Three-compartment model with permanent immunity after recovery.
Arguments
- init
Named numeric vector.
Rdefaults to 0 if omitted, e.g.c(S = 990, I = 10).- params
Named numeric vector, e.g.
c(beta = 0.002, gamma = 0.1).- beta
Transmission rate.
- gamma
Recovery rate (1/gamma = mean infectious period).
- N
Total population size. Optional — defaults to
sum(init).- b
Birth rate (per capita per time). Optional.
- d
Death rate (per capita per time). Optional.
- times
Numeric vector of output times.
- ...
Additional arguments passed to
ode_solver.
Details
Closed population (default): $$\frac{dS}{dt} = -\beta \frac{S I}{N}$$ $$\frac{dI}{dt} = \beta \frac{S I}{N} - \gamma I$$ $$\frac{dR}{dt} = \gamma I$$
Open population (when b / d are provided):
$$\frac{dS}{dt} = b N - \beta \frac{S I}{N} - d S$$
$$\frac{dI}{dt} = \beta \frac{S I}{N} - \gamma I - d I$$
$$\frac{dR}{dt} = \gamma I - d R$$
In the closed case population size \(N = S + I + R\) is conserved. The basic reproduction number is \(R_0 = \beta / \gamma\).
Examples
# Closed population
model_sir(
init = c(S = 990, I = 10),
params = c(beta = 0.002, gamma = 0.1),
times = seq(0, 50, by = 0.1)
)
#> time S I R
#> 1 0.0 990.0000 10.00000000 0.00000000
#> 2 0.1 989.9980 9.90245840 0.09951193
#> 3 0.2 989.9961 9.80586820 0.19805324
#> 4 0.3 989.9941 9.71022067 0.29563284
#> 5 0.4 989.9922 9.61550610 0.39226063
#> 6 0.5 989.9903 9.52171532 0.48794596
#> 7 0.6 989.9885 9.42883938 0.58269798
#> 8 0.7 989.9866 9.33686935 0.67652577
#> 9 0.8 989.9848 9.24579641 0.76943835
#> 10 0.9 989.9829 9.15561180 0.86144466
#> 11 1.0 989.9811 9.06630686 0.95255352
#> 12 1.1 989.9794 8.97787300 1.04277370
#> 13 1.2 989.9776 8.89030174 1.13211386
#> 14 1.3 989.9758 8.80358466 1.22058258
#> 15 1.4 989.9741 8.71771342 1.30818837
#> 16 1.5 989.9724 8.63267978 1.39493964
#> 17 1.6 989.9707 8.54847556 1.48084473
#> 18 1.7 989.9690 8.46509268 1.56591189
#> 19 1.8 989.9673 8.38252312 1.65014930
#> 20 1.9 989.9657 8.30075895 1.73356504
#> 21 2.0 989.9640 8.21979232 1.81616713
#> 22 2.1 989.9624 8.13961545 1.89796352
#> 23 2.2 989.9608 8.06022062 1.97896205
#> 24 2.3 989.9592 7.98160023 2.05917051
#> 25 2.4 989.9577 7.90374670 2.13859661
#> 26 2.5 989.9561 7.82665256 2.21724798
#> 27 2.6 989.9546 7.75031041 2.29513217
#> 28 2.7 989.9530 7.67471290 2.37225667
#> 29 2.8 989.9515 7.59985278 2.44862888
#> 30 2.9 989.9500 7.52572285 2.52425616
#> 31 3.0 989.9485 7.45231600 2.59914575
#> 32 3.1 989.9471 7.37962516 2.67330486
#> 33 3.2 989.9456 7.30764335 2.74674062
#> 34 3.3 989.9442 7.23636366 2.81946007
#> 35 3.4 989.9428 7.16577925 2.89147021
#> 36 3.5 989.9413 7.09588331 2.96277795
#> 37 3.6 989.9399 7.02666916 3.03339015
#> 38 3.7 989.9386 6.95813012 3.10331358
#> 39 3.8 989.9372 6.89025962 3.17255498
#> 40 3.9 989.9358 6.82305113 3.24112098
#> 41 4.0 989.9345 6.75649821 3.30901819
#> 42 4.1 989.9332 6.69059444 3.37625311
#> 43 4.2 989.9318 6.62533351 3.44283222
#> 44 4.3 989.9305 6.56070914 3.50876190
#> 45 4.4 989.9292 6.49671513 3.57404850
#> 46 4.5 989.9280 6.43334532 3.63869829
#> 47 4.6 989.9267 6.37059362 3.70271747
#> 48 4.7 989.9254 6.30845401 3.76611220
#> 49 4.8 989.9242 6.24692052 3.82888857
#> 50 4.9 989.9230 6.18598723 3.89105261
#> 51 5.0 989.9217 6.12564830 3.95261030
#> 52 5.1 989.9205 6.06589791 4.01356754
#> 53 5.2 989.9193 6.00673034 4.07393020
#> 54 5.3 989.9182 5.94813989 4.13370407
#> 55 5.4 989.9170 5.89012094 4.19289490
#> 56 5.5 989.9158 5.83266791 4.25150837
#> 57 5.6 989.9147 5.77577529 4.30955012
#> 58 5.7 989.9135 5.71943760 4.36702573
#> 59 5.8 989.9124 5.66364944 4.42394071
#> 60 5.9 989.9113 5.60840544 4.48030053
#> 61 6.0 989.9102 5.55370030 4.53611061
#> 62 6.1 989.9091 5.49952876 4.59137632
#> 63 6.2 989.9080 5.44588561 4.64610295
#> 64 6.3 989.9069 5.39276571 4.70029577
#> 65 6.4 989.9059 5.34016394 4.75395999
#> 66 6.5 989.9048 5.28807526 4.80710076
#> 67 6.6 989.9038 5.23649465 4.85972319
