Epidemic.ipynb

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    "# Epidemic modeling using various fully-mixed compartmental models such as the SIR, SIS and SIRS models in Julia\n",
    "\n",
    "#### These are traditional models that assumes a fully mixed population and therefore of lower fidelity compared to more modern techniques. However these models are helpful in providing a general understanding of how diseases spread.\n",
    "\n",
    "#### Ref: \"Networks\" by Mark Newman, Print ISBN-13: 9780198805090\n",
    "\n",
    "@copyright Dr. Srijith Rajamohan under the Creative Commons Attribution license CC BY 4.0\n",
    "(Rights given to reproduce this as desired under the license agreement listed)"
   ]
  },
  {
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   "source": [
    "##        Contents\n",
    "\n",
    "##       [1. Introduction](#I1)\n",
    "##       [2. SIR Model](#I2)\n",
    "##       [3. SIS Model](#I3)\n",
    "##       [4. SIRS Model](#I4)\n",
    "##       [5. Social distancing](#I5)\n",
    "##       [6. Coronavirus vs. Flu](#I6)\n",
    "##       [7. Size of Outbreak](#I7)"
   ]
  },
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   "source": [
    "using Pkg"
   ]
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      "\u001b[32m\u001b[1m  Building\u001b[22m\u001b[39m Sundials → `~/.julia/packages/Sundials/SKP8f/deps/build.log`\n"
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   "source": [
    "using DifferentialEquations\n",
    "using Plots"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<a id='I1'></a>"
   ]
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    "## The following applies to the models below\n",
    "\n",
    "#### s corresponds to the percentage of susceptible people in the population\n",
    "#### x corresponds to the percentage of infected people in the population\n",
    "#### r corresponds to the percentage of recovered people in the population\n",
    "#### beta measures the average number of contacts a person has in unit time ( implications for social distancing )\n",
    "#### gamma corresponds to the rate at which individuals recover or die, i.e. the fraction of recovery one experiences in unit time \n",
    "#### R0 is the the basic reproduction number that represents the average number of people that an infected person 'A' infects at the beginning of the epidemic, before the person 'A' recovers. ( This is different from r0 which is the initial condition for r in the equations below)\n",
    "\n",
    "#### R0 = beta / gamma"
   ]
  },
  {
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   "source": [
    "## Using the SIR model\n",
    "\n",
    "<a id='I2'></a>"
   ]
  },
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       "  263.64,1386.65 265.752,1386.4 267.865,1386.13 269.977,1385.85 272.089,1385.56 274.202,1385.25 276.314,1384.93 278.426,1384.6 280.539,1384.25 282.651,1383.88 \n",
       "  284.763,1383.49 286.876,1383.09 288.988,1382.66 291.101,1382.22 293.213,1381.75 295.325,1381.27 297.438,1380.76 299.55,1380.23 301.662,1379.67 303.775,1379.09 \n",
       "  305.887,1378.48 307.999,1377.84 310.112,1377.17 312.224,1376.47 314.336,1375.74 316.449,1374.98 318.561,1374.18 320.673,1373.35 322.786,1372.48 324.898,1371.57 \n",
       "  327.01,1370.62 329.123,1369.63 331.235,1368.59 333.348,1367.51 335.46,1366.38 337.572,1365.21 339.685,1363.98 341.797,1362.7 343.909,1361.37 346.022,1359.98 \n",
       "  348.134,1358.53 350.246,1357.02 352.359,1355.45 354.471,1353.82 356.583,1352.12 358.696,1350.35 360.808,1348.51 362.92,1346.6 365.033,1344.62 367.145,1342.56 \n",
       "  369.258,1340.42 371.37,1338.2 373.482,1335.9 375.595,1333.51 377.707,1331.04 379.819,1328.48 381.932,1325.82 384.044,1323.08 386.156,1320.24 388.269,1317.31 \n",
       "  390.381,1314.28 392.493,1311.15 394.606,1307.92 396.718,1304.59 398.83,1301.16 400.943,1297.63 403.055,1293.99 405.168,1290.24 407.28,1286.39 409.392,1282.43 \n",
       "  411.505,1278.37 413.617,1274.19 415.729,1269.91 417.842,1265.52 419.954,1261.03 422.066,1256.43 424.179,1251.72 426.291,1246.91 428.403,1241.99 430.516,1236.96 \n",
       "  432.628,1231.84 434.74,1226.61 436.853,1221.29 438.965,1215.86 441.077,1210.34 443.19,1204.73 445.302,1199.02 447.415,1193.23 449.527,1187.34 451.639,1181.38 \n",
       "  453.752,1175.32 455.864,1169.19 457.976,1162.98 460.089,1156.7 462.201,1150.35 464.313,1143.93 466.426,1137.44 468.538,1130.89 470.65,1124.28 472.763,1117.61 \n",
       "  474.875,1110.9 476.987,1104.13 479.1,1097.32 481.212,1090.46 483.325,1083.56 485.437,1076.63 487.549,1069.66 489.662,1062.66 491.774,1055.63 493.886,1048.58 \n",
       "  495.999,1041.51 498.111,1034.42 500.223,1027.31 502.336,1020.19 504.448,1013.07 506.56,1005.93 508.673,998.787 510.785,991.643 512.897,984.499 515.01,977.358 \n",
       "  517.122,970.222 519.235,963.094 521.347,955.975 523.459,948.867 525.572,941.774 527.684,934.696 529.796,927.636 531.909,920.596 534.021,913.576 536.133,906.58 \n",
       "  538.246,899.608 540.358,892.663 542.47,885.745 544.583,878.856 546.695,871.997 548.807,865.17 550.92,858.376 553.032,851.616 555.144,844.891 557.257,838.202 \n",
       "  559.369,831.55 561.482,824.937 563.594,818.362 565.706,811.827 567.819,805.333 569.931,798.88 572.043,792.468 574.156,786.1 576.268,779.774 578.38,773.492 \n",
       "  580.493,767.254 582.605,761.061 584.717,754.912 586.83,748.809 588.942,742.752 591.054,736.741 593.167,730.776 595.279,724.858 597.392,718.986 599.504,713.162 \n",
       "  601.616,707.384 603.729,701.654 605.841,695.971 607.953,690.335 610.066,684.747 612.178,679.207 614.29,673.714 616.403,668.269 618.515,662.871 620.627,657.521 \n",
       "  622.74,652.218 624.852,646.962 626.964,641.754 629.077,636.593 631.189,631.479 633.302,626.412 635.414,621.391 637.526,616.417 639.639,611.49 641.751,606.609 \n",
       "  643.863,601.774 645.976,596.985 648.088,592.241 650.2,587.543 652.313,582.89 654.425,578.282 656.537,573.719 658.65,569.2 660.762,564.726 662.874,560.295 \n",
       "  664.987,555.908 667.099,551.565 669.211,547.265 671.324,543.007 673.436,538.793 675.549,534.62 677.661,530.49 679.773,526.401 681.886,522.354 683.998,518.347 \n",
       "  686.11,514.382 688.223,510.457 690.335,506.573 692.447,502.728 694.56,498.923 696.672,495.157 698.784,491.43 700.897,487.741 703.009,484.091 705.121,480.479 \n",
       "  707.234,476.905 709.346,473.368 711.459,469.868 713.571,466.405 715.683,462.978 717.796,459.588 719.908,456.233 722.02,452.914 724.133,449.63 726.245,446.38 \n",
       "  728.357,443.166 730.47,439.985 732.582,436.839 734.694,433.726 736.807,430.646 738.919,427.599 741.031,424.585 743.144,421.604 745.256,418.654 747.369,415.736 \n",
       "  749.481,412.85 751.593,409.995 753.706,407.17 755.818,404.376 757.93,401.613 760.043,398.879 762.155,396.176 764.267,393.501 766.38,390.856 768.492,388.239 \n",
       "  770.604,385.651 772.717,383.091 774.829,380.56 776.941,378.055 779.054,375.579 781.166,373.129 783.278,370.707 785.391,368.31 787.503,365.941 789.616,363.597 \n",
       "  791.728,361.279 793.84,358.986 795.953,356.719 798.065,354.477 800.177,352.26 802.29,350.067 804.402,347.898 806.514,345.753 808.627,343.632 810.739,341.535 \n",
       "  812.851,339.461 814.964,337.409 817.076,335.381 819.188,333.375 821.301,331.391 823.413,329.43 825.526,327.49 827.638,325.572 829.75,323.675 831.863,321.799 \n",
       "  833.975,319.944 836.087,318.11 838.2,316.296 840.312,314.503 842.424,312.73 844.537,310.976 846.649,309.242 848.761,307.528 850.874,305.832 852.986,304.156 \n",
       "  855.098,302.498 857.211,300.859 859.323,299.239 861.436,297.636 863.548,296.052 865.66,294.485 867.773,292.936 869.885,291.404 871.997,289.89 874.11,288.392 \n",
       "  876.222,286.912 878.334,285.448 880.447,284 882.559,282.569 884.671,281.154 886.784,279.755 888.896,278.372 891.008,277.004 893.121,275.652 895.233,274.314 \n",
       "  897.345,272.992 899.458,271.685 901.57,270.393 903.683,269.115 905.795,267.852 907.907,266.603 910.02,265.368 912.132,264.148 914.244,262.94 916.357,261.747 \n",
       "  918.469,260.567 920.581,259.401 922.694,258.247 924.806,257.107 926.918,255.98 929.031,254.865 931.143,253.763 933.255,252.674 935.368,251.597 937.48,250.532 \n",
