Consider a class with $100$ enrolled students. None of the students were ill at the beginning of the school year. On September $30,$ a total of $5$ students reported having diarrhea. All $5$ continued to be ill on October $1,$ but all $5$ recovered within $3$ days. On October $14,$ another $3$ students developed diarrhea. All of these students continued to be ill on October $15,$ but all $3$ recovered $5$ days later. In this example, assume that a person cannot get diarrhea more than once. Calculate the prevalence of diarrhea in the class on October $1.$