© 2016 IEEE. Routing in previous research works for bounded delay in Delay-Tolerant Networks (DTNs) tried their best to impose bounded delay on the message delivery. However, the existing routing protocols had a lot of issues due to the infinite buffer size and bandwidth assumption, where these resources are limited in realistic environment. In this paper, we modify and extend epidemic routing using ordinary differential equations (ODEs) to distribute message copies among the potential relay nodes to achieve the assigned threshold delay. In particular, we consider the delay requirements in DTNs by analyzing the relationship among the finite buffer threshold delivery delay and the selected relay nodes for packet forwarding. In addition, the proposed scheme defined a utility value in its message field for every forwarding relay node to update remaining TTL value each time. Furthermore, the scheme also utilize the buffer management model to further improve its performance. The considered performance metrics are delivery delay, delivery ratio and buffer occupancy (usage). Extensive simulation results show that the proposed scheme can minimize the delivery delay and uses less resource than epidemic routing to deliver the message timely in DTN environment.