ehpi is an application programming interface (for the Shiny application that
motivated its creation) or a convenient function epi() for
interactive use in epidemic modelling. The arguments provided to the function
determine compartments used in the model that is run.
Usage
epi(
population,
susceptible,
exposed = 0,
infected,
recovered,
dead = 0,
beta,
gamma,
sigma = 0,
delta = 0,
xi = 0,
muBirth = 0,
muDeath = 0,
vitalDynamics = FALSE,
trueMassAction = FALSE,
timesteps = 25,
timeIncrement = 1
)Arguments
- population
The initial number of persons in the simulation; if
vitalDynamics = TRUEandmuBirthandmuDeathare equal, then it is constant.- susceptible
The initial number of people who are susceptible to the disease (they have no natural immunity to it).
- exposed
The initial number of people who have been exposed to the infection and may become infectious themselves.
- infected
The initial number of people who are infectious and may infect others.
- recovered
The initial number of people whom are no longer infectious and not a risk to others.
- dead
The initial number of fatalities (people killed by the disease).
- beta
The rate of becoming infectious (if
sigma = 0, the default), or the rate of susceptible people being exposed to the infectious.- gamma
The rate of becoming recovered (no longer infectious, if
sigma = 0, the default), or the rate of becoming infectious after exposure.- sigma
The rate of becoming recovered (no longer infectious).
- delta
The rate of fatality.
- xi
The rate of loss of learned immunity.
- muBirth
The rate of birth.
- muDeath
The rate of death not due to the disease under consideration.
- vitalDynamics
Whether vital dynamics is enabled (if
FALSE, the default, implies muBirth and muDeath of zero).- trueMassAction
Whether the force of infection is scaled by the population or not.
- timesteps
The number of output rows in the simulation.
- timeIncrement
The number of increments of "time" that pass between each output row.
Value
A dataframe with a column for each enabled compartment and a column
for time, with timesteps rows. Each row contains the values of the
differential equations for each point in time of the model (per function
arguments).
Details
The convention for pronouncing epi() is "epi epi" as if you were
saying "epidemic" twice. If you do want to refer to the package alone, and
not the singular function it packages, you can say "eh pee eye", like "API".
"eh pee eye" is a tongue-in-cheek Canadianism. You might understand it as, "Eh, P.I.?", or just a Canadian slant on API. The latter was thought of first.
While ehpi provides "an API" for the Episim Shiny application, it won't enable the development of applications that provide any feature episim doesn't already provide by wrapping this package. The function is packages was extracted from that project to allow associating well-formatted documentation, vignettes, and examples with the function because that function is expected to be used outside of Episim, interactively, during workshops run by Dr. Krishnamurthy.
