Difference between revisions of "Dynamical systems driven by ODEs"

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(A basic model)
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\begin{equation}
 
\begin{equation}
 
\label{vk1}
 
\label{vk1}
{
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\left\{
 
\begin{array}{lll}
 
\begin{array}{lll}
 
   \dot{N}(t) & = & s - \beta \, N(t) V(t) - d N(t)   
 
   \dot{N}(t) & = & s - \beta \, N(t) V(t) - d N(t)   
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   \dot{V}(t) & = & p I(t) - c \, V(t)
 
   \dot{V}(t) & = & p I(t) - c \, V(t)
 
\end{array}
 
\end{array}
left}
+
\right.
 
\end{equation}
 
\end{equation}
  

Revision as of 13:38, 23 January 2013

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:

  1. let $A_0 = A(t_0)$ be the initial condition of the system defined at the initial time $t_0$,
  2. 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:

ID TIME VL
1 -5 6.5
1 -2 7.1
1 1 6.3
1 5 4.2
1 12 2.1
1 20 0.9
$\vdots$ $\vdots$ $\vdots$



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} \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} \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):


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, $\verb{T_0 = 0}$ means that the system is constant and is $A^{\star}$, defined in the script by $<span style="font-family:'Tahorma';">(N_0, I_0, V_0)</span>$, for any $t<0$.
    • Remark2: If the initial condition is not given in the model, it is assumed to be 0.

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*}

Example:  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 \texttt{EVENT} is necessary in the dataset to describe this information EVENT is an extension of the \texttt{EVID} (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 $T_{Stop}=200$:


align="left" 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 \verb"$EQUATION" and using the statement \verb"SWITCH" with MLXTRAN). We only show the blocks \verb"$VARIABLES" and \verb"$EQUATION" of the code: \hspace*{2cm} \begin{minipage}[b]{10cm} \begin{verbatim} $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 DDT_V = p*(1-epsilon)*I - c*V 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 \end{verbatim} \end{minipage} \noindent{\bf Remark 1:} Here, \verb"EVENT" is a reserved variable name. Then, the information in the column \verb"EVENT" is recognized as a succession of events. Furthermore, the times of the events \verb"Start1", \verb"Start2" and \verb"Stop" are automatically created as \verb"T_Start1", \verb"T_Start2" and \verb"T_Stop". \noindent{\bf 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: \hspace*{2cm} \begin{minipage}[b]{10cm} \begin{verbatim} $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 \end{verbatim} \end{minipage}

Dynamical systems with source terms