Stiff ordinary differential equations play an important role in mathematical modeling of many biological, physical and chemical phenomena. In this talk, we consider the numerical simulation of stiff initial value problems on semi-infinite intervals. For a class of $\text{ Runge-Kutta }$ methods applied to strictly $\text{dissipative}$ systems, a long-time convergence result on semi-infinite time intervals is obtained, in which the error bounds of the numerical methods are independent of the length of the integration interval. Applications of our results to some classes of high-order implicit $\text{ Runge-Kutta }$ methods are provided, and numerical experiments are also presented to confirm the theoretical results.