We develop a mathematical model to study the effects of non-pharmaceutical interventions (NPIs) on the dynamics of an epidemic. The level of intervention is assessed as a fraction of the population being isolated and depends on the level of incidence of the epidemic in the population. We perform the mathematical analysis of the model and show that, depending on the choice of the prevalence-dependent isolation function, it is possible to create new endemic equilibria and to change the stability of the disease-free equilibrium for which the epidemic vanishes. The model is then applied to the case of the covid-19 pandemic. Several NPI management strategies are considered. In the case of a NPI intensity increasing with the level of infection, it is possible to avoid the initial epidemic peak of great amplitude that would have occurred without intervention and to stabilize the epidemic at a chosen and sufficiently low endemic level. In the case of a NPI intensity decreasing with the level of infection, the epidemic can be driven to extinction by generating an “Allee” effect: when the incidence is below a given level, the epidemic goes extinct while above it, the epidemic will still be able take hold at a lower endemic level. Simulations illustrate that appropriate NPIs could make the Covid-19 vanish relatively fast. We show that in the context of the covid-19 pandemic, most countries have not chosen to use the most efficient strategies.