15) − α ) , < α < − ln (cid:15) + (cid:15) + O ( (cid:15) ) , α = 1 (cid:15) − α α − − (cid:104) π sin( − απ ) (cid:105) + O ( (cid:15) − α ) , < α < (cid:15) − + ln (cid:15) + O ( (cid:15) ) , α = 2 (cid:15) − α α − − (cid:15) − α α − + O (1) , α > , (39)where the identity Γ(1 − α )Γ( α ) = π sin( απ ) ( α / ∈ Z ) was invoked. B. APPENDIX B
In this Appendix, the asymptotic regime of the integral A ( ϕ ) := (cid:90) ∞ m d xx β
11 + λ (1 − ϕ x ) (40)is examined. We will omit the dependence on the parameters β and λ to keep the notation simpler (similar commentsfor the forthcoming functions). Here, the parameters obey β > β = γ − ω ), λ > ϕ is restricted to 0 ≤ ϕ <
1. It is important to remember that ϕ is strictly smaller than 1, because we are searchingfor the nontrivial solution of (26), as explained in section V. Introducing a change of variable y = − x ln ϕ , the integral(40) can be written as A ( ϕ, λ ) = ( − ln ϕ ) β − λ (cid:90) ∞− m ln ϕ d yy β λ − + 1 − e − y . (41)One should now to examine the behaviour of the integral A ( ϕ ) := (cid:90) ∞− m ln ϕ d yy β λ − + 1 − e − y ( β >
1) (42)for ϕ ∼ − . The general scheme consists of integrating by parts until the exponent of the algebraic term in theintegrand (in (42), it is β ) falls in the interval (0 , ϕ = 1 − without diverging, and the correction terms can be easily managed. We illustrate thisprocedure by dealing with the following cases: (i) 1 < β <
2, (ii) 2 < β < β > < β <
2. One should first note that the integral in (42) diverges for ϕ →
1. By performing an integrationby parts, one has A ( ϕ ) := 1 β − (cid:34) ( − m ln ϕ ) − β λ − + 1 − ϕ m − A ( ϕ ) (cid:35) , (43)where A ( ϕ ) := (cid:90) ∞− m ln ϕ d yy β − e − y ( λ − + 1 − e − y ) . (44)For 1 < β <
2, the integral A ( ϕ ) converges at ϕ = 1 − . Then, A ( ϕ ) = A (1) − (cid:90) − m ln ϕ d yy β − (cid:34) e − y ( λ − + 1 − e − y ) (cid:35) = A (1) − (cid:90) − m ln ϕ d yy β − (cid:2) λ + O ( y ) (cid:3) = A (1) − λ ( − m ln ϕ ) − β − β + O (ln − β ϕ ) . (45)2Therefore, from (41) to (45), we have A ( ϕ ) = 1 β − (cid:34) m − β λ (1 − ϕ m ) − ( − ln ϕ ) β − λ A (1) + λ m − β ( − ln ϕ )2 − β (cid:35) + O (ln ϕ ) (1 < β < . (46)(ii) Case 2 < β <
3. The starting point is the integral (44). Contrary to the previous case, now the integral A diverges for ϕ → − . Then, integrating it by parts leads to A ( ϕ ) = 1 β − (cid:34) ( − m ln ϕ ) − β ϕ m ( λ − + 1 − ϕ m ) − A ( ϕ ) (cid:35) , (47)where A ( ϕ ) := (cid:90) ∞− m ln ϕ d yy β − e − y (cid:0) λ − + 1 + e − y (cid:1) ( λ − + 1 − e − y ) . (48)For 2 < β < A converges for ϕ → − . Then, A ( ϕ ) = A (1) − (cid:90) − m ln ϕ d yy β − (cid:34) e − y (cid:0) λ − + 1 + e − y (cid:1) ( λ − + 1 − e − y ) (cid:35) = A (1) − (cid:90) − m ln ϕ d yy β − (cid:2) λ (1 + 2 λ ) + O ( y ) (cid:3) = A (1) − λ (1 + 2 λ ) ( − m ln ϕ ) − β − β + O (ln − β ϕ ) . (49)From (41), (42), (43), (47) and (49), one has A ( ϕ ) = 1 λ ( β − (cid:34) m − β λ − + 1 − ϕ m − m − β ( − ln ϕ ) ϕ m ( β −
2) ( λ − + 1 − ϕ m ) + ( − ln ϕ ) β − β − A (1) + O (ln ϕ ) (cid:35) (2 < β < . (50)(iii) Case β >
