A geometric analysis of the SIR, SIRS and SIRWS epidemiological models
We study fast–slow versions of the SIR, SIRS, and SIRWS epidemiological models. The multiple time scale behaviour is introduced to account for large differences between some of the rates of the epidemiological pathways. Our main purpose is to show that the fast–slow models, even though in nonstandar...
Saved in:
| Published in | Nonlinear analysis: real world applications Vol. 58; p. 103220 |
|---|---|
| Main Authors | , , , |
| Format | Journal Article |
| Language | English |
| Published |
Elsevier Ltd
01.04.2021
|
| Subjects | |
| Online Access | Get full text |
Cover
Loading…
| Abstract | We study fast–slow versions of the SIR, SIRS, and SIRWS epidemiological models. The multiple time scale behaviour is introduced to account for large differences between some of the rates of the epidemiological pathways. Our main purpose is to show that the fast–slow models, even though in nonstandard form, can be studied by means of Geometric Singular Perturbation Theory (GSPT). In particular, without using Lyapunov’s method, we are able to not only analyse the stability of the endemic equilibria but also to show that in some of the models limit cycles arise. We show that the proposed approach is particularly useful in more complicated (higher dimensional) models such as the SIRWS model, for which we provide a detailed description of its dynamics by combining analytic and numerical techniques. |
|---|---|
| AbstractList | We study fast–slow versions of the SIR, SIRS, and SIRWS epidemiological models. The multiple time scale behaviour is introduced to account for large differences between some of the rates of the epidemiological pathways. Our main purpose is to show that the fast–slow models, even though in nonstandard form, can be studied by means of Geometric Singular Perturbation Theory (GSPT). In particular, without using Lyapunov’s method, we are able to not only analyse the stability of the endemic equilibria but also to show that in some of the models limit cycles arise. We show that the proposed approach is particularly useful in more complicated (higher dimensional) models such as the SIRWS model, for which we provide a detailed description of its dynamics by combining analytic and numerical techniques. |
| ArticleNumber | 103220 |
| Author | Sensi, Mattia Jardón-Kojakhmetov, Hildeberto Pugliese, Andrea Kuehn, Christian |
| Author_xml | – sequence: 1 givenname: Hildeberto orcidid: 0000-0001-8708-7409 surname: Jardón-Kojakhmetov fullname: Jardón-Kojakhmetov, Hildeberto organization: Technische Universität München (TUM), Germany – sequence: 2 givenname: Christian orcidid: 0000-0002-7063-6173 surname: Kuehn fullname: Kuehn, Christian organization: Technische Universität München (TUM), Germany – sequence: 3 givenname: Andrea orcidid: 0000-0002-3512-8560 surname: Pugliese fullname: Pugliese, Andrea organization: Università degli Studi di Trento, Italy – sequence: 4 givenname: Mattia surname: Sensi fullname: Sensi, Mattia email: mattia.sensi@unitn.it organization: Università degli Studi di Trento, Italy |
| BookMark | eNqFkM1KAzEUhYNUsK2-gYs8gFPz05mkLoRSrBYKgi24DCG5qSkzk5IEpW_vDOPKhW7uPZzLOXC_CRq1oQWEbimZUUKr--OsM-KXnjHCeoszRi7QmEohi1LQxajT80oWlFF5hSYpHQmhgnI6RuslPkBoIEdvsG51fU4-4eBw_gC827zd9WPXXWwv3ncYTt5C40MdDt7oGjfBQp2u0aXTdYKbnz1F-_XTfvVSbF-fN6vltjCcVLkowRknBOHSWkmc4MyCltzwigIrHQErJRcO5rRkZEG1AMGBWcN5VTIOfIrmQ62JIaUITp2ib3Q8K0pUj0Id1YBC9SjUgKKLPfyKGZ919qHNUfv6v_DjEO7ehE8PUSXjoTVgfQSTlQ3-74JvKot9Sw |
| CitedBy_id | crossref_primary_10_1016_j_automatica_2023_111303 crossref_primary_10_1137_23M1565632 crossref_primary_10_1007_s11071_023_08498_4 crossref_primary_10_1007_s00285_021_01664_5 crossref_primary_10_1016_j_cnsns_2023_107675 crossref_primary_10_1007_s00285_021_01605_2 crossref_primary_10_1109_TCSS_2022_3201061 crossref_primary_10_1016_j_chaos_2024_115063 crossref_primary_10_1016_j_mbs_2023_109067 crossref_primary_10_1016_j_nonrwa_2023_103986 crossref_primary_10_1016_j_chaos_2025_116209 crossref_primary_10_1137_19M1295507 crossref_primary_10_1007_s10884_023_10266_2 crossref_primary_10_1142_S1793524523500444 crossref_primary_10_1063_5_0040518 crossref_primary_10_1142_S0218339022500243 crossref_primary_10_1016_j_physd_2023_133887 crossref_primary_10_1002_mma_9186 crossref_primary_10_1080_17513758_2022_2109766 crossref_primary_10_1186_s13662_021_03633_0 crossref_primary_10_1142_S179352452150056X crossref_primary_10_1007_s40574_023_00368_6 crossref_primary_10_3390_math10183347 crossref_primary_10_11948_20220433 crossref_primary_10_1016_j_chaos_2024_115104 crossref_primary_10_23919_JSEE_2023_000080 crossref_primary_10_3390_math10030402 crossref_primary_10_3390_math11244918 |
