Well, the red spots around your eyes can be something related with irritants. In my opinion, it can be because of the eye infection which leads to your situation. So for your situation, you should not touch your eyes with your dirty hand. And then, just have some cool teabag under your eyes. And it can be effective. And generally speaking, for the causes, it may be related to different lifestyles or diets and are certainly not infectious or caused by poor hygiene. And if it is serious, you need to go and see the eye doctor.
According to your description, it can be the enlarged or broken blood vessel because there are a multiple of blood vessels in our eyes. If they are broken or enlarged, our eyes can appear red. You should take it easy because it usually go away on its own under such circumstances after a short period of time without any treatment. But it occurs to me that it can be a symptom of pink eye. But if you suffer from pink eye, you would feel itching. I think you can observe your situation for a time before you decide to go to see a doctor.
The red spot in the corner of your eye may be the blood redness. It may mean that you spend a lot of time in front of the computers and cause the tiredness in the eyes. You need to take it relax. You should not use the eyes too much time every day. You could use some eye drops to release the symptom.
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