Journal Article

Bifurcations of an epidemic model with non-linear incidence and infection-dependent removal rate

Seyed M. Moghadas and Murray E. Alexander

in Mathematical Medicine and Biology: A Journal of the IMA

Published on behalf of Institute of Mathematics and its Applications

Volume 23, issue 3, pages 231-254
Published in print September 2006 | ISSN: 1477-8599
Published online September 2006 | e-ISSN: 1477-8602 | DOI:
Bifurcations of an epidemic model with non-linear incidence and infection-dependent removal rate

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  • Applied Mathematics
  • Biomathematics and Statistics


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An epidemic model with a generalized non-linear incidence is extended to incorporate the effect of an infection-dependent removal strategy, which is defined as a function of the number of infected individuals. It is assumed that the removal rate decreases from a maximum capacity for removing infected individuals as their number increases. The existence and stability of the associated equilibria are analysed, and the basic reproductive number (

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) is formulated. It is shown that

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is independent of the functional form of the incidence, but depends on the removal rate. Normal forms are derived to show the different types of bifurcation the model undergoes, including transcritical, generalized Hopf (Bautin), saddle-node and Bogdanov–Takens. A degenerate Hopf bifurcation at the Bautin point, where the first Lyapunov coefficient vanishes, is discussed. Sotomayor's theorem is applied to establish a saddle-node bifurcation at the turning point of backward bifurcation. The Bogdanov–Takens normal form is derived, from which the local bifurcation curve for a family of homoclinic orbits is formulated. Bifurcation diagrams and numerical simulations, using parameter values estimated for some infectious diseases, are also presented to provide more intuition to the theoretical findings. The results show that sufficiently increasing the removal rate can reduce

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below a subthreshold domain, which leads to disease eradication.

Keywords: epidemic models; non-linear incidence; removal rate; Hopf bifurcation; saddle-node bifurcation; Bogdanov–Takens bifurcation

Journal Article.  0 words. 

Subjects: Applied Mathematics ; Biomathematics and Statistics

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