3. The starting point is the integral (48). Contrary to the previous case, now the integral A divergesfor ϕ → − . Then, integrating it by parts leads to A ( ϕ ) = 1 β − (cid:34) ( − m ln ϕ ) − β ϕ m (cid:0) λ − + 1 + ϕ m (cid:1) ( λ − + 1 − ϕ m ) − A ( ϕ ) (cid:35) , (51)where A ( ϕ ) is an integral that appears after the procedure above. We will not show its explicit form, since we havesufficient terms in the expansion of A ( ϕ ) and the contribution of A ( ϕ ) will be negligible. If 3 < β < A is oforder O (1), while A ( ϕ ) = O (ln − β ϕ ) if β >
4; then, A ( ϕ ) = O (1) for β >
3. Hence, from (41), (42), (43), (47) and(51), one finally has A ( ϕ ) = 1 λ ( β − (cid:34) m − β λ − + 1 − ϕ m − m − β ( − ln ϕ ) ϕ m ( β −
2) ( λ − + 1 − ϕ m ) + m − β ( − ln ϕ ) ϕ m (cid:0) λ − + 1 + ϕ m (cid:1) ( β −
2) ( β −
3) ( λ − + 1 − ϕ m ) ++ O (ln ϕ, ln β − ϕ ) (cid:35) ( β > . (52)Summarizing the results above, we have A ( ϕ ) = λ ( β − (cid:34) m − β λ − +1 − ϕ m − ( − ln ϕ ) β − A (1) + λ m − β ( − ln ϕ )2 − β (cid:35) + O (ln ϕ ) , < β < λ ( β − (cid:34) m − β λ − +1 − ϕ m − m − β ( − ln ϕ ) ϕ m ( β − λ − +1 − ϕ m ) + ( − ln ϕ ) β − β − A (1) (cid:35) + O (ln ϕ ) , < β < λ ( β − (cid:34) m − β λ − +1 − ϕ m − m − β ( − ln ϕ ) ϕ m ( β − λ − +1 − ϕ m ) + m − β ( − ln ϕ ) ϕ m ( λ − +1+ ϕ m ) ( β − β − λ − +1 − ϕ m ) (cid:35) + O (ln ϕ, ln β − ϕ ) , β > . (53)3Translating the result (53) into the variable Θ = 1 − ϕ , we have A (Θ) = β − (cid:104) m − β − A (1) λ Θ β − (cid:105) + O (Θ) , < β < β − (cid:34) m − β − λ (cid:16) β − β − (cid:17) m − β Θ + A (1) λ ( β − Θ β − (cid:35) + O (Θ ) , < β < β − (cid:34) m − β − λ (cid:16) β − β − (cid:17) m − β Θ + λm − β ( β − (cid:104) m (1+2 λ ) β − − β − (cid:105) Θ (cid:35) + O (Θ , Θ β − ) , β > . (54)when Θ ∼
0. We also note that A (1) ∼ λ Γ(2 − β ) for λ ∼
0, which is the neighbourhood of the null infectionthreshold when β < [1] Hethcote H W 1989 Three Basic Epidemiological Models Applied Mathematical Ecology ed L Gross, T G Hallam and S ALevin (Berlin: Springer-Verlag) pp 119-144[2] Anderson R M and May R M 1992
Infectious Diseases of Humans (Oxford: Oxford University Press)[3] Hamer W H 1906 The Milroy Lectures on Epidemic Disease in England - The Evidence of Variability and of Persistencyof Type
The Lancet
Rev. Mod. Phys. Evolution of Networks: From Biological Nets to the Internet and WWW (Oxford:Oxford University Press)[6] Newman M E J 2010
Networks (Oxford: Oxford University Press)[7] Pastor-Satorras R, Castellano C, Van Mieghem P and Vespignani A 2015 Epidemic processes in complex networks
Rev.Mod. Phys. Proc.Natl. Acad. Sci.
J. Stat. Mech.
P07043[10] Chu X, Zhang Z, Guan J and Zhou S 2010 Epidemic spreading with nonlinear infectivity in weighted scale-free networks
Physica A
Sci. Rep. Phys. Rev. E Phys. Lett. A
J. Phys. A Austral. J. Statist.
Int. J. Epidemiol. Phys. Rev. Lett. Phys. Rev. E Ann. Prob. Gonorrhea Transmission Dynamics andControl ed S Levin (Berlin: Springer-Verlag) Ch 3[21] Wang Y, Chakrabarti D, Wang C, Faloutsos C 2003 Epidemic Spreading in Real Networks: An Eigenvalue Viewpoint
Proceedings 22nd International Symposium on Reliable Distributed System
Phys. Rev. Lett.
Phys. Rev. Lett.
J. Phys. A J. Phys. A Nat. Phys. J. Theor. Biol.
Phys. Rev. E Statistical Mechanics of Complex Networks ed R Pastor-Satorras, M Rubi and A Diaz-Guilera (Berlin: Springer-Verlag) [30] Newman M E J 2002 Assortative Mixing in Networks Phys. Rev. Lett. Science286