| Cites_doi | 10.1007/BF00305757 10.1016/0022-0396(79)90152-9 10.1137/S0036144500371907 10.3934/dcdss.2009.2.851 10.2307/2341437 10.1007/s00332-015-9235-z 10.1088/0951-7715/22/1/011 10.1016/j.jde.2007.10.023 10.1007/BF02566923 10.3934/mbe.2016032 10.1080/13873950701742754 10.1080/17513758.2012.716454 10.1016/j.aml.2009.11.014 10.1098/rspa.2017.0499 10.1016/j.jde.2016.01.008 10.1016/S0167-2789(00)00187-1 10.1056/NEJMoa064040 10.1063/1.3615231 10.1137/120876642 10.1007/s00033-015-0552-8 10.1186/s12879-017-2716-y 10.1016/j.mbs.2016.07.003 10.1016/j.epidem.2011.12.002 10.1007/s00285-015-0905-0 10.1073/pnas.1014394108 10.1088/0951-7715/19/3/009 10.1016/0025-5564(93)90041-8 10.1109/TCT.1960.1086720 10.1016/0025-5564(76)90132-2 10.1137/18M1186617 10.3934/dcdss.2009.2.807 10.1007/s12080-013-0185-5 10.1017/S1446181113000023 10.1080/00207160.2015.1050961 10.1088/1361-6544/ab6a77 10.3934/dcdss.2020073 |
| ContentType | Journal Article |
| Copyright | 2020 Elsevier Ltd |
| Copyright_xml | – notice: 2020 Elsevier Ltd |
| DBID | AAYXX CITATION |
| DOI | 10.1016/j.nonrwa.2020.103220 |
| DatabaseName | CrossRef |
| DatabaseTitle | CrossRef |
| DatabaseTitleList | |
| DeliveryMethod | fulltext_linktorsrc |
| Discipline | Mathematics |
| EISSN | 1878-5719 |
| ExternalDocumentID | 10_1016_j_nonrwa_2020_103220 S1468121820301383 |
| GroupedDBID | --K --M -~X .~1 0R~ 123 1B1 1~. 1~5 29N 4.4 457 4G. 5VS 6OB 7-5 71M 8P~ AACTN AAEDT AAEDW AAIAV AAIKJ AAKOC AALRI AAOAW AAQFI AAQXK AAXUO ABAOU ABEFU ABMAC ABXDB ABYKQ ACAZW ACDAQ ACGFS ACIWK ACNNM ACRLP ADBBV ADEZE ADGUI ADMUD ADTZH AEBSH AECPX AEKER AFKWA AFTJW AGHFR AGUBO AGYEJ AHJVU AIEXJ AIGVJ AIKHN AITUG AJBFU AJOXV ALMA_UNASSIGNED_HOLDINGS AMFUW AMRAJ ARUGR ASPBG AVWKF AXJTR AZFZN BJAXD BKOJK BLXMC CS3 EBS EFJIC EFLBG EJD EO8 EO9 EP2 EP3 F5P FDB FEDTE FGOYB FIRID FNPLU FYGXN G-Q GBLVA HVGLF HZ~ IHE J1W J9A JJJVA KOM M41 MHUIS MO0 N9A O-L O9- OAUVE OZT P-8 P-9 P2P PC. PQQKQ Q38 R2- RIG ROL RPZ SDF SDG SDP SES SEW SPC SPCBC SST SSW SSZ T5K XPP YQT ZMT ~G- AATTM AAXKI AAYWO AAYXX ABWVN ACRPL ADNMO AEIPS AFJKZ AFXIZ AGCQF AGQPQ AGRNS AIIUN ANKPU APXCP BNPGV CITATION SSH |
| ID | FETCH-LOGICAL-c306t-5efcf77038dd80f732dea83c361e25f0ed8837fe4152091a7e73e2dc336523e3 |
| IEDL.DBID | .~1 |
| ISSN | 1468-1218 |
| IngestDate | Tue Jul 01 04:03:13 EDT 2025 Thu Apr 24 22:59:07 EDT 2025 Fri Feb 23 02:48:12 EST 2024 |
| IsPeerReviewed | true |
| IsScholarly | true |
| Keywords | Entry–exit function Fast–slow system Epidemic model Numerical continuation Non-standard form Bifurcation analysis |
| Language | English |
| LinkModel | DirectLink |
| MergedId | FETCHMERGED-LOGICAL-c306t-5efcf77038dd80f732dea83c361e25f0ed8837fe4152091a7e73e2dc336523e3 |
| ORCID | 0000-0002-7063-6173 0000-0001-8708-7409 0000-0002-3512-8560 |
| ParticipantIDs | crossref_primary_10_1016_j_nonrwa_2020_103220 crossref_citationtrail_10_1016_j_nonrwa_2020_103220 elsevier_sciencedirect_doi_10_1016_j_nonrwa_2020_103220 |
| ProviderPackageCode | CITATION AAYXX |
| PublicationCentury | 2000 |
| PublicationDate | April 2021 2021-04-00 |
| PublicationDateYYYYMMDD | 2021-04-01 |
| PublicationDate_xml | – month: 04 year: 2021 text: April 2021 |
| PublicationDecade | 2020 |
| PublicationTitle | Nonlinear analysis: real world applications |
| PublicationYear | 2021 |
| Publisher | Elsevier Ltd |
| Publisher_xml | – name: Elsevier Ltd |
