A Durham nursing home has been fined $2,000 for lapses in patient care in connection with an outbreak of viral respiratory illnesses at the home in April, in which four patients died of pneumonia. Nearly half of the home’s 43 residents were sickened during the outbreak, state records show. The state Department of Public Health released details of the citation and fine against Twin Maples Health Care Facility on Friday. It also released details of fines against an East Hampton home and a Danbury home. Twin Maples officials did not report the illnesses as an outbreak until April 24, nine days after 10 percent of the residents had symptoms, the state citation said.