docs/notebooks/SIR Real Data - Time Varying Parameters.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Os Dados"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [
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       "<table border=\"1\" class=\"dataframe\">\n",
       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>date</th>\n",
       "      <th>country</th>\n",
       "      <th>state</th>\n",
       "      <th>city</th>\n",
       "      <th>newDeaths</th>\n",
       "      <th>deaths</th>\n",
       "      <th>newCases</th>\n",
       "      <th>totalCases</th>\n",
       "      <th>deathsMS</th>\n",
       "      <th>totalCasesMS</th>\n",
       "      <th>deaths_per_100k_inhabitants</th>\n",
       "      <th>totalCases_per_100k_inhabitants</th>\n",
       "      <th>deaths_by_totalCases</th>\n",
       "      <th>recovered</th>\n",
       "      <th>suspects</th>\n",
       "      <th>tests</th>\n",
       "      <th>tests_per_100k_inhabitants</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>0</th>\n",
       "      <td>2020-02-25</td>\n",
       "      <td>Brazil</td>\n",
       "      <td>SP</td>\n",
       "      <td>TOTAL</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.00218</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>1</th>\n",
       "      <td>2020-02-25</td>\n",
       "      <td>Brazil</td>\n",
       "      <td>TOTAL</td>\n",
       "      <td>TOTAL</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.00048</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>2</th>\n",
       "      <td>2020-02-26</td>\n",
       "      <td>Brazil</td>\n",
       "      <td>SP</td>\n",
       "      <td>TOTAL</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.00218</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>3</th>\n",
       "      <td>2020-02-26</td>\n",
       "      <td>Brazil</td>\n",
       "      <td>TOTAL</td>\n",
       "      <td>TOTAL</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.00048</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>4</th>\n",
       "      <td>2020-02-27</td>\n",
       "      <td>Brazil</td>\n",
       "      <td>SP</td>\n",
       "      <td>TOTAL</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0</td>\n",
       "      <td>1</td>\n",
       "      <td>0.0</td>\n",
       "      <td>0.00218</td>\n",
       "      <td>0.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
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      "text/plain": [
       "         date country  state   city  newDeaths  deaths  newCases  totalCases  \\\n",
       "0  2020-02-25  Brazil     SP  TOTAL          0       0         1           1   \n",
       "1  2020-02-25  Brazil  TOTAL  TOTAL          0       0         1           1   \n",
       "2  2020-02-26  Brazil     SP  TOTAL          0       0         0           1   \n",
       "3  2020-02-26  Brazil  TOTAL  TOTAL          0       0         0           1   \n",
       "4  2020-02-27  Brazil     SP  TOTAL          0       0         0           1   \n",
       "\n",
       "   deathsMS  totalCasesMS  deaths_per_100k_inhabitants  \\\n",
       "0         0             0                          0.0   \n",
       "1         0             0                          0.0   \n",
       "2         0             1                          0.0   \n",
       "3         0             1                          0.0   \n",
       "4         0             1                          0.0   \n",
       "\n",
       "   totalCases_per_100k_inhabitants  deaths_by_totalCases  recovered  suspects  \\\n",
       "0                          0.00218                   0.0        NaN       NaN   \n",
       "1                          0.00048                   0.0        NaN       NaN   \n",
       "2                          0.00218                   0.0        NaN       NaN   \n",
       "3                          0.00048                   0.0        NaN       NaN   \n",
       "4                          0.00218                   0.0        NaN       NaN   \n",
       "\n",
       "   tests  tests_per_100k_inhabitants  \n",
       "0    NaN                         NaN  \n",