       "  939.593,249.479 941.705,248.438 943.817,247.409 945.93,246.392 948.042,245.386 950.154,244.392 952.267,243.409 954.379,242.437 956.491,241.476 958.604,240.526 \n",
       "  960.716,239.587 962.828,238.658 964.941,237.74 967.053,236.833 969.165,235.936 971.278,235.049 973.39,234.172 975.503,233.305 977.615,232.448 979.727,231.601 \n",
       "  981.84,230.763 983.952,229.935 986.064,229.117 988.177,228.308 990.289,227.508 992.401,226.717 994.514,225.935 996.626,225.162 998.738,224.398 1000.85,223.642 \n",
       "  1002.96,222.895 1005.08,222.157 1007.19,221.427 1009.3,220.706 1011.41,219.992 1013.52,219.287 1015.64,218.59 1017.75,217.901 1019.86,217.219 1021.97,216.546 \n",
       "  1024.09,215.88 1026.2,215.222 1028.31,214.571 1030.42,213.928 1032.54,213.292 1034.65,212.663 1036.76,212.042 1038.87,211.427 1040.99,210.82 1043.1,210.22 \n",
       "  1045.21,209.626 1047.32,209.039 1049.43,208.459 1051.55,207.886 1053.66,207.319 1055.77,206.759 1057.88,206.205 1060,205.657 1062.11,205.116 1064.22,204.581 \n",
       "  1066.33,204.052 1068.45,203.529 1070.56,203.012 1072.67,202.501 1074.78,201.996 1076.9,201.496 1079.01,201.003 1081.12,200.515 1083.23,200.032 1085.34,199.555 \n",
       "  1087.46,199.084 1089.57,198.618 1091.68,198.157 1093.79,197.702 1095.91,197.251 1098.02,196.806 1100.13,196.366 1102.24,195.931 1104.36,195.502 1106.47,195.077 \n",
       "  1108.58,194.656 1110.69,194.241 1112.81,193.831 1114.92,193.425 1117.03,193.024 1119.14,192.627 1121.25,192.235 1123.37,191.847 1125.48,191.464 1127.59,191.086 \n",
       "  1129.7,190.711 1131.82,190.341 1133.93,189.975 1136.04,189.614 1138.15,189.256 1140.27,188.903 1142.38,188.554 1144.49,188.208 1146.6,187.867 1148.72,187.53 \n",
       "  1150.83,187.196 1152.94,186.866 1155.05,186.54 1157.16,186.218 1159.28,185.9 1161.39,185.585 1163.5,185.274 1165.61,184.966 1167.73,184.662 1169.84,184.361 \n",
       "  1171.95,184.064 1174.06,183.77 1176.18,183.48 1178.29,183.193 1180.4,182.909 1182.51,182.629 1184.63,182.352 1186.74,182.078 1188.85,181.807 1190.96,181.539 \n",
       "  1193.07,181.274 1195.19,181.012 1197.3,180.754 1199.41,180.498 1201.52,180.245 1203.64,179.995 1205.75,179.748 1207.86,179.504 1209.97,179.263 1212.09,179.024 \n",
       "  1214.2,178.788 1216.31,178.555 1218.42,178.325 1220.54,178.097 1222.65,177.872 1224.76,177.649 1226.87,177.429 1228.98,177.212 1231.1,176.997 1233.21,176.784 \n",
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       "  1932.4,158.933 1934.51,158.929 1936.62,158.924 1938.73,158.92 1940.85,158.915 1942.96,158.911 1945.07,158.907 1947.18,158.903 1949.3,158.898 1951.41,158.894 \n",
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      ]
     },
     "execution_count": 259,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function sir(;s0,  beta, gamma, r0, title=\"Epidemic simulation using the SIR model\")\n",
    "    f(r,p,t) = gamma * (1 - r - s0 * exp(-beta * r / gamma))\n",
    "\n",
    "    tspan = (0.0,60.0)\n",
    "    prob = ODEProblem(f,r0,tspan)\n",
    "    r = solve(prob, Tsit5(), reltol=1e-8, abstol=1e-8)   # Fifth order Tsitouras method\n",
    "    s = s0 * exp.( -beta .* r ./ gamma)\n",
    "    x = 1 .- s .- r\n",
    "\n",
    "    plot(r,linewidth=5,title=title,\n",
    "         xaxis=\"Time (t)\",yaxis=\"% of population)\",label=\"Recovered\") # legend=false\n",
    "    plot!(r.t, s,lw=3,ls=:dash,label=\"Susceptible\")\n",
    "    plot!(r.t, x,lw=3,label=\"Infected\")\n",
    "end\n",
    "\n",
    "sir(s0 = 0.99, beta = 1.0, gamma = 0.2, r0 = 0.0001)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Using the SIS model\n",
    "\n",
    "<a id='I3'></a>\n",
    "\n",
    "#### Unlike the SIR model, there is no assumption that those who are infected gain immunity"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 144,
   "metadata": {},
   "outputs": [
    {
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       "  284.763,1387.47 286.876,1387.38 288.988,1387.29 291.101,1387.19 293.213,1387.1 295.325,1387 297.438,1386.89 299.55,1386.78 301.662,1386.67 303.775,1386.55 \n",
       "  305.887,1386.43 307.999,1386.31 310.112,1386.18 312.224,1386.05 314.336,1385.91 316.449,1385.77 318.561,1385.62 320.673,1385.46 322.786,1385.31 324.898,1385.14 \n",
       "  327.01,1384.97 329.123,1384.8 331.235,1384.61 333.348,1384.43 335.46,1384.23 337.572,1384.03 339.685,1383.82 341.797,1383.6 343.909,1383.38 346.022,1383.15 \n",
       "  348.134,1382.91 350.246,1382.66 352.359,1382.4 354.471,1382.14 356.583,1381.86 358.696,1381.58 360.808,1381.29 362.92,1380.98 365.033,1380.67 367.145,1380.34 \n",
       "  369.258,1380 371.37,1379.65 373.482,1379.29 375.595,1378.92 377.707,1378.53 379.819,1378.14 381.932,1377.72 384.044,1377.3 386.156,1376.85 388.269,1376.4 \n",
       "  390.381,1375.93 392.493,1375.44 394.606,1374.93 396.718,1374.41 398.83,1373.87 400.943,1373.31 403.055,1372.74 405.168,1372.14 407.28,1371.53 409.392,1370.89 \n",
       "  411.505,1370.23 413.617,1369.55 415.729,1368.85 417.842,1368.12 419.954,1367.37 422.066,1366.6 424.179,1365.8 426.291,1364.97 428.403,1364.12 430.516,1363.24 \n",
       "  432.628,1362.33 434.74,1361.39 436.853,1360.42 438.965,1359.42 441.077,1358.39 443.19,1357.32 445.302,1356.22 447.415,1355.09 449.527,1353.92 451.639,1352.71 \n",
       "  453.752,1351.47 455.864,1350.18 457.976,1348.86 460.089,1347.5 462.201,1346.09 464.313,1344.64 466.426,1343.15 468.538,1341.61 470.65,1340.03 472.763,1338.4 \n",
       "  474.875,1336.72 476.987,1334.99 479.1,1333.21 481.212,1331.38 483.325,1329.49 485.437,1327.56 487.549,1325.56 489.662,1323.52 491.774,1321.41 493.886,1319.25 \n",
       "  495.999,1317.02 498.111,1314.74 500.223,1312.4 502.336,1309.99 504.448,1307.52 506.56,1304.99 508.673,1302.39 510.785,1299.72 512.897,1296.99 515.01,1294.19 \n",
       "  517.122,1291.32 519.235,1288.39 521.347,1285.38 523.459,1282.3 525.572,1279.16 527.684,1275.94 529.796,1272.64 531.909,1269.28 534.021,1265.84 536.133,1262.34 \n",
       "  538.246,1258.75 540.358,1255.1 542.47,1251.37 544.583,1247.57 546.695,1243.7 548.807,1239.76 550.92,1235.74 553.032,1231.66 555.144,1227.5 557.257,1223.27 \n",
       "  559.369,1218.98 561.482,1214.62 563.594,1210.19 565.706,1205.7 567.819,1201.14 569.931,1196.52 572.043,1191.84 574.156,1187.1 576.268,1182.31 578.38,1177.46 \n",
       "  580.493,1172.55 582.605,1167.6 584.717,1162.59 586.83,1157.54 588.942,1152.45 591.054,1147.32 593.167,1142.14 595.279,1136.93 597.392,1131.69 599.504,1126.42 \n",
       "  601.616,1121.12 603.729,1115.8 605.841,1110.46 607.953,1105.1 610.066,1099.73 612.178,1094.34 614.29,1088.95 616.403,1083.55 618.515,1078.16 620.627,1072.76 \n",
       "  622.74,1067.37 624.852,1061.99 626.964,1056.62 629.077,1051.27 631.189,1045.93 633.302,1040.62 635.414,1035.33 637.526,1030.07 639.639,1024.84 641.751,1019.64 \n",
       "  643.863,1014.48 645.976,1009.36 648.088,1004.28 650.2,999.25 652.313,994.263 654.425,989.326 656.537,984.44 658.65,979.608 660.762,974.832 662.874,970.114 \n",
       "  664.987,965.455 667.099,960.857 669.211,956.322 671.324,951.851 673.436,947.446 675.549,943.108 677.661,938.837 679.773,934.636 681.886,930.504 683.998,926.442 \n",
       "  686.11,922.452 688.223,918.533 690.335,914.687 692.447,910.912 694.56,907.211 696.672,903.582 698.784,900.026 700.897,896.542 703.009,893.131 705.121,889.793 \n",
       "  707.234,886.527 709.346,883.332 711.459,880.209 713.571,877.156 715.683,874.174 717.796,871.261 719.908,868.417 722.02,865.641 724.133,862.933 726.245,860.291 \n",
       "  728.357,857.715 730.47,855.204 732.582,852.756 734.694,850.372 736.807,848.049 738.919,845.787 741.031,843.586 743.144,841.443 745.256,839.357 747.369,837.329 \n",
       "  749.481,835.356 751.593,833.437 753.706,831.572 755.818,829.76 757.93,827.998 760.043,826.287 762.155,824.624 764.267,823.01 766.38,821.442 768.492,819.92 \n",
       "  770.604,818.443 772.717,817.009 774.829,815.618 776.941,814.268 779.054,812.959 781.166,811.689 783.278,810.457 785.391,809.263 787.503,808.106 789.616,806.984 \n",
       "  791.728,805.896 793.84,804.842 795.953,803.82 798.065,802.831 800.177,801.872 802.29,800.943 804.402,800.044 806.514,799.173 808.627,798.329 810.739,797.512 \n",
