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We are concerned in this paper with dynamic behaviors and an optimal control problem of a new delayed epidemic model. There are three major ingredients. The first one is the dynamic behaviors of the state system. The locally asymptotic stability of the disease-free equilibrium and the endemic equilibrium are investigated and the effect of time delay on stability is also discussed. It is also found that the Hopf bifurcation appears at a specific time delay. The second, which is the main new ingredient of this paper, is an optimal control problem. Applying vaccine strategy in the system, an optimal control problem is proposed to minimize the total number of infected individuals as much as possible, maximize the total number of the uninfected individuals, and reduce the total control cost. In view of Pontryagin's maximum principle, the specific characteristics of the optimal control policy are given. The third ingredient is the numerical simulations of the theoretical results. • The model considers time delay, competition and spread of disease. • It studies dynamics and an optimal control problem of a delayed model. • Examples and numerical simulations are given to verify the theoretical results. [ABSTRACT FROM AUTHOR]

In this paper, we present a complete parametric analysis on a nonlinear predator–prey system with the generalized Holling type III functional response and a strong Allee effect. We apply the hierarchical parametric analysis to derive explicit conditions for the existence and stability of equilibrium solutions in a 5-dimensional parameter space. Specifically, a detailed study on Hopf bifurcation is given to show bifurcation of multiple limit cycles, and complex dynamics of multistable states including bistable, tristable and tetrastable phenomena. We also conduct simulations and provide a biological interpretation of the multistable states under various conditions. • A predator–prey generalized Holling type III functional response and a strong Allee effect is investigated. • The hierarchical parametric analysis is applied to study the existence and stability of equilibrium solutions in a 5-dimensional parameter space. • A detailed study on Hopf bifurcation is given to show bifurcation of multiple limit cycles. • Complex dynamics of multistable states including bistable, tristable and tetrastable phenomena are explored. [ABSTRACT FROM AUTHOR]