       "1    NaN                         NaN  \n",
       "2    NaN                         NaN  \n",
       "3    NaN                         NaN  \n",
       "4    NaN                         NaN  "
      ]
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   ],
   "source": [
    "\n",
    "import pandas as pd\n",
    "\n",
    "# reading csv file  \n",
    "BRdata = pd.read_csv(\"./PG_IMT/DadosEpidemia/CoVid19.csv\") \n",
    "BRdata.head()\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Filtrando e limpando os dados"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
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       "  <thead>\n",
       "    <tr style=\"text-align: right;\">\n",
       "      <th></th>\n",
       "      <th>date</th>\n",
       "      <th>country</th>\n",
       "      <th>state</th>\n",
       "      <th>city</th>\n",
       "      <th>newDeaths</th>\n",
       "      <th>deaths</th>\n",
       "      <th>newCases</th>\n",
       "      <th>totalCases</th>\n",
       "      <th>deathsMS</th>\n",
       "      <th>totalCasesMS</th>\n",
       "      <th>deaths_per_100k_inhabitants</th>\n",
       "      <th>totalCases_per_100k_inhabitants</th>\n",
       "      <th>deaths_by_totalCases</th>\n",
       "      <th>recovered</th>\n",
       "      <th>suspects</th>\n",
       "      <th>tests</th>\n",
       "      <th>tests_per_100k_inhabitants</th>\n",
       "    </tr>\n",
       "  </thead>\n",
       "  <tbody>\n",
       "    <tr>\n",
       "      <th>321</th>\n",
       "      <td>2020-03-23</td>\n",
       "      <td>Brazil</td>\n",
       "      <td>TOTAL</td>\n",
       "      <td>TOTAL</td>\n",
       "      <td>9</td>\n",
       "      <td>34</td>\n",
       "      <td>358</td>\n",
       "      <td>1952</td>\n",
       "      <td>34</td>\n",
       "      <td>1891</td>\n",
       "      <td>0.01618</td>\n",
       "      <td>0.92887</td>\n",
       "      <td>0.01742</td>\n",
       "      <td>8.0</td>\n",
       "      <td>15867.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>349</th>\n",
       "      <td>2020-03-24</td>\n",
       "      <td>Brazil</td>\n",
       "      <td>TOTAL</td>\n",
       "      <td>TOTAL</td>\n",
       "      <td>13</td>\n",
       "      <td>47</td>\n",
       "      <td>303</td>\n",
       "      <td>2255</td>\n",
       "      <td>46</td>\n",
       "      <td>2201</td>\n",
       "      <td>0.02237</td>\n",
       "      <td>1.07306</td>\n",
       "      <td>0.02084</td>\n",
       "      <td>20.0</td>\n",
       "      <td>17700.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>377</th>\n",
       "      <td>2020-03-25</td>\n",
       "      <td>Brazil</td>\n",
       "      <td>TOTAL</td>\n",
       "      <td>TOTAL</td>\n",
       "      <td>12</td>\n",
       "      <td>59</td>\n",
       "      <td>311</td>\n",
       "      <td>2566</td>\n",
       "      <td>57</td>\n",
       "      <td>2433</td>\n",
       "      <td>0.02808</td>\n",
       "      <td>1.22105</td>\n",
       "      <td>0.02299</td>\n",
       "      <td>27.0</td>\n",
       "      <td>27227.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>405</th>\n",
       "      <td>2020-03-26</td>\n",
       "      <td>Brazil</td>\n",
       "      <td>TOTAL</td>\n",
       "      <td>TOTAL</td>\n",
       "      <td>18</td>\n",
       "      <td>77</td>\n",
       "      <td>424</td>\n",
       "      <td>2990</td>\n",
       "      <td>77</td>\n",
       "      <td>2915</td>\n",
       "      <td>0.03664</td>\n",
       "      <td>1.42281</td>\n",
       "      <td>0.02575</td>\n",
       "      <td>42.0</td>\n",
       "      <td>48793.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "    <tr>\n",
       "      <th>433</th>\n",
       "      <td>2020-03-27</td>\n",
       "      <td>Brazil</td>\n",
       "      <td>TOTAL</td>\n",
       "      <td>TOTAL</td>\n",
       "      <td>16</td>\n",
       "      <td>93</td>\n",
       "      <td>486</td>\n",
       "      <td>3476</td>\n",
       "      <td>92</td>\n",
       "      <td>3417</td>\n",
       "      <td>0.04425</td>\n",
       "      <td>1.65408</td>\n",
       "      <td>0.02675</td>\n",
