Dynamical systems driven by ODEs
Sommaire 
Autonomous dynamical systems
Consider a timevarying system $A(t)=(A_1(t),A_2(t),\ldots A_J(t))$ defined by a system of Ordinary Differential Equations (ODE)

(1.1) 
where $\dot{A}(t)$ denotes the vector of derivatives of $A(t)$ with respect to $t$:

(1.2) 
Notations:
 let $A_0 = A(t_0)$ be the initial condition of the system defined at the initial time $t_0$,
 let $A^{\star}$ be the solution of the system at equilibrium: $F(A^{\star}) =0$
A basic model
We assume here that the system doesn't have any input:
 \(\begin{align} A(t_0) & = &A_0 \\ \dot{A}(t) & = &F(A(t)) \, \ t \geq t_0 \end{align}\)


Consider a basic VK model with $A=(N,I,V)$ where $N$ is the number of non infected target cells, $C$ the number of infected target cells and $V$ the number of virus.
After infection and before treatment, the dynamics of the system is described by this ODE system:

(1.3) 
The equilibrium state of this system is $A^{\star} = (N^{\star} , I^{\star} , V^{\star})$, where

(1.4) 
Assume that the system has reached the equilibrium state $A^{\star}$ when the treatment starts at time $t_0=0$.
The treatment inhibits the infection of the target cells and blocks the production of virus. The dynamics of the new system is described with the new ODE system (1.5).
This system, and the initial conditions, can be exactly reproduced using the MLXTRAN macros (block EQUATION)

(1.5) 
$EQUATION T_0 = 0 N_0 = delta*c/(beta*p); I_0 = (sd*N)/delta V_0 = p*I/c DDT_N = s  beta*(1eta)*N*V  d*N DDT_I = beta*(1eta)*N*V  delta*I DDT_V = p*(1epsilon)*I  c*V 
 Remark1
 Here, $T_0 = 0$ means that the system is constant and is $A^{\star}$, defined in the script by $(N_0, I_0, V_0)$, for any $t<0$.
 Remark2
 If the initial condition is not given in the model, it is assumed to be 0.
Piecewise defined dynamical systems
More generally, we can consider inputless systems which are piecewise defined: there exists a sequence of times $t_0< t_1< ...<t_K$ and functions $F^{(1)}, F^{(2)},\ldots,F^{(K)}$ such that
 \(\begin{align} A(t_0) &= A_0 \\ \dot{A}(t) &= F_k(A(t)) \ , \ t_{k1} \leq t \leq t_{k} \end{align}\)


We will consider the same viral kinetics model defined above (the switching times is still given in the data set). This system is now piecewise defined:

(1.6) (1.7) (1.8) 
VARIABLES ID, TIME, VL use=DV, EVENT list=(Start1, Start2, Stop) $EQUATION SWITCH CASE T < T_Start1 N = delta*c/(beta*p); I = (sd*N)/delta V = p*I/c CASE T_Start1 < T < T_Start2 DDT_N = s  beta*N*V  d*N DDT_I = beta*N*V  delta*I CASE T_Start2 < T < T_Stop DDT_N = s  beta*(1eta)*N*V  d*N DDT_I = beta*(1eta)*N*V  delta*I DDT_V = p*(1epsilon)*I  c*V CASE T > T_Stop DDT_N = s  beta*(1eta*exp(k1*(TT_Stop)))*N*V  d*N DDT_I = beta*(1eta*exp(k1*(TT_Stop)))*N*V  delta*I DDT_V = p*(1epsilon*exp(k2*(TT_Stop)))*I  c*V END 
 Remark 1
 Here, EVENT is a reserved variable name. Then, the information in the column EVENT is recognized as a succession of events. Furthermore, the times of the events Start1,Start2 and Stop are automatically created as T_Start1, T_Start2 and T_Stop.
 Remark 2
 In this particular example, the dynamical system is described by parameters $\beta$ and $p$ whose definition switches. Then, the same model could be encoded as follows:
T_0 = T_Start1 N_0 = delta*c/(beta*p) I_0 = (sd*N)/delta V_0 = p*I/c SWITCH CASE T_Start1 < T < T_Start2 be = beta pe = p*(1epsilon) CASE T_Start2 < T < T_Stop be = beta*(1eta) pe = p*(1epsilon) CASE T > T_Stop be = beta*(1eta*exp(k1*(TT_Stop))) pe = p*(1epsilon*exp(k2*(TT_Stop))) END DDT_N = s  be*N*V  d*N DDT_I = be*N*V  delta*I DDT_V = pe*I  c*V
Dynamical systems with source terms
Consider now the system (1.1) with inputs:

