Analysis of a time-delayed vector-host epidemic model with nonlinear incidences

In this study, we investigate a time-delayed vector-host epidemic model with nonlinear incidence rates to gain a deeper understanding of the dynamics of vector-borne diseases, particularly those transmitted by vectors like mosquitoes. The model incorporates a constant human recruitment rate, exponential natural death, and an asymptotically constant vector population. We rigorously analyze the stability of both the disease-free and endemic equilibria, introducing a threshold parameter, the basic reproduction number R-0, to determine the long-term behavior of the epidemic. For R-0 < 1, the disease-free equilibrium is globally asymptotically stable, indicating disease eradication. When R-0 > 1, the endemic equilibrium emerges, and its stability is assessed through local stability analysis. The impact of time delay on the system dynamics is examined, revealing conditions under which a Hopf bifurcation occurs, leading to sustained periodic oscillations. This phenomenon highlights the critical role of time delay in influencing the epidemiology and control of vector-borne diseases. Our findings provide valuable insights into the complex interplay between time delays and nonlinear transmission dynamics, offering implications for effective disease management and control strategies.