#> 68 6.7 989.9028 5.18541717 4.91183233
#> 69 6.8 989.9017 5.13483791 4.96343319
#> 70 6.9 989.9007 5.08475199 5.01453073
#> 71 7.0 989.8997 5.03515463 5.06512986
#> 72 7.1 989.8987 4.98604104 5.11523544
#> 73 7.2 989.8977 4.93740651 5.16485228
#> 74 7.3 989.8968 4.88924637 5.21398515
#> 75 7.4 989.8958 4.84155599 5.26263877
#> 76 7.5 989.8949 4.79433079 5.31081782
#> 77 7.6 989.8939 4.74756622 5.35852693
#> 78 7.7 989.8930 4.70125781 5.40577067
#> 79 7.8 989.8920 4.65540109 5.45255359
#> 80 7.9 989.8911 4.60999166 5.49888018
#> 81 8.0 989.8902 4.56502516 5.54475490
#> 82 8.1 989.8893 4.52049727 5.59018214
#> 83 8.2 989.8884 4.47640372 5.63516629
#> 84 8.3 989.8875 4.43274025 5.67971165
#> 85 8.4 989.8867 4.38950268 5.72382251
#> 86 8.5 989.8858 4.34668686 5.76750311
#> 87 8.6 989.8850 4.30428867 5.81075764
#> 88 8.7 989.8841 4.26230404 5.85359026
#> 89 8.8 989.8833 4.22072893 5.89600509
#> 90 8.9 989.8824 4.17955935 5.93800619
#> 91 9.0 989.8816 4.13879134 5.97959762
#> 92 9.1 989.8808 4.09842099 6.02078335
#> 93 9.2 989.8800 4.05844441 6.06156735
#> 94 9.3 989.8792 4.01885777 6.10195354
#> 95 9.4 989.8784 3.97965727 6.14194579
#> 96 9.5 989.8776 3.94083913 6.18154795
#> 97 9.6 989.8768 3.90239963 6.22076383
#> 98 9.7 989.8761 3.86433507 6.25959720
#> 99 9.8 989.8753 3.82664180 6.29805177
#> 100 9.9 989.8746 3.78931619 6.33613126
#> 101 10.0 989.8738 3.75235466 6.37383931
#> 102 10.1 989.8731 3.71575366 6.41117955
#> 103 10.2 989.8723 3.67950967 6.44815557
#> 104 10.3 989.8716 3.64361921 6.48477093
#> 105 10.4 989.8709 3.60807883 6.52102913
#> 106 10.5 989.8702 3.57288512 6.55693366
#> 107 10.6 989.8695 3.53803468 6.59248797
#> 108 10.7 989.8688 3.50352419 6.62769548
#> 109 10.8 989.8681 3.46935031 6.66255958
#> 110 10.9 989.8674 3.43550977 6.69708360
#> 111 11.0 989.8667 3.40199932 6.73127087
#> 112 11.1 989.8661 3.36881573 6.76512468
#> 113 11.2 989.8654 3.33595582 6.79864827
#> 114 11.3 989.8647 3.30341642 6.83184486
#> 115 11.4 989.8641 3.27119443 6.86471765
#> 116 11.5 989.8634 3.23928672 6.89726980
#> 117 11.6 989.8628 3.20769025 6.92950443
#> 118 11.7 989.8622 3.17640198 6.96142463
#> 119 11.8 989.8615 3.14541890 6.99303348
#> 120 11.9 989.8609 3.11473803 7.02433402
#> 121 12.0 989.8603 3.08435642 7.05532924
#> 122 12.1 989.8597 3.05427116 7.08602213
#> 123 12.2 989.8591 3.02447936 7.11641564
#> 124 12.3 989.8585 2.99497815 7.14651269
#> 125 12.4 989.8579 2.96576470 7.17631616
#> 126 12.5 989.8573 2.93683620 7.20582893
#> 127 12.6 989.8568 2.90818987 7.23505383
#> 128 12.7 989.8562 2.87982297 7.26399366
#> 129 12.8 989.8556 2.85173275 7.29265121
#> 130 12.9 989.8551 2.82391654 7.32102923
#> 131 13.0 989.8545 2.79637164 7.34913045
#> 132 13.1 989.8539 2.76909543 7.37695756
#> 133 13.2 989.8534 2.74208527 7.40451324
#> 134 13.3 989.8529 2.71533857 7.43180014
#> 135 13.4 989.8523 2.68885276 7.45882088
#> 136 13.5 989.8518 2.66262529 7.48557806
#> 137 13.6 989.8513 2.63665366 7.51207424
#> 138 13.7 989.8508 2.61093535 7.53831198
#> 139 13.8 989.8502 2.58546790 7.56429378
#> 140 13.9 989.8497 2.56024887 7.59002216
#> 141 14.0 989.8492 2.53527583 7.61549958
#> 142 14.1 989.8487 2.51054638 7.64072849
#> 143 14.2 989.8482 2.48605814 7.66571131
#> 144 14.3 989.8477 2.46180876 7.69045045
#> 145 14.4 989.8473 2.43779591 7.71494828
#> 146 14.5 989.8468 2.41401729 7.73920715
#> 147 14.6 989.8463 2.39047061 7.76322940
#> 148 14.7 989.8458 2.36715361 7.78701733
#> 149 14.8 989.8454 2.34406405 7.81057323
#> 150 14.9 989.8449 2.32119970 7.83389936
#> 151 15.0 989.8444 2.29855837 7.85699796
#> 152 15.1 989.8440 2.27613790 7.87987126
#> 153 15.2 989.8435 2.25393611 7.90252145
#> 154 15.3 989.8431 2.23195088 7.92495071
#> 155 15.4 989.8427 2.21018010 7.94716118
#> 156 15.5 989.8422 2.18862168 7.96915502
#> 157 15.6 989.8418 2.16727354 7.99093432
#> 158 15.7 989.8414 2.14613363 8.01250118
#> 159 15.8 989.8409 2.12519993 8.03385768
