If a member of the fire department shall die, before retirement, from causes other than an injury received in or illness caused by the performance of duty, or regardless of cause, if no allowance shall be payable under Section 8.572 or 8.573 preceding, a death benefit shall be paid to his estate or designated beneficiary, the amount of which and the conditions for the payment of which shall be determined in the manner prescribed by the Board of Supervisors for the death benefit of other members of the Retirement System. Upon the death of a member after retirement and regardless of the cause of death, a death benefit shall be paid to his estate or designated beneficiary the amount of which and the conditions for the payment of which shall be determined in the manner prescribed by the Board of Supervisors for the death benefit of other members of the Retirement System.