A Dynamics and Control Study of the New H1N1 Influenza with Two Roots of Infection: The Impact of Optimal Vaccination and Treatment

H1N1 influenza, also known as swine flu, is a subtype of the influenza A virus that can infect humans, pigs, and birds. Sensitivity analysis and optimal control studies play a crucial role in understanding the dynamics of H1N1 influenza. In this study, we have derived a mathematical model incorporating both symptomatic and asymptomatic infections, as well as vaccination, to assess the impact of key parameters on disease transmission. Also, we have assumed a density-dependent infection transmission in the modeling process of H1N1 dynamics. We determine the basic reproduction number using the next-generation matrix method and found that the disease-free equilibrium is stable when the basic reproduction number $R_0<1$ and the endemic equilibrium exists and is stable globally when $R_0>1$. By performing sensitivity analysis, the most influential factors affecting infection spread are identified, aiding in targeted intervention strategies. Optimal control techniques are then applied to determine the best approaches to minimize infections while considering resource constraints. The findings provide valuable insights for public health policies, offering effective strategies for mitigating H1N1 outbreaks and enhancing disease management efforts using optimal vaccination.