标题： 导航流行病学数学：探索生物学中数学建模的工具 显示英文标题

标题： Navigating Epidemic Mathematics: Exploring Tools for Mathematical Modelling in Biology

摘要： 疾病研究这个不断变化的世界严重依赖于数学模型。它们在发现和控制传染病方面起着关键作用。我们旨在探索用于研究生物学中疾病传播的这些数学工具。SEIR 模型是我们关注的重点。它是一种以灵活性和实用性著称的重要工具。我们研究了修改后的 SEIR 模型的设计与分析。我们深入探讨了构成修改后 SEIR 模型的核心部分，例如使其运作的方程、设定参数身份，并检查其解的正性和极限。 本研究从详细考察修改后的 SEIR 模型的设计与分析开始，展示了其角度。我们深入研究模型的核心，处理驱动修改后 SEIR 模型的方程、建立参数身份以及确保其解的正性和有界性等关键问题。 基本再生数标志着一个重要里程碑。我们研究了局部稳定性、DFE 和 EE。通过使用李雅普诺夫稳定性定理，对全局稳定性进行了仔细检查，这是理解系统长期行为的关键考虑因素。分岔分析分类并阐明了其中的基本概念。一维分岔和前向及后向分岔分析被细致地研究，提供了对动态行为和基本概念的全面理解。 总之，我们提供了对 SEIR 模型的透彻描述和分析，也为流行病学中的数学建模奠定了基础。通过将理论见解与实际应用相结合，本研究努力使研究人员和政策制定者深入了解传染病动力学，从而为有针对性的公共卫生策略做出贡献。 显示英文摘要

摘要： The ever-changing world of disease study heavily relies on mathematical models. They are key in finding and controlling infectious diseases. We aim to explore these mathematical tools used for studying disease spread in biology. The SEIR model holds our focus. It is a super important tool known for being flexible and useful. We look at the modified SEIR models' design and analysis. We dive right into vital parts like the equations that make the modified SEIR model work, setting parameter identities, and then checking its solutions' positivity and limits. The study begins with a detailed examination of the design and analysis of a modified SEIR model, demonstrating its angularity. We delve into the model's heart, dealing with critical issues such as the equations that drive the modified SEIR model, establishing parameter identities, and ensuring the positivity and boundlessness of its solutions. Basic Reproduction Number marks a significant milestone. We investigate the local stability, DFE, and EE. Global stability, a paramount consideration in understanding the long-term behaviors of the systems, is scrutinized by employing the Lyapunov stability theorem. The bifurcation analysis classifies and elucidates the fundamental concepts therein. One-dimensional bifurcation and forward and backward bifurcation analyses are intricately examined, providing a comprehensive understanding of the dynamical behavior and basic concepts. In summary, we offer a thorough description and analysis of the SEIR model but also lay the groundwork for advancing mathematical modeling in epidemiology. By bridging theoretical insights with practical implications, this study strives to empower researchers and policymakers with a deep understanding of infectious disease dynamics, thereby contributing to targeted public health strategies.