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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| Vydáno v: | Chaos, solitons and fractals Ročník 194; s. 116209 |
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| Médium: | Journal Article |
| Jazyk: | angličtina |
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Elsevier Ltd
01.05.2025
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| ISSN: | 0960-0779 |
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| Abstract | 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 |
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| AbstractList | 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 |
| ArticleNumber | 116209 |
| Author | Woldegerima, Woldegebriel Aseefa Tran, Joe |
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| Cites_doi | 10.1016/j.isci.2022.105421 10.1080/00207160.2013.783208 10.1371/journal.pcbi.1007734 10.1016/j.nonrwa.2020.103220 10.1016/j.jaut.2016.02.006 10.1016/0022-0396(79)90152-9 10.1093/humupd/dmx024 10.1097/IPC.0000000000000654 10.1159/000211976 10.1140/epjp/s13360-022-02368-5 10.1016/S0025-5564(02)00108-6 10.1007/s00285-009-0266-7 10.1007/978-3-319-05140-6 10.1016/j.pbiomolbio.2007.07.019 10.1080/17513758.2017.1367849 |
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| Keywords | Tikhonov’s theorem Critical manifold Singular perturbation Zika virus model Fast-slow system Geometric singular perturbation theory Multi-time scale models |
| Language | English |
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article-title: A geometric analysis of the SIR, SIRS and SIRWS epidemiological models publication-title: Nonlinear Anal Real World Appl doi: 10.1016/j.nonrwa.2020.103220 – volume: 2020 start-page: 1687 issue: 589 year: 2020 ident: 10.1016/j.chaos.2025.116209_b26 article-title: A new mathematical model for Zika virus transmission publication-title: Adv Difference Equ – volume: 68 start-page: 1 year: 2016 ident: 10.1016/j.chaos.2025.116209_b12 article-title: The Zika outbreak of the 21st century publication-title: J Autoimmun doi: 10.1016/j.jaut.2016.02.006 – year: 2012 ident: 10.1016/j.chaos.2025.116209_b28 – start-page: 309 year: 2024 ident: 10.1016/j.chaos.2025.116209_b23 – volume: 31 start-page: 53 issue: 1 year: 1979 ident: 10.1016/j.chaos.2025.116209_b22 article-title: Geometric singular perturbation theory for ordinary differential equations publication-title: J Differential Equations doi: 10.1016/0022-0396(79)90152-9 – year: 2024 ident: 10.1016/j.chaos.2025.116209_b16 – year: 2018 ident: 10.1016/j.chaos.2025.116209_b17 article-title: Aggregation methods in analysis of complex multiple scale systems – year: 2023 ident: 10.1016/j.chaos.2025.116209_b25 article-title: Entry-exit functions in fast-slow systems with intersecting eigenvalues publication-title: J Dynam Differential Equations – year: 2018 ident: 10.1016/j.chaos.2025.116209_b7 – volume: 23 start-page: 629 issue: 6 year: 2017 ident: 10.1016/j.chaos.2025.116209_b10 article-title: Zika virus and reproduction: facts, questions and current management publication-title: Hum Reprod Update doi: 10.1093/humupd/dmx024 – volume: 26 start-page: e80 issue: 6 year: 2016 ident: 10.1016/j.chaos.2025.116209_b13 article-title: Zika virus–associated Guillain-Barré syndrome in a returning US traveler publication-title: Infect Dis Clin Pr ( Balt Md) doi: 10.1097/IPC.0000000000000654 – ident: 10.1016/j.chaos.2025.116209_b15 – volume: 287 year: 2016 ident: 10.1016/j.chaos.2025.116209_b4 article-title: Multi-timescale systems and fast-slow analysis publication-title: Math Biosci – volume: 19 start-page: 240 issue: 4 year: 1973 ident: 10.1016/j.chaos.2025.116209_b14 article-title: Life-span of mosquitoes (Culicidae, Diptera) under laboratory conditions publication-title: Gerontology doi: 10.1159/000211976 – year: 1985 ident: 10.1016/j.chaos.2025.116209_b19 – volume: 216 start-page: S860 issue: Suppl 10 year: 2017 ident: 10.1016/j.chaos.2025.116209_b8 article-title: History and emergence of Zika virus publication-title: J Infect Dis – volume: 137 year: 2022 ident: 10.1016/j.chaos.2025.116209_b11 article-title: Mathematical modeling for the transmission potential of Zika virus with optimal control strategies publication-title: Eur Phys J Plus doi: 10.1140/epjp/s13360-022-02368-5 – year: 2016 ident: 10.1016/j.chaos.2025.116209_b31 – volume: 180 start-page: 29 issue: 1–2 year: 2002 ident: 10.1016/j.chaos.2025.116209_b32 article-title: Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission publication-title: Math Biosci doi: 10.1016/S0025-5564(02)00108-6 – volume: 4 year: 2019 ident: 10.1016/j.chaos.2025.116209_b20 article-title: A singular perturbation approach to epidemics of vector-transmitted diseases publication-title: Infect Dis Model – year: 2017 ident: 10.1016/j.chaos.2025.116209_b21 – volume: 60 start-page: 347 year: 2009 ident: 10.1016/j.chaos.2025.116209_b24 article-title: Geometric singular perturbation theory in biological practice publication-title: J Math Biol doi: 10.1007/s00285-009-0266-7 – year: 2014 ident: 10.1016/j.chaos.2025.116209_b1 doi: 10.1007/978-3-319-05140-6 – year: 1982 ident: 10.1016/j.chaos.2025.116209_b27 – volume: 96 start-page: 60 issue: 1–3 year: 2008 ident: 10.1016/j.chaos.2025.116209_b3 article-title: Multi-scale computational modelling in biology and physiology publication-title: Prog Biophys Mol Biol doi: 10.1016/j.pbiomolbio.2007.07.019 – volume: 11 start-page: 378 issue: 1 year: 2017 ident: 10.1016/j.chaos.2025.116209_b6 article-title: A complete categorization of multiscale models of infectious disease systems publication-title: J Biol Dyn doi: 10.1080/17513758.2017.1367849 – year: 2024 ident: 10.1016/j.chaos.2025.116209_b30 |
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| SubjectTerms | Critical manifold Fast-slow system Geometric singular perturbation theory Multi-time scale models Singular perturbation Tikhonov’s theorem Zika virus model |
| Title | Singular perturbation analysis of a two-time scale model of vector-borne disease: Zika virus model as a case study |
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