| References | Smith (b10) 1983; 17 Diekmann, Heesterbeek, Britton (b5) 2013 LaSalle (b35) 1960; 7 Breda, Diekmann, De Graaf, Pugliese, Vermiglio (b14) 2012; 6 Hethcote (b34) 2005 Wechselberger (b23) 2020 Martcheva (b6) 2015 Jelbart, Wechselberger (b22) 2020; 33 Hethcote (b9) 1976; 28 Kuehn, Szmolyan (b21) 2015; 25 Taghvafard, Jardón-Kojakhmetov, Szmolyan, Cao (b37) 2019 Swart, Tomasi, Kretzschmar, Havelaar, Diekmann (b17) 2012; 4 Keeling, Rohani (b4) 2008 Bertram, Rubin (b26) 2017; 287 Desroches, Krauskopf, Osinga (b44) 2009; 2 Hsu, Ruan (b33) 2019 Taghvafard, Jardón-Kojakhmetov, Cao (b41) 2018; 474 Jardón-Kojakhmetov, Kuehn (b31) 2019 Chaves, Gargiullo, Zhang, Civen, Guris, Mascola, Seward (b18) 2007; 356 Woolthuis, Wallinga, van Boven (b19) 2017; 17 O’Regan, Drake (b48) 2013; 6 Anderson, May (b2) 1991 Ersöz, Desroches, Guillamon, Rinzel, Tabak (b47) 2020 Iuorio, Kuehn, Szmolyan (b46) 2020; 13 Dafilis, Frascoli, Wood, McCaw (b16) 2012; 54 Hethcote (b3) 2000; 42 Soper (b7) 1929; 92 Kermack, Mckendrick (b8) 1932; 138A Keeling, Rohani, Grenfell (b11) 2001; 148 Kosiuk, Szmolyan (b20) 2016; 72 Haiduc (b39) 2009; 22 Kuehn (b27) 2015 Andreasen (b12) 1993; 114 O’ Regan, Kelly, Korobeinikov, O’ Callaghan, Pokrovskii (b25) 2010; 23 Widder, Kuehn (b49) 2016; 13 De Maesschalck, Schecter (b24) 2016; 260 Guckenheimer, Wechselberger, Young (b38) 2006; 19 Rocha, Mateus, Skwara, Aguiar, Stollenwerk (b13) 2016; 93 Kermack, McKendrick (b1) 1927; 115 Lavine, King, Bjørnstad (b15) 2011; 108 Dhooge, Govaerts, Kuznetsov, Meijer, Sautois (b43) 2008; 14 Shuai, van den Driessche (b36) 2013; 73 De Maesschalck (b32) 2008; 244 Thom (b40) 1954; 28 Kuehn (b42) 2011; 21 Fenichel (b30) 1979; 31 Letson, Rubin (b29) 2018; 17 Benoît, Brøns, Desroches, Krupa (b28) 2015; 66 Guckenheimer, Kuehn (b45) 2009; 2 Smith (10.1016/j.nonrwa.2020.103220_b10) 1983; 17 Thom (10.1016/j.nonrwa.2020.103220_b40) 1954; 28 Shuai (10.1016/j.nonrwa.2020.103220_b36) 2013; 73 Keeling (10.1016/j.nonrwa.2020.103220_b4) 2008 Soper (10.1016/j.nonrwa.2020.103220_b7) 1929; 92 Dhooge (10.1016/j.nonrwa.2020.103220_b43) 2008; 14 Guckenheimer (10.1016/j.nonrwa.2020.103220_b45) 2009; 2 Taghvafard (10.1016/j.nonrwa.2020.103220_b41) 2018; 474 Haiduc (10.1016/j.nonrwa.2020.103220_b39) 2009; 22 Hethcote (10.1016/j.nonrwa.2020.103220_b3) 2000; 42 Swart (10.1016/j.nonrwa.2020.103220_b17) 2012; 4 Diekmann (10.1016/j.nonrwa.2020.103220_b5) 2013 Desroches (10.1016/j.nonrwa.2020.103220_b44) 2009; 2 Kuehn (10.1016/j.nonrwa.2020.103220_b42) 2011; 21 Letson (10.1016/j.nonrwa.2020.103220_b29) 2018; 17 Hethcote (10.1016/j.nonrwa.2020.103220_b34) 2005 Lavine (10.1016/j.nonrwa.2020.103220_b15) 2011; 108 Hsu (10.1016/j.nonrwa.2020.103220_b33) 2019 Keeling (10.1016/j.nonrwa.2020.103220_b11) 2001; 148 Taghvafard (10.1016/j.nonrwa.2020.103220_b37) 2019 Kuehn (10.1016/j.nonrwa.2020.103220_b21) 2015; 25 Iuorio (10.1016/j.nonrwa.2020.103220_b46) 2020; 13 Kermack (10.1016/j.nonrwa.2020.103220_b8) 1932; 138A LaSalle (10.1016/j.nonrwa.2020.103220_b35) 1960; 7 Widder (10.1016/j.nonrwa.2020.103220_b49) 2016; 13 Guckenheimer (10.1016/j.nonrwa.2020.103220_b38) 2006; 19 Jelbart (10.1016/j.nonrwa.2020.103220_b22) 2020; 33 De Maesschalck (10.1016/j.nonrwa.2020.103220_b32) 2008; 244 O’Regan (10.1016/j.nonrwa.2020.103220_b48) 2013; 6 Anderson (10.1016/j.nonrwa.2020.103220_b2) 1991 Breda (10.1016/j.nonrwa.2020.103220_b14) 2012; 6 Rocha (10.1016/j.nonrwa.2020.103220_b13) 2016; 93 O’ Regan (10.1016/j.nonrwa.2020.103220_b25) 2010; 23 Martcheva (10.1016/j.nonrwa.2020.103220_b6) 2015 Fenichel (10.1016/j.nonrwa.2020.103220_b30) 1979; 31 Chaves (10.1016/j.nonrwa.2020.103220_b18) 2007; 356 Andreasen (10.1016/j.nonrwa.2020.103220_b12) 1993; 114 Bertram (10.1016/j.nonrwa.2020.103220_b26) 2017; 287 Benoît (10.1016/j.nonrwa.2020.103220_b28) 2015; 66 De Maesschalck (10.1016/j.nonrwa.2020.103220_b24) 2016; 260 Kuehn (10.1016/j.nonrwa.2020.103220_b27) 2015 Dafilis (10.1016/j.nonrwa.2020.103220_b16) 2012; 54 Kosiuk (10.1016/j.nonrwa.2020.103220_b20) 2016; 72 Jardón-Kojakhmetov (10.1016/j.nonrwa.2020.103220_b31) 2019 Kermack (10.1016/j.nonrwa.2020.103220_b1) 1927; 115 Hethcote (10.1016/j.nonrwa.2020.103220_b9) 1976; 28 Wechselberger (10.1016/j.nonrwa.2020.103220_b23) 2020 Woolthuis (10.1016/j.nonrwa.2020.103220_b19) 2017; 17 Ersöz (10.1016/j.nonrwa.2020.103220_b47) 2020 |
