Figure 4.33:

Species rcDNA versus time for hepatitis B virus model; deterministic and average stochastic models.

Code for Figure 4.33

Text of the GNU GPL.

main.m

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% Copyright (C) 2001, James B. Rawlings and John G. Ekerdt
%
% This program is free software; you can redistribute it and/or
% modify it under the terms of the GNU General Public License as
% published by the Free Software Foundation; either version 2, or (at
% your option) any later version.
%
% This program is distributed in the hope that it will be useful, but
% WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
% General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with this program; see the file COPYING.  If not, write to
% the Free Software Foundation, 59 Temple Place - Suite 330, Boston,
% MA 02111-1307, USA.
%
%
% Hepatitis B model, stochastic simulation

% This m-file loads data file 'hbv_binary.mat'.
% Compile .mex file from 'stochsim.c'.
if exist('stochsim') ~= 3
    mex('-v', 'stochsim.c');
end

[p, tsolver, x] = hbv_common();

% already ran simulation, just load file and write a plotting file
load hbv_binary.mat;


% save for plotting
%
% nullseed(121) generates a nice figure
%
rand('seed', nullseed(121))
xnull = stochsim(p.x0, p.k, p.stoiT, p.tout);

table = [p.tout, x, xavg', xout', xnull'];
save -ascii hbv_stoch_rcdna.dat table;

if (~ strcmp (getenv ('OMIT_PLOTS'), 'true')) % PLOTTING
plot (table(:,1),[table(:,3),table(:,6)]);
axis ([0,200,0,250]);
% TITLE
end % PLOTTING

hbv_common.m

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% Copyright (C) 2001, James B. Rawlings and John G. Ekerdt
%
% This program is free software; you can redistribute it and/or
% modify it under the terms of the GNU General Public License as
% published by the Free Software Foundation; either version 2, or (at
% your option) any later version.
%
% This program is distributed in the hope that it will be useful, but
% WITHOUT ANY WARRANTY; without even the implied warranty of
% MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
% General Public License for more details.
%
% You should have received a copy of the GNU General Public License
% along with this program; see the file COPYING.  If not, write to
% the Free Software Foundation, 59 Temple Place - Suite 330, Boston,
% MA 02111-1307, USA.

% Hepatitis B model, stochastic simulation.  Data and functions defined
% in this file are used in all the hbv_stoch.m and hbv_binary.m

function [p, tsolver, x] = hbv_common()

k = [1; 0.025; 1000; 0.25; 2; 7.5e-6];
p.k = k;
stoi = [0, 1, 0; 1, -1, 0;
        0, 0, 1; -1, 0, 0;
        0, 0, -1; 0, -1, -1];
x0  = [1; 0; 0];
stoiT = stoi';
tfin  = 200;
nts   = 200;
tout = linspace(0,tfin,nts)';

p.x0 = x0; p.k = k; p.stoiT = stoiT; p.tout = tout; p.tfin = tfin;
p.nts = nts;

% steady state analysis
cas = ( k(1)-k(4) ) /   ...
      ( k(6) *k(4)/k(2)*( (-k(1)+k(4) +k(3)) / (k(5))) );
cbs = k(4)/k(2)* cas;
ccs = ( k(1)-k(4) ) / ( k(6) *k(4)/k(2) );
Jac = [-k(4), k(2), 0;
        k(1), -k(2)-k(6)*ccs, -k(6)*cbs;
        k(3), -k(6)*ccs, -k(5)-k(6)*cbs];
eig(Jac);

% function rhs = hbv_deriv(t, x, p);
%   rhs = [p.k(2)*x(2) - p.k(4)*x(1);
% 	 p.k(1)*x(1) - p.k(2)*x(2) - p.k(6)*x(2)*x(3);
% 	 p.k(3)*x(1) - p.k(5)*x(3) - p.k(6)*x(2)*x(3)];
% end%function


% deterministic dynamics
opts = odeset ('AbsTol', sqrt (eps), 'RelTol', sqrt (eps));
[tsolver, x] = ode15s(@(t,x) hbv_deriv(t,x,p), tout, x0, opts);

% Although stochsim is a .cc file for speed,
% list the function here as an m file so it's easier to
% read what it's doing.

