Systems of Differential Equations
/*--------------------------------------------------------------
SAS Sample Library
Name: modex10.sas
Description: Example program from SAS/ETS User's Guide,
The MODEL Procedure
Title: Systems of Differential Equations
Product: SAS/ETS Software
Keys: nonlinear simultaneous equation models
PROC: MODEL
Notes:
--------------------------------------------------------------*/
data fit;
length type $ 8;
input type $ time f ;
datalines;
assoc 0 95.3156
assoc 10 97.8261
assoc 20 102.851
assoc 30 101.506
assoc 40 102.737
assoc 50 103.932
assoc 60 105.481
assoc 70 106.458
assoc 80 107.287
assoc 90 106.778
assoc 100 108.601
assoc 110 111.473
assoc 120 115.147
assoc 130 111.667
assoc 140 114.835
assoc 150 113.911
assoc 160 113.759
assoc 170 116.336
assoc 180 116.574
assoc 190 116.567
assoc 200 117.409
assoc 210 121.065
assoc 220 119.332
assoc 230 120.39
assoc 240 126.192
assoc 250 122.187
assoc 260 122.927
assoc 270 123.114
assoc 280 124.94
assoc 290 127.959
assoc 300 129.619
assoc 310 126.638
assoc 320 126.27
assoc 330 128.662
assoc 340 129.566
assoc 350 131.738
assoc 360 131.98
assoc 370 131.361
assoc 380 134.137
assoc 390 133.676
assoc 400 132.277
assoc 410 133.478
assoc 420 135.748
assoc 430 135.881
assoc 440 134.653
assoc 450 135.946
assoc 460 137.99
assoc 470 137.034
assoc 480 140.372
assoc 490 139.477
assoc 500 139.563
assoc 510 140.564
assoc 520 142.086
assoc 530 140.779
assoc 540 140.389
assoc 550 144.758
assoc 560 148.054
assoc 570 142.608
assoc 580 146.63
assoc 590 147.299
assoc 600 144.674
assoc 610 147.017
assoc 620 148.054
assoc 630 148.512
assoc 640 147.753
assoc 650 151.589
assoc 660 150.636
assoc 670 151.332
assoc 680 150.099
assoc 690 150.497
assoc 700 150.895
assoc 710 151.705
assoc 720 153.556
assoc 730 151.709
assoc 740 157.416
assoc 750 153.535
assoc 760 152.944
assoc 770 153.816
assoc 780 154.755
assoc 790 158.671
assoc 800 155.613
assoc 810 155.237
assoc 820 157.842
assoc 830 156.842
assoc 840 157.126
assoc 850 158.536
assoc 860 158.95
assoc 870 159.514
assoc 880 158.838
assoc 890 162.257
assoc 900 162.525
assoc 910 159.284
assoc 920 164.124
assoc 930 161.183
assoc 940 162.96
assoc 950 162.346
assoc 960 171.145
assoc 970 163.673
assoc 980 165.25
assoc 990 164.323
assoc 1000 164.919
assoc 1010 166.521
assoc 1020 164.93
assoc 1030 166.189
assoc 1040 174.801
assoc 1050 168.71
assoc 1060 167.583
assoc 1070 172.601
assoc 1080 166.945
assoc 1090 168.205
assoc 1100 170.792
assoc 1110 174.314
assoc 1120 169.344
assoc 1130 170.462
assoc 1140 171.963
assoc 1150 172.162
assoc 1160 170.211
assoc 1170 171.167
assoc 1180 170.612
assoc 1190 172.059
assoc 1200 176.471
assoc 1210 173.598
assoc 1220 173.896
assoc 1230 175.865
assoc 1240 176.173
assoc 1250 175.497
assoc 1260 176.433
assoc 1270 172.532
assoc 1280 175.219
assoc 1290 175.07
assoc 1300 174.613
assoc 1310 174.751
assoc 1320 175.179
assoc 1330 176.77
assoc 1340 181.655
assoc 1350 178.105
assoc 1360 174.831
assoc 1370 176.513
assoc 1380 178.623
assoc 1390 176.737
assoc 1400 177.368
assoc 1410 178.847
assoc 1420 178.095
assoc 1430 182.049
assoc 1440 177.725
assoc 1450 179.253
assoc 1460 181.2
assoc 1470 182.122
assoc 1480 180.92
assoc 1490 179.435
assoc 1500 180.661
assoc 1510 180.382
assoc 1520 181.768
assoc 1530 180.342
assoc 1540 181.648
assoc 1550 181.178
assoc 1560 183.903
assoc 1570 182.538
assoc 1580 181.243
assoc 1590 183.426
assoc 1600 186.311
assoc 1610 184.986
assoc 1620 184.56
assoc 1630 183.473
assoc 1640 182.797
assoc 1650 183.016
assoc 1660 182.876
assoc 1670 183.612
assoc 1680 187.733
assoc 1690 184.976
assoc 1700 186.317
assoc 1710 186.696
assoc 1720 185.016
assoc 1730 191.803
assoc 1740 190.652
assoc 1750 186.834
assoc 1760 186.056
assoc 1770 185.311
assoc 1780 185.72
assoc 1790 187.316
assoc 1800 187.126
assoc 1810 188.341
assoc 1820 187.734
assoc 1830 187.41
assoc 1840 191.155
assoc 1850 190.326
assoc 1860 187.938
assoc 1870 188.124
assoc 1880 190.667
assoc 1890 188.345
assoc 1900 191.922
assoc 1910 188.809
assoc 1920 190.362
assoc 1930 192.464
assoc 1940 190.092
assoc 1950 189.901
assoc 1960 191.202
assoc 1970 192.834
assoc 1980 194.99
assoc 1990 189.845
assoc 2000 192.008
assoc 2010 190.895
assoc 2020 190.642
assoc 2030 190.83
assoc 2040 190.837
assoc 2050 191.898
assoc 2060 192.644
assoc 2070 193.474
assoc 2080 194.946
assoc 2090 193.121
assoc 2100 192.384
assoc 2110 193.209
assoc 2120 192.644
assoc 2130 194.561
assoc 2140 195.312
assoc 2150 196.94
assoc 2160 195.35
assoc 2170 192.86
assoc 2180 194.497
assoc 2190 195.129
assoc 2200 197.658
assoc 2210 195.149
assoc 2220 196.149
assoc 2230 194.4
assoc 2240 195.635
assoc 2250 197.024
assoc 2260 195.287
assoc 2270 199.246
assoc 2280 197.62
assoc 2290 195.04
assoc 2300 195.628
assoc 2310 196.539
assoc 2320 196.461
assoc 2330 196.582
assoc 2340 198.768
assoc 2350 195.822
assoc 2360 196.264
assoc 2370 199.489
assoc 2380 197.161
assoc 2390 198.202
assoc 2400 200.118
assoc 2410 198.228
assoc 2420 201.065
assoc 2430 199.882
assoc 2440 197.442
assoc 2450 197.082
assoc 2460 198.814
assoc 2470 197.586
assoc 2480 203.957
assoc 2490 199.327
assoc 2500 198.97
assoc 2510 199.327
assoc 2520 198.851
assoc 2530 198.456
assoc 2540 203.135
assoc 2550 207.656
assoc 2560 200.119
assoc 2570 203.257
assoc 2580 200.636
assoc 2590 200.199
assoc 2600 200.119
assoc 2610 199.563
assoc 2620 202.982
assoc 2630 200.596
assoc 2640 200.278
assoc 2650 199.722
assoc 2660 201.314
assoc 2670 198.886
assoc 2680 202.787
assoc 2690 201.193
assoc 2700 202.31
assoc 2710 203.701
assoc 2720 200.636
assoc 2730 205.763
assoc 2740 203.26
assoc 2750 203.221
assoc 2760 201.073
assoc 2770 204.138
assoc 2780 202.589
assoc 2790 200.517
assoc 2800 201.475
assoc 2810 199.283
assoc 2820 202.432
assoc 2830 206.978
assoc 2840 201.433
assoc 2850 202.466
assoc 2860 207.895
assoc 2870 204.823
assoc 2880 202.273
assoc 2890 203.708
assoc 2900 202.911
assoc 2910 205.425
assoc 2920 202.791
assoc 2930 204.143
assoc 2940 204.625
assoc 2950 203.911
assoc 2960 207.365
