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.

Periodicity and stationary distribution of two novel stochastic epidemic models with infectivity in the latent period and household quarantine

Two types of stochastic epidemic models are formulated, in which both infectivity in the latent period and household quarantine on the susceptible are incorporated. With the help of Lyapunov functions and Has'minskii's theory, we derive that, for the nonautonomous periodic version with white noises, it owns a positive periodic solution. For the other version with white and telephone noises, we construct stochastic Lyapunov function with regime switching to present easily verifiable sufficient criteria for the existence of ergodic stationary distribution. Also, we introduce a series of numerical simulations to support our analytical findings. At last, a brief discussion of our theoretical results shows that the stochastic perturbations and household quarantine measures can significantly affect both periodicity and stationary distribution.