# Analytical Solutions and Differential Equations

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 `DEProblem`

s. 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 Solutions

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 Central1`

declares the use of the `Central1`

. Analytical solutions have preset names which are used in the internal model. These parameters must be given values in the `pre`

block.

`Central1`

The `Central1`

model corresponds to the following `@dynamics`

block:

```
@dynamics begin
Central' = -(CL/Vc)*Central
end
```

The variables `CL`

and `Vc`

are required to be defined in the `@pre`

block.

`Depots1Central1`

The `Depots1Central1`

model corresponds to the following `@dynamics`

block:

```
@dynamics begin
Depot' = -Ka*Depot
Central' = Ka*Depot - (CL/Vc)*Central
end
```

The variables `Ka`

, `CL`

and `Vc`

are required to be defined in the `@pre`

block.

`Depots2Central1`

The `Depots2Central1`

model corresponds to the following `@dynamics`

block:

```
@dynamics begin
Depot1' = -Ka1*Depot1
Depot2' = -Ka2*Depot2
Central' = Ka1*Depot1 + Ka2*Depot2 - (CL/Vc)*Central
end
```

`Central1Periph1`

The `Central1Periph1`

model corresponds to the following `@dynamics`

block:

```
@dynamics begin
Central' = -(CL+Q)/Vc*Central + Q/Vp*Peripheral
Peripheral' = Q/Vc*Central - Q/Vp*Peripheral
end
```

The variables `CL`

, `Vc`

, `Q`

, and `Vp`

are required to be defined in the `@pre`

block.

`Depots1Central1Periph1`

The `Depots1Central1Periph1`

model corresponds to the following `@dynamics`

block:

```
@dynamics begin
Depot' = -Ka*Depot
Central' = Ka*Depot -(CL+Q)/Vc*Central + Q/Vp*Peripheral
Peripheral' = Q/Vc*Central - Q/Vp*Peripheral
end
```

The variables `Ka`

, `CL`

, `Vc`

, `Q`

, and `Vp`

are required to be defined in the `@pre`

block.

`Central1Periph1Meta1`

The `Central1Periph1Meta1`

model corresponds to the following `@dynamics`

block:

```
@dynamics begin
Central' = -(CL+Q+CLfm)/Vc*Central + Q/Vp*CPeripheral
CPeripheral' = Q/Vc*Central - Q/Vp*CPeripheral
Metabolite' = -CLm/Vm*Metabolite + CLfm/Vc*Central
end
```

The variables `CL`

, `CLm`

, `Vc`

, `Vp`

, `Vm`

, `Q`

, and `CLfm`

are required to be defined in the `@pre`

block.

`Central1Periph1Meta1Periph1`

The `Central1Periph1Meta1Periph1`

model corresponds to the following `@dynamics`

block:

```
@dynamics begin
Central' = -(CL+Q+CLfm)/Vc*Central + Q/Vp*CPeripheral
CPeripheral' = Q/Vc*Central - Q/Vp*CPeripheral
Metabolite' = -(CLm+Qm)/Vm*Metabolite + Qm/Vmp*MPeripheral + CLfm/Vc*Central
MPeripheral' = Qm/Vm*Metabolite - Qm/Vmp*MPeripheral
end
```

The variables `CL`

, `CLm`

, `Vc`

, `Vp`

, `Vmp`

, `Q`

, `Qm`

, and `CLfm`

are required to be defined in the `@pre`

block.

`LinearODE`

The `LinearODE`

dynamic model specification is used for general piece-wise linear ODEs with constant coefficients. These models are solved with a matrix exponential which is faster and more stable than numerical ODE integration. Hence, the `LinearODE`

dynamic model specification is to be preffered over the equation based specification when the method is applicable.

The coefficients of the ODE are specified in the `@pre`

block as a special matrix named `A`

corresponding the $A$ matrix in the linear system of ODEs $u' = Au$. The names of the ODE variables as well as their initial values should be declared in the `@init`

block. Hence, the `Central1Periph1`

could equivalently be defined as

```
@pre begin
...
A = [-(CL+Q)/Vc Q/Vp
Q/Vc -Q/Vp]
end
@init begin
Central = 0.0
Preriph = 0.0
end
@dynamics LinearODE
```

## Differential Equations `DEProblem`

`DEProblem`

s 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. The models are defined by writing each of the differential equations in the system, e.g.

```
@dynamics begin
Central' = -(CL/Vc)*Central
end
```

for a simple one compartment model and

```
@dynamics begin
Depot' = -Ka*Depot
Central' = Ka*Depot - (CL/Vc)*Central
end
```

for a one compartment model with first order absorption.

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.