Abstract
We introduce stochasticity into a multigroup SIR model with nonlinear incidence. We prove that when the intensity of white noise is small, the solution of stochastic system converges weakly to a singular measure (i.e., a distribution) ifℛ0≤1and there exists an invariant distribution which is ergodic ifℛ0>1. This is the same situation as the corresponding deterministic case. When the intensity of white noise is large, white noise controls this system. This means that the disease will extinct exponentially regardless of the magnitude ofℛ0.
Highlights
We introduce stochasticity into a multigroup SIR model with nonlinear incidence
Considering different contact patterns, distinct number of sexual partners, or different geography, and so forth, individual hosts are often divided into groups in modeling epidemic diseases
Multigroup models have been proposed in the literature to describe the transmission dynamics of infectious diseases in heterogeneous host populations
Summary
Considering different contact patterns, distinct number of sexual partners, or different geography, and so forth, individual hosts are often divided into groups in modeling epidemic diseases. We will prove that when the intensity of white noise is small, the solution of system (3) converges weakly to a singular measure (i.e., a distribution) if R0 ≤ 1 and there exists an invariant distribution which is ergodic if R0 > 1. This is the same situation as in Proposition 1. Beretta et al [17] considered a stochastic SIR model with time delays and obtained asymptotic mean square stability conditions for positive equilibrium. X(t) ≡ 0 is a solution of (5), called the trivial solution or equilibrium position
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