Hopf bifurcation and other dynamical behaviors for a fourth order differential equation in models of infectious disease

Periodic solutions of Hopf type and other dynamical behaviors for a four-order differential equation which occurs in the model of infections disease are investigated. The extended theorem about the conditions for the existence of Hopf bifurcation is proved in higher-order differential equations with several parameters. The Hopf bifurcation value is given through the medium of the corresponding coordinate at the Hopf bifurcation point, and depends on one parameter. The paper reveals that the model of Holt and Picker has periodic solutions, and proves the reliability of the numerical solution which is given by Liu Winmin.