Although the question may more properly be one of etiology than etymology, the affliction of 'sequelitis' can be observed in evidence from the popular press corpus of Newspapers+ Publishers Extra as early as 1920. The original sufferer appears to have been Sir Harry Johnston, as remarked in an article datelined "London March 13" in The Washington Post (Washington, District of Columbia)
A strange malady has attacked Sir Harry Johnston, the famous explorer, zoologist and writer. "Sequelitis" has got Sir Harry. He is now 61, and, until a few months ago, his many publications had consisted wholly of books of travel and the like. All of a sudden, however, he published a novel! And what did this novel prove to be but a sequel to "Dombey and Son," one of the most famous and popular of the romances of Charles Dickens? He finished it during his last visit to the United States and afterward submitted it to three London publishers, all of whom promptly turned it down.
Given the origin in Dickens, the lesser known 'serialitis' malady might be thought to have somehow mutated into 'sequelitis'; however, the first evidence of the former does not appear until 1940, in a Lincoln Journal Star (Lincoln, Nebraska) column concerning radio programming and titled, appropriately, "Your Problems". The 'serialitis' disease is mentioned in a letter submitted by "Old Batch" to the columnist, Mary Gordon:
If I had a wife afflicted with serialitis, I would make it a point to be home as little as possible.
In light of the timing, then, and taking into account Sir Harry's preoccupations with exploration and zoology, the origin of 'sequelitis' more likely is zoonotic than a progressive mutation of the more infrequent 'serialitis'.
After the 1920 initial presentation, 'sequelitis' goes undiagnosed until a more-or-less simultaneous 05 Jul 1949 re-appearance on both US coasts, in The Bakersfield Californian and the Evening Courier (Camden, New Jersey):
Bob Hope must have sequelitis. I heard a few days ago that he's going to do a follow-up on "Paleface".
This strain of 'sequelitis', associated with the churning Hollywood film industry, is now the classic strain. It was rare initially. After making several further appearances in 1951, it seems to have hidden, dormant between film frames, for another seven years, until 1958, then broke out again in 1965, before slipping away into four more years of quiescence, until 1969. After 1969, although still infreqent, the malady was diagnosed with some regularity (1970, 1972, 1974 – 8). In 1978 a minor epidemic broke out; the spread of the contagion was hastened by dissemination through syndicated columns. Since then, clusters of disease have been observed and recorded in nearly unbroken sequence.