# sundialr - An Interface to ‘SUNDIALS’ Ordinary Differential Equation (ODE) Solvers

#### Satyaprakash Nayak

#### 2021-05-16

## Introduction

Ordinary Differential Equations (ODEs) describe the rate of change of dependent variables with respect to a single independent variable and are used in many fields to model behavior of the system. There are many good `C` libraries available to solve (i.e., integrate systems of ODEs) and SUNDIALS available from the Lawrence Livermore National Laboratory is a one of the most popular and well-respected `C` library for solving non-stiff and stiff systems of ODEs.

Currently, this package provides an interface to the `CVODE` and `CVODES` function (serial version) in the library which is used to solve ODEs (or Initial Value Problems) and calculate sensitivities.

The four exported functions from the package are:

- `cvode` - An interface to the `CVODE` function in `SUNDIALS` to solve a system of ODEs.

- `cvodes` - An interface to the `CVODES` function in `SUNDIALS` to calculate forward sensitivites with respect to parameters of the ODE system.

- `ida` - An interface to the `IDA` function in `SUNDIALS` to solve a system of Differential-Algebraic Equations (DAEs).

- `cvsolve` - A convenient interface to solve a system of ODEs with discontinuities in solution.

In future, we plan to provide interface for the other solvers (i.e., `IDA/IDAS` and `ARCODE` in the library also. Right now, this package serves as a test case for providing an interface to the `SUNDIALS` library for `R` users.

One of the advantage of using this package is that all the source code of the `SUNDIALS` library is bundled with the package itself, so it does not require the `SUNDIALS` library to be installed on the machine separately (which is sometimes non trivial on a Windows machine).

## System of ODEs

As described in the link above, the problem is from chemical kinetics, and consists of the following three rate equations:

\[ \begin{aligned} \frac{dy_1}{dt} &= -.04 \times y_1 + 10^4 \times y_2 \times y_3 \\ \frac{dy_2}{dt} &= .04 \times y_1 - 10^4 \times y_2 \times y_3 - 3 \times 10^7 \times y_2^2 \\ \frac{dy_3}{dt} &= 3 \times 10^7 \times y_2^2 \end{aligned} \]

with time interval from \(t = 0.0\) to \(t = 4 \times 10^{10}\) and initial conditions: \[ y_1 = 1.0 , ~y_2 = y_3 = 0 \]

The problem is stiff.

The original example , While integrating the system, also uses the rootfinding feature to find the points at which

\[ y_1 = 1 \times 10^{-4} \] or at which \[ y_3 = 0.01 \] but currently root-finding is not supported in this version. As in the original example, this package also solves the problem with the `BDF` method, Newton iteration with the `SUNDENSE` dense linear solver, however, without a user-supplied Jacobian routine (unlike the original example). The future versions may include an ability to provide Jacobian calculated analytically or via automatic differentiation. `CVODE` uses a scalar relative tolerance and a vector absolute tolerance (which can be provided as an input). Output is printed in decades from \(t = 0.4\) to \(t = 4 \times 10^{10}\) in this example.

## Writing the Differential Equations

### Using R

Differential equations can be written as an `R` function or as an `Rcpp` function. Differential equations function must be written as

    function(t, y, p){
      # code to write differential equations
      # using parameter vector (p) and state/entity vector (y)
      # should return `ydot`, the vector representing
      # rate of change of entities in `y`
      # length of `ydot` must be equal to `y1
    }

where `t` represents time, `y` is the vector describing the values of states/entities of the ODE system at time `t` and `p` is the vector of parameters used to define the ODEs. The output of this function is a vector of rate of change of entities of `y`.

