Although the destruction order was not implemented pending the appeal, the dog has since been euthanised because of its ill-health, the judgement said.
Epiha appealed the conviction to the High Court arguing that the section of the Dog Control Act under which he was convicted was not an offence involving strict liability.
His appeal was dismissed by Justice Woodhouse on May 12 this year who ruled that the offence Epiha was convicted of was a strict liability offence under the Act.
In a written judgement dated November 13, the three Court of Appeal justices declined Epiha's request saying they agreed with Justice Woodhouse's ruling.
The justices said Epiha's leave application was governed by section 253 of the Criminal Procedure Act 2011.
"This is a classic public welfare offence directed at protecting the public interest...Once the prosecution has proved the defendant is the owner of the dog that has attacked a person, the onus shifts to the defendant to prove total absence of fault on the balance of probabilities.
"The proposed appeal does not raise any matter of general or public importance justifying a second appeal. Nor is there any indication that justice has been miscarried. The application is accordingly declined," they said.