Stability and bifurcation analysis of a reaction-diffusion SIRS epidemic model with the general saturated incidence rate. (English) Zbl 07923960
Summary: In this paper, we are concerned with the dynamics of a reaction-diffusion SIRS epidemic model with the general saturated nonlinear incidence rates. Firstly, we show the global existence and boundedness of the in-time solutions for the parabolic system. Secondly, for the ODEs system, we analyze the existence and stability of the disease-free equilibrium solution, the endemic equilibrium solutions as well as the bifurcating periodic solution. In particular, in the language of the basic reproduction number, we are able to address the existence of the saddle-node-like bifurcation and the secondary bifurcation (Hopf bifurcation). Our results also suggest that the ODEs system has a Allee effect, i.e., one can expect either the coexistence of a stable disease-free equilibrium and a stable endemic equilibrium solution, or the coexistence of a stable disease-free equilibrium solution and a stable periodic solution. Finally, for the PDEs system, we are capable of deriving the Turing instability criteria in terms of the diffusion rates for both the endemic equilibrium solutions and the Hopf bifurcating periodic solution. The onset of Turing instability can bring out multi-level bifurcations and manifest itself as the appearance of new spatiotemporal patterns. It seems also interesting to note that \(p\) and \(k\), appearing in the saturated incidence rate \(kSI^p /(1+\alpha I^p)\), tend to play far reaching roles in the spatiotemporal pattern formations.
MSC:
| 34C23 | Bifurcation theory for ordinary differential equations |
| 34C25 | Periodic solutions to ordinary differential equations |
| 34C45 | Invariant manifolds for ordinary differential equations |
| 34D10 | Perturbations of ordinary differential equations |
| 35B32 | Bifurcations in context of PDEs |
| 35B36 | Pattern formations in context of PDEs |
| 35K57 | Reaction-diffusion equations |
| 35K58 | Semilinear parabolic equations |
Keywords:
the diffusive SIRS model; general saturated incidence rate; saddle-node-like bifurcation and the multi-level bifurcation; Turing instability of both the equilibrium solutions and the periodic solutionReferences:
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