A dynamic advertising model in a vaccination campaign

Vaccines save thousands of lives every year, but many people remain unvaccinated because serious adverse neurological disorders are wrongly attributed to vaccination. The “urban myth” of a relevant vaccine-associated risk is sustained by anti-vaccination groups and it is spread by word-of-mouth communication. We face the problem of increasing the vaccination coverage using an approach which draws some elements from the theory of dynamic advertising models. We propose a dynamic model for the evolution of the number of unvaccinated people and assume that a policy-maker can control this dynamics through advertising. From a mathematical point of view, we state and analyze an optimal control problem with a pure state constraint. We find the unique optimal solution, which minimizes a cost functional, but may fail to be satisfactory from the different viewpoint of moving towards eradication of the disease. Our analysis suggests that we modify the problem statement in order to consider explicitly the goal of reducing the number of unvaccinated people, to a level which guarantees the herd immunity. Hence we introduce an upper bound to the final number of unvaccinated people. From the solutions to the two problems we obtain some prescriptions for the policy-maker.