%function xout = stochsim(x0, k, stoiT, tout)
%    nts  = length(tout);
%    tfin = tout(nts);
%    time = tout(1);
%    x = x0;
%    iout = 1;

%    while (time < tfin)
%        r(1) = k(1)*x(1);
%        r(2) = k(2)*x(2);
%        r(3) = k(3)*x(1);
%        r(4) = k(4)*x(1);
%        r(5) = k(5)*x(3);
%        r(6) = k(6)*x(2)*x(3);
%        rtot = sum(r);

%        p=rand(2,1);
%        % choose likely time of next reaction
%        tau = -log(p(1))/rtot;
%        timenew = time + tau;
%        % choose which reaction (mth) is likely to occur
%        rcum = 0;
%        m = 0;

%        while ( rcum <= p(2)*rtot)
%            m = m + 1;
%            rcum = rcum + r(m);
%        end

%        xnew  = x + stoiT(:,m);
%        % check if passing time of requested output

%        while ( tout(iout) <= timenew )
%            xout(:,iout) = x;
%            iout = iout + 1;

%            if (iout > nts)
%                break;
%            end
%        end

%        time = timenew;
%        x = xnew;
%    end
%end%function

%xplot1 = stochsim(x0, k, stoiT, tout);

end%function

hbv_binary.mat

stochsim.c (Matlab mex file source)

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/*

Copyright (C) 2001, James B. Rawlings

This program is free software; you can redistribute it and/or
modify it under the terms of the GNU General Public License as
published by the Free Software Foundation; either version 2, or (at
your option) any later version.

This program is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
General Public License for more details.

You should have received a copy of the GNU General Public License
along with this program; see the file COPYING.  If not, write to
the Free Software Foundation, 59 Temple Place - Suite 330, Boston,
MA 02111-1307, USA.

*/

#include <math.h>
#include <stdio.h>

#include <mex.h>

static double
get_rand (void)
{
  double rnd;

  mxArray *plhs[1];

  mexCallMATLAB (1, plhs, 0, 0, "rand");

  rnd = (mxGetPr (plhs[0]))[0];

  mxDestroyArray (plhs[0]);

  return rnd;
}

void
mexFunction (int nlhs, mxArray* plhs[], int nrhs, const mxArray* prhs[])
{
  int nargin = nrhs;

  const mxArray *xin;    double *pxin;
  const mxArray *k;      double *pk;
  const mxArray *stoiT;  double *pstoiT;  int stoiT_nr;
  const mxArray *tout;   double *ptout;

  mxArray *xout;   double *pxout;

  int nts, nx;
  double tfin, time;
  int iout = 0;
  double x0, x1, x2;
  double k0, k1, k2, k3, k4, k5;
  int pxout_offset = 0;

  if (nargin < 4 || nargin > 4)
    mexErrMsgTxt ("usage: stochsim (x0, k, stoiT, tout)");

  /* Maybe we should check that these are numeric, to avoid crashing
     Matlab if someone passes something else?  */
  xin = prhs[0];
  k = prhs[1];
  stoiT = prhs[2];
  tout = prhs[3];

  pxin = mxGetPr (xin);
  pk = mxGetPr (k);
  pstoiT = mxGetPr (stoiT);
  ptout = mxGetPr (tout);

  stoiT_nr = mxGetM (stoiT);

  nts = mxGetNumberOfElements (tout);
  tfin = ptout[nts-1];

  time = ptout[0];

  nx = mxGetNumberOfElements (xin);

  if (nx != 3)
    mexErrMsgTxt ("stochsim: expecting X to be a 3-element vector");

  x0 = pxin[0];
  x1 = pxin[1];
  x2 = pxin[2];

  xout = mxCreateDoubleMatrix (nx, nts, mxREAL);
  pxout = mxGetPr (xout);

  k0 = pk[0];
  k1 = pk[1];
  k2 = pk[2];
  k3 = pk[3];
  k4 = pk[4];
  k5 = pk[5];

  while (time < tfin) 
    {
      double r0 = k0 * x0;
      double r1 = k1 * x1;
      double r2 = k2 * x0;
      double r3 = k3 * x0;
      double r4 = k4 * x2;
      double r5 = k5 * x1 * x2;

      double rtot = r0 + r1 + r2 + r3 + r4 + r5;

      double timenew, rnd, rcum, tmp;
      int m = 0;
      double xnew0, xnew1, xnew2;
      
      /* return if system has extinguished */
      if (rtot == 0.0)
	{
	  while (iout < nts)
	    {
	      pxout[pxout_offset++] = x0;
	      pxout[pxout_offset++] = x1;
	      pxout[pxout_offset++] = x2;

	      iout++;
	    }
	  break;
	}

      /* choose likely time of next reaction */
      timenew = time - log (get_rand ()) / rtot;

      /* choose which reaction (mth) is likely to occur */
      rnd = get_rand ();
      rcum = r0;
      m = 0;
      tmp = rnd*rtot;
      if (rcum <= tmp)
	{
	  ++m; rcum += r1;

	  if (rcum <= tmp)
	    {
	      ++m; rcum += r2;

	      if (rcum <= tmp)
		{
		  ++m; rcum += r3;

		  if (rcum <= tmp)
		    {
		      ++m; rcum += r4;

		      if (rcum <= tmp)
			++m; rcum += r5;
		    }
		}
	    }
	}

      xnew0 = x0 + pstoiT[stoiT_nr*m];
      xnew1 = x1 + pstoiT[stoiT_nr*m+1];
      xnew2 = x2 + pstoiT[stoiT_nr*m+2];

