Turing instability in a network-organized epidemic model with delay.

In this paper, we show the impact of both network and time delays on Turing instability and demarcate the role of diffusion in the epidemic. The stability and bifurcation of equilibrium points are analyzed to reveal the epidemic state, which is the precondition of pattern formation. The network could lead to the transition from the endemic to the periodic outbreak via negative wavenumber, which provides a way to prevent significant harm or decrease the damage of the epidemic to humans by the delay, the connection rate, and the infection rate. Also, the threshold value of time delay is proportional to the minimum eigenvalue of the network matrix, which provides a way to control the periodic behavior. Finally, numerical simulations validate these analytical results and the mechanisms of frequent outbreaks. • Turing instability is investigated through wavenumber. • The transition mechanism of the epidemic from the endemic is showed. • The critical value of delay is proportional to the least eigenvalue of network. [ABSTRACT FROM AUTHOR]