Dynamical Behaviour of Dengue: An SIR Epidemic Model

In this chapter, we have demonstrated the dynamical behaviour of dengue using an SIR epidemic model, spanning both distributed and discrete time delays. The existence of boundary and interior equilibrium points has been studied. Furthermore, we have discussed the local stability of the equilibrium points. The disease-free equilibrium point is locally asymptotically stable if R0 β1 + β2e−bτ and unstable for R0 > 1. The endemic equilibrium point is locally asymptotically stable for [0,τ′), and it undergoes Hopf bifurcation at τ = τ′. The direction and stability of Hopf bifurcation have been established using the normal form theory and the centre manifold theorem, and lastly the analytical results are verified numerically, and further, sensitivity analysis is conducted to show how the periodic solution of the system is dependent upon delay, rate of infection, and birth and death rate.