This paper studies efficient algorithms for dynamic curing policies and the corresponding network design problems to guarantee the fast extinction of epidemic spread in a susceptible-infected-susceptible (SIS) model. We consider a Markov process-based SIS epidemic model. We provide a computationally efficient curing algorithm based on the curing policy proposed by Drakopoulos, Ozdaglar, and Tsitsiklis (2014). Since the corresponding optimization problem is NP-hard, finding optimal policies is intractable for large graphs. We provide approximation guarantees on the curing budget of the proposed dynamic curing algorithm. We also present a curing algorithm fair to demographic groups. When the total infection rate is high, the original curing policy includes a waiting period in which no measurements are taken to mitigate the spread until the rate slows down. By utilizing network design strategies that either delete edges or reduce their weights, we limit the total infection rate, allowing the curing process to proceed continuously without requiring a waiting period. We provide algorithms with provable guarantees for the considered network design problems. In summary, the proposed curing and network design algorithms together provide an effective and computationally efficient approach that mitigates SIS epidemic spread in networks.