Given a flu on an island of 10000 people. Every day 15% of healthy people become infected and have a less illness (not require hospitalization), 12% of healthy people become infected and have a more illness (need to get hospitalized). At the same time, 12% of people with a less illness recover completely, and 15% go into the category of more ill patients. In the category of more ill patients, the situation is as follows: 20% go into the category of patients with a less form of the illness and 10% completely recover. Recovered patients may become infected again. At the initial time on the island, 500 patients were identified in a less illness of the disease and 100 patients in a more illness. Find mathematical of the number of patients in less and more illness with respect to time.
I formulated the problem by making four states: Infected, Less Illness, More Illness and Recovered. The transitions between them were modelled as part of the above problem statement. For getting the state of the model with respect to time, i am thinking of using the page rank algorithm to calculate the same. For less and more illness, state probabilities can be calculated at the initial time by the given data but for recovered and infected state how to calculate the same? Also for the transition probabilities would it be $\frac{1}{d_{i}}$ * percentage given in the problem statement where $d_{i}$ is the outdegree of node $i$ for the above model?
I have done it by using Markov chain but have a question should i consider damping factor in the above formulation while calculating the transition matrix?