HAVANA (AP) — Tropical Storm Fay lashed southeastern Cuba with downpours and heavy winds Sunday and was expected to churn its way over the center of the island before heading toward Florida.

Authorities evacuated dozens of sparsely populated, low-lying communities and ordered Cubans to pay close attention to the storm, which they said could spark strong storm surges, flooding and mudslides.

Jose Rubiera, Cuba's chief meteorologist, said Fay had brought gusts of wind of up to 70 mph (110 kph) per hour as its center roared close to two coastal communities on the island's southern tip. Still, he said civil defense officials were mostly concerned about the effects of heavy rains.

The storm was expected to gain force and could be near hurricane strength when it moves over Cuba late Sunday and zeros in on Florida, where officials declared a state of emergency. It has already killed at least four people, after battering Haiti and the Dominican Republic with weekend torrential rains and floods.

Cuba's government issued a hurricane warning from Havana eastward to central Sancti Spiritus province. A hurricane warning was also in effect for the Florida keys, where authorities ordered visitors to leave.

The storm was centered about 75 miles (125 kilometers) southwest of Camaguey, Cuba, or about 350 miles (560 kilometers) southeast of Key West, Florida on Sunday, according to the U.S. National Hurricane Center in Miami. It had maximum sustained winds near 50 mph (85 kph), was moving west-northwest at 13 mph (20 kph) and was expected to turn to the northwest by late Monday.

Authorities in four eastern and central Cuban provinces evacuated residents, pulled fishing boats out of the water and moved farm animals to higher ground, while setting up temporary shelters and food distribution centers, civil defense authorities said. In Havana, dark clouds rolled in by midday, but rainfall was light and there was little sign of concern on the streets.

But Fay already was drenching the southern coast. In the city of Niquero, authorities converted a downtown hotel into a shelter for evacuees.

"It's raining intensely, but the wind comes and goes," said a front desk clerk at the Hotel Niquero, who said he was not authorized to have his name appear in the foreign press. He said there had been no major flooding and that "we hope the worst is over."

A man died Saturday in Haiti while trying to cross a river in Leogane, south of Port-au-Prince.

Fay also damaged crucial farmland as Haiti contends with a food crisis that has sparked deadly riots, although it was unclear how many acres were affected.

In neighboring Dominican Republic, flood waters drowned a woman and her 13-year-old niece and 5-year-old nephew, after they tried to cross a swollen river in a car, civil defense agency director Luis Luna Paulino said.

The hurricane center said Fay could be near hurricane strength as it approaches the Florida Keys late Monday or early Tuesday and could make landfall somewhere along the western coast of Florida.

A light stream of traffic headed out of Key West on Sunday morning as officials urged visitors to leave the string of low-lying islands. Most of the islands sit at sea level and could be flooded by Fay's storm surge.

A tropical storm watch was in effect for much of southern Florida. A watch means storm conditions are possible within 36 hours.