Most chronic viral infections are managed with small molecule therapies that inhibit replication but are not curative because non-replicating viral forms can persist despite decades of suppressive treatment. There are therefore numerous strategies in development to eradicate all non-replicating viruses from the body. We are currently engineering DNA cleavage enzymes that specifically target hepatitis B virus covalently closed circular DNA (HBV cccDNA), the episomal form of the virus that persists despite potent antiviral therapies. DNA cleavage enzymes, including homing endonucleases or meganucleases, zinc-finger nucleases (ZFNs), TAL effector nucleases (TALENs), and CRISPR-associated system 9 (Cas9) proteins, can disrupt specific regions of viral DNA. Because DNA repair is error prone, the virus can be neutralized after repeated cleavage events when a target sequence becomes mutated. DNA cleavage enzymes will be delivered as genes within viral vectors that enter hepatocytes. Here we develop mathematical models that describe the delivery and intracellular activity of DNA cleavage enzymes. Model simulations predict that high vector to target cell ratio, limited removal of delivery vectors by humoral immunity, and avid binding between enzyme and its DNA target will promote the highest level of cccDNA disruption. Development of de novo resistance to cleavage enzymes may occur if DNA cleavage and error prone repair does not render the viral episome replication incompetent: our model predicts that concurrent delivery of multiple enzymes which target different vital cccDNA regions, or sequential delivery of different enzymes, are both potentially useful strategies for avoiding multi-enzyme resistance. The underlying dynamics of cccDNA persistence are unlikely to impact the probability of cure provided that antiviral therapy is given concurrently during eradication trials. We conclude by describing experiments that can be used to validate the model, which will in turn provide vital information for dose selection for potential curative trials in animals and ultimately humans.