一类具有广义饱和发生率与饱和治疗函数的随机SIR模型动力学分析

Dynamic Analysis of a Stochastic SIR Model with Generalized Saturated Incidence Rate and Saturated Treatment Function

摘要: 本文建立并研究了一类具有广义饱和发病率与饱和治疗函数的随机SIR传染病模型，其中饱和 发病率采用形式$$g(S,I)=\frac{\beta SI}{1+{\alpha }_{1}S+{\alpha }_{2}I}$$ ，以更全面地描述疾病传播过程中的心理抑制效应与接触饱BSI和现象，我们证明了随机SIR模型存在唯一的全局正解，并分析了其平衡点的稳定性. 通过构造Lyapunov函数，我们得到了疾病灭绝(当R₀ ≤1)与持续存在(当R₀ >1)的充分条件.进一步，我们探讨了白噪声对系统动态的影响，通过数值模拟验证了理论结果的有效性，本研究为随机环境下的传染病控制提供了理论依据与数值支持。

Abstract: In this paper, we establish and investigate a class of stochastic SIR epidemic models with a generalized saturated incidence rate and a saturated treatment function. The saturated incidence rate takes the form $$g(S,I)=\frac{\beta SI}{1+{\alpha }_{1}S+{\alpha }_{2}I}$$, which provides a more comprehensive description of psychological inhibition effects and contact saturation during disease transmission. We prove the existence and uniqueness of a global positive solution to the stochastic SIR model and analyze the stability of its equilibrium points. By constructing Lyapunov functions, we derive sufficient conditions for disease extinction (when R₀ ≤ 1) and persistence (when R₀ > 1) .Furthermore, we explore the impact of white noise on the system dynamics and validate the theoretical results through numerical simulations. This study provides theoretical foundations and numerical support for epidemic control in stochastic environments.