Benzoic acid is mainly used in the preparation of sodium benzoate preservatives, as well as in the synthesis of drugs and dyes. Therefore, a thorough understanding of its properties is of utmost importance. This paper is mainly concerned with the existence of solutions for a class of Hadamard type fractional differential systems with $p$-Laplacian operators on benzoic acid graphs. Meanwhile, the Hyers-Ulam stability of the systems is also proved. Furthermore, an example is presented on a formaldehyde graph to demonstrate the applicability of the conclusions obtained. The novelty of this paper lies in the integration of fractional differential equations with graph theory, utilizing the formaldehyde graph as a specific case for numerical simulation, and providing an approximate solution graph after iterations.

This paper constructs a Susceptible-Exposed-Infectious-Recovered (SEIR) model that takes into account the dynamic changes in the birth population and includes both vaccinated and unvaccinated populations. The basic reproduction number $R_0$ is derived, and its global stability is analyzed. It is proven that when $R_0<1$, the disease-free equilibrium point is globally stable. When $R_0>1$, there exist an unstable disease-free equilibrium and a unique endemic equilibrium. Subsequently, we conduct the numerical simulations to validate theoretical results and plot data visualizations for various scenarios. In addition, the parameter sensitivity analysis and the forward bifurcation of the model are performed to examine how each system parameter specifically affects disease transmission. Our results offer practical insights for public health planning, particularly in optimizing vaccination strategies in populations with high birth rates or waning immunity.