One way to draft the letter would be to point out the incredible opportunity that staying for another week in Oman would give to your son.
I'm assuming that your son speaks English, and that many of the people he would encounter in Oman do not. If this is true then speaking with those around...
Perhaps his reaction to his physical illness is a symptom of dealing with major depression.
Another possibility is that he is seriously physically ill with more than one illness, such as diabetes, heart disease, etc. and this is all that he is willing to show you.
Regardless of what is...