       "      <td>42.0</td>\n",
       "      <td>50684.0</td>\n",
       "      <td>NaN</td>\n",
       "      <td>NaN</td>\n",
       "    </tr>\n",
       "  </tbody>\n",
       "</table>\n",
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      "text/plain": [
       "           date country  state   city  newDeaths  deaths  newCases  \\\n",
       "321  2020-03-23  Brazil  TOTAL  TOTAL          9      34       358   \n",
       "349  2020-03-24  Brazil  TOTAL  TOTAL         13      47       303   \n",
       "377  2020-03-25  Brazil  TOTAL  TOTAL         12      59       311   \n",
       "405  2020-03-26  Brazil  TOTAL  TOTAL         18      77       424   \n",
       "433  2020-03-27  Brazil  TOTAL  TOTAL         16      93       486   \n",
       "\n",
       "     totalCases  deathsMS  totalCasesMS  deaths_per_100k_inhabitants  \\\n",
       "321        1952        34          1891                      0.01618   \n",
       "349        2255        46          2201                      0.02237   \n",
       "377        2566        57          2433                      0.02808   \n",
       "405        2990        77          2915                      0.03664   \n",
       "433        3476        92          3417                      0.04425   \n",
       "\n",
       "     totalCases_per_100k_inhabitants  deaths_by_totalCases  recovered  \\\n",
       "321                          0.92887               0.01742        8.0   \n",
       "349                          1.07306               0.02084       20.0   \n",
       "377                          1.22105               0.02299       27.0   \n",
       "405                          1.42281               0.02575       42.0   \n",
       "433                          1.65408               0.02675       42.0   \n",
       "\n",
       "     suspects  tests  tests_per_100k_inhabitants  \n",
       "321   15867.0    NaN                         NaN  \n",
       "349   17700.0    NaN                         NaN  \n",
       "377   27227.0    NaN                         NaN  \n",
       "405   48793.0    NaN                         NaN  \n",
       "433   50684.0    NaN                         NaN  "
      ]
     },
     "execution_count": 2,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "\n",
    "is_SP =  BRdata['state']=='TOTAL'\n",
    "SPdata = BRdata[is_SP]\n",
    "\n",
    "# SPdata = SPdata.fillna(0)\n",
    "\n",
    "SPLimpo = SPdata[SPdata.recovered.notnull()]\n",
    "SPLimpo.head()\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "\n",
    "from datetime import datetime\n",
    "\n",
    "first_date = SPLimpo[\"date\"].iloc[0]\n",
    "first_date = datetime.fromisoformat(first_date)\n",
    "\n",
    "SPsir = SPLimpo[[\"totalCases\", \"deaths\", \"recovered\"]].to_numpy()\n",
    "\n",
    "SPsir[:,0:]\n",
    "SPI = SPsir[:,0]\n",
    "auxM = SPsir[:,1]\n",
    "SPM = auxM\n",
    "auxR = SPsir[:,2]\n",
    "SPR = auxR\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Visualizando a evolução"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
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       "     * Register the mime type and append_mime function with output_area\n",
       "     */\n",
       "    OutputArea.prototype.register_mime_type(EXEC_MIME_TYPE, append_mime, {\n",
       "      /* Is output safe? */\n",
       "      safe: true,\n",
       "      /* Index of renderer in `output_area.display_order` */\n",
       "      index: 0\n",
       "    });\n",
       "  }\n",
       "\n",
       "  // register the mime type if in Jupyter Notebook environment and previously unregistered\n",
       "  if (root.Jupyter !== undefined) {\n",
       "    var events = require('base/js/events');\n",
       "    var OutputArea = require('notebook/js/outputarea').OutputArea;\n",
       "\n",
       "    if (OutputArea.prototype.mime_types().indexOf(EXEC_MIME_TYPE) == -1) {\n",
       "      register_renderer(events, OutputArea);\n",
       "    }\n",
       "  }\n",
       "\n",
       "  \n",
       "  if (typeof (root._bokeh_timeout) === \"undefined\" || force === true) {\n",
       "    root._bokeh_timeout = Date.now() + 5000;\n",
