How to predict next visit date based on this data
If you plot days since the first visit vs the visit number, this ends up being a simple regression problem.

import pandas as pd

data = pd.read_csv('data.txt', '|')
data['visit_date'] = pd.to_datetime(data['visit_date'])
data['days_since_first_visit'] = (data['visit_date'] - data['visit_date'][0]).map(lambda delta: delta.days)
data = data.reset_index()

data.plot('index', 'days_since_first_visit')




I would recommend using a library like scikit-learn or statsmodels to do the regression.