       "  812.851,796.721 814.964,795.956 817.076,795.214 819.188,794.497 821.301,793.803 823.413,793.131 825.526,792.48 827.638,791.851 829.75,791.242 831.863,790.653 \n",
       "  833.975,790.083 836.087,789.532 838.2,788.999 840.312,788.483 842.424,787.984 844.537,787.501 846.649,787.035 848.761,786.584 850.874,786.147 852.986,785.725 \n",
       "  855.098,785.317 857.211,784.923 859.323,784.542 861.436,784.173 863.548,783.816 865.66,783.472 867.773,783.139 869.885,782.817 871.997,782.506 874.11,782.205 \n",
       "  876.222,781.914 878.334,781.633 880.447,781.362 882.559,781.099 884.671,780.846 886.784,780.601 888.896,780.364 891.008,780.135 893.121,779.914 895.233,779.7 \n",
       "  897.345,779.494 899.458,779.295 901.57,779.102 903.683,778.916 905.795,778.736 907.907,778.562 910.02,778.394 912.132,778.232 914.244,778.075 916.357,777.924 \n",
       "  918.469,777.777 920.581,777.636 922.694,777.5 924.806,777.368 926.918,777.24 929.031,777.117 931.143,776.999 933.255,776.884 935.368,776.773 937.48,776.666 \n",
       "  939.593,776.562 941.705,776.462 943.817,776.366 945.93,776.273 948.042,776.183 950.154,776.096 952.267,776.012 954.379,775.931 956.491,775.852 958.604,775.777 \n",
       "  960.716,775.704 962.828,775.633 964.941,775.565 967.053,775.499 969.165,775.435 971.278,775.374 973.39,775.315 975.503,775.257 977.615,775.202 979.727,775.149 \n",
       "  981.84,775.097 983.952,775.047 986.064,774.999 988.177,774.953 990.289,774.908 992.401,774.865 994.514,774.823 996.626,774.782 998.738,774.743 1000.85,774.706 \n",
       "  1002.96,774.669 1005.08,774.634 1007.19,774.6 1009.3,774.567 1011.41,774.536 1013.52,774.505 1015.64,774.476 1017.75,774.447 1019.86,774.42 1021.97,774.393 \n",
       "  1024.09,774.367 1026.2,774.343 1028.31,774.319 1030.42,774.296 1032.54,774.273 1034.65,774.252 1036.76,774.231 1038.87,774.211 1040.99,774.191 1043.1,774.173 \n",
       "  1045.21,774.155 1047.32,774.137 1049.43,774.12 1051.55,774.104 1053.66,774.088 1055.77,774.073 1057.88,774.058 1060,774.044 1062.11,774.031 1064.22,774.017 \n",
       "  1066.33,774.005 1068.45,773.992 1070.56,773.98 1072.67,773.969 1074.78,773.958 1076.9,773.947 1079.01,773.937 1081.12,773.927 1083.23,773.917 1085.34,773.908 \n",
       "  1087.46,773.899 1089.57,773.89 1091.68,773.882 1093.79,773.874 1095.91,773.866 1098.02,773.858 1100.13,773.851 1102.24,773.844 1104.36,773.837 1106.47,773.831 \n",
       "  1108.58,773.824 1110.69,773.818 1112.81,773.812 1114.92,773.807 1117.03,773.801 1119.14,773.796 1121.25,773.791 1123.37,773.786 1125.48,773.781 1127.59,773.776 \n",
       "  1129.7,773.772 1131.82,773.768 1133.93,773.764 1136.04,773.76 1138.15,773.756 1140.27,773.752 1142.38,773.748 1144.49,773.745 1146.6,773.741 1148.72,773.738 \n",
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     },
     "execution_count": 144,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "function sis(;x0,  beta)\n",
    "    f(x,p,t) = (beta - x - beta * x) * x\n",
    "\n",
    "    tspan = (0.0,35.0)\n",
    "    prob = ODEProblem(f,x0,tspan)\n",
    "    x = solve(prob, Tsit5(), reltol=1e-8, abstol=1e-8)   # Fifth order Tsitouras method\n",
    "    \n",
    "    plot(x,linewidth=5,title=\"Epidemic simulation using the SIS model\",\n",
    "         xaxis=\"Time (t)\",yaxis=\"% of population\",label=\"Infected\") # legend=false\n",
    "    plot!(x.t,  1 .- x,lw=3,ls=:dash,label=\"Susceptible\")\n",
    "    \n",
    "end\n",
    "\n",
    "sis(x0 = 0.001, beta = 1.0)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Using the SIRS model\n",
    "\n",
    "<a id='I4'></a>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 216,
   "metadata": {},
   "outputs": [
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    "function sirs(; beta, gamma, delta)\n",
    "    function f!(du,u,p,t)\n",
    "        du[1] = gamma * (1 - u[1] - u[2]) - beta * u[1] * u[2]\n",
    "        du[2] = beta * u[1] * u[2] - gamma * u[2]\n",
    "    end\n",
    "\n",
    "    u0 = [0.99; 0.0001]\n",
    "    tspan = (0.0,60.0)\n",
    "    prob = ODEProblem(f!,u0,tspan)\n",
    "    u = solve(prob, Tsit5(), reltol=1e-8, abstol=1e-8)   # Fifth order Tsitouras method\n",
    "    u0 = [ elem[1] for elem in u.u]\n",
    "    u1 = [ elem[2] for elem in u.u]\n",
    "\n",
    "    plot(u.t, u0,linewidth=5,title=\"Epidemic simulation using the SIRS model\",\n",
    "         xaxis=\"Time (t)\",yaxis=\"% of population)\",label=\"Recovered\") # legend=false\n",
    "    plot!(u.t, u1,lw=3,ls=:dash,label=\"Susceptible\")\n",
    "end\n",
    "\n",
    "sirs( beta = 2.0, gamma = 0.2, delta = 0.1)"
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    "# Setup colormap\n",
    "C(g::ColorGradient) = RGB[g[z] for z=linspace(0,1,30)]\n",
    "g = :amp\n",
    "cgrad(g)[0]"
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   "source": [
    "## The influence of social distancing on the total infection rate as shown by the changing the parameter beta using the SIR model\n",
    "\n",
    "<a id='I5'></a>"
   ]
  },
  {
   "cell_type": "code",
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       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:48px; text-anchor:end;\" transform=\"rotate(0, 191.754, 426.304)\" x=\"191.754\" y=\"426.304\">0.6</text>\n",
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       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:84px; text-anchor:middle;\" transform=\"rotate(0, 1284.25, 73.2)\" x=\"1284.25\" y=\"73.2\">Epidemic simulation using the SIR model</text>\n",
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       "<text style=\"fill:#000000; fill-opacity:1; font-family:Arial,Helvetica Neue,Helvetica,sans-serif; font-size:66px; text-anchor:middle;\" transform=\"rotate(0, 1284.25, 1559.48)\" x=\"1284.25\" y=\"1559.48\">Time (t)</text>\n",
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       "  595.994,1189.74 616.257,1166.2 636.897,1141.73 658.136,1116.69 680.327,1091.47 704.153,1066.42 731.807,1041.31 760.383,1021.12 786.07,1008.75 812.378,1002.02 \n",
       "  838.289,1001.16 864.749,1005.72 891.759,1015.29 919.892,1029.56 949.65,1048.27 982.24,1071.61 1022.5,1102.59 1058.47,1130.72 1093.97,1157.81 1128.83,1183.09 \n",
       "  1163.5,1206.54 1198,1227.99 1232.53,1247.49 1267.17,1265.09 1302.06,1280.9 1337.28,1295.04 1372.94,1307.64 1409.12,1318.82 1445.93,1328.7 1483.44,1337.41 \n",
       "  1521.74,1345.05 1560.93,1351.73 1601.09,1357.55 1642.31,1362.59 1684.69,1366.94 1728.33,1370.68 1773.32,1373.87 1819.79,1376.58 1867.86,1378.86 1917.65,1380.78 \n",
       "  1969.31,1382.38 2022.99,1383.71 2078.87,1384.79 2137.13,1385.68 2198,1386.39 2261.71,1386.96 2292.27,1387.18 \n",
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       "  334.478,1323.69 342.992,1310.13 351.567,1293.89 360.147,1274.69 368.693,1252.27 377.179,1226.43 385.597,1197.02 393.947,1163.98 402.242,1127.35 410.506,1087.26 \n",
       "  418.77,1043.97 427.079,997.844 435.493,949.377 444.089,899.173 452.981,847.927 462.35,796.373 472.549,745.043 484.888,692.143 496.648,652.965 508.086,626.162 \n",
       "  519.742,610.138 531.637,604.603 544.024,608.92 557.076,622.588 571.097,645.286 586.53,677.077 604.438,719.544 628.852,781.824 647.263,829.242 668.064,881.303 \n",
       "  687.961,928.547 708.676,974.468 729.414,1016.83 750.625,1056.39 772.125,1092.69 794.053,1126 816.381,1156.34 839.172,1183.89 862.439,1208.79 886.224,1231.23 \n",
       "  910.556,1251.37 935.471,1269.38 961.005,1285.44 987.197,1299.7 1014.09,1312.32 1041.72,1323.45 1070.13,1333.21 1099.38,1341.75 1129.51,1349.19 1160.59,1355.63 \n",
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       "  1587.21,1385.87 1636.62,1386.57 1688.52,1387.13 1743.16,1387.56 1800.8,1387.89 1861.78,1388.14 1926.48,1388.32 1995.34,1388.45 2068.88,1388.55 2147.73,1388.61 \n",
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       "  316.55,1310.01 322.224,1292.18 327.914,1270.66 333.586,1245.05 339.219,1214.99 344.8,1180.24 350.328,1140.62 355.808,1096.13 361.254,1046.89 366.689,993.197 \n",
       "  372.139,935.524 377.645,874.505 383.258,810.93 389.048,745.718 395.124,679.844 401.683,614.14 409.261,548.023 417.903,488.138 425.415,450.44 433.528,424.265 \n",
