More than 30,800 people have died from coronavirus, with the number of cases worldwide exceeding 660,000.
Shevchenko, who was a child at the time of the Chernobyl disaster in 1986 that reportedly killed 54 people and is said to have led to thousands of deaths, said he saw similarities.
"I lived a moment very similar to this. At the time, I was nine years old," the Ukraine coach said.
"The only thing we have to do is avoid stupid behaviour. None of us can be sure that we have no contracted the virus.
"We must not go out, we must not think only of us, but above all of those people for whom the infection would be a greater problem."
Shevchenko thanked doctors and nurses and urged people to respect guidance from their governments.
Italy, where Shevchenko played with AC Milan from 1999-2006, has seen more than 10,000 deaths due to coronavirus.
"The only solution is to respect the rules of the government – that is to stay at home and give doctors the opportunity to do their job, which is really huge," he said.
"Give them volunteer nurses, they are the heroes of our time. They must be thanked for all they are doing."