assoc 2970 204.982
assoc 2980 203.947
assoc 2990 202.154
assoc 3000 203.149
assoc 3010 204.629
assoc 3020 206.223
assoc 3030 203.992
assoc 3040 202.43
assoc 3050 204.79
assoc 3060 202.714
assoc 3070 205.465
assoc 3080 204.703
assoc 3090 205.502
assoc 3100 203.349
assoc 3110 204.622
assoc 3120 203.866
assoc 3130 204.109
assoc 3140 204.266
assoc 3150 202.595
assoc 3160 204.109
assoc 3170 203.831
assoc 3180 204.663
assoc 3190 205.022
assoc 3200 205.299
assoc 3210 205.54
assoc 3220 205.104
assoc 3230 204.948
assoc 3240 207.339
assoc 3250 206.302
assoc 3260 206.741
assoc 3270 206.975
assoc 3280 205.791
assoc 3290 207.228
assoc 3300 204.544
assoc 3310 207.057
assoc 3320 206.14
assoc 3330 204.663
assoc 3340 206.497
assoc 3350 204.358
assoc 3360 206.434
assoc 3370 205.762
assoc 3380 208.538
assoc 3390 205.505
assoc 3400 208.763
assoc 3410 206.352
assoc 3420 205.359
assoc 3430 206.984
assoc 3440 206.163
assoc 3450 207.282
assoc 3460 204.976
assoc 3470 206.476
assoc 3480 205.761
assoc 3490 207.441
assoc 3500 206.17
assoc 3510 208.038
assoc 3520 205.412
assoc 3530 208.194
assoc 3540 207.723
assoc 3550 207.402
assoc 3560 205.615
assoc 3570 206.656
assoc 3580 205.347
assoc 3590 206.807
assoc 3600 206.494
assoc 3610 213.426
assoc 3620 205.255
assoc 3630 207.137
assoc 3640 206.741
assoc 3650 208.45
assoc 3660 207.41
assoc 3670 207.3
assoc 3680 207.146
assoc 3690 206.587
assoc 3700 208.227
assoc 3710 206.385
assoc 3720 207.06
assoc 3730 206.225
assoc 3740 206.621
assoc 3750 206.576
assoc 3760 209.051
assoc 3770 208.892
assoc 3780 209.497
assoc 3790 208.011
assoc 3800 207.49
assoc 3810 209.773
assoc 3820 207.094
assoc 3830 207.766
assoc 3840 210.008
assoc 3850 210.606
assoc 3860 208.895
assoc 3870 209.015
assoc 3880 205.706
assoc 3890 207.018
assoc 3900 209.651
assoc 3910 209.268
assoc 3920 208.048
assoc 3930 208.37
assoc 3940 206.651
assoc 3950 207.903
assoc 3960 207.686
assoc 3970 208.367
assoc 3980 209.011
assoc 3990 209.326
assoc 4000 210.251
disassoc 0 294.573
disassoc 10 294.493
disassoc 20 294.02
disassoc 30 292.277
disassoc 40 293.548
disassoc 50 293.499
disassoc 60 294.596
disassoc 70 292.038
disassoc 80 290.898
disassoc 90 294.043
disassoc 100 290.694
disassoc 110 289.905
disassoc 120 290.232
disassoc 130 288.845
disassoc 140 292.18
disassoc 150 287.869
disassoc 160 288.14
disassoc 170 288.595
disassoc 180 286.963
disassoc 190 288.806
disassoc 200 288.136
disassoc 210 284.148
disassoc 220 283.556
disassoc 230 281.607
disassoc 240 283.846
disassoc 250 283.918
disassoc 260 283.918
disassoc 270 281.578
disassoc 280 279.354
disassoc 290 280
disassoc 300 282.446
disassoc 310 280.569
disassoc 320 281.725
disassoc 330 280.299
disassoc 340 277.83
disassoc 350 278.215
disassoc 360 279.969
disassoc 370 283.124
disassoc 380 279.283
disassoc 390 276.784
disassoc 400 278.312
disassoc 410 276.025
disassoc 420 277.129
disassoc 430 275.226
disassoc 440 275.868
disassoc 450 275.819
disassoc 460 272.86
disassoc 470 277.371
disassoc 480 272.452
