The effect of constant and pulse vaccination on SIR epidemic model with horizontal and vertical transmission

A SIR epidemic model with horizontal and vertical transmission is presented. The dynamics of this disease model under constant and pulse vaccination are analyzed. Theoretical results show that under constant vaccination, the dynamic behavior is similar to no vaccination. But under a planned pulse vaccination regime, the system converges to a stable periodic solution with the number of infectious individuals equal to zero. Further, the global stability of an infection-free periodic solution is analyzed. And the existence of a nontrivial periodic solution (endemic) is considered by using bifurcation theory. Our theoretical results are confirmed by numerical simulations. The effectiveness of constant and pulse vaccination policies are compared.