       "    root._bokeh_failed_load = false;\n",
       "  }\n",
       "\n",
       "  var NB_LOAD_WARNING = {'data': {'text/html':\n",
       "     \"<div style='background-color: #fdd'>\\n\"+\n",
       "     \"<p>\\n\"+\n",
       "     \"BokehJS does not appear to have successfully loaded. If loading BokehJS from CDN, this \\n\"+\n",
       "     \"may be due to a slow or bad network connection. Possible fixes:\\n\"+\n",
       "     \"</p>\\n\"+\n",
       "     \"<ul>\\n\"+\n",
       "     \"<li>re-rerun `output_notebook()` to attempt to load from CDN again, or</li>\\n\"+\n",
       "     \"<li>use INLINE resources instead, as so:</li>\\n\"+\n",
       "     \"</ul>\\n\"+\n",
       "     \"<code>\\n\"+\n",
       "     \"from bokeh.resources import INLINE\\n\"+\n",
       "     \"output_notebook(resources=INLINE)\\n\"+\n",
       "     \"</code>\\n\"+\n",
       "     \"</div>\"}};\n",
       "\n",
       "  function display_loaded() {\n",
       "    var el = document.getElementById(\"1001\");\n",
       "    if (el != null) {\n",
       "      el.textContent = \"BokehJS is loading...\";\n",
       "    }\n",
       "    if (root.Bokeh !== undefined) {\n",
       "      if (el != null) {\n",
       "        el.textContent = \"BokehJS \" + root.Bokeh.version + \" successfully loaded.\";\n",
       "      }\n",
       "    } else if (Date.now() < root._bokeh_timeout) {\n",
       "      setTimeout(display_loaded, 100)\n",
       "    }\n",
       "  }\n",
       "\n",
       "\n",
       "  function run_callbacks() {\n",
       "    try {\n",
       "      root._bokeh_onload_callbacks.forEach(function(callback) {\n",
       "        if (callback != null)\n",
       "          callback();\n",
       "      });\n",
       "    } finally {\n",
       "      delete root._bokeh_onload_callbacks\n",
       "    }\n",
       "    console.debug(\"Bokeh: all callbacks have finished\");\n",
       "  }\n",
       "\n",
       "  function load_libs(css_urls, js_urls, callback) {\n",
       "    if (css_urls == null) css_urls = [];\n",
       "    if (js_urls == null) js_urls = [];\n",
       "\n",
       "    root._bokeh_onload_callbacks.push(callback);\n",
       "    if (root._bokeh_is_loading > 0) {\n",
       "      console.debug(\"Bokeh: BokehJS is being loaded, scheduling callback at\", now());\n",
       "      return null;\n",
       "    }\n",
       "    if (js_urls == null || js_urls.length === 0) {\n",
       "      run_callbacks();\n",
       "      return null;\n",
       "    }\n",
       "    console.debug(\"Bokeh: BokehJS not loaded, scheduling load and callback at\", now());\n",
       "    root._bokeh_is_loading = css_urls.length + js_urls.length;\n",
       "\n",
       "    function on_load() {\n",
       "      root._bokeh_is_loading--;\n",
       "      if (root._bokeh_is_loading === 0) {\n",
       "        console.debug(\"Bokeh: all BokehJS libraries/stylesheets loaded\");\n",
       "        run_callbacks()\n",
       "      }\n",
       "    }\n",
       "\n",
       "    function on_error() {\n",
       "      console.error(\"failed to load \" + url);\n",
       "    }\n",
       "\n",
       "    for (var i = 0; i < css_urls.length; i++) {\n",
       "      var url = css_urls[i];\n",
       "      const element = document.createElement(\"link\");\n",
       "      element.onload = on_load;\n",
       "      element.onerror = on_error;\n",
       "      element.rel = \"stylesheet\";\n",
       "      element.type = \"text/css\";\n",
       "      element.href = url;\n",
       "      console.debug(\"Bokeh: injecting link tag for BokehJS stylesheet: \", url);\n",
       "      document.body.appendChild(element);\n",
       "    }\n",
       "\n",
       "    const hashes = {\"https://cdn.bokeh.org/bokeh/release/bokeh-2.0.1.min.js\": \"JpP8FXbgAZLkfur7LiK3j9AGBhHNIvF742meBJrjO2ShJDhCG2I1uVvW+0DUtrmc\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-2.0.1.min.js\": \"xZlADit0Q04ISQEdKg2k3L4W9AwQBAuDs9nJL9fM/WwzL1tEU9VPNezOFX0nLEAz\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-2.0.1.min.js\": \"4BuPRZkdMKSnj3zoxiNrQ86XgNw0rYmBOxe7nshquXwwcauupgBF2DHLVG1WuZlV\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-2.0.1.min.js\": \"Dv1SQ87hmDqK6S5OhBf0bCuwAEvL5QYL0PuR/F1SPVhCS/r/abjkbpKDYL2zeM19\"};\n",
