Periodic Incidence in a Discrete-Time SIS Epidemic Model - Mathematical Methods and Models in Biomedicine - page 412

As in [ 8 - 11 ], our model framework allows for oscillatory host population
dynamics. What is ℜ 0 when the host population exhibits oscillatory (non-

equilibrium) dynamics? Can ℜ 0 be used to control the spread of an infectious

disease in a strongly fluctuating host population? What is the relationship between
the host population attractor and the infective population attractor?
In periodic environments, Franke and Yakubu obtained that the demographic
dynamics ( N - dynamics ) does not always drive the disease dynamics ( I - dynamics

)
[ 18 ]. Our extended model results support this prediction. Furthermore, we showed
that it is possible for I - dynamics and S - dynamics to follow the N - dynamics as the
N - dynamics undergoes period doubling bifurcations route to chaos. In this case, our
numerical explorations seem to suggest that the differences between the N - dynamics
and I - dynamics are limited to their variation in “amplitude.” Qualitative proofs of
these results that include the case where individuals (infected and susceptibles) do
not have equal probability of surviving one generation are welcome.

Acknowledgments This research has been partially supported by the National Marine Fisheries
Service, Northeast Fisheries Science Center (Woods Hole, MA 02543), Department of Homeland
Security, DIMACS and CCICADA of Rutgers University, Mathematical Biosciences Institute
of the Ohio State University and National Science Foundation under grants DMS 0931642 and
0832782.

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