Bifurcation Analysis of an SIR Model with Logistic Growth, Nonlinear Incidence, and Saturated Treatment

There is a wide range of works that have proposed mathematical models to describe the spread of infectious diseases within human populations. Based on such models, researchers can evaluate the effect of applying different strategies for the treatment of diseases. In this article, we generalize previ...

Full description

Saved in:
Bibliographic Details
Published inComplexity (New York, N.Y.) Vol. 2019; no. 2019; pp. 1 - 21
Main Authors Perez, Angel G.C, Avila-Vales, Eric, Garcia-Almeida, Gerardo Emilio
Format Journal Article
LanguageEnglish
Published Cairo, Egypt Hindawi Publishing Corporation 01.01.2019
Hindawi
Hindawi Limited
Wiley
Subjects
Analysis
Care and treatment
Communicable diseases
Computer simulation
Disease transmission
Dynamical systems
Economic models
Epidemics
Growth rate
Hopf bifurcation
Incidence
Infectious diseases
Mathematical models
Population
Therapeutics
Online AccessGet full text

Cover

More Information
Summary:There is a wide range of works that have proposed mathematical models to describe the spread of infectious diseases within human populations. Based on such models, researchers can evaluate the effect of applying different strategies for the treatment of diseases. In this article, we generalize previous models by studying an SIR epidemic model with a nonlinear incidence rate, saturated Holling type II treatment rate, and logistic growth. We compute the basic reproduction number and determine conditions for the local stability of equilibria and the existence of backward bifurcation and Hopf bifurcation. We also show that, when the disease transmission rate and treatment parameter are varied, our model undergoes a Bogdanov-Takens bifurcation of codimension 2 or 3. Simulations of the solutions and numerical continuation of equilibria are carried out to generate 2D and 3D bifurcation diagrams, as well as several related phase portraits that illustrate our results. Our work shows that incorporating these factors into epidemic models can lead to very complex dynamics.
ISSN:1076-2787
1099-0526
DOI:10.1155/2019/9876013