       "\n",
       "    for (var i = 0; i < js_urls.length; i++) {\n",
       "      var url = js_urls[i];\n",
       "      var element = document.createElement('script');\n",
       "      element.onload = on_load;\n",
       "      element.onerror = on_error;\n",
       "      element.async = false;\n",
       "      element.src = url;\n",
       "      if (url in hashes) {\n",
       "        element.crossOrigin = \"anonymous\";\n",
       "        element.integrity = \"sha384-\" + hashes[url];\n",
       "      }\n",
       "      console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n",
       "      document.head.appendChild(element);\n",
       "    }\n",
       "  };var element = document.getElementById(\"1001\");\n",
       "  if (element == null) {\n",
       "    console.error(\"Bokeh: ERROR: autoload.js configured with elementid '1001' but no matching script tag was found. \")\n",
       "    return false;\n",
       "  }\n",
       "\n",
       "  function inject_raw_css(css) {\n",
       "    const element = document.createElement(\"style\");\n",
       "    element.appendChild(document.createTextNode(css));\n",
       "    document.body.appendChild(element);\n",
       "  }\n",
       "\n",
       "  \n",
       "  var js_urls = [\"https://cdn.bokeh.org/bokeh/release/bokeh-2.0.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-2.0.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-2.0.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-2.0.1.min.js\"];\n",
       "  var css_urls = [];\n",
       "  \n",
       "\n",
       "  var inline_js = [\n",
       "    function(Bokeh) {\n",
       "      Bokeh.set_log_level(\"info\");\n",
       "    },\n",
       "    function(Bokeh) {\n",
       "    \n",
       "    \n",
       "    }\n",
       "  ];\n",
       "\n",
       "  function run_inline_js() {\n",
       "    \n",
       "    if (root.Bokeh !== undefined || force === true) {\n",
       "      \n",
       "    for (var i = 0; i < inline_js.length; i++) {\n",
       "      inline_js[i].call(root, root.Bokeh);\n",
       "    }\n",
       "    if (force === true) {\n",
       "        display_loaded();\n",
       "      }} else if (Date.now() < root._bokeh_timeout) {\n",
       "      setTimeout(run_inline_js, 100);\n",
       "    } else if (!root._bokeh_failed_load) {\n",
       "      console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n",
       "      root._bokeh_failed_load = true;\n",
       "    } else if (force !== true) {\n",
       "      var cell = $(document.getElementById(\"1001\")).parents('.cell').data().cell;\n",
       "      cell.output_area.append_execute_result(NB_LOAD_WARNING)\n",
       "    }\n",
       "\n",
       "  }\n",
       "\n",
       "  if (root._bokeh_is_loading === 0) {\n",
       "    console.debug(\"Bokeh: BokehJS loaded, going straight to plotting\");\n",
       "    run_inline_js();\n",
       "  } else {\n",
       "    load_libs(css_urls, js_urls, function() {\n",
       "      console.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n",
       "      run_inline_js();\n",
       "    });\n",
       "  }\n",
       "}(window));"
      ],
      "application/vnd.bokehjs_load.v0+json": "\n(function(root) {\n  function now() {\n    return new Date();\n  }\n\n  var force = true;\n\n  if (typeof root._bokeh_onload_callbacks === \"undefined\" || force === true) {\n    root._bokeh_onload_callbacks = [];\n    root._bokeh_is_loading = undefined;\n  }\n\n  \n\n  \n  if (typeof (root._bokeh_timeout) === \"undefined\" || force === true) {\n    root._bokeh_timeout = Date.now() + 5000;\n    root._bokeh_failed_load = false;\n  }\n\n  var NB_LOAD_WARNING = {'data': {'text/html':\n     \"<div style='background-color: #fdd'>\\n\"+\n     \"<p>\\n\"+\n     \"BokehJS does not appear to have successfully loaded. If loading BokehJS from CDN, this \\n\"+\n     \"may be due to a slow or bad network connection. Possible fixes:\\n\"+\n     \"</p>\\n\"+\n     \"<ul>\\n\"+\n     \"<li>re-rerun `output_notebook()` to attempt to load from CDN again, or</li>\\n\"+\n     \"<li>use INLINE resources instead, as so:</li>\\n\"+\n     \"</ul>\\n\"+\n     \"<code>\\n\"+\n     \"from bokeh.resources import INLINE\\n\"+\n     \"output_notebook(resources=INLINE)\\n\"+\n     \"</code>\\n\"+\n     \"</div>\"}};\n\n  function display_loaded() {\n    var el = document.getElementById(\"1001\");\n    if (el != null) {\n      el.textContent = \"BokehJS is loading...