Hajja, Yemen — A ceasefire is being put to the test in a key port city in Yemen Monday night. Four years of fierce fighting between the Saudi-backed government and Houthi rebels has pushed millions to the brink of starvation.

CBS News has seen the desperation firsthand. Having fled from Saudi airstrikes and vicious fighting, people in remote northern Yemen are now up against another killer: malnutrition.

Inside one tent, a woman named Fatima was frozen with grief and shock. Nurse Makia Mehdi explains her baby starved to death just last week.

A teenager named Sara is also deeply traumatized. In 2015, Saudi warplanes bombed another camp in the area and 43 people died. Sara saw it all and hasn't said a word since.

Makia guessed that 15,000 people in this district alone are marooned in misery and utterly broke. The cruel paradox is that a few miles away on the main road, there's plenty of food the desperate can't dream of buying.

In this rugged landscape, the Aslam Clinic is a beacon of hope. Inside, Makia was coaxing 25 malnourished babies back from the brink. Assam is 2 years old and weighs half of what he should. But he won't eat the high-nutrition paste Makia offers him, because he can't keep it down.

International aid groups led by the U.N. are battling political and logistical obstacles to reach the civilian victims of this war. But this is an emergency and time is running out.