We in this paper explore a novel model of influence minimization for the need to effectively prevent the outbreak of epidemic-prone spread on networks. The current network-blocking models usually report the expected number of infected nodes under the limited number of cutting edges. However, to control the epidemic-prone spread such as dengue fever, epidemiologists tend to deploy a cost-effective intervention with low outbreak risk, but the outbreak risk cannot be estimated based on the expectation of infected count. We in this paper explore the first solution to estimate the probability that can successfully bound the infected count below the out-of-control threshold, which can be logically mapped to the outbreak risk and can facilitate the authority to adaptively adjust the intervention cost for the need of risk control. We elaborate upon the proposed MCP (standing for Maximization of Containment Probability) problem and show that it is a NP-hard challenge without the submodular property. We further devise an effective measurement of sufficient number of Monte Carlo iterations based on the relative error of Monte Carol integration. The experimental results show that our proposed algorithm with small iterations can deliver the qualified guarantee of containment probability, demonstrating its feasibility for real applications.