  Many complex networks exhibit vulnerability to spreading of epidemics, and
such vulnerability relates to the viral strain as well as to the network
characteristics. For instance, the structure of the network plays an important
role in spreading of epidemics. Additionally, properties of previous epidemic
models require prior knowledge of the complex network structure, which means
the models are limited to only well-known network structures. In this paper, we
propose a new epidemiological SIR model based on the continuous time Markov
chain, which is generalized to any type of network. The new model is capable of
evaluating the states of every individual in the network. Through mathematical
analysis, we prove an epidemic threshold exists below which an epidemic does
not propagate in the network. We also show that the new epidemic threshold is
inversely proportional to the spectral radius of the network. In particular, we
employ the new epidemic model as a novel measure to assess the vulnerability of
networks to the spread of epidemics. The new measure considers all possible
effective infection rates that an epidemic might possess. Next, we apply the
measure to correlated networks to evaluate the vulnerability of disassortative
and assortative scalefree networks. Ultimately, we verify the accuracy of the
theoretical epidemic threshold through extensive numerical simulations. Within
the set of tested networks, the numerical results show that disassortative
scale-free networks are more vulnerable to spreading of epidemics than
assortative scale-free networks.
