Dynamical systems driven by ODEs
Contents
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, $T_0 = 0$ 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 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 EVENT is necessary in the dataset to describe this information EVENT is an extension of the 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 | 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 $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:
{| align=left style="width: 700px" cellspacing="0" style="border: 1px solid darkgray;"
| $VARIABLES ID, TIME, VL use=DV, EVENT list=(Start1, Start2, Stop)
|-
|-
| $EQUATION
|-
|-
| SWITCH
|-
|align="right" width="50%"| CASE T < T_Start1
|-
| align="right" width="40%"| N = delta*c/(beta*p);
|-
| align="right" width="40%"| I = (s-d*N)/delta
|-
| align="right" width="40%"|V = p*I/c
|-
|-
|align="right" width="50%"| CASE T_Start1 < T < T_Start2
|-
|align="right" width="40%"|DDT_N = s - beta*N*V - d*N
|-
|align="right" width="40%"|DDT_I = beta*N*V - delta*I
|-
|align="right" width="40%"|DDT_V = p*(1-epsilon)*I - c*V
|-
|-
|align="right" width="40%"|CASE T_Start2 < T < T_Stop
|-
|align="right" width="40%"|DDT_N = s - beta*(1-eta)*N*V - d*N
|-
|align="right" width="40%"|DDT_I = beta*(1-eta)*N*V - delta*I
|-
|align="right" width="40%"|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:
{| align=left style="width: 700px" cellspacing="0" style="border: 1px solid darkgray;"
! $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 |}
