On a common fixed point theorem in vector-valued $b$-metric spaces: Its consequences and application

We introduce a Ćirić type contraction principle in a vector-valued $b$-metric space that generalizes Perov's contraction principle. We investigate the possible conditions on the mappings $W,E:G\to G$ ( $G$ is a non-empty set), for which these mappings admit a unique common fixed point in $G$ subject to a nonlinear operator $\mathbf{F}:{\mathbb{P}}^m\to {\mathbb{R}}^m$. We illustrate the hypothesis of our findings with examples. We consider an infectious disease model represented by the system of delay integro-differential equations and apply the obtained fixed point theorem to show the existence of a solution to this model.