Photo: NICK WAGNER, MBR / TNS Photo: Bob Owen, Staff-photographer / San Antonio Express-News

Texas Gov. Greg Abbott painted a positive picture of the state’s battle against COVID-19 on Tuesday, but said it’s not yet time to fully reopen the economy.

Abbott said Tuesday the number of people testing positive for the coronavirus is “leveling off” and has remained mostly under 1,000 new cases a day even as testing has accelerated. He said that on April 9, Texas had more than 1,000 people test positive for COVID-19, but hasn’t hit that number in 12 straight days.

“It’s not because COVID-19 is suddenly dispersed from the geography of the state of Texas,” Abbott said. “The reason why it is leveling off is because our fellow Texans have done such a great job of reducing their interactions with others.”

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The state continues to see about a 10 percent infection rate for those tested, and the death toll has now surpassed 500 people. The time for total cases to double has grown to 15 days, another good sign. Those hospitalized for the infection remain level at around 1,400 per day.

The governor said he is in touch with private sector employers and health professionals and is hoping to have an announcement soon on reopening more parts of the economy that have been closed since he issued his initial stay-at-home order on March 31.

Last week, he announced a team of more than four dozen advisers — many of them wealthy campaign donors — to help guide a phased reopening. The governor plans to unveil their first round of recommendations on Monday, including a decision on whether to roll back the stay-at-home order.

Abbott continued to highlight the state’s improved testing capabilities, though Texas has been among the lowest in testing per capita.. Texas has now tested over 205,000 people — about 15,000 more than were reported on Monday. The testing rate fluctuates by day, and in the past has typically been higher earlier in the week.

“We continue this ongoing increase in the number of people who are tested,” Abbott said.

But he noted that some of the drive-thru testing sites have been empty at times, according to satellite images.

"I don't know if it's because people aren't feeling like they have any symptoms, or if inadequate information is being given," Abbott said.

Companies including Walgreen’s have recently opened new sites in Harris and Dallas counties, but are taking patients by appointment only.