Abstract
We focus on the node-based epidemic modeling for networks, introduce the propagation medium, and propose a node-based Susceptible-Infected-Recovered-Susceptible (SIRS) epidemic model with infective media. Theoretical investigations show that the endemic equilibrium is globally asymptotically stable. Numerical examples of three typical network structures also verify the theoretical results. Furthermore, comparison between network node degree and its infected percents implies that there is a strong positive correlation between both; namely, the node with bigger degree is infected with more percents. Finally, we discuss the impact of the epidemic spreading rate of media as well as the effective recovered rate on the network average infected state. Theoretical and numerical results show that (1) network average infected percents go up (down) with the increase of the infected rate of media (the effective recovered rate); (2) the infected rate of media has almost no influence on network average infected percents for the fully connected network and NW small-world network; (3) network average infected percents decrease exponentially with the increase of the effective recovered rate, implying that the percents can be controlled at low level by an appropriate large effective recovered rate.
Highlights
With the development of network science, the mathematical modeling of epidemic spreading has involved in a research area across many disciplines including mathematical biology, physics, social science, computer and information science, and so on
The theoretical studies of epidemic spreading models in complex networks rely mostly on the mean-field theory approaches, especially on degree-based mean-field (DBMF) theory which was the first theoretical approach presented for the analysis of general dynamical processes on complex networks [9]
The epidemic spreading model based on DBMF theory depends in general on the statistical topological properties of the underlying networks instead of the whole network structure, resulting into the loss of detailed features of network topologies such that it is difficult to deeply understand the effect of network structures on the disease propagation
Summary
With the development of network science, the mathematical modeling of epidemic spreading has involved in a research area across many disciplines including mathematical biology, physics, social science, computer and information science, and so on. The theoretical studies of epidemic spreading models in complex networks rely mostly on the mean-field theory approaches, especially on degree-based mean-field (DBMF) theory which was the first theoretical approach presented for the analysis of general dynamical processes on complex networks [9]. This approach assumes that all nodes of degree k are statistically equivalent, and any given vertex of degree k is connected with the same probability to any node of degree k.
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