\";\n    }\n    if (root.Bokeh !== undefined) {\n      if (el != null) {\n        el.textContent = \"BokehJS \" + root.Bokeh.version + \" successfully loaded.\";\n      }\n    } else if (Date.now() < root._bokeh_timeout) {\n      setTimeout(display_loaded, 100)\n    }\n  }\n\n\n  function run_callbacks() {\n    try {\n      root._bokeh_onload_callbacks.forEach(function(callback) {\n        if (callback != null)\n          callback();\n      });\n    } finally {\n      delete root._bokeh_onload_callbacks\n    }\n    console.debug(\"Bokeh: all callbacks have finished\");\n  }\n\n  function load_libs(css_urls, js_urls, callback) {\n    if (css_urls == null) css_urls = [];\n    if (js_urls == null) js_urls = [];\n\n    root._bokeh_onload_callbacks.push(callback);\n    if (root._bokeh_is_loading > 0) {\n      console.debug(\"Bokeh: BokehJS is being loaded, scheduling callback at\", now());\n      return null;\n    }\n    if (js_urls == null || js_urls.length === 0) {\n      run_callbacks();\n      return null;\n    }\n    console.debug(\"Bokeh: BokehJS not loaded, scheduling load and callback at\", now());\n    root._bokeh_is_loading = css_urls.length + js_urls.length;\n\n    function on_load() {\n      root._bokeh_is_loading--;\n      if (root._bokeh_is_loading === 0) {\n        console.debug(\"Bokeh: all BokehJS libraries/stylesheets loaded\");\n        run_callbacks()\n      }\n    }\n\n    function on_error() {\n      console.error(\"failed to load \" + url);\n    }\n\n    for (var i = 0; i < css_urls.length; i++) {\n      var url = css_urls[i];\n      const element = document.createElement(\"link\");\n      element.onload = on_load;\n      element.onerror = on_error;\n      element.rel = \"stylesheet\";\n      element.type = \"text/css\";\n      element.href = url;\n      console.debug(\"Bokeh: injecting link tag for BokehJS stylesheet: \", url);\n      document.body.appendChild(element);\n    }\n\n    const hashes = {\"https://cdn.bokeh.org/bokeh/release/bokeh-2.0.1.min.js\": \"JpP8FXbgAZLkfur7LiK3j9AGBhHNIvF742meBJrjO2ShJDhCG2I1uVvW+0DUtrmc\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-2.0.1.min.js\": \"xZlADit0Q04ISQEdKg2k3L4W9AwQBAuDs9nJL9fM/WwzL1tEU9VPNezOFX0nLEAz\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-2.0.1.min.js\": \"4BuPRZkdMKSnj3zoxiNrQ86XgNw0rYmBOxe7nshquXwwcauupgBF2DHLVG1WuZlV\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-2.0.1.min.js\": \"Dv1SQ87hmDqK6S5OhBf0bCuwAEvL5QYL0PuR/F1SPVhCS/r/abjkbpKDYL2zeM19\"};\n\n    for (var i = 0; i < js_urls.length; i++) {\n      var url = js_urls[i];\n      var element = document.createElement('script');\n      element.onload = on_load;\n      element.onerror = on_error;\n      element.async = false;\n      element.src = url;\n      if (url in hashes) {\n        element.crossOrigin = \"anonymous\";\n        element.integrity = \"sha384-\" + hashes[url];\n      }\n      console.debug(\"Bokeh: injecting script tag for BokehJS library: \", url);\n      document.head.appendChild(element);\n    }\n  };var element = document.getElementById(\"1001\");\n  if (element == null) {\n    console.error(\"Bokeh: ERROR: autoload.js configured with elementid '1001' but no matching script tag was found. \")\n    return false;\n  }\n\n  function inject_raw_css(css) {\n    const element = document.createElement(\"style\");\n    element.appendChild(document.createTextNode(css));\n    document.body.appendChild(element);\n  }\n\n  \n  var js_urls = [\"https://cdn.bokeh.org/bokeh/release/bokeh-2.0.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-widgets-2.0.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-tables-2.0.1.min.js\", \"https://cdn.bokeh.org/bokeh/release/bokeh-gl-2.0.1.min.js\"];\n  var css_urls = [];\n  \n\n  var inline_js = [\n    function(Bokeh) {\n      Bokeh.set_log_level(\"info\");\n    },\n    function(Bokeh) {\n    \n    \n    }\n  ];\n\n  function run_inline_js() {\n    \n    if (root.Bokeh !