#> 160 15.9 989.8405 2.10447041 8.05500586
#> 161 16.0 989.8401 2.08394309 8.07594776
#> 162 16.1 989.8397 2.06361600 8.09668539
#> 163 16.2 989.8393 2.04348719 8.11722074
#> 164 16.3 989.8389 2.02355471 8.13755579
#> 165 16.4 989.8385 2.00381665 8.15769248
#> 166 16.5 989.8381 1.98427113 8.17763276
#> 167 16.6 989.8377 1.96491625 8.19737854
#> 168 16.7 989.8373 1.94575016 8.21693172
#> 169 16.8 989.8369 1.92677103 8.23629417
#> 170 16.9 989.8366 1.90797701 8.25546775
#> 171 17.0 989.8362 1.88936632 8.27445432
#> 172 17.1 989.8358 1.87093716 8.29325569
#> 173 17.2 989.8354 1.85268776 8.31187366
#> 174 17.3 989.8351 1.83461637 8.33031003
#> 175 17.4 989.8347 1.81672124 8.34856658
#> 176 17.5 989.8344 1.79900067 8.36664504
#> 177 17.6 989.8340 1.78145295 8.38454717
#> 178 17.7 989.8336 1.76407639 8.40227467
#> 179 17.8 989.8333 1.74686933 8.41982926
#> 180 17.9 989.8330 1.72983010 8.43721262
#> 181 18.0 989.8326 1.71295708 8.45442641
#> 182 18.1 989.8323 1.69624864 8.47147231
#> 183 18.2 989.8319 1.67970317 8.48835193
#> 184 18.3 989.8316 1.66331910 8.50506691
#> 185 18.4 989.8313 1.64709483 8.52161884
#> 186 18.5 989.8310 1.63102882 8.53800933
#> 187 18.6 989.8306 1.61511952 8.55423994
#> 188 18.7 989.8303 1.59936540 8.57031224
#> 189 18.8 989.8300 1.58376495 8.58622776
#> 190 18.9 989.8297 1.56831667 8.60198805
#> 191 19.0 989.8294 1.55301907 8.61759460
#> 192 19.1 989.8291 1.53787069 8.63304892
#> 193 19.2 989.8288 1.52287007 8.64835251
#> 194 19.3 989.8285 1.50801576 8.66350681
#> 195 19.4 989.8282 1.49330635 8.67851330
#> 196 19.5 989.8279 1.47874041 8.69337342
#> 197 19.6 989.8276 1.46431655 8.70808859
#> 198 19.7 989.8273 1.45003339 8.72266022
#> 199 19.8 989.8270 1.43588954 8.73708972
#> 200 19.9 989.8267 1.42188366 8.75137847
#> 201 20.0 989.8265 1.40801439 8.76552785
#> 202 20.1 989.8262 1.39428041 8.77953921
#> 203 20.2 989.8259 1.38068038 8.79341390
#> 204 20.3 989.8256 1.36721302 8.80715326
#> 205 20.4 989.8254 1.35387702 8.82075860
#> 206 20.5 989.8251 1.34067109 8.83423123
#> 207 20.6 989.8248 1.32759399 8.84757245
#> 208 20.7 989.8246 1.31464443 8.86078354
#> 209 20.8 989.8243 1.30182119 8.87386576
#> 210 20.9 989.8241 1.28912303 8.88682038
#> 211 21.0 989.8238 1.27654873 8.89964863
#> 212 21.1 989.8236 1.26409708 8.91235176
#> 213 21.2 989.8233 1.25176688 8.92493098
#> 214 21.3 989.8231 1.23955696 8.93738750
#> 215 21.4 989.8228 1.22746613 8.94972252
#> 216 21.5 989.8226 1.21549324 8.96193722
#> 217 21.6 989.8223 1.20363713 8.97403277
#> 218 21.7 989.8221 1.19189667 8.98601034
#> 219 21.8 989.8219 1.18027073 8.99787109
#> 220 21.9 989.8216 1.16875819 9.00961614
#> 221 22.0 989.8214 1.15735795 9.02124662
#> 222 22.1 989.8212 1.14606890 9.03276367
#> 223 22.2 989.8209 1.13488997 9.04416837
#> 224 22.3 989.8207 1.12382008 9.05546183
#> 225 22.4 989.8205 1.11285817 9.06664513
#> 226 22.5 989.8203 1.10200318 9.07771935
#> 227 22.6 989.8201 1.09125407 9.08868555
#> 228 22.7 989.8198 1.08060981 9.09954478
#> 229 22.8 989.8196 1.07006938 9.11029809
#> 230 22.9 989.8194 1.05963176 9.12094651
#> 231 23.0 989.8192 1.04929595 9.13149106
#> 232 23.1 989.8190 1.03906096 9.14193276
#> 233 23.2 989.8188 1.02892580 9.15227262
#> 234 23.3 989.8186 1.01888950 9.16251161
#> 235 23.4 989.8184 1.00895109 9.17265073
#> 236 23.5 989.8182 0.99910963 9.18269096
#> 237 23.6 989.8180 0.98936416 9.19263324
#> 238 23.7 989.8178 0.97971375 9.20247856
#> 239 23.8 989.8176 0.97015748 9.21222783
#> 240 23.9 989.8174 0.96069441 9.22188202
#> 241 24.0 989.8172 0.95132365 9.23144203
#> 242 24.1 989.8170 0.94204429 9.24090879
#> 243 24.2 989.8169 0.93285545 9.25028322
#> 244 24.3 989.8167 0.92375623 9.25956620
#> 245 24.4 989.8165 0.91474577 9.26875864
#> 246 24.5 989.8163 0.90582320 9.27786141
#> 247 24.6 989.8161 0.89698767 9.28687539
#> 248 24.7 989.8160 0.88823831 9.29580145
#> 249 24.8 989.8158 0.87957430 9.30464044
#> 250 24.9 989.8156 0.87099479 9.31339322
