Stability and Bifurcation Analysis of a Reaction–Diffusion SIRS Epidemic Model with the General Saturated Incidence Rate

• Published in: Journal of nonlinear science 2024-12, Vol.34 (6), Article 101
• Main Authors: She, Gaoyang, Yi, Fengqi
• Format: Article
• Language: English
• Subjects: Analysis; Classical Mechanics; Diffusion rate; Economic Theory/Quantitative Economics/Mathematical Methods; Epidemics; Equilibrium; Hopf bifurcation; Mathematical and Computational Engineering; Mathematical and Computational Physics; Mathematics; Mathematics and Statistics; Nonlinear dynamics; Stability criteria; Theoretical
• ISSN: 0938-8974; 1432-1467
• DOI: 10.1007/s00332-024-10081-z

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 k S I p / ( 1 + α I p ) , tend to play far reaching roles in the spatiotemporal pattern formations.