A few ways this could go. Don't know about the EU, most of which is responding in a retarded manner tbh. The USA (which is also retarded) will avoid full scale martial law at all costs. It would be viewed as an act of war by almost half the population, who are also heavily armed. They eventually may have no choice though.
If they do have to unveil a mass quarantine where the National Guard or Army is called in to enforce law, they will not use those words ("martial law"). They will try to put a happy face on whatever they do.
The fact the US is so heavily armed is a catch 22 in a situation like this. On one hand, the gvt won't be able to enforce directives with too heavy a hand. On the other, you could end up with situations where amidst empty supermarkets and mass unrest, Chad and his gang start going door to door while locked and loaded searching for food or supplies. Not good.
I don't think most normies realize how thin of ice western society is standing on right now. Could go full out mad max by the first few weeks in April.
