**QUESTION**

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A dangerous virus spreads on an island with a population of 10,000. Every day, island authorities collect statistics and want to understand if they have enough health system resources. Doctors report that every day $15 \%$ of healthy people become infected and have a mild illness (which does not require hospitalization), $12 \%$ of healthy people become infected and have a difficult illness (they need to get to the hospital). At the same time, $12 \%$ of people with a mild form of the disease recover completely, and $15 \%$ go into the category of seriously ill patients. In the category of seriously ill patients, the situation is as follows: $20 \%$ go into the category of patients with a mild form of the disease and $10 \%$ completely recover. Recovered patients may become infected again. At the initial time on the island, 500 patients were identified in a mild form of the disease and 100 patients in a severe form. Luckily, the virus is not lethal. How will the number of patients behave with increasing time? From a mathematical point of view, find the limits of the number of patients in mild and severe forms, if they exist. Please do not forget that with real viruses everything is not so simple.