       "  441.695,411.664 450.429,411.076 459.757,422.135 470.093,444.836 482.013,480.237 499.529,542.237 513.162,593.936 528.65,652.953 543.982,709.762 560.214,766.961 \n",
       "  576.828,821.736 594.073,874.318 611.801,923.85 630.073,970.292 648.846,1013.44 668.139,1053.33 687.945,1090 708.277,1123.56 729.143,1154.15 750.559,1181.92 \n",
       "  772.539,1207.04 795.105,1229.69 818.278,1250.02 842.082,1268.22 866.547,1284.44 891.702,1298.86 917.58,1311.62 944.218,1322.86 971.656,1332.74 999.937,1341.37 \n",
       "  1029.11,1348.88 1059.23,1355.39 1090.35,1361.01 1122.53,1365.82 1155.84,1369.92 1190.37,1373.4 1226.19,1376.33 1263.4,1378.79 1302.1,1380.82 1342.41,1382.5 \n",
       "  1384.45,1383.88 1428.38,1384.99 1474.35,1385.88 1522.56,1386.59 1573.21,1387.14 1626.55,1387.57 1682.85,1387.9 1742.44,1388.15 1805.69,1388.33 1873.04,1388.46 \n",
       "  1945.01,1388.55 2022.21,1388.62 2105.4,1388.66 2195.48,1388.69 2292.27,1388.71 \n",
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       "  307.635,1298.24 311.948,1276.69 316.264,1250.57 320.557,1219.39 324.815,1182.76 329.032,1140.39 333.208,1092.16 337.352,1038.16 341.479,978.682 345.609,914.223 \n",
       "  349.771,845.514 354.003,773.484 358.356,699.224 362.907,623.917 367.785,548.657 373.292,473.606 381.6,385.093 387.12,343.769 393.838,311.25 400.291,296.067 \n",
       "  407.497,294.173 415.277,305.656 424.427,331.651 440.959,397.832 450.898,443.108 463.932,504.026 476.331,561.259 490.128,622.529 504.338,682.117 519.449,741.192 \n",
       "  535.152,797.85 551.549,852.039 568.531,903.128 586.114,951.029 604.259,995.594 622.966,1036.85 642.226,1074.84 662.04,1109.67 682.413,1141.47 703.353,1170.39 \n",
       "  724.872,1196.6 746.986,1220.27 769.712,1241.56 793.072,1260.64 817.091,1277.69 841.795,1292.86 867.216,1306.32 893.387,1318.2 920.345,1328.65 948.132,1337.8 \n",
       "  976.793,1345.78 1006.38,1352.71 1036.94,1358.7 1068.54,1363.85 1101.24,1368.25 1135.12,1371.98 1170.25,1375.14 1206.73,1377.79 1244.66,1380 1284.13,1381.83 \n",
       "  1325.29,1383.33 1368.26,1384.55 1413.21,1385.53 1460.3,1386.31 1509.73,1386.93 1561.74,1387.41 1616.59,1387.77 1674.57,1388.05 1736.04,1388.26 1801.41,1388.41 \n",
       "  1871.15,1388.52 1945.85,1388.59 2026.19,1388.65 2113,1388.68 2207.3,1388.7 2292.27,1388.71 \n",
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       "  302.236,1287.31 305.74,1262.31 309.24,1231.95 312.719,1195.67 316.168,1153.04 319.582,1103.82 322.967,1047.97 326.331,985.7 329.69,917.501 333.065,844.121 \n",
       "  336.485,766.567 339.988,686.055 343.631,603.939 347.503,521.563 351.785,439.73 357.271,353.127 362.128,295.242 367.252,252.801 372.533,226.839 378.246,215.481 \n",
       "  384.559,218.401 392.037,236.596 401.989,276 410.242,315.782 419.918,366.017 429.829,418.551 440.785,475.818 452.511,534.872 465.222,595.531 478.791,656.115 \n",
       "  493.207,715.774 508.377,773.517 524.257,828.783 540.793,881.133 557.954,930.336 575.714,976.264 594.057,1018.89 612.971,1058.24 632.452,1094.41 652.501,1127.51 \n",
       "  673.123,1157.67 694.325,1185.06 716.121,1209.84 738.527,1232.17 761.561,1252.22 785.246,1270.17 809.607,1286.17 834.672,1300.39 860.474,1312.96 887.048,1324.05 \n",
       "  914.433,1333.77 942.671,1342.28 971.81,1349.67 1001.9,1356.08 1033,1361.59 1065.18,1366.32 1098.5,1370.35 1133.03,1373.77 1168.87,1376.64 1206.11,1379.04 \n",
       "  1244.85,1381.04 1285.21,1382.68 1327.32,1384.02 1371.32,1385.11 1417.39,1385.97 1465.72,1386.66 1516.51,1387.2 1570,1387.62 1626.5,1387.94 1686.31,1388.17 \n",
       "  1749.82,1388.35 1817.48,1388.47 1889.82,1388.56 1967.47,1388.62 2051.19,1388.67 2141.9,1388.69 2240.76,1388.71 2292.27,1388.71 \n",
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       "  276.235,1370.14 277.161,1368.66 278.633,1366.09 280.311,1362.75 282.37,1357.99 284.659,1351.69 287.195,1343.24 289.892,1332.23 292.72,1317.95 295.625,1299.76 \n",
       "  298.575,1276.9 301.538,1248.64 304.496,1214.27 307.433,1173.21 310.344,1125.03 313.228,1069.53 316.091,1006.79 318.941,937.19 321.797,861.436 324.678,780.539 \n",
       "  327.615,695.782 330.649,608.644 333.844,520.679 337.313,433.189 341.365,345.43 345.982,267.277 350.12,217.126 354.686,181.479 359.456,162.599 364.836,158.579 \n",
       "  371.245,170.557 378.869,199.63 385.877,234.326 393.8,278.036 402.129,325.934 411.345,378.972 421.396,435.372 432.472,494.862 444.548,556.087 457.605,617.912 \n",
       "  471.543,679.034 486.285,738.52 501.758,795.655 517.913,849.985 534.708,901.213 552.115,949.18 570.111,993.822 588.684,1035.15 607.826,1073.22 627.535,1108.14 \n",
       "  647.812,1140.04 668.665,1169.07 690.104,1195.38 712.143,1219.15 734.798,1240.55 758.091,1259.73 782.044,1276.87 806.685,1292.13 832.042,1305.66 858.15,1317.62 \n",
       "  885.046,1328.14 912.769,1337.35 941.364,1345.4 970.88,1352.38 1001.37,1358.41 1032.9,1363.6 1065.53,1368.03 1099.32,1371.8 1134.38,1374.99 1170.77,1377.67 \n",
       "  1208.6,1379.9 1247.98,1381.74 1289.03,1383.26 1331.88,1384.49 1376.7,1385.48 1423.66,1386.27 1472.95,1386.9 1524.8,1387.38 1579.48,1387.76 1637.27,1388.04 \n",
       "  1698.53,1388.25 1763.66,1388.4 1833.14,1388.51 1907.54,1388.59 1987.54,1388.64 2073.97,1388.68 2167.83,1388.7 2270.37,1388.71 2292.27,1388.71 \n",
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   "source": [
    "gamma_val = 0.2\n",
    "r0_val = 0.0001\n",
    "\n",
    "function sir(;s0,  beta, gamma, r0)\n",
    "    f(r,p,t) = gamma * (1 - r - s0 * exp(-beta * r / gamma))\n",
    "\n",
    "    tspan = (0.0,60.0)\n",
    "    prob = ODEProblem(f,r0,tspan)\n",
    "    r = solve(prob, Tsit5(), reltol=1e-8, abstol=1e-8)   # Fifth order Tsitouras method\n",
    "    s = s0 * exp.( -beta .* r ./ gamma)\n",
    "    x = 1 .- s .- r\n",
    "    return(x, r.t)\n",
    "    \n",
    "end\n",
    "\n",
    "x,t = sir(s0 = 0.99, beta = 0.5, gamma = gamma_val, r0 = r0_val)\n",
    "plot(t, x,linewidth=5,title=\"Epidemic simulation using the SIR model\", linecolor=\"steelblue\",\n",
    "         xaxis=\"Time (t)\",yaxis=\"% of population that is infected\",label=\"Baseline of beta = 0.5\") # legend=false\n",
    "\n",
    "count = 0.1\n",
    "for i in 1.0: 0.5 :3.0\n",
    "    x,t = sir(s0 = 0.99, beta = i, gamma = gamma_val, r0 = r0_val)\n",
    "    #print(t)\n",
    "    a = plot!(t, x,lw=3,label=i, linecolor=cgrad(g)[count]) # legend=false\n",
    "    count += 0.1\n",
    "end\n",
    "display(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Beta is a measure of how many people a sick person can infect, with social distancing this number decreases. For e.g with beta going from 3.0 to 0.5 the total percentage of the population that is infected reduces from 80% to 20%."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Simulation using the SIR model for the Coronavirus and the Flu from estimated parameters (These numbers most certainly need to be updated!)\n",
    "\n",
    "<a id='I6'></a>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 264,
   "metadata": {},
   "outputs": [
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       "  559.369,799.564 561.482,792.98 563.594,786.443 565.706,779.955 567.819,773.517 569.931,767.13 572.043,760.796 574.156,754.516 576.268,748.29 578.38,742.12 \n",
       "  580.493,736.006 582.605,729.95 584.717,723.952 586.83,718.013 588.942,712.134 591.054,706.314 593.167,700.556 595.279,694.858 597.392,689.222 599.504,683.649 \n",
       "  601.616,678.137 603.729,672.688 605.841,667.303 607.953,661.98 610.066,656.721 612.178,651.525 614.29,646.392 616.403,641.323 618.515,636.318 620.627,631.376 \n",
       "  622.74,626.497 624.852,621.682 626.964,616.93 629.077,612.24 631.189,607.614 633.302,603.05 635.414,598.549 637.526,594.11 639.639,589.732 641.751,585.416 \n",
       "  643.863,581.161 645.976,576.966 648.088,572.832 650.2,568.758 652.313,564.743 654.425,560.788 656.537,556.89 658.65,553.052 660.762,549.27 662.874,545.546 \n",
       "  664.987,541.879 667.099,538.267 669.211,534.712 671.324,531.211 673.436,527.765 675.549,524.372 677.661,521.034 679.773,517.748 681.886,514.514 683.998,511.332 \n",