disassoc 490 277.257
disassoc 500 275.187
disassoc 510 272.003
disassoc 520 272.566
disassoc 530 269.574
disassoc 540 273.028
disassoc 550 271.457
disassoc 560 272.942
disassoc 570 268.903
disassoc 580 269.98
disassoc 590 271.164
disassoc 600 268.664
disassoc 610 269.11
disassoc 620 269.173
disassoc 630 267.64
disassoc 640 267.021
disassoc 650 270.004
disassoc 660 268.309
disassoc 670 265.708
disassoc 680 270.484
disassoc 690 264.564
disassoc 700 266.759
disassoc 710 272.652
disassoc 720 266.43
disassoc 730 267.141
disassoc 740 264.43
disassoc 750 265.113
disassoc 760 264.283
disassoc 770 263.316
disassoc 780 263.083
disassoc 790 263.636
disassoc 800 264.531
disassoc 810 262.095
disassoc 820 261.797
disassoc 830 259.684
disassoc 840 259.945
disassoc 850 262.426
disassoc 860 259.311
disassoc 870 257.905
disassoc 880 258.017
disassoc 890 260.158
disassoc 900 260.008
disassoc 910 258.225
disassoc 920 258.294
disassoc 930 257.075
disassoc 940 258.879
disassoc 950 256.174
disassoc 960 256.748
disassoc 970 255.512
disassoc 980 255.88
disassoc 990 255.665
disassoc 1000 258.007
disassoc 1010 255.424
disassoc 1020 256.788
disassoc 1030 256.719
disassoc 1040 253.855
disassoc 1050 255.31
disassoc 1060 254.427
disassoc 1070 255.867
disassoc 1080 252.867
disassoc 1090 251.798
disassoc 1100 251.303
disassoc 1110 254.721
disassoc 1120 252.453
disassoc 1130 252.552
disassoc 1140 250.534
disassoc 1150 253.081
disassoc 1160 252.27
disassoc 1170 253.557
disassoc 1180 253.218
disassoc 1190 253.218
disassoc 1200 249.487
disassoc 1210 253.515
disassoc 1220 252.64
disassoc 1230 251.323
disassoc 1240 248.399
disassoc 1250 249.426
disassoc 1260 250.771
disassoc 1270 250.237
disassoc 1280 250.869
disassoc 1290 248.837
disassoc 1300 248.276
disassoc 1310 249.686
disassoc 1320 247.147
disassoc 1330 249.094
disassoc 1340 246.809
disassoc 1350 247.217
disassoc 1360 247.31
disassoc 1370 245.71
disassoc 1380 247.845
disassoc 1390 245.71
disassoc 1400 246.066
disassoc 1410 247.843
disassoc 1420 245.965
disassoc 1430 247.129
disassoc 1440 248.082
disassoc 1450 248.003
disassoc 1460 244.48
disassoc 1470 245.429
disassoc 1480 249.723
disassoc 1490 246.455
disassoc 1500 246.26
disassoc 1510 242.557
disassoc 1520 242.501
disassoc 1530 244.264
disassoc 1540 245.368
disassoc 1550 244.114
disassoc 1560 242.863
disassoc 1570 247.466
disassoc 1580 244.123
disassoc 1590 242.931
disassoc 1600 242.535
disassoc 1610 243.361
disassoc 1620 244.097
disassoc 1630 241.035
disassoc 1640 240.285
disassoc 1650 242.682
disassoc 1660 242.178
disassoc 1670 243.743
disassoc 1680 240.988
disassoc 1690 241.551
disassoc 1700 238.924
disassoc 1710 242.241
disassoc 1720 241.97
disassoc 1730 237.89
disassoc 1740 241.404
disassoc 1750 239.597
disassoc 1760 241.304
disassoc 1770 237.643
disassoc 1780 239.636
disassoc 1790 239.628
disassoc 1800 238.005
disassoc 1810 240.055
disassoc 1820 240.119
disassoc 1830 238.033
disassoc 1840 239.352
disassoc 1850 241.36
disassoc 1860 237.737
disassoc 1870 237.999
