Turing pattern induced by the directed ER network and delay.

Infectious diseases generally spread along with the asymmetry of social network propagation because the asymmetry of urban development and the prevention strategies often affect the direction of the movement. But the spreading mechanism of the epidemic remains to explore in the directed network. In this paper, the main effect of the directed network and delay on the dynamic behaviors of the epidemic is investigated. The algebraic expressions of Turing instability are given to show the role of the directed network in the spread of the epidemic, which overcomes the drawback that undirected networks cannot lead to the outbreaks of infectious diseases. Then, Hopf bifurcation is analyzed to illustrate the dynamic mechanism of the periodic outbreak, which is consistent with the transmission of COVID-19. Also, the discrepancy ratio between the imported and the exported is proposed to explain the importance of quarantine policies and the spread mechanism. Finally, the theoretical results are verified by numerical simulation.

Turing instability in the fractional-order system with random network

The epidemic often spreads along social networks and shows the effect of memorability on the outbreak. But the dynamic mechanism remains to be illustrated in the fractional-order epidemic system with a network. In this paper, Turing instability induced by the network and the memorability of the epidemic are investigated in a fractional-order epidemic model. A method is proposed to analyze the stability of the fractional-order model with a network through the Laplace transform. Meanwhile, the conditions of Turing instability and Hopf bifurcation are obtained to discuss the role of fractional order in the pattern selection and the Hopf bifurcation point. These results prove the fractional-order epidemic model may describe dynamical behavior more accurately than the integer epidemic model, which provides the bridge between Turing instability and the outbreak of infectious diseases. Also, the early warning area is discussed, which can be treated as a controlled area to avoid the spread of infectious diseases. Finally, the numerical simulation of the fractional-order system verifies the academic results is qualitatively consistent with the instances of COVID-19.

Pattern mechanism in stochastic SIR networks with ER connectivity.

The diffusion of the susceptible and infected is a vital factor in spreading infectious diseases. However, the previous SIR networks cannot explain the dynamical mechanism of periodic behavior and endemic diseases. Here, we incorporate the diffusion and network effect into the SIR model and describes the mechanism of periodic behavior and endemic diseases through wavenumber and saddle-node bifurcation. We also introduce the standard network structured entropy (NSE) and demonstrate diffusion effect could induce the saddle-node bifurcation and Turing instability. Then we reveal the mechanism of the periodic outbreak and endemic diseases by the mean-field method. We provide the Turing instability condition through wavenumber in this network-organized SIR model. In the end, the data from COVID-19 authenticated the theoretical results.