Examples
epi(population = 500,
susceptible = 489,
exposed = 10,
infected = 1,
recovered = 0,
dead = 29,
beta = 0.005,
gamma = 0.0009,
sigma = 0.000001,
delta = 0.01,
timesteps = 230)
#> time N S E I R D
#> 1 1 500.0000 4.890000e+02 10.00000 1.000000 0.000000e+00 29.00000
#> 2 2 499.9900 4.865614e+02 12.42847 1.000094 9.998648e-07 29.01000
#> 3 3 499.9800 4.841322e+02 14.84546 1.002357 2.000908e-06 29.02001
#> 4 4 499.9699 4.817070e+02 17.25618 1.006758 3.005288e-06 29.03005
#> 5 5 499.9598 4.792809e+02 19.66567 1.013273 4.015128e-06 29.04015
#> 6 6 499.9497 4.768490e+02 22.07881 1.021883 5.032533e-06 29.05033
#> 7 7 499.9394 4.744065e+02 24.50032 1.032571 6.059588e-06 29.06060
#> 8 8 499.9290 4.719489e+02 26.93480 1.045327 7.098365e-06 29.07098
#> 9 9 499.9185 4.694716e+02 29.38672 1.060143 8.150928e-06 29.08151
#> 10 10 499.9078 4.669704e+02 31.86041 1.077018 9.219338e-06 29.09219
#> 11 11 499.8969 4.644409e+02 34.36012 1.095951 1.030565e-05 29.10306
#> 12 12 499.8859 4.618790e+02 36.88996 1.116947 1.141193e-05 29.11412
#> 13 13 499.8746 4.592806e+02 39.45398 1.140015 1.254023e-05 29.12540
#> 14 14 499.8631 4.566418e+02 42.05611 1.165166 1.369265e-05 29.13693
#> 15 15 499.8513 4.539587e+02 44.70018 1.192416 1.487127e-05 29.14871
#> 16 16 499.8392 4.512275e+02 47.38996 1.221782 1.607819e-05 29.16078
#> 17 17 499.8268 4.484444e+02 50.12911 1.253287 1.731554e-05 29.17316
#> 18 18 499.8141 4.456060e+02 52.92120 1.286955 1.858548e-05 29.18585
#> 19 19 499.8011 4.427085e+02 55.76975 1.322813 1.989018e-05 29.19890
#> 20 20 499.7877 4.397486e+02 58.67815 1.360892 2.123185e-05 29.21232
#> 21 21 499.7739 4.367229e+02 61.64973 1.401225 2.261272e-05 29.22613
#> 22 22 499.7596 4.336281e+02 64.68771 1.443847 2.403506e-05 29.24035
#> 23 23 499.7450 4.304609e+02 67.79524 1.488796 2.550119e-05 29.25501
#> 24 24 499.7299 4.272184e+02 70.97536 1.536113 2.701344e-05 29.27013
#> 25 25 499.7143 4.238974e+02 74.23101 1.585841 2.857422e-05 29.28574
#> 26 26 499.6981 4.204951e+02 77.56501 1.638024 3.018594e-05 29.30186
#> 27 27 499.6815 4.170087e+02 80.98009 1.692710 3.185110e-05 29.31851
#> 28 28 499.6643 4.134355e+02 84.47884 1.749948 3.357222e-05 29.33572
#> 29 29 499.6465 4.097729e+02 88.06373 1.809787 3.535186e-05 29.35352
#> 30 30 499.6281 4.060187e+02 91.73709 1.872281 3.719267e-05 29.37193
#> 31 31 499.6090 4.021704e+02 95.50111 1.937482 3.909733e-05 29.39097
#> 32 32 499.5893 3.982260e+02 99.35782 2.005448 4.106856e-05 29.41069
#> 33 33 499.5689 3.941836e+02 103.30908 2.076233 4.310916e-05 29.43109
#> 34 34 499.5478 3.900413e+02 107.35656 2.149894 4.522198e-05 29.45222
#> 35 35 499.5259 3.857976e+02 111.50178 2.226492 4.740993e-05 29.47410
#> 36 36 499.5032 3.814511e+02 115.74601 2.306083 4.967597e-05 29.49676
