{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<div style=\"width: 100%; overflow: hidden;\">\n",
    "    <div style=\"width: 150px; float: left;\"> <img src=\"data/D4Sci_logo_ball.png\" alt=\"Data For Science, Inc\" align=\"left\" border=\"0\" width=160px> </div>\n",
    "    <div style=\"float: left; margin-left: 10px;\"> <h1>Epidemiology 202</h1>\n",
    "<h1>Network Models</h1>\n",
    "        <p>Bruno Gonçalves<br/>\n",
    "        <a href=\"http://www.data4sci.com/\">www.data4sci.com</a><br/>\n",
    "            @bgoncalves, @data4sci</p></div>\n",
    "</div>"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "from collections import Counter\n",
    "from pprint import pprint\n",
    "\n",
    "import pandas as pd\n",
    "import numpy as np\n",
    "\n",
    "import matplotlib\n",
    "import matplotlib.pyplot as plt \n",
    "\n",
    "import networkx as nx\n",
    "\n",
    "import scipy\n",
    "from scipy.optimize import curve_fit\n",
    "\n",
    "import tqdm as tq\n",
    "from tqdm.notebook import tqdm\n",
    "\n",
    "import watermark\n",
    "\n",
    "import epidemik\n",
    "from epidemik import EpiModel, NetworkEpiModel\n",
    "\n",
    "%load_ext watermark\n",
    "%matplotlib inline"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We start by print out the versions of the libraries we're using for future reference"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Python implementation: CPython\n",
      "Python version       : 3.13.2\n",
      "IPython version      : 8.32.0\n",
      "\n",
      "Compiler    : Clang 16.0.0 (clang-1600.0.26.6)\n",
      "OS          : Darwin\n",
      "Release     : 24.3.0\n",
      "Machine     : arm64\n",
      "Processor   : arm\n",
      "CPU cores   : 16\n",
      "Architecture: 64bit\n",
      "\n",
      "Git hash: 6d9f1a6eb4084a40d0e306ad0ba8a2eaa55e5c85\n",
      "\n",
      "watermark : 2.5.0\n",
      "epidemik  : 0.1.2\n",
      "matplotlib: 3.10.0\n",
      "networkx  : 3.4.2\n",
      "pandas    : 2.2.3\n",
      "tqdm      : 4.67.1\n",
      "numpy     : 2.2.2\n",
      "scipy     : 1.15.1\n",
      "\n"
     ]
    }
   ],
   "source": [
    "%watermark -n -v -m -g -iv"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Load default figure style"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.style.use('./d4sci.mplstyle')\n",
    "colors = plt.rcParams['axes.prop_cycle'].by_key()['color']"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# A simple Model"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Let us start with a network where eveyrone is connected to everybody else"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [],
   "source": [
    "np.random.seed(1234)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [],
   "source": [
    "N = 300\n",
    "beta = 0.05\n",
    "G_full = nx.erdos_renyi_graph(N, p=1.)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And an SI model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "ename": "TypeError",
     "evalue": "must be real number, not str",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
