ch03/stratified_r0/R_0 of stratified system.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "source": [
    "| **Chapter**                   | 3: Host factors                                                    |\n",
    "| ----------------------------- |--------------------------------------------------------------------|\n",
    "| **Computational Note**        | 3.1: Calculating the $\\mathfrak{R}_0$ of complex stratified models |\n",
    "| **Package prerequisites**     | [SymPy](https://www.sympy.org/en/index.html)                       |\n",
    "\n",
    "# Calculating the $\\mathfrak{R}_0$ of complex stratified models"
   ],
   "attachments": {},
   "metadata": {
    "datalore": {
     "type": "MD",
     "hide_input_from_viewers": false,
     "hide_output_from_viewers": false
    },
    "pycharm": {
     "name": "#%% md\n"
    }
   }
  },
  {
   "cell_type": "code",
   "source": [
    "import numpy as np\n",
    "from sympy.interactive.printing import init_printing\n",
    "init_printing(use_unicode=True, wrap_line=True)\n",
    "from sympy.matrices import Matrix\n",
    "from sympy import symbols\n",
    "from sympy import factor\n",
    "from sympy import latex\n",
    "from sympy import lambdify"
   ],
   "execution_count": 1,
   "outputs": [],
   "metadata": {
    "datalore": {
     "type": "CODE",
     "hide_input_from_viewers": false,
     "hide_output_from_viewers": false
    },
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "outputs": [
    {
     "name": "stdout",
     "output_type": "stream",
     "text": [
      "Author: Chris von Csefalvay\n",
      "\n",
      "Github username: chrisvoncsefalvay\n",
      "\n",
      "Python implementation: CPython\n",
      "Python version       : 3.10.2\n",
      "IPython version      : 8.4.0\n",
      "\n",
      "sympy: 1.10.1\n",
      "\n",
      "numpy: 1.22.4\n",
      "\n"
     ]
    }
   ],
   "source": [
    "%load_ext watermark\n",
    "%watermark --iversions -v -a \"Chris von Csefalvay\" -gu \"chrisvoncsefalvay\" -p sympy"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "code",
   "source": [
    "I_H, I_L, beta_HH, beta_HL, beta_LH, beta_LL, gamma_H, gamma_L, n_H, n_L = symbols(\"I_H I_L beta_HH beta_HL beta_LH beta_LL gamma_H gamma_L n_H n_L\")"
   ],
   "execution_count": 3,
   "outputs": [],
   "metadata": {
    "datalore": {
     "type": "CODE",
     "hide_input_from_viewers": false,
     "hide_output_from_viewers": false
    },
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "code",
   "source": [
    "infectious_system = Matrix(np.array([I_H, I_L]).T)"
   ],
   "execution_count": 4,
   "outputs": [],
   "metadata": {
    "datalore": {
     "type": "CODE",
     "hide_input_from_viewers": false,
     "hide_output_from_viewers": false
    },
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "code",
   "source": [
    "d_Is = Matrix(np.array([(beta_HH * n_H - gamma_H) * I_H + (beta_HL * n_H) * I_L,\n",
    "                        beta_LH * n_L * I_H + (beta_LL * n_L - gamma_L) * I_L]).T)"