#> 37 37 499.4798 3.770007e+02 120.09033 2.388728 5.202311e-05 29.52023
#> 38 38 499.4555 3.724453e+02 124.53561 2.474486 5.445446e-05 29.54454
#> 39 39 499.4303 3.677844e+02 129.08244 2.563417 5.697314e-05 29.56973
#> 40 40 499.4042 3.630174e+02 133.73118 2.655581 5.958237e-05 29.59582
#> 41 41 499.3771 3.581441e+02 138.48193 2.751036 6.228540e-05 29.62285
#> 42 42 499.3491 3.531648e+02 143.33448 2.849842 6.508556e-05 29.65086
#> 43 43 499.3201 3.480797e+02 148.28835 2.952055 6.798622e-05 29.67986
#> 44 44 499.2901 3.428895e+02 153.34274 3.057732 7.099082e-05 29.70991
#> 45 45 499.2590 3.375954e+02 158.49655 3.166929 7.410286e-05 29.74103
#> 46 46 499.2267 3.321986e+02 163.74832 3.279699 7.732587e-05 29.77326
#> 47 47 499.1934 3.267009e+02 169.09626 3.396092 8.066347e-05 29.80663
#> 48 48 499.1588 3.211043e+02 174.53824 3.516159 8.411928e-05 29.84119
#> 49 49 499.1230 3.154113e+02 180.07174 3.639946 8.769702e-05 29.87697
#> 50 50 499.0860 3.096245e+02 185.69388 3.767496 9.140043e-05 29.91400
#> 51 51 499.0477 3.037473e+02 191.40143 3.898850 9.523328e-05 29.95233
#> 52 52 499.0080 2.977831e+02 197.19074 4.034046 9.919941e-05 29.99199
#> 53 53 498.9670 2.917360e+02 203.05779 4.173115 1.033027e-04 30.03303
#> 54 54 498.9245 2.856102e+02 208.99818 4.316088 1.075469e-04 30.07547
#> 55 55 498.8806 2.794104e+02 215.00712 4.462989 1.119362e-04 30.11936
#> 56 56 498.8353 2.731419e+02 221.07944 4.613838 1.164742e-04 30.16474
#> 57 57 498.7883 2.668100e+02 227.20959 4.768650 1.211651e-04 30.21165
#> 58 58 498.7399 2.604206e+02 233.39166 4.927435 1.260129e-04 30.26013
#> 59 59 498.6898 2.539801e+02 239.61940 5.090197 1.310213e-04 30.31021
#> 60 60 498.6381 2.474948e+02 245.88618 5.256934 1.361946e-04 30.36195
#> 61 61 498.5846 2.409718e+02 252.18508 5.427639 1.415365e-04 30.41537
#> 62 62 498.5295 2.344182e+02 258.50886 5.602298 1.470512e-04 30.47051
#> 63 63 498.4726 2.278415e+02 264.85002 5.780890 1.527424e-04 30.52742
#> 64 64 498.4139 2.212495e+02 271.20077 5.963389 1.586143e-04 30.58614
#> 65 65 498.3533 2.146502e+02 277.55313 6.149759 1.646705e-04 30.64671
#> 66 66 498.2908 2.080518e+02 283.89889 6.339962 1.709151e-04 30.70915
#> 67 67 498.2265 2.014627e+02 290.22971 6.533948 1.773517e-04 30.77352
#> 68 68 498.1602 1.948912e+02 296.53710 6.731664 1.839842e-04 30.83984
#> 69 69 498.0918 1.883461e+02 302.81249 6.933047 1.908162e-04 30.90816
#> 70 70 498.0215 1.818360e+02 309.04728 7.138027 1.978515e-04 30.97851
#> 71 71 497.9491 1.753695e+02 315.23284 7.346530 2.050935e-04 31.05093
#> 72 72 497.8745 1.689553e+02 321.36059 7.558472 2.125457e-04 31.12546
#> 73 73 497.7979 1.626019e+02 327.42201 7.773764 2.202115e-04 31.20212
#> 74 74 497.7191 1.563178e+02 333.40875 7.992308 2.280943e-04 31.28094
#> 75 75 497.6380 1.501112e+02 339.31257 8.214002 2.361972e-04 31.36197
#> 76 76 497.5548 1.439903e+02 345.12550 8.438737 2.445233e-04 31.44523
