Grammatically inversion is required. But performing inversion in this sentence reveals just how poorly it is written. The inversion required is:
"It also may explain why only until the economics was relaxed after a difficult period could the policies to solve EVD crisis attempt to get openly involved in the response to EVD."
"Only until" could be used in the past tense in a case like:
"Only until the government outlawed the practice were companies able to shift costs incurred in one jurisdiction to another."
In the example sentence, it appears that "only after" is more appropriate. Moreover, "the policies to solve EVD crisis could attempt to get openly involved in the response to EVD" is garbled because in the normal course of things policies cannot "get openly involved" in something. People get openly involved, not policies. Depending on what you mean, the sentence could possibly be rewritten as:
"It also may explain why only after the economics was relaxed after a difficult period was it possible to attempt to get policies to solve the EVD crisis openly involved in the response to EVD."
Unfortunately the sentence suffers from other problems. "The economics was relaxed" is poor English and is difficult to interpret. And the awkward phrasing of "possible to attempt to get policies to solve the EVD crisis openly involved in the response to EVD" could also be improved.