#> 251 25.0 989.8154 0.86249898 9.32206062
#> 252 25.1 989.8153 0.85408603 9.33064347
#> 253 25.2 989.8151 0.84575514 9.33914261
#> 254 25.3 989.8149 0.83750552 9.34755885
#> 255 25.4 989.8148 0.82933636 9.35589299
#> 256 25.5 989.8146 0.82124689 9.36414584
#> 257 25.6 989.8144 0.81323632 9.37231819
#> 258 25.7 989.8143 0.80530389 9.38041083
#> 259 25.8 989.8141 0.79744883 9.38842453
#> 260 25.9 989.8140 0.78967039 9.39636006
#> 261 26.0 989.8138 0.78196782 9.40421819
#> 262 26.1 989.8137 0.77434039 9.41199967
#> 263 26.2 989.8135 0.76678735 9.41970524
#> 264 26.3 989.8134 0.75930799 9.42733566
#> 265 26.4 989.8132 0.75190158 9.43489165
#> 266 26.5 989.8131 0.74456742 9.44237393
#> 267 26.6 989.8129 0.73730480 9.44978323
#> 268 26.7 989.8128 0.73011301 9.45712026
#> 269 26.8 989.8126 0.72299138 9.46438573
#> 270 26.9 989.8125 0.71593921 9.47158032
#> 271 27.0 989.8123 0.70895583 9.47870474
#> 272 27.1 989.8122 0.70204056 9.48575967
#> 273 27.2 989.8121 0.69519275 9.49274578
#> 274 27.3 989.8119 0.68841173 9.49966374
#> 275 27.4 989.8118 0.68169686 9.50651423
#> 276 27.5 989.8117 0.67504748 9.51329790
#> 277 27.6 989.8115 0.66846296 9.52001540
#> 278 27.7 989.8114 0.66194267 9.52666737
#> 279 27.8 989.8113 0.65548598 9.53325446
#> 280 27.9 989.8111 0.64909227 9.53977730
#> 281 28.0 989.8110 0.64276092 9.54623652
#> 282 28.1 989.8109 0.63649134 9.55263273
#> 283 28.2 989.8108 0.63028290 9.55896655
#> 284 28.3 989.8106 0.62413502 9.56523859
#> 285 28.4 989.8105 0.61804711 9.57144945
#> 286 28.5 989.8104 0.61201859 9.57759973
#> 287 28.6 989.8103 0.60604886 9.58369001
#> 288 28.7 989.8101 0.60013737 9.58972090
#> 289 28.8 989.8100 0.59428354 9.59569295
#> 290 28.9 989.8099 0.58848680 9.60160676
#> 291 29.0 989.8098 0.58274661 9.60746288
#> 292 29.1 989.8097 0.57706241 9.61326188
#> 293 29.2 989.8096 0.57143365 9.61900431
#> 294 29.3 989.8094 0.56585980 9.62469073
#> 295 29.4 989.8093 0.56034032 9.63032169
#> 296 29.5 989.8092 0.55487467 9.63589772
#> 297 29.6 989.8091 0.54946234 9.64141936
#> 298 29.7 989.8090 0.54410280 9.64688714
#> 299 29.8 989.8089 0.53879554 9.65230159
#> 300 29.9 989.8088 0.53354004 9.65766323
#> 301 30.0 989.8087 0.52833581 9.66297256
#> 302 30.1 989.8086 0.52318234 9.66823011
#> 303 30.2 989.8085 0.51807914 9.67343638
#> 304 30.3 989.8084 0.51302571 9.67859186
#> 305 30.4 989.8083 0.50802158 9.68369705
#> 306 30.5 989.8082 0.50306626 9.68875245
#> 307 30.6 989.8081 0.49815927 9.69375854
#> 308 30.7 989.8080 0.49330015 9.69871580
#> 309 30.8 989.8079 0.48848842 9.70362470
#> 310 30.9 989.8078 0.48372363 9.70848572
#> 311 31.0 989.8077 0.47900532 9.71329933
#> 312 31.1 989.8076 0.47433303 9.71806598
#> 313 31.2 989.8075 0.46970631 9.72278614
#> 314 31.3 989.8074 0.46512472 9.72746026
#> 315 31.4 989.8073 0.46058782 9.73208879
#> 316 31.5 989.8072 0.45609518 9.73667216
#> 317 31.6 989.8071 0.45164635 9.74121084
#> 318 31.7 989.8071 0.44724092 9.74570524
#> 319 31.8 989.8070 0.44287847 9.75015580
#> 320 31.9 989.8069 0.43855856 9.75456295
#> 321 32.0 989.8068 0.43428079 9.75892711
#> 322 32.1 989.8067 0.43004475 9.76324870
#> 323 32.2 989.8066 0.42585003 9.76752814
#> 324 32.3 989.8065 0.42169622 9.77176584
#> 325 32.4 989.8065 0.41758293 9.77596220
#> 326 32.5 989.8064 0.41350976 9.78011763
#> 327 32.6 989.8063 0.40947632 9.78423253
#> 328 32.7 989.8062 0.40548223 9.78830729
#> 329 32.8 989.8061 0.40152709 9.79234230
#> 330 32.9 989.8061 0.39761053 9.79633796
#> 331 33.0 989.8060 0.39373218 9.80029464
#> 332 33.1 989.8059 0.38989165 9.80421273
#> 333 33.2 989.8058 0.38608859 9.80809260
#> 334 33.3 989.8057 0.38232262 9.81193462
#> 335 33.4 989.8057 0.37859339 9.81573917
#> 336 33.5 989.8056 0.37490053 9.81950661
#> 337 33.6 989.8055 0.37124369 9.82323730
#> 338 33.7 989.8054 0.36762252 9.82693161
#> 339 33.8 989.8054 0.36403667 9.83058987
#> 340 33.9 989.8053 0.36048580 9.83421246
#> 341 34.0 989.8052 0.35696957 9.83779970
