Stability and Bifurcation Analysis on a Fractional Model of Disease Spreading with Different Time Delays

This paper investigates the problem of stabilization and Hopf bifurcation for a fractional order disease spreading model simulated by small-world networks via two types of time delays, that is, the linear and nonlinear time delays. It is worth mentioning that the study of making delay parameter pairs as the bifurcation parameter is unprecedented. Firstly, we build a fractional disease transmission network model based on the Caputo fractional derivative, and fix one type of time delay and use another one as the bifurcation parameter to get criterions of stability and Hopf bifurcation. Then, numerical fitting obtains clear stable region and bifurcation boundary curve, and Hopf bifurcation occurs at the equilibrium when select time delay parameter pairs are in the area through the bifurcation boundary curve. Finally, some numerical examples verify the effectiveness of theories. In addition, simulation examples of regular lattices show that the new model covers the extreme conditions of regula and random networks, and it presents more flexible internal nonlinear interactions than previous models.