| dc.description.abstract | Stochastic epidemic models are useful in modelling the duration of epidemic
outbreaks. It has been observed that the behaviour of the extinction
time of epidemics changes across some point (or domain in
multi-dimensional spaces) in the parameter space, known as the `criticality':
generally speaking, epidemics in the subcritical regime tend to
end quickly, whereas epidemics in the supercritical regime tend to prevail
around the quasi-stationary state for a long time before extinction. In
recent years, there has been substantial interest in the phase transition
window around the criticality, called the `critical regime'. We expect
to observe the critical behaviour not only at the criticality point, but
across the entire critical regime, and the boundary of the critical regime
is expected to be approaching the criticality as the population size tends
to in nity. However, while this phenomenon is well-discussed for onedimensional
epidemic models like SIS, there is little work done on two
or higher-dimensional models.
This thesis is concerned with the scaling behaviour in and around the
phase transition window of the extinction time of a class of two-dimensional
stochastic epidemic models named SIRS. The stochastic SIRS model is
a continuous-time Markov chain modelling the spread of infectious diseases
with temporary immunity, in a homogeneously-mixing population
of xed size N. More speci cally, we study the asymptotic distributions
of the extinction time of SIRS models as N tends to in nity, with
both the parameter space and the initial state of the model treated as
functions of N. Our results provide a comprehensive picture of various
possible scalings and the corresponding limit distributions within the
subcritical and the critical regimes. Our approach also provides us with
descriptions of the entire trajectory of SIRS epidemics. Simulations are
implemented to verify our results. | en_US |