       "  686.11,508.201 688.223,505.12 690.335,502.09 692.447,499.109 694.56,496.176 696.672,493.292 698.784,490.455 700.897,487.666 703.009,484.922 705.121,482.225 \n",
       "  707.234,479.572 709.346,476.964 711.459,474.401 713.571,471.88 715.683,469.402 717.796,466.967 719.908,464.573 722.02,462.22 724.133,459.908 726.245,457.635 \n",
       "  728.357,455.402 730.47,453.207 732.582,451.051 734.694,448.932 736.807,446.851 738.919,444.806 741.031,442.796 743.144,440.822 745.256,438.883 747.369,436.979 \n",
       "  749.481,435.108 751.593,433.27 753.706,431.465 755.818,429.693 757.93,427.952 760.043,426.242 762.155,424.563 764.267,422.914 766.38,421.295 768.492,419.705 \n",
       "  770.604,418.144 772.717,416.611 774.829,415.106 776.941,413.629 779.054,412.178 781.166,410.754 783.278,409.356 785.391,407.983 787.503,406.636 789.616,405.313 \n",
       "  791.728,404.015 793.84,402.74 795.953,401.489 798.065,400.262 800.177,399.056 802.29,397.874 804.402,396.713 806.514,395.573 808.627,394.455 810.739,393.358 \n",
       "  812.851,392.281 814.964,391.224 817.076,390.187 819.188,389.169 821.301,388.17 823.413,387.19 825.526,386.229 827.638,385.285 829.75,384.359 831.863,383.451 \n",
       "  833.975,382.559 836.087,381.685 838.2,380.827 840.312,379.985 842.424,379.159 844.537,378.348 846.649,377.553 848.761,376.773 850.874,376.008 852.986,375.257 \n",
       "  855.098,374.521 857.211,373.798 859.323,373.089 861.436,372.394 863.548,371.712 865.66,371.043 867.773,370.386 869.885,369.742 871.997,369.111 874.11,368.491 \n",
       "  876.222,367.884 878.334,367.288 880.447,366.703 882.559,366.129 884.671,365.567 886.784,365.015 888.896,364.474 891.008,363.943 893.121,363.422 895.233,362.912 \n",
       "  897.345,362.411 899.458,361.92 901.57,361.438 903.683,360.966 905.795,360.502 907.907,360.048 910.02,359.602 912.132,359.165 914.244,358.736 916.357,358.316 \n",
       "  918.469,357.903 920.581,357.499 922.694,357.102 924.806,356.713 926.918,356.332 929.031,355.958 931.143,355.591 933.255,355.231 935.368,354.879 937.48,354.533 \n",
       "  939.593,354.193 941.705,353.861 943.817,353.534 945.93,353.215 948.042,352.901 950.154,352.593 952.267,352.292 954.379,351.996 956.491,351.706 958.604,351.421 \n",
       "  960.716,351.142 962.828,350.869 964.941,350.601 967.053,350.338 969.165,350.08 971.278,349.827 973.39,349.579 975.503,349.336 977.615,349.097 979.727,348.864 \n",
       "  981.84,348.634 983.952,348.41 986.064,348.189 988.177,347.973 990.289,347.761 992.401,347.553 994.514,347.35 996.626,347.15 998.738,346.954 1000.85,346.762 \n",
       "  1002.96,346.574 1005.08,346.389 1007.19,346.208 1009.3,346.031 1011.41,345.857 1013.52,345.686 1015.64,345.519 1017.75,345.355 1019.86,345.194 1021.97,345.036 \n",
       "  1024.09,344.882 1026.2,344.73 1028.31,344.581 1030.42,344.436 1032.54,344.293 1034.65,344.153 1036.76,344.015 1038.87,343.881 1040.99,343.749 1043.1,343.619 \n",
       "  1045.21,343.492 1047.32,343.368 1049.43,343.246 1051.55,343.126 1053.66,343.009 1055.77,342.894 1057.88,342.781 1060,342.671 1062.11,342.562 1064.22,342.456 \n",
       "  1066.33,342.352 1068.45,342.25 1070.56,342.15 1072.67,342.052 1074.78,341.955 1076.9,341.861 1079.01,341.768 1081.12,341.678 1083.23,341.589 1085.34,341.502 \n",
       "  1087.46,341.416 1089.57,341.332 1091.68,341.25 1093.79,341.17 1095.91,341.091 1098.02,341.013 1100.13,340.937 1102.24,340.863 1104.36,340.79 1106.47,340.719 \n",
       "  1108.58,340.648 1110.69,340.58 1112.81,340.512 1114.92,340.446 1117.03,340.381 1119.14,340.318 1121.25,340.256 1123.37,340.195 1125.48,340.135 1127.59,340.076 \n",
       "  1129.7,340.018 1131.82,339.962 1133.93,339.907 1136.04,339.853 1138.15,339.799 1140.27,339.747 1142.38,339.696 1144.49,339.646 1146.6,339.597 1148.72,339.549 \n",
       "  1150.83,339.502 1152.94,339.456 1155.05,339.41 1157.16,339.366 1159.28,339.322 1161.39,339.279 1163.5,339.237 1165.61,339.196 1167.73,339.156 1169.84,339.117 \n",
       "  1171.95,339.078 1174.06,339.04 1176.18,339.003 1178.29,338.966 1180.4,338.931 1182.51,338.896 1184.63,338.861 1186.74,338.828 1188.85,338.795 1190.96,338.762 \n",
       "  1193.07,338.73 1195.19,338.699 1197.3,338.669 1199.41,338.639 1201.52,338.61 1203.64,338.581 1205.75,338.553 1207.86,338.525 1209.97,338.498 1212.09,338.471 \n",
       "  1214.2,338.445 1216.31,338.42 1218.42,338.395 1220.54,338.37 1222.65,338.346 1224.76,338.323 1226.87,338.3 1228.98,338.277 1231.1,338.255 1233.21,338.233 \n",
       "  1235.32,338.212 1237.43,338.191 1239.55,338.17 1241.66,338.15 1243.77,338.13 1245.88,338.111 1248,338.092 1250.11,338.074 1252.22,338.055 1254.33,338.037 \n",
       "  1256.45,338.02 1258.56,338.003 1260.67,337.986 1262.78,337.969 1264.89,337.953 1267.01,337.937 1269.12,337.922 1271.23,337.907 1273.34,337.892 1275.46,337.877 \n",
       "  1277.57,337.863 1279.68,337.849 1281.79,337.835 1283.91,337.821 1286.02,337.808 1288.13,337.795 1290.24,337.782 1292.36,337.77 1294.47,337.758 1296.58,337.746 \n",
       "  1298.69,337.734 1300.8,337.722 1302.92,337.711 1305.03,337.7 1307.14,337.689 1309.25,337.678 1311.37,337.668 1313.48,337.658 1315.59,337.648 1317.7,337.638 \n",
       "  1319.82,337.628 1321.93,337.619 1324.04,337.609 1326.15,337.6 1328.27,337.591 1330.38,337.583 1332.49,337.574 1334.6,337.566 1336.71,337.558 1338.83,337.549 \n",
       "  1340.94,337.542 1343.05,337.534 1345.16,337.526 1347.28,337.519 1349.39,337.511 1351.5,337.504 1353.61,337.497 1355.73,337.49 1357.84,337.484 1359.95,337.477 \n",
       "  1362.06,337.471 1364.18,337.464 1366.29,337.458 1368.4,337.452 1370.51,337.446 1372.62,337.44 1374.74,337.434 1376.85,337.429 1378.96,337.423 1381.07,337.418 \n",
       "  1383.19,337.412 1385.3,337.407 1387.41,337.402 1389.52,337.397 1391.64,337.392 1393.75,337.387 1395.86,337.383 1397.97,337.378 1400.09,337.373 1402.2,337.369 \n",
       "  1404.31,337.365 1406.42,337.36 1408.53,337.356 1410.65,337.352 1412.76,337.348 1414.87,337.344 1416.98,337.34 1419.1,337.336 1421.21,337.333 1423.32,337.329 \n",
       "  1425.43,337.325 1427.55,337.322 1429.66,337.318 1431.77,337.315 1433.88,337.312 1436,337.309 1438.11,337.305 1440.22,337.302 1442.33,337.299 1444.44,337.296 \n",
       "  1446.56,337.293 1448.67,337.29 1450.78,337.288 1452.89,337.285 1455.01,337.282 1457.12,337.279 1459.23,337.277 1461.34,337.274 1463.46,337.272 1465.57,337.269 \n",
       "  1467.68,337.267 1469.79,337.265 1471.91,337.262 1474.02,337.26 1476.13,337.258 1478.24,337.256 1480.35,337.254 1482.47,337.251 1484.58,337.249 1486.69,337.247 \n",
       "  1488.8,337.245 1490.92,337.243 1493.03,337.242 1495.14,337.24 1497.25,337.238 1499.37,337.236 1501.48,337.234 1503.59,337.233 1505.7,337.231 1507.82,337.229 \n",
       "  1509.93,337.228 1512.04,337.226 1514.15,337.225 1516.26,337.223 1518.38,337.222 1520.49,337.22 1522.6,337.219 1524.71,337.217 1526.83,337.216 1528.94,337.215 \n",
       "  1531.05,337.213 1533.16,337.212 1535.28,337.211 1537.39,337.209 1539.5,337.208 1541.61,337.207 1543.73,337.206 1545.84,337.205 1547.95,337.203 1550.06,337.202 \n",
       "  1552.17,337.201 1554.29,337.2 1556.4,337.199 1558.51,337.198 1560.62,337.197 1562.74,337.196 1564.85,337.195 1566.96,337.194 1569.07,337.193 1571.19,337.192 \n",
       "  1573.3,337.192 1575.41,337.191 1577.52,337.19 1579.64,337.189 1581.75,337.188 1583.86,337.187 1585.97,337.187 1588.08,337.186 1590.2,337.185 1592.31,337.184 \n",
       "  1594.42,337.184 1596.53,337.183 1598.65,337.182 1600.76,337.181 1602.87,337.181 1604.98,337.18 1607.1,337.179 1609.21,337.179 1611.32,337.178 1613.43,337.178 \n",
       "  1615.55,337.177 1617.66,337.176 1619.77,337.176 1621.88,337.175 1623.99,337.175 1626.11,337.174 1628.22,337.174 1630.33,337.173 1632.44,337.173 1634.56,337.172 \n",
       "  1636.67,337.172 1638.78,337.171 1640.89,337.171 1643.01,337.17 1645.12,337.17 1647.23,337.169 1649.34,337.169 1651.46,337.168 1653.57,337.168 1655.68,337.168 \n",
       "  1657.79,337.167 1659.9,337.167 1662.02,337.166 1664.13,337.166 1666.24,337.166 1668.35,337.165 1670.47,337.165 1672.58,337.165 1674.69,337.164 1676.8,337.164 \n",
       "  1678.92,337.164 1681.03,337.163 1683.14,337.163 1685.25,337.163 1687.37,337.162 1689.48,337.162 1691.59,337.162 1693.7,337.161 1695.81,337.161 1697.93,337.161 \n",