      "File \u001b[0;32m~/Library/Caches/pypoetry/virtualenvs/epidemiology101-UtPQcdCX-py3.13/lib/python3.13/site-packages/IPython/core/formatters.py:770\u001b[0m, in \u001b[0;36mPlainTextFormatter.__call__\u001b[0;34m(self, obj)\u001b[0m\n\u001b[1;32m    763\u001b[0m stream \u001b[38;5;241m=\u001b[39m StringIO()\n\u001b[1;32m    764\u001b[0m printer \u001b[38;5;241m=\u001b[39m pretty\u001b[38;5;241m.\u001b[39mRepresentationPrinter(stream, \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39mverbose,\n\u001b[1;32m    765\u001b[0m     \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39mmax_width, \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39mnewline,\n\u001b[1;32m    766\u001b[0m     max_seq_length\u001b[38;5;241m=\u001b[39m\u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39mmax_seq_length,\n\u001b[1;32m    767\u001b[0m     singleton_pprinters\u001b[38;5;241m=\u001b[39m\u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39msingleton_printers,\n\u001b[1;32m    768\u001b[0m     type_pprinters\u001b[38;5;241m=\u001b[39m\u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39mtype_printers,\n\u001b[1;32m    769\u001b[0m     deferred_pprinters\u001b[38;5;241m=\u001b[39m\u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39mdeferred_printers)\n\u001b[0;32m--> 770\u001b[0m \u001b[43mprinter\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43mpretty\u001b[49m\u001b[43m(\u001b[49m\u001b[43mobj\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m    771\u001b[0m printer\u001b[38;5;241m.\u001b[39mflush()\n\u001b[1;32m    772\u001b[0m \u001b[38;5;28;01mreturn\u001b[39;00m stream\u001b[38;5;241m.\u001b[39mgetvalue()\n",
      "File \u001b[0;32m~/Library/Caches/pypoetry/virtualenvs/epidemiology101-UtPQcdCX-py3.13/lib/python3.13/site-packages/IPython/lib/pretty.py:419\u001b[0m, in \u001b[0;36mRepresentationPrinter.pretty\u001b[0;34m(self, obj)\u001b[0m\n\u001b[1;32m    408\u001b[0m                         \u001b[38;5;28;01mreturn\u001b[39;00m meth(obj, \u001b[38;5;28mself\u001b[39m, cycle)\n\u001b[1;32m    409\u001b[0m                 \u001b[38;5;28;01mif\u001b[39;00m (\n\u001b[1;32m    410\u001b[0m                     \u001b[38;5;28mcls\u001b[39m \u001b[38;5;129;01mis\u001b[39;00m \u001b[38;5;129;01mnot\u001b[39;00m \u001b[38;5;28mobject\u001b[39m\n\u001b[1;32m    411\u001b[0m                     \u001b[38;5;66;03m# check if cls defines __repr__\u001b[39;00m\n\u001b[0;32m   (...)\u001b[0m\n\u001b[1;32m    417\u001b[0m                     \u001b[38;5;129;01mand\u001b[39;00m \u001b[38;5;28mcallable\u001b[39m(_safe_getattr(\u001b[38;5;28mcls\u001b[39m, \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124m__repr__\u001b[39m\u001b[38;5;124m\"\u001b[39m, \u001b[38;5;28;01mNone\u001b[39;00m))\n\u001b[1;32m    418\u001b[0m                 ):\n\u001b[0;32m--> 419\u001b[0m                     \u001b[38;5;28;01mreturn\u001b[39;00m \u001b[43m_repr_pprint\u001b[49m\u001b[43m(\u001b[49m\u001b[43mobj\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;28;43mself\u001b[39;49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mcycle\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m    421\u001b[0m     \u001b[38;5;28;01mreturn\u001b[39;00m _default_pprint(obj, \u001b[38;5;28mself\u001b[39m, cycle)\n\u001b[1;32m    422\u001b[0m \u001b[38;5;28;01mfinally\u001b[39;00m:\n",
      "File \u001b[0;32m~/Library/Caches/pypoetry/virtualenvs/epidemiology101-UtPQcdCX-py3.13/lib/python3.13/site-packages/IPython/lib/pretty.py:794\u001b[0m, in \u001b[0;36m_repr_pprint\u001b[0;34m(obj, p, cycle)\u001b[0m\n\u001b[1;32m    792\u001b[0m \u001b[38;5;250m\u001b[39m\u001b[38;5;124;03m\"\"\"A pprint that just redirects to the normal repr function.\"\"\"\u001b[39;00m\n\u001b[1;32m    793\u001b[0m \u001b[38;5;66;03m# Find newlines and replace them with p.break_()\u001b[39;00m\n\u001b[0;32m--> 794\u001b[0m output \u001b[38;5;241m=\u001b[39m \u001b[38;5;28;43mrepr\u001b[39;49m\u001b[43m(\u001b[49m\u001b[43mobj\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m    795\u001b[0m lines \u001b[38;5;241m=\u001b[39m output\u001b[38;5;241m.\u001b[39msplitlines()\n\u001b[1;32m    796\u001b[0m \u001b[38;5;28;01mwith\u001b[39;00m p\u001b[38;5;241m.\u001b[39mgroup():\n",
