双斑块耐药结核病动力学模型的稳定性分析

Stability Analysis of Drug-Resistant Tuberculosis Model in a Two-Patch Environment

摘要: 为了研究人口迁移对耐药结核病传播造成的影响，建立了一个含有迁移的耐药结核病双斑块动力学模型。首先，利用下一代矩阵的方法得到了基本再生数；其次，证明了无病平衡点的全局渐近稳定性，并通过构造李雅普诺夫函数证明了地方病平衡点的全局渐近稳定性；最后，对模型进行了基本再生数的敏感性分析和数值模拟。模拟结果表明，迁移率和耐药性转化率的增大均会导致耐药结核病人数的增多，且迁移率相对于耐药性转化率来说对耐药结核病人数的影响更明显。因此，在合理控制迁移率的基础上降低药物敏感结核病向耐药结核病的转化率，能更有效控制结核病的传播。

Abstract: A two-patch drug-resistant tuberculosis model is formulated to investigate the impact of population migration on the spread of drug-resistant tuberculosis. Firstly, by using the next generation matrix method, the basic reproduction number was obtained. Secondly, the global asymptotic stability of disease-free equilibrium point is proved, by using the theory of stability of differential equations. And the global asymptotic stability of endemic equilibrium point by constructing the Lyapunov function is proved. Finally, a sensitivity analysis of the basic reproduction number and numerical simulation of the model. The numerical simulation shows that the increase in migration rate and drug-resistance conversion rate will lead to an increase in the number of patients with resistance to resistance. The increase in migration rate and drug resistance at the same time will have a negative impact on the number of patients with drug-resistant tuberculosis. Therefore, to reduce the conversion rate of Drug-Sensitive tuberculosis to Drug-Resistant tuberculosis on the basis of reasonable control of migration, which can more effectively control the spread of tuberculosis.