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Bifurcation Analysis in Models for Vector-Borne Diseases with Logistic Growth

We establish and study vector-borne models with logistic and exponential growth of vector and host populations, respectively. We discuss and analyses the existence and stability of equilibria. The model has backward bifurcation and may have no, one, or two positive equilibria when the basic reproduc...

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Autores principales: Li, Guihua, Jin, Zhen
Formato: Online Artículo Texto
Lenguaje:English
Publicado: Hindawi Publishing Corporation 2014
Materias:
Acceso en línea:https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3984867/
https://www.ncbi.nlm.nih.gov/pubmed/24790552
http://dx.doi.org/10.1155/2014/195864
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author Li, Guihua
Jin, Zhen
author_facet Li, Guihua
Jin, Zhen
author_sort Li, Guihua
collection PubMed
description We establish and study vector-borne models with logistic and exponential growth of vector and host populations, respectively. We discuss and analyses the existence and stability of equilibria. The model has backward bifurcation and may have no, one, or two positive equilibria when the basic reproduction number R (0) is less than one and one, two, or three endemic equilibria when R (0) is greater than one under different conditions. Furthermore, we prove that the disease-free equilibrium is stable if R (0) is less than 1, it is unstable otherwise. At last, by numerical simulation, we find rich dynamical behaviors in the model. By taking the natural death rate of host population as a bifurcation parameter, we find that the system may undergo a backward bifurcation, saddle-node bifurcation, Hopf bifurcation, Bogdanov-Takens bifurcation, and cusp bifurcation with the saturation parameter varying. The natural death rate of host population is a crucial parameter. If the natural death rate is higher, then the host population and the disease will die out. If it is smaller, then the host and vector population will coexist. If it is middle, the period solution will occur. Thus, with the parameter varying, the disease will spread, occur periodically, and finally become extinct.
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spelling pubmed-39848672014-04-30 Bifurcation Analysis in Models for Vector-Borne Diseases with Logistic Growth Li, Guihua Jin, Zhen ScientificWorldJournal Research Article We establish and study vector-borne models with logistic and exponential growth of vector and host populations, respectively. We discuss and analyses the existence and stability of equilibria. The model has backward bifurcation and may have no, one, or two positive equilibria when the basic reproduction number R (0) is less than one and one, two, or three endemic equilibria when R (0) is greater than one under different conditions. Furthermore, we prove that the disease-free equilibrium is stable if R (0) is less than 1, it is unstable otherwise. At last, by numerical simulation, we find rich dynamical behaviors in the model. By taking the natural death rate of host population as a bifurcation parameter, we find that the system may undergo a backward bifurcation, saddle-node bifurcation, Hopf bifurcation, Bogdanov-Takens bifurcation, and cusp bifurcation with the saturation parameter varying. The natural death rate of host population is a crucial parameter. If the natural death rate is higher, then the host population and the disease will die out. If it is smaller, then the host and vector population will coexist. If it is middle, the period solution will occur. Thus, with the parameter varying, the disease will spread, occur periodically, and finally become extinct. Hindawi Publishing Corporation 2014-03-25 /pmc/articles/PMC3984867/ /pubmed/24790552 http://dx.doi.org/10.1155/2014/195864 Text en Copyright © 2014 G. Li and Z. Jin. https://creativecommons.org/licenses/by/3.0/ This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
spellingShingle Research Article
Li, Guihua
Jin, Zhen
Bifurcation Analysis in Models for Vector-Borne Diseases with Logistic Growth
title Bifurcation Analysis in Models for Vector-Borne Diseases with Logistic Growth
title_full Bifurcation Analysis in Models for Vector-Borne Diseases with Logistic Growth
title_fullStr Bifurcation Analysis in Models for Vector-Borne Diseases with Logistic Growth
title_full_unstemmed Bifurcation Analysis in Models for Vector-Borne Diseases with Logistic Growth
title_short Bifurcation Analysis in Models for Vector-Borne Diseases with Logistic Growth
title_sort bifurcation analysis in models for vector-borne diseases with logistic growth
topic Research Article
url https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3984867/
https://www.ncbi.nlm.nih.gov/pubmed/24790552
http://dx.doi.org/10.1155/2014/195864
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