Matstat seminarium fredagen 7 december 2001 kl 13:15 i MH:227

Mikael Andersson, Matematisk statistik, Stockholms universitet:

Almost Reducible Multitype Epidemic Processes


ABSTRACT

When analysing multitype population dynamics models like branching and 
epidemic processes, it is often assumed that the process is irreducible, i.e. 
that the transition matrix describing transitions between types is 
irreducible. For branching processes this means that individuals of any type 
can generate individuals of any other type and for epidemic processes that 
infective individuals of any type may infect susceptible individuals of any 
other type. This assumption is often made in order to facilitate derivations 
of asymptotic results about the extinction probability and the final size 
distribution. For instance, this implies the well known bifurcation 
phenomenon, i.e. that the process either dies out quickly or starts to grow 
exponentially for all types of individuals. Relaxing this assumption to allow 
reducible processes does not present too much additional difficulties as long 
as the exact structure of the transition matrix is considered. Basically, it 
means that some types may die out quickly while others grow. However, it 
turns out that when allowing a sequence of irreducible epidemic processes to 
converge to a reducible limit in a certain way (the almost reducible case), a 
somewhat different kind of asymptotic behaviour emerges, e.g. multiple 
outbreaks.