Incorporating economic constraints for optimal control of immunizing infections

It is well-known that the interruption of transmission of a disease can be achieved, provided the vaccinated population reaches a threshold depending on, among others, the efficacy of vaccines. The purpose of this paper is to address the optimal vaccination strategy by imposing the economic constraints. In particular, an S--(I,V)--S model used to describe the spreading of the disease in a well-mixed population and a cost function consisting of vaccination and infection costs are proposed. The well-definedness of the above-described modeling is provided. We were then able to provide an optimal strategy to minimize the cost for all parameters. In particular, the optimal vaccination level to minimize the cost can be completely characterized for all parameters. For instance, the optimal vaccination level can be classified by the magnitude of the failure rate of the vaccine with other parameters being given. Under these circumstances, the optimal strategy to minimize the cost is roughly to eliminate the disease locally (respectively, choose an economic optimum resulting in not to wipe out the disease completely or take no vaccination for anyone) provided the vaccine failure rate is relatively small (respectively, intermediate or large). Numerical simulations to illustrate our main results are also provided. Moreover, the data collected at the height of the Covid-19 pandemic in Taiwan are also numerically simulated to provide the corresponding optimal vaccination strategy.