DOI: 10.62796/pijst.2024v1i8002

  DOI URL: https://doi.org/10.62796/pijst.2024v1i8002

Abstract- Epidemiological models play a crucial role in understanding the spread and control of infectious diseases. By employing mathematical techniques, such as stability and bifurcation analysis, researchers can gain insights into the dynamics of these models, predicting disease outbreaks and evaluating intervention strategies. This paper delves into the mathematical underpinnings of epidemiological models, focusing on classical frameworks like the SIR and SEIR models. Explore the significance of stability analysis in identifying equilibrium states and determining the conditions under which these states are stable or unstable. Bifurcation theory is also examined, highlighting its role in uncovering critical parameter thresholds where qualitative changes in disease dynamics occur. Through both analytical and numerical methods, analyze how variations in model parameters, such as transmission and recovery rates, can lead to different stability scenarios and bifurcation phenomena, such as saddle-node and Hopf bifurcations. Real-world case studies, including those from recent pandemics, are presented to demonstrate the practical applications of these mathematical techniques in predicting disease behavior and informing public health policies. Additionally, the paper discusses the challenges associated with applying these techniques to complex, nonlinear models, and suggests future directions for enhancing the robustness of epidemiological modeling.