Stochastic epidemic SIR models with hidden states

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.