Fares usually come in one of three forms - fully refundable, refundable after payment of a fee and totally non-refundable. You need to determine which of those three covers the fare on the ticket you purchased.
Even if the fare is non refundable the taxes/fees/charges associated with the ticket (generally those levied by the airport or government) are theoretically refundable. However, many airlines charge an administration fee which can exceed the refund value (gotcha!).
Of course, if you have insurance you need to consider what condtions that covers. Based on your brief description I would certainly think it worthwhile pursuing a claim on your policy (you did buy travel insurance, didn't you?).
A request (rather than a demand) to the airline may produce a result but I wouldn't hold out much hope.