Dynamic behaviors and optimal control of a new delayed epidemic model

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.