Doug Ducey, Kate Gallego

Phoenix Mayor Kate Gallego and Gov. Doug Ducey at a 2019 news conference.

PHOENIX — With a new spike in COVID-19 cases, the Democratic mayors of four cities renewed their demand Friday that Gov. Doug Ducey impose a statewide mask mandate even though they concede that their police are not enforcing their own local orders.

In an online press briefing, the mayors also said they want a mandatory 14-day quarantine of people who visit the state or proof of a negative test for COVID-19.

The governor’s chief of staff said there’s nothing new in their arguments. Daniel Scarpinato said his boss sees no reason to back away from his belief that the best way to get people to mask up is through messaging on the importance and securing voluntary compliance.

Whether that message is working is in serious question.

The Department of Health Services reported 4,471 new cases of infection from the coronavirus. That compares with a peak of 5,450 at the end of June when Ducey imposed new restrictions on business. There were fewer than 600 a day in September before the new upswing. There also were 43 deaths reported, bringing the statewide tally to 6,427.

Meanwhile, a separate metric shows that the rate of spread in Arizona continues to rise.

The Institute for Health Metrics Evaluation predicts that, absent some change in policies — whether masks or other restrictions — the demand for beds in intensive-care units will exceed capacity sometime this coming month.

Tucson Mayor Regina Romero said the messaging about masks being done by the governor’s office, backed by a paid public relations and media campaign, isn’t enough.

“What we need is decisive statewide action,’’ she said. “Unfortunately, we have yet to see that from Gov. Ducey.’’

In fact, Ducey did not want any sort of mask mandate at all, even issuing an executive order barring local officials from imposing their own. It was only after Romero and others threatened to go ahead anyway — and provoke a legal fight — the governor backed down.

Ducey is using that option for local action to explain why more is not needed, claiming that existing ordinances cover 90% of Arizonans. Romero scoffed at that as an answer.

“COVID-19 does not see county or city limits,’’ she said.

Phoenix Mayor Kate Gallego cited the predictions that hospitals could run out of ICU beds. And that, in turn, could lead to implementation of the Crisis Standards of Care that would permit medical personnel to essentially triage patients to determine who gets the limited care available.

She also pointed out that Ducey, at his most recent press conference, said he believes that masks do help stop the spread. But that, Gallego said, is not enough.

“You can’t say you believe in it and not implement the policy,’’ she said.

That’s not how Ducey sees it.

At a Wednesday press event, the governor noted officials in some communities, facing local opposition, have opted not to impose mask mandate or have repealed the ones that were in place. That sort of controversy, the governor said, doesn’t help build compliance.

“What I want to avoid is some of the division and politics that have happened around this issue,’’ Ducey said, saying he prefers “participation and cooperation.’’

That goes to the question of enforcement.

All four cities — Tucson, Phoenix, Tolleson and Flagstaff — have mask mandates. Romero conceded there are people at events in her city who are not complying.

Citations for violating the city ordinance are not being issued.

“Pima County Health Department is responsible for enforcement,’’ Romero said.

That’s not exactly true. When Ducey amended his executive order allowing local mask-up ordinances, he also gave cities the authority to compel compliance.

Romero said it’s up to individuals who see violations to call the health department to report not just the local mask ordinance but also situations like crowding in bars which is prohibited under the statewide executive order.

“We know and realize there are bad actors,’’ she said.

What of the role of Tucson police?

“TPD doesn’t go into a business and take it upon themselves to enforce that mask mandate,’’ Romero said. “What we’re doing is coordinating and cooperating with Pima County Health Department.’’

That raises the question of what would be the point of a statewide mask mandate if enforcement won’t be any stricter than it is now. Romero said a single mandate with a single message is better than the “patchwork’’ set of ordinances that exist now.

“That’s why it’s important that Gov. Ducey lead,’’ she said. Romero said it also means that local health departments can work directly with the state.

Gallego conceded the issue of curbing the virus may be less tied to people wearing masks in public than other forms of transmission. She said there is an issue of spread in “small groups of people who know each other, including family members.

“It is not going to be possible to have enforcement at people’s homes if they are having big Thanksgivings in their dining room,’’ she said.

Gallego said her city’s police department has had “hundreds of educational contacts with our residents talking about the importance of masks.’’

“We believe that arresting people and putting them in jail — that would be one of the most likely areas for transmission — is not the way to get through this,’’ she said.

The issue of mandatory testing of new arrivals goes a step beyond the directive by Ducey on Wednesday for the state health department to set up sites at Sky Harbor, Tucson International and Phoenix Gateway airports at which people can get free saliva testing and results within 48 hours. The governor said anyone who tests positive would be expected to self-quarantine.

The governor issued a similar edict in April on visitors from the New York City area after there was a spike in cases there.