Retrovirus HTLV-1 gene circuit is characterized by positive and negative feedback phenomena, thus candidating it as a potential relaxation oscillator deliverable into eukaryotes. Here we describe a model of HTLV-1 which, by providing predictions of genes and proteins kinetics, can be helpful for designing gene circuits for eukaryotes, or for optimizing gene therapy approaches which are currently carried out by means of lentiviral vectors or re-engineered adenoviruses. Oscillatory patterns of HTLV-1 gene circuit are predicted when positive feedback is faster than negative feedback. Techniques to mutate the retroviral genome in order to implement practically the above conditions are discussed. Finally, the effect of stochasticity on the system behavior is tested by means of Gillespie algorithm. Simulations show the difficulties to preserve synchronization in viral expression for a multiplicity of cells, while the long tail of the density probability function of the master regulator gene tax/rex, due to its steady state fluctuations, suggests an activation mechanism of HTLV-1 similar to that recently proposed for HIV(1): the virus tends to latency but under certain circumstances, the master regulator gene reaches high values of expression, whose persistence induces the viral replication.