#> 342 34.1 989.8052 0.35348763 9.84135196
#> 343 34.2 989.8051 0.35003966 9.84486957
#> 344 34.3 989.8050 0.34662532 9.84835287
#> 345 34.4 989.8050 0.34324428 9.85180219
#> 346 34.5 989.8049 0.33989622 9.85521786
#> 347 34.6 989.8048 0.33658082 9.85860022
#> 348 34.7 989.8048 0.33329776 9.86194959
#> 349 34.8 989.8047 0.33004672 9.86526628
#> 350 34.9 989.8046 0.32682739 9.86855063
#> 351 35.0 989.8046 0.32363947 9.87180293
#> 352 35.1 989.8045 0.32048264 9.87502352
#> 353 35.2 989.8044 0.31735660 9.87821269
#> 354 35.3 989.8044 0.31426106 9.88137075
#> 355 35.4 989.8043 0.31119570 9.88449801
#> 356 35.5 989.8042 0.30816025 9.88759477
#> 357 35.6 989.8042 0.30515441 9.89066132
#> 358 35.7 989.8041 0.30217789 9.89369795
#> 359 35.8 989.8041 0.29923040 9.89670497
#> 360 35.9 989.8040 0.29631166 9.89968266
#> 361 36.0 989.8039 0.29342139 9.90263130
#> 362 36.1 989.8039 0.29055931 9.90555118
#> 363 36.2 989.8038 0.28772515 9.90844258
#> 364 36.3 989.8038 0.28491863 9.91130577
#> 365 36.4 989.8037 0.28213949 9.91414104
#> 366 36.5 989.8037 0.27938746 9.91694865
#> 367 36.6 989.8036 0.27666227 9.91972888
#> 368 36.7 989.8036 0.27396366 9.92248199
#> 369 36.8 989.8035 0.27129138 9.92520824
#> 370 36.9 989.8034 0.26864516 9.92790790
#> 371 37.0 989.8034 0.26602475 9.93058123
#> 372 37.1 989.8033 0.26342991 9.93322848
#> 373 37.2 989.8033 0.26086037 9.93584991
#> 374 37.3 989.8032 0.25831590 9.93844577
#> 375 37.4 989.8032 0.25579624 9.94101631
#> 376 37.5 989.8031 0.25330117 9.94356178
#> 377 37.6 989.8031 0.25083043 9.94608242
#> 378 37.7 989.8030 0.24838379 9.94857847
#> 379 37.8 989.8030 0.24596101 9.95105017
#> 380 37.9 989.8029 0.24356187 9.95349777
#> 381 38.0 989.8029 0.24118613 9.95592149
#> 382 38.1 989.8028 0.23883357 9.95832157
#> 383 38.2 989.8028 0.23650395 9.96069824
#> 384 38.3 989.8028 0.23419705 9.96305172
#> 385 38.4 989.8027 0.23191265 9.96538225
#> 386 38.5 989.8027 0.22965054 9.96769005
#> 387 38.6 989.8026 0.22741049 9.96997534
#> 388 38.7 989.8026 0.22519230 9.97223833
#> 389 38.8 989.8025 0.22299574 9.97447925
#> 390 38.9 989.8025 0.22082060 9.97669832
#> 391 39.0 989.8024 0.21866668 9.97889574
#> 392 39.1 989.8024 0.21653377 9.98107172
#> 393 39.2 989.8024 0.21442167 9.98322648
#> 394 39.3 989.8023 0.21233017 9.98536022
#> 395 39.4 989.8023 0.21025907 9.98747315
#> 396 39.5 989.8022 0.20820817 9.98956547
#> 397 39.6 989.8022 0.20617727 9.99163738
#> 398 39.7 989.8021 0.20416619 9.99368908
#> 399 39.8 989.8021 0.20217472 9.99572077
#> 400 39.9 989.8021 0.20020267 9.99773264
#> 401 40.0 989.8020 0.19824986 9.99972489
#> 402 40.1 989.8020 0.19631610 10.00169770
#> 403 40.2 989.8019 0.19440121 10.00365128
#> 404 40.3 989.8019 0.19250499 10.00558579
#> 405 40.4 989.8019 0.19062726 10.00750144
#> 406 40.5 989.8018 0.18876786 10.00939840
#> 407 40.6 989.8018 0.18692658 10.01127685
#> 408 40.7 989.8018 0.18510327 10.01313699
#> 409 40.8 989.8017 0.18329775 10.01497898
#> 410 40.9 989.8017 0.18150983 10.01680300
#> 411 41.0 989.8017 0.17973936 10.01860923
#> 412 41.1 989.8016 0.17798615 10.02039785
#> 413 41.2 989.8016 0.17625005 10.02216901
#> 414 41.3 989.8015 0.17453088 10.02392290
#> 415 41.4 989.8015 0.17282847 10.02565969
#> 416 41.5 989.8015 0.17114268 10.02737953
#> 417 41.6 989.8014 0.16947333 10.02908260
#> 418 41.7 989.8014 0.16782026 10.03076905
#> 419 41.8 989.8014 0.16618331 10.03243905
#> 420 41.9 989.8013 0.16456233 10.03409277
#> 421 42.0 989.8013 0.16295717 10.03573035
#> 422 42.1 989.8013 0.16136766 10.03735197
#> 423 42.2 989.8012 0.15979365 10.03895776
#> 424 42.3 989.8012 0.15823500 10.04054789
#> 425 42.4 989.8012 0.15669155 10.04212251
#> 426 42.5 989.8012 0.15516316 10.04368177
#> 427 42.6 989.8011 0.15364967 10.04522582
#> 428 42.7 989.8011 0.15215095 10.04675481
#> 429 42.8 989.8011 0.15066684 10.04826889
#> 430 42.9 989.8010 0.14919722 10.04976820
#> 431 43.0 989.8010 0.14774192 10.05125288
