Dynamic analysis and application of a dengue transmission model with vaccination and incubation delays

In this paper, we investigate the threshold dynamics of a dengue epidemic mathematical model with vaccination, standard incidence and incubation delays. We obtain the expression of the basic reproduction number. The local asymptotic stability of the disease-free equilibrium is studied by analyzing the distribution of roots of the corresponding characteristic equation. We determine the condition that model undergoes backward bifurcation by using the theory of centre manifold. By constructing suitable Lyapunov functional and using LaSalle's invariance principle, the global stability of the endemic equilibrium in the case of zero mortality from disease is proved. In addition, we fit the cumulative weekly deaths reported in the earlier stage of the dengue outbreak in Brazil in 2019. Elasticity analysis is carried out to show the effects of vaccine-related parameters on the basic reproduction number. Furthermore, we use the Pontryagin's minimum principle with delay to determine the optimal control strategies for dengue transmission.