AUSTIN — Gov. Greg Abbott on Monday told Texans they should in coming weeks expect “an exponential increase” in the number of state residents who test positive for the new coronavirus.

Abbott also said that by the end of this week, all Texans who need a COVID-19 test will be able to get one, if they have symptoms and a doctor’s order. By Friday, Abbott estimated more than 10,000 people would be tested for the highly contagious virus each week.

The promise of expanded testing comes as Texans have reported being denied even with a doctor’s order. Commercial labs are expected to carry out the bulk of new testing, since the state’s network of public health labs still require patients meet stringent criteria for testing.

Speaking at a news conference at a state emergency management center in San Antonio, Abbott acknowledged that an estimate of how many Texans have been tested at public laboratories – more than 200 -- hasn’t been updated since Friday.

“That number likely will be increased by the end of the day,” he said, before seeking to prepare state residents for sobering spikes in the statistics in the near future, once testing becomes more available.

The press conference began around the time the White House advised against gatherings larger than 10 people, lowering the earlier recommendation of 50 people. Fifteen officials crowded onto the stage around Abbott. A few -- including the Republican governor -- passed a microphone back and forth as he had two state and two local officials join him in speaking. Abbott’s office said each attendee was screened for a fever before entering the room, and the governor didn’t have advance notice of President Donald Trump’s crowd-size advisory.

Abbott spoke as San Antonio and state officials stood up a new drive-through testing center in the Alamo City.

“Over the coming weeks, … you’re going to see an exponential increase in the number of people who test positive on a daily basis,” Abbott said.

“So people just need to be prepared and not shocked by the mathematical reality that once wide scale testing [happens,] which is now being implemented, there will be a lot more people who identify as testing positive for COVID-19.”

Not everyone will be granted a test, he warned.

"You can't just drive up like you would drive up to a Whataburger," Abbott said, referring to drive-through testing sites that are beginning to open around the state. "You have to have some explanation."

Guidelines for testing will differ based on which lab is carrying it out: commercial or government, Abbott said.

In a briefing Monday with Trump, Vice President Mike Pence and other federal officials, Abbott said he was told “they want to test as many people as possible, but they are not going to test someone with no symptoms whatsoever.”

Private labs should be testing more people, regardless of a patient’s travel history or whether they came into contact with someone known to have the virus, Abbott said. First priority will go to frontline health care workers and emergency responders, Abbott said, followed by seniors and other vulnerable populations likely to develop severe cases of the disease.

Those with symptoms including fever or cough can be referred to testing through a doctor’s order, Abbott said.

In recent weeks, however, a doctor’s order has not guaranteed patients a test at a public health lab run by the state or a county government. With limited capacity to test roughly 273 people a day, those labs have been requiring patients to meet strict criteria, including travel to an area with an outbreak, contact with someone who tested positive or sickness so severe it results in hospitalization.

It’s not clear whether those government lab guidelines will be changing.

It is not clear how many clinics or hospitals are currently contracting with commercial labs, such as LabCorp or Quest Diagnostics, to test patients for COVID-19. Several physicians have said a nationwide shortage of face masks, gowns and other protective equipment could hinder their ability to collect swabs from patients’ throats and noses to submit for testing.

Abbott said testing locations will proliferate rapidly.

Some will be set up by private health care companies; some, in "pods" erected by the Federal Emergency Management Agency.

When statewide plans become fully operational later this week, Texas will be testing far more than 10,000 people per week, he said.

In other actions Monday, state government: