Stability analysis and optimal vaccination approach for a COVID-19 model with memory effects

Abstract

In this article, we have used fractional differential equations to better describe the dynamics of the COVID-19 virus. First, we have presented a local stability study, which revealed that in the case of the basic reproduction rate $R_0\leq 1,$ the disease-free equilibrium point is locally asymptotically stable. In the other case, the endemic equilibrium point is locally asymptotically stable. Second, a better vaccination strategy to limit the spread of the disease was found for a fractional order derivative. In this way, it is possible to reduce the number of infected and susceptible people and maximize the number of recovered people. We have obtained numerical simulations using Pontryagin's maximum principle. They show how the optimal adoption of available control measures can reduce the infected population. We have observed that the objective function of our optimal control problem decreases under the effects of vaccination and for different values of the order of the derivative.
In addition, we found a control strategy for a non-integer order derivative to be more efficient than the control strategy that uses an integer order derivative.