{
"cells": [
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Modeling the COVID-19 Pandemic"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Reading\n",
"\n",
"1. Bregman D.J., A.D. Langmuir, 1990. Farr's law applied to AIDS projections. _JAMA_, 263:1522–1525. doi: 10.1001/jama.263.11.1522. \n",
"\n",
"1. King, A.A., M.D. de Cellès, F.M.G. Magpantay, and P. Rohani, 2015. Avoidable errors in the modelling of outbreaks of emerging pathogens, with special reference to Ebola, _Proc. R. Soc. B._ 28220150347 http://doi.org/10.1098/rspb.2015.0347.\n",
"\n",
"1. Murray, C.J.L., and IHME COVID-19 health service utilization forecasting team, 2020. Forecasting COVID-19 impact on hospital bed-days, ICU-days, ventilator-days and deaths by US state in the next 4 months,\n",
"https://www.medrxiv.org/content/10.1101/2020.03.27.20043752v1.full-text\n",
"\n",
"1. Jewell, N.P., J.A. Lewnard, and B.L. Jewell, 2020.\n",
"Predictive Mathematical Models of the COVID-19 Pandemic: Underlying Principles and Value of Projections,\n",
"_JAMA_, 323(19):1893-1894. doi:10.1001/jama.2020.6585\n",
"\n",
"1. Froese, H., 2020. Infectious Disease Modelling: Fit Your Model to Coronavirus Data, Towards Data Science, https://towardsdatascience.com/infectious-disease-modelling-fit-your-model-to-coronavirus-data-2568e672dbc7; https://github.com/henrifroese/infectious_disease_modelling \n",
"\n",
"1. Smith, D., and L. Moore, 2004. The SIR model for spread of disease, _Convergence_, https://www.maa.org/press/periodicals/loci/joma/the-sir-model-for-spread-of-disease-introduction\n",
"\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Models of epidemics\n",
"\n",
"This notebook explores two basic approaches to modeling the spread of a communicable disease through a population. There are more complicated approaches than are considered here, but these are common, and the predictions of COVID-19 infections, ICU beds, and deaths are often based on variations of these models."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# SIR models\n",
"\n",
"Mathematical models of epidemics date back to the early 1900s.\n",
"\n",
"One of the simplest models of epidemics is the SIR model, which partitions the population into three disjoint groups: Susceptible, Infected, and Recovered (or Removed).\n",
"\n",
"The SIR model treats the population as fixed: no births, immigration, emigration, or deaths from other causes.\n",
"It ignores the possibility of carriers, population heterogeneity, or geographic heterogeneity.\n",
"It treats everyone as equivalent and assumes the population \"mixes\" perfectly.\n",
"\n",
"Here are the state variables of the SIR model:\n",
"\n",
"+ $N$: initial population size\n",
"+ $S(t)$: number of susceptible individuals in the population at time $t$\n",
"+ $I(t)$: number of infected individuals in the population at time $t$\n",
"+ $R(t)$: number \"removed\" from risk (dead or recovered) in the population at time $t$\n",
"\n",
"Assumptions:\n",
"\n",
"+ Everyone is either susceptible, infected, or removed (recovered or dead). (There are more complex models with more \"compartments.\")\n",
"+ There are no births, immigration, emigration, or deaths from other causes.\n",
"+ If a susceptible person is infected, that person eventually dies or recovers. \n",
"+ While a person is infected, the person is infectious.\n",
"+ There is no incubation period between being infected and being infectious.\n",
"+ If an infected person gets \"close enough\" to a susceptible person, the susceptible person becomes infected.\n",
"+ Every unit of time, every infected person exposes $\\beta$ people.\n",
"+ Every susceptible person exposed to an infected person becomes infected.\n",
"+ Every infected person exposes a disjoint group of people.\n",
