Hybrid stochastic epidemic SIR models with hidden states

Abstract

This paper focuses on realistic hybrid SIR models that take into account stochasticity. 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, this paper looks at a system of stochastic differential equations that also includes a class of hidden state individuals, for which the infection status is unknown. Assuming that the direct observation of the percentage of hidden state individuals being infected, α(t), is not given and only a noise-corrupted observation process is available. Using nonlinear filtering techniques in conjunction with an invasion type analysis, this paper shows that the long-term behavior of the disease is governed by a threshold λ∈R that depends on the model parameters. It turns out that if λ<0 the number I(t) of infected individuals converges to zero exponentially fast (extinction). However, if λ>0, the infection is endemic and the system is persistent. We showcase our theorems by applying them in some illuminating examples.