#> 432 43.1 989.8010 0.14630083 10.05272308
#> 433 43.2 989.8009 0.14487378 10.05417895
#> 434 43.3 989.8009 0.14346066 10.05562061
#> 435 43.4 989.8009 0.14206132 10.05704820
#> 436 43.5 989.8009 0.14067564 10.05846188
#> 437 43.6 989.8008 0.13930346 10.05986176
#> 438 43.7 989.8008 0.13794468 10.06124799
#> 439 43.8 989.8008 0.13659914 10.06262070
#> 440 43.9 989.8008 0.13526673 10.06398002
#> 441 44.0 989.8007 0.13394732 10.06532608
#> 442 44.1 989.8007 0.13264078 10.06665901
#> 443 44.2 989.8007 0.13134698 10.06797894
#> 444 44.3 989.8006 0.13006580 10.06928599
#> 445 44.4 989.8006 0.12879712 10.07058029
#> 446 44.5 989.8006 0.12754081 10.07186197
#> 447 44.6 989.8006 0.12629676 10.07313115
#> 448 44.7 989.8005 0.12506484 10.07438795
#> 449 44.8 989.8005 0.12384494 10.07563249
#> 450 44.9 989.8005 0.12263693 10.07686489
#> 451 45.0 989.8005 0.12144071 10.07808527
#> 452 45.1 989.8005 0.12025616 10.07929374
#> 453 45.2 989.8004 0.11908316 10.08049043
#> 454 45.3 989.8004 0.11792161 10.08167544
#> 455 45.4 989.8004 0.11677138 10.08284890
#> 456 45.5 989.8004 0.11563238 10.08401091
#> 457 45.6 989.8003 0.11450448 10.08516158
#> 458 45.7 989.8003 0.11338759 10.08630103
#> 459 45.8 989.8003 0.11228159 10.08742937
#> 460 45.9 989.8003 0.11118637 10.08854670
#> 461 46.0 989.8002 0.11010184 10.08965313
#> 462 46.1 989.8002 0.10902789 10.09074877
#> 463 46.2 989.8002 0.10796442 10.09183373
#> 464 46.3 989.8002 0.10691132 10.09290810
#> 465 46.4 989.8002 0.10586849 10.09397199
#> 466 46.5 989.8001 0.10483583 10.09502550
#> 467 46.6 989.8001 0.10381324 10.09606874
#> 468 46.7 989.8001 0.10280063 10.09710180
#> 469 46.8 989.8001 0.10179790 10.09812478
#> 470 46.9 989.8001 0.10080495 10.09913779
#> 471 47.0 989.8000 0.09982168 10.10014091
#> 472 47.1 989.8000 0.09884800 10.10113425
#> 473 47.2 989.8000 0.09788382 10.10211790
#> 474 47.3 989.8000 0.09692905 10.10309196
#> 475 47.4 989.8000 0.09598359 10.10405652
#> 476 47.5 989.7999 0.09504735 10.10501166
#> 477 47.6 989.7999 0.09412024 10.10595749
#> 478 47.7 989.7999 0.09320218 10.10689410
#> 479 47.8 989.7999 0.09229307 10.10782157
#> 480 47.9 989.7999 0.09139283 10.10873999
#> 481 48.0 989.7998 0.09050137 10.10964945
#> 482 48.1 989.7998 0.08961861 10.11055005
#> 483 48.2 989.7998 0.08874445 10.11144185
#> 484 48.3 989.7998 0.08787882 10.11232496
#> 485 48.4 989.7998 0.08702164 10.11319946
#> 486 48.5 989.7998 0.08617282 10.11406542
#> 487 48.6 989.7997 0.08533227 10.11492294
#> 488 48.7 989.7997 0.08449993 10.11577210
#> 489 48.8 989.7997 0.08367570 10.11661297
#> 490 48.9 989.7997 0.08285952 10.11744564
#> 491 49.0 989.7997 0.08205129 10.11827019
#> 492 49.1 989.7997 0.08125095 10.11908669
#> 493 49.2 989.7996 0.08045842 10.11989523
#> 494 49.3 989.7996 0.07967361 10.12069588
#> 495 49.4 989.7996 0.07889647 10.12148873
#> 496 49.5 989.7996 0.07812690 10.12227384
#> 497 49.6 989.7996 0.07736484 10.12305129
#> 498 49.7 989.7996 0.07661021 10.12382116
#> 499 49.8 989.7996 0.07586294 10.12458352
#> 500 49.9 989.7995 0.07512296 10.12533844
#> 501 50.0 989.7995 0.07439020 10.12608600
# Open population with births and deaths
model_sir(
init = c(S = 990, I = 10),
params = c(beta = 0.002, gamma = 0.1, b = 0.01, d = 0.01),
times = seq(0, 200, by = 1)
)
#> time S I R
#> 1 0 990.0000 1.000000e+01 0.0000000
#> 2 1 990.0808 8.976098e+00 0.9430736
#> 3 2 990.1628 8.057034e+00 1.7802035
#> 4 3 990.2456 7.232074e+00 2.5223290
#> 5 4 990.3291 6.491583e+00 3.1792710
#> 6 5 990.4132 5.826912e+00 3.7598411
#> 7 6 990.4978 5.230298e+00 4.2719511
#> 8 7 990.5825 4.694771e+00 4.7227008
#> 9 8 990.6675 4.214077e+00 5.1184613
#> 10 9 990.7524 3.782602e+00 5.4649509
#> 11 10 990.8374 3.395305e+00 5.7673016
#> 12 11 990.9222 3.047664e+00 6.0301190
#> 13 12 991.0068 2.735618e+00 6.2575361
#> 14 13 991.0912 2.455522e+00 6.4532621
#> 15 14 991.1753 2.204105e+00 6.6206254
#> 16 15 991.2590 1.978431e+00 6.7626130