"+ Every unit of time, a fraction $\\gamma$ of infected people recover or die.\n",
"+ People who recover or die are not infectious.\n",
"+ People who recover or die never become susceptible again.\n",
"\n",
"Then $S + I + R = N$ is constant, the initial population size.\n",
"\n",
"The basic reproductive number is $R_0 = \\beta/\\gamma$.\n",
"Initially, when essentially the entire population is susceptible (the proportion who are infected or recovered is small compared to $N$), this is the number of people each infected person infects before recovering or dying. \n",
"(Note that $R_0$ has no connection to $R$: the notation is unfortunate.)\n",
"\n",
"Define $s(t) \\equiv S(t)/N$, $i(t) \\equiv I(t)/N$, and $r(t) \\equiv R(t)/N$.\n",
"Then $s+i+r = 1$ for all $t$.\n",
"There is an epidemic if $s(0)\\beta/\\gamma > 1$: the rate at which susceptible people are\n",
"getting infected is greater than the rate at which infected people are recoving. \n",
"\n",
"The parameters of the model are:\n",
"\n",
"+ $\\beta$, the number of exposed people per unit of time per infected persion\n",
"+ $\\gamma$, the fraction of infected people who are removed (recover or die) per unit time\n",
"\n",
"The state of the population evolves according to three coupled differential equations:\n",
"\n",
"+ $dS/dt = -\\beta I S/N$\n",
"+ $dI/dt = \\beta I S/N - \\gamma I$ (equivalently, $di/dt = \\beta is - \\gamma i$)\n",
"+ $dR/dt = \\gamma I$\n",
"\n",
"Another interesting variable is the cumulative number of infections, $C$.\n",
"\n",
"+ $dC/dt = \\beta I S/N$; $C(0) = I(0)$.\n",
"\n",
"The SIR model has some built-in features that might not match reality.\n",
"First, note that the derivative of $R$ is always non-negative and proportional to $I$.\n",
"Since $R \\le N$ and $N$ is finite, $I$ must eventually be zero; that is, the SIR implies that eventually\n",
"the entire population will be disease-free. \n",
"Once $I=0$, it can never grow again, since the derivative of $I$ is proportional to $I$.\n",
"(If nobody is infected, nobody can infect anyone.)\n",
"Epidemics that follow the SIR model are necessarily self-limiting.\n",
"\n",
"That isn't the case if immunity wears off over time (or if the population is not isolated).\n",
"In that case, we might we parametrize the transition from recovered to susceptible by \n",
"assuming that a fraction $\\delta$ of $R$ return to $S$ in each time period. \n",
"(There are other ways we could do this, for instance, having a variable \"lag\" between recovery and becoming susceptible again.)\n",
"That yields:\n",
"\n",
"+ $dS/dt = -\\beta I S/N + \\delta R$\n",
"+ $dI/dt = \\beta I S/N - \\gamma I$ (equivalently, $di/dt = \\beta is - \\gamma i$)\n",
"+ $dR/dt = \\gamma I - \\delta R$\n",
"\n",
"In this model, $I$ does not necessarily go to zero eventually.\n",
"\n",
"Other generalizations include separate \"compartments\" for dead versus recovered), needing an ICU bed, etc., as well as time-varying values of $\\beta$ to account for lockdowns or other policy interventions, heterogeneity in the population, and more."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Let's run this model (in discrete time) for a population of $N=1,000,000$ people of whom 0.05% are initially infected.\n",
"We will take the time interval to be 1 day, the period of infectiousness to be $1/\\gamma = 14$ days,\n",
"and $R_0 = 2 = \\beta/\\gamma$, so $\\beta = 2 \\gamma = 1/7$. We will start with $\\delta=0$ (no re-infections)."
]
},
{
"cell_type": "code",
"execution_count": 1,
"metadata": {},
"outputs": [],
"source": [
"# boilerplate\n",
"import numpy as np\n",
"import scipy as sp\n",
"from scipy.integrate import odeint\n",
"import matplotlib.pyplot as plt\n",
"import matplotlib.dates as mdates\n",
"\n",
"from ipywidgets import interact, interactive, fixed, interact_manual\n",
