After several months of "lockdown" as the sole answer to the COVID-19 pandemic, balancing the re-opening of society against the implementation of non-pharmaceutical measures needed for minimizing interpersonal contacts has become important. Here, we present a stochastic model that examines this problem. In our model, people are allowed to move between discrete positions on a one-dimensional grid with viral infection possible when two people are collocated at the same site. Our model features three sets of adjustable parameters, which characterize (i) viral transmission, (ii) viral detection, and (iii) degree of personal mobility, and as such, it is able to provide a qualitative assessment of the potential for second-wave infection outbreaks based on the timing, extent, and pattern of the lockdown relaxation strategies. Our results suggest that a full lockdown will yield the lowest number of infections (as anticipated) but we also found that when personal mobility exceeded a critical level, infections increased, quickly reaching a plateau that depended solely on the population density. Confinement was not effective if not accompanied by a detection/quarantine capacity surpassing 40% of the symptomatic patients. Finally, taking action to ensure a viral transmission probability of less than 0.4, which, in real life, may mean actions such as social distancing or mask-wearing, could be as effective as a soft lockdown.