Complex dynamics of a fractional-order delayed epidemic model incorporating waning immunity and optimal control

To explore the effect of memory, we first incorporate the fractional-order derivative in our defined model, which is a SIR-type epidemic model with logistic growth in susceptible and incubation delay in saturated incidence rate. Based on the value of a threshold parameter \(R_0\), called basic reproduction number, there exist two equilibria. Also, depending on that threshold value, stability and Hopf bifurcation analysis were performed in our formulated model. To study the effects of vaccination and treatment on Hopf bifurcation, we include these measures in our model and derive that these measures may increase the length of the critical delay. We also looked into a fractional-order optimal control problem to better understand the optimal role of treatment and vaccination in reducing disease prevalence and lowering associated costs. We have run simulations to verify the analytical results, considering the model’s feasible parameter values. Finally, to study the uncertainty analysis, we have used the partial rank correlation coefficient technique.