  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 $\mathcal{R}_0$.