"import ipywidgets as widgets\n",
"from IPython.display import clear_output, display, HTML"
]
},
{
"cell_type": "code",
"execution_count": 2,
"metadata": {},
"outputs": [],
"source": [
"def sir_model(N, i_0, beta=1, gamma=1, delta=0, steps=365):\n",
" '''\n",
" Run SIR model as coupled finite-difference equations.\n",
" \n",
" Assumes that initially a fraction i_0 of the population of N people is infected and \n",
" everyone else is susceptible (none has died or recovered).\n",
" \n",
" Parameters\n",
" -----------\n",
" N : int\n",
" population size\n",
" i_0 : double in (0, 1)\n",
" initial fraction of population infected\n",
" beta : double in [0, infty)\n",
" number of encounters each infected person has per time step\n",
" gamma : double in (0, 1)\n",
" fraction of infecteds who recover or die per time step\n",
" delta : double in (0, 1)\n",
" fraction of recovereds who become susceptible per time step\n",
" steps : int\n",
" number of steps in time to run the model\n",
" \n",
" Returns\n",
" --------\n",
" S, I, R, C \n",
" S : list\n",
" time history of susceptibles\n",
" I : list\n",
" time history of infecteds\n",
" R : list\n",
" time history of recovered/dead \n",
" C : list\n",
" cumulative number of infections over time\n",
" '''\n",
" assert i_0 > 0, 'initial rate of infection is zero'\n",
" assert i_0 <= 1, 'infection rate greater than 1'\n",
" assert beta >= 0, 'beta must be nonnegative'\n",
" assert gamma > 0, 'gamma must be positive'\n",
" assert gamma < 1, 'gamma must be less than 1'\n",
" assert delta >= 0, 'delta must be nonnegative'\n",
" assert delta < 1, 'delta must be less than 1'\n",
" S = np.zeros(steps)\n",
" I = np.zeros(steps)\n",
" R = np.zeros(steps)\n",
" C = np.zeros(steps)\n",
" I[0] = int(N*i_0)\n",
" C[0] = I[0]\n",
" S[0] = N-I[0]\n",
" for i in range(steps-1):\n",
" new_i = beta*I[i]*S[i]/N\n",
" S[i+1] = max(0, S[i] - new_i + delta*R[i])\n",
" I[i+1] = max(0, I[i] + new_i - gamma*I[i])\n",
" R[i+1] = max(0, R[i] + gamma*I[i] - delta*R[i])\n",
" C[i+1] = C[i] + new_i\n",
" return S, I, R, C"
]
},
{
"cell_type": "code",
"execution_count": 3,
"metadata": {},
"outputs": [],
"source": [
"def plot_sir(N, i_0, beta=2/14, gamma=1/14, delta=0, steps=365, verbose=False):\n",
" '''\n",
" Plot the time history of an SIR model.\n",
" \n",
" Parameters\n",
" -----------\n",
" N : int\n",
" population size\n",
" i_0 : double in (0, 1)\n",
" infection rate at time 0\n",
" beta : double in [0, infty)\n",
" number of encounters each infected person has per time step\n",
" gamma : double in (0, 1)\n",
" fraction of infecteds who recover or die per time step\n",
" delta : double in (0, 1)\n",
" fraction of recovereds who become susceptible per time step\n",
" steps : int\n",
" number of time steps to run the model\n",
" verbose : Boolean\n",
" if True, return the model predictions\n",
" \n",
" Returns (if verbose == True)\n",
" --------\n",
" S : list\n",
" susceptibles as a function of time\n",
" I : list\n",
" infecteds as a function of time\n",
" R : list\n",
" recovereds as a function of time\n",
" C : list\n",
" cumulative incidence as a function of time\n",
" '''\n",
" S, I, R, C = sir_model(N, i_0, beta=beta, gamma=gamma, delta=delta, steps=steps)\n",
" times = list(range(steps))\n",
" fig, ax = plt.subplots(nrows=1, ncols=1)\n",
" ax.plot(times, I, linestyle='--', color='r', label='Infected')\n",
" ax.plot(times, S, linestyle='-', color='b', label='Susceptible')\n",
" ax.plot(times, R, linestyle=':', color='g', label='Recovered')\n",
" ax.plot(times, C, linestyle='-.', color='k', label='Tot infect.')\n",
" ax.legend(loc='best')\n",
" plt.show()\n",
" if verbose:\n",
" return S, I, R, C"
]
},
{
"cell_type": "code",
"execution_count": 4,
"metadata": {},
"outputs": [
{