The key aspect to keep in mind is that the signature of the function must be `function(t,y,p)`. As an example, we try to solve the `cv_Roberts_dns.c` problem described above, the original code can be found here. An example of an `R` function is as follows:

    ODE_R <- function(t, y, p){

       ## initialize the derivative vector
       ydot <- vector(mode = "numeric", length = length(y))

       ## p (parameter vector input) is  [-0.04 1e04 3e07]

       ydot[1] = p[1]*y[1] + p[2]*y[2]*y[3]
       ydot[2] = -p[1]*y[1] - p[2]*y[2]*y[3] - p[3]*y[2]*y[2]
       ydot[3] = p[3]*y[2]*y[2]

       ydot      ## return ydot
    }

where `p` is a parameter vector with the values `[-0.04 1e04 3e07]`.

### Using Rcpp

Also, since this package using `Rcpp` to bundle the C code, we can use the notation used in `Rcpp` to describe the system of ODEs. The `cv_Roberts_dns` problem describe above can be described in an `Rcpp` function as follows (indices in `C++` start from `0`, functions need to declare their return type, here `NumericVector` and every expression ends in a semicolon, `;`) :

    #include <Rcpp.h>
    using namespace Rcpp;

    // [[Rcpp::export]]
    NumericVector ODE_Rcpp (double t, NumericVector y){

      // Initialize ydot filled with zeros
      NumericVector ydot(y.length());

      // p (parameter vector) is [-0.04 1e04 3e07]
      ydot[0] = p[0] * y[0] + p[1] * y[1] * y[2];
      ydot[1] = -p[0]*y[0] - p[1]*y[1]*y[2] - p[2]*y[1]*y[1]
      ydot[2] = p[2] * y[1] * y[1];

      return ydot;

    }

The above is a re-write of the `cvRoberts_dns.c` example in the documentation of `CVODE`.

### Putting everything together

The entire `R` file to create right hand side of ODE function (which calculates rates of change) is as follows (also found here):

    # ODEs described by an R function
    ODE_R <- function(t, y, p){

       ## initialize the derivative vector
       ydot <- vector(mode = "numeric", length = length(y))

       ## p (parameter vector) is  [-0.04 1e04 3e07]

       ydot[1] = p[1]*y[1] + p[2]*y[2]*y[3]
       ydot[2] = -p[1]*y[1] - p[2]*y[2]*y[3] - p[3]*y[2]*y[2]
       ydot[3] = p[3]*y[2]*y[2]

       ydot      ## return ydot
    }

    # ODEs can also be described using Rcpp
    Rcpp::sourceCpp(code = '

    #include <Rcpp.h>
    using namespace Rcpp;

    // [[Rcpp::export]]
    NumericVector ODE_Rcpp (double t, NumericVector y){

      // Initialize ydot filled with zeros
      NumericVector ydot(y.length());

      // p (parameter vector) is [-0.04 1e04 3e07]
      ydot[0] = p[0] * y[0] + p[1] * y[1] * y[2];
      ydot[1] = -p[0]*y[0] - p[1]*y[1]*y[2] - p[2]*y[1]*y[1]
      ydot[2] = p[2] * y[1] * y[1];

      return ydot;

    }')

    # Generate time vector, IC and call cvode to solve the equations
    # R code to genrate time vector, IC and solve the equations
    time_vec <- c(0.0, 0.4, 4.0, 40.0, 4E2, 4E3, 4E4, 4E5, 4E6, 4E7, 4E8, 4E9, 4E10)
    IC <- c(1,0,0)
    params <- c(0.04, 10000, 30000000)
    reltol <- 1e-04
    abstol <- c(1e-8,1e-14,1e-6)

    ## Solving the ODEs using cvode function
    df1 <- cvode(time_vec, IC, ODE_R , params, reltol, abstol)           ## using R
    df2 <- cvode(time_vec, IC, ODE_Rcpp , params, reltol, abstol)        ## using Rcpp

    ## Check that both solutions are identical
    # identical(df1, df2)

The final output is the `df1` matrix in which first column is time, second, third and fourth column are the values of `y1`, `y2` and `y3` respectively.

