Optimal Vaccination Patterns in Age-Structured Populations

Within the framework of homogeneous age-structured SIR-models, optimal vaccination strategies are considered. There are two concepts for a vaccination strategy: the effort (the number of vaccinations per time) and the effect (the reproduction number). There are also two problems: to find the strategy with minimal reproduction number at given costs and to find the strategy with minimal costs at given reproduction number. For a fixed vaccination strategy, the susceptible population tends asymptotically to a certain pattern of susceptibles and infected persons. If a disease invades the vaccinated population, this pattern is crucial for the transmission of disease. Hence the set of susceptible populations is studied instead of the set of vaccination strategies, or equivalently, the set of vaccination strategies is endowed with a metric induced by the susceptible population. The resulting mathematical framework leads to existence results for optimal strategies and insights in their structure.