Impact of vaccination and nonlinear incidence rate on the dynamic of influenza strains epidemic model: Optimal control approach

In this study, we develop and analyze a mathematical model describing the transmission dynamics of influenza strains, including both drug-resistant and non-resistant variants. Our model generalizes several existing epidemic models by incorporating key biological factors. We first prove the existence, uniqueness, positivity, and boundedness of global solutions to ensure the model is well-posed under natural conditions. Using the next-generation matrix method, we calculate two basic reproduction numbers associated with the model. We identify four biologically relevant equilibrium points: the disease-free equilibrium, the resistant endemic equilibrium, the non-resistant endemic equilibrium, and the coexistence endemic equilibrium. The local stability of these equilibrium is established through standard stability analysis techniques. Furthermore, we propose two optimal control strategies, treatment and media awareness campaigns based on Pontryagin’s Maximum Principle to reduce the spread of infection. Numerical simulations performed in Matlab illustrate and support our theoretical results. Finally, the paper concludes with a discussion and suggestions for future research directions.