# Immigration-Death

The models in this section deal with the immigration-death process, which is comprised of the immigration reaction 0 → X and the death reaction X → 0.

## dsmts-002-01

The immigration and death reactions have mass-action kinetic laws with rate parameters Alpha = 1 and Mu = 0.1, respectively. Use the open button in the tool bar to import the file `examples/sbml/dsmts31/dsmts-002-01.xml`. The model is imported correctly and the rate laws are recognized as having mass-action kinetics. Add a method in the methods list. In the methods editor select `Time Homogeneous` and `Time Series, Uniform` with the default solver and options. Set the recording time to 50 and the number of frames to 51. To see the result of these modifications open the file `examples/cain/dsmts31/dsmts-002-01.xml`. Below are the plots of the mean and standard deviation of the species population calculated from 10,000 trajectories.

Note that this problem has a steady state solution. Select `SteadyState` in the methods list panel. In calculating the steady state solution, we let the process equilibrate for 100 time units and then record the state for 100,000 time units. We generate four trajectories and plot the steady state histogram, shown below.

With the steady state output selected, click the table button and then select `Estimated Error`. The estimated error in the probability distribution is 0.0045.

## dsmts-002-02

For this model, the value of the Alpha is changed to 10. Below are plots of the mean and standard deviation of the species population.

## dsmts-002-03

This model is the same as `dsmts-002-02`, except that the global parameter Alpha = 10 is overridden by the local parameter Alpha = 5. Cain does not support overriding global parameters. You will need to change the value of Alpha in the parameters editor.

## dsmts-002-04

This model is the same as `dsmts-002-02`, except that Alpha = 1,000. This increases the reaction rates and makes the simulation more expensive. Thus we only generate 1,000 trajectories in calculating the mean and standard deviation of the species populations.

## dsmts-002-05

This model is the same as `dsmts-002-02`, except that there are two additional species Source and Sink, which are both boundary conditions. Cain does not support boundary conditions.

## dsmts-002-06

This model is the same as `dsmts-002-05`, except that Sink is not a boundary condition.

## dsmts-002-07

This model is the same as `dsmts-002-05`, except that Sink is a constant boundary condition.

## dsmts-002-08

This model tests parameter loading, which is not supported in Cain.

## dsmts-002-09

This model is the same as `dsmts-002-01`, except that an event is triggered at time 25 which sets the value of X to 50. Although Cain supports events, it will not parse events in SBML files. Thus one must enter the events in either the time events editor or the trigger events panel. Open the file `examples/cain/dsmts31/dsmts-002-09.xml` to see how to enter the appropriate time event. For models that utilize events, we must use solvers in the `Use Events` category, which are much slower than the other solvers. Thus we only generate 100 trajectories to calculate the mean and standard deviation plotted below.

## dsmts-002-10

This model is the same as `dsmts-002-09`, except that the event is triggered at time 22.5 and X is assigned the value 20.