  The extinction and persistence of infective individuals are closely related
to the random change of the environment. In this paper, via the
random/stochastic SIRS models, we analyze qualitatively and quantitatively the
impact caused by the random change of the environment. Our contributions
consist in (i) giving some sufficient conditions on extinction (persistence) of
the infectious individuals even though they are persistent (resp. extinct) in
certain fixed environments; (ii) revealing the influence of random switching of
incidence functions on extinction for the infectious individuals, which has not
been studied before; (iii) establishing a criterion to judge extinction of the
infectious individuals for a range of random/stochastic SIRS models with
state-dependent switching via a stochastic comparison for functionals of jump
processes. Moreover, some examples are set to illustrate the applications of
our theory.
