{
"cells": [
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"from __future__ import print_function\n",
"%matplotlib inline\n",
"%precision 16\n",
"import numpy\n",
"import matplotlib.pyplot as plt\n",
"from scipy.integrate import solve_ivp\n",
"\n",
"from stability_diagrams import plot_stability_ssms\n",
"from stability_diagrams import plot_stability_lmm"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Question 1 - The SIR model for Infectious diseases\n",
"\n",
"ODE based models are a classical approach to understanding the spread of infectious diseases.\n",
"\n",
"One of the pioneering models is the SIR model of [Kermack and McKendrick, (1927)](https://royalsocietypublishing.org/doi/10.1098/rspa.1927.0118) which is a mean-field theory for large populations that assumes the general population can be divided into three subgroups\n",
"\n",
"* $S$: the number of susceptible patients\n",
"* $I$: the number of infected patients\n",
"* $R$: the number of recovered (or otherwise) patients\n",
"\n",
"Given these definitions, the simplest dynamical systems for the three populations can be written\n",
"\n",
"$$\n",
"\\begin{aligned}\n",
"\\frac{d S}{dt} &= -\\alpha SI \\\\\n",
"\\frac{d I}{dt} &= \\alpha SI -\\beta I\\\\\n",
"\\frac{d R}{dt} &= \\beta I \\\\\n",
"\\end{aligned}\n",
"$$\n",
"\n",
"where $\\alpha$ is the infection rate per person, and $\\beta$ is the recovery rate (hint: check the units) \n",
"\n",
"This is the original Kermack and McKendrick model. We will discuss a host of themes and variations later. If you are interested [Siettos and Russo, (2013)](https://doi.org/10.4161/viru.24041) provides a short overview of the much larger field of epidemiology modeling (although this paper has some significant mistakes with respect to the analysis of the Kermack and McKendrick model).\n",
"\n",
"We will explore this problem through a combination of Analysis and Numerical methods and to use it to test different methods for ODE solvers.\n",
"\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**(a)** Explain in words, what this system of ODE's describes"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"### Some analysis. \n",
"\n",
"Here we will conduct some elementary dimensional analysis to reduce the parameter space. More detailed analysis of this system can be found Steve Strogatz' excellent text on [dynamical systems](https://smile.amazon.com/Nonlinear-Dynamics-Chaos-Applications-Nonlinearity/dp/0813349109/ref=dp_ob_image_bk)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**(b)** Show that the SIR equations imply that $S + I + R = N$ where $N$ is a constant (the total population of susceptible people that can all interact)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"**(c)** Show that under the change of variables (linear scaling of the variables)\n",
"\n",
"$$\n",
"\\begin{matrix}\n",
" S = N s, & I = Ni, & R = Nr, & t = \\tau\\frac{1}{\\alpha N} \\\\\n",
"\\end{matrix}\n",
"$$\n",
"\n",
"The original equations can be rewritten as the scaled equations \n",
"\n",
"$$\n",
"\\begin{align}\n",
" \\frac{d s}{d\\tau} &= -si\\\\\n",
" \\frac{d i}{d\\tau} &= (s - s^*)i \\\\\n",
" \\frac{d r}{d\\tau} & = s^* i \\\\\n",
"\\end{align}\n",
"$$\n",
"\n",
"where \n",
"$$\n",
" s^* = \\frac{\\beta}{\\alpha N}\n",
"$$\n",
"\n",
"\n",
"\n",
"**Notes**, \n",
"* if we write $s = S/N$ etc. then $s$ is simply the fraction of the total population that is susceptible. \n",
"* Also show that $s + i + r = 1$.\n",
"* this scaling reduces a two-parameter problem to a one parameter problem where the only variable is $s^*$\n",
"\n",
"\n"
]

Prove that for any positive integer n, n!