#> 17 16 991.3422 1.775863e+00 6.8819049
#> 18 17 991.4251 1.594036e+00 6.9809063
#> 19 18 991.5074 1.430827e+00 7.0617749
#> 20 19 991.5892 1.284328e+00 7.1264469
#> 21 20 991.6705 1.152829e+00 7.1766595
#> 22 21 991.7512 1.034794e+00 7.2139711
#> 23 22 991.8314 9.288442e-01 7.2397798
#> 24 23 991.9109 8.337426e-01 7.2553399
#> 25 24 991.9898 7.483784e-01 7.2617764
#> 26 25 992.0681 6.717544e-01 7.2600983
#> 27 26 992.1458 6.029757e-01 7.2512108
#> 28 27 992.2228 5.412392e-01 7.2359252
#> 29 28 992.2992 4.858237e-01 7.2149696
#> 30 29 992.3749 4.360821e-01 7.1889964
#> 31 30 992.4500 3.914334e-01 7.1585906
#> 32 31 992.5244 3.513561e-01 7.1242766
#> 33 32 992.5981 3.153823e-01 7.0865244
#> 34 33 992.6712 2.830916e-01 7.0457553
#> 35 34 992.7435 2.541072e-01 7.0023466
#> 36 35 992.8153 2.280903e-01 6.9566364
#> 37 36 992.8863 2.047372e-01 6.9089274
#> 38 37 992.9567 1.837752e-01 6.8594907
#> 39 38 993.0265 1.649594e-01 6.8085690
#> 40 39 993.0956 1.480700e-01 6.7563796
#> 41 40 993.1640 1.329099e-01 6.7031167
#> 42 41 993.2317 1.193020e-01 6.6489540
#> 43 42 993.2989 1.070873e-01 6.5940469
#> 44 43 993.3653 9.612324e-02 6.5385342
#> 45 44 993.4312 8.628173e-02 6.4825399
#> 46 45 993.4964 7.744785e-02 6.4261745
#> 47 46 993.5609 6.951843e-02 6.3695370
#> 48 47 993.6249 6.240086e-02 6.3127152
#> 49 48 993.6882 5.601202e-02 6.2557875
#> 50 49 993.7509 5.027731e-02 6.1988237
#> 51 50 993.8130 4.512974e-02 6.1418859
#> 52 51 993.8745 4.050920e-02 6.0850292
#> 53 52 993.9353 3.636173e-02 6.0283025
#> 54 53 993.9956 3.263890e-02 5.9717491
#> 55 54 994.0553 2.929723e-02 5.9154073
#> 56 55 994.1144 2.629769e-02 5.8593110
#> 57 56 994.1729 2.360525e-02 5.8034899
#> 58 57 994.2308 2.118848e-02 5.7479704
#> 59 58 994.2882 1.901915e-02 5.6927754
#> 60 59 994.3450 1.707192e-02 5.6379250
#> 61 60 994.4012 1.532405e-02 5.5834367
#> 62 61 994.4569 1.375514e-02 5.5293258
#> 63 62 994.5120 1.234686e-02 5.4756053
#> 64 63 994.5666 1.108276e-02 5.4222865
#> 65 64 994.6207 9.948083e-03 5.3693790
#> 66 65 994.6742 8.929578e-03 5.3168910
#> 67 66 994.7272 8.015351e-03 5.2648292
#> 68 67 994.7796 7.194725e-03 5.2131992
#> 69 68 994.8315 6.458117e-03 5.1620055
#> 70 69 994.8830 5.796924e-03 5.1112517
#> 71 70 994.9339 5.203427e-03 5.0609406
#> 72 71 994.9843 4.670692e-03 5.0110742
#> 73 72 995.0342 4.192501e-03 4.9616536
#> 74 73 995.0836 3.763267e-03 4.9126797
#> 75 74 995.1325 3.377980e-03 4.8641526
#> 76 75 995.1809 3.032138e-03 4.8160721
#> 77 76 995.2288 2.721705e-03 4.7684373
#> 78 77 995.2763 2.443054e-03 4.7212473
#> 79 78 995.3233 2.192933e-03 4.6745005
#> 80 79 995.3698 1.968419e-03 4.6281952
#> 81 80 995.4159 1.766891e-03 4.5823295
#> 82 81 995.4615 1.585995e-03 4.5369012
#> 83 82 995.5067 1.423620e-03 4.4919079
#> 84 83 995.5514 1.277869e-03 4.4473469
#> 85 84 995.5956 1.147040e-03 4.4032156
#> 86 85 995.6395 1.029606e-03 4.3595110
#> 87 86 995.6828 9.241941e-04 4.3162303
#> 88 87 995.7258 8.295747e-04 4.2733702
#> 89 88 995.7683 7.446425e-04 4.2309277
#> 90 89 995.8104 6.684058e-04 4.1888995
#> 91 90 995.8521 5.999742e-04 4.1472823
#> 92 91 995.8934 5.385488e-04 4.1060727
#> 93 92 995.9342 4.834121e-04 4.0652674
#> 94 93 995.9747 4.339203e-04 4.0248629
#> 95 94 996.0148 3.894956e-04 3.9848558
#> 96 95 996.0544 3.496191e-04 3.9452425
#> 97 96 996.0937 3.138252e-04 3.9060197
#> 98 97 996.1325 2.816958e-04 3.8671837
#> 99 98 996.1710 2.528559e-04 3.8287312
#> 100 99 996.2091 2.269686e-04 3.7906585
#> 101 100 996.2468 2.037317e-04 3.7529622
#> 102 101 996.2842 1.828737e-04 3.7156389
#> 103 102 996.3212 1.641512e-04 3.6786849
#> 104 103 996.3578 1.473459e-04 3.6420968
#> 105 104 996.3940 1.322615e-04 3.6058713
#> 106 105 996.4299 1.187214e-04 3.5700047
#> 107 106 996.4654 1.065674e-04 3.5344938
#> 108 107 996.5006 9.565774e-05 3.4993350
#> 109 108 996.5354 8.586490e-05 3.4645251
#> 110 109 996.5699 7.707460e-05 3.4300606
