Optimal Vaccination Strategies in an SIR Epidemic Model with Time Scales

Abstract

Childhood related diseases such as measles are characterised by short periodic outbreaks lasting about 2 weeks. This means therefore that the timescale at which such diseases operate is much shorter than the time scale of the human population dynamics. We analyse a compartmental model of the SIR type with periodic coefficients and di ff erent time scales for 1 ) disease dynamics and 2 ) human population dynamics. Interest is to determine the optimal vaccination strategy for such diseases. In a model with time scales, Singular Perturbation theory is used to determine stability condition for the disease free state . The stability condition is here referred to as instantaneous stability condition, and implies vaccination is done only when an instantaneous threshold condition is met. We make a comparison of disease control using the instantaneous condition to two other scenarios: one where vaccination is done constantly over time (constant vaccination strategy) and another where vaccination is done when a periodic threshold condition is satis fi ed (orbital stability from Floquet theory). Results show that when time scales of the disease and human population match, we see a di ff erence in the performance of the vaccination strategies and above all, both the two threshold strategies outperform a constant vaccination strategy.