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The transcritical Bogdanov Takens bifurcation with boundary due to the risk perception on a recruitment epidemiological model

Jocelyn A. Castro-Echeverría, Fernando Verduzco, Jorge X. Velasco-Hernández

TL;DR

This model undergoes the transcritical Bogdanov-Takens bifurcation with boundary, where the system experiences the transcritical bifurcation between the disease-free equilibrium point and the endemic equilibrium point.

Abstract

We analyze an epidemiological model with treatment and recruitment considering the risk perception. In this model, we consider an exponential function as a recruitment rate. We have found that this model undergoes the transcritical Bogdanov-Takens bifurcation with boundary, where the system experiences the transcritical bifurcation between the disease-free equilibrium point and the endemic equilibrium point. The Hopf bifurcation also arises at the endemic equilibrium point, this is, the appearance or disappearance of a limit cycle, and finally, the Homoclinic bifurcation which transforms the limit cycle into a homoclinic cycle, starting and ending at the disease-free equilibrium point.

The transcritical Bogdanov Takens bifurcation with boundary due to the risk perception on a recruitment epidemiological model

TL;DR

This model undergoes the transcritical Bogdanov-Takens bifurcation with boundary, where the system experiences the transcritical bifurcation between the disease-free equilibrium point and the endemic equilibrium point.

Abstract

We analyze an epidemiological model with treatment and recruitment considering the risk perception. In this model, we consider an exponential function as a recruitment rate. We have found that this model undergoes the transcritical Bogdanov-Takens bifurcation with boundary, where the system experiences the transcritical bifurcation between the disease-free equilibrium point and the endemic equilibrium point. The Hopf bifurcation also arises at the endemic equilibrium point, this is, the appearance or disappearance of a limit cycle, and finally, the Homoclinic bifurcation which transforms the limit cycle into a homoclinic cycle, starting and ending at the disease-free equilibrium point.
Paper Structure (9 sections, 5 theorems, 35 equations, 5 figures, 2 tables)

This paper contains 9 sections, 5 theorems, 35 equations, 5 figures, 2 tables.

Table of Contents

  1. Introduction
  2. Model setup
  3. Desease free equilibrium point
  4. Endemic equilibrium point
  5. Bifurcation analysis
  6. Transcritical bifurcation (Forward bifurcation).
  7. Transcritical Bogdanov-Takens bifurcation
  8. Center manifold analysis
  9. Simulations

Key Result

Theorem 1

The point $E_0=(S_0,I_0,U_0)=\left(\frac{T}{\mu+1},\;0,\;0\right)$ is the desease-free equilibrium point of system (sistema-reducido), this equilibrium is always present and its stability is given as follows: where $R_0=\frac{\beta}{\mu+\tau}$ is the basic reproductive number.

Figures (5)

  • Figure 1: Diagram for system (\ref{['sistema-original']}).
  • Figure 2: Simulation of system (\ref{['sistema-reducido']}) at the invariant plane $I=0$.
  • Figure 3: tBt Bifurcation with boundary diagram presented in castro2020bifurcation.
  • Figure 4: Simulation at the $S$-$I$ plane, a transcritical bifurcation ocurrs between the disease-free and the endemic equilibrium points, meanwhile the cycle arises, converging to the homoclinic cycle.
  • Figure 5: 3D simulation of the cycle around the endemic equlibrium converging to the homoclinic curve at the disease-free equilibrium.

Theorems & Definitions (6)

  • Remark 1
  • Theorem 1
  • Proposition 1
  • Theorem 2
  • Theorem 3
  • Lemma 1