#> 111 110 996.6040 6.918420e-05 3.3959382
#> 112 111 996.6378 6.210158e-05 3.3621546
#> 113 112 996.6712 5.574403e-05 3.3287064
#> 114 113 996.7044 5.003732e-05 3.2955905
#> 115 114 996.7372 4.491484e-05 3.2628035
#> 116 115 996.7696 4.031676e-05 3.2303423
#> 117 116 996.8018 3.618940e-05 3.1982037
#> 118 117 996.8336 3.248458e-05 3.1663845
#> 119 118 996.8651 2.915904e-05 3.1348815
#> 120 119 996.8963 2.617394e-05 3.1036916
#> 121 120 996.9272 2.349443e-05 3.0728119
#> 122 121 996.9577 2.108924e-05 3.0422391
#> 123 122 996.9880 1.893028e-05 3.0119703
#> 124 123 997.0180 1.699221e-05 2.9820025
#> 125 124 997.0477 1.525255e-05 2.9523326
#> 126 125 997.0770 1.369102e-05 2.9229579
#> 127 126 997.1061 1.228936e-05 2.8938753
#> 128 127 997.1349 1.103120e-05 2.8650819
#> 129 128 997.1634 9.901879e-06 2.8365749
#> 130 129 997.1916 8.888196e-06 2.8083514
#> 131 130 997.2196 7.978290e-06 2.7804087
#> 132 131 997.2472 7.161534e-06 2.7527439
#> 133 132 997.2746 6.428347e-06 2.7253543
#> 134 133 997.3018 5.770215e-06 2.6982372
#> 135 134 997.3286 5.179474e-06 2.6713899
#> 136 135 997.3552 4.649212e-06 2.6448096
#> 137 136 997.3815 4.173237e-06 2.6184937
#> 138 137 997.4076 3.746002e-06 2.5924397
#> 139 138 997.4334 3.362515e-06 2.5666448
#> 140 139 997.4589 3.018288e-06 2.5411066
#> 141 140 997.4842 2.709300e-06 2.5158224
#> 142 141 997.5092 2.431927e-06 2.4907898
#> 143 142 997.5340 2.182948e-06 2.4660063
#> 144 143 997.5585 1.959464e-06 2.4414693
#> 145 144 997.5828 1.758860e-06 2.4171765
#> 146 145 997.6069 1.578793e-06 2.3931253
#> 147 146 997.6307 1.417161e-06 2.3693135
#> 148 147 997.6543 1.272076e-06 2.3457386
#> 149 148 997.6776 1.141845e-06 2.3223982
#> 150 149 997.7007 1.024946e-06 2.2992901
#> 151 150 997.7236 9.200156e-07 2.2764118
#> 152 151 997.7462 8.258273e-07 2.2537612
#> 153 152 997.7687 7.412818e-07 2.2313360
#> 154 153 997.7909 6.653917e-07 2.2091339
#> 155 154 997.8128 5.972711e-07 2.1871527
#> 156 155 997.8346 5.361245e-07 2.1653903
#> 157 156 997.8562 4.812381e-07 2.1438443
#> 158 157 997.8775 4.319708e-07 2.1225128
#> 159 158 997.8986 3.877474e-07 2.1013934
#> 160 159 997.9195 3.480513e-07 2.0804843
#> 161 160 997.9402 3.124192e-07 2.0597831
#> 162 161 997.9607 2.804349e-07 2.0392880
#> 163 162 997.9810 2.517251e-07 2.0189968
#> 164 163 998.0011 2.259544e-07 1.9989074
#> 165 164 998.0210 2.028220e-07 1.9790180
#> 166 165 998.0407 1.820579e-07 1.9593264
#> 167 166 998.0602 1.634195e-07 1.9398308
#> 168 167 998.0795 1.466893e-07 1.9205292
#> 169 168 998.0986 1.316719e-07 1.9014196
#> 170 169 998.1175 1.181919e-07 1.8825002
#> 171 170 998.1362 1.060919e-07 1.8637690
#> 172 171 998.1548 9.523064e-08 1.8452242
#> 173 172 998.1731 8.548134e-08 1.8268640
#> 174 173 998.1913 7.673014e-08 1.8086864
#> 175 174 998.2093 6.887484e-08 1.7906896
#> 176 175 998.2271 6.182374e-08 1.7728720
#> 177 176 998.2448 5.549450e-08 1.7552316
#> 178 177 998.2622 4.981322e-08 1.7377668
#> 179 178 998.2795 4.471357e-08 1.7204757
#> 180 179 998.2966 4.013600e-08 1.7033567
#> 181 180 998.3136 3.602706e-08 1.6864080
#> 182 181 998.3304 3.233877e-08 1.6696280
#> 183 182 998.3470 2.902808e-08 1.6530149
#> 184 183 998.3634 2.605632e-08 1.6365671
#> 185 184 998.3797 2.338879e-08 1.6202830
#> 186 185 998.3958 2.099436e-08 1.6041609
#> 187 186 998.4118 1.884505e-08 1.5881993
#> 188 187 998.4276 1.691579e-08 1.5723964
#> 189 188 998.4432 1.518403e-08 1.5567508
#> 190 189 998.4587 1.362956e-08 1.5412609
#> 191 190 998.4741 1.223423e-08 1.5259251
#> 192 191 998.4893 1.098175e-08 1.5107419
#> 193 192 998.5043 9.857494e-09 1.4957098
#> 194 193 998.5192 8.848332e-09 1.4808272
#> 195 194 998.5339 7.942484e-09 1.4660927
#> 196 195 998.5485 7.129372e-09 1.4515049
#> 197 196 998.5629 6.399503e-09 1.4370621
#> 198 197 998.5772 5.744354e-09 1.4227631
#> 199 198 998.5914 5.156275e-09 1.4086064
#> 200 199 998.6054 4.628403e-09 1.3945905
#> 201 200 998.6193 4.154571e-09 1.3807141