Difference between revisions of "Dynamical systems driven by ODEs"
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− | The initial condition and the dynamical system are described in the MDL <span style="font-family:'Tahorma' | + | The initial condition and the dynamical system are described in the MDL <span style="font-family:'Tahorma';">EQUATION</span> with MLXTRAN): |
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</nowiki> | </nowiki> | ||
− | **Remark1: Here, $\verb{T_0 = 0}$ means that the system is constant and is $\ | + | **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 $t<0$. |
\noindent{\bf Remark2:} If the initial condition is not given in the model, it is assumed to be 0. | \noindent{\bf Remark2:} If the initial condition is not given in the model, it is assumed to be 0. | ||
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\noindent \underline{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 $T_{Stop}$. | \noindent \underline{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 | + | 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 $T_{Stop}=200$: |
{| | {| | ||
− | !ID | + | !ID || TIME || VL || EVENT |
− | 1 | + | |- |
− | 1 | + | 1 || -5 || 6.5 || . |
− | 1 | + | |- |
− | 1 | + | 1 || -2 || 7.1 || . |
− | \vdots | + | |- |
− | 1 | + | 1 || 0 || . || Start1 |
− | 1 | + | |- |
− | 1 | + | 1 || 5 || 5.2 || . |
− | \vdots | + | |- |
− | 1 | + | \vdots || \vdots \vdots || \vdots |
− | 1 | + | |- |
− | 1 | + | 1 || 18 || 4.6 || . |
− | \vdots | + | |- |
+ | 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 | ||
} | } | ||
Revision as of 16:08, 22 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:
- 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} \begin{array}{lll} \dot{N}(t) & = & s - \beta \, \it{N}(t) \it{V}(t) - d\it{N}(t) \\ \dot{I}(t) & = & \beta \, \it{N}(t)\, \it{V}(t) - \delta \, \it{I}(t) \\ \dot{V}(t) & = & p \it{I}(t) - c \, \it{V}(t) \end{array} \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 </nowiki>
- Remark1: Here, $\verb{T_0 = 0}$ means that the system is constant and is $\A^{\star}$, defined in the script by $(N_0, I_0, V_0)$, for any $t<0$.
\noindent{\bf Remark2:} If the initial condition is not given in the model, it is assumed to be 0.
\subsubsection{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 \\
\dA{A}{} &= &F_k(A(t)) \ , \ t_{k-1} \leq t \leq t_{k}
\end{eqnarray*}
\noindent \underline{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 $T_{Stop}=200$:
Dynamical systems with source terms
ID | TIME | VL | EVENT |
---|