    > df1
           [,1]         [,2]         [,3]       [,4]
     [1,] 0e+00 1.000000e+00 0.000000e+00 0.00000000
     [2,] 4e-01 9.851641e-01 3.386242e-05 0.01480205
     [3,] 4e+00 9.055097e-01 2.240338e-05 0.09446793
     [4,] 4e+01 7.158016e-01 9.185043e-06 0.28418924
     [5,] 4e+02 4.505209e-01 3.222826e-06 0.54947590
     [6,] 4e+03 1.832217e-01 8.943516e-07 0.81677741
     [7,] 4e+04 3.898091e-02 1.621669e-07 0.96101893
     [8,] 4e+05 4.936971e-03 1.984450e-08 0.99506301
     [9,] 4e+06 5.170103e-04 2.069098e-09 0.99948299
    [10,] 4e+07 5.204927e-05 2.082078e-10 0.99994795
    [11,] 4e+08 5.184946e-06 2.073989e-11 0.99999482
    [12,] 4e+09 5.246212e-07 2.098486e-12 0.99999948
    [13,] 4e+10 6.043000e-08 2.417200e-13 0.99999994

## Solving a system of Differential-Algebraic Equations

An interface to the `IDA` solver is also provided to solve a system of Differential-Algebraic equations. A system of differential-algebraic equations is a system of equations containing both differential and algebraic equations and can be written as \[ F(\dot{x}(t), x(t), t) = 0 \] The equations for such a system are written in terms of residuals and require both the value of \(x_0\) and \(\dot{x}_0\) as the initial conditions. Writing the previously solved equations as a system of DAEs, we have, \[ \begin{aligned} \dot{y}_1 &= -p_1y_1 + p_2y_2y_3 \\ \dot{y}_2 &= p_1y_1 - p_2y_2y_3 - p_3y_2^2 \\ 1 &= y_1 + y_2 + y_3 \end{aligned} \] The above system of DAEs can be written in terms of residuals as

\[ \begin{aligned} res_1 &= -p_1y_1 + p_2y_2y_3 - \dot{y}_1 \\ res_2 &= p_1y_1 - p_2y_2y_3 - p_3y_2^2 - \dot{y}_2\\ res_3 &= y_1 + y_2 + y_3 - 1 \end{aligned} \] Here is the complete code for solving this system of DAEs,

    DAE_R <- function(t, y, ydot, p){

      # vector containing the residuals
      res = vector(mode = "numeric", length = length(y))

      # R indices start from 1
      res[1] <- -0.04 * y[1] + 10000 * y[2] * y[3] - ydot[1]
      res[2] <- -res[1] - 30000000 * y[2] * y[2] - ydot[2]
      res[3] <- y[1] + y[2] + y[3] - 1.0

      res
    }

    # R code to genrate time vector, IC and solve the equations
    time_vec <- c(0.0, 0.4, 4.0, 40.0, 4E2, 4E3, 4E4, 4E5, 4E6, 4E7, 4E8, 4E9, 4E10)
    IC <- c(1,0,0)                  # Initial value of y
    IRes <- c(-0.4, 0.4, 0)         # Initial value of ydot
    params <- c(0.04, 10000, 30000000)
    reltol <- 1e-04
    abstol <- c(1e-8,1e-14,1e-6)

    ## Solving the DAEs using the ida function
    df1 <- sundialr::ida(time_vec, IC, IRes, DAE_R , params, reltol, abstol)

## Solving an ODE system with discontinuties in solution

The `cvsolve` function defined in `sundialr` package provides a convenience interface to solve ODEs with one or more discontinuities in solution. An example of such a system of ODEs would be pharmacokinetics of a drug with repeated bolus administration. Let’s look at a simple example of multiple doses of a drug with a first-order degradation administered intravenously. The ODE system for the drug is \[ \frac{dC}{dt} = -k_{el} * C \] where \(C\) is the concentration of the drug and \(k_{el}\) is the elimination rate of the drug. The \(R\) code for such a system is

    ODErepeated_R <- function(t, y, p){

      # vector containing the right hand side gradients
      ydot = vector(mode = "numeric", length = length(y))

      # R indices start from 1
      ydot[1] = -p[1]*y[1]

      ydot

    }

We also need to define when the multiple doses are given and the state to which they are to be applied (here to be applied to the only state in the model, \(C\)). This is provided via the \(Events\) dataframe (here, \(TDOSE\) or the dosing dataframe).

