Hey Hazem, how are you, my friend! I hope you are doing just fine! :)
Firstly you should know that after any preposition (mostly ''of''), you have to use a verb in the -ing form. The ''having'' used in the context given is considered to be a gerund form that refers to the actual state in which you are at the moment and that which was affected by a condition or an experience you had once in the past. This Grammar Patter is what we call ''The Perfect Participle''.
* ''The Perfect Participle'' may be considered as a reduced form of explaining a certain situation that occurred (in the past) whose consequence is still ongoing. You can tell that it is the Perfect form of the Past Participle because it tells you the reason for the action that follows. If anything, it is used to indicate an action that happens long before the action in the main clause.
E.g.: ''Having spent ten years in Italy, he could speak Italian fluently''. (After he had spent ten years in Italy, he could speak Italian fluently)
E.g.: ''Having attended this course before, Tom knew what to expect.'' (Because Tom had attended this course before, he knew what to expect)
* As regards the sentence you provided, you could understand it in other ways, as follows:
''Because these people had had (once in the past) the disease, the RTD spoke with them.
''After having had the disease (once in the past), these people still live with its stigma (they continue to be treated as if they still had the disease)
Hope that helps!!!