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Bifurcation Analysis of SIRS Model with Nonlinear Incidence Rate

  • Xiaoling Wang [1] ; Shimin Li [2] ; Kuilin Wu [1]
    1. [1] Guizhou University

      Guizhou University

      China

    2. [2] Hangzhou Normal University

      Hangzhou Normal University

      China

  • Localización: Qualitative theory of dynamical systems, ISSN 1575-5460, Vol. 25, Nº 4, 2026
  • Idioma: inglés
  • Enlaces
  • Resumen

    • The bifurcation phenomena of the SIRS model with a general nonlinear incidence rate f (I )S = βI (1 + υ I k−1)S is investigated (βI S represents the new infections caused by single contacts and f (I )S = βIυ I k−1S is the new infective individuals arising from k exposures). Under the demographic constraint governed by the total population equation ˙N = b−dN, the long-term dynamics are confined to the asymptotically invariant plane S + I + R = with = b/d. We study the existence and stability of equilibria of the epidemic model, which is influenced by k and R0 (the basic reproductive number). When R0 < 1, the SIRS model has at most two (three) endemic equilibria for 1 < k ≤ 2 (k > 2). The three endemic equilibria can coincide into a nilpotent focus of codimension 3 or a nilpotent equilibrium of codimension at least 4. Local bifurcation theory is applied to explore the rich variety of dynamical behavior of the model. Normal forms of the epidemic model are applied to explore for different types of complex bifurcation phenomena, such as saddle-node bifurcation, Hopf bifurcation and Bogdanov-Takens including cusp bifurcation of codimension 3 when 1 < k ≤ 2 (codimension 4 when k > 2) and nilpotent equilibrium bifurcation of at least codimension 3. It is showed that the SIRS model can admit two (three) limit cycles coexisting for 1 < k ≤ 2 (k > 2), due to the occurrence of Hopf bifurcation.

      The theoretical results and numerical simulations indicate that various complex bifurcation phenomena depend on parameters k and R0.

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