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 equation...

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Bibliographic Details
Published in:	Chaos, solitons and fractals Vol. 194; p. 116209
Main Authors:	Tran, Joe, Woldegerima, Woldegebriel Aseefa
Format:	Journal Article
Language:	English
Published:	Elsevier Ltd 01.05.2025
Subjects:	Critical manifold Fast-slow system Geometric singular perturbation theory Multi-time scale models Singular perturbation Tikhonov’s theorem Zika virus model Tikhonov’s theorem Critical manifold Singular perturbation Zika virus model Fast-slow system Geometric singular perturbation theory Multi-time scale models
ISSN:	0960-0779
Online Access:	Get full text
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Summary:	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. •Studied singular perturbation analysis of two-time scale models•We considered an SIR-SI type mathematical model of Zika Virus with two-time scales•Geometric Singular Perturbation Theory (GSPT) and Tikhonov’s Theorem Applied•Fast–slow models, even though in nonstandard form, can be studied by means of GSPT•Illustrated qualitative behaviours of the reduced system under singular perturbation
ISSN:	0960-0779
DOI:	10.1016/j.chaos.2025.116209