*(English)*Zbl 1154.92033

Summary: One major drawback associated with the use of anti-retroviral drugs in curtailing HIV spread in a population is the emergence and transmission of HIV strains that are resistant to these drugs. This paper presents a deterministic HIV treatment model, which incorporates a wild (drug sensitive) and a drug-resistant strain, for gaining insights into the dynamical features of the two strains, and determining effective ways to control HIV spread under this situation. Rigorous qualitative analysis of the model reveals that it has a globally asymptotically stable disease-free equilibrium whenever a certain epidemiological threshold (${\mathcal{R}}_{0}^{t}$) is less than unity and that the disease will persist in the population when this threshold exceeds unity. Further, for the case where ${\mathcal{R}}_{0}^{t}>1$, it is shown that the model can have two co-existing endemic equilibria, and the competitive exclusion phenomenon occurs whenever the associated reproduction number of the resistant strain (${\mathcal{R}}_{\text{r}}^{t}$) is greater than that of the wild strain (${\mathcal{R}}_{\text{w}}^{t}$).

Unlike in the treatment model, it is shown that the model without treatment can have a family of infinitely many endemic equilibria when its associated epidemiological threshold $\left({\mathcal{R}}_{0}\right)$ exceeds unity. For the case when ${\mathcal{R}}_{{\text{w}}^{t}}>{\mathcal{R}}_{\text{r}}^{t}$ and ${R}_{\text{w}}^{t}>1$, it is shown that the widespread use of treatment against the wild strain can lead to its elimination from the community if the associated reduction in infectiousness of infected individuals (treated for the wild strain) does not exceed a certain threshold value (in this case, the use of treatment is expected to make ${\mathcal{R}}_{{\text{w}}^{t}}>{\mathcal{R}}_{\text{r}}^{t}$).