Multiple endemic states in age-structured $SIR$ epidemic models
$SIR$ age-structured models are very often used as a basic model ofepidemic spread. Yet, their behaviour, under generic assumptions on contactrates between different age classes, is not completely known, and, in the mostdetailed analysis so far, Inaba (1990) was able to prove uniqueness of the endem...
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AIMS Press
2012-06-01
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| Series: | Mathematical Biosciences and Engineering |
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| Online Access: | https://www.aimspress.com/article/doi/10.3934/mbe.2012.9.577 |
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| author | Andrea Franceschetti Andrea Pugliese Dimitri Breda |
| author_facet | Andrea Franceschetti Andrea Pugliese Dimitri Breda |
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| description | $SIR$ age-structured models are very often used as a basic model ofepidemic spread. Yet, their behaviour, under generic assumptions on contactrates between different age classes, is not completely known, and, in the mostdetailed analysis so far, Inaba (1990) was able to prove uniqueness of the endemicequilibrium only under a rather restrictive condition.   Here, we show anexample in the form of a $3 \times 3$ contact matrix in which multiplenon-trivial steady states exist. This instance of non-uniqueness of positiveequilibria differs from most existing ones for epidemic models, since it arisesnot from a backward transcritical bifurcation at the disease free equilibrium,but through two saddle-node bifurcations of the positive equilibrium.The dynamical behaviour of the model is analysed numerically around the range wheremultiple endemic equilibria exist; many other features are shown to occur, from coexistence of multiple attractive periodic solutions, some with extremely long period, to quasi-periodic and chaotic attractors.   It is also shown that, if thecontact rates are in the form of a $2 \times 2$ WAIFW matrix, uniqueness of non-trivial steady states always holds, so that 3 is theminimum dimension of the contact matrix to allow for multiple endemic equilibria. |
| format | Article |
| id | doaj-art-1a1fba684b6742d7ab686876d8b36cd2 |
| institution | Kabale University |
| issn | 1551-0018 |
| language | English |
| publishDate | 2012-06-01 |
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| series | Mathematical Biosciences and Engineering |
| spelling | doaj-art-1a1fba684b6742d7ab686876d8b36cd22025-01-24T02:07:00ZengAIMS PressMathematical Biosciences and Engineering1551-00182012-06-019357759910.3934/mbe.2012.9.577Multiple endemic states in age-structured $SIR$ epidemic modelsAndrea Franceschetti0Andrea Pugliese1Dimitri Breda2Dept. Mathematics, Università di Trento, Via Sommarive 14, 38123 Povo (TN)Dept. Mathematics, Università di Trento, Via Sommarive 14, 38123 Povo (TN)Dept. Mathematics, Università di Trento, Via Sommarive 14, 38123 Povo (TN)$SIR$ age-structured models are very often used as a basic model ofepidemic spread. Yet, their behaviour, under generic assumptions on contactrates between different age classes, is not completely known, and, in the mostdetailed analysis so far, Inaba (1990) was able to prove uniqueness of the endemicequilibrium only under a rather restrictive condition.   Here, we show anexample in the form of a $3 \times 3$ contact matrix in which multiplenon-trivial steady states exist. This instance of non-uniqueness of positiveequilibria differs from most existing ones for epidemic models, since it arisesnot from a backward transcritical bifurcation at the disease free equilibrium,but through two saddle-node bifurcations of the positive equilibrium.The dynamical behaviour of the model is analysed numerically around the range wheremultiple endemic equilibria exist; many other features are shown to occur, from coexistence of multiple attractive periodic solutions, some with extremely long period, to quasi-periodic and chaotic attractors.   It is also shown that, if thecontact rates are in the form of a $2 \times 2$ WAIFW matrix, uniqueness of non-trivial steady states always holds, so that 3 is theminimum dimension of the contact matrix to allow for multiple endemic equilibria.https://www.aimspress.com/article/doi/10.3934/mbe.2012.9.577periodic and chaotic solutionsage-structured epidemic modelmultiple endemic equilibriafixed point index.numerical bifurcation analysis |
| spellingShingle | Andrea Franceschetti Andrea Pugliese Dimitri Breda Multiple endemic states in age-structured $SIR$ epidemic models Mathematical Biosciences and Engineering periodic and chaotic solutions age-structured epidemic model multiple endemic equilibria fixed point index. numerical bifurcation analysis |
| title | Multiple endemic states in age-structured $SIR$ epidemic models |
| title_full | Multiple endemic states in age-structured $SIR$ epidemic models |
| title_fullStr | Multiple endemic states in age-structured $SIR$ epidemic models |
| title_full_unstemmed | Multiple endemic states in age-structured $SIR$ epidemic models |
| title_short | Multiple endemic states in age-structured $SIR$ epidemic models |
| title_sort | multiple endemic states in age structured sir epidemic models |
| topic | periodic and chaotic solutions age-structured epidemic model multiple endemic equilibria fixed point index. numerical bifurcation analysis |
| url | https://www.aimspress.com/article/doi/10.3934/mbe.2012.9.577 |
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