Stability and Hopf Bifurcations Analysis in a Three‐Phase Dengue Diffusion Model With Time Delay in Fractional Derivative and Laplace–Adomian Decomposition Numerical Approach

ABSTRACT This study examines the complex dynamics of dengue transmission by incorporating time delay into a comprehensive model. The model is designed to capture several essential components, including steady‐state events, immune waning, recuperation from infection, and partial shielding in human populations. To further refine our understanding, we introduce a fractional framework that provides a more precise perspective on the finer dynamics of the model. Through the framework of the stability theory of delayed differential equations, this study closely analyzes the stability of local and disease‐free equilibria. As the fundamental reproducibility number ( ) exceeds unity, the experiment exhibits instability. This provides the basis for delay‐parameterized Hopf bifurcation analysis. The stability conditions for local equilibrium are explained, and reliable numerical simulations confirm the mathematical framework and provide strong support for our conclusions. Additionally, the study uses fixed‐point theory to verify that the model has a unique solution. In addressing the subtleties of fractional‐order differential equations, we use the Laplace–Adomian decomposition method, assigning a unique fractional order ( ) to each segment. The resulting approximate solutions are visualized through graphical depictions, providing valuable insights into the influence of fractional parameters ( ) on the complex dynamics underlying dengue transmission. This research, with its holistic overview and innovative methodology, makes a significant contribution to our understanding of dengue dynamics. The implications of these findings extend to the area of public health strategies, improving our ability to plan interventions and reduce the impact of dengue on vulnerable populations.