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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

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 [

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 [

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 [

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

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

Let

Open triple and closed triple.

Open triple

Closed triple

We denote

In degree uncorrelated networks, the dynamical mean-field (MF) reaction rate equations of disease transmission are established:

Suppose that the initial relative infected densities

According to system (

System (

The equilibrium point of system (

If the basic reproduction number

First, we analyse the global stability of the disease-free equilibrium point.

If

Next, the local stability of the endemic equilibrium is analyzed.

If

Let

Let us consider the linear systems

Suppose that the initial relative infected densities

Similar to proving the global stability of the disease-free equilibrium, the following equation can be obtained:

Suppose that the initial relative infected densities

In order to prove this theorem, we first prove that the limit

On the other hand, we consider the function

In this section, we present some numerical simulations of system (

In Figure

(a) In the ER network, the infection fractions with clustering coefficients

The infection fractions of ER network

The infection fractions of BA network

Comparison of infection fractions between ER network and BA network

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.

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

The authors declare that they have no conflicts of interest.

This work is supported by National Natural Science Foundation of China under Grant 11701528, 11571324 and Shanxi Province Youth Natural Science Foundation (201601D021015).