== undefined || force === true) {\n      \n    for (var i = 0; i < inline_js.length; i++) {\n      inline_js[i].call(root, root.Bokeh);\n    }\n    if (force === true) {\n        display_loaded();\n      }} else if (Date.now() < root._bokeh_timeout) {\n      setTimeout(run_inline_js, 100);\n    } else if (!root._bokeh_failed_load) {\n      console.log(\"Bokeh: BokehJS failed to load within specified timeout.\");\n      root._bokeh_failed_load = true;\n    } else if (force !== true) {\n      var cell = $(document.getElementById(\"1001\")).parents('.cell').data().cell;\n      cell.output_area.append_execute_result(NB_LOAD_WARNING)\n    }\n\n  }\n\n  if (root._bokeh_is_loading === 0) {\n    console.debug(\"Bokeh: BokehJS loaded, going straight to plotting\");\n    run_inline_js();\n  } else {\n    load_libs(css_urls, js_urls, function() {\n      console.debug(\"Bokeh: BokehJS plotting callback run at\", now());\n      run_inline_js();\n    });\n  }\n}(window));"
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       "  <div class=\"bk-root\" id=\"0e5b9881-aba0-4563-ac5f-4af36ba9adfb\" data-root-id=\"1002\"></div>\n"
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       "id": "1002"
      }
     },
     "output_type": "display_data"
    }
   ],
   "source": [
    "\n",
    "from bokeh.models   import Legend, ColumnDataSource, RangeTool, LinearAxis, Range1d\n",
    "from bokeh.palettes import brewer, Inferno256\n",
    "from bokeh.plotting import figure, show\n",
    "from bokeh.layouts  import column\n",
    "from bokeh.io       import output_notebook\n",
    "\n",
    "output_notebook()\n",
    "\n",
    "from datetime import timedelta\n",
    "\n",
    "# Criando o vetor de tempo\n",
    "t_num = range(len(SPM))\n",
    "date_vec = [ first_date + timedelta(days=k) for k in t_num]\n",
    "\n",
    "\n",
    "# Criando a figura \n",
    "p = figure(\n",
    "    tools=\"hover\",\n",
    "    title=\"Evolução do COVID\",\n",
    "    x_axis_type='datetime',\n",
    "    y_axis_type=\"log\",\n",
    "    plot_width=650, \n",
    "    plot_height=500\n",
    ")\n",
    "\n",
    "# Incluindo os dados de mortes\n",
    "p.line(\n",
    "    date_vec, SPM,\n",
    "    legend_label=\"Mortes\",\n",
    "    line_width=4,\n",
    "    line_cap=\"round\",\n",
    "    color=\"#de425b\"\n",
    ")\n",
    "\n",
    "# Incluindo os dados de infectados\n",
    "p.line(\n",
    "    date_vec, SPI,\n",
    "    legend_label=\"Infectados\",\n",
    "    line_width=4,\n",
    "    line_cap=\"round\",\n",
    "    color=\"#ffd885\"\n",
    ")\n",
    "\n",
    "# Incluindo os dados de recuperados\n",
    "p.line(\n",
    "    date_vec, SPR,\n",
    "    legend_label=\"Removidos\",\n",
    "    line_width=4,\n",
    "    line_cap=\"round\",\n",
    "    color=\"#99d594\"\n",
    ")\n",
    "\n",
    "\n",
    "p.grid.grid_line_alpha = 0\n",
    "p.ygrid.band_fill_color = \"olive\"\n",
    "p.ygrid.band_fill_alpha = 0.1\n",
    "p.yaxis.axis_label = \"Indivíduos\"\n",
    "p.xaxis.axis_label = \"Dias\"\n",
    "p.legend.click_policy=\"hide\"\n",
    "p.legend.location = \"bottom_right\"\n",
    "\n",
    "show(p)\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Análise de um modelo com $\\beta$ variável no tempo\n",
    "\n",
    "Uma outra forma de abordar a modelagme de dados é assumindo que a taxa de transmissão $\\beta$ é variante no tempo. Vamos, neste sentido, utilizar a abordagem elaborada por *Mark Pollicott, Hao Wang & Howard (Howie) Weiss em 2012 - Extracting the time-dependent transmission rate from infection data via solution of an inverse ODE problem*\n",
    "\n",
    "Reescrevendo o conjunto de equações diferenciais que caracteriza o modelo é descrito abaixo. No modelo $\\beta(t) - \\text{representa a taxa de transmissão ou taxa efetiva de contato} $  e $r - \\text{a taxa de remoção ou recuperação.}$ \n",
    "\n",
    "\n",
    "$$ \\begin{split}\n",
    "   \\frac{dS(t)}{dt} & = -\\mathbf{\\beta(t)} S(t) I(t) \\\\\n",
