Dynamical systems driven by ODEs
Autonomous dynamical systems
Consider a time-varying system $A(t)=(A_1(t),A_2(t),\ldots A_J(t))$ defined by a system of Ordinary Differential Equations (ODE)
\begin{equation} \label{ode1_model} \dot{A} = F(A(t)) \end{equation}
where $\dot{A}(t)$ denotes the vector of derivatives of $A(t)$ with respect to $t$: \begin{equation} \label{ode2_model} \left\{ \begin{array}{lll} \dot{A}_1(t) & = & F_1(A(t)) \\ \dot{A}_2(t) & = & F_2(A(t)) \\ \vdots & \vdots & \vdots \\ \dot{A}_L(t) & = & F_L(A(t)) \end{array} \right. \end{equation}
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 there is no input: \begin{eqnarray*} A(t_0) &= &A_0 \\ \dot{A}(t) &= &F(A(t)) \, \ t \geq t_0 \end{eqnarray*}
| Example: A viral kinetic (VK) model.
In this example, the data file contains the viral load: |
|
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: \begin{equation} \label{vk1} \left\{ \begin{array}{lll} \dot{N}(t) & = & s - \beta \, N(t) V(t) - d N(t) \\ \dot{I}(t) & = & \beta \, N(t)\, V(t) - \delta \, I(t) \\ \dot{V}(t) & = & p I(t) - c \, V(t) \\ \end{array} \right. \end{equation}
The equilibrium state of this system is $A^{\star} = (N^{\star} , I^{\star} , V^{\star})$, where
\begin{equation} \label{eq1} N^{\star} = \frac{\delta \, c}{ \beta \, p} \quad ; \quad I^{\star} = \frac{s - d\, N^{\star}}{ \delta} \quad ; \quad V^{\star} = \frac{ p \, I^{\star} }{c}. \end{equation}
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 this new ODE system:
\begin{equation} \label{vk2} \left\{ \begin{array}{lll} \dot{N}(t) & = & s - \beta(1-\eta) \, N(t) \, V(t) - d\, N(t) \\ \dot{I}(t) & = & \beta(1-\eta) \, N(t) \, V(t) - \delta \, I(t) \\ \dot{V}(t) & = & p(1-\varepsilon) \, I(t) - c \, V(t) \\ \end{array} \right. \end{equation}
where $0<\varepsilon <1$ and $0 < \eta < 1$.
The initial condition and the dynamical system are described in the MDL EQUATION with MLXTRAN):
$EQUATION
|-
|T_0 = 0
|-
|N_0 = delta*c/(beta*p);
|-
|I_0 = (s-d*N)/delta
|-
|V_0 = p*I/c
|-
|DDT_N = s - beta*(1-eta)*N*V - d*N
|-
|DDT_I = beta*(1-eta)*N*V - delta*I
|-
|DDT_V = p*(1-epsilon)*I - c*V
|}
* ''Remark1:'' Here, <span style="font-family:'Monospaced';">$T_0 = 0$</span> means that the system is constant and is $A^{\star}$, defined in the script by <span style="font-family:'Monospaced';">$(N_0, I_0, V_0)$</span>, for any $t<0$.
* ''Remark2:'' If the initial condition is not given in the model, it is assumed to be 0.
==='"`UNIQ--h-2--QINU`"'''Piecewise defined dynamical systems''===
More generally, we can consider input-less 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{eqnarray*}
A(t_0) &= &A_0 \\
\dot{A}(t) &= &F_k(A(t)) \ , \ t_{k-1} \leq t \leq t_{k} \\
\end{eqnarray*}
{| align=left; style="border: 1px solid darkgray;"
| <u>Example:</u> viral kinetic model. We assume here that a first treatment which blocks the production of virus starts first at time $T_{Start1}$, then a second treatment which inhibits the infection of the target cells starts at time $T_{Start2}$. Both treatments stop at time $T_{Stop}$.
|}
The values of the switching times $(T_{Start1},T_{Start2},T_{Stop})$ are part of the data and then should be contained in the
datafile itself. Using the NONMEM format for example, a column <span style="font-family:'Monospaced'">EVENT</span> is necessary in the dataset to describe this
information EVENT is an extension of the <span style="font-family:'Monospaced'">EVID</span> (Event Identification) column used by NONMEM and which is limited to some very specific events). In the following example, $T_{Start1}=0$ is used as the reference time, $T_{Start2}=20$ and $T_{Stop}=200$:
{| class="wikitable" style="text-align:center; width:400px;"
! ID || TIME || VL || EVENT
|-
|1 || -5 || 6.5 || .
|-
|1 || -2 || 7.1 || .
|-
|1 || 0 || . || Start1
|-
|1 || 5 || 5.2 || .