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\n",
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"i_0 = 0.0005\n",
"N = int(10**6)\n",
"steps = 365\n",
"\n",
"plot_sir(N, i_0)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"For these parameters, the number of infections is shaped like a bell curve and the cumulative number of infections is shaped like a CDF: it is \"sigmoidal\" (vaguely \"S\"-shaped). You might imagine fitting a scaled CDF to the curve. Essentially, the IHME model fits a scaled Gaussian CDF to the total number of infections.\n",
"(More on this below.)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"What if immunity wears off over time? Let's try a positive value of $\\delta$:"
]
},
{
"cell_type": "code",
"execution_count": 5,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"plot_sir(N, i_0, delta=0.01)"
]
},
{
"cell_type": "code",
"execution_count": 6,
"metadata": {},
"outputs": [
{
"data": {
"application/vnd.jupyter.widget-view+json": {
"model_id": "da8e0e4dea6143c2932312d32071b4b6",
"version_major": 2,
"version_minor": 0
},
"text/plain": [
"interactive(children=(FloatSlider(value=0.0005, description='i_0', max=1.0), FloatSlider(value=0.1428571428571…"
]
},
"metadata": {},
"output_type": "display_data"
},
{
"data": {
"text/plain": [
"<function __main__.plot_sir(N, i_0, beta=0.14285714285714285, gamma=0.07142857142857142, delta=0, steps=365, verbose=False)>"
]
},
"execution_count": 6,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"interact(plot_sir, N=fixed(N),\n",
" i_0 = widgets.FloatSlider(min=0, max=1, value=i_0),\n",
" beta=widgets.FloatSlider(value=2/14, min=0.01, max=100),\n",
" gamma=widgets.FloatSlider(value=1/14, min=0, max=1, step=0.01),\n",
" delta=widgets.FloatSlider(value=0, min=0, max=1, step=0.01),\n",
" steps=fixed(steps),\n",
" verbose=fixed(False))"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Fitting the SIR model to data\n",
"\n",
"Now we will use least squares to fit the SIR model to a time history of infection data collected by researchers at Johns Hopkins University. See https://github.com/CSSEGISandData\n",
"\n",
"\n",
"This exercise is for illustration. There are many reasons to be wary of this modeling:\n",
"\n",
"+ There are serious issues with the data quality: these are cases that were _reported_ according to the rules and circumstances of where they were detected. For mortality (rather than incidence), excess mortality might give a more accurate measure.\n",
"+ The model is a cartoon, not \"physics\" of epidemics, and it omits many factors that plausibly matter. It is not clear what estimates of the parameters mean when the model is wrong.\n",
"+ Absent a trustworthy generative model for the data, it is not clear how to assess or interpret the uncertainty of the estimates.\n",
"+ If the estimated model is to be used for prediction, there is no obvious way to assign meaningful uncertainties to the predictions.\n",
"+ The optimization problem to fit the parameters has no statistical content or motivation. It is not clear that it yields an estimate that is \"good\" in a useful sense. \n",
"+ The objective function is not convex in the model parameters, and the optimization algorithm is not guaranteed to solve the optimization problem.\n",
"\n",
"The data record total \"confirmed\" cases as a function of time, deaths, and recoveries: $C(t)$"
]
},
{
"cell_type": "code",
"execution_count": 7,
"metadata": {},
"outputs": [],
"source": [
"from scipy.optimize import curve_fit # nonlinear least squares\n",
"\n",
"def f(x, beta, gamma):\n",
" '''\n",
" Model cumulative infections\n",
" \n",
" This is just a wrapper for a call to sir_model, to generate a set of\n",
" predictions of the cumulative incidence for curve fitting\n",
" ''' \n",