    TDOSE <- data.frame(ID = 1, TIMES = c(0, 10, 20, 30, 40, 50), VAL = 100)
    TDOSE
    #>   ID TIMES VAL
    #> 1  1     0 100
    #> 2  1    10 100
    #> 3  1    20 100
    #> 4  1    30 100
    #> 5  1    40 100
    #> 6  1    50 100

`TDOSE` is a data frame with the index of the state to which discontinuity is to be applied (represented by `ID`), the times at which the discontinuity is to be applied (represented by `TIMES`) at the value to be added to the value of the state at that time-point. A typical example would be addition of the dose amount to the value of the concentration at the specified times, e.g., the `TDOSE` data frame says that a value of `100` is to be added to the concentration of the 1st state (the only state in this system) at the times specified by the `TIMES` column. In summary,

- `ID` represents the index of the state with discontinuity

- `TIMES` represents the times at which discontinutiy is applied

- `VAL` represents the value added to the value of the state at specified times.

The complete code for simulating such a system is

    # Example of solving a set of ODEs with multiple discontinuities using cvsolve
    # A simple One dimensional equation, y = -0.1 * y
    # ODEs described by an R function
    ODE_R <- function(t, y, p){

      # vector containing the right hand side gradients
      ydot = vector(mode = "numeric", length = length(y))

      # R indices start from 1
      ydot[1] = -p[1]*y[1]

      ydot

    }

    # R code to generate time vector, IC and solve the equations
    TSAMP <- seq(from = 0, to = 100, by = 0.1)      # sampling time points
    IC <- c(1)
    params <- c(0.1)

    # A dataset describing the dosing at times at which additions to y[1] are to be done
    # Names of the columns don't matter, but they MUST be in the order of state index,
    # times and Values at discontinuity.
    TDOSE <- data.frame(ID = 1, TIMES = c(0, 10, 20, 30, 40, 50), VAL = 100)

    df1 <- sundialr::cvsolve(TSAMP, c(1), ODE_R, params)         # solving without any discontinuity
    df2 <- sundialr::cvsolve(TSAMP, c(1), ODE_R, params, TDOSE)  # solving with discontinuity

    ## Plot the solution with discontinuities
    ## first column is time, second column is the state
    time <- df2[,1]
    y1 <- df2[,2]
    plot(time, y1, type = "l", lty = 1, main = "An ODE system with discontinuties", frame.plot = F)

Note that in the example above, `TSAMP` is the sampling time at which the solution is desired. Also, even though an Initial Value of \(y_1\) of 1 is provided by the `IC` parameter, it is overwritten by the value of 100 provided in the `TDOSE` data frame. In general, the values in the initial conditions are overwritten by values in the `Events` input.

## System of ODEs for Parameter Sensitivities

Sensitivity with respect to the parameters of the ODE system can be calculated using `CVODES` function. This package implements Forward Sensitivity Analysis from `CVODES` function (see the example `cvRoberts_FSA_dns.c` from the link here). Briefly, given the ODE system as described below

