  This paper focuses on and analyzes realistic SIR models that take
stochasticity into account. The proposed systems are applicable to most
incidence rates that are used in the literature including the bilinear
incidence rate, the Beddington-DeAngelis incidence rate, and a Holling type II
functional response. Given that many diseases can lead to asymptomatic
infections, we look at a system of stochastic differential equations that also
includes a class of hidden state individuals, for which the infection status is
unknown. We assume that the direct observation of the percentage of hidden
state individuals that are infected, $\alpha(t)$, is not given and only a
noise-corrupted observation process is available. Using the nonlinear filtering
techniques in conjunction with an invasion type analysis (or analysis using
Lyapunov exponents from the dynamical system point of view), this paper proves
that the long-term behavior of the disease is governed by a threshold
$\lambda\in \mathbb{R}$ that depends on the model parameters. It turns out that
if $\lambda<0$ the number $I(t)$ of infected individuals converges to zero
exponentially fast, or the extinction happens. In contrast, if $\lambda>0$, the
infection is endemic and the system is permanent. We showcase our results by
applying them in specific illuminating examples. Numerical simulations are also
given to illustrate our results.
