In this article, mathematical modeling and stability analysis of Lyme disease and its transmission dynamics using Caputo fractional-order derivatives is presented. The compartmental model has been formulated to analyze the spread of Borrelia burgdorferi virus through tick vectors and mammalian hosts. The feasible region is established, and the boundedness of the model is verified. Analytically, the disease-free equilibrium and the basic reproduction number $\left({\Re }_0\right)$ has been determined to assess outbreak potential. By virtue of the fixed-point theory, the existence and uniqueness of solutions has been established. The numerical simulations are obtained via the Runge–Kutta 4 method, demonstrating the model’s ability to capture realistic disease progression. Finally, sensitivity analysis and control strategies (tick population reduction, host vaccination, public awareness, and early treatment) are evaluated, revealing that integrated control measures significantly reduce infection rates and enhance recovery.

The aim of this research article is to explore the framework of tvs-cone b-metric spaces and establish some generalized common fixed point theorems for self mappings under generalized contractive conditions. To extend this theory, we introduce the notion of a generalized Hausdorff distance function in the framework of tvs-cone b-metric spaces and prove some novel fixed point results for multi-valued mappings. Our findings naturally encompass and generalize several known results in tvs-cone metric spaces, cone b-metric spaces, and cone metric spaces. To illustrate the applicability of the developed framework, we apply the main results to Fredholm integral equations.

The primary objective of this study is to establish some new fixed point results in the framework of perturbed $\mathcal{F}$-metric spaces. Specifically, we develop fixed point theorems for three classes of contractions: Banach-type, rational-type, and interpolative contractions. To demonstrate the validity and applicability of the obtained results, some nontrivial examples are provided. Moreover, our findings yield various consequences that extend existing results in both $\mathcal{F}$-metric spaces and perturbed metric spaces. As an application of the main result, we examine the existence of solutions for a Caputo fractional initial value problem.