With the prevalence of COVID-19, the modeling of epidemic propagation and its analyses have played a significant role in controlling epidemics. However, individual behaviors, in particular the self-protection and migration, which have a strong influence on epidemic propagation, were always neglected in previous studies. In this paper, we mainly propose two models from the individual and population perspectives. In the first individual model, we introduce the individual protection degree that effectively suppresses the epidemic level as a stochastic variable to the SIRS model. In the alternative population model, an open Markov queueing network is constructed to investigate the individual number of each epidemic state, and we present an evolving population network via the migration of people. Besides, stochastic methods are applied to analyze both models. In various simulations, the infected probability, the number of individuals in each state and its limited distribution are demonstrated.

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Epidemic spreading processes on dynamic multiplex networks provide a more accurate description of natural spreading processes than those on single layered networks. To describe the influence of different individuals in the awareness layer on epidemic spreading, we propose a two-layer network-based epidemic spreading model, including some individuals who neglect the epidemic, and we explore how individuals with different properties in the awareness layer will affect the spread of epidemics. The two-layer network model is divided into an information transmission layer and a disease spreading layer. Each node in the layer represents an individual with different connections in different layers. Individuals with awareness will be infected with a lower probability compared to unaware individuals, which corresponds to the various epidemic prevention measures in real life. We adopt the micro-Markov chain approach to analytically derive the threshold for the proposed epidemic model, which demonstrates that the awareness layer affects the threshold of disease spreading. We then explore how individuals with different properties would affect the disease spreading process through extensive Monte Carlo numerical simulations. We find that individuals with high centrality in the awareness layer would significantly inhibit the transmission of infectious diseases. Additionally, we propose conjectures and explanations for the approximately linear effect of individuals with low centrality in the awareness layer on the number of infected individuals.

With the outbreak of COVID-19, great loss and damage were brought to human society, making the study of epidemic spreading become a significant topic nowadays. To analyze the spread of infectious diseases among different areas, e.g., communities, cities, or countries, we construct a network, based on the epidemic model and the network coupling, whose nodes denote areas, and edges represent population migrations between two areas. Each node follows its dynamic, which describes an epidemic spreading among individuals in an area, and the node also interacts with other nodes, which indicates the spreading among different areas. By giving mathematical proof, we deduce that our model has a stable solution despite the network structure. We propose the peak infected ratio (PIR) as a property of infectious diseases in a certain area, which is not independent of the network structure. We find that increasing the population mobility or the disease infectiousness both cause higher peak infected population all over different by simulation. Furthermore, we apply our model to real-world data on COVID-19 and after properly adjusting the parameters of our model, the distribution of the peak infection ratio in different areas can be well fitted.

Ever since the Barabási-Albert (BA) scale-free network has been proposed, network modeling has been studied intensively in light of the network growth and the preferential attachment (PA). However, numerous real systems are featured with a dynamic evolution including network reduction in addition to network growth. In this paper, we propose a novel mechanism for evolving networks from the perspective of vertex degree. We construct a queueing system to describe the increase and decrease of vertex degree, which drives the network evolution. In our mechanism, the degree increase rate is regarded as a function positively correlated to the degree of a vertex, ensuring the preferential attachment in a new way. Degree distributions are investigated under two expressions of the degree increase rate, one of which manifests a "long tail", and another one varies with different values of parameters. In simulations, we compare our theoretical distributions with simulation results and also apply them to real networks, which presents the validity and applicability of our model.