Advertising in a vaccination campaign: A variable time control problem

The best way to prevent a highly contagious infection such as measles is active immunization with a suitable vaccine. Unfortunately, in the last 10 years in EU there has been a decrease in vaccinations, so that their coverage goes under the minimum level to obtain herd immunity. This is a sanitary emergency and it is connected with the “urban myth” that serious adverse neurological disorders should be attributed to vaccination. Anti-vaccination groups spread this “urban myth” by word-of-mouth communication. We model the evolution of the number of unvaccinated people assuming that a policy-maker can control this dynamics through advertising. The idea of connecting dynamic advertising models and vaccination campaign is recent and it represents some aspects of the issue that have never been explored before. From a mathematical point of view we study a variable final time optimal control problem in which the policy-maker wants to minimize the costs in order to obtain herd immunity in due time. We find the optimal solution and we obtain some prescriptions for the policy-maker.