\[ \begin{aligned} \frac{dy_1}{dt} &= -p_1y_1 + p_2y_2y_3 \\ \frac{dy_2}{dt} &= p_1y_1 - p_2y_2y_3 - p_3y_2^2 \\ \frac{dy_3}{dt} &= p_3y_2^2 \end{aligned} \] with the same initial conditions as above (i.e., \(y_1 = 0, y_2 = y_3 = 0\)) and \(p_1 = 0.04, \quad p_2 = 10^4, \quad p_3 = 3\times10^7\). The system of Sensitivity equations (taken from `cvs_guide.pdf`) that is solved can be given by \[ \begin{aligned} \frac{ds}{dt} = \left[\begin{array} {ccc} -p_1 & p_2y_3 & p_2y_2 \\ p_1 & -p_2y_3-2p_3y_2 & -p_2y_2 \\ 0 & 2p_3y_2 & 0 \end{array}\right]s_i + \frac{\partial f}{\partial p_i}, \quad s_i(t_0) = \left[\begin{array} {c} 0 \\ 0 \\ 0 \end{array}\right], \quad i = 1, 2, 3 \end{aligned} \] where \[ \frac{\partial f}{\partial p_1} = \left[\begin{array} {c} -y_1 \\ y_1 \\ 0 \end{array}\right], \quad \frac{\partial f}{\partial p_2} = \left[\begin{array} {c} y_2y_3 \\ -y_2y_3 \\ 0 \end{array}\right], \quad \frac{\partial f}{\partial p_3} = \left[\begin{array} {c} 0 \\ -y_2^2 \\ y_2^2 \end{array}\right] \] In the original `CVODES` interface from `SUNDIALS`, the sensitivity equations can either be provided by the user or can be calculated using numerical interpolation by the solver. Here, I have only included the numerical interpolation version and currently the user cannot specify the sensitivity equations. However, in the future versions I will provide an ability to specify user-defined Jacobian as well as user-defined sensitivity equations.

Also, currently, forward sensitivities are calculated with respect to all parameters of the system. I plan to provide in future, an ability to specify specific particular parameters for which sensitivity is desired. Currently, `SIMULATENOUS` and `STAGGERED` methods of sensitivity calculations from the `SUNDIALS` library are supported in this package.

## Calculation of Sensitivities using `CVODES`

Once, the system of ODEs has been defined using the instructions provided above, sensitivities can be easily calculated using the `cvodes` function using the function call below (the entire code can be found at this link)

    df1 <- cvodes(time_vec, IC, ODE_R , params, reltol, abstol,"STG",F)  ## using R
    df2 <- cvodes(time_vec, IC, ODE_Rcpp , params, reltol, abstol,"STG",F)  ## using Rcpp

The additional arguments in `cvodes` specify the senstivity calculation method to be used (`STG` for `STAGGERED` or `SIM` for `SIMULATENOUS`) and flag for error control (either `T` or `F`).

The output of `cvodes` is a matrix with number of rows equal to the length of the time vector (`time_vec`) and the number of columns being equal to length of (`y` \(\times\) `p` + 1). The first columns is for time. Currently, the sensitivity of every enitity is calculated with respect to every parameter in model. For example, for the current model with `3` entities (ODEs) and `3` parameters, a total of `9` sensitivities are calculated at each output time, i.e. `y1` w.r.t `p1`, `p2`, `p3`, `y2` w.r.t. `p1`, `p2`, `p3` and so on. The first 3 (`length(y)`) columns give sensitivity w.r.t the first parameter, the next 3 (`length(y)`) columns give sensitivity w.r.t the second parameter and so on.

In the Sensitivity Matrix output for the systems of equations described above, the first column gives output time, the next `3` columns provide sensitivity of `y1`, `y2` and `y3` w.r.t first parameter (say `p1`), the next three columns provide sensitivity of `y1`, `y2` and `y3` w.r.t. the second parameter (`p2`) and so on. The output Sensitivity Matrix is given below. The sensitivity values match with the values provided in the `CVODES` documentation.