   ],
   "execution_count": 5,
   "outputs": [],
   "metadata": {
    "datalore": {
     "type": "CODE",
     "hide_input_from_viewers": false,
     "hide_output_from_viewers": false
    },
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "code",
   "source": [
    "coeffs = d_Is.jacobian(infectious_system)\n",
    "coeffs"
   ],
   "execution_count": 6,
   "outputs": [
    {
     "data": {
      "text/plain": "⎡β_HH⋅n_H - γ_H     β_HL⋅n_H   ⎤\n⎢                              ⎥\n⎣   β_LH⋅n_L     β_LL⋅n_L - γ_L⎦",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle \\left[\\begin{matrix}\\beta_{HH} n_{H} - \\gamma_{H} & \\beta_{HL} n_{H}\\\\\\beta_{LH} n_{L} & \\beta_{LL} n_{L} - \\gamma_{L}\\end{matrix}\\right]$"
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "metadata": {
    "datalore": {
     "type": "CODE",
     "hide_input_from_viewers": false,
     "hide_output_from_viewers": false
    },
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "code",
   "source": [
    "beta = Matrix([[beta_HH, beta_HL], [beta_LH, beta_LL]])\n",
    "beta"
   ],
   "execution_count": 7,
   "outputs": [
    {
     "data": {
      "text/plain": "⎡β_HH  β_HL⎤\n⎢          ⎥\n⎣β_LH  β_LL⎦",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle \\left[\\begin{matrix}\\beta_{HH} & \\beta_{HL}\\\\\\beta_{LH} & \\beta_{LL}\\end{matrix}\\right]$"
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "metadata": {
    "datalore": {
     "type": "CODE",
     "hide_input_from_viewers": false,
     "hide_output_from_viewers": false
    },
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "code",
   "source": [
    "J = coeffs.subs({beta_HH: 10,\n",
    "                 beta_HL: 0.5,\n",
    "                 beta_LH: 0.5,\n",
    "                 beta_LL: 2,\n",
    "                 n_H: 0.2,\n",
    "                 n_L: 0.8,\n",
    "                 gamma_H: 0.05,\n",
    "                 gamma_L: 0.05})"
   ],
   "execution_count": 9,
   "outputs": [],
   "metadata": {
    "datalore": {
     "type": "CODE",
     "hide_input_from_viewers": false,
     "hide_output_from_viewers": false
    },
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "code",
   "execution_count": 13,
   "outputs": [
    {
     "data": {
      "text/plain": "⎡1.95  0.1 ⎤\n⎢          ⎥\n⎣0.4   1.55⎦",
      "image/png": "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\n",
      "text/latex": "$\\displaystyle \\left[\\begin{matrix}1.95 & 0.1\\\\0.4 & 1.55\\end{matrix}\\right]$"
     },
     "execution_count": 13,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "J"
   ],
   "metadata": {
    "collapsed": false,
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "code",
   "source": [
    "R_0 = max([abs(i) for i in J.eigenvals()])"
   ],
   "execution_count": 14,
   "outputs": [],
   "metadata": {
    "datalore": {
     "type": "CODE",
     "hide_input_from_viewers": false,
     "hide_output_from_viewers": false
    },
    "pycharm": {
     "name": "#%%\n"
    }
