1. A machine-implemented method for notifying at least one care-concerned person such as a medical service provider or a medical assistant to a health or medical care-related situation constituted by at least one of the following notification worthy situations: (0.1) completion of a care-related task where said task completion calls for relatively prompt attention thereto; (0.2) in-completion of a care-related communication delivery-attempt where said in-completion of the delivery-attempt calls for relatively prompt attention thereto; (0.3) in-completion of a care-related task where said in-completion of the care-related task calls for relatively prompt attention thereto; (0.4) an unexpected change or a lack of expected change in a medical condition of a given patient where said unexpected change or lack of expected change calls for relatively prompt attention thereto; and (0.5) an apparent failure to notify a primary medical service provider, or a primary medical assistant as may be appropriate, of one of the above notification-worthy situations; where said machine-implemented alerting method comprises: (a) monitoring communications related to progress or lack of progress in timely completion of pre-scheduled, care-related tasks and related to timely completion of pre-scheduled, care-related communication delivery-attempts and related to medical conditions of respective ones of plural patients; (b) identifying among the monitored communications those which are indicative of at least one of said care-related situations that call for relatively prompt attention thereto by a care-concerned person such as the medical service provider and the medical assistant, where criteria for determining whether or not a given situation is one that calls for relatively prompt attention thereto are machine-defined and where timeliness of pre-scheduled ones of said care-related tasks and care-related communication delivery-attempts are machine-defined; and (c) for an identified one of the monitored communications which is indicative of at least one of said care-related situations that call for relatively prompt attention thereto, initiating a delivery-attempt for delivering to at least one targeted recipient in a group comprised of said medical service provider and medical assistant an alert signal for notifying the targeted recipient of the monitored and identified communication.