TITLE:
Optimal Vaccination Strategies in an SIR Epidemic Model with Time Scales
AUTHORS:
Onyango Nelson Owuor, Müller Johannes, Moindi Stephen Kibet
KEYWORDS:
Singular Perturbation Theory; Optimization; Vaccination Strategies
JOURNAL NAME:
Applied Mathematics,
Vol.4 No.10B,
October
4,
2013
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
different 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 satisfied (orbital stability
from Floquet theory). Results show that when time scales of the
disease and human population match, we see a difference in the
performance of the vaccination strategies and above all, both the two threshold
strategies outperform a constant vaccination strategy.