       "  1700.04,337.161 1702.15,337.16 1704.26,337.16 1706.38,337.16 1708.49,337.16 1710.6,337.159 1712.71,337.159 1714.83,337.159 1716.94,337.159 1719.05,337.158 \n",
       "  1721.16,337.158 1723.28,337.158 1725.39,337.158 1727.5,337.158 1729.61,337.157 1731.72,337.157 1733.84,337.157 1735.95,337.157 1738.06,337.157 1740.17,337.156 \n",
       "  1742.29,337.156 1744.4,337.156 1746.51,337.156 1748.62,337.156 1750.74,337.156 1752.85,337.155 1754.96,337.155 1757.07,337.155 1759.19,337.155 1761.3,337.155 \n",
       "  1763.41,337.155 1765.52,337.154 1767.63,337.154 1769.75,337.154 1771.86,337.154 1773.97,337.154 1776.08,337.154 1778.2,337.154 1780.31,337.154 1782.42,337.153 \n",
       "  1784.53,337.153 1786.65,337.153 1788.76,337.153 1790.87,337.153 1792.98,337.153 1795.1,337.153 1797.21,337.153 1799.32,337.152 1801.43,337.152 1803.54,337.152 \n",
       "  1805.66,337.152 1807.77,337.152 1809.88,337.152 1811.99,337.152 1814.11,337.152 1816.22,337.152 1818.33,337.152 1820.44,337.152 1822.56,337.151 1824.67,337.151 \n",
       "  1826.78,337.151 1828.89,337.151 1831.01,337.151 1833.12,337.151 1835.23,337.151 1837.34,337.151 1839.45,337.151 1841.57,337.151 1843.68,337.151 1845.79,337.151 \n",
       "  1847.9,337.151 1850.02,337.15 1852.13,337.15 1854.24,337.15 1856.35,337.15 1858.47,337.15 1860.58,337.15 1862.69,337.15 1864.8,337.15 1866.91,337.15 \n",
       "  1869.03,337.15 1871.14,337.15 1873.25,337.15 1875.36,337.15 1877.48,337.15 1879.59,337.15 1881.7,337.15 1883.81,337.15 1885.93,337.15 1888.04,337.149 \n",
       "  1890.15,337.149 1892.26,337.149 1894.38,337.149 1896.49,337.149 1898.6,337.149 1900.71,337.149 1902.82,337.149 1904.94,337.149 1907.05,337.149 1909.16,337.149 \n",
       "  1911.27,337.149 1913.39,337.149 1915.5,337.149 1917.61,337.149 1919.72,337.149 1921.84,337.149 1923.95,337.149 1926.06,337.149 1928.17,337.149 1930.29,337.149 \n",
       "  1932.4,337.149 1934.51,337.149 1936.62,337.149 1938.73,337.149 1940.85,337.149 1942.96,337.149 1945.07,337.149 1947.18,337.148 1949.3,337.148 1951.41,337.148 \n",
       "  1953.52,337.148 1955.63,337.148 1957.75,337.148 1959.86,337.148 1961.97,337.148 1964.08,337.148 1966.2,337.148 1968.31,337.148 1970.42,337.148 1972.53,337.148 \n",
       "  1974.64,337.148 1976.76,337.148 1978.87,337.148 1980.98,337.148 1983.09,337.148 1985.21,337.148 1987.32,337.148 1989.43,337.148 1991.54,337.148 1993.66,337.148 \n",
       "  1995.77,337.148 1997.88,337.148 1999.99,337.148 2002.11,337.148 2004.22,337.148 2006.33,337.148 2008.44,337.148 2010.55,337.148 2012.67,337.148 2014.78,337.148 \n",
       "  2016.89,337.148 2019,337.148 2021.12,337.148 2023.23,337.148 2025.34,337.148 2027.45,337.148 2029.57,337.148 2031.68,337.148 2033.79,337.148 2035.9,337.148 \n",
       "  2038.02,337.148 2040.13,337.148 2042.24,337.148 2044.35,337.148 2046.46,337.148 2048.58,337.148 2050.69,337.148 2052.8,337.148 2054.91,337.148 2057.03,337.148 \n",
       "  2059.14,337.148 2061.25,337.148 2063.36,337.148 2065.48,337.148 2067.59,337.148 2069.7,337.148 2071.81,337.148 2073.93,337.148 2076.04,337.148 2078.15,337.148 \n",
       "  2080.26,337.148 2082.37,337.148 2084.49,337.148 2086.6,337.148 2088.71,337.148 2090.82,337.148 2092.94,337.148 2095.05,337.147 2097.16,337.147 2099.27,337.147 \n",
       "  2101.39,337.147 2103.5,337.147 2105.61,337.147 2107.72,337.147 2109.84,337.147 2111.95,337.147 2114.06,337.147 2116.17,337.147 2118.28,337.147 2120.4,337.147 \n",
       "  2122.51,337.147 2124.62,337.147 2126.73,337.147 2128.85,337.147 2130.96,337.147 2133.07,337.147 2135.18,337.147 2137.3,337.147 2139.41,337.147 2141.52,337.147 \n",
       "  2143.63,337.147 2145.75,337.147 2147.86,337.147 2149.97,337.147 2152.08,337.147 2154.19,337.147 2156.31,337.147 2158.42,337.147 2160.53,337.147 2162.64,337.147 \n",
       "  2164.76,337.147 2166.87,337.147 2168.98,337.147 2171.09,337.147 2173.21,337.147 2175.32,337.147 2177.43,337.147 2179.54,337.147 2181.66,337.147 2183.77,337.147 \n",
       "  2185.88,337.147 2187.99,337.147 2190.1,337.147 2192.22,337.147 2194.33,337.147 2196.44,337.147 2198.55,337.147 2200.67,337.147 2202.78,337.147 2204.89,337.147 \n",
       "  2207,337.147 2209.12,337.147 2211.23,337.147 2213.34,337.147 2215.45,337.147 2217.57,337.147 2219.68,337.147 2221.79,337.147 2223.9,337.147 2226.01,337.147 \n",
       "  2228.13,337.147 2230.24,337.147 2232.35,337.147 2234.46,337.147 2236.58,337.147 2238.69,337.147 2240.8,337.147 2242.91,337.147 2245.03,337.147 2247.14,337.147 \n",
       "  2249.25,337.147 2251.36,337.147 2253.48,337.147 2255.59,337.147 2257.7,337.147 2259.81,337.147 2261.92,337.147 2264.04,337.147 2266.15,337.147 2268.26,337.147 \n",
       "  2270.37,337.147 2272.49,337.147 2274.6,337.147 2276.71,337.147 2278.82,337.147 2280.94,337.147 2283.05,337.147 2285.16,337.147 2287.27,337.147 2289.39,337.147 \n",
       "  2291.5,337.147 2293.61,337.147 2295.72,337.147 2297.83,337.147 2299.95,337.147 2302.06,337.147 2304.17,337.147 2306.28,337.147 2308.4,337.147 2310.51,337.147 \n",
       "  2312.62,337.147 2314.73,337.147 2316.85,337.147 2318.96,337.147 2321.07,337.147 2323.18,337.147 2325.3,337.147 2327.41,337.147 2329.52,337.147 2331.63,337.147 \n",
       "  2333.74,337.147 2335.86,337.147 2337.97,337.147 2340.08,337.147 2342.19,337.147 2344.31,337.147 2346.42,337.147 2348.53,337.147 2350.64,337.147 2352.76,337.147 \n",
       "  \n",
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       "  304.444,199.752 314.156,210.533 324.052,223.186 334.069,237.897 344.156,254.843 354.285,274.193 364.438,296.1 374.618,320.704 384.838,348.13 395.128,378.489 \n",
       "  405.534,411.889 416.126,448.452 427.007,488.351 438.342,531.918 450.444,579.953 464.23,635.457 479.528,696.463 492.496,746.473 506.011,795.954 519.115,840.698 \n",
       "  532.503,882.698 546.068,921.225 560.147,956.935 574.959,989.99 591.114,1021.16 611.02,1053.36 628.628,1076.91 645.994,1096.32 662.958,1112.19 679.795,1125.41 \n",
       "  696.504,1136.45 713.195,1145.74 729.911,1153.59 746.721,1160.24 763.675,1165.91 780.825,1170.74 798.219,1174.87 815.905,1178.41 833.927,1181.43 852.33,1184.02 \n",
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     "execution_count": 264,
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   ],
   "source": [
    "# Estimates for gamma and beta for COVID obtained from https://arxiv.org/pdf/2002.06563.pdf\n",
    "# Epidemic analysis of COVID-19 in China by dynamical modeling\n",
    "# Liangrong Peng, Wuyue Yang, Dongyan Zhang, Changjing Zhuge, Liu Hong\n",
    "# medRxiv 2020.02.16.20023465; doi: https://doi.org/10.1101/2020.02.16.20023465\n",
    "# Estimate for gamma for flu was not available\n",
    "\n",
    "beta_val(;R0, gamma) = R0 * gamma\n",
    "cv = beta_val(R0 = 2.2, gamma = 0.5) \n",
    "fl = beta_val(R0 = 1.3, gamma = 0.5)\n",
    "\n",
    "sir(s0 = 0.99, beta = cv, gamma = 0.5, r0 = 0.0001, title=\"Coronavirus - SIR simulation with estimated parameters\")"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 263,
   "metadata": {},
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       "  559.369,1284.99 561.482,1283.93 563.594,1282.86 565.706,1281.79 567.819,1280.72 569.931,1279.64 572.043,1278.55 574.156,1277.47 576.268,1276.37 578.38,1275.27 \n",
       "  580.493,1274.17 582.605,1273.06 584.717,1271.95 586.83,1270.83 588.942,1269.71 591.054,1268.58 593.167,1267.45 595.279,1266.31 597.392,1265.17 599.504,1264.03 \n",
       "  601.616,1262.88 603.729,1261.72 605.841,1260.56 607.953,1259.4 610.066,1258.23 612.178,1257.06 614.29,1255.89 616.403,1254.71 618.515,1253.52 620.627,1252.34 \n",
       "  622.74,1251.14 624.852,1249.95 626.964,1248.75 629.077,1247.54 631.189,1246.33 633.302,1245.12 635.414,1243.9 637.526,1242.68 639.639,1241.46 641.751,1240.23 \n",
       "  643.863,1239 645.976,1237.76 648.088,1236.52 650.2,1235.28 652.313,1234.04 654.425,1232.79 656.537,1231.53 658.65,1230.28 660.762,1229.01 662.874,1227.75 \n",
       "  664.987,1226.48 667.099,1225.21 669.211,1223.94 671.324,1222.66 673.436,1221.38 675.549,1220.1 677.661,1218.82 679.773,1217.53 681.886,1216.23 683.998,1214.94 \n",
       "  686.11,1213.64 688.223,1212.34 690.335,1211.04 692.447,1209.73 694.56,1208.42 696.672,1207.11 698.784,1205.8 700.897,1204.48 703.009,1203.17 705.121,1201.84 \n",