#> 77 77 497.4692 1.379628e+02 350.83977 8.666397 2.530757e-04 31.53076
#> 78 78 497.3814 1.320363e+02 356.44796 8.896862 2.618571e-04 31.61857
#> 79 79 497.2913 1.262181e+02 361.94295 9.130005 2.708703e-04 31.70870
#> 80 80 497.1988 1.205148e+02 367.31800 9.365696 2.801179e-04 31.80118
#> 81 81 497.1040 1.149331e+02 372.56679 9.603799 2.896025e-04 31.89602
#> 82 82 497.0067 1.094788e+02 377.68344 9.844174 2.993263e-04 31.99326
#> 83 83 496.9071 1.041576e+02 382.66253 10.086678 3.092915e-04 32.09292
#> 84 84 496.8050 9.897438e+01 387.49913 10.331164 3.195003e-04 32.19500
#> 85 85 496.7005 9.393379e+01 392.18885 10.577482 3.299545e-04 32.29954
#> 86 86 496.5934 8.903980e+01 396.72782 10.825482 3.406558e-04 32.40656
#> 87 87 496.4839 8.429587e+01 401.11271 11.075010 3.516059e-04 32.51606
#> 88 88 496.3719 7.970490e+01 405.34076 11.325911 3.628063e-04 32.62806
#> 89 89 496.2574 7.526925e+01 409.40976 11.578030 3.742582e-04 32.74258
#> 90 90 496.1404 7.099071e+01 413.31807 11.831213 3.859627e-04 32.85963
#> 91 91 496.0208 6.687051e+01 417.06458 12.085303 3.979209e-04 32.97921
#> 92 92 495.8987 6.290934e+01 420.64876 12.340148 4.101336e-04 33.10134
#> 93 93 495.7740 5.910737e+01 424.07060 12.595593 4.226014e-04 33.22601
#> 94 94 495.6468 5.546421e+01 427.33061 12.851488 4.353249e-04 33.35325
#> 95 95 495.5170 5.197902e+01 430.42980 13.107683 4.483045e-04 33.48304
#> 96 96 495.3846 4.865044e+01 433.36966 13.364033 4.615403e-04 33.61540
#> 97 97 495.2497 4.547669e+01 436.15212 13.620394 4.750325e-04 33.75033
#> 98 98 495.1122 4.245553e+01 438.77954 13.876626 4.887811e-04 33.88781
#> 99 99 494.9721 3.958436e+01 441.25469 14.132592 5.027857e-04 34.02786
#> 100 100 494.8295 3.686018e+01 443.58068 14.388161 5.170461e-04 34.17046
#> 101 101 494.6844 3.427970e+01 445.76095 14.643203 5.315618e-04 34.31562
#> 102 102 494.5367 3.183929e+01 447.79925 14.897597 5.463323e-04 34.46332
#> 103 103 494.3864 2.953508e+01 449.69957 15.151221 5.613568e-04 34.61357
#> 104 104 494.2337 2.736296e+01 451.46616 15.403964 5.766345e-04 34.76634
#> 105 105 494.0784 2.531863e+01 453.10342 15.655715 5.921644e-04 34.92164
#> 106 106 493.9205 2.339763e+01 454.61593 15.906371 6.079455e-04 35.07946
#> 107 107 493.7602 2.159537e+01 456.00841 16.155832 6.239767e-04 35.23977
#> 108 108 493.5974 1.990715e+01 457.28564 16.404007 6.402568e-04 35.40257
#> 109 109 493.4322 1.832820e+01 458.45249 16.650805 6.567843e-04 35.56784
#> 110 110 493.2644 1.685375e+01 459.51385 16.896145 6.735579e-04 35.73558
#> 111 111 493.0942 1.547897e+01 460.47463 17.139948 6.905761e-04 35.90576
#> 112 112 492.9216 1.419907e+01 461.33971 17.382143 7.078372e-04 36.07837
#> 113 113 492.7466 1.300929e+01 462.11393 17.622660 7.253398e-04 36.25340
#> 114 114 492.5692 1.190493e+01 462.80207 17.861439 7.430820e-04 36.43082
#> 115 115 492.3894 1.088137e+01 463.40883 18.098421 7.610621e-04 36.61062
