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It is a good question.
It is actually a challenge between someone's feeling and/or emotions and his/her career and here, your skills will have the upper hand over your experience. If you have the full control over your emotions and feelings and you do have the skills to control that, regardless of how many years of experience do you have, you will be able to focus on your work and once you leave, you can start worrying about your family issue but if you can't do that, of course, you will have very hard time and I completely understand how hard that is, not easy to achieve it but I do know for a fact that there are some people who actually managed to do so.
From personal experience, I can say that personal circumstanses have a vital role on your performance as a project manager (as for any other job as well), as you tend to loose focus at some critical times, and you need to balance your work and life (especially for illness for long periods), and at many times both are very demanding, and both require consuming a lot of your time towards them, hence, you may not be able to deliver the constant quality you previously, and you will need support from your colleagues, first, you need to meet with your team, and from your prior evaluation, you would know who would be able to take up some additional tasks, and who was exposed to project management concepts, so that you would delegate some tasks to her/him, or could check your work in troubled times, and try to get some counselation (don't know if this word exists, derives from counselling) to be able to get over this situation, and be able to perform well enough for project success.
Work and private life as far as possible should be kept apart but that is not always the case.
If it is an employee’s relative then we should try to do all we can to ease the problem and prevent it escalating into a major problem, i.e. time off or change of working hours
If it is my relative I would weigh up all the pro's and con's to asses my best course of action, i.e. private medical care if affordable, could a relative take care of them in my absence, and if I let this problem affect me to the point of dismissal - what would I do then.
Always a difficult situation but quite often, alternative solutions can be found.
The reality is, as the project manager it will affect the course of projects, dending on severity and options availbale, it needs to be handled smartly and, communicate, delegate and reassure the rest of the team, if the illness is severe then judgement call needs to be made on where the priority will be.
What mean of ill, if it mean seak then I will isolated the member who is seak and tranfer him/her to health care, if it mean have a bad effect on the team I will use with him/her negative reinforcemant, corecive action, and equity system, if he/she become in severtity level I will fire him/her.