| References_xml | – year: 2013 ident: b5 article-title: Mathematical Tools for Understanding Infectious Disease Dynamics – volume: 31 start-page: 53 year: 1979 end-page: 98 ident: b30 article-title: Geometric singular perturbation theory for ordinary differential equations publication-title: J. Differ. Equ. – volume: 66 start-page: 2255 year: 2015 end-page: 2270 ident: b28 article-title: Extending the zero-derivative principle for slow–fast dynamical systems publication-title: Z. Angew. Math. Phys. – volume: 33 start-page: 2364 year: 2020 ident: b22 article-title: Two-stroke relaxation oscillators publication-title: Nonlinearity – volume: 260 start-page: 6697 year: 2016 end-page: 6715 ident: b24 article-title: The entry–exit function and geometric singular perturbation theory publication-title: J. Differ. Equ. – volume: 17 start-page: 1 year: 2017 end-page: 11 ident: b19 article-title: Variation in loss of immunity shapes influenza epidemics and the impact of vaccination publication-title: BMC Infect. Dis. – volume: 14 start-page: 147 year: 2008 end-page: 175 ident: b43 article-title: New features of the software MatCont for bifurcation analysis of dynamical systems publication-title: Math. Comput. Model. Dyn. Syst. – volume: 17 start-page: 163 year: 1983 end-page: 177 ident: b10 article-title: Subharmonic bifurcation in an SIR epidemic model publication-title: J. Math. Biol. – volume: 2 start-page: 851 year: 2009 end-page: 872 ident: b45 article-title: Homoclinic orbits of the FitzHugh–Nagumo equation: The singular limit publication-title: Discrete Contin. Dyn. Syst. S – volume: 474 year: 2018 ident: b41 article-title: Parameter-robustness analysis for a biochemical oscillator model describing the social-behaviour transition phase of myxobacteria publication-title: Proc. R. Soc. A – volume: 22 start-page: 213 year: 2009 end-page: 237 ident: b39 article-title: Horseshoes in the forced van der Pol system publication-title: Nonlinearity – volume: 28 start-page: 17 year: 1954 end-page: 86 ident: b40 article-title: Quelques propriétés globales des variétés différentiables publication-title: Comment. Math. Helv. – year: 2020 ident: b47 article-title: Canard-induced complex oscillations in an excitatory network publication-title: J. Math. Biol. – year: 2019 ident: b37 article-title: Geometric analysis of Oscillations in the Frzilator model – volume: 93 start-page: 1405 year: 2016 end-page: 1422 ident: b13 article-title: Understanding dengue fever dynamics: a study of seasonality in vector-borne disease models publication-title: Int. J. Comput. Math. – volume: 19 start-page: 701 year: 2006 end-page: 720 ident: b38 article-title: Chaotic attractors of relaxation oscillations publication-title: Nonlinearity – year: 2008 ident: b4 article-title: Modeling Infectious Diseases in Humans and Animals – volume: 7 start-page: 520 year: 1960 end-page: 527 ident: b35 article-title: Some extensions of Liapunov’s second method publication-title: IRE Trans. Circuit Theory – volume: 25 start-page: 583 year: 2015 end-page: 629 ident: b21 article-title: Multiscale geometry of the Olsen model and non-classical relaxation oscillations publication-title: J. Nonlinear Sci. – year: 2015 ident: b27 article-title: Multiple Time Scale Dynamics, Vol. 191 – volume: 17 start-page: 2414 year: 2018 end-page: 2445 ident: b29 article-title: A new frame for an old (phase) portrait: Finding rivers and other flow features in the plane publication-title: SIAM J. Appl. Dyn. Syst. – volume: 114 start-page: 29 year: 1993 end-page: 58 ident: b12 article-title: The effect of age-dependent host mortality on the dynamics of an endemic disease publication-title: Math. Biosci. – year: 2019 ident: b33 article-title: Relaxation oscillations and the entry-exit function in multi-dimensional slow-fast systems – volume: 4 start-page: 43 year: 2012 end-page: 47 ident: b17 article-title: The protective effects of temporary immunity under imposed infection pressure publication-title: Epidemics – volume: 73 start-page: 1513 year: 2013 end-page: 1532 ident: b36 article-title: Global stability of infectious disease models using Lyapunov functions publication-title: SIAM J. Appl. Math. – start-page: 1 year: 2005 end-page: 135 ident: b34 article-title: The basic epidemiology models: Models, expressions for