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Bifurcation analysis for a SIRS model with a nonlinear incidence rate

Xiaoling Wang, Kuilin Wu

TL;DR

The paper addresses a SIRS model with a nonlinear incidence $f(I)S=\beta I(1+\upsilon I^{k-1})S$ and performs a thorough local bifurcation analysis to reveal a rich repertoire of dynamical behaviors, including saddle-node, Bogdanov–Takens, Nilpotent focus, and Hopf bifurcations, with high-codimension scenarios and multiple limit cycles.Using a reduced two-dimensional formulation and center-manifold/normal-form techniques, the authors derive explicit conditions for equilibria, degenerate states, and the unfolding of bifurcations, along with calculations of the first four focal values to classify Hopf bifurcations.Key findings show that, depending on $k$ and parameter combinations (notably the basic reproduction number $R_0$ and the reproduction parameters $\Lambda_0$, $\gamma$), the system can exhibit up to four limit cycles emanating from Hopf bifurcations, and various degenerate cases lead to intricate bifurcation structures.The results underscore the sensitivity of epidemic dynamics to nonlinear incidence and provide theoretical tools for predicting and controlling complex outbreak patterns in public health contexts.

Abstract

In this paper, the main purpose is to explore an SIRS epidemic model with a general nonlinear incidence rate $f(I)S=βI(1+\upsilon I^{k-1})S$ ($k>0$). We analyzed the existence and stability of equilibria of the epidemic model. Local bifurcation theory is applied to explore the rich variety of dynamical behavior of the model. Normal forms of the epidemic model are derived for different types of bifurcation, including Bogdanov-Takens bifurcation, Nilpotent focus bifurcation and Hopf bifurcation. The first four focal values are computed to determine the codimension of the Hopf bifurcation, which can be undergo some limit cycles. Some numerical results and simulations are presented to illustrate these theoretical results.

Bifurcation analysis for a SIRS model with a nonlinear incidence rate

TL;DR

The paper addresses a SIRS model with a nonlinear incidence $f(I)S=\beta I(1+\upsilon I^{k-1})S$ and performs a thorough local bifurcation analysis to reveal a rich repertoire of dynamical behaviors, including saddle-node, Bogdanov–Takens, Nilpotent focus, and Hopf bifurcations, with high-codimension scenarios and multiple limit cycles.Using a reduced two-dimensional formulation and center-manifold/normal-form techniques, the authors derive explicit conditions for equilibria, degenerate states, and the unfolding of bifurcations, along with calculations of the first four focal values to classify Hopf bifurcations.Key findings show that, depending on $k$ and parameter combinations (notably the basic reproduction number $R_0$ and the reproduction parameters $\Lambda_0$, $\gamma$), the system can exhibit up to four limit cycles emanating from Hopf bifurcations, and various degenerate cases lead to intricate bifurcation structures.The results underscore the sensitivity of epidemic dynamics to nonlinear incidence and provide theoretical tools for predicting and controlling complex outbreak patterns in public health contexts.

Abstract

In this paper, the main purpose is to explore an SIRS epidemic model with a general nonlinear incidence rate (). We analyzed the existence and stability of equilibria of the epidemic model. Local bifurcation theory is applied to explore the rich variety of dynamical behavior of the model. Normal forms of the epidemic model are derived for different types of bifurcation, including Bogdanov-Takens bifurcation, Nilpotent focus bifurcation and Hopf bifurcation. The first four focal values are computed to determine the codimension of the Hopf bifurcation, which can be undergo some limit cycles. Some numerical results and simulations are presented to illustrate these theoretical results.

Paper Structure

This paper contains 17 sections, 18 theorems, 106 equations, 17 figures.

Table of Contents

  1. Introduction
  2. Existence of equilibria
  3. Disease free equilibrium
  4. Endemic equilibria
  5. Degenerate equilibria of system \ref{['1.5']}
  6. Nilpotent bifurcations
  7. Saddle-node bifurcation
  8. Bogdanov-Takens bifurcation
  9. Hopf bifurcation
  10. $1<k<2$
  11. $R_i=0$ ($i=0, 1, 2, 3$)
  12. $R_i\neq0$ ($i=0, 1, 2, 3$)
  13. $k=2$
  14. $k>2$
  15. $R_i=0$ ($i=0, 2, 3$)
  16. ...and 2 more sections

Key Result

Lemma 2.1

The disease free equilibrium $E_0(0,0)$ (i.e., boundary equilibrium) is a stable node (saddle) if $R_0<1$ ($R_0>1$), see Fig.1(a) (Fig.1(b)). If $R_0=1$, then $E_0$ is a saddle-node with the parabolic sector on the right half-plane, see Fig.1(c).

Figures (17)

  • Figure 2.1: The disease free equilibrium $E_0$ of system \ref{['1.5']}: (a) $E_0$ is a stable node for $R_0<1$; (b) $E_0$ is a saddle for $R_0>1$; (c) $E_0$ is a saddle-node for $R_0=1$.
  • Figure 2.2: The curve of $H'(x)$ when $k>2$.
  • Figure 2.3: The curve of $H(x)$ when $H'(\bar{x}_c)\leq0$.
  • Figure 2.4: Graph of $H(x)$ for $H'(\bar{x}_c)>0$ and $R_0\leq1$.
  • Figure 2.5: Graph of $H(x)$ for $H'(\bar{x}_c)>0$ and $R_0>1$.
  • ...and 12 more figures

Theorems & Definitions (29)

  • Lemma 2.1
  • Lemma 2.2
  • Lemma 2.3
  • Lemma 2.4
  • Theorem 3.1
  • proof
  • Theorem 3.2
  • Lemma 3.1
  • proof
  • Lemma 3.2
  • ...and 19 more