2d:null 2e:["$","title",null,{"children":"A Simple SEIR model | Blake Elias | Observable"}] 33:T58f,chart = { //const svg = d3.select(DOM.svg(width, height)); const svg = d3.create("svg") .attr("viewBox", [0, 0, width, height+10]); svg.append("g") .call(xAxis); svg.append("g") .call(yAxis); var colors = d3.scaleOrdinal().domain(['S', 'E', 'I', 'R']) .range(['green', 'yellow', 'red', 'blue']); const path = svg.append("g") .attr("fill", "none") .attr("stroke-width", 2) .attr("stroke-linejoin", "round") .attr("stroke-linecap", "round") .selectAll("path") .data(plot_data.series) .enter() .append("path") .attr("d", d => line(d.values)) .style("mix-blend-mode", "multiply") .style('stroke', function(d) { return colors(d.name); }); svg.append("text") .attr("text-anchor", "middle") .attr("font-family", "sans-serif") .attr("font-size", 10) .attr("x", width/2) .attr("y", height) .text("Arbitrary units of time"); svg.append("text") .attr("text-anchor", "left") .attr("font-family", "sans-serif") .attr("font-size", 10) .attr("x", 0) .attr("y", 10) .text("Percent of the population"); return svg.node(); }34:[] 31:{"id":"defd484b7db63297","slug":"a-simple-seir-model","title":"A Simple SEIR model","thumbnail":"a21e03afd5b13776f51d734188fdd85468786902215d766af2cdffe7e7ad6fe9","access_level":"public","create_time":"$D2020-04-14T01:17:04.387Z","update_time":"$D2020-04-15T19:50:58.100Z","view_time":null,"first_public_time":"$D2020-04-15T19:51:01.629Z","has_unpublished_edits":null,"fork":{"id":"97ea264453e667ed","slug":"a-simple-sir-model","title":"A Simple SIR model","owner":{"id":"1339992d0aa77f13","login":"kevinxperese","name":"Kevin Perese"}},"stars":0,"starred":false,"owner":{"type":"team","id":"614157e65006a2d4","login":"blakeelias","name":"Blake Elias","avatar_url":"https://avatars-next.observableusercontent.com/bcb2d8974dfe29de8246d628c79d40f80dd37cb59dc6d869a9030f5158e1b5ca","tier":"starter_2024"},"version":901,"publish_version":901,"role":null,"authored":false,"authors":[{"type":"individual","id":"01ac8a6a43bd3e70","login":"blakeelias","name":"Blake Elias","avatar_url":"https://avatars-next.observableusercontent.com/bcb2d8974dfe29de8246d628c79d40f80dd37cb59dc6d869a9030f5158e1b5ca","tier":null}],"members":[{"type":"individual","id":"01ac8a6a43bd3e70","login":"blakeelias","name":"Blake Elias","avatar_url":"https://avatars-next.observableusercontent.com/bcb2d8974dfe29de8246d628c79d40f80dd37cb59dc6d869a9030f5158e1b5ca","tier":null,"role":"editor"}],"comments":[],"custom_thumbnail":false,"trashed":false,"lapsed":null,"subscribed":null,"forks":1,"collection":null,"type":"notebook","body":{"title":"A Simple SEIR model","cells":[{"id":0,"value":"md`# A Simple SEIR model`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":822,"value":"md` \n## First: A VERY IMPORTANT CAVEAT:`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":826,"value":"md`# I AM NOT AN EPIDEMIOLOGIST.\nI made this just to try to better understand the models being discussed in the news. \nThis is a gross simplification, it is likely that there are errors, and nothing here applies to any real-world situation.\n\nI was originally inspired to build this interactive model by this video: \n[The Coronavirus Curve - Numberphile - Mar 25, 2020](https://www.youtube.com/watch?v=k6nLfCbAzgo)\n\nOther good resources include: \nhttps://www.maa.org/press/periodicals/loci/joma/the-sir-model-for-spread-of-disease-introduction\n\nOriginal source: \n[Kermack, W. O. and McKendrick, A. G. \"*A Contribution to the Mathematical Theory of Epidemics.*\" Proceedings of the Royal Society London, Series A, Vol. 115, 1927, pp. 700-721](https://royalsocietypublishing.org/doi/pdf/10.1098/rspa.1927.0118)\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":39,"value":"md`----\n## Adjustable model parameters`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":58,"value":"viewof transmission_rate = slider({\n min: 0,\n max: 2,\n value: 0.5,\n step: 0.01,\n description: \"transmission rate\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":887,"value":"viewof infection_rate = slider({\n min: 0,\n max: 2,\n value: 0.5,\n step: 0.01,\n description: \"infection rate\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":15,"value":"viewof removal_rate = slider({\n min: 0,\n max: 1,\n value: 0.2,\n step: 0.01,\n description: \"removal rate\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":512,"value":"R0 = infection_rate * transmission_rate / removal_rate","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":792,"value":"md`Susceptible (S) \nExposed (E) \nInfected (I) \nRemoved (R)\n`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":197,"value":"$33","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":61,"value":"viewof max_duration = slider({\n min: 10,\n max: 100,\n step: 10,\n value: 80,\n description: \"max duration\"})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":426,"value":"md`---- \n## Chart Code`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":357,"value":"line = d3.line()\n .defined(d => !isNaN(d))\n .x(function(d,i) { return x(i) })\n .y(function(d) { return y(d) })","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":352,"value":"x = d3.scaleLinear()\n .domain([0, results._data.length])\n .range([margin.left, width - margin.right]);","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":354,"value":"y = d3.scaleLinear()\n .domain([0, 1])\n .range([height - margin.bottom, margin.top])","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":350,"value":"xAxis = g => g\n .attr(\"transform\", `translate(0,${height - margin.bottom})`)\n .call(d3.axisBottom(x).ticks(width / 80).tickSizeOuter(0))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":348,"value":"/*\nyAxis = g => g\n .attr(\"transform\", `translate(${margin.left},0)`)\n .call(d3.axisLeft(y))\n*/\n\nyAxis = g => g\n .attr(\"transform\", `translate(${margin.left}, 0)`)\n .call(d3.axisRight(y)\n .tickSize(width - margin.left - margin.right))\n .call(g => g.select(\".domain\")\n .remove())\n .call(g => g.selectAll(\".tick:not(:first-of-type) line\")\n .attr(\"stroke-opacity\", 0.15)\n .attr(\"stroke-dasharray\", \"2,2\"))\n .call(g => g.selectAll(\".tick text\")\n .attr(\"x\", -20)\n .attr(\"dy\", 3))","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":193,"value":"margin = ({top: 20, right: 50, bottom: 30, left: 25})","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":368,"value":"width = 500","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":191,"value":"height = 300","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":42,"value":"md`## Set model starting values`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":62,"value":"total_pop = 1","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":96,"value":"infected_0 = 0.01","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":4,"value":"// share_susceptible is total_pop minus share_infected and share_removed\n// Assumes share_removed are removed because of death or they attain permanent immunity\nsusceptible_0 = total_pop - infected_0 - removed_0","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":853,"value":"exposed_0 = 0.0","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":67,"value":"removed_0 = 0","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":34,"value":"md`## Set up differential equations`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":872,"value":"tex`\\beta = \\text{transmission rate}\\\\\na = \\text{infection rate}\\\\\n\\gamma = \\text{recovery rate}`","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":17,"value":"function s_prime(transmision_rate, infection_rate, recovery_rate, share_susceptible, share_exposed, share_infected, share_removed) {\n const n = share_susceptible + share_exposed + share_infected + share_removed;\n return -1 * transmision_rate * share_infected * share_susceptible / n;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":862,"value":"function e_prime(transmision_rate, infection_rate, recovery_rate, share_susceptible, share_exposed, share_infected, share_removed) {\n const n = share_susceptible + share_exposed + share_infected + share_removed;\n return transmision_rate * share_infected * share_susceptible / n - infection_rate * share_exposed;\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":19,"value":"function i_prime(transmision_rate, infection_rate, recovery_rate, share_susceptible, share_exposed, share_infected, share_removed) {\n const n = share_susceptible + share_exposed + share_infected + share_removed;\n return infection_rate * share_exposed - recovery_rate * share_infected\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":24,"value":"function r_prime(transmision_rate, infection_rate, recovery_rate, share_susceptible, share_exposed, share_infected, share_removed) {\n const n = share_susceptible + share_exposed + share_infected + share_removed;\n return recovery_rate * share_infected\n}","pinned":true,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":146,"value":"md`## Numerical Solver for Ordinary Differential Equations\nAdapted from: [https://mathjs.org/examples/browser/rocket_trajectory_optimization.html.html](https://mathjs.org/examples/browser/rocket_trajectory_optimization.html.html)`\n","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":145,"value":"function ndsolve(functions, rates, x0s, max_duration) {\n \n const num_functions = functions.length; // Number of functions\n const x = math.clone(x0s); // Current values of variables\n const dxdt = []; // Temporary variable to hold derivative values\n const result = []; // Contains entire solution\n const step_size = 0.10; // Set a \"reasonable\"(?) step size\n const num_steps = max_duration / step_size // Number of time steps\n \n for(let t = 0; t < num_steps; t++) {\n \n // Loop through each function\n for(let f = 0; f < num_functions; f++) {\n \n // Compute values of derivative functions\n dxdt[f] = functions[f].apply(null, Array.from(rates.concat(x)));\n \n // Update x's; Note: assuming that dt = 1\n //x[f] += dxdt[f]; \n x[f] += dxdt[f] * step_size\n \n }\n result.push(math.clone(x));\n }\n return math.matrix(result);\n}","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":170,"value":"results = ndsolve([s_prime, e_prime, i_prime, r_prime], [transmission_rate, infection_rate, removal_rate], [susceptible_0, exposed_0, infected_0, removed_0], max_duration) ","pinned":false,"hidden":false,"mode":"ojs","author":"$undefined","summary":"$undefined","output":"$undefined","database":"$undefined"},{"id":391,"value":"// Get the data into shape\nplot_data = {\n var results_trans = math.transpose(results)\n var names = ['S', 'E', 'I', 'R']\n \n var plot_data = {};\n plot_data['series'] = []\n\n for (var i = 0; i