In this paper, a theoretical analysis model is proposed for the linear growth of the Richtmyer-Meshkov instability in elastoplastic solid medium-vacuum interface under the explosion shock wave loading. The analysis of the dynamic evolution of small perturbations shows that after the initial phase inversion, some perturbations would stop growing after they have reached their maximum amplitude, some others would continue to grow and then form jetting from the solid-vacuum interfaces. Numerical simulations show excellent agreement with the experimental results of explosively-driven Richtmyer-Meshkov instability in the sample of copper. The effects of two physical factors on the maximum amplitude of spikes are also studied numerically. The first physical factor is the initial configuration of the perturbation, which is expressed as the time values of the initial wave number and initial amplitude. With increasing the value of the initial configuration, the maximum amplitudes of the spikes would become greater while the growth of perturbations is suppressed. On the other hand, the maximum amplitudes of spikes would become smaller in the solid which has a higher yield strength when the initial configuration keeps unchanged. Further investigations show that the boundary of the stage division between the stable growth and the unstable growth is revealed by a combination parameter form of the two physical factors, which is expressed as the ratio of initial configuration to yield strength. In the stable stage, the linear relation between the non-dimensional maximum amplitude and the non-dimensional maximum growth rate of the spikes is fitted with the coefficient value 0.30, which is very close to 0.29, a theoretical prediction based on the Newton's second law analysis. Considering the shock Hugoniot relations in the elastoplastic medium and the maximum growth rate equation of the Richtmyer-Meshkov instability in ideal fluid, the linear model is improved to add the effects of the loading shockwave pressure and the compression acoustic impedance of the material on the amplitude growth of the spike to the analytical model proposed by the former researchers. Extensive numerical simulations are performed to show that the linear model could accurately describe the growth factor of the spikes in the stable cases in different metal materials, such as copper, aluminum, and stain-less steels. In the numerical analysis of the scope of application of the linear model, a rough estimation of the stage division boundary between the stable and unstable growth is given as 0.8 GPa-1. When the ratio of initial configuration to yield strength is lower than the division boundary, the perturbation growth would be stable and the linear model could describe the growth law of the spikes.