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This study provides a detailed exposition of in-hospital community-acquired methicillin-resistant S. aureus (CA-MRSA) which is a new strain of MRSA, and hospital-acquired methicillin-resistant S. aureus (HA-MRSA) employing Caputo fractional operator. These two strains of MRSA, referred to as staph, have been a serious problem in hospitals and it is known that they give rise to more deaths per year than AIDS. Hence, the transmission dynamics determining whether the CA-MRSA overtakes HA-MRSA is analyzed by means of a non-local fractional derivative. We show the existence and uniqueness of the solutions of the fractional staph infection model through fixed-point theorems. Moreover, stability analysis and iterative solutions are furnished by the recursive procedure. We make use of the parameter values obtained from the Beth Israel Deaconess Medical Center. Analysis of the model under investigation shows that the disease-free equilibrium existing for all parameters is globally asymptotically stable when both R 0 H and R 0 C are less than one. We also carry out the sensitivity analysis to identify the most sensitive parameters for controlling the spread of the infection. Additionally, the solution for the above-mentioned model is obtained by the Laplace-Adomian decomposition method and various simulations are performed by using convenient fractional-order α ., (© Korean Society for Informatics and Computational Applied Mathematics 2021.)