Bifurcation analysis of an influenza A (H1N1) model with treatment and vaccination.

This research uses numerical simulations and mathematical theories to simulate and analyze the spread of the influenza virus. The existence, uniqueness, positivity, and boundedness of the solution are established. We investigate the fundamental reproduction number guaranteeing the asymptotic stability of equilibrium points that are endemic and disease-free. We also examine the qualitative behavior of the models. Using the Lyapunov method, Routh-Hurwitz, and other criteria, we explore the local and global stability of these states and present our findings graphically. Our research assesses control policies and proposes alternatives, performing bifurcation analyses to establish prevention strategies. We investigate transcritical, Hopf, and backward bifurcations analytically and numerically to demonstrate disease transmission dynamics, which is novel to our study. Contour plots, box plots, and phase portraits highlight key characteristics for controlling epidemics. The disease's persistence depends on its fundamental reproduction quantity. To validate our outcomes, we fit the model to clinical data from influenza cases in Mexico and Colombia (October 1, 2020, to March 31, 2023), aiming to analyze trends, identify critical factors, and forecast influenza trajectories at national levels. Additionally, we assess the efficacy of implemented control policies.