Dynamic analysis of a communicable disease fractional order model incorporating vaccination and multiple time delays

This work presents a delayed fractional-order model that explores the impact of vaccination and multiple time-delays on the infectious disease dynamics. The model explicitly incorporates two key time delays: one representing the recovery period, and another accounting for the delay in incubation. We perform a qualitative analysis of the equilibrium points, investigating the local stability conditions. Additionally, we calculate R0, providing insights into the transmissibility of the disease. Furthermore, we analyze different scenarios by considering cases with zero delay, single delay, and both delays, leading to periodic population behaviors due to Hopf-bifurcation. Our study reveals that when time delays exceeds the critical values, Hopf bifurcation occurs. The global stability is demonstrated through the construction of an appropriate Lyapunov function. Furthermore, we observe that incorporating fractional order α and vaccination parameter ρ enhances the model dynamics, and the time-delays have a major impact on Hopf bifurcation and system stability. To verify our theory, we conduct numerical simulations.