    "   \\frac{dI(t)}{dt} & = \\mathbf{\\beta(t)} S(t) I(t) - rI(t)  \\\\\n",
    "   \\frac{dR(t)}{dt} & = r I(t)\n",
    "   \\end{split}$$\n",
    "\n",
    "   Gostaríamos de identificar quais parâmetros $\\beta(t)$ e $r$ resultam num melhor ajuste do modelo para os dados de **S**,**I** e **R**"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Proposta de algoritmo para extração do $\\beta(t)$ a partir dos dados\n",
    "\n",
    "1.   Interpolar os dados para suvizar eventuais variações gerando uma função $f(t)$\n",
    "2.   **Verificar a seguinte condição**: $\\frac{f'(t)}{f(t)}>-r$. Neste algoritmo a taza de romação $r$ é considerada constante.\n",
    "\n",
    "3.   Determinando a função $p(t) = \\frac{f''(t)f(t)-f'^2(t)}{f(t)(f'(t)+rf(t))}$.\n",
    "\n",
    "4.   Escolhendo um valor inicial para $\\beta(0)$ e determinando a integral $P(t) = \\int_0^t p(\\tau) d\\tau$.\n",
    "\n",
    "5.   **Verificar a seguinte condição:** $\\beta(0) <\\frac{1}{\\int_0^t e^{P(s)}f(s) ds}$.\n",
    "\n",
    "6.   O parâmetro $\\mathbf{\\beta(t)}$ pode ser calculado por: $\\beta(t) = \\frac{1}{\\frac{e^{-P(t)}}{\\beta(0)}-e^{-P(t)}\\int_0^t e^{P(s)}f(s) ds}$.\n",
    "\n",
    "\n"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "### 1) Interpolar os dados para suvizar eventuais variações gerando uma função $f(t)$"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "[  67.47775863 -501.31086778 4785.98379386]\n"
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      "application/vnd.bokehjs_exec.v0+json": ""
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     "metadata": {
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       "id": "1261"
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     "output_type": "display_data"
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   ],
   "source": [
    "\n",
    "import numpy as np\n",
    "from scipy.interpolate import Rbf, UnivariateSpline\n",
    "\n",
    "# Generating weights for polynomial function with degree =2\n",
    "weights = np.polyfit(t_num, SPI, 2)\n",
    "rbf = Rbf(t_num, SPI, function='gaussian')\n",
    "spline = UnivariateSpline(t_num, SPI)\n",
    "print(weights)\n",
    "\n",
    "\n",
    "# Generating model with the given weights\n",
    "model = np.poly1d(weights)\n",
    "\n",
    "t_sim = np.linspace(t_num[0], t_num[-1], 100)\n",
    "\n",
    "# Prediction on validation set\n",
    "pred = model(t_sim)\n",
    "pred_rbf = rbf(t_sim)\n",
    "pred_spline = spline(t_sim)\n",
    "\n",
    "# We will plot the graph for 70 observations only\n",
    "\n",
    "# Criando a figura \n",
    "p = figure(\n",
    "    tools=\"hover\",\n",
    "    title=\"Dados interpolados\",\n",
    "    x_axis_type='datetime',\n",
    "#     y_axis_type=\"log\",\n",
    "    plot_width=650, \n",
    "    plot_height=500\n",
    ")\n",
    "\n",
    "# Incluindo os dados de mortes\n",
    "p.line(\n",
    "    t_num, SPI,\n",
    "    legend_label=\"Medidas\",\n",
    "    line_width=4,\n",
    "    line_cap=\"round\",\n",
    "    color=\"#de425b\"\n",
    ")\n",
    "\n",
    "# Incluindo os dados de infectados\n",
    "p.line(\n",
    "    t_sim, pred,\n",
    "    legend_label=\"Interpolação\",\n",
    "    line_width=4,\n",
    "    line_cap=\"round\",\n",
    "    color=\"#1e88e5\"\n",
    ")\n",
    "\n",
    "p.line(\n",
    "    t_sim, pred_rbf,\n",
    "    legend_label=\"Interpolação - RBF\",\n",
    "    line_width=3,\n",
    "    line_dash=\"dashed\",\n",
    "    line_cap=\"round\",\n",
    "    color=\"#512da8\"\n",
    ")\n",
    "\n",
    "p.line(\n",
    "    t_sim, pred_spline,\n",
    "    legend_label=\"Interpolação - Spline\",\n",
    "    line_width=3,\n",
    "    line_dash=\"dashed\",\n",
    "    line_cap=\"round\",\n",
    "    color=\"#00796b\"\n",
    ")\n",
    "\n",
    "p.grid.grid_line_alpha = 0\n",
    "p.ygrid.band_fill_color = \"olive\"\n",
    "p.ygrid.band_fill_alpha = 0.1\n",
    "p.yaxis.axis_label = \"Indivíduos\"\n",
    "p.xaxis.axis_label = \"Dias\"\n",
    "p.legend.click_policy=\"hide\"\n",
    "p.legend.location = \"bottom_right\"\n",
    "\n",
    "show(p)\n",
    "\n"
   ]
  }
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