  Real epidemic spreading networks often composed of several kinds of networks
interconnected with each other, and the interrelated networks have the
different topologies and epidemic dynamics. Moreover, most human diseases are
derived from animals, and the zoonotic infections always spread on
interconnected networks. In this paper, we consider the epidemic spreading on
one-way circular-coupled network consist of three interconnected subnetworks.
Here, two one-way three-layer circular interactive networks are established by
introducing the heterogeneous mean-field approach method, then we get the basic
reproduction numbers and prove the global stability of the disease-free
equilibrium and endemic equilibrium of the models. Through mathematical
analysis and numerical simulations, it is found that the basic reproduction
numbers $R_0$ of the two models are dependent on the infection rates, infection
periods, average degrees and degree ratios. In the first model, the network
structures of the inner contact patterns have a bigger impact on $R_0$ than
that of the cross contact patterns. Under the same contact pattern, the
internal infection rates have greater influence on $R_0$ than the
cross-infection rates. In the second model, the disease prevails in a
heterogeneous network has a greater impact on $R_0$ than the disease from a
homogeneous network, and the infections among the three subnetworks all play a
important role in the propagation process. Numerical examples verify and expand
these theoretical results very well.