      "File \u001b[0;32m~/Library/Caches/pypoetry/virtualenvs/epidemiology101-UtPQcdCX-py3.13/lib/python3.13/site-packages/epidemik/EpiModel.py:562\u001b[0m, in \u001b[0;36mEpiModel.__repr__\u001b[0;34m(self)\u001b[0m\n\u001b[1;32m    560\u001b[0m text \u001b[38;5;241m+\u001b[39m\u001b[38;5;241m=\u001b[39m \u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mParameters:\u001b[39m\u001b[38;5;130;01m\\n\u001b[39;00m\u001b[38;5;124m\"\u001b[39m\n\u001b[1;32m    561\u001b[0m \u001b[38;5;28;01mfor\u001b[39;00m rate, value \u001b[38;5;129;01min\u001b[39;00m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39mparams\u001b[38;5;241m.\u001b[39mitems():\n\u001b[0;32m--> 562\u001b[0m     text \u001b[38;5;241m+\u001b[39m\u001b[38;5;241m=\u001b[39m \u001b[38;5;124;43m\"\u001b[39;49m\u001b[38;5;124;43m  \u001b[39;49m\u001b[38;5;132;43;01m%s\u001b[39;49;00m\u001b[38;5;124;43m : \u001b[39;49m\u001b[38;5;132;43;01m%f\u001b[39;49;00m\u001b[38;5;130;43;01m\\n\u001b[39;49;00m\u001b[38;5;124;43m\"\u001b[39;49m\u001b[43m \u001b[49m\u001b[38;5;241;43m%\u001b[39;49m\u001b[43m \u001b[49m\u001b[43m(\u001b[49m\u001b[43mrate\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mvalue\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m    563\u001b[0m text \u001b[38;5;241m+\u001b[39m\u001b[38;5;241m=\u001b[39m \u001b[38;5;124m\"\u001b[39m\u001b[38;5;130;01m\\n\u001b[39;00m\u001b[38;5;130;01m\\n\u001b[39;00m\u001b[38;5;124mTransitions:\u001b[39m\u001b[38;5;130;01m\\n\u001b[39;00m\u001b[38;5;124m\"\u001b[39m\n\u001b[1;32m    565\u001b[0m \u001b[38;5;28;01mfor\u001b[39;00m edge \u001b[38;5;129;01min\u001b[39;00m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39mtransitions\u001b[38;5;241m.\u001b[39medges(data\u001b[38;5;241m=\u001b[39m\u001b[38;5;28;01mTrue\u001b[39;00m):\n",
      "\u001b[0;31mTypeError\u001b[0m: must be real number, not str"
     ]
    }
   ],
   "source": [
    "SI_full = NetworkEpiModel(G_full)\n",
    "SI_full.add_interaction('S', 'I', 'I', beta=beta)\n",
    "SI_full"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "kavg= 299.0\n",
      "[('S', 'I', {'agent': 'I', 'rate': 'rate'})]\n"
     ]
    }
   ],
   "source": [
    "print(\"kavg=\", SI_full.kavg_)\n",
    "print(SI_full.transitions.edges(data=True))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We perform 100 runs"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [],
   "source": [
    "def simulate_runs(model, Nruns):\n",
    "    values = []\n",
    "\n",
    "    for i in tqdm(range(Nruns), total=Nruns):\n",
    "        model.simulate(100, seeds={30: 'I'})\n",
    "        values.append(model.I)\n",
    "\n",
    "    values =  pd.DataFrame(values).T\n",
    "    values.columns = np.arange(values.shape[1])\n",
    "    \n",
    "    return values"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "application/vnd.jupyter.widget-view+json": {
       "model_id": "5c484df21d494171b0937331f3c59ea5",
       "version_major": 2,
       "version_minor": 0
      },
      "text/plain": [
       "  0%|          | 0/100 [00:00<?, ?it/s]"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    },
    {
     "ename": "TypeError",
