Mathematical Problems in Engineering

Volume 2015 (2015), Article ID 316092, 12 pages

http://dx.doi.org/10.1155/2015/316092

## Epidemic Spreading Characteristics and Immunity Measures Based on Complex Network with Contact Strength and Community Structure

^{1}College of Information System and Management, National University of Defense Technology, Changsha, Hunan 410073, China^{2}College of Electronic Science and Engineering, National University of Defense Technology, Changsha, Hunan 410073, China

Received 3 July 2015; Accepted 22 October 2015

Academic Editor: Xiaobo Qu

Copyright © 2015 Xueting Zhang et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

#### Abstract

Middle East Respiratory Syndrome (MERS), bursting in South Korea from May 2015 and mainly spreading within the hospitals at the beginning, has caused a large scale of public panic. Aiming at this kind of epidemic spreading swiftly by intimate contact within community structure, we first established a spreading model based on contact strength and SI model, and a weighted network with community structure based on BBV network model. Meanwhile, the sufficient conditions were deduced to ensure the optimal community division. Next, after the verification by the real data of MERS, it is found that the spreading rate is closely related to the average weight of network but not the number of communities. Then, as the further study shows, the final infection proportion declines with the decreases both in isolation delay and in average weight; however, this proportion can only be postponed rather than decreased with respect to sole average weight reduction without isolation. Finally, the opportunities to take action can be found to restrain the epidemic spreading to the most extent.

#### 1. Introduction

Middle East Respiratory Syndrome (MERS), first identified in Saudi Arabia in 2012, is a viral respiratory disease caused by a novel coronavirus. According to the World Health Organization (WHO), on May 14, 2013, there were 38 MERS cases, which grew further to 1150 after two years (i.e., May 31, 2015). For human-to-human transmission, the virus does not appear to pass easily from person to person unless they have close contact, such as providing unprotected care to or living together with infected patients.

The epidemic status in South Korea from May 20, 2015, caused national public panic and worldwide attention. At the beginning, it spread evidently amongst the infectors and the patients in the same sickroom. This paper thus focuses on the spreading characteristics of intimate contact with community structure.

Nowadays, the studies of epidemics spreading are twofold: the spreading model of differential equation and the complex network theory. There are three spreading models widely used in the study of virus transmission, namely, SIR model, SIS model, and SI model [1–4], and the solving algorithms are mainly based on percolation theory [5, 6], mean field theory [7, 8], and Markov chain theory [9, 10]. For the complex network, Erdös [11] proposed the random network, while Watts and Strogatz [12] presented the small world network model with smaller average shortest path length and bigger clustering coefficient. Barabási and Albert [13] put forward the scale-free network model with both adding points and preferential attachment. According to the real network, the connections in many networks are not merely binary entities (i.e., either present or not) but have associated weights that record their strengths relative to one another. Thus, Barrat et al. [14] created the BBV model where point weight and edge weight evolve dramatically.

With the further study on the network topology, it is widely recognized that closely connected nodes and communities in social networks play an important role in topological properties and functional dynamics of involved complex networks [15, 16]. As a result, there are many community division algorithms and accuracy indices such as modularity [17–20]. Based on the aforementioned models, Liu and Hu [21] researched on the epidemic spreading through the network with small world effect by SIS model, while Smieszek et al. [22] studied that by SIR model. Salathé and Jones [23] focused on the influence of community structure to virus transmission. In order to know about the virus transmission mechanism better, the dynamic models endeavor to slow down the outbreak rate and control the spreading range. Many different immunization strategies are proposed, such as random immunization [1], target immunization [8], and acquaintance immunization [24].

SI model is often applied to study on the epidemic dynamics at the early outbreak stages [25]. At the beginning, MERS in South Korea were mainly concentrated in three hospitals with obvious characteristics of community structure. In reality, the infection probability increases with the raise of contact time between infectious people and susceptible people, which must be highlighted in the epidemic spreading models. Edge weight is essential to describe the contact intimacy. In order to study the epidemic spreading characteristics and the optimal opportunity to take measures of MERS in South Korea, it is supposed that there is a linear relationship between contact strength and contact time. In this paper, spreading model is based on SI model with contact strength, and weighted network with community structure is based on BBV network model. The spreading characteristics are obtained by simulation according to the aforementioned models. Hence, they are verified by the real data in the South Korea MERS epidemic.

The remainder of this paper is organized as follows. Section 2 demonstrates the spreading model based on SI model in view of contact strength. Section 3 establishes the weighted network with community structure based on BBV model and analyzes the characteristics. Then, the spreading characteristics are studied to find the effective factors in Section 4, where the epidemic spreading process is divided into five stages. Section 5 studies the controlling measures (such as how to execute isolation and reduce the average weight of network) and the optimal opportunity to carry them out. Finally, case study based on MERS in South Korea is demonstrated in Section 6 to verify the models and the measure effects. Overall, both theoretical analysis and simulation results focus on the spreading characteristics and measure effects.

#### 2. Epidemic Spreading Model

In this section, we suppose that the longer time the susceptible person contacts with the infectious ones, the larger probability the susceptible person will get infectious, at the beginning stage of the epidemic outbreak. In order to study the epidemic spreading process, we propose the spreading model based on contact strength and the computer simulation flowchart.

##### 2.1. SI Model Based on Contact Strength

During the study of epidemic diffusion theory, the models are always based on some assumptions that the infectious unit is the node in the network and the epidemic can only spread through the links. The individuals are divided into 3 types: (Susceptible) means the healthy state which is likely to be infectious, (Infected) indicates the illness state which has already been infectious, and (Removed) signifies the immune state which has always been recovered or dead.

In terms of some epidemic bursting suddenly without valid control, such as SARS, H1N1 [26], and especially MERS, SI model is often used to study the spreading characteristics at the beginning period of diffusion process. Overall, prompt prevention measures would reduce the detrimental influence, which are of theoretical value and reality significance.

When the contact strength to an infectious person is , the noninfectious probability of susceptible person is defined as

Thus, when the contact strength to an infectious person is , the infectious probability of susceptible person is defined as

One susceptible person contacts an infectious person with strength, then leaves for a while, and then contact the same infectious person with strength. If he does not get infection at the first time, he would seem as a healthy susceptible person at the second moment. That is, the two contacts are independent, and the noninfectious probability of susceptible person after second contact is

If one susceptible point contacts two infectious points with strengths and , respectively, the noninfectious probability of susceptible person after two contacts is

For susceptible point , the infection probability iswhere indicates the edge strength between point and infectious point and means the sum of edge strength between point and its adjacent infectious points.

is defined as the average of each point in the whole network.

When the whole network is stable, the ratio of susceptible points is , and the ratio of infectious points is , then

In the actual infection process, the longer time the susceptible person contacts the infectious person, the larger probability of susceptible person to get infection. In other words, the edge strength between them is magnified. In conclusion, is a formula that depends on time .

For simplicity, it is assumed that there is a linear relationship between and :where indicates the coefficient of contact strength, , means the average , and signifies the average point degree of network. Thus,where .

The solution iswhere is the ratio of infectious points at time 0, and when , .

##### 2.2. Simulation of Spreading Model

According to the above spreading model, the simulation flowchart is shown in Figure 1.