Import\Git\Project from git (using smart import)
Obviously dependencies need to be installed and since this is not a Maven or Ant project it must be checked what Eclipse installs.
Apache's implementation of RK4 is used: org.apache.commons.math3.ode.nonstiff.RungeKuttaIntegrator
To add Apache commons to an Eclipse project:
To use RungeKuttaIntegrator and other solvers from Apache Commons:
https://commons.apache.org/proper/commons-math/userguide/ode.html
"The user should describe his problem in his own classes which should implement the FirstOrderDifferentialEquations interface. Then they should pass it to the integrator they prefer among all the classes that implement the FirstOrderIntegrator interface."
So in the very first example SIR_ODE implements FirstOrderDifferentialEquations.
The SIR_ODE.getDimension method return the size of the array holding the cardinality of each compartment. So for this simple SIR example it is 3 because the compartments are S, I and R with as many differential equations for dS/dt dI/dt and dR/dt.
It is not mandatory to create a Maven project to use XChart
Check that you have correctly configured Git by following this procedure
Then, follow this step
Import\Git\Project from git (using smart import)
Both kinds of incidence are only equivalent if N is fixed. Their assumptions are different which matters when the population gets large.
Mass action assumes transmission is unbounded when the number of infected people grows.
dS/dt = - beta IS
Caveat: beta typically has to be changed when N is changed.
Standard incidence assumes transmisson depends on the number of contacts with infected individuals and is bounded when N grows.
dS/dt = - beta/N IS
Note that bata/N is automatically adapted when N changes.
Some modes rely on propotions instead of cardinalities. For the sake of condistency it is assumed, in JKendrick, that they are reformulated in terms of quotients of cardinalities
For instance the SIR example below has these values
Initial_Susceptibles 0.999999
Initial_Infecteds 0.000001
This is reformulated as
N = 1 000 000
S = 999 999
I = 1
and implicitly
R = 0
STATUS : not yet enforced ^^
We use RK4 for deterministic simulations and Gillespie's Direct Method and Tau leap dor stochastic ones.
For stochastic models, following the Keeling book Section 4.1.1 we assume that "in small time period the number of events that occur is Poisson distributed".
The transition rates matrix is composed of the functional transitions rates between compartments. A transition is set by giving the compartment from which the transition starts, the compartment toward which it ends, and the rate at which the transition occurs. Each concern has it’s own transition rates matrix.
When concerns are merged into a scenario, a new transition rates matrix is created using the tensor sum of the concern’s matrices. (Not yet implemented)
Each not null transition of the matrix is considered as a possible event for stochastic simulations. For deterministic simulations, the equations are generated using the transition rates matrix.
Each concern has a name and compartments names. When concerns are merged in a scenario, new compartment names are created this way : concern1Name:compartemntName_concern2Name:compartmentName_(...)concernNName:compartmentName_ It's composed of attributes, themselves composed of the first concern name, then ":", then the first compartment name of the first concern, and then "_". There are as many attributes by names than there are concerns by scenario.
Here is an example with 2 concerns : a first one with the name "status" and the compartments "S","I","R", and a second one with the name "species" and the compartments "human","bird". The names used in the scenario would be :
status:S_species:human_
status:S_species:bird_
status:I_species:human_
status:I_species:bird_
status:R_species:human_
status:R_species:bird_
The attributes of a concern name are stored in a map, so the order of attributes doesn't matter. status:S_species:human_ is equivalent to species:human_status:S_
This solver uses ordinary differential equations generated via the scenario transition rate matrix. It uses the Runge-Katta method.
These solvers run multiple times and then return the median path.
This solver has a randomized time step, depending on the rate at which any event may occur. Then, an event is randomly selected in the list of possible events, using a roulette wheel taking in consideration the rate of each event.
For each time step, this solver generate a random number of occurrences of each event, following a Poisson distribution around (the step size*the rate of the event).
Here are the steps needed for a simulation :
- Define concerns, with a concern name, compartments and parameters that would be useful to get transition rates.
- For each concern, define every transition possible between compartments, with origin compartment, destination compartment and rate.
- Define a scenario with the list of your concerns.
- In this scenario, define value for every parameter (including population of every compartment).
- Define a model with the scenario, the size of a time step, the time of the last step and the number of iterations if you want a stochastic simulation.
- Define a solver with this model.
- Define a simulation with the solver, a visualization tool and a title.
- Run the simulation.
As mentionned above, the results are currently displayed using XChart. More precisely, the jKendrick.IHM package currently holds 2 kinds of classes : Main classes with a proper main method running a full example and the Visualization class with two methods : getChart which should probably be static and xchartExample is a simple one to play with XChart. Both methods rely on XYChart.
Visialization displays the results of a simulation using this method of the ISolver interface double[][][] getResult() in which the leftmost dimension is the number of cycles, the middle one is the number of steps and the rightmost one is the number of compartments.
The example is adapted from https://homepages.warwick.ac.uk/~masfz/ModelingInfectiousDiseases/Chapter2/Program_2.1/index.html.
https://github.com/KendrickOrg/kendrick/wiki/Basic-SIR-(Benchmark-Model-1)
https://github.com/KendrickOrg/kendrick/blob/ba11051d2c8976bb27a12da0e3e51650cee99e51/src/Kendrick-Examples/KEDeterministicExamples.class.st#L1183
We use mass action for this model and simulate it using RK4 and Gillespie's Direct Method.