Motivated by the recent medical experiments on vectored immunoprophylaxis for protection against HIV infection and the success of some theoretical models in simulating the observed behavior, a simple kinetic model is proposed to explain both the T cell and viral dynamics after pathogenic attack. The model, described through a system of just three coupled differential equations, is successful in faithfully simulating the autocatalytic augmentations of both T-cells and auto-inhibition of virions. A long-time coexistence of both virions and T-cells, either with damping or growing, or with sustained oscillations, as evidences of some complicated dynamical behavior is observed in the proposed model. The possible physiological consequences that can account for a variety of in vivo experimental data related to viral attacks is thoroughly assessed. Paradoxical situations showing a fall off of T-cell count in spite of enhanced immune activity is highlighted over a wide range of values of system parameters. © 2017, The National Academy of Sciences, India.