According to the assumption that the infectious disease exhibits a large immune failure rate within the population, a slow–fast SIRS epidemic model with a nonlinear incidence rate is studied in this paper. Based on the geometric singular perturbation theory, the multi-scale dynamics of the model are investigated. It is shown that the system has a globally stable positive equilibrium for <InlineEquation ID="IEq1"> <EquationSource Format="TEX">$n<n_{\epsilon }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo><</mo> <msub> <mi>n</mi> <mi>ϵ</mi> </msub> </mrow> </math></EquationSource> </InlineEquation> and a hyperbolically stable relaxation oscillation cycle for <InlineEquation ID="IEq2"> <EquationSource Format="TEX">$n>n_{\epsilon }$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <mi>n</mi> <mo>></mo> <msub> <mi>n</mi> <mi>ϵ</mi> </msub> </mrow> </math></EquationSource> </InlineEquation>, indicating that the infectious disease is uniformly persistent. Furthermore, by choosing parameter n as the bifurcation parameter, we prove that the system undergoes a singular Hopf bifurcation and a canard explosion bifurcation as n crosses the singular Hopf bifurcation curve <InlineEquation ID="IEq3"> <EquationSource Format="TEX">$n_H(\sqrt{\epsilon })$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>n</mi> <mi>H</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msqrt> <mi>ϵ</mi> </msqrt> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation> and the canard explosion curve <InlineEquation ID="IEq4"> <EquationSource Format="TEX">$n_c(\sqrt{\epsilon })$</EquationSource> <EquationSource Format="MATHML"><math> <mrow> <msub> <mi>n</mi> <mi>c</mi> </msub> <mrow> <mo stretchy="false">(</mo> <msqrt> <mi>ϵ</mi> </msqrt> <mo stretchy="false">)</mo> </mrow> </mrow> </math></EquationSource> </InlineEquation>. Moreover, the canard cycles bifurcated from the singular slow–fast cycles are hyperbolically stable. Finally, some numerical simulations are presented to illustrate our theoretical results and demonstrate that the infectious disease persists in the form of a positive coexistent steady state or positive periodic coexistent oscillations for different positive initial populations.

Multi-scale dynamics of a slow–fast SIRS model with nonlinear incidence rate

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发表日期：

2025年12月5日

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Xiao Wu,
Ping Zhang,
Shuying Lu,
Feng Xie

摘要

According to the assumption that the infectious disease exhibits a large immune failure rate within the population, a slow–fast SIRS epidemic model with a nonlinear incidence rate is studied in this paper. Based on the geometric singular perturbation theory, the multi-scale dynamics of the model are investigated. It is shown that the system has a globally stable positive equilibrium for $n<n_{\epsilon }$ $n < n_{ϵ}$ and a hyperbolically stable relaxation oscillation cycle for $n>n_{\epsilon }$ $n > n_{ϵ}$ , indicating that the infectious disease is uniformly persistent. Furthermore, by choosing parameter n as the bifurcation parameter, we prove that the system undergoes a singular Hopf bifurcation and a canard explosion bifurcation as n crosses the singular Hopf bifurcation curve $n_H(\sqrt{\epsilon })$ $n_{H} (\sqrt{ϵ})$ and the canard explosion curve $n_c(\sqrt{\epsilon })$ $n_{c} (\sqrt{ϵ})$ . Moreover, the canard cycles bifurcated from the singular slow–fast cycles are hyperbolically stable. Finally, some numerical simulations are presented to illustrate our theoretical results and demonstrate that the infectious disease persists in the form of a positive coexistent steady state or positive periodic coexistent oscillations for different positive initial populations.