Bifurcation Analysis of an Influenza A (H1N1) Model with Treatment and Vaccination

This study focuses on the modeling, mathematical analysis, developing theories, and numerical simulation of Influenza virus transmission. We have proved the existence, uniqueness, positivity, and boundedness of the solutions. Also, investigate the qualitative behavior of the models and find the basic reproduction number $(\mathcal{R}_0)$ that guarantees the asymptotic stability of the disease-free and endemic equilibrium points. The local and global asymptotic stability of the disease free state and endemic equilibrium of the system is analyzed with the Lyapunov method, Routh-Hurwitz, and other criteria and presented graphically. This study helps to investigate the effectiveness of control policy and makes suggestions for alternative control policies. Bifurcation analyses are carried out to determine prevention strategies. Transcritical, Hopf, and backward bifurcation analyses are displayed analytically and numerically to show the dynamics of disease transmission in different cases. Moreover, analysis of contour plot, box plot, relative biases, phase portraits are presented to show the influential parameters to curtail the disease outbreak. We are interested in finding the nature of $\mathcal{R}_0$, which determines whether the disease dies out or persists in the population. The findings indicate that the dynamics of the model are determined by the threshold parameter $\mathcal{R}_0$.