publication-title: Epidemiology, Vol. 9 – volume: 138A start-page: 55 year: 1932 end-page: 83 ident: b8 article-title: Contributions to the mathematical theory of epidemics-II . The problem of endemicity publication-title: Proc. R. Soc. A – volume: 115 start-page: 700 year: 1927 end-page: 721 ident: b1 article-title: A contribution to the mathematical theory of epidemics publication-title: Proc. R. Soc. A – volume: 108 start-page: 7259 year: 2011 end-page: 7264 ident: b15 article-title: Natural immune boosting in pertussis dynamics and the potential for long-term vaccine failure publication-title: Proc. Natl. Acad. Sci. USA – volume: 21 year: 2011 ident: b42 article-title: On decomposing mixed-mode oscillations and their return maps publication-title: Chaos – year: 1991 ident: b2 article-title: Infectious Diseases of Humans, Vol. 1 – volume: 6 start-page: 103 year: 2012 end-page: 117 ident: b14 article-title: On the formulation of epidemic models (an appraisal of Kermack and McKendrick) publication-title: J. Biol. Dyn. – volume: 244 start-page: 1448 year: 2008 end-page: 1466 ident: b32 article-title: Smoothness of transition maps in singular perturbation problems with one fast variable publication-title: J. Differ. Equ. – year: 2015 ident: b6 article-title: An Introduction to Mathematical Epidemiology, Vol. 61 – volume: 28 start-page: 335 year: 1976 end-page: 356 ident: b9 article-title: Qualitative analyses of communicable disease models publication-title: Math. Biosci. – volume: 92 start-page: 34 year: 1929 end-page: 73 ident: b7 article-title: The interpretation of periodicity in disease prevalence publication-title: J. R. Stat. Soc. – volume: 356 start-page: 1121 year: 2007 end-page: 1129 ident: b18 article-title: Loss of vaccine-induced immunity to varicella over time publication-title: New Engl. J. Med. – volume: 13 start-page: 1269 year: 2020 end-page: 1290 ident: b46 article-title: Geometry and numerical continuation of multiscale orbits in a nonconvex variational problem publication-title: Discrete Contin. Dyn. Syst. S – year: 2020 ident: b23 article-title: Geometric Singular Perturbation Theory Beyond the Standard Form – volume: 42 start-page: 599 year: 2000 end-page: 653 ident: b3 article-title: The mathematics of infectious diseases publication-title: SIAM Rev. – year: 2019 ident: b31 article-title: A survey on the blow-up method for fast-slow systems – volume: 72 start-page: 1337 year: 2016 end-page: 1368 ident: b20 article-title: Geometric analysis of the Goldbeter minimal model for the embryonic cell cycle publication-title: J. Math. Biol. – volume: 13 start-page: 1093 year: 2016 end-page: 1118 ident: b49 article-title: Heterogeneous population dynamics and scaling laws near epidemic outbreaks publication-title: Math. Biosci. Eng. – volume: 6 start-page: 333 year: 2013 end-page: 357 ident: b48 article-title: Theory of early warning signals of disease emergence and leading indicators of elimination publication-title: Theor. Ecol. – volume: 23 start-page: 446 year: 2010 end-page: 448 ident: b25 article-title: Lyapunov Functions for SIR and SIRS epidemic models publication-title: Appl. Math. Lett. – volume: 287 start-page: 105 year: 2017 end-page: 121 ident: b26 article-title: Multi-timescale systems and fast-slow analysis publication-title: Math. Biosci. – volume: 54 start-page: 50 year: 2012 end-page: 63 ident: b16 article-title: The influence of increasing life expectancy on the dynamics of SIRS systems with immune boosting publication-title: ANZIAM J. – volume: 2 start-page: 807 year: 2009 end-page: 827 ident: b44 article-title: The geometry of mixed-mode oscillations in the Olsen model for the perioxidase-oxidase reaction publication-title: Discrete Contin. Dyn. Syst. S – volume: 148 start-page: 317 year: 2001 end-page: 335 ident: b11 article-title: Seasonally forced disease dynamics explored as switching between attractors publication-title: Physica D – volume: 17 start-page: 163 year: 1983 ident: 10.1016/j.nonrwa.2020.103220_b10 article-title: Subharmonic bifurcation in an SIR epidemic model publication-title: J. Math. Biol. doi: 10.1007/BF00305757 – volume: 31 start-page: 53 issue: 1 year: 1979 