" return sir_model(N, i_0, beta, gamma, delta=0, steps=len(x))[3] # C"
]
},
{
"cell_type": "code",
"execution_count": 8,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([ 500. , 571.39285714, 647.87464122, 729.80689646,\n",
" 817.5766761 , 911.59831599, 1012.31532767, 1120.20241833,\n",
" 1235.76764552, 1359.55471484, 1492.14542917, 1634.1622985 ,\n",
" 1786.27131969, 1949.18493587, 2123.66518564, 2310.5270524 ,\n",
" 2510.64202461, 2724.94187786, 2954.42268994, 3200.14910031,\n",
" 3463.25882519, 3744.96743963, 4046.57343789, 4369.46358264,\n",
" 4715.11855381])"
]
},
"execution_count": 8,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# test f\n",
"N = int(10**6)\n",
"i_0 = 0.0005\n",
"x = range(25)\n",
"y = f(x, 1/7, 1/14 )\n",
"y"
]
},
{
"cell_type": "code",
"execution_count": 9,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([0.14285714, 0.07142857])"
]
},
"execution_count": 9,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# test curve_fit\n",
"popt, pvoc = curve_fit(f, x, y, p0=[1, 0.5], bounds = (0, [np.inf, 1]))\n",
"popt"
]
},
{
"cell_type": "code",
"execution_count": 10,
"metadata": {},
"outputs": [],
"source": [
"# try some real data from the JHU site\n",
"import pandas as pd\n",
"# data for countries. THIS IS A \"LIVE\" DATASET THAT IS UPDATED FREQUENTLY\n",
"url = \"https://raw.githubusercontent.com/CSSEGISandData/COVID-19/master/csse_covid_19_data/csse_covid_19_time_series/time_series_covid19_confirmed_global.csv\"\n",
"df = pd.read_csv(url, sep=\",\")"
]
},
{
"cell_type": "code",
"execution_count": 11,
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"<div>\n",
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" }\n",
"</style>\n",
"<table border=\"1\" class=\"dataframe\">\n",
" <thead>\n",
" <tr style=\"text-align: right;\">\n",
" <th></th>\n",
" <th>Province/State</th>\n",
" <th>Country/Region</th>\n",
" <th>Lat</th>\n",
" <th>Long</th>\n",
" <th>1/22/20</th>\n",
" <th>1/23/20</th>\n",
" <th>1/24/20</th>\n",
" <th>1/25/20</th>\n",
" <th>1/26/20</th>\n",
" <th>1/27/20</th>\n",
" <th>...</th>\n",
" <th>3/4/22</th>\n",
" <th>3/5/22</th>\n",
" <th>3/6/22</th>\n",
" <th>3/7/22</th>\n",
" <th>3/8/22</th>\n",
" <th>3/9/22</th>\n",
" <th>3/10/22</th>\n",
" <th>3/11/22</th>\n",
" <th>3/12/22</th>\n",
" <th>3/13/22</th>\n",
" </tr>\n",
" </thead>\n",
" <tbody>\n",
" <tr>\n",
" <th>0</th>\n",
" <td>NaN</td>\n",
" <td>Afghanistan</td>\n",
" <td>33.93911</td>\n",
" <td>67.709953</td>\n",
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" <td>176201</td>\n",
" </tr>\n",
" <tr>\n",
" <th>1</th>\n",
" <td>NaN</td>\n",
" <td>Albania</td>\n",
" <td>41.15330</td>\n",
" <td>20.168300</td>\n",
" <td>0</td>\n",
" <td>0</td>\n",
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" <td>272337</td>\n",
" <td>272412</td>\n",
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" <td>272552</td>\n",
" <td>272621</td>\n",
" <td>272663</td>\n",
" </tr>\n",
" <tr>\n",
" <th>2</th>\n",
" <td>NaN</td>\n",
" <td>Algeria</td>\n",
" <td>28.03390</td>\n",
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" <td>265432</td>\n",
" </tr>\n",
" <tr>\n",
" <th>3</th>\n",
" <td>NaN</td>\n",
" <td>Andorra</td>\n",
" <td>42.50630</td>\n",
" <td>1.521800</td>\n",
" <td>0</td>\n",
" <td>0</td>\n",
" <td>0</td>\n",
" <td>0</td>\n",
" <td>0</td>\n",
" <td>0</td>\n",
" <td>...</td>\n",
" <td>38434</td>\n",
" <td>38434</td>\n",
" <td>38434</td>\n",
" <td>38620</td>\n",
" <td>38710</td>\n",
" <td>38794</td>\n",
" <td>38794</td>\n",
" <td>38794</td>\n",
" <td>38794</td>\n",
" <td>38794</td>\n",
" </tr>\n",
" <tr>\n",
" <th>4</th>\n",
" <td>NaN</td>\n",
" <td>Angola</td>\n",
" <td>-11.20270</td>\n",
" <td>17.873900</td>\n",
" <td>0</td>\n",
" <td>0</td>\n",
" <td>0</td>\n",
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" </tr>\n",
" </tbody>\n",