      /* check if passing time of requested output */
      while (ptout[iout] <= timenew)
	{
	  pxout[pxout_offset++] = x0;
	  pxout[pxout_offset++] = x1;
	  pxout[pxout_offset++] = x2;

	  iout++;

	  if (iout >= nts)
	    break;
	}

      time = timenew;

      x0 = xnew0;
      x1 = xnew1;
      x2 = xnew2;
    }

  plhs[0] = xout;
}

hbv_deriv.m

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function rhs = hbv_deriv(t, x, p);
  rhs = [p.k(2)*x(2) - p.k(4)*x(1);
	 p.k(1)*x(1) - p.k(2)*x(2) - p.k(6)*x(2)*x(3);
	 p.k(3)*x(1) - p.k(5)*x(3) - p.k(6)*x(2)*x(3)];
end%function

stochsim.cc (Octave oct file source)

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/*

Copyright (C) 2001, James B. Rawlings

This program is free software; you can redistribute it and/or
modify it under the terms of the GNU General Public License as
published by the Free Software Foundation; either version 2, or (at
your option) any later version.

This program is distributed in the hope that it will be useful, but
WITHOUT ANY WARRANTY; without even the implied warranty of
MERCHANTABILITY or FITNESS FOR A PARTICULAR PURPOSE.  See the GNU
General Public License for more details.

You should have received a copy of the GNU General Public License
along with this program; see the file COPYING.  If not, write to
the Free Software Foundation, 59 Temple Place - Suite 330, Boston,
MA 02111-1307, USA.

*/

#include <octave/oct.h>
#include <octave/parse.h>
#include <octave/variables.h>
#include <octave/oct-map.h>
#include <octave/oct-rand.h>
#include <octave/quit.h>

DEFUN_DLD (stochsim, args, ,
	   "stochsim(x0, k, stoiT, tout)")

{
  octave_value retval;

  int nargin  (args.length ());

  if (nargin < 4 || nargin > 4)
    {
      print_usage ("stochsim");
      return retval;
    }

  const ColumnVector xin (args(0) . vector_value ());
  const ColumnVector k (args(1) . vector_value ());
  const Matrix stoiT (args(2) . matrix_value ());
  const ColumnVector tout (args(3) . vector_value ());

  // if (error_state)
  //   {
  //     error ("stochsim: invalid arguments");
  //     return retval;
  //   }

  const int nts  = tout.numel ();
  const double tfin = tout(nts-1);

  double time = tout(0);
  int iout = 0;

  const int nx = xin.numel ();

  if (nx != 3)
    {
      error ("stochsim: expecting X to be a 3-element vector");
      return retval;
    }

  double x0 = xin(0);
  double x1 = xin(1);
  double x2 = xin(2);

  Matrix xout (nx, nts, 0.0);

  double *pxout = xout.fortran_vec ();

  int pxout_offset = 0;

  double k0 = k(0);
  double k1 = k(1);
  double k2 = k(2);
  double k3 = k(3);
  double k4 = k(4);
  double k5 = k(5);

  while (time < tfin) 
    {
      // Allow this loop to be interrupted.
      OCTAVE_QUIT;

      double r0 = k0 * x0;
      double r1 = k1 * x1;
      double r2 = k2 * x0;
      double r3 = k3 * x0;
      double r4 = k4 * x2;
      double r5 = k5 * x1 * x2;

      double rtot = r0 + r1 + r2 + r3 + r4 + r5;

      // return if system has extinguished
      if (rtot == 0.0)
	{
	  while (iout < nts)
	    {
	      pxout[pxout_offset++] = x0;
	      pxout[pxout_offset++] = x1;
	      pxout[pxout_offset++] = x2;

	      iout++;
	    }
	  break;
	}

      // choose likely time of next reaction
      double timenew = time - log (octave::rand::scalar ()) / rtot;

      // choose which reaction (mth) is likely to occur
      double rnd = octave::rand::scalar ();
      double rcum = r0;
      int m = 0;
      double tmp = rnd*rtot;
      if (rcum <= tmp)
	{
	  ++m; rcum += r1;

	  if (rcum <= tmp)
	    {
	      ++m; rcum += r2;

	      if (rcum <= tmp)
		{
		  ++m; rcum += r3;

		  if (rcum <= tmp)
		    {
		      ++m; rcum += r4;

		      if (rcum <= tmp)
			++m; rcum += r5;
		    }
		}
	    }
	}


      double xnew0 = x0 + stoiT(0,m);
      double xnew1 = x1 + stoiT(1,m);
      double xnew2 = x2 + stoiT(2,m);

      // check if passing time of requested output
      while (tout(iout) <= timenew)
	{
	  pxout[pxout_offset++] = x0;
	  pxout[pxout_offset++] = x1;
	  pxout[pxout_offset++] = x2;

	  iout++;

	  if (iout >= nts)
	    break;
	}

      time = timenew;

      x0 = xnew0;
      x1 = xnew1;
      x2 = xnew2;
    }

  retval = xout;

  return retval;
}