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:

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} \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 (in a block EQUATION with MLXTRAN):

\begin{minipage}[b]{10cm} \begin{verbatim} '"`UNIQ-MathJax22-QINU`"'\As'"`UNIQ-MathJax23-QINU`"'t<0'"`UNIQ-MathJax24-QINU`"'t_0< t_1< ...<t_K'"`UNIQ-MathJax25-QINU`"'F^{(1)}, F^{(2)},\ldots,F^{(K)}'"`UNIQ-MathJax26-QINU`"'T_{Start1}'"`UNIQ-MathJax27-QINU`"'T_{Start2}'"`UNIQ-MathJax28-QINU`"'T_{Stop}'"`UNIQ-MathJax29-QINU`"'(T_{Start1},T_{Start2},T_{Stop})'"`UNIQ-MathJax30-QINU`"'T_{Start1}=0'"`UNIQ-MathJax31-QINU`"'T_{Start2}=20'"`UNIQ-MathJax32-QINU`"'T_{Stop}=200'"`UNIQ-MathJax33-QINU`"'T_{Start1}'"`UNIQ-MathJax34-QINU`"'A(t) = \As'"`UNIQ-MathJax35-QINU`"'\As'"`UNIQ-MathJax36-QINU`"'T_{Start1}'"`UNIQ-MathJax37-QINU`"'T_{Start2}'"`UNIQ-MathJax38-QINU`"'T_{Start2}'"`UNIQ-MathJax39-QINU`"'T_{Stop}'"`UNIQ-MathJax40-QINU`"'T_{Stop}'"`UNIQ-MathJax41-QINU`"'EQUATION" and using the statement \verb"SWITCH" with MLXTRAN). We only show the blocks \verb"'"`UNIQ-MathJax42-QINU`"'EQUATION" of the code: \hspace*{2cm} \begin{minipage}[b]{10cm} \begin{verbatim} '"`UNIQ-MathJax43-QINU`"'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}