ident: 10.1016/j.nonrwa.2020.103220_b30 article-title: Geometric singular perturbation theory for ordinary differential equations publication-title: J. Differ. Equ. doi: 10.1016/0022-0396(79)90152-9 – year: 2015 ident: 10.1016/j.nonrwa.2020.103220_b27 – year: 1991 ident: 10.1016/j.nonrwa.2020.103220_b2 – year: 2019 ident: 10.1016/j.nonrwa.2020.103220_b33 – volume: 42 start-page: 599 issue: 4 year: 2000 ident: 10.1016/j.nonrwa.2020.103220_b3 article-title: The mathematics of infectious diseases publication-title: SIAM Rev. doi: 10.1137/S0036144500371907 – volume: 2 start-page: 851 issue: 4 year: 2009 ident: 10.1016/j.nonrwa.2020.103220_b45 article-title: Homoclinic orbits of the FitzHugh–Nagumo equation: The singular limit publication-title: Discrete Contin. Dyn. Syst. S doi: 10.3934/dcdss.2009.2.851 – volume: 92 start-page: 34 year: 1929 ident: 10.1016/j.nonrwa.2020.103220_b7 article-title: The interpretation of periodicity in disease prevalence publication-title: J. R. Stat. Soc. doi: 10.2307/2341437 – volume: 25 start-page: 583 issue: 3 year: 2015 ident: 10.1016/j.nonrwa.2020.103220_b21 article-title: Multiscale geometry of the Olsen model and non-classical relaxation oscillations publication-title: J. Nonlinear Sci. doi: 10.1007/s00332-015-9235-z – volume: 22 start-page: 213 year: 2009 ident: 10.1016/j.nonrwa.2020.103220_b39 article-title: Horseshoes in the forced van der Pol system publication-title: Nonlinearity doi: 10.1088/0951-7715/22/1/011 – volume: 244 start-page: 1448 issue: 6 year: 2008 ident: 10.1016/j.nonrwa.2020.103220_b32 article-title: Smoothness of transition maps in singular perturbation problems with one fast variable publication-title: J. Differ. Equ. doi: 10.1016/j.jde.2007.10.023 – volume: 28 start-page: 17 issue: 1 year: 1954 ident: 10.1016/j.nonrwa.2020.103220_b40 article-title: Quelques propriétés globales des variétés différentiables publication-title: Comment. Math. Helv. doi: 10.1007/BF02566923 – volume: 13 start-page: 1093 issue: 5 year: 2016 ident: 10.1016/j.nonrwa.2020.103220_b49 article-title: Heterogeneous population dynamics and scaling laws near epidemic outbreaks publication-title: Math. Biosci. Eng. doi: 10.3934/mbe.2016032 – start-page: 1 year: 2005 ident: 10.1016/j.nonrwa.2020.103220_b34 article-title: The basic epidemiology models: Models, expressions for R0, parameter estimation, and applications – volume: 14 start-page: 147 issue: 2 year: 2008 ident: 10.1016/j.nonrwa.2020.103220_b43 article-title: New features of the software MatCont for bifurcation analysis of dynamical systems publication-title: Math. Comput. Model. Dyn. Syst. doi: 10.1080/13873950701742754 – volume: 6 start-page: 103 issue: supl. 2 year: 2012 ident: 10.1016/j.nonrwa.2020.103220_b14 article-title: On the formulation of epidemic models (an appraisal of Kermack and McKendrick) publication-title: J. Biol. Dyn. doi: 10.1080/17513758.2012.716454 – volume: 23 start-page: 446 issue: 4 year: 2010 ident: 10.1016/j.nonrwa.2020.103220_b25 article-title: Lyapunov Functions for SIR and SIRS epidemic models publication-title: Appl. Math. Lett. doi: 10.1016/j.aml.2009.11.014 – year: 2013 ident: 10.1016/j.nonrwa.2020.103220_b5 – volume: 474 issue: 2209 year: 2018 ident: 10.1016/j.nonrwa.2020.103220_b41 article-title: Parameter-robustness analysis for a biochemical oscillator model describing the social-behaviour transition phase of myxobacteria publication-title: Proc. R. Soc. A doi: 10.1098/rspa.2017.0499 – year: 2015 ident: 10.1016/j.nonrwa.2020.103220_b6 – volume: 260 start-page: 6697 issue: 8 year: 2016 ident: 10.1016/j.nonrwa.2020.103220_b24 article-title: The entry–exit function and geometric singular perturbation theory publication-title: J. Differ. Equ. doi: 10.1016/j.jde.2016.01.008 – volume: 148 start-page: 317 year: 2001 ident: 10.1016/j.nonrwa.2020.103220_b11 article-title: Seasonally forced disease dynamics explored as switching between attractors publication-title: Physica D doi: 10.1016/S0167-2789(00)00187-1 – volume: 356 start-page: 1121 issue: 11 year: 2007 ident: 10.1016/j.nonrwa.2020.103220_b18 article-title: Loss of vaccine-induced immunity to varicella over time publication-title: New