       "  707.234,1200.52 709.346,1199.2 711.459,1197.87 713.571,1196.54 715.683,1195.21 717.796,1193.87 719.908,1192.54 722.02,1191.2 724.133,1189.86 726.245,1188.52 \n",
       "  728.357,1187.17 730.47,1185.83 732.582,1184.48 734.694,1183.13 736.807,1181.78 738.919,1180.43 741.031,1179.08 743.144,1177.73 745.256,1176.37 747.369,1175.02 \n",
       "  749.481,1173.66 751.593,1172.3 753.706,1170.94 755.818,1169.58 757.93,1168.22 760.043,1166.85 762.155,1165.49 764.267,1164.13 766.38,1162.76 768.492,1161.39 \n",
       "  770.604,1160.03 772.717,1158.66 774.829,1157.29 776.941,1155.93 779.054,1154.56 781.166,1153.19 783.278,1151.82 785.391,1150.45 787.503,1149.08 789.616,1147.71 \n",
       "  791.728,1146.34 793.84,1144.98 795.953,1143.61 798.065,1142.24 800.177,1140.87 802.29,1139.5 804.402,1138.13 806.514,1136.77 808.627,1135.4 810.739,1134.03 \n",
       "  812.851,1132.67 814.964,1131.3 817.076,1129.94 819.188,1128.57 821.301,1127.21 823.413,1125.85 825.526,1124.49 827.638,1123.13 829.75,1121.77 831.863,1120.41 \n",
       "  833.975,1119.05 836.087,1117.7 838.2,1116.34 840.312,1114.99 842.424,1113.64 844.537,1112.29 846.649,1110.94 848.761,1109.59 850.874,1108.24 852.986,1106.9 \n",
       "  855.098,1105.56 857.211,1104.22 859.323,1102.88 861.436,1101.54 863.548,1100.21 865.66,1098.87 867.773,1097.54 869.885,1096.21 871.997,1094.88 874.11,1093.56 \n",
       "  876.222,1092.24 878.334,1090.92 880.447,1089.6 882.559,1088.28 884.671,1086.97 886.784,1085.66 888.896,1084.35 891.008,1083.04 893.121,1081.74 895.233,1080.44 \n",
       "  897.345,1079.14 899.458,1077.84 901.57,1076.55 903.683,1075.26 905.795,1073.97 907.907,1072.69 910.02,1071.4 912.132,1070.13 914.244,1068.85 916.357,1067.58 \n",
       "  918.469,1066.31 920.581,1065.04 922.694,1063.78 924.806,1062.52 926.918,1061.26 929.031,1060.01 931.143,1058.76 933.255,1057.51 935.368,1056.27 937.48,1055.03 \n",
       "  939.593,1053.79 941.705,1052.55 943.817,1051.32 945.93,1050.1 948.042,1048.88 950.154,1047.66 952.267,1046.44 954.379,1045.23 956.491,1044.02 958.604,1042.82 \n",
       "  960.716,1041.62 962.828,1040.42 964.941,1039.22 967.053,1038.04 969.165,1036.85 971.278,1035.67 973.39,1034.49 975.503,1033.32 977.615,1032.15 979.727,1030.98 \n",
       "  981.84,1029.82 983.952,1028.66 986.064,1027.51 988.177,1026.36 990.289,1025.21 992.401,1024.07 994.514,1022.93 996.626,1021.8 998.738,1020.67 1000.85,1019.55 \n",
       "  1002.96,1018.43 1005.08,1017.31 1007.19,1016.2 1009.3,1015.09 1011.41,1013.99 1013.52,1012.89 1015.64,1011.79 1017.75,1010.7 1019.86,1009.62 1021.97,1008.54 \n",
       "  1024.09,1007.46 1026.2,1006.39 1028.31,1005.32 1030.42,1004.25 1032.54,1003.19 1034.65,1002.14 1036.76,1001.09 1038.87,1000.04 1040.99,999 1043.1,997.962 \n",
       "  1045.21,996.93 1047.32,995.902 1049.43,994.878 1051.55,993.859 1053.66,992.844 1055.77,991.834 1057.88,990.829 1060,989.828 1062.11,988.831 1064.22,987.839 \n",
       "  1066.33,986.852 1068.45,985.869 1070.56,984.891 1072.67,983.918 1074.78,982.949 1076.9,981.985 1079.01,981.025 1081.12,980.07 1083.23,979.119 1085.34,978.173 \n",
       "  1087.46,977.232 1089.57,976.295 1091.68,975.363 1093.79,974.435 1095.91,973.513 1098.02,972.594 1100.13,971.681 1102.24,970.771 1104.36,969.867 1106.47,968.967 \n",
       "  1108.58,968.072 1110.69,967.181 1112.81,966.295 1114.92,965.413 1117.03,964.536 1119.14,963.664 1121.25,962.796 1123.37,961.933 1125.48,961.074 1127.59,960.22 \n",
       "  1129.7,959.37 1131.82,958.525 1133.93,957.685 1136.04,956.849 1138.15,956.017 1140.27,955.191 1142.38,954.368 1144.49,953.55 1146.6,952.737 1148.72,951.928 \n",
       "  1150.83,951.124 1152.94,950.324 1155.05,949.528 1157.16,948.737 1159.28,947.951 1161.39,947.169 1163.5,946.391 1165.61,945.618 1167.73,944.849 1169.84,944.084 \n",
       "  1171.95,943.324 1174.06,942.569 1176.18,941.817 1178.29,941.07 1180.4,940.328 1182.51,939.589 1184.63,938.855 1186.74,938.126 1188.85,937.4 1190.96,936.679 \n",
       "  1193.07,935.963 1195.19,935.25 1197.3,934.542 1199.41,933.838 1201.52,933.138 1203.64,932.442 1205.75,931.751 1207.86,931.064 1209.97,930.381 1212.09,929.702 \n",
       "  1214.2,929.027 1216.31,928.356 1218.42,927.69 1220.54,927.027 1222.65,926.369 1224.76,925.715 1226.87,925.065 1228.98,924.418 1231.1,923.776 1233.21,923.138 \n",
       "  1235.32,922.504 1237.43,921.874 1239.55,921.248 1241.66,920.625 1243.77,920.007 1245.88,919.393 1248,918.782 1250.11,918.176 1252.22,917.573 1254.33,916.974 \n",
       "  1256.45,916.379 1258.56,915.788 1260.67,915.201 1262.78,914.617 1264.89,914.037 1267.01,913.461 1269.12,912.889 1271.23,912.321 1273.34,911.756 1275.46,911.195 \n",
       "  1277.57,910.637 1279.68,910.083 1281.79,909.533 1283.91,908.987 1286.02,908.444 1288.13,907.905 1290.24,907.369 1292.36,906.837 1294.47,906.308 1296.58,905.783 \n",
       "  1298.69,905.261 1300.8,904.743 1302.92,904.229 1305.03,903.718 1307.14,903.21 1309.25,902.706 1311.37,902.205 1313.48,901.707 1315.59,901.213 1317.7,900.722 \n",
       "  1319.82,900.235 1321.93,899.751 1324.04,899.27 1326.15,898.793 1328.27,898.319 1330.38,897.848 1332.49,897.38 1334.6,896.916 1336.71,896.454 1338.83,895.996 \n",
       "  1340.94,895.541 1343.05,895.09 1345.16,894.641 1347.28,894.195 1349.39,893.753 1351.5,893.314 1353.61,892.878 1355.73,892.444 1357.84,892.014 1359.95,891.587 \n",
       "  1362.06,891.163 1364.18,890.742 1366.29,890.324 1368.4,889.909 1370.51,889.496 1372.62,889.087 1374.74,888.681 1376.85,888.277 1378.96,887.876 1381.07,887.478 \n",
       "  1383.19,887.083 1385.3,886.691 1387.41,886.302 1389.52,885.915 1391.64,885.531 1393.75,885.15 1395.86,884.772 1397.97,884.396 1400.09,884.024 1402.2,883.653 \n",
       "  1404.31,883.286 1406.42,882.921 1408.53,882.559 1410.65,882.199 1412.76,881.842 1414.87,881.488 1416.98,881.136 1419.1,880.787 1421.21,880.44 1423.32,880.096 \n",
       "  1425.43,879.754 1427.55,879.415 1429.66,879.078 1431.77,878.744 1433.88,878.412 1436,878.083 1438.11,877.756 1440.22,877.431 1442.33,877.109 1444.44,876.789 \n",
       "  1446.56,876.472 1448.67,876.157 1450.78,875.844 1452.89,875.534 1455.01,875.225 1457.12,874.92 1459.23,874.616 1461.34,874.315 1463.46,874.016 1465.57,873.719 \n",
       "  1467.68,873.424 1469.79,873.132 1471.91,872.842 1474.02,872.553 1476.13,872.268 1478.24,871.984 1480.35,871.702 1482.47,871.423 1484.58,871.145 1486.69,870.87 \n",
       "  1488.8,870.597 1490.92,870.325 1493.03,870.056 1495.14,869.789 1497.25,869.524 1499.37,869.261 1501.48,869 1503.59,868.74 1505.7,868.483 1507.82,868.228 \n",
       "  1509.93,867.975 1512.04,867.723 1514.15,867.474 1516.26,867.226 1518.38,866.98 1520.49,866.736 1522.6,866.494 1524.71,866.254 1526.83,866.016 1528.94,865.779 \n",
       "  1531.05,865.545 1533.16,865.312 1535.28,865.081 1537.39,864.851 1539.5,864.624 1541.61,864.398 1543.73,864.174 1545.84,863.951 1547.95,863.731 1550.06,863.512 \n",
       "  1552.17,863.294 1554.29,863.079 1556.4,862.865 1558.51,862.653 1560.62,862.442 1562.74,862.233 1564.85,862.025 1566.96,861.82 1569.07,861.615 1571.19,861.413 \n",
       "  1573.3,861.211 1575.41,861.012 1577.52,860.814 1579.64,860.617 1581.75,860.422 1583.86,860.229 1585.97,860.037 1588.08,859.847 1590.2,859.658 1592.31,859.47 \n",
       "  1594.42,859.284 1596.53,859.1 1598.65,858.917 1600.76,858.735 1602.87,858.555 1604.98,858.376 1607.1,858.198 1609.21,858.022 1611.32,857.847 1613.43,857.674 \n",
       "  1615.55,857.502 1617.66,857.331 1619.77,857.162 1621.88,856.994 1623.99,856.827 1626.11,856.662 1628.22,856.498 1630.33,856.335 1632.44,856.174 1634.56,856.013 \n",
       "  1636.67,855.854 1638.78,855.697 1640.89,855.54 1643.01,855.385 1645.12,855.231 1647.23,855.078 1649.34,854.926 1651.46,854.776 1653.57,854.627 1655.68,854.479 \n",
       "  1657.79,854.332 1659.9,854.186 1662.02,854.042 1664.13,853.898 1666.24,853.756 1668.35,853.615 1670.47,853.475 1672.58,853.336 1674.69,853.198 1676.8,853.061 \n",
       "  1678.92,852.926 1681.03,852.791 1683.14,852.657 1685.25,852.525 1687.37,852.394 1689.48,852.263 1691.59,852.134 1693.7,852.006 1695.81,851.878 1697.93,851.752 \n",
       "  1700.04,851.627 1702.15,851.503 1704.26,851.379 1706.38,851.257 1708.49,851.136 1710.6,851.015 1712.71,850.896 1714.83,850.778 1716.94,850.66 1719.05,850.544 \n",
