Singular perturbation analysis of a two-time scale model of vector-borne disease: Zika virus model as a case study

Biological systems evolve across different spatial and temporal scales. For instance, multi-time scale epidemiological models on two-time scales are important in the study of infectious disease models. Particularly, vector-borne disease models are often described using ordinary differential equations with multiple time scales, which can involve singular perturbations—situations where rapid transitions or significant changes in system behavior occur due to small parameter variations or the interaction between fast and slow dynamics. To analyze these multi-time scale problems, we employ tools such as Geometric Singular Perturbation Theory (GSPT), Tikhonov’s Theorem, and Fenichel’s Theory. These methods provide insights into complex phenomena, including the loss of normal hyperbolicity and other intricate behaviors. Particularly, applying singular perturbation theory to vector-borne diseases allows us to reduce the dynamics to a one-time scale and understand their dynamics. To illustrate this, we present a Zika virus model and apply Tikhonov’s theorem and GSPT to investigate the model’s asymptotic behavior. Additionally, we conduct a bifurcation analysis to explore how the system’s behavior changes with variations in the parameter ɛ. We illustrate the various qualitative scenarios of the reduced system under singular perturbation. We will show that the fast–slow models, even though in nonstandard form, can be studied by means of GSPT.