Mathematical models of amperometric biosensors at three basic types of enzyme kinetics in nonstationary diffusion conditions are discussed. The models are based on nonstationary diffusion equations containing a linear term related to the first-order and nonlinear term related to the Michaelis–Menten and ping–pong of the enzymatic reaction mechanism. In this paper, we obtain approximate closed-form analytical solutions for the nonlinear equations under steady-state condition by using the homotopy analysis method. Analytical expressions for concentrations of substrate and cosubstrate and corresponding current response have been derived for all possible values of parameters. Furthermore, in this work, the numerical simulation of the problem is also reported using Scilab/MATLAB program. An agreement between analytical and numerical results is noted.