   }
  },
  {
   "cell_type": "code",
   "source": [
    "R_0"
   ],
   "execution_count": 15,
   "outputs": [
    {
     "data": {
      "text/plain": "2.03284271247462",
      "image/png": "iVBORw0KGgoAAAANSUhEUgAAAJ8AAAAPCAYAAAD6fR2jAAAACXBIWXMAAA7EAAAOxAGVKw4bAAAGVUlEQVRoBe2Z3ZEVNxBGL9QGADgDyAB2IwAyABMBkIEp3nijIANDBIAzAEeAIQOTgc1mgM8RaqHRaGZ0p7ZcfrCqdFtqff0jqdWamXvp27dvh//LcSvw7Nmz60jcob6lfX6c9L+H/q/7ecngy04+yctyCv2b+gT+59GlAvsiY/+C3qC+gPells92HmfeFaibKO5DB7fpD3LqeErV5k9U++9affBmBcxNmI+h4U/C0NenVdtg7oF5lwT6P+dgrrZD8LSp3C3as6CFt2m71Rn9rHsyH3i7/FRnT1/YCgom9jxYb+CVmKG9Op+TDDAA7ocG2ir9BL1LnQRGYGoK5hP959Df5EMNgpBPAZh5BnTZbNqPwL2H3qeGbATkiD/6XfRl2wbfldAnb6EYBGWhxCAzavssyzo3D2pdzIgRvOp0LV5TxXmwtTErR9ieyWbGbD7wh/3sKO3pS7Ds63s6BnuKD6h4q4nnkDGrcXUCzkB7qEAUBA0SA0NlsxMcOGnGTTYbniffYPqVelccRX2P4BtsKdDov6WKMXsFb8gfdKjPoG+Lc/mdGvra8QOyv8yY3xlDtoE631utDngpsKDFNm0zXDpItLVr9uuVUdsz2ax3xocx7GctvKIvYMbFB3B1YvKQ1Tfd5nwuI+BJ/YoiheuiYp3vntQK6MJOMkge+wi9U+kV40ZYU2GstIMHHfXHE5Y2tZLdbGLTzZ/4UQmN2vbU98osE/dAC7xR2xPxjfkc7eeGvgPjXuWuYcnuOgTfWzISjazN+Rh8BtkXBHuBoJIr/qwUjbRXj/DQ57jOeVKuSu1baDsRi9kvyqg/LqzBbSatffTE1fpCb9AH4F9Fp6FDtpEvmS3k4Wn3efR30CHbHb2L89np56K+bNtbypst9rfjUmJtzucEJUvZI10PjPeyWtLOWL3pS05c6w0ga1C6YT43lGCgPeQPOIPZIDCAzdyeRLNhfa3T/VHAeO0tBibjQ7Z/aPzeQs7b4Wb2oR0e6u+xjczqfFrDW34O6jMuPoOVPqDGC+bkRY/xzbU8aR20nxW7oJPU2sFGYPVOQWTDSYBm3QbeGdXA/qOjd8Ja8ge+LyoGsJsgVZ90VsA5H09s/Vwyw7UM8C7y1lpos2u31XdMf832zvks+jmiD0y9l6f0S3zQNgE8pM5uhZgzY5O1vBwDDfWB8hXglw1/T9dPIKWg01PzkurJeEP1rTiu34JrGl1/Kjkznmneyf0JP131jY5Jhm3G1rpd2yGALQPzHlT7F13WbB81nwE/j9Fnli+3VZ60L4+v4dcB2q7HZD6z4EPYa8krbfIJo9WS+5HdesORFU3L3YINT4lZM30e6YGW/IHv264Pub6Z+8zqw26k+om+jF28bnt25SE3shaukxn3Qsua7Z3zWfRzVB8498rSuz388mDgnQpoS28+k+DLThygI4EnLpzRaFuClxwF62kxM7Ulrt1ZtgJvgB2gPX+8QkrazziD2Syo7aQPWTOTb+29BWOoX8Cv2a6FzNprh7DGDrXXbO+dD4a7fu7Q5zrGvvfm43pPytJ8TgIFQOduQMtGZ8cO0LWN87qZGYQXmc8sajCkb3K0feNdcx5oCrhFfxh2sw2omR54ZkGDMOzr2xk8U35dPAjXM1+ZEsi0F20zVtaCtvNS/4VlvgHbe+az5uex+py/MkulrI+Atfmk4APgRrhBZQOyZjehvdvzUCFuau9h24+wPt+lAMnUQGwD5jRrKs9MYFb9UUeuBs9kslmXi530MS4tuvP4Af5X+dC4qtMQ/VXbIZ9p+H4hmW/ENpij5rPl5w597nXv+6H77b6Utd6azwkAo9gAciPq56Jr8Lwq00sH1A11wwyo8nWfti8mPnf50G3GOUDF/ky9bT+XNrDFGdxifdg9Fwcd8geoQeOz3e2QzfJel35u6QWlkCjatZZyhO2QCfnkezD30B22WzOz+VSAPX529eGncWL1g3raU6hY99t/l1KBt7mPZj4DT2B6xkmSP37KdYIyo9oNjWe0Hyj+KKfzNBuUf0Y1KGp5g/QOtQ5w7bb/H4/64wI4Wd+w6swz+d7E+KRk+5Gx9MdTbLB6yIZsVwqd3zn1Y8WbNdGtXjco7PqGr+zHbFeZY20rc0De9Qy97XwShp8hPwWP6APjnhl8sZcmqsl+09+czz8yydgyfyhwgAAAAABJRU5ErkJggg==\n",
      "text/latex": "$\\displaystyle 2.03284271247462$"
     },
     "execution_count": 15,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "metadata": {
    "datalore": {
     "type": "CODE",
     "hide_input_from_viewers": false,
     "hide_output_from_viewers": false
    },
    "pycharm": {
     "name": "#%%\n"
    }
   }
  }
 ],
 "metadata": {
  "datalore": {
   "version": 1,
   "computation_mode": "JUPYTER",
   "package_manager": "pip",
   "base_environment": "default",
   "packages": []
  }
 },
 "nbformat": 4,
 "nbformat_minor": 4
}