    > df1
           [,1]          [,2]          [,3]         [,4]         [,5]          [,6]          [,7]          [,8]          [,9]        [,10]
     [1,] 0e+00  0.000000e+00  0.000000e+00 0.000000e+00 0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00  0.000000e+00 0.000000e+00
     [2,] 4e-01 -3.561085e-01  3.902252e-04 3.557183e-01 9.483149e-08 -2.132509e-10 -9.461823e-08 -1.573297e-11 -5.289692e-13 1.626194e-11
     [3,] 4e+00 -1.876130e+00  1.792229e-04 1.875951e+00 2.961233e-06 -5.830758e-10 -2.960650e-06 -4.932970e-10 -2.762408e-13 4.935732e-10
     [4,] 4e+01 -4.247395e+00  4.592812e-05 4.247349e+00 1.372964e-05 -2.357270e-10 -1.372941e-05 -2.288274e-09 -1.138015e-13 2.288387e-09
     [5,] 4e+02 -5.958192e+00  3.545986e-06 5.958189e+00 2.273754e-05 -2.260807e-11 -2.273752e-05 -3.789554e-09 -4.994795e-14 3.789604e-09
     [6,] 4e+03 -4.750132e+00 -5.991971e-06 4.750138e+00 1.880937e-05  2.312156e-11 -1.880939e-05 -3.134824e-09 -1.875976e-14 3.134843e-09
     [7,] 4e+04 -1.574902e+00 -2.761679e-06 1.574905e+00 6.288404e-06  1.100645e-11 -6.288415e-06 -1.047876e-09 -4.536508e-15 1.047881e-09
     [8,] 4e+05 -2.363168e-01 -4.584043e-07 2.363173e-01 9.450741e-07  1.832930e-12 -9.450760e-07 -1.574929e-10 -6.362045e-16 1.574935e-10
     [9,] 4e+06 -2.566355e-02 -5.105587e-08 2.566361e-02 1.026491e-07  2.042044e-13 -1.026493e-07 -1.711080e-11 -6.851356e-17 1.711087e-11
    [10,] 4e+07 -2.597859e-03 -5.190342e-09 2.597864e-03 1.039134e-08  2.076100e-14 -1.039136e-08 -1.732552e-12 -6.930923e-18 1.732559e-12
    [11,] 4e+08 -2.601996e-04 -5.199259e-10 2.602002e-04 1.040802e-09  2.079717e-15 -1.040804e-09 -1.737821e-13 -6.951356e-19 1.737828e-13
    [12,] 4e+09 -2.648142e-05 -5.616896e-11 2.648147e-05 1.059193e-10  2.246502e-16 -1.059195e-10 -1.804535e-14 -7.218146e-20 1.804542e-14
    [13,] 4e+10 -2.899376e-06 -7.759920e-12 2.899383e-06 1.159764e-11  3.104024e-17 -1.159768e-11 -1.727574e-15 -6.910296e-21 1.727581e-15

In future, I intend to provide options to select specific entities and parameters with respect to which sensitivities are to be computed as the sensitivity matrix can get very large for medium to large models.

## Summary

- The package `sundialr` provides a way to interface with the famous `SUNDIALS` C library (provided by Lawerence Livermore National Security) to solver initial value problems.

- The package allows the system of differential equations to be written in `R` or using `Rcpp`. Function `cvode` is used to solve initial value problems with a single initialization, but problems with multiple discontinuities in the solution can be solved using the `cvsolve` interface.

- For sensitivities, currently, calculation of forward sensitivities for all entities with respect to all the parameters in the model is implemented in the `cvodes` function. An ability to select specific entities and parameters for which sensitivities is to be calculated will be added soon.

- To solve a system of differential algebraic systems, the `ida` function is provided which is an interface to the `IDA` function in `SUNDIALS`.

As a note, since this package is under active development, the interfaces of both `CVODE` and `CVODES` (i.e., the function signatures) may change in the future versions. Please keep this mind if you intend to use `sundialr` in your applications. In near future, interface for other solvers from the `C` library such as`IDAS` and `ARKODE` may also be added.