A simple approach for studying stability properties of an SEIRS epidemic model

In this work, we study stability properties of a well-known integer-order SEIRS model with nonlinear incidence and vertical transmission. Firstly, we introduce a simple approach to the analysis of global asymptotic stability (GAS) of the integer-order model. This approach is based on general quadratic Lyapunov functions and characteristic of quadratic forms associated with real matrices. The result is that the GAS of disease-free and disease-endemic equilibrium points is completely established. This provides an important improvement for results constructed in two previous works. Secondly, we generalize the integer-order SEIRS model by considering it in the context of the Caputo fractional-order derivative. After that, the present approach is utilized to investigate the GAS of the proposed fractional-order model. As an important consequence, not only the GAS but also the uniform stability of the fractional-order model are determined fully. Therefore, the applicability of the approach is shown. Finally, a series of numerical experiments is conducted to illustrate and support the theoretical findings.