一类非线性感染率的HIV空间扩散模型的研究

Study on an HIV Spatial-Diffusion Model with Nonlinear Infection Rate

摘要: 建立一类非线性感染率的HIV空间扩散模型。首先，通过偏微分方程理论分析模型解适定性，进一步地计算基本再生数$R_0$ 和分析地方病平衡点$E^{*}$ 的存在性；其次，采用特征方程方法，分析$E_0$ 和$E^{*}$ 两个平衡点的局部稳定性：当$R_0$ < 1时，系统(1)无病平衡点$E_0$ 是局部渐近稳定的，意味着HIV传播难以持续扩散；当$R_0$ > 1时，系统(1)的地方病平衡点$E^{*}$ 为局部渐近稳定。最后，通过构造Lyapunov泛函数证明系统(1)的两个平衡点$E_0$ 和$E^{*}$ 的全局渐近稳定性，当$R_0$ < 1时，系统(1)的无病平衡点$E_0$ 是全局渐近稳定的，当$R_0$ > 1时，系统(1)的地方病平衡点$E^{*}$ 是全局渐近稳定的。

Abstract: An HIV spatial diffusion model with nonlinear infection rate is established. Firstly, the well-posedness of model solutions is analyzed by partial differential equation theory. The basic reproduction number $R_0$ is calculated, and the existence of the endemic equilibrium $E^{*}$ is investigated. Secondly, the characteristic equation method is adopted to analyze the local stability of the two equilibria $E_0$ and $E^{*}$ . When $R_0$ < 1, the disease-free equilibrium $E_0$ of system (1) is locally asymptotically stable, indicating that HIV transmission cannot spread sustainably. When $R_0$ > 1, the endemic equilibrium $E^{*}$ of system (1) is locally asymptotically stable. Finally, Lyapunov functionals are constructed to prove the global asymptotic stability of $E_0$ and $E^{*}$ for system (1). If $R_0$ < 1, the disease-free equilibrium $E_0$ of system (1) is globally asymptotically stable; if $R_0$ > 1, the endemic equilibrium $E^{*}$ of system (1) is globally asymptotically stable.