时至今日传染病依旧是危害人类生命安全的一个重大问题，探索传染病的传播机理对其制定控制策略具有重要意义。本文以新发传染病新冠肺炎为例进行研究，利用非线性动力学理论和方法，针对新冠肺炎的传播特点，建立符合实际情况的传染病动力学模型来分析疾病传播机理，预测其传播趋势，并探究有效抑制疾病传播的最优混合控制策略。本文的主要研究内容和创新点如下：

    (1) 考虑到新冠肺炎传播过程中存在疫苗接种者的情况，构建了一种SVEIQR传染病模型。计算了该模型的基本再生数和平衡点，证明了模型的局部渐近稳定性，分析了模型的跨临界分岔现象。为了准确预测疾病的发展趋势，考虑到参数随时间变化的特性，将SVEIQR模型参数修改为时变参数，使用机器学习方法对模型参数进行估计，进一步预测新冠肺炎传播趋势。实验结果表明，该模型的拟合优度R²值为0.9988，说明该模型预测精度较高，可以较为准确预测新冠肺炎传播趋势。

    (2) 考虑到新冠肺炎传播过程中存在病毒变异者的情况，构建了一种SEIMQR传染病模型。计算了该模型的基本再生数和平衡点，证明了模型的局部渐近稳定性和全局渐进稳定性，分析了平衡点的跨临界分岔现象和重要参数的灵敏度。根据英国疫情真实数据集，使用最小二乘法对该模型参数进行分段估计，进一步预测新冠肺炎传播趋势。实验结果表明，英国疫情三个不同阶段的平均相对误差分别为2.24%，2.20%和6.54%，说明该模型预测精度较高，可以为研究新冠肺炎传播趋势提供技术支撑。

    (3) 综合考虑疫苗接种者和变异者群体状态，构建了一种SVEIMQR传染病控制模型。首先，分析了该模型的动力学行为，探究了疫苗接种者和变异者群体共同作用对疫情发展趋势的影响。其次，基于该模型考虑保护策略和阻断策略结合提出一种混合控制策略，并以疫情管控所产生的成本花费和感染者总人数最小化为目标，建立传染病最优控制模型。利用Hamiltion函数和Pontryagin最小值原理分析并求解最优控制变量，最后将最优控制策略与其他控制策略进行仿真对比，验证了最优控制策略的有效性。

    本文的研究成果不仅有助于丰富新冠肺炎等重大传染病科学防控与管理的理论体系，同时也为未来应对新发传染病提供了技术支撑。

    Infectious diseases are still a major problem endangering human life safety, and it is of great significance to explore the transmission mechanism of infectious diseases to formulate their control strategies. This thesis conducts research by taking the novel coronavirus pneumonia (COVID-19) as an example and using nonlinear dynamics theory and approach. According to the transmission characteristics of COVID-19, a practical infectious disease dynamics model has been established to analyze the disease transmission mechanism, predict its transmission trend, and explore the optimal mixed control strategy for effectively inhibiting the spread of the disease. The main research contents and innovations points this thesis are as follows:

    (1) Considering the vaccinated individuals in the process of the COVID-19 pandemic, we construct a type of SVEIQR infectious disease model. We calculate the equilibria and basic reproduction number of the proposed model, proving the local asymptotic stability of the model and simulating the model's transcritical bifurcation phenomenon. To accurately predict the development trend of diseases, taking into account the time-varying characteristics of parameters, the parameters of the SVEIQR model are modified into time-varying parameters. The model parameters are estimated by using machine learning method, and then COVID-19 transmission tendency is predicted. The simulation results show that the value of the goodness-of-fit R² of the model is 0.9988, suggesting that the model has a high prediction accuracy and can accurately predict the trend of the spread of COVID-19.

    (2) Considering the variant individuals in the process of the COVID-19 pandemic, we construct a type of SEIMQR infectious disease model. We calculate equilibria and basic reproduction number, and prove the local asymptotic stability and global asymptotic stability of the proposed model. The transcritical bifurcation of the equilibria and sensitivity of important parameters are analyzed. According to the real data in Britain, the least square method is employed to estimate model parameters, and then COVID-19 transmission tendency is predicted. Simulation results show that the average relative errors of the three different phases in Britain are 2.24%, 2.20% and 6.54%, respectively, indicating that the model has a high prediction accuracy and can provide technical support for exploring the trend of the spread of COVID-19.

    (3) Taking into account both the vaccinated and variant individuals, we construct a type of SVEIMQR infectious disease control model. The dynamics of the proposed model were analyzed to explore the impact of the vaccinated and variant individuals on the trend of epidemic development. On the basis of this model, a mixed control strategy model is proposed that simultaneously includes protective strategies and blocked strategies. The optimal control model is established with the aim of minimizing the economic costs generated by epidemic control and the total number of infected individuals. The optimal control variables are analyzed and solved by using Hamiltion function and Pontryagin's minimum principle, and finally the optimal control strategy is compared with other control strategies in simulation to verify the effectiveness of the optimal control strategy.

     The research findings of this thesis not only contribute to enriching the theoretical system of scientific prevention and control of major infectious diseases such as COVID-19, but also provide technical support for future responses to emerging infectious diseases.