     "evalue": "'<' not supported between instances of 'float' and 'str'",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mTypeError\u001b[0m                                 Traceback (most recent call last)",
      "Cell \u001b[0;32mIn[10], line 2\u001b[0m\n\u001b[1;32m      1\u001b[0m Nruns \u001b[38;5;241m=\u001b[39m \u001b[38;5;241m100\u001b[39m\n\u001b[0;32m----> 2\u001b[0m values_full \u001b[38;5;241m=\u001b[39m \u001b[43msimulate_runs\u001b[49m\u001b[43m(\u001b[49m\u001b[43mSI_full\u001b[49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mNruns\u001b[49m\u001b[43m)\u001b[49m\n",
      "Cell \u001b[0;32mIn[9], line 5\u001b[0m, in \u001b[0;36msimulate_runs\u001b[0;34m(model, Nruns)\u001b[0m\n\u001b[1;32m      2\u001b[0m values \u001b[38;5;241m=\u001b[39m []\n\u001b[1;32m      4\u001b[0m \u001b[38;5;28;01mfor\u001b[39;00m i \u001b[38;5;129;01min\u001b[39;00m tqdm(\u001b[38;5;28mrange\u001b[39m(Nruns), total\u001b[38;5;241m=\u001b[39mNruns):\n\u001b[0;32m----> 5\u001b[0m     \u001b[43mmodel\u001b[49m\u001b[38;5;241;43m.\u001b[39;49m\u001b[43msimulate\u001b[49m\u001b[43m(\u001b[49m\u001b[38;5;241;43m100\u001b[39;49m\u001b[43m,\u001b[49m\u001b[43m \u001b[49m\u001b[43mseeds\u001b[49m\u001b[38;5;241;43m=\u001b[39;49m\u001b[43m{\u001b[49m\u001b[38;5;241;43m30\u001b[39;49m\u001b[43m:\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;124;43m'\u001b[39;49m\u001b[38;5;124;43mI\u001b[39;49m\u001b[38;5;124;43m'\u001b[39;49m\u001b[43m}\u001b[49m\u001b[43m)\u001b[49m\n\u001b[1;32m      6\u001b[0m     values\u001b[38;5;241m.\u001b[39mappend(model\u001b[38;5;241m.\u001b[39mI)\n\u001b[1;32m      8\u001b[0m values \u001b[38;5;241m=\u001b[39m  pd\u001b[38;5;241m.\u001b[39mDataFrame(values)\u001b[38;5;241m.\u001b[39mT\n",
      "File \u001b[0;32m~/Library/Caches/pypoetry/virtualenvs/epidemiology101-UtPQcdCX-py3.13/lib/python3.13/site-packages/epidemik/NetworkEpiModel.py:112\u001b[0m, in \u001b[0;36mNetworkEpiModel.simulate\u001b[0;34m(self, timesteps, seeds, **kwargs)\u001b[0m\n\u001b[1;32m    109\u001b[0m prob \u001b[38;5;241m=\u001b[39m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39mrng\u001b[38;5;241m.\u001b[39mrandom()\n\u001b[1;32m    111\u001b[0m rate \u001b[38;5;241m=\u001b[39m \u001b[38;5;28mself\u001b[39m\u001b[38;5;241m.\u001b[39mparams[infections[state_i][state_j][\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mrate\u001b[39m\u001b[38;5;124m\"\u001b[39m]]\n\u001b[0;32m--> 112\u001b[0m \u001b[38;5;28;01mif\u001b[39;00m \u001b[43mprob\u001b[49m\u001b[43m \u001b[49m\u001b[38;5;241;43m<\u001b[39;49m\u001b[43m \u001b[49m\u001b[43mrate\u001b[49m:\n\u001b[1;32m    113\u001b[0m     new_state \u001b[38;5;241m=\u001b[39m infections[state_i][state_j][\u001b[38;5;124m\"\u001b[39m\u001b[38;5;124mtarget\u001b[39m\u001b[38;5;124m\"\u001b[39m]\n\u001b[1;32m    114\u001b[0m     population[t, node_j] \u001b[38;5;241m=\u001b[39m new_state\n",
      "\u001b[0;31mTypeError\u001b[0m: '<' not supported between instances of 'float' and 'str'"
     ]
    }
   ],
   "source": [
    "Nruns = 100\n",
    "values_full = simulate_runs(SI_full, Nruns)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "And plot them. Each run has it's own stochastic path, despite the strong connectivity constraint"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "ename": "NameError",
     "evalue": "name 'values_full' is not defined",
     "output_type": "error",
     "traceback": [
      "\u001b[0;31m---------------------------------------------------------------------------\u001b[0m",
      "\u001b[0;31mNameError\u001b[0m                                 Traceback (most recent call last)",