#> 116 116 492.2072 9.934082e+00 463.93880 18.333554 7.792782e-04 36.79278
#> 117 117 492.0227 9.058648e+00 464.39648 18.566788 7.977285e-04 36.97729
#> 118 118 491.8359 8.250773e+00 464.78622 18.798080 8.164111e-04 37.16411
#> 119 119 491.6468 7.506297e+00 465.11224 19.027391 8.353240e-04 37.35324
#> 120 120 491.4553 6.821206e+00 465.37860 19.254685 8.544652e-04 37.54465
#> 121 121 491.2617 6.191633e+00 465.58923 19.479931 8.738327e-04 37.73833
#> 122 122 491.0658 5.613871e+00 465.74789 19.703100 8.934244e-04 37.93424
#> 123 123 490.8676 5.084371e+00 465.85817 19.924168 9.132382e-04 38.13238
#> 124 124 490.6673 4.599752e+00 465.92348 20.143115 9.332720e-04 38.33272
#> 125 125 490.4648 4.156793e+00 465.94709 20.359922 9.535237e-04 38.53524
#> 126 126 490.2601 3.752441e+00 465.93210 20.574576 9.739912e-04 38.73991
#> 127 127 490.0533 3.383808e+00 465.88141 20.787064 9.946722e-04 38.94672
#> 128 128 489.8444 3.048165e+00 465.79780 20.997377 1.015565e-03 39.15565
#> 129 129 489.6333 2.742943e+00 465.68385 21.205509 1.036666e-03 39.36666
#> 130 130 489.4203 2.465731e+00 465.54201 21.411454 1.057975e-03 39.57975
#> 131 131 489.2051 2.214265e+00 465.37456 21.615212 1.079488e-03 39.79488
#> 132 132 488.9880 1.986431e+00 465.18364 21.816781 1.101205e-03 40.01205
#> 133 133 488.7688 1.780255e+00 464.97125 22.016164 1.123121e-03 40.23121
#> 134 134 488.5476 1.593897e+00 464.73923 22.213364 1.145236e-03 40.45236
#> 135 135 488.3245 1.425648e+00 464.48933 22.408385 1.167547e-03 40.67547
#> 136 136 488.0995 1.273923e+00 464.22313 22.601234 1.190052e-03 40.90052
#> 137 137 487.8725 1.137255e+00 463.94212 22.791920 1.212749e-03 41.12749
#> 138 138 487.6436 1.014287e+00 463.64767 22.980449 1.235635e-03 41.35635
#> 139 139 487.4129 9.037675e-01 463.34105 23.166834 1.258709e-03 41.58709
#> 140 140 487.1803 8.045446e-01 463.02341 23.351084 1.281968e-03 41.81968
#> 141 141 486.9459 7.155595e-01 462.69582 23.533212 1.305411e-03 42.05411
#> 142 142 486.7097 6.358405e-01 462.35926 23.713230 1.329034e-03 42.29034
#> 143 143 486.4716 5.644975e-01 462.01464 23.891151 1.352836e-03 42.52836
#> 144 144 486.2318 5.007164e-01 461.66276 24.066990 1.376816e-03 42.76816
#> 145 145 485.9903 4.437536e-01 461.30439 24.240761 1.400970e-03 43.00970
#> 146 146 485.7470 3.929316e-01 460.94020 24.412479 1.425296e-03 43.25296
#> 147 147 485.5021 3.476333e-01 460.57082 24.582160 1.449794e-03 43.49794
#> 148 148 485.2554 3.072978e-01 460.19681 24.749820 1.474460e-03 43.74460
#> 149 149 485.0071 2.714162e-01 459.81868 24.915474 1.499293e-03 43.99293
#> 150 150 484.7571 2.395270e-01 459.43691 25.079139 1.524290e-03 44.24290
#> 151 151 484.5055 2.112126e-01 459.05190 25.240833 1.549451e-03 44.49451
#> 152 152 484.2523 1.860957e-01 458.66404 25.400571 1.574771e-03 44.74771
#> 153 153 483.9975 1.638355e-01 458.27368 25.558372 1.600251e-03 45.00251
