Dynamics Modeling and Analysis of SIS Epidemic Spreading in Cluster Networks

In this paper, we propose and study an SIS epidemic model with clustering characteristics based on networks. Using the method of the existence of positive equilibrium point, we obtain the formula of the basic reproduction number R0. Furthermore, by constructing Lyapunov function, we also prove that the disease-free equilibrium of the model is globally asymptotically stable when R0 < 1. When R0 > 1, there is only one positive equilibrium point which is globally asymptotically stable. It is also shown that the infection proportion and the basic reproduction number R0 increases as the clustering coefficient increases when the average degree of networks is fixed.


Introduction
A large number of complex systems in nature and human society can be described by complex networks. At present, the structure of complex networks and mathematical models based on networks in the field of biology and other fields has been deeply studied [1][2][3]. In the biological field, the spread of infectious diseases can affect people's physical and mental health. The spread of infectious diseases is not only related to the transmission mechanisms of diseases, but also related to the topology of the complex network. A lot of researches have shown that two topologies of the network can profoundly affect the dynamics of infectious diseases: one is the degree distribution (the number distribution of contact neighbors per individual) [4][5][6]; the other is contact clusters (such as households or school) [7,8]. The effect of degree distribution on the spread of diseases in networks has been widely discussed and fully understood [9,10]. The results of these studies clearly show that the basic reproduction number of diseases tends to infinity when the variance of the degree distribution tends to infinity in scale-free networks. This means that the disease can easily spread in scale-free networks regardless of how fast the epidemic spreads [9].
Clustering in a complex network means that two neighboring nodes of a given node also have a tendency to become neighbors, so a triangle is formed in the network. In the contact network, these triangles called clusters mean that two friends of one person are also friends with each other. The clustering coefficient is an indicator for measuring the level of clustering in the network. The average value of the clustering coefficients of all nodes in the network is called the clustering coefficient of the network [11]. Generally, when the degree distribution is fixed, the number of network clusters (namely, the number of triangles) can significantly increase as the clustering coefficient of the network increases. Network clusters not only affect the network structure, but also affect the dynamics of the disease transmission on the network.
In the past decade, some researchers have used different methods to study the impact of clustering on the spread of epidemics in weak clustering networks. Eames [12] studied the spread of epidemics in random networks when the number of triangles is given. The results show that sufficient clustering can increase the epidemic threshold. However, at the small and moderate levels, clustering appears not to change the final size of epidemics significantly. Miller found that clusters reduced the basic reproduction number and the final scale of the epidemic in weak clustering networks [13,14]. Trapman constructed a random graph when degree distribution and the expected number of triangles were given [15] and studied the effects of the degree distribution and the expected number of triangles on disease transmissions in the network. The results show that clusters reduce the basic reproduction number and the final size of the epidemic. Newman found that, when the average degree of weak clustering network is fixed, the basic reproduction number of diseases increases and the final scale of epidemics decreases as the clustering coefficient increases [11]. In 2011, Volz et al. took advantage of the method of dynamical probability generating function based nodes in weak clustering networks to find that clustering always slows the spread of the epidemic, but simultaneously increasing clustering and the variance of the degree distribution can increase the final infection scale [8].
Li proposed a new SIS model that includes network clusters. It is pointed out that, due to the heterogeneity of infection, clusters always promote the spread of diseases in the network [16].
In this paper, we establish an SIS dynamical model on a class of clustered networks to further study how degree distribution and clustering influence the spread of disease. In Section 2, considering the infectivity heterogeneity of infective nodes located at different sites (at the end of single edge or the edge in the triangle), we derive an SIS dynamical model based on the mean-field method describing the transmission of diseases in networks with arbitrary degree distributions and clustering coefficients. In Section 3, we calculate the reproduction number 0 of diseases and prove the local and global stability of disease-free equilibrium and the endemic equilibrium. In Section 4, the impacts of degree distributions and clusters on disease in the network are analyzed by simulations. The results show that the reproduction number and the relation size of infection individual always increase as the clustering coefficient increases.