disassoc 1880 237.811
disassoc 1890 239.439
disassoc 1900 236.611
disassoc 1910 238.871
disassoc 1920 237.332
disassoc 1930 235.57
disassoc 1940 235.869
disassoc 1950 235.918
disassoc 1960 235.243
disassoc 1970 235.836
disassoc 1980 236.169
disassoc 1990 238.221
disassoc 2000 235.152
disassoc 2010 235.836
disassoc 2020 234.903
disassoc 2030 235.152
disassoc 2040 237.717
disassoc 2050 235.289
disassoc 2060 234.673
disassoc 2070 237.954
disassoc 2080 234.874
disassoc 2090 234.254
disassoc 2100 234.597
disassoc 2110 234.741
disassoc 2120 234.913
disassoc 2130 234.242
disassoc 2140 232.517
disassoc 2150 234.532
disassoc 2160 234.255
disassoc 2170 235.336
disassoc 2180 231.75
disassoc 2190 236.371
disassoc 2200 234.349
disassoc 2210 235.338
disassoc 2220 233.016
disassoc 2230 233.941
disassoc 2240 234.423
disassoc 2250 234.939
disassoc 2260 233.162
disassoc 2270 232.648
disassoc 2280 233.544
disassoc 2290 230.988
disassoc 2300 232.527
disassoc 2310 232.317
disassoc 2320 235.166
disassoc 2330 234.453
disassoc 2340 232.055
disassoc 2350 230.198
disassoc 2360 232.57
disassoc 2370 234.599
disassoc 2380 235.18
disassoc 2390 231.29
disassoc 2400 233.175
disassoc 2410 230.946
disassoc 2420 234.086
disassoc 2430 234.007
disassoc 2440 231.683
disassoc 2450 230.653
disassoc 2460 231.389
disassoc 2470 229.149
disassoc 2480 229.109
disassoc 2490 228.73
disassoc 2500 229.027
disassoc 2510 229.712
disassoc 2520 229.704
disassoc 2530 229.854
disassoc 2540 231.292
disassoc 2550 229.905
disassoc 2560 230.435
disassoc 2570 230.684
disassoc 2580 234.375
disassoc 2590 228.956
disassoc 2600 229.77
disassoc 2610 229.053
disassoc 2620 230.206
disassoc 2630 229.798
disassoc 2640 229.703
disassoc 2650 229.465
disassoc 2660 228.113
disassoc 2670 229.647
disassoc 2680 229.041
disassoc 2690 227.615
disassoc 2700 228.147
disassoc 2710 230.599
disassoc 2720 226.574
disassoc 2730 229.274
disassoc 2740 230.412
disassoc 2750 230.916
disassoc 2760 228.022
disassoc 2770 228.271
disassoc 2780 229.211
disassoc 2790 226.585
disassoc 2800 227.348
disassoc 2810 228.781
disassoc 2820 227.118
disassoc 2830 228.639
disassoc 2840 228.922
disassoc 2850 227.428
disassoc 2860 227.431
disassoc 2870 226.247
disassoc 2880 225.791
disassoc 2890 227.01
disassoc 2900 228.577
disassoc 2910 226.12
disassoc 2920 226.437
disassoc 2930 226.416
disassoc 2940 226.931
disassoc 2950 226.318
disassoc 2960 228
disassoc 2970 225.416
disassoc 2980 228.532
disassoc 2990 227.273
disassoc 3000 226.664
disassoc 3010 227.021
disassoc 3020 227.24
disassoc 3030 225.942
disassoc 3040 228.962
disassoc 3050 227.208
disassoc 3060 226.894
disassoc 3070 226.347
disassoc 3080 231.761
disassoc 3090 225.01
disassoc 3100 225.149
disassoc 3110 224.465
disassoc 3120 224.395
disassoc 3130 225.119
disassoc 3140 224.245
disassoc 3150 225.139
disassoc 3160 226.45
disassoc 3170 223.879
disassoc 3180 226.801
disassoc 3190 228
disassoc 3200 227.208
disassoc 3210 227.377
disassoc 3220 224.614
disassoc 3230 225.05
disassoc 3240 224.911