       "  1721.16,850.428 1723.28,850.313 1725.39,850.199 1727.5,850.087 1729.61,849.975 1731.72,849.864 1733.84,849.753 1735.95,849.644 1738.06,849.536 1740.17,849.428 \n",
       "  1742.29,849.321 1744.4,849.216 1746.51,849.111 1748.62,849.007 1750.74,848.903 1752.85,848.801 1754.96,848.699 1757.07,848.598 1759.19,848.498 1761.3,848.399 \n",
       "  1763.41,848.301 1765.52,848.203 1767.63,848.106 1769.75,848.01 1771.86,847.915 1773.97,847.82 1776.08,847.726 1778.2,847.633 1780.31,847.541 1782.42,847.45 \n",
       "  1784.53,847.359 1786.65,847.269 1788.76,847.179 1790.87,847.091 1792.98,847.003 1795.1,846.916 1797.21,846.829 1799.32,846.744 1801.43,846.658 1803.54,846.574 \n",
       "  1805.66,846.49 1807.77,846.407 1809.88,846.325 1811.99,846.243 1814.11,846.162 1816.22,846.082 1818.33,846.002 1820.44,845.923 1822.56,845.845 1824.67,845.767 \n",
       "  1826.78,845.69 1828.89,845.613 1831.01,845.537 1833.12,845.462 1835.23,845.387 1837.34,845.313 1839.45,845.239 1841.57,845.166 1843.68,845.094 1845.79,845.022 \n",
       "  1847.9,844.951 1850.02,844.881 1852.13,844.811 1854.24,844.741 1856.35,844.672 1858.47,844.604 1860.58,844.536 1862.69,844.469 1864.8,844.402 1866.91,844.336 \n",
       "  1869.03,844.271 1871.14,844.206 1873.25,844.141 1875.36,844.077 1877.48,844.014 1879.59,843.951 1881.7,843.888 1883.81,843.826 1885.93,843.765 1888.04,843.704 \n",
       "  1890.15,843.644 1892.26,843.584 1894.38,843.524 1896.49,843.465 1898.6,843.407 1900.71,843.349 1902.82,843.291 1904.94,843.234 1907.05,843.178 1909.16,843.121 \n",
       "  1911.27,843.066 1913.39,843.01 1915.5,842.956 1917.61,842.901 1919.72,842.847 1921.84,842.794 1923.95,842.741 1926.06,842.688 1928.17,842.636 1930.29,842.585 \n",
       "  1932.4,842.533 1934.51,842.482 1936.62,842.432 1938.73,842.382 1940.85,842.332 1942.96,842.283 1945.07,842.234 1947.18,842.186 1949.3,842.138 1951.41,842.09 \n",
       "  1953.52,842.043 1955.63,841.996 1957.75,841.949 1959.86,841.903 1961.97,841.858 1964.08,841.812 1966.2,841.767 1968.31,841.723 1970.42,841.678 1972.53,841.634 \n",
       "  1974.64,841.591 1976.76,841.548 1978.87,841.505 1980.98,841.462 1983.09,841.42 1985.21,841.379 1987.32,841.337 1989.43,841.296 1991.54,841.255 1993.66,841.215 \n",
       "  1995.77,841.175 1997.88,841.135 1999.99,841.096 2002.11,841.056 2004.22,841.018 2006.33,840.979 2008.44,840.941 2010.55,840.903 2012.67,840.866 2014.78,840.828 \n",
       "  2016.89,840.792 2019,840.755 2021.12,840.719 2023.23,840.683 2025.34,840.647 2027.45,840.611 2029.57,840.576 2031.68,840.541 2033.79,840.507 2035.9,840.473 \n",
       "  2038.02,840.439 2040.13,840.405 2042.24,840.371 2044.35,840.338 2046.46,840.305 2048.58,840.273 2050.69,840.24 2052.8,840.208 2054.91,840.177 2057.03,840.145 \n",
       "  2059.14,840.114 2061.25,840.083 2063.36,840.052 2065.48,840.021 2067.59,839.991 2069.7,839.961 2071.81,839.931 2073.93,839.902 2076.04,839.872 2078.15,839.843 \n",
       "  2080.26,839.814 2082.37,839.786 2084.49,839.758 2086.6,839.729 2088.71,839.702 2090.82,839.674 2092.94,839.646 2095.05,839.619 2097.16,839.592 2099.27,839.566 \n",
       "  2101.39,839.539 2103.5,839.513 2105.61,839.487 2107.72,839.461 2109.84,839.435 2111.95,839.41 2114.06,839.384 2116.17,839.359 2118.28,839.334 2120.4,839.31 \n",
       "  2122.51,839.285 2124.62,839.261 2126.73,839.237 2128.85,839.213 2130.96,839.19 2133.07,839.166 2135.18,839.143 2137.3,839.12 2139.41,839.097 2141.52,839.074 \n",
       "  2143.63,839.052 2145.75,839.03 2147.86,839.008 2149.97,838.986 2152.08,838.964 2154.19,838.942 2156.31,838.921 2158.42,838.9 2160.53,838.879 2162.64,838.858 \n",
       "  2164.76,838.837 2166.87,838.817 2168.98,838.796 2171.09,838.776 2173.21,838.756 2175.32,838.736 2177.43,838.716 2179.54,838.697 2181.66,838.678 2183.77,838.658 \n",
       "  2185.88,838.639 2187.99,838.62 2190.1,838.602 2192.22,838.583 2194.33,838.565 2196.44,838.546 2198.55,838.528 2200.67,838.51 2202.78,838.492 2204.89,838.475 \n",
       "  2207,838.457 2209.12,838.44 2211.23,838.423 2213.34,838.406 2215.45,838.389 2217.57,838.372 2219.68,838.355 2221.79,838.339 2223.9,838.322 2226.01,838.306 \n",
       "  2228.13,838.29 2230.24,838.274 2232.35,838.258 2234.46,838.242 2236.58,838.226 2238.69,838.211 2240.8,838.196 2242.91,838.18 2245.03,838.165 2247.14,838.15 \n",
       "  2249.25,838.135 2251.36,838.121 2253.48,838.106 2255.59,838.092 2257.7,838.077 2259.81,838.063 2261.92,838.049 2264.04,838.035 2266.15,838.021 2268.26,838.007 \n",
       "  2270.37,837.994 2272.49,837.98 2274.6,837.967 2276.71,837.953 2278.82,837.94 2280.94,837.927 2283.05,837.914 2285.16,837.901 2287.27,837.888 2289.39,837.876 \n",
       "  2291.5,837.863 2293.61,837.85 2295.72,837.838 2297.83,837.826 2299.95,837.814 2302.06,837.802 2304.17,837.79 2306.28,837.778 2308.4,837.766 2310.51,837.754 \n",
       "  2312.62,837.743 2314.73,837.731 2316.85,837.72 2318.96,837.709 2321.07,837.697 2323.18,837.686 2325.3,837.675 2327.41,837.664 2329.52,837.654 2331.63,837.643 \n",
       "  2333.74,837.632 2335.86,837.622 2337.97,837.611 2340.08,837.601 2342.19,837.59 2344.31,837.58 2346.42,837.57 2348.53,837.56 2350.64,837.55 2352.76,837.54 \n",
       "  \n",
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   "source": [
    "sir(s0 = 0.99, beta = fl, gamma = 0.5, r0 = 0.0001, title=\"Flu - SIR simulation with estimated parameters\")"
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   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Size of the outbreak (S) as a percentage of the population vs. transmission probability of the disease(phi)\n",
    "\n",
    "<a id='I7'></a>"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### The size of the outbreak (S), not to be confused with 's' above, as a percentage of the population vs. transmission probability (function of the transmission rate) of the disease(phi). The mean degree (c) of the network is the average number of connections a person has. This is modelled as a SIR assuming a Poisson random network model and 'c' corresponds to the mean degree of the network. "
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 211,
   "metadata": {},
   "outputs": [
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     "metadata": {},
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    }
   ],
   "source": [
    "using Optim\n",
    "\n",
    "# c is the mean degree of the random network\n",
    "c_baseline = 1.5\n",
    "c_end = 4.0\n",
    "\n",
    "# Newton's solve \n",
    "function solve_S(;c,phi)\n",
    "    f(S) = 1 - exp(-phi*c*S) - S# Function\n",
    "    df(S) = c * phi * exp(-phi * c * S) - 1\n",
    "\n",
    "    S_iter = 0.5\n",
    "    for i in 1 : 300\n",
    "        delS = -f(S_iter) / df(S_iter)\n",
    "        S_iter += delS \n",
    "    end\n",
    "    \n",
    "    return(S_iter)\n",
    "end\n",
    "\n",
    "r = [ solve_S(c = c_baseline, phi = i) for i in 0:0.01:1]\n",
    "plot([0: 0.01: 1], r,linewidth=5,title=\"Outbreak size for varying 'mean degree' of the network\", linecolor=\"steelblue\",\n",
    "         xaxis=\"Phi or the transmission probability\",yaxis=\"% of population that is infected\",label=\"Baseline of c = 1.5\") # legend=false\n",
    "\n",
    "count = 0.1\n",
    "for c in c_baseline + 0.5: 0.5: c_end\n",
    "    r = [ solve_S(c = c, phi = i) for i in 0 : 0.01 : 1]\n",
    "    a = plot!([0: 0.01: 1], r, lw=3, label=c, linecolor=cgrad(g)[count]) # legend=false\n",
    "    count += 0.1\n",
    "end\n",
    "display(a)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "#### Diseases tend to have a certain transmission and disease 'A' may have a higher transmission probability (e.g. 0.55) compared to disease 'B' (e.g. 0.75). If we pick a curve corresponding to our network structure, let's say c = 2.0, we can see that drawing a line up to this curve 'c = 2.0' intersects it at around 20% for A and around 70% for B. As the degree of the network or the connectedness of the network increases, we can see that both diseases tend to infect a larger percentage of the population.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  }
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