Dynamic Modeling
We consider a class of weakly clustered network where there are no common edges between any two triangles. We assume that each node has some lines (or single edges) and triangles in the network. For convenience, we assume the numbers of lines and triangles of every node are independent. In the current clustered network, we consider that each individual exists only in two discrete states: S-susceptible and I-infected. At each time step, each susceptible (healthy) node is infected if it is contacted by one infected individual; at the same time, infected nodes are cured and become again susceptible with rate . Without lack of generality, we can set = 1. It is worth noting that each susceptible node is infected by its infected neighbors which are connected by the line or edge in triangles. If a susceptible node is not infected, then it must be not infected by any of all infected neighbors to which it is connected, neither by lines nor by triangles. Let 1 be the infection probability that a susceptible node is infected by the random edge in triangles and 2 be the infection probability that a susceptible node is infected by a random infected neighbor connected by a line.
Let represents the total number of nodes in the network. ( ) and ( ) ( = 1, 2, . . . ), respectively, represent the number of susceptible and infected persons with degree at time .
represents the total number of nodes with degree and is a constant in the static network. It is obvious that the degree distribution is given by = / . Then, the average degree ⟨ ⟩ is given by Figure 1(a)). If nodes V and V are also connected by an edge, the triple is called a closed triple or triangle (see Figure 1(b)).
We denote △ as the number of triangles around a node of degree k and 3 as the number of connected all triples (open triple and triangle) around a node of degree . The clustering coefficient of nodes with degree is defined by = △ / 3 = 2 △ / ( − 1) [17]. Then, the number of edges connected with the nodes of degree to the nodes in the triangle is ( −1) . The number of lines around a node with degree is − ( −1) . In degree uncorrelated networks, is independent of and so local clustering coefficient ( ) is equal with global clustering coefficient ( ) [17,18]. Obviously, where 1 is the probability that any given edge points to an infected node in the triangles and 2 is the probability that any given link points to an infected node. In degree uncorrelated networks, the dynamical meanfield (MF) reaction rate equations of disease transmission are established: Discrete Dynamics in Nature and Society 3 Due to + = 1, then we can rewrite system (2) as Obviously, in real life, the probability that a susceptible node is infected by an infected node in the family is greater than the probability that a susceptible node is infected by an infected stranger. So we assume that 1 > 2 and 2 = 1 (0 < < 1). (3) can be rewritten aṡ

Analysis of the Feasible Region
Proof. According to system (4),̃( ) satisfies the following equation: where > 0. System (4) can be rewritten as Sincẽ( ) > 0, we have For the above inequality from 0 → integral, we get On the other hand, it can be verified that the function 1− ( ) satisfies the equation For equation (11) from 0 → integral, we get Thus, as > 0, it follows that 0 < ( ) < 1. Further,

Theorem 2.
If the basic reproduction number 0 < 1, system ( ) has a disease-free equilibrium point 0 ; if 0 > 1, system ( ) has a disease-free equilibrium point and a unique endemic equilibrium point * .

Stability of Equilibrium Points
First, we analyse the global stability of the disease-free equilibrium point. Theorem 3. If 0 < 1, the disease-free equilibrium point 0 of system ( ) is globally asymptotically stable within Ω.
Next, the local stability of the endemic equilibrium is analyzed.

(24)
Since matrices and have the same characteristic roots, we only need to consider the eigenvalues of matrix . The characteristic equation of matrix is Obviously, when 0 > 1, the real part of all the eigenvalues of matrix is negative. The eigenvalues of are also the eigenvalues of the Jacobian matrix of system (4) at * , so the endemic equilibrium point * of system (4) is locally asymptotically stable.
Similarly, it can be obtained that By induction, we know that for each , the sequence ( ) is decreasing, so its limit exists and is denoted by lim →∞ ( ) = . Letting → ∞ on both sides of formulas (32) and (39), we deduce that lim →∞ ( ) = satisfies the following stability equation: Then, lim On the other hand, we consider the function By simple calculations, we obtain By the definition of derivatives, if > 0 is sufficiently small, then ( ) > (0) = 0. According to Lemma 5 and formula (44), we can take (1) such that We define the following sequence: By formulas (46) and (48), we obtain If for all , ( +1) > ( ) , it follows from formula (46) that Thus, by induction, we know that, for each , the sequence ( ) , ≥ 2 is increasing, so its limit exists and is denoted by Letting → ∞ on both sides of formulas Discrete Dynamics in Nature and Society 7 (46) and (47), we deduce that the limit = lim →∞ ( ) satisfies the following relations: By formulas (41)   That is, lim →+∞ ( ) = , and Theorem 6 is proved.

Numerical Simulation and Conclusion
In this section, we present some numerical simulations of system (3) in a random network and a scale-free network to study the effect of clustering coefficients on disease transmission. We first consider the degree distribution of the network. In the random network, obeying Poisson distribution is expressed as = − / !, where indicates the average degree of nodes in the network. In the scale-free network, obeying a Power-law distribution is expressed as = 2 2 / 3 , where represents the number of edges generated for each new node introduced in the network.
In Figure 2, we take the total number of nodes in the network = 1000, 1 = 0.22, and 2 = 0.18. Figures 2(a) and 2(b) show that the infection fractions and the basic reproduction number 0 will be increasing with the clustering coefficient in the ER random network and the BA scale-free network. This means that the increase in clustering coefficient can easily cause some nodes to connect with their neighbor's neighbors. So some triangles are formed; the final epidemic size will be increasing. From Figure 2(c), we compare the infection fractions in the ER random network and the BA scale-free network where the average degree is fixed and the clustering coefficient is changed. It is found that the increase in clustering coefficient always promotes the disease spreading.
In conclusion, the network model we presented more accurately depicts the special local relationships between individuals in the contact network. We study the influence of clustering coefficients on the basic reproduction number and the infection fractions in the network. The basic reproduction number can change larger as the clustering coefficient increases. Thus, the disease is more easy to spread. Simulations indicate that the final infection fraction can increase when the clustering coefficient is larger. From the perspective of sociology and biology, the reduction of household or school clusters will effectively impede the disease spreading.

Data Availability
No data were used to support this study. The values of parameters that appeared in the simulations are assumed by us.

Conflicts of Interest
The authors declare that they have no conflicts of interest.