disassoc 3250 224.861
disassoc 3260 225.554
disassoc 3270 224.038
disassoc 3280 225.813
disassoc 3290 222.76
disassoc 3300 223.595
disassoc 3310 224.614
disassoc 3320 225.674
disassoc 3330 224.693
disassoc 3340 224.098
disassoc 3350 224.405
disassoc 3360 226.278
disassoc 3370 223.266
disassoc 3380 223.233
disassoc 3390 226.886
disassoc 3400 224.881
disassoc 3410 224.027
disassoc 3420 223.007
disassoc 3430 223.728
disassoc 3440 221.477
disassoc 3450 225.762
disassoc 3460 222.306
disassoc 3470 226.685
disassoc 3480 225.04
disassoc 3490 222.495
disassoc 3500 224.574
disassoc 3510 226.111
disassoc 3520 223.94
disassoc 3530 225.208
disassoc 3540 222.606
disassoc 3550 222.558
disassoc 3560 223.046
disassoc 3570 226.825
disassoc 3580 225.357
disassoc 3590 223.732
disassoc 3600 224.157
disassoc 3610 225.268
disassoc 3620 223.662
disassoc 3630 221.086
disassoc 3640 221.737
disassoc 3650 222.477
disassoc 3660 223.702
disassoc 3670 222.451
disassoc 3680 223.422
disassoc 3690 222.61
disassoc 3700 221.094
disassoc 3710 221.429
disassoc 3720 226.071
disassoc 3730 222.827
disassoc 3740 221.318
disassoc 3750 220.159
disassoc 3760 221.579
disassoc 3770 222.408
disassoc 3780 223.541
disassoc 3790 221.57
disassoc 3800 220.946
disassoc 3810 224.553
disassoc 3820 221.851
disassoc 3830 221.264
disassoc 3840 221.207
disassoc 3850 223.522
disassoc 3860 221.599
disassoc 3870 221.318
disassoc 3880 219.778
disassoc 3890 221.9
disassoc 3900 220.707
disassoc 3910 219.992
disassoc 3920 222.421
disassoc 3930 225.099
disassoc 3940 221.584
disassoc 3950 220.853
disassoc 3960 220.499
disassoc 3970 219.27
disassoc 3980 222.949
disassoc 3990 221.927
disassoc 4000 221.998
;
title1 'Systems of Differential Equations Example';
proc sort data=fit;
by type time;
run;
%let k1f = 6.85e6 ;
%let k1r = 3.43e-4 ;
%let k2f = 1.8e7 ;
%let k2r = 2.1e-2 ;
%let qf = 2.1e8 ;
%let qb = 4.0e9 ;
%let dt = 5.0e-7 ;
%let et = 5.0e-8 ;
%let it = 8.05e-6 ;
proc model data=fit;
parameters qf = 2.1e8
qb = 4.0e9
k2f = 1.8e5
k2r = 2.1e-3
l = 0;
k1f = 6.85e6;
k1r = 3.43e-4;
/* Initial values for concentrations */
control dt 5.0e-7
et 5.0e-8
it 8.05e-6;
/* Association initial values --------------*/
if type = 'assoc' and time=0 then do;
ed = 0;
/* solve quadratic equation ----------*/
a = 1;
b = -(&it+&et+(k2r/k2f));
c = &it*&et;
ei = (-b-(((b**2)-(4*a*c))**.5))/(2*a);
d = &dt-ed;
i = &it-ei;
e = &et-ed-ei;
end;
/* Disassociation initial values ----------*/
if type = 'disassoc' and time=0 then do;
ei = 0;
a = 1;
b = -(&dt+&et+(&k1r/&k1f));
c = &dt*&et;
ed = (-b-(((b**2)-(4*a*c))**.5))/(2*a);
d = &dt-ed;
i = &it-ei;
e = &et-ed-ei;
end;
if time ne 0 then do;
dert.d = k1r* ed - k1f *e *d;
dert.ed = k1f* e *d - k1r*ed;
dert.e = k1r* ed - k1f* e * d + k2r * ei - k2f * e *i;
dert.ei = k2f* e *i - k2r * ei;
dert.i = k2r * ei - k2f* e *i;
end;
/* L - offset between curves */
if type = 'disassoc' then
F = (qf*(d-ed)) + (qb*ed) -L;
else
F = (qf*(d-ed)) + (qb*ed);
fit F / method=marquardt;
run;