      "Cell \u001b[0;32mIn[11], line 3\u001b[0m\n\u001b[1;32m      1\u001b[0m fig, ax \u001b[38;5;241m=\u001b[39m plt\u001b[38;5;241m.\u001b[39msubplots(\u001b[38;5;241m1\u001b[39m)\n\u001b[0;32m----> 3\u001b[0m \u001b[43mvalues_full\u001b[49m\u001b[38;5;241m.\u001b[39mmedian(axis\u001b[38;5;241m=\u001b[39m\u001b[38;5;241m1\u001b[39m)\u001b[38;5;241m.\u001b[39mplot(ax\u001b[38;5;241m=\u001b[39max, color\u001b[38;5;241m=\u001b[39mcolors[\u001b[38;5;241m0\u001b[39m])\n\u001b[1;32m      4\u001b[0m values_full\u001b[38;5;241m.\u001b[39mplot(ax\u001b[38;5;241m=\u001b[39max, color\u001b[38;5;241m=\u001b[39mcolors[\u001b[38;5;241m0\u001b[39m], lw\u001b[38;5;241m=\u001b[39m\u001b[38;5;241m.5\u001b[39m)\n\u001b[1;32m      5\u001b[0m ax\u001b[38;5;241m.\u001b[39mget_legend()\u001b[38;5;241m.\u001b[39mremove()\n",
      "\u001b[0;31mNameError\u001b[0m: name 'values_full' is not defined"
     ]
    },
    {
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",
      "text/plain": [
       "<Figure size 3840x2640 with 1 Axes>"
      ]
     },
     "metadata": {},
     "output_type": "display_data"
    }
   ],
   "source": [
    "fig, ax = plt.subplots(1)\n",
    "\n",
    "values_full.median(axis=1).plot(ax=ax, color=colors[0])\n",
    "values_full.plot(ax=ax, color=colors[0], lw=.5)\n",
    "ax.get_legend().remove()\n",
    "ax.set_xlim(0, 20)\n",
    "ax.legend(['Median Run', 'Individual Runs'])\n",
    "ax.set_xlabel('Time')\n",
    "ax.set_ylabel('Population')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "kmean = SI_full.kavg_\n",
    "print(kmean)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can obtain a specific average degree by setting: $$p=\\frac{\\langle k\\rangle}{N}$$ We choose this specific value so that we match the average degree of the BA model below"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "G_small = nx.erdos_renyi_graph(N, p=0.0198)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "SI_small = NetworkEpiModel(G_small)\n",
    "SI_small.add_interaction('S', 'I', 'I', beta)\n",
    "SI_small"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "SI_small.kavg_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "values_small = simulate_runs(SI_small, Nruns)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(1)\n",
    "\n",
    "values_small.median(axis=1).plot(ax=ax, color=colors[1])\n",
    "values_small.plot(ax=ax, color=colors[1], lw=.3)\n",
    "\n",
    "ax.get_legend().remove()\n",
    "ax.legend(['Median Run', 'Individual Runs'])\n",
    "ax.set_xlabel('Time')\n",
    "ax.set_ylabel('Population')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "time = np.arange(0, 100)\n",
    "yt = lambda time, beta: N/(1+(N-1)*np.exp(-beta*N*time))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "coef, covariance = curve_fit(yt, time, values_small.median(axis=1).values)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "coef"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(1)\n",
    "\n",
    "ax.plot(time, yt(time, *coef), color=colors[5])\n",
    "values_small.median(axis=1).plot(ax=ax, color=colors[1])\n",
    "values_small.plot(ax=ax, color=colors[1], lw=.3)\n",
    "\n",
    "ax.get_legend().remove()\n",
    "ax.legend(['Fit', 'Median Run', 'Individual Runs'])\n",
    "ax.set_xlabel('Time')\n",
    "ax.set_ylabel('Population')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We can compare the two scenarios above by plotting the median number of infected node as a function of time. Not surprisingly, the network with the smallest average connectivity is slower"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(1)\n",
