Edit in JSFiddle

angular.module('myApp',[]).
    controller("listController", ["$scope", function($scope) {
        $scope.patients = [
            {name: "Jean Dupont"},
            {name: "Pierre Merti"},
            {name: "Jacques Egard"},
            {name: "Philippe Petit"},
            {name: "Albert Partin"}
        ];
        $scope.patient;
        $scope.addPatient = function() {
            $scope.patients.push({name: $scope.patient});
            $scope.patient = null;
        };
        $scope.removePatient = function(i) {
            $scope.patients.splice(i,1);
        };
    }]);

<div ng-controller="listController">
  <ul>
    <li ng-repeat="p in patients">{{p.name}} <button ng-click="removePatient($index)">remove</button></li>
  </ul>
  <input type="text" ng-model="patient"><button ng-click="addPatient()">add</button>
</div>