Dengue fever is a rapidly spreading vector-borne disease, posing severe health challenges worldwide due to its complex transmission dynamics between humans and mosquitoes. Traditional mathematical models often fail to capture memory effects and environmental influences that significantly affect disease progression. To address this, we formulated a fractional-order dengue transmission model incorporating environmental factors using the Caputo derivative. This approach allows the system dynamics to account for hereditary effects and delayed responses, providing a more realistic representation of dengue outbreaks. We analyzed the model's stability, bifurcation, and sensitivity, and conducted numerical simulations to examine how varying the fractional order and key epidemiological parameters impacts the disease dynamics. Our findings show that lower fractional orders slow the epidemic's progression, while higher orders approach classical integer-order behavior. The mosquito biting rate emerged as the most influential factor affecting the basic reproduction ( $R_0$), whereas disease-induced human mortality had minimal impact. These results highlight critical parameters for controlling dengue's transmission and demonstrate that fractional-order models offer enhanced predictive power and insight compared with traditional models, supporting more effective public health interventions.

In this paper, we considered some systems of rational difference equations of higher order as follows

$$\begin{array}{rcl}{u}_{n+1}& =& \frac{v_{n-6}}{1\pm v_n{u}_{n-1}{v}_{n-2}{u}_{n-3}{v}_{n-4}{u}_{n-5}{v}_{n-6}},\\ v_{n+1}& =& \frac{u_{n-6}}{1\pm u_n{v}_{n-1}{u}_{n-2}{v}_{n-3}{u}_{n-4}{v}_{n-5}{u}_{n-6}},\end{array}$$

where the initial conditions $u_{0,}$ $u_{-1},$ $u_{-2},$ $u_{-3},$ $u_{-4},$ $u_{-5},$ $u_{-6},$ $v_{0,}$ $v_{-1},$ $v_{-2},$ $v_{-3},$ $v_{-4},$ $v_{-5}$ and $v_{-6}$ were arbitrary real numbers. We obtained a closed form of the solutions for each considered system and also some periodic solutions of some systems were found. We presented some numerical examples to explain the obtained theoretical results.