Bifurcations of an SIRS model with generalized non-monotone incidence rate
Abstract We consider an SIRS epidemic model with a more generalized non-monotone incidence: χ(I)=κIp1+Iq $\chi(I)=\frac{\kappa I^{p}}{1+I^{q}}$ with 0<p<q $0< p< q$, describing the psychological effect of some serious diseases when the number of infective individuals is getting larger. B...
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doaj-d6f7dfcb19424fe4a511ce2719d34f992020-11-25T00:40:22ZengSpringerOpenAdvances in Difference Equations1687-18472018-06-012018112110.1186/s13662-018-1675-yBifurcations of an SIRS model with generalized non-monotone incidence rateJinhui Li0Zhidong Teng1School of Mathematics and Statistics, Central China Normal UniversityCollege of Mathematics and System Sciences, Xinjiang UniversityAbstract We consider an SIRS epidemic model with a more generalized non-monotone incidence: χ(I)=κIp1+Iq $\chi(I)=\frac{\kappa I^{p}}{1+I^{q}}$ with 0<p<q $0< p< q$, describing the psychological effect of some serious diseases when the number of infective individuals is getting larger. By analyzing the existence and stability of disease-free and endemic equilibrium, we show that the dynamical behaviors of p<1 $p<1$, p=1 $p=1$ and p>1 $p>1$ distinctly vary. On one hand, the number and stability of disease-free and endemic equilibrium are different. On the other hand, when p≤1 $p\leq1$, there do not exist any closed orbits and when p>1 $p>1$, by qualitative and bifurcation analyses, we show that the model undergoes a saddle-node bifurcation, a Hopf bifurcation and a Bogdanov–Takens bifurcation of codimension 2. Besides, for p=2 $p=2$, q=3 $q=3$, we prove that the maximal multiplicity of weak focus is at least 2, which means at least 2 limit cycles can arise from this weak focus. And numerical examples of 1 limit cycle, 2 limit cycles and homoclinic loops are also given.http://link.springer.com/article/10.1186/s13662-018-1675-yEpidemic modelNon-monotone incidenceHopf bifurcationBogdanov–Takens bifurcation |
collection |
DOAJ |
language |
English |
format |
Article |
sources |
DOAJ |
author |
Jinhui Li Zhidong Teng |
spellingShingle |
Jinhui Li Zhidong Teng Bifurcations of an SIRS model with generalized non-monotone incidence rate Advances in Difference Equations Epidemic model Non-monotone incidence Hopf bifurcation Bogdanov–Takens bifurcation |
author_facet |
Jinhui Li Zhidong Teng |
author_sort |
Jinhui Li |
title |
Bifurcations of an SIRS model with generalized non-monotone incidence rate |
title_short |
Bifurcations of an SIRS model with generalized non-monotone incidence rate |
title_full |
Bifurcations of an SIRS model with generalized non-monotone incidence rate |
title_fullStr |
Bifurcations of an SIRS model with generalized non-monotone incidence rate |
title_full_unstemmed |
Bifurcations of an SIRS model with generalized non-monotone incidence rate |
title_sort |
bifurcations of an sirs model with generalized non-monotone incidence rate |
publisher |
SpringerOpen |
series |
Advances in Difference Equations |
issn |
1687-1847 |
publishDate |
2018-06-01 |
description |
Abstract We consider an SIRS epidemic model with a more generalized non-monotone incidence: χ(I)=κIp1+Iq $\chi(I)=\frac{\kappa I^{p}}{1+I^{q}}$ with 0<p<q $0< p< q$, describing the psychological effect of some serious diseases when the number of infective individuals is getting larger. By analyzing the existence and stability of disease-free and endemic equilibrium, we show that the dynamical behaviors of p<1 $p<1$, p=1 $p=1$ and p>1 $p>1$ distinctly vary. On one hand, the number and stability of disease-free and endemic equilibrium are different. On the other hand, when p≤1 $p\leq1$, there do not exist any closed orbits and when p>1 $p>1$, by qualitative and bifurcation analyses, we show that the model undergoes a saddle-node bifurcation, a Hopf bifurcation and a Bogdanov–Takens bifurcation of codimension 2. Besides, for p=2 $p=2$, q=3 $q=3$, we prove that the maximal multiplicity of weak focus is at least 2, which means at least 2 limit cycles can arise from this weak focus. And numerical examples of 1 limit cycle, 2 limit cycles and homoclinic loops are also given. |
topic |
Epidemic model Non-monotone incidence Hopf bifurcation Bogdanov–Takens bifurcation |
url |
http://link.springer.com/article/10.1186/s13662-018-1675-y |
work_keys_str_mv |
AT jinhuili bifurcationsofansirsmodelwithgeneralizednonmonotoneincidencerate AT zhidongteng bifurcationsofansirsmodelwithgeneralizednonmonotoneincidencerate |
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1725290592933511168 |