Monotone dynamical systems are systems where partial ordering of states is maintained, and a sub-class of monotone systems are cooperative systems, in which the states reinforce each other through positive feedback. Many biological models fall under this category. Optimal control problems of networked dynamical systems involve state transfer in prescribed time while minimizing a certain cost function. Given the final state, it is required to be verified if it belongs to the reachable set in the given time. In this paper, we compute the smallest interval bounds containing the reachable set for a particular class of cooperative dynamical systems. We solve the reachability problem by formulating it as an optimal control problem and show that the boundary points of the bounding interval are obtained by the application of extreme values of the control input. As an application, we compute the interval bounds on reachable set for a Susceptible-Infected-Susceptible (SIS) epidemic spreading model. © 2019 IEEE.