    "values_small.median(axis=1).plot(color=colors[1], ax=ax, label='ER Network')\n",
    "values_full.median(axis=1).plot(color=colors[0], ax=ax, label='Full Network')\n",
    "ax.legend()\n",
    "ax.set_xlabel('Time')\n",
    "ax.set_ylabel('Population')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.plot(dict(G_small.degree()).values())\n",
    "\n",
    "ax = plt.gca()\n",
    "ax.set_xlabel('Node')\n",
    "ax.set_ylabel('Degree')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "But what about two networks with exactly the same average degree, but different topologies? Let's run a Barabasi albert model"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "BA = nx.barabasi_albert_graph(N, m=3)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "SI_BA = NetworkEpiModel(BA)\n",
    "SI_BA.add_interaction('S', 'I', 'I', beta)\n",
    "SI_BA"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "The average degree is exactly the same"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "SI_BA.kavg_"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "values_BA = simulate_runs(SI_BA, Nruns)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(1)\n",
    "\n",
    "values_BA.median(axis=1).plot(color=colors[2], ax=ax)\n",
    "values_BA.plot(ax=ax, color=colors[2], lw=.1)\n",
    "\n",
    "ax.get_legend().remove()\n",
    "ax.legend(['Median Run', 'Individual Runs'])\n",
    "\n",
    "ax.set_xlabel('Time')\n",
    "ax.set_ylabel('Population')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "coef, covariance = curve_fit(yt, time, values_BA.median(axis=1).values)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "coef"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(1)\n",
    "\n",
    "ax.plot(time, yt(time, *coef), color=colors[5])\n",
    "values_BA.median(axis=1).plot(ax=ax, color=colors[2])\n",
    "values_BA.plot(ax=ax, color=colors[2], lw=.3)\n",
    "\n",
    "ax.get_legend().remove()\n",
    "ax.legend(['Fit', 'Median Run', 'Individual Runs'])\n",
    "ax.set_xlabel('Time')\n",
    "ax.set_ylabel('Population')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "plt.plot(dict(BA.degree()).values())\n",
    "ax = plt.gca()\n",
    "ax.set_xlabel('Node')\n",
    "ax.set_ylabel('Degree')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "## Topology comparison"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(1)\n",
    "values_small.median(axis=1).plot(color=colors[1], ax=ax, label='ER Network')\n",
    "values_BA.median(axis=1).plot(color=colors[2], ax=ax, label='BA Network')\n",
    "ax.legend()\n",
    "\n",
    "ax.set_xlabel('Time')\n",
    "ax.set_ylabel('Population')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "fig, ax = plt.subplots(1)\n",
    "values_full.median(axis=1).plot(color=colors[0], ax=ax, label='Full Network')\n",
    "values_small.median(axis=1).plot(color=colors[1], ax=ax, label='ER Network')\n",
    "values_BA.median(axis=1).plot(color=colors[2], ax=ax, label='BA Network')\n",
    "ax.legend()\n",
    "\n",
    "ax.set_xlabel('Time')\n",
    "ax.set_ylabel('Population')"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "<center>\n",
    "     <img src=\"data/D4Sci_logo_full.png\" alt=\"Data For Science, Inc\" align=\"center\" border=\"0\" width=300px> \n",
    "</center>"
   ]
  }
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