Navigating Epidemic Mathematics: Exploring Tools for Mathematical Modelling in Biology

Abstract: 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.