COVID-19 SIR model: Bifurcation analysis and optimal control

This paper proposes a SIR epidemic model with vital dynamics to control or eliminate the spread of the COVID-19 epidemic considering the constant population, saturated treatment, and direct-indirect transmission rate of the model. We demonstrate positivity, boundness and calculate the disease -free equilibrium point and basic reproduction number from the model. We use the Jacobian matrix and the Lyapunov function to analyze the local and global stability, respectively. It is observed that indirect infection increases the basic reproduction number and gives rise to multiple endemic diseases. We perform transcritical, forward, backward, and Hopf bifurcation analyses. We propose two control parameters (Use of face mask, hand sanitizer, social distancing, and vaccination) to minimize the spread of the coronavirus. We use Pontryagin's maximum principle to solve the optimal control problem and demonstrate the results numerically.