"</table>\n",
"<p>5 rows × 786 columns</p>\n",
"</div>"
],
"text/plain": [
" Province/State Country/Region Lat Long 1/22/20 1/23/20 \\\n",
"0 NaN Afghanistan 33.93911 67.709953 0 0 \n",
"1 NaN Albania 41.15330 20.168300 0 0 \n",
"2 NaN Algeria 28.03390 1.659600 0 0 \n",
"3 NaN Andorra 42.50630 1.521800 0 0 \n",
"4 NaN Angola -11.20270 17.873900 0 0 \n",
"\n",
" 1/24/20 1/25/20 1/26/20 1/27/20 ... 3/4/22 3/5/22 3/6/22 3/7/22 \\\n",
"0 0 0 0 0 ... 174214 174331 174582 175000 \n",
"1 0 0 0 0 ... 272030 272030 272210 272250 \n",
"2 0 0 0 0 ... 265186 265227 265265 265297 \n",
"3 0 0 0 0 ... 38434 38434 38434 38620 \n",
"4 0 0 0 0 ... 98796 98796 98806 98806 \n",
"\n",
" 3/8/22 3/9/22 3/10/22 3/11/22 3/12/22 3/13/22 \n",
"0 175353 175525 175893 175974 176039 176201 \n",
"1 272337 272412 272479 272552 272621 272663 \n",
"2 265323 265346 265366 265391 265410 265432 \n",
"3 38710 38794 38794 38794 38794 38794 \n",
"4 98829 98855 98855 98855 98909 98927 \n",
"\n",
"[5 rows x 786 columns]"
]
},
"execution_count": 11,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"df.head()"
]
},
{
"cell_type": "code",
"execution_count": 12,
"metadata": {},
"outputs": [
{
"name": "stdout",
"output_type": "stream",
"text": [
"746 [ 1 1 3 4 4 6 10 10 23\n",
" 23 35 90 262 442 615 801 827 864\n",
" 914 977 1057 1151 1255 1326 1395 1450 1591\n",
" 1724 1877 2046 2201 2395 2577 2860 3107 3386\n",
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" 6174 6318 6511 6681 6879 7073 7242 7384 7515\n",
" 7695 7912 8073 8210 8445 8575 8698 8851 9008\n",
" 9158 9311 9407 9523 9670 9821 9938 10083 10218\n",
" 10319 10429 10513 10591 10667 10713 10791 10858 10927\n",
" 10968 11044 11117 11182 11230 11289 11360 11387 11428\n",
" 11480 11512 11593 11633 11669 11699 11734 11771 11811\n",
" 11875 11924 11948 11962 12001 12016 12035 12099 12139\n",
" 12193 12217 12250 12294 12344 12391 12391 12391 12527\n",
" 12561 12615 12636 12675 12675 12675 12751 12768 12794\n",
" 12815 12832 12832 12832 12878 12888 12900 12916 12946\n",
" 12946 12946 13037 13061 13092 13124 13173 13173 13173\n",
" 13262 13302 13350 13390 13438 13438 13438 13547 13577\n",
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" 14442 14442 14442 14815 14959 15070 15214 15379 15483\n",
" 15617 15740 15855 15940 16056 16127 16239 16317 16397\n",
" 16480 16537 16627 16700 16779 16891 16985 17084 17195\n",
" 17374 17547 17736 17883 18113 18356 18607 18924 19216\n",
" 19557 19890 20237 20571 20571 21393 21847 22436 22905\n",
" 23323 23799 24357 24916 25594 26213 26637 27072 27464\n",
" 27998 28396 28932 29302 29680 30057 30379 30710 31156\n",
" 31638 32082 32422 32811 33101 33593 34023 34441 34941\n",
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" 353744 354068 354393 354645 354913 355257 355603 355944 356326\n",
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" 579275 589274 600468 609062 617274 627356 647934 661320 673807\n",
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" 2864063 2876288 2891763 2908721 2923030 2936856 2943947 2950605]\n"
]
}
],
"source": [
"# data for Denmark\n",
"DK = df.loc[(df[\"Country/Region\"] == \"Denmark\") & df[\"Province/State\"].isna()] # Denmark\n",
"DK = DK.drop(['Province/State','Country/Region','Lat','Long'], axis=1).T # remove fields we don't need\n",
"y = DK.to_numpy() # turn series into a vector\n",
"y = y[np.nonzero(y)] # remove the data before the first detected case\n",
"x = range(len(y)) # for model-fitting and plotting\n",
"print(len(y),y)"
]
},
{
"cell_type": "code",
"execution_count": 13,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"Text(0.5, 0, 'Days since first detected case')"
]
},
"execution_count": 13,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"image/png": 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\n",
