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Most of previous research on survivable networks has been related to unicast communications. In this paper we propose a new approach to network survivability of anycast communications. To the best of our knowledge, anycast defined as one-to-one-of-many technique has not received much attention recently. To provide network survivability, we use the single backup path approach, i.e. each demand has two node-disjoint paths to provide protection against a single node failure. We formulate new ILP models to find the optimal paths for anycast and unicast connections. To test the proposed approach, we run extensive numerical experiments using CPLEX solver on four example network topologies with different scenarios of number of replica servers, and the proportion between unicast and anycast traffic. Obtained results show the efficiency of our solution in terms of various parameters including cost, path length, resource utilization, and time of computation.