I would look at the Ecumenical Councils, particularly the First Council of Nicaea in 325 and the Second Council of Constantinople in 502. The First Council of Nicaea anathematized Arianism, making it pretty unpopular for fourth century Christians to harbor any sort of Arianism. This is when the decline really started. Around the year 400, Theophilius, along with Jerome and Epiphanius of Salamis, went on a campaign to rid the Mediterranean of Arianism and Origenism (metaphysics in Christianity), opting for Scripture-based Christianity. Because they had Church authority at their back, they were able to really stomp out a lot of it by the time they were both formally anathematized (again) at Constantinople II.

I study the history of theology and deal mostly with the application of Greek thought onto Christianity, so my knowledge in this area is somewhat limited, but if I understand correctly, the Arians were pretty unpopular to both Latin and Eastern Christians and the Caliphate didn't really like them too much, either, so they were kind of a minority from the fourth century until they eventually disappeared.

It's also worth noting that the term Arianism became pejorative after Nicaea I, so the labeling of some group or individual as an "Arian" might not be as accurate as we might like...

I hope this helps.

There are several clues to the presence of Arianism after the Byzantine reconquest, but we can say nothing definitive about their popularity or influence. Despite North Africa being a relatively wealthy province, we have very few literary sources from that region. I think the only major sources written in Africa in the Byzantine period are a chronicle by Victor of Tunnuna, a poem by Corippus and some letters by Maximus Confessor. They were all written by Nicene Christians, and Victor and Maximus both fulminated against heretics, but their targets were not Arians, but the Byzantine state for making compromises with Miaphysite heretics, which was an issue that arose out of the Council of Chalcedon in 451. Indeed, if we are to look at the Byzantine period as a whole, it would appear that North Africa was firmly Chalcedonian in allegiance, which caused a whole host of issues with the emperors Justinian I (527-565) and Constans II (641-668), the two emperors who tried the hardest to unite the divided Christian communities within their empire (spoiler alert, both failed).

So what do we actually know about African Arians? Apparently soon after the reconquest of Africa Justinian was keen to keep the locals happy, which meant the toleration of Arianism. Nicene Christians however responded in force and held a synod condemning this, demanding that Arians (along with Donatists and Jews) should not be allowed to hold public office. I'm not sure how to interpret this, since we have no idea about how many Arians there were even under the Vandals and this may just well be a rhetorical thing from Nicenes eager to secure their own position after being persecuted for decades (maybe, the extent of the Arian persecution of Nicenes is I think also debatable). In any case, Justinian responded with a law in 535 that accepted their demands.

There was also a mutiny by Byzantine soldiers in 536 that was possibly sparked by Arianism, as allegedly Byzantine soldiers who married Vandal women thought that they were being cheated out of their property (Justinian had issued a law that returned land to anyone deprived of property during the Vandalic period). This didn't involve just the Vandals though; many Byzantine soldiers were Arians too, as the army often recruited auxiliaries from outside the empire. This was a practise followed until at least Tiberius II (574-582) and their Arian faith was tolerated, albeit uneasily during periods of religious tension, so it is possible that Arian soldiers of all ethnic backgrounds found the discrimination against the once-powerful Arian Church distasteful, causing a military revolt in North Africa. As for the mutiny, it triggered a series of revolts in Africa for the next decade, but there were no more mentions of Arianism. Again I'm not too sure of what this means. It could be that the sources were silent on this because they tried to airbrush the Arians out of history, or it may simply because Arianism had lost its prominence in a world dominated by Nicenes; the dominant theological debate in the sixth century had shifted to that of Chalcedonianism versus Miaphysitism after all.

Arianism did live on in other regions of the empire though. As I said, there were Arian Gothic soldiers in Constantinople under Tiberius II, whilst Byzantine Italy had been invaded by the Lombards in 568, who were a mixture of Arians, Nicenes and pagans. Ravenna, the bureaucratic centre of Byzantine Italy, still had Arian churches until the 560s, suggesting that local Arians reached an accommodation with the new regime. So even without any literary evidence, I think I would be correct to say that many Arians remained in Africa - it was simply impossible for them to disappear so quickly, especially as Donatism, a heresy with an even older pedigree, survived in North Africa until at least the time of Gregory the Great (590-604). As for how long they lasted for, I don't think we'll ever know, though archaeologists are always finding more stuff, so we might be able to have a better idea about when and where they flourished in the future.

Now to address a few of /u/DenysTheAreopagite's points:

You are right that the Ecumenical Councils were important, but they didn't really matter with regards to Arianism after Constantinople I. Ephesus dealt with Nestorius' followers, Chalcedon with Ephesus' consequences and Constantinople II with the fallout from Chalcedon. Each of these councils dealt with issues within the Eastern Roman Empire, and as Arianism was never a big movement in the east after Theodosius I, it wasn't a priority there. In the west though, Arianism continued to be a force to be reckoned with, as aside from the Vandals, the Visigoths, Ostrogoths and Burgundians all followed this brand of Christianity - even the Franks may have experimented with it, as there are grounds to doubt the Frankish claim that Clovis converted from paganism straight to Nicene Christianity. The Ostrogoths and the Vandals were defeated by the Byzantines, but the Visigoths only converted under King Reccared I (586-601) after his father had persecuted the Nicenes. The Lombards meanwhile were partially Arian until the eighth century. You are probably correct to say that the Arians were a minority in the west, but it definitely wasn't a slow decline from the fourth century onwards - by the end of the fifth century Arian kingdoms dominated Western Europe, with nary a Nicene king in sight.

You are also right that Arian isn't the best word to use to describe their version of Christianity. I know that Homoian describes the brand of Christianity that Valentinian II (375-392) followed, but I still haven't found out what the appropriate term for Arians in the fifth century and beyond is yet. If anyone has any ideas, please let me know! :)