Analytical and numerical investigation of influenza transmission with vaccination and treatment: stability, sensitivity and bifurcation perspective

Abstract

The impact of parameters on influenza dynamics deeper insights to understand and control the spread of the disease. The present study examines the impact of parameters on the transmission dynamics of the influenza virus through a nonlinear SVITR (Susceptible–Vaccinated–Infected–Treated–Recovered) model. The Basic Reproduction Number R₀ is studied using Next Generation Metrix. Furthermore, the model exhibits transcritical bifurcation, where stability shifts between equilibria as critical parameter thresholds, particularly the transmission rate, are crossed. Numerical simulations validate the theoretical results, demonstrating the sensitivity of disease dynamics to variations in the model and identifying key parameters that significantly influence disease spread. Based on different parameter values, stability analysis is performed for disease-free and endemic equilibria. The model's positivity, boundedness, and uniqueness are also established to ensure biological feasibility. These findings offer valuable insights for designing effective control strategies against influenza outbreaks.