Intense use of antibiotics for the treatment of diseases such as tuberculosis, malaria, Staphylococcus aureus or gonorrhea has led to rapidly increasing population levels of drug resistance. This has generally necessitated a switch to new drugs and the discontinuation of older ones, after which resistance often only declines slowly or even persists indefinitely. These long-term effects are usually ascribed to low fitness costs of resistance in absence of the drug. Here we show that structure in the host population, in particular heterogeneity in number of contacts, also plays an important role in the reversion dynamics. Host contact structure acts both during the phase of intense treatment, leading to non-random distributions of the resistant strain among the infected population, and after the discontinuation of the drug, by affecting the competition dynamics resulting in a mitigation of fitness advantages. As a consequence, we observe both a lower rate of reversion and a lower probability that reversion to sensitivity on the population level occurs after treatment is stopped. Our simulations show that the impact of heterogeneity in the host structure is maximal in the biologically most plausible parameter range, namely when fitness costs of resistance are small.