Dynamical Problem Types

# Dynamical Problem Types

The dynamical problem types specify the dynamical models that are the nonlinear transformation of the NLME model. There are two major types of dynamical models: analytical models and DEProblems. An analytical model is a small differential equation with an analytical solution. This analytical solution is used by the solvers to greatly enhance the performance. On the other hand, DEProblem is a specification of a differential equation for numerical solution by DifferentialEquations.jl. This is used for specifying dynamical equations which do not have an analytical solution, such as many nonlinear ordinary differential equations (ODEs), or the myriad of differential equation types supported by DifferentialEquations.jl, such as delay differential equations (DDEs) and stochastic differential equations (SDEs).

## Analytical Problems

Analytical problems are a predefined ODE with an analytical solution. While limited in flexibility, the analytical solutions can be much faster for simulation and estimation. In the @model DSL, an analytical solution is declared by name. For example:

@dynamics ImmediateAbsorptionModel

declares the use of the ImmediateAbsorptionModel. Analytical solutions have preset names which are used in the internal model. These parameters must be given values in the pre block.

### ImmediateAbsorptionModel

The ImmediateAbsorptionModel corresponds to the ODE:

\begin{align} Central' &= -(CL/V)*Central\\ \end{align}

### OneCompartmentModel

The OneCompartmentModel corresponds to the ODE:

\begin{align} Depot' &= -Ka*Depot\\ Central' &= Ka*Depot - (CL/V)*Central \end{align}

### OneCompartmentParallelModel

The OneCompartmentParallelModel corresponds to the ODE:

\begin{align} Depot1' &= -Ka1*Depot1\\ Depot2' &= -Ka2*Depot2\\ Central' &= Ka1*Depot1 + Ka2*Depot2 - (CL/V)*Central \end{align}

## DEProblem

DEProblems are types from DifferentialEquations.jl which are used to specify differential equations to be solved numerically via the solvers of the package. In the @model interface, the DEProblem is set to be an ODEProblem defining an ODE. In the function-based interface, any DEProblem can be used, which includes:

• Discrete equations (function maps, discrete stochastic (Gillespie/Markov) simulations)
• Ordinary differential equations (ODEs)
• Split and Partitioned ODEs (Symplectic integrators, IMEX Methods)
• Stochastic ordinary differential equations (SODEs or SDEs)
• Random differential equations (RODEs or RDEs)
• Differential algebraic equations (DAEs)
• Delay differential equations (DDEs)
• Mixed discrete and continuous equations (Hybrid Equations, Jump Diffusions)

The problem type that is given can use sentinel values for the initial condition, timespan, and parameters which will be overridden by Pumas during the simulation chain.