Global dynamics of treatment models with time delay

The problem of the asymptomatic dynamics of a treatment model with time delay is considered, subject to two incidence functions, namely standard incidence and Holling type II (saturated) incidence function. Rigorous qualitative analysis of the model shows that for each of the two incidence functions, the model has a globally-asymptotically stable disease-free equilibrium whenever the associated reproduction threshold quantity is less than unity. Further, it has a unique endemic equilibrium when the threshold quantity exceeds unity. For the case with Holling type II incidence function, it is shown that the unique endemic equilibrium of the model is globally-asymptotically stable for a special case. Finally, for each of the two incidence functions, the disease burden decreases with increasing time delay (incubation period). In summary the results in this article is similar to those established in Safi and Gumel (Nonlinear Anal Ser B Real World Appl 12:215–235, 2011) (i.e., treatment models considered here have the same dynamics of quarantine-isolation models in Safi and Gumel (Nonlinear Anal Ser B Real World Appl 12:215–235, 2011).