Backward bifurcation for pulse vaccination

We investigate an SIVS model with pulse vaccination strategy. First we compute the disease free periodic solution and prove its global asymptotic stability in the disease free subspace. We identify the corresponding control reproduction number RcRc and prove that the disease free periodic solution is locally asymptotically stable if Rc<1Rc<1, and under some additional conditions it is globally asymptotically stable as well. For Rc>1Rc>1 we prove the uniform persistence of the disease. Our main result is that nontrivial endemic periodic solutions are bifurcating from the disease free periodic solution as RcRc is passing through the threshold value one. A complete bifurcation analysis is provided for the associated nonlinear fixed point equation. We show that backward bifurcation of periodic orbits is possible for suitable parameter values, and give explicit conditions to determine whether the bifurcation is backward or forward. The main mathematical tools are comparison principles and Lyapunov–Schmidt reduction. Finally, we compare the pulse vaccination strategy with continuous vaccination, and illustrate that backward bifurcation occurs in more realistic models as well when pulse vaccination is applied.