(2.1) 
where:

(2.2) 
The input $u(t)=(u_1(t),u_2(t),\ldots,u_J(t))$ of the system can be defined either in the datafile (doses in a PK model for instance) or in the model.
Piecewise constant inputs
We assume an additive model for the inputs: for any $1\leq \ell \leq L$
 \( \dot{A_{\ell}}(t) = F_\ell(A(t)) + u_\ell(t) \)
Then, there exists a sequence of times $(\tau_{\ell,j})$, durations $(d_{\ell,j})$ and amounts $(a_{\ell,j})$ such that

(2.3) 
Consider an IV infusion with two compartments, i.e. a very basic situation with only one type of administration. Such infusion can be described by the following system of equations:

If there is no need of any additional column DPT (depot compartment) in the datafile to distinguish different target depots for different types of administration, the model reduces to:
$VARIABLES ID, TIME, AMT, TINF, DV $EQUATION DDT_Ac = k*Ac  k12*Ac + k21*Ap DDT_Ap = k12*Ac  k21*Ap
If no additional information about the input is given in the model, the default is to assume that the input goes to the first component of the ODE system. Assume now that the target compartment is not the first component of the system. Then, it is mandatory to associate in the model a component with the target compartment. As an example, consider the same datafile as before, but assume that the ODE system has been permuted, then the model should define the second component of the system as the target component:
$EQUATION INPUT(CMT=2) DDT_Ap = k12*Ac  k21*Ap <br> DDT_Ac = k*Ac  k12*Ac + k21*Ap 
Spike inputs
We consider an inputless dynamical system: for any $1\leq \ell \leq L$,
 \( \dot{A_\ell}(t) = F_\ell(A(t)) \)
Spike inputs means that there exists a sequence of times $(\tau_{\ell,j})$ and amounts $(a_{\ell,j})$ such that
 \( A_\ell(\tau_{\ell,j}) = A_\ell(\tau_{\ell,j}^{}) + a_{\ell,j} \)
In other words, the amount $a_{\ell,j}$ is added to the component $A_\ell$ at time $\tau_{\ell,j}$.


Inputs defined in the model
Only some very basic inputs can be directly derived from the information in the datafile. More complex inputs should be defined in the model, or using some external forcing function

$EQUATION DDT_Ac = k*Ac  k12*Ac + k21*Ap + a*exp(b*T) DDT_Ap = k12*Ac  k21*Ap
$EQUATION INPUT(CMT=1,RATE=ExpInput(a,b)) DDT_Ac = k*Ac  k12*Ac + k21*Ap DDT_Ap = k12*Ac  k21*Ap
$EQUATION INPUT(CMT=1,RATE=inline('a*exp(b*T)')) DDT_Ac = k*Ac  k12*Ac + k21*Ap DDT_Ap = k12*Ac  k21*Ap 
Multiple inputs
Different inputs in different components of the ODE system can easily be combined.

$EQUATION INPUT(DPT='GUT1', CMT=1) INPUT(DPT='GUT2', CMT=2) INPUT(DPT='SP', CMT=4) INPUT(DPT='SC', CMT=3) DDT_Ad1 = ka1*Ad1 DDT_Ad2 = ka2*Ad2 DDT_Ac = ka1*Ad1 + ka2*Ad2  k*Ac DDT_As = ks*As ... ...
$EQUATION INPUT(DPT='GUT1', CMT=1 , Tlag= TLAG1 , P=F1) INPUT(DPT='GUT2', CMT=2 , P=F2) ... 
Remark 1: The datafile only contains information about the administration of the drugs, not about the PK model. The link between the depot compartments and the components of the ODES is defined in the model:
Remark 2: Bioavailability or lagtimes can easily be taken into account in the model: