Around the world, the people are simultaneously susceptible to or infected with several infections. This work aims at the analysis of the dynamics of transmission of two deadly viruses, COVID-19 and Influenza, using a co-infection epidemiological model by applying the Caputo fractional derivative. Fractional differential equations are currently used worldwide to model physical and biological phenomena. Our comprehension of complicated phenomena is improved when fractional-order derivatives are used to model systems with memory effects and long-range interactions. Mathematical depictions of infectious disease dynamics and dissemination across communities are provided by epidemiological models, which are essential resources for understanding and controlling infectious diseases. These models support informed decision making to prevent outbreaks, evaluate intervention measures, and help researchers and policymakers understand how diseases spread. A subclass of epidemiological models called co-infection models focuses on studying the dynamics of several infectious illnesses that occur in the same population at the same time. They are especially useful in situations where people are simultaneously susceptible to or infected with several infections. Co-infection models provide information on the development of effective control techniques, the progression of disease, and the interactions between several pathogens. The qualitative study via stability analysis is discussed at equilibrium, the reproduction number $R_0$ is computed, and the results are simulated using the Laplace Adomian Decomposition Method (LADM) for Fractional Differential Equations. We employ MATLAB R2023a for graphical presentations and numerical simulations.

Respiratory infectious diseases that affect the respiratory system continue to pose a serious global threat to public health. Influenza and severe acute respiratory syndrome coronavirus 2 (SARS-CoV-2) are viral infections that affect the respiratory system and cause respiratory complications and poor lung function. Co-infection occurs because of the contagious nature of both diseases, which exacerbates their severity. The novelty of this study lies in the use of fuzzy parameters and the ABC fractional differentiation to model the respiratory co-infection dynamics caused by SARS-CoV-2 and influenza. Fuzzy parameters were used to represent imprecise and uncertain system quantities more realistically. This approach demonstrated that combining fractional derivatives with fuzzy parameterization yielded stronger and more flexible representations of system dynamics, facilitating a more precise understanding of behavior under uncertainty. The basic reproduction number was determined. For respiratory co-infections caused by coronavirus disease 2019 (COVID-19) and influenza, the crisp model yielded reproduction numbers of 1.779898 and 0.004849, respectively, whereas the trapezoidal fuzzy model yielded defuzzified values of 1.779867 and 0.004849. A low level of uncertainty regarding the estimated transmission dynamics was indicated by the average values of 1.779867 (SD = 0.000458) and 0.004849 (SD = 0.000001) obtained from the fuzzy reproduction numbers. The provided fuzzy model used trapezoidal fuzzy parameters to account for uncertainty while reproducing values close to those of the crisp model. The resilience of transmission dynamics to parameter uncertainty was demonstrated by the low variance of the fuzzy reproduction numbers. The qualitative behavior of the modified model was studied using stability analysis at the equilibrium point. Fractional differential equations of the model based on the ABC derivative were solved numerically using the Laplace Adomian Decomposition Method (LADM), and a graphical representation of the model variables at various fractional powers was obtained for comparison.