Abstract The curve characteristics of the breakthrough models were constrained by the mathematical laws that the models themselves followed. The existing breakthrough models such as Bohart–Adams, Thomas, Yoon–Nelson, Clark and modified dose-response models may be not enough to describe all adsorption systems. This work established the Gudermannian and error models only containing the two parameters. The fractal-like Gudermannian and fractal-like error models could describe the diffusion-limited process on the heterogeneous surfaces. The applicability of these models was evaluated by the adjusted coefficient of determination (Adj. R2), reduced chi-square value (χ2) and residual plot. The introduction of the fractal-like kinetics made the Gudermannian and error models have the ability to describe the asymmetric curves. Among these models, the fractal-like Gudermannian model provided the best fit for norfloxacin adsorption on granular activated carbon (Adj. R2 = 0.9991 and χ2 = 1.26 × 10−4) and methylene blue adsorption on porous silica microsphere (Adj. R2 = 0.9997 and χ2 = 6.35 × 10−5). The breakthrough models proposed in this work were an important supplement of adsorption model studies.