#> 154 154 483.7411 1.441249e-01 457.88112 25.714251 1.625887e-03 45.25887
#> 155 155 483.4832 1.266875e-01 457.48664 25.868227 1.651679e-03 45.51679
#> 156 156 483.2238 1.112746e-01 457.09050 26.020317 1.677623e-03 45.77623
#> 157 157 482.9628 9.766308e-02 456.69291 26.170537 1.703719e-03 46.03719
#> 158 158 482.7004 8.565257e-02 456.29407 26.318907 1.729964e-03 46.29964
#> 159 159 482.4364 7.506374e-02 455.89418 26.465442 1.756356e-03 46.56356
#> 160 160 482.1711 6.573608e-02 455.49338 26.610160 1.782894e-03 46.82894
#> 161 161 481.9042 5.752612e-02 455.09183 26.753079 1.809576e-03 47.09576
#> 162 162 481.6360 5.030579e-02 454.68965 26.894216 1.836400e-03 47.36400
#> 163 163 481.3664 4.396087e-02 454.28695 27.033588 1.863364e-03 47.63364
#> 164 164 481.0953 3.838963e-02 453.88385 27.171213 1.890466e-03 47.90466
#> 165 165 480.8229 3.350152e-02 453.48042 27.307106 1.917706e-03 48.17706
#> 166 166 480.5492 2.921608e-02 453.07675 27.441286 1.945080e-03 48.45080
#> 167 167 480.2741 2.546184e-02 452.67292 27.573769 1.972588e-03 48.72588
#> 168 168 479.9977 2.217542e-02 452.26898 27.704572 2.000227e-03 49.00227
#> 169 169 479.7200 1.930064e-02 451.86500 27.833712 2.027996e-03 49.27996
#> 170 170 479.4411 1.678776e-02 451.46101 27.961205 2.055894e-03 49.55894
#> 171 171 479.1608 1.459281e-02 451.05708 28.087067 2.083918e-03 49.83918
#> 172 172 478.8793 1.267691e-02 450.65322 28.211316 2.112067e-03 50.12067
#> 173 173 478.5966 1.100576e-02 450.24949 28.333966 2.140340e-03 50.40340
#> 174 174 478.3127 9.549086e-03 449.84590 28.455034 2.168735e-03 50.68735
#> 175 175 478.0275 8.280232e-03 449.44249 28.574536 2.197250e-03 50.97250
#> 176 176 477.7412 7.175718e-03 449.03928 28.692488 2.225883e-03 51.25883
#> 177 177 477.4537 6.214895e-03 448.63628 28.808905 2.254634e-03 51.54634
#> 178 178 477.1650 5.379613e-03 448.23353 28.923803 2.283501e-03 51.83501
#> 179 179 476.8752 4.653937e-03 447.83103 29.037196 2.312481e-03 52.12481
#> 180 180 476.5843 4.023882e-03 447.42879 29.149102 2.341574e-03 52.41574
#> 181 181 476.2922 3.477192e-03 447.02683 29.259533 2.370779e-03 52.70779
#> 182 182 475.9991 3.003128e-03 446.62516 29.368506 2.400093e-03 53.00093
#> 183 183 475.7048 2.592293e-03 446.22379 29.476034 2.429515e-03 53.29515
#> 184 184 475.4096 2.236466e-03 445.82272 29.582134 2.459045e-03 53.59045
#> 185 185 475.1132 1.928464e-03 445.42197 29.686818 2.488679e-03 53.88679
#> 186 186 474.8158 1.662016e-03 445.02154 29.790102 2.518418e-03 54.18418
#> 187 187 474.5174 1.431647e-03 444.62143 29.892000 2.548259e-03 54.48259
#> 188 188 474.2180 1.232585e-03 444.22165 29.992526 2.578201e-03 54.78201
#> 189 189 473.9176 1.060672e-03 443.82220 30.091693 2.608244e-03 55.08244
#> 190 190 473.6162 9.122870e-04 443.42309 30.189516 2.638384e-03 55.38384
#> 191 191 473.3138 7.842792e-04 443.02432 30.286009 2.668622e-03 55.68622
#> 192 192 473.0104 6.739090e-04 442.62588 30.381184 2.698956e-03 55.98956
