The narratives by Beckett and Moraine drag on a bit, and I found myself wanting to skim them to the dialogue rather than actually read it all. Also, I'm curious as to why you're showing us what's happening from the outsider's view of the quarantine rather than the insider's-- I think there would be more emotional impact if were able to see from Teyla's or Ford's or even McKay's eyes what was happening to their team, to Sheppard, to Elizabeth. And unless Kavanaugh's recalcitrance pertains to the plot somehow, I don't understand why you're including it. It seems like a pointless aside to the tension that's building with Elizabeth's escalating illness, breaking the flow of the narrative unpleasantly.
I understand that you don't want to leave out the repercussion of Elizabeth's illness on the rest of the city, but if anything, you need to reverse how you've been expressing what's going on. The main action should happen in the sickroom and quarantine areas, and the "telling" instead of "showing" that you do (i.e. simply informing the reader what's going on instead of showing the actual playing-out of it) should be relegated to Moraine, Kavanaugh, and Beckett's internal musings.
I think that leaving out the actual experience of John locating the memory of the child in Elizabeth's mind really hurts the story-- you could be showing us his frustration, his patience, his hope, his relief, and instead of you leave it all out. I find that you're doing a lot of telling instead of showing throughout this chapter, and it's really compromising the impact it could/should be having on the reader.
The end, while sweet, makes me wonder why they love each other. Simply liking and respecting each other isn't a natural segue into romantic love. I think you could add a little something to each of John's and Elizabeth's introspections about how the more they learned of each other, the more they found there was to admire rather than shrink from. After all, we all put on public facades, and who we really are is sometimes completely different from how we appear to others.
We all have expectations, based on how we were raised, of how "good" people and "bad" people think and behave. What if once they had access to each others' thoughts, they realized they didn't actually respect/admire who the other was on the inside? Extreme example, but-- what if Elizabeth were secretly a racist? Even if she didn't express racist views in public, learning that she actually was a bigot might well repel John rather than attract him. Or what if Elizabeth learned that John had actually left his fellow soldiers to die, then covered it up and made it look like he was going back to save them to preserve his reputation and keep himself out of jail? Discovering that about John could be quite unpleasant and repulsive to her.
One assumes that neither of them have learned any such dark, ugly secrets. Just the opposite, that the more they learn, the more delighted they are to find that the other suits their concepts of "attractive" and "satisfactory". But I don't get a sense of that appreciation (or even relief) on their parts, that they'd explored the other's mind and found it acceptable.
Sorry to nitpick about it like this, but it really struck me quite a lot.