Optimal therapies of a virus replication model with pharmacological delays based on reverse transcriptase inhibitors and protease inhibitors

A short delay in the pharmacological effect on account of the time required for drug absorption, distribution, and penetration into target cells after application of any anti-viral drug, is defined by the pharmacological delay (Herz et al 1996 Proc. Natl Acad. Sci. USA 93 7247-51). In this paper, a virus replication model with Beddington-DeAngelis incidence rate and the pharmacological and intracellular delays is presented to describe the treatment to cure the virus infection. The optimal controls represent the efficiency of reverse transcriptase inhibitors and protease inhibitors in suppressing viral production and prohibiting new infections. Due to the fact that both the control and state variables contain delays, we derive a necessary conditions for our optimal problem. Based on these results, numerical simulations are implemented not only to show the optimal therapeutic schedules for different infection and release rates, but also to compare the effective of three treatment programs. Furthermore, comparison of therapeutic effects under different maximum tolerable dosages is shown. Our research indicates that (1) the proper and specific treatment program should be determined according to the infection rates of different virus particles; (2) the optimal combined drug treatment is the most efficient; (3) the appropriate proportion of medicament must be formulated during the therapy due to the non-monotonic relationship between maximum tolerable dosages and therapeutic effects; (4) the therapeutic effect is advantageous when the pharmacological delay is considered.