Hi Darshan
This is how I understand it.
The Poisson models only allows decrements [e.g. alive to dead] and cannot model increments [dead to alive].
In the example I have given the Poisson model makes sense when you have a model with only these states [Alive, Dead]. Note it is just a fancy way of saying the transitions only happen in one direction.
When you look at a Poisson model, by definition we fully define a Poisson process as having mean lambda * t.
This lambda parameter by definition is constant -> by assumption. It is not even a good assumption unless the time period is small enough to make this reasonable!
With a two state model [example with Healthy and Sick], of course we can go from Healthy to Sick [Decrement] and Sick to Healthy [increment]. But I do not think it is necessary to have increments for the model to be two-state. In fact, surely the Alive -> Dead model is poisson and two-state with the right assumptions but will still be two-state if you remove the assumption of constant mu, etc.
With the second point, in a Poisson model, because the exposed to risk has not been accounted for in the model, it is surely an assumption that it does not matter what the exposed to risk is, it will not effect the mortality. So if mu = no of deaths / exposed to risk. Both of these are either constant or change proportionately to give the same constant parameter of mu, and therefore, lambda.
mu = 1/lambda => lambda = exposed to risk / number of deaths
With a two state model, of course we can have a situation where both of these need to be estimated since we do not actually know in advance the number of decrements or the central exposed to risk.
Regards
Last edited by a moderator: Mar 20, 2019