Despite efforts made to model and predict COVID-19 transmission, large predictive uncertainty remains. Failure to understand the dynamics of the nonlinear pandemic prediction model is an important reason. To this end, local and multiple global sensitivity analysis approaches are synthetically applied to analyze the sensitivities of parameters and initial state variables and community size (N) in susceptible-infected-recovered (SIR) and its variant susceptible-exposed-infected-recovered (SEIR) models and basic reproduction number (R0), aiming to provide prior information for parameter estimation and suggestions for COVID-19 prevention and control measures. We found that N influences both the maximum number of actively infected cases and the date on which the maximum number of actively infected cases is reached. The high effect of N on maximum actively infected cases and peak date suggests the necessity of isolating the infected cases in a small community. The protection rate and average quarantined time are most sensitive to the infected populations, with a summation of their first-order sensitivity indices greater than 0.585, and their interactions are also substantial, being 0.389 and 0.334, respectively. The high sensitivities and interaction between the protection rate and average quarantined time suggest that protection and isolation measures should always be implemented in conjunction and started as early as possible. These findings provide insights into the predictability of the pandemic models by estimating influential parameters and suggest how to effectively prevent and control epidemic transmission.