|-
|$\vdots$ || $\vdots$ || $\vdots$ || $\vdots$
|-
|1 || 18 || 4.6 || .
|-
|1 || 20 || . || Start2
|-
|1 || 25 || 2.3 || .
|-
|$\vdots$ || $\vdots$ || $\vdots$ || $\vdots$
|-
|1 || 175 || 1.4 || .
|-
|1 || 200 || . || Stop
|-
|1 || 250 || 2.8 || .
|-
|$\vdots$ || $\vdots$ || $\vdots$ || $\vdots$
|}
We will consider the same viral kinetics model defined above. This system is now piecewise defined:
* before $T_{Start1}$, $A(t) = A^{\star}$, where $A^{\star}$ is the equilibrium state defined in (\ref{eq1})
* between $T_{Start1}$ and $T_{Start2}$,
\begin{equation}
\label{vk3}
\left\{
\begin{array}{lll}
\dot{N}(t) & = & s - \beta \, N(t) \, V(t) - d\, N(t) \\
\dot{I}(t) & = & \beta \, N(t) \, V(t) - \delta \, I(t) \\
\dot{V}(t) & = & p(1-\varepsilon) \, I(t) - c \, V(t) \\
\end{array}
\right.
\end{equation}
* between $T_{Start2}$ and $T_{Stop}$, the system is governed by the ODES described in (\ref{vk2})
\begin{equation}
\label{vk3bis}
\left\{
\begin{array}{lll}
\dot{N}(t) & = & s - \beta(1-\eta) \, N(t) \, V(t) - d\,N(t) \\
\dot{I}(t) & = & \beta(1-\eta) \, N(t) \, V(t) - \delta \, I(t) \\
\dot{V}(t) & = & p(1-\varepsilon) \, I(t) - c \, V(t)\\
\end{array}
\right.
\end{equation}
* after $T_{Stop}$, the system smoothly returns to its original state governed by the ODES described in (\ref{vk1})
\begin{equation}
\label{vk4}
\left\{
\begin{array}{lll}
\dot{N}(t) & = & s - \beta (1-\eta \, e^{-k_1 (t-T_{Stop})})\, N(t) \, V(t) - d\,N(t) \\
\dot{I}(t) & = & \beta (1-\eta \, e^{-k_1 (t-T_{Stop})}) \, N(t) \, V(t) - \delta \, I(t) \\
\dot{V}(t) & = & p(1-\varepsilon\, e^{-k_2 (t-T_{Stop})}) \, I(t) - c \, V(t) \\
\end{array}
\right.
\end{equation}
We have seen that the information about switching times is given in the data set.
The different dynamical systems are described in the MDL (in a block <span style="font-family:'Monospaced'">EQUATION<span></span> and using the statement <span style="font-family:'Monospaced'">SWITCH<span></span> with MLXTRAN).
We only show the blocks <span style="font-family:'Monospaced'">VARIABLES<span></span> and <span style="font-family:'Monospaced'">EQUATION<span></span> of the code:
$VARIABLES ID, TIME, VL use=DV, EVENT list=(Start1, Start2, Stop)
$EQUATION
SWITCH
CASE T < T_Start1
N = delta*c/(beta*p);
I = (s-d*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*(1-eta)*N*V - d*N
DDT_I = beta*(1-eta)*N*V - delta*I
DDT_V = p*(1-epsilon)*I - c*V
CASE T > T_Stop
DDT_N = s - beta*(1-eta*exp(-k1*(T-T_Stop)))*N*V - d*N
DDT_I = beta*(1-eta*exp(-k1*(T-T_Stop)))*N*V - delta*I
DDT_V = p*(1-epsilon*exp(-k2*(T-T_Stop)))*I - c*V
END
* ''Remark 1:'' Here, <span style="font-family:'Monospaced'">EVENT<span></span> is a reserved variable name. Then, the information in the column <span style="font-family:'Monospaced'">EVENT<span></span> is recognized as a succession of events. Furthermore, the times of the events \verb"Start1", \verb"Start2" and <span style="font-family:'Monospaced'">Stop<span></span> are automatically created as <span style="font-family:'Monospaced'">T_Start1<span></span>, <span style="font-family:'Monospaced'">T_Start2<span></span> and <span style="font-family:'Monospaced'">T_Stop<span></span>.
* ''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:
$EQUATION
T_0 = T_Start1
N_0 = delta*c/(beta*p);
I_0 = (s-d*N)/delta
V_0 = p*I/c
SWITCH
CASE T_Start1 < T < T_Start2
be = beta
pe = p*(1-epsilon)
CASE T_Start2 < T < T_Stop
be = beta*(1-eta)
pe = p*(1-epsilon)
CASE T > T_Stop
be = beta*(1-eta*exp(-k1*(T-T_Stop)))
pe = p*(1-epsilon*exp(-k2*(T-T_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 |