Engl. J. Med. doi: 10.1056/NEJMoa064040 – volume: 21 issue: 3 year: 2011 ident: 10.1016/j.nonrwa.2020.103220_b42 article-title: On decomposing mixed-mode oscillations and their return maps publication-title: Chaos doi: 10.1063/1.3615231 – volume: 73 start-page: 1513 issue: 4 year: 2013 ident: 10.1016/j.nonrwa.2020.103220_b36 article-title: Global stability of infectious disease models using Lyapunov functions publication-title: SIAM J. Appl. Math. doi: 10.1137/120876642 – volume: 66 start-page: 2255 issue: 5 year: 2015 ident: 10.1016/j.nonrwa.2020.103220_b28 article-title: Extending the zero-derivative principle for slow–fast dynamical systems publication-title: Z. Angew. Math. Phys. doi: 10.1007/s00033-015-0552-8 – volume: 17 start-page: 1 issue: 1 year: 2017 ident: 10.1016/j.nonrwa.2020.103220_b19 article-title: Variation in loss of immunity shapes influenza epidemics and the impact of vaccination publication-title: BMC Infect. Dis. doi: 10.1186/s12879-017-2716-y – volume: 287 start-page: 105 year: 2017 ident: 10.1016/j.nonrwa.2020.103220_b26 article-title: Multi-timescale systems and fast-slow analysis publication-title: Math. Biosci. doi: 10.1016/j.mbs.2016.07.003 – volume: 4 start-page: 43 issue: 1 year: 2012 ident: 10.1016/j.nonrwa.2020.103220_b17 article-title: The protective effects of temporary immunity under imposed infection pressure publication-title: Epidemics doi: 10.1016/j.epidem.2011.12.002 – volume: 72 start-page: 1337 issue: 5 year: 2016 ident: 10.1016/j.nonrwa.2020.103220_b20 article-title: Geometric analysis of the Goldbeter minimal model for the embryonic cell cycle publication-title: J. Math. Biol. doi: 10.1007/s00285-015-0905-0 – volume: 108 start-page: 7259 issue: 17 year: 2011 ident: 10.1016/j.nonrwa.2020.103220_b15 article-title: Natural immune boosting in pertussis dynamics and the potential for long-term vaccine failure publication-title: Proc. Natl. Acad. Sci. USA doi: 10.1073/pnas.1014394108 – volume: 19 start-page: 701 year: 2006 ident: 10.1016/j.nonrwa.2020.103220_b38 article-title: Chaotic attractors of relaxation oscillations publication-title: Nonlinearity doi: 10.1088/0951-7715/19/3/009 – volume: 114 start-page: 29 issue: 1 year: 1993 ident: 10.1016/j.nonrwa.2020.103220_b12 article-title: The effect of age-dependent host mortality on the dynamics of an endemic disease publication-title: Math. Biosci. doi: 10.1016/0025-5564(93)90041-8 – volume: 7 start-page: 520 issue: 4 year: 1960 ident: 10.1016/j.nonrwa.2020.103220_b35 article-title: Some extensions of Liapunov’s second method publication-title: IRE Trans. Circuit Theory doi: 10.1109/TCT.1960.1086720 – year: 2020 ident: 10.1016/j.nonrwa.2020.103220_b23 – volume: 28 start-page: 335 issue: 3–4 year: 1976 ident: 10.1016/j.nonrwa.2020.103220_b9 article-title: Qualitative analyses of communicable disease models publication-title: Math. Biosci. doi: 10.1016/0025-5564(76)90132-2 – volume: 17 start-page: 2414 issue: 4 year: 2018 ident: 10.1016/j.nonrwa.2020.103220_b29 article-title: A new frame for an old (phase) portrait: Finding rivers and other flow features in the plane publication-title: SIAM J. Appl. Dyn. Syst. doi: 10.1137/18M1186617 – volume: 2 start-page: 807 issue: 4 year: 2009 ident: 10.1016/j.nonrwa.2020.103220_b44 article-title: The geometry of mixed-mode oscillations in the Olsen model for the perioxidase-oxidase reaction publication-title: Discrete Contin. Dyn. Syst. S doi: 10.3934/dcdss.2009.2.807 – year: 2020 ident: 10.1016/j.nonrwa.2020.103220_b47 article-title: Canard-induced complex oscillations in an excitatory network publication-title: J. Math. Biol. – volume: 115 start-page: 700 issue: 772 year: 1927 ident: 10.1016/j.nonrwa.2020.103220_b1 article-title: A contribution to the mathematical theory of epidemics publication-title: Proc. R. Soc. A – volume: 6 start-page: 333 issue: 3 year: 2013 ident: 10.1016/j.nonrwa.2020.103220_b48 article-title: Theory of early warning signals of disease emergence and leading indicators of elimination publication-title: Theor. Ecol. doi: 10.1007/s12080-013-0185-5 – year: 2019 ident: 10.1016/j.nonrwa.2020.103220_b37 – year: 2008 ident: 10.1016/j.nonrwa.2020.103220_b4 – volume: 54 start-page: 50 issue: 1–2 year: 2012 ident: 10.1016/j.nonrwa.2020.103220_b16 article-title: The influence of increasing life expectancy on the dynamics of SIRS systems with immune boosting publication-title: ANZIAM J. doi: 10.1017/S1446181113000023 – volume: 93 start-page: 1405 issue: 8 year: 2016 ident: 10.1016/j.nonrwa.2020.103220_b13 article-title: Understanding dengue fever dynamics: a study of seasonality in vector-borne disease models publication-title: Int. J. Comput. Math. doi: 10.1080/00207160.2015.1050961 – year: 2019 ident: 10.1016/j.nonrwa.2020.103220_b31 – volume: 33 start-page: 2364 issue: 5 year: 2020 ident: 10.1016/j.nonrwa.2020.103220_b22 article-title: Two-stroke relaxation oscillators publication-title: Nonlinearity doi: 10.1088/1361-6544/ab6a77 – volume: 138A start-page: 55 year: 1932 ident: 10.1016/j.nonrwa.2020.103220_b8 article-title: Contributions to the mathematical theory of epidemics-II . The problem of endemicity publication-title: Proc. R. Soc. A – volume: 13 start-page: 1269 issue: 2 year: 2020 ident: 10.1016/j.nonrwa.2020.103220_b46 article-title: Geometry and numerical continuation of multiscale orbits in a nonconvex variational problem publication-title: Discrete Contin. Dyn. Syst. S doi: 10.3934/dcdss.2020073 |
| SSID | ssj0017131 |
| Score | 2.5269394 |
| Snippet | We study fast–slow versions of the SIR, SIRS, and SIRWS epidemiological models. The multiple time scale behaviour is introduced to account for large... |
| SourceID | crossref elsevier |
| SourceType | Enrichment Source Index Database Publisher |
| StartPage | 103220 |
| SubjectTerms | Bifurcation analysis Entry–exit function Epidemic model Fast–slow system Non-standard form Numerical continuation |
| Title | A geometric analysis of the SIR, SIRS and SIRWS epidemiological models |
| URI | https://dx.doi.org/10.1016/j.nonrwa.2020.103220 |
| Volume | 58 |
| hasFullText | 1 |
| inHoldings | 1 |
| isFullTextHit | |
| isPrint | |
| link | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV3NS8MwFA9jXvQgfuL8GDl4NG5pkqYex3Bs6oasE3crbT5kom5sE2_-7eal7VQQBS9taRNIf0lf3qPv_X4InV5wkabMhoQxzQmnGSeR0IykaSYCbiWTDGqH-4Owe8evxmJcQe2yFgbSKgvbn9t0b62LO40CzcZsMmnEUDREPQE5g99twPjJuYRVfv6-SvOgLgijZYURtC7L53yOl4uw52_APhT46vMAVL9_2p6-bDmdLbRZ-Iq4lQ9nG1XMyw7a6K-IVhe7qNPCD2b6DLJYCqcFwQieWuza4Lg3PIND7J5ouLiPsflUhIXpwV4JZ7GHRp3LUbtLCmkEopyPvyTCWGWl-1ojraOmgzTQJo2YYiE1gbBNoyMXeVoD27PzCFJpJDOBVoyFLvI0bB9V3bubA4RDITUNlfVauNyqzAqaqaZiijLpwu4aYiUgiSpow0G94ikp88MekxzGBGBMchhriKx6zXLajD_ayxLr5Nv0J86y_9rz8N89j9B6AAkqPg3nGFWX81dz4jyMZVb3S6iO1lrt4c0tnHvX3cEHdpTPow |
| linkProvider | Elsevier |
| linkToHtml | http://utb.summon.serialssolutions.com/2.0.0/link/0/eLvHCXMwnV07T8MwELZKGYAB8RTl6YER0yZnx2GsKqoCbQdSRDcr8QMVAa3aIjZ-O7aTFJAQSCxRlNxJyef4HsrdfQidXlCWpmAiAqAooUFGScwUkDTNWEgNBw6ud7jXjzp39HrIhhXUKnthXFllYftzm-6tdXGlXqBZn4xG9cQ1DQV-ADm4322whJap3b6OxuD8fVHnEdgsLChbjJx42T_ni7xsij19c-OHQt9-Hjra75_80xef095A60WwiJv582yiin7ZQmu9xaTV2TZqN_GDHj87XiyJ02LCCB4bbGVwcnV75g6JvaPcyX2C9SclrFsf7KlwZjto0L4ctDqk4EYg0gb5c8K0kYbb7RorFTcspqHSaQwSokCHzDS0im3qabTzzzYkSLnmoEMlASKbemrYRVX77noP4YhxFUTSeDJcamRmWJDJhgQZALd5dw1BCYiQxdxwR1_xJMoCsUeRwygcjCKHsYbIQmuSz834Q56XWItv6y-saf9Vc__fmidopTPodUX3qn9zgFZDV63ia3IOUXU-fdVHNtyYZ8f-c_oA5urPnA |
| openUrl | ctx_ver=Z39.88-2004&ctx_enc=info%3Aofi%2Fenc%3AUTF-8&rfr_id=info%3Asid%2Fsummon.serialssolutions.com&rft_val_fmt=info%3Aofi%2Ffmt%3Akev%3Amtx%3Ajournal&rft.genre=article&rft.atitle=A+geometric+analysis+of+the+SIR%2C+SIRS+and+SIRWS+epidemiological+models&rft.jtitle=Nonlinear+analysis%3A+real+world+applications&rft.au=Jard%C3%B3n-Kojakhmetov%2C+Hildeberto&rft.au=Kuehn%2C+Christian&rft.au=Pugliese%2C+Andrea&rft.au=Sensi%2C+Mattia&rft.date=2021-04-01&rft.issn=1468-1218&rft.volume=58&rft.spage=103220&rft_id=info:doi/10.1016%2Fj.nonrwa.2020.103220&rft.externalDBID=n%2Fa&rft.externalDocID=10_1016_j_nonrwa_2020_103220 |
| thumbnail_l | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/lc.gif&issn=1468-1218&client=summon |
| thumbnail_m | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/mc.gif&issn=1468-1218&client=summon |
| thumbnail_s | http://covers-cdn.summon.serialssolutions.com/index.aspx?isbn=/sc.gif&issn=1468-1218&client=summon |