I’ve come across the Latin word desiderium in a few different contexts recently. If you look it up in a dictionary, you’ll find something like the following:
desiderium, desideri(i) N. desire/longing/want/requirement; desire/grief/regret for dead/absent/loss; favorite, object of desire; pleasure, that desired/needed; petition, request;
You can see from the different definitions it’s a bit of a tricky word to translate. It’s often translated simply as ‘desire’, but I think it’s real sense is hard to get at, because we don’t really have an English equivalent for the feeling this word describes. Here’s the word in a few different contexts so you can see what I mean.
Quis desiderio sit pudor aut modus
tam cari capitis? Praecipe lububris
cantus, Melpomene, cui liquidam pater
vocem cum cithara dedit.
What moderation or limit can there be to our regret for the loss
of so dear a life? Direct our mournful
songs, Melpomene, to whom our Father
has given a clear voice along with the lyre.
[Horace, Odes I.24]
Quae ubi omnia foras versa vidit nec in partem aliam ferre, confusus atque incertus animi ex loco infesto agere porro armentum occepit. Inde cum actae boves quaedam ad desiderium, ut fit, relictarum mugissent, reddita inclusarum ex spelunca boum vox Herculem convertit.
When Hercules saw that all the footprints were turned away from the cave, and led to no particular place, confused and uncertain in his mind, he began to drive the herd onward from that hostile place. Then, when some of the cows, being driven away, out of longing for those left behind, as often happens, mooed, the answering call of the imprisoned cows from out of the cave attracted Hercules attention.
[Livy, Ab urbe condita I.7]
nec sibi enim quisquam tum se vitamque requiret,
cum pariter mens et corpus sopita quiescunt;
nam licet aeternum per nos sic esse soporem,
nec desiderium nostri nos adficit ullum…
For no one misses either himself or his life
at the time when the sleepy mind and body rest equally;
indeed, if we could sleep like this forever,
no feeling of loss for ourselves would touch us…
[Lucretius, De rerum natura III. 919ff]
[Interestingly I couldn’t find a citation from Virgil]
From these contexts I would suggest the meaning of the word comes closest to "the feeling you have when you miss someone ", kind of like homesickness for a person. Horace misses his dead friend, Hercules' cattle miss the cows which Cacus has stolen, and Lucretius' point is that (since the soul cannot survive death) you don't miss yourself when you're dead.