“It’s wishful thinking to say al-Qaida is on the brink of defeat,” says Seth Jones, a Rand analyst and adviser to U.S. special operations forces. “They have increased global presence, the number of attacks by affiliates has risen, and in some places like Yemen, they’ve expanded control of territory.”

It’s a complicated, somewhat murky picture for Americans to grasp.

U.S. officials say bin Laden’s old team is all but dismantled. But they say new branches are hitting Western targets and U.S. allies overseas, and still aspire to match their parent organization’s milestone of Sept. 11, 2001.

The deadliest is in Yemen.

“They are continuing to try to again, carry out an attack against U.S. persons inside of Yemen, as well as against the homeland,” White House counterterrorism advisor John Brennan said Sunday on ABC’s “This Week.”

“We’re working very closely with our Yemeni partners to track down all these leads,” he said.

Brennan says there’s no sign of an active revenge plot against U.S. targets, but U.S. citizens in Pakistan and beyond are being warned to be vigilant ahead of the May 2 anniversary of the night raid. U.S. helicopters swooped down on bin Laden’s compound in the Pakistani army town of Abbottabad, killing him, one of his sons, two couriers and their wives.

The last view for Americans of the mastermind behind the Sept. 11 attacks was that of a wizened old man sitting in front of an old television, wrapped in a blanket.

The world may never see photographic proof of his death. U.S. District Judge James E. Boasberg in Washington ruled last week that the Obama administration, under the Freedom of Information Act, would not have to turn over images of bin Laden during or after the raid.

“Verbal descriptions of the death and burial of Osama Bin Laden will have to suffice,” Boasberg wrote in his ruling on the lawsuit by the public interest group Judicial Watch.

Bin Laden’s killing and al-Qaida’s stumbling efforts to regroup are now the national security centerpiece of President Barack Obama’s re-election campaign.

The White House frequently cites the president’s decision to approve the raid, with only a 50-50 chance that bin Laden was even at the compound. Obama could have gone down in history as the man who put the Navy SEALs and the relationship with Pakistan in jeopardy, while failing to catch the al-Qaida leader.

“Al-Qaida was and is our No. 1 enemy,” White House spokesman Jay Carney said last week. “So it’s a part of his foreign policy record, obviously, but it’s also part of a very serious endeavor to keep our country safe.”

How safe remains in question.

U.S. officials say al-Qaida is less able to carry out a complex attack like Sept. 11 and they rule out al-Qaida’s ability to attack with weapons of mass destruction in the coming year. These officials spoke on condition of anonymity because they say publicly identifying themselves could make them a target of the terrorist group.

U.S. counterterrorist forces have killed roughly half of al-Qaida’s top 20 leaders since the raid. That includes U.S.-born cleric Anwar al-Awlaki, killed by a drone in Yemen last September, less than six months after bin Laden’s death.

Only a few of the original al-Qaida team remain, and most of the new names on the U.S. target lists are relative unknowns, officials say.

“The last terror attack (in the West) was seven years ago in London and they haven’t had any major attacks in the U.S.” says Peter Bergen, an al-Qaida expert who once met bin Laden. “They are recruiting no-hopers and dead-enders.”

Yet Zawahri is still out there. Though constantly hunted, he has managed to release 13 audio and video messages to followers since bin Laden’s death, a near record-rate of release according to the IntelCenter, a private intelligence firm. He has urged followers to seize on the unrest left by the Arab Spring to build organizations and influence in Egypt, Libya and elsewhere, and back rebels in Syria — a call that U.S. intelligence officials say is being heeded.

U.S. attempts to deliver a “knockout punch” to Zawahri and his followers in Pakistan have been hamstrung by a breakdown in relations with Pakistan’s government over the bin Laden raid.

Pakistani officials saw the raid as a violation of their sovereignty, made worse by a U.S. friendly fire attack that killed almost two dozen Pakistani troops on the border with Afghanistan last fall. Pakistan’s parliament called for a redrafting of what the U.S. is allowed to do, and where.

CIA drone strikes in Pakistan’s border area continue, but are limited to a relatively small area of the tribal region.

“Our efforts are focused on one small kill box and, we’ve hit them hard, but they still maintain a vital network throughout Pakistan” says Bill Roggio, editor of the Long War Journal, which tracks U.S. counterterrorism efforts worldwide.

Al-Qaida also takes shelter in Pakistan’s urban areas, as shown by the bin Laden raid, and the CIA’s efforts to search those areas is often blocked by the Pakistani intelligence service.

U.S. officials say they believe factions within the agency shelter and even fund al-Qaida’s senior leaders and related militant groups such as the Haqqani network, which attacks U.S. troops in Afghanistan, from their Pakistani safe haven. Pakistan denies the charge.

Afghanistan is the temporary home to up to 100 al-Qaida fighters at any single time, U.S. officials say, adding that a steady series of U.S. special operations raids is essential to keeping them out. With the withdrawal of U.S. forces, U.S. counterterrorism officials fear al-Qaida could return.

By the numbers, al-Qaida’s greatest presence is still greatest in Iraq, where intelligence officials estimate up to a 1,000 fighters have refocused their campaign from striking now-absent U.S. troops to hitting the country’s Shiite-dominated government.

Yemen’s al-Qaida of the Arabian Peninsula (AQAP) is becoming a major draw for foreign fighters as it carves out a stronghold in the south of the country, easily defeating Yemeni forces preoccupied battling tribal and political unrest. The White House recently agreed to expanded drone strikes to give the CIA and the military greater leeway to target militant leaders.

This al-Qaida group has been a major threat since 2009, when one of its adherents tried to bring down a jetliner over Detroit.

Al-Qaida affiliates such as al-Shabab in Somalia are struggling to carry out attacks in the face of a stepped up CIA-U.S. military campaign, and a loss of popular support after blocking U.N. food aid to some 4 million starving Somalis, officials say.

But the group is kept afloat by a stream of cash, partly from piracy and kidnapping of the Somali coast. Brennan, the White House counterterrorism chief, told an audience of CIA officers that total ransom payments paid to Somali pirates increased from approximately $80 million in 2010 to $140 million in 2011, according to remarks obtained by The Associated Press.

Cutting off those finances by persuading companies and U.S. to stop paying up is now central to the terrorism-fighting effort.

So, too, is the strategy of fighting small, smartly and covertly, avoiding land invasions such as those in Iraq and Afghanistan that caused Muslim outrage and helped draw fresh recruits, says Rand’s Jones, in his new book “Hunting in the Shadows,” a comprehensive history of the counterterror search since Sept. 11.

Many U.S. officials cite the Yemen model as the way ahead: a small network of U.S. intelligence and military forces working with local forces to selectively target militants.

“The key challenge will be balancing aggressive counterterrorism operations with the risk of exacerbating the anti-Western global agenda” of al-Qaida and its affiliates, says Robert Cardillo, a senior official in the Office of the Director of National Intelligence.

In other words, adds Jones, “it is a war in which the side that kills the most civilians loses.”

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Kimberly Dozier can be reached at Twitter (at)kimberlydozier