The analysis of a fractional network-based epidemic model with saturated treatment function and fuzzy transmission

For understanding the influence of malware attacking on complex heterogeneous networks, this work studies a fractional network-based SIRS epidemic model with fuzzy transmission and saturated treatment function. Firstly, we apply the next-generation method to obtain the basic reproductive ratio \mathcal{R}_۰, that is an important threshold value in the investigation of asymptotic behavior of the proposed epidemic model. The obtained theoretical results indicates that the value \mathcal{R}_۰ significantly depends on the topology structure of the underlying network and the malware load. In addition, we give a threshold value \tilde{\mathcal{R}}_۰>\mathcal{R}_۰ that not only determines the existence of endemic equilibrium  \mathbf{E}_\ast but also ensures the clean of malware programs on the network. At last, the sensitivity analysis of the threshold value \mathcal{R}_۰ and some graphical simulations are presented to illustrate for the theoretical results.