#> 193 193 472.7062 5.787972e-04 442.22780 30.475055 2.729384e-03 56.29384
#> 194 194 472.4009 4.968774e-04 441.83005 30.567636 2.759906e-03 56.59906
#> 195 195 472.0948 4.263560e-04 441.43265 30.658940 2.790519e-03 56.90519
#> 196 196 471.7878 3.656778e-04 441.03560 30.748979 2.821223e-03 57.21223
#> 197 197 471.4798 3.134951e-04 440.63890 30.837767 2.852016e-03 57.52016
#> 198 198 471.1710 2.686404e-04 440.24255 30.925316 2.882898e-03 57.82898
#> 199 199 470.8613 2.301034e-04 439.84655 31.011640 2.913867e-03 58.13867
#> 200 200 470.5508 1.970102e-04 439.45090 31.096749 2.944921e-03 58.44921
#> 201 201 470.2394 1.686052e-04 439.05560 31.180658 2.976060e-03 58.76060
#> 202 202 469.9272 1.442355e-04 438.66065 31.263378 3.007282e-03 59.07282
#> 203 203 469.6141 1.233375e-04 438.26606 31.344921 3.038586e-03 59.38586
#> 204 204 469.3003 1.054246e-04 437.87181 31.425300 3.069971e-03 59.69971
#> 205 205 468.9856 9.007739e-05 437.47792 31.504526 3.101436e-03 60.01436
#> 206 206 468.6702 7.693518e-05 437.08438 31.582610 3.132980e-03 60.32980
#> 207 207 468.3540 6.568565e-05 436.69119 31.659566 3.164601e-03 60.64601
#> 208 208 468.0370 5.605956e-05 436.29836 31.735404 3.196299e-03 60.96299
#> 209 209 467.7193 4.782614e-05 435.90587 31.810135 3.228072e-03 61.28072
#> 210 210 467.4008 4.078684e-05 435.51374 31.883772 3.259919e-03 61.59919
#> 211 211 467.0816 3.477091e-05 435.12196 31.956325 3.291839e-03 61.91839
#> 212 212 466.7617 2.963165e-05 434.73053 32.027805 3.323831e-03 62.23831
#> 213 213 466.4411 2.524304e-05 434.33945 32.098223 3.355894e-03 62.55894
#> 214 214 466.1197 2.149690e-05 433.94873 32.167591 3.388027e-03 62.88027
#> 215 215 465.7977 1.830040e-05 433.55835 32.235918 3.420229e-03 63.20229
#> 216 216 465.4750 1.557393e-05 433.16833 32.303216 3.452499e-03 63.52499
#> 217 217 465.1516 1.324924e-05 432.77866 32.369496 3.484835e-03 63.84835
#> 218 218 464.8276 1.126784e-05 432.38933 32.434767 3.517237e-03 64.17237
#> 219 219 464.5030 9.579663e-06 432.00036 32.499040 3.549704e-03 64.49704
#> 220 220 464.1777 8.141814e-06 431.61173 32.562325 3.582235e-03 64.82235
#> 221 221 463.8517 6.917608e-06 431.22346 32.624633 3.614829e-03 65.14829
#> 222 222 463.5252 5.875658e-06 430.83553 32.685973 3.647484e-03 65.47484
#> 223 223 463.1980 4.989132e-06 430.44796 32.746355 3.680200e-03 65.80200
#> 224 224 462.8702 4.235098e-06 430.06073 32.805790 3.712976e-03 66.12976
#> 225 225 462.5419 3.593967e-06 429.67385 32.864286 3.745811e-03 66.45811
#> 226 226 462.2130 3.049009e-06 429.28732 32.921854 3.778705e-03 66.78705
#> 227 227 461.8835 2.585945e-06 428.90113 32.978503 3.811655e-03 67.11655
#> 228 228 461.5534 2.192593e-06 428.51530 33.034243 3.844661e-03 67.44661
#> 229 229 461.2228 1.858561e-06 428.12981 33.089083 3.877723e-03 67.77723
#> 230 230 460.8916 1.574990e-06 427.74466 33.143032 3.910839e-03 68.10839