This presentation motivates the studies of the existence and stability of Hopf bifurcation at infection-free and endemic states of the SIR and SIRS epidemic models involving bilinear incidence rate. We try to determine the algebraic invariant planes and investigate Hopf bifurcation on the invariant planes. We also analyze the existence and the stability of limit cycles of the SIR and SIRS models with bilinear incidence rate, at the equilibrium point where infection disappears and at the endemic equilibrium point where the system exhibits oscillatory behavior. We illustrate the algebraic results with numerical simulations for parameter values demonstrating distinct states, satisfying the conditions for both models’ infection-free and endemic states.