Robust optimal control of a network-based SIVS epidemic model with time delay

In this paper, we investigate robust optimal control of a network-based SIVS epidemic system with an uncertain time delay, which contains time-dependent and degree-dependent vaccination controls. The dynamic behaviours of the delayed system with time-independent controls are discussed. We propose a robust optimal control problem, in which the accumulated density of infected individuals and control costs, and the sensitivity to the uncertain time delay are all involved. Since the objective contains a non-standard sensitivity term, it is difficult to solve this problem using conventional optimization techniques. By introducing an auxiliary initial value system, we transform this problem into a standard form. Furthermore, a numerical scheme which combines the control parametrization scheme with the interior point algorithm is constructed. Finally, numerical simulations with different time delay and weight factor are carried out. We find that, in the case of the basic reproduction number being smaller than one, the optimal control schemes can effectively shorten the disease duration for larger time delays; in face of the disease pandemic, greater control efforts should be made in time at the beginning of disease outbreak, especially for no time delay; in either case, smaller weight factors contribute to minor vibrations of optimal controls and lesser disease survivals. These results can provide a new insight into controlling diseases.