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"plt.plot(x,y, linestyle='-', color='r')\n",
"plt.title('\"Confirmed\" COVID-19 Cases in Denmark')\n",
"plt.xlabel('Days since first detected case')"
]
},
{
"cell_type": "code",
"execution_count": 14,
"metadata": {},
"outputs": [],
"source": [
"N = 5792202 # estimated population of DK in 2020\n",
"i_0 = y[0]/N # initial prevalence\n",
"x = np.array(range(len(y)))"
]
},
{
"cell_type": "code",
"execution_count": 15,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([2.11506788e-02, 4.12605498e-16])"
]
},
"execution_count": 15,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# fit the model by nonlinear least squares\n",
"popt, pvoc = curve_fit(f, x, y, p0=[1, 0.5], bounds = (0, [500, 1]), maxfev=10000)\n",
"popt"
]
},
{
"cell_type": "code",
"execution_count": 16,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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5ciQiwp49e6hbty5RUVGMHz+e7OxsAKpVq8a+ffuOLVdQOWNMBZSaCpMmsfyaa8jKyuKss84qsU1F7hl6mBw6dIiUlBSysrKIiYlhyJAh/PnPfwbgjjvu4PLLL+fDDz+kV69eVK1aFYB27doRExND+/btGTp0aIHljDEV0NixUKkSC1u1AijRhB62Z4p26tRJ814oXLVqFa28nTahYZ+pMWF29tlQpQrDk5OZNGkSGRkZJ1WHLiILVbVToHlW5WKMMSVl8WKYNw/at2fhwoWcddZZJdolhyV0Y4wpKc88A0D2wIGkp6eXaHULWB26McaUjAMHYPJkuPVWonv2ZNu2bSXe+6mdoRtjTEn4/HM4eBAGDQJcP1El3UDCEroxxpSEDz6A00+H88/nqaee4r777ivxTVqVizHGhNrevTBlCtxyC0RHs3HjRgI91CfU7Aw9jyeeeII2bdrQrl07UlJSmDvXPZypZ8+ex/pjSUpK4swzz6Rdu3b06NGDn3/+OeC6kpKSOO+883JNS0lJoW3bvM8HKZz/tk+mjDGmlHz6KWRmHqtuefXVV/nwww9LfLOW0P3Mnj2bzz77jEWLFpGamsq0adNo2LBhwLLTp08nNTWVnj178vjjjxe4zn379rFpk3s636pVq0okbmNMhJk6FWrXhm7dSnWzltD9bN26ldq1a1OpUiUAateuTb169Qpdplu3bvzyyy8Fzr/qqqv44IMPAJg4cSKDBw8+Ni8zM5Mbb7yRM888kw4dOjB9+nTA3a169dVX065dOwYNGsShQ4eOLfP111/TrVs3OnbsyJVXXsn+/fsxxkSY2bNdMo+K4q233qJdu3bs3LmzxDcb0XXoPXv2LLLMxRdfzD333HOs/NChQxk6dCg7duzgiiuuyFV2xowZha6rb9++PProozRv3pw+ffowaNAgevToUegyX375JQMHDixw/hVXXMHQoUO55557+PTTT5kwYQLjx48H4JVXXgFg2bJl/PTTT/Tt25c1a9bw2muvUaVKFVJTU0lNTaVjx44A7Nixg8cff5xp06ZRtWpVnn76aZ577jkeeuihQmM0xpSiHTtgzRq48UYA5s6dy+bNm6lZs2aJb9rO0P0kJCSwcOFCxowZQ506dRg0aBDjxo0LWLZXr14kJiYybdo0rrnmmgLXeeqpp1KzZk3ef/99WrVqRZUqVY7NmzlzJkOGDAGgZcuWNG7cmDVr1vD9999z3XXXAa6fmHbt2gEwZ84cVq5cSffu3UlJSeGdd94psP7eGBMmc+a4V6+6ZcGCBSV+h6hPRJ+hF3VGXVj52rVrF3t5cG1Fe/bsSc+ePTnzzDN55513GDp0aL5y06dPp2rVqgwdOpSHHnqI5557rsB1Dho0iBEjRuT751BYPzqBvnxV5YILLmDixIlB748xppTNng3R0dCpE4cPH2b58uXHOvgraXaG7mf16tWsXbv22PslS5bQuHHjAstXrlyZ559/nnfffbfQ+rE//OEP3HvvvfTr1y/X9PPPP58JEyYAsGbNGjZu3EiLFi1yTV++fDmpqakAdO3alVmzZpGWlga4JyPZwzOMiTCzZ0P79lC1KsuWLSMrK4tOnQL2pRVyltD97N+/nxtuuIHWrVvTrl07Vq5cycMPP1zoMnXr1mXw4MHH6sMDqVatGvfddx9xcXG5pt9xxx1kZ2dz5plnHqveqVSpErfffjv79++nXbt2PPPMM3Tp0gWAOnXqMG7cOAYPHky7du3o2rUrP/3000nvtzEmRI4edZ1xnXMOAAsXLgRKtstcf9Z9bjlnn6kxpWjxYujYESZMgGuuYfjw4SHpMtefdZ9rjDGlYfZs9+p3QbRTp06lckEULKEbY0zozJ0LiYmQlHTsgmhpVbdABCb0cFUBlUf2WRpTyubPh86dQYQDBw5w66235msMUZKKTOgi0lBEpovIKhFZISJ/ClCmp4jsEZEl3nBCd7rEx8eTkZFhiSgEVJWMjAzi4+PDHYoxFcO+ffDTTy6h4+5Beemll4K6QTJUgmmHfhS4W1UXiUg1YKGITFXVlXnK/aCqF59MMA0aNGDz5s2l0itZRRAfH0+DBg3CHYYxFcOiRaB6LKFv2bKFxMREYmJK73afIrekqluBrd74PhFZBdQH8ib0kxYbG0uTJk1CvVpjjCl53v0idOgAwIABA6hXrx6ff/55qYVQrH8dIpIEdADmBpjdTUSWAluAe1R1RYDlhwPDARo1alTsYI0xJmKtXg3Vq7uHWgAPPPAACQkJpRpC0O3QRSQB+A54QlU/zjOvOpCjqvtF5CLgBVVNLmx9gdqhG2NMmdW3L+ze7W4sKkEn3Q5dRGKBj4AJeZM5gKruVdX93vgUIFZEap9EzMYYU7asXg3NmwOuB9UFCxaUegOPYFq5CPAWsEpVA/ZAJSKne+UQkS7eejNCGagxxkSsQ4dg40Zo0QKAZ555ht///veldkORTzB16N2BIcAyEVniTXsQaASgqq8DVwC3i8hR4BBwtVrbQ2NMReHr1M87Q58/fz6dvdYupSmYVi4zgUL/zajqy8DLoQrKGGPKlNWr3WuLFuzZs4fVq1cfe6ZBaYq4O0WNMabM8XVjnZx8rIfFcJyhW0I3xpiTtXo1NGgAVasyf/58gFLrA92fJXRjjDlZa9YcuyA6f/58mjZtSq1atUo9DEvoxhhzMlRzNVkM1wVRsIRujDEnZ9Mmd0NRmzZs27aNjRs3WkI3xpgyyaszp3PnY/Xn4UropdcNmDHGlEe+5/q2bUvf2FgWL15MC68+vbRZQjfGmJORng6nnQZVqhALpKSkhC0Uq3IxxpiTsWEDJCWhqtx///38+OOPYQvFEroxxpyM9HRo0oTt27fz0ksvsWTJkrCFYlUuxhhzorKzXadcV15JYmIie/fuJSsrK2zhWEI3xpgTtWULZGVBUhIA0dHRREdHhy0cq3IxxpgTlZ7uXps0YcSIETz77LNhDccSujHGnKgNGwDIadSI9957j/Xr14c1HEvoxhhzorwz9LVHjrB3796w3VDkYwndGGNOVHo61KvHXK9ly9lnnx3WcCyhG2PMifLaoM+ZM4fq1avTsmXLsIZjCd0YY06U1wZ9zpw5dOnShaio8KZUS+jGGHMijh6FTZs4WL8+qampdO3aNdwRWUI3xpgTsnkzZGezUJXs7Oyw15+DJXRjjDkxXguXObt3A+G/IAqW0I0x5sR4Cf2Uxo255JJLqFOnTnjjwW79N8aYE7NhA4gwfNQohsfFhTsawM7QjTHmxKSnk1W/PjkxkXNebAndGGNOxIYN/C8hgRo1arB69epwRwMEkdBFpKGITBeRVSKyQkT+FKCMiMiLIpImIqki0rFkwjXGmAixbh1JTZty/fXX06RJk3BHAwR3hn4UuFtVWwFdgREi0jpPmf5AsjcMB14LaZTGGBNJdu6ELVvo1LMnL7/8MnFlpQ5dVbeq6iJvfB+wCqifp9ilwLvqzAFqiEjdkEdrjDGRYMUKsoDUqlXJzs4OdzTHFKsOXUSSgA7A3Dyz6gOb/N5vJn/SR0SGi8gCEVmwffv2YoZqjDERYs0aUoH2I0bw0UcfhTuaY4JO6CKSAHwE3KWqe/PODrCI5pugOkZVO6lqp0hos2mMMSckLY25Xr8tkXBDkU9Q7W1EJBaXzCeo6scBimwGGvq9bwBsOfnwjDEmAq1bx5yqVTm9alUaNWoU7miOCaaViwBvAatU9bkCik0Grvdau3QF9qjq1hDGaYwxkSMtjTle/y0uRUaGYM7QuwNDgGUissSb9iDQCEBVXwemABcBacBB4MaQR2qMMZFAlW1r1rD24EFuPueccEeTS5EJXVVnEriO3L+MAiNCFZQxxkSs7dv58cABALp37x7mYHKzO0WNMaY40tKYBcTFxHDWWWeFO5pcLKEbY0xxeAm9c7t2xMfHhzuaXCyhG2NMMRxatYoFQPdevcIdSj6R002YMcaUAXEbNjCzXj1Ove22cIeSjyV0Y4wphuj16+nSti2ccUa4Q8nHqlyMMaYY/rViBVMrVQp3GAFZQjfGmCBpRgaPHjzI+7/9Fu5QArIqF2OMCZKsW8c6YN8f/xjuUAKyM3RjjAlWWhpxQK2OkfkMH0voxhgTpCfeeotHASLkCUV5WZWLMcYE6Z3582lduTJE2A1FPnaGbowxQdi2bRtr9+2je4MG4Q6lQJbQjTEmCN9PmwbAeRHWw6I/S+jGGBOE7yZPpipwVr9+4Q6lQJbQjTEmCDNmz+YcILZFi3CHUiBL6MYYU4QdO3awfONGegI0bRrmaApmCd0YY4rwww8/ANCjWjWoUSO8wRTCEroxxhThu+++o3JUFJ0jsEMuf5bQjTGmCKeddhrXJCQQl5wc7lAKZQndGGOK8MC99/LmoUMRXX8OdqeoMcYU6sCBA1Teto2orKyIT+h2hm6MMYX429/+RsPOnckGS+jGGFOW9enThzt79CAaIj6hW5WLMcYU4qKLLuKiWbMgOhoaNgx3OIUq8gxdRMaKyDYRWV7A/J4iskdElnjDQ6EP0xhjSl96ejorV65E162Dxo0hJrLPgYOpchkHXFhEmR9UNcUbHj35sIwxJvxeeuklOnbsSOa6dRFf3QJBJHRV/R7YWQqxGGNMRJk2bRrnnnsulTdsKB8JPUjdRGSpiHwhIm0KKiQiw0VkgYgs2L59e4g2bYwxoffbb7+RmppKn3PPhYwMaNYs3CEVKRQJfRHQWFXbAy8BnxRUUFXHqGonVe1Up06dEGzaGGNKxjfffANAH9/doRH62Dl/J53QVXWvqu73xqcAsSJS+6QjM8aYMJo2bRo1a9akQ3S0mxDht/1DCBK6iJwuIuKNd/HWmXGy6zXGmHBRVaZNm0bv3r2JXrfOTYzwjrkguGaLE4HZQAsR2SwiN4nIbSJym1fkCmC5iCwFXgSuVlUtuZCNMaZkrV27lk2bNnHBBRfA2rVQty4kJIQ7rCIV2ahSVQcXMf9l4OWQRWSMMWE2zXt+aJ8+feDdd6F58zBHFBy79d8YY/L47rvvSEpKomnTpu4MvQzUn4Pd+m+MMfmMHz+en3/+Gdm7F7ZvLzMJ3c7QjTEmj7i4OJKTk93ZOVhCN8aYsmjMmDE88MADqCqsWeMmlpGEblUuxhjjZ9myZSxbtgwROX6GXgbuEgVL6MYYk8tLL71ETk6Oe7N2resyt3Ll8AYVJKtyMcYYjy+RR0V5qbEMtXABS+jGGHPM9ddfz+WXX358giV0Y4wpe44ePcqUKVOoXr26m5CRAbt2WUI3xpiyZt68eezatYv+/fu7CWWsySJYQjfGGACmTJlCdHS0678Fjif0MnLbP1hCN8YYAL744gvOOeccatas6SasXQtRUWXiSUU+ltCNMRXe1q1bWbRo0fHqFoDVq92DoePiwhdYMVlCN8ZUeJMnTwbg97///fGJK1ZAmwKfqBmRLKEbYyq8//73v5xxxhm0bt3aTcjKcrf9t20b3sCKyRK6MaZC27NnD99++y0DBw7Ee/iaqz/PyipzZ+h2678xpkKLjY1l7NixdOjQ4fjEFSvcqyV0Y4wpO6pUqcJ1112Xe+LChRAbCy1bhieoE2RVLsaYCuvw4cO88MILbN26NfeMmTOhU6cy0ymXjyV0Y0yFNWfOHO666y4WL158fGJmJsyfD+eeG77ATpBVuRhjKqwePXqQlpZGgwYNjk+cPx+OHLGEbowxZU2zvA+vmDnTvXbvXvrBnCSrcjHGVEgzZ87kyiuvZPPmzbln/PADtG4NtWqFJ7CTYAndGFMhTZgwgSlTphzvuwUgOxt+/LFMVreAJXRjTAV09OhRJk2axCWXXELVqlWPz1i+HPbsgfPOC19wJ6HIhC4iY0Vkm4gsL2C+iMiLIpImIqki0jH0YRpjTOh8++237Nixg0GDBuWe8cMP7rUcn6GPAy4sZH5/INkbhgOvnXxYxhhTcj744AOqVauWu3dFgClTXHe5jRuHJ7CTVGRCV9XvgZ2FFLkUeFedOUANEakbqgCNMSaUjhw5wscff8zAgQOJj48/PmP/fvj2W7j0UvD16VLGhKIOvT6wye/9Zm9aPiIyXEQWiMiC7du3h2DTxhhTPF9//TW7d+/OX90ydSocPgyXXBKewEIgFAk90L8yDVRQVceoaidV7VSnTp0QbNoYY4pn/Pjx1KpV6/ij5nwmT4YaNcps/TmEJqFvBhr6vW8AbAnBeo0xJqR27drF//73PwYPHkyc/5OIsrPh88/hootcp1xlVCgS+mTgeq+1S1dgj6puLWohY4wpbVFRUTz22GPccsstuWdMngzbt8Nll4UnsBAp8tZ/EZkI9ARqi8hm4O9ALICqvg5MAS4C0oCDwI0lFawxxpyMU045hVGjRuWf8eGHcPrpMHBgqccUSkUmdFUdXMR8BUaELCJjjCkBGzZsYNasWVx++eVU9u8WVxVmzIDevSE6OmzxhYLdKWqMqRD+85//cMMNN7Br167cM9auha1boUeP8AQWQpbQjTEVwqhRo1i8eDH16tXLPWPGDPfas2dphxRyltCNMRVCVFQU7dq1yz9jxgyoWxeSk0s9plCzhG6MKfduuOEGHn/88fwzfPXnPXuW2btD/VlCN8aUaxs3buS9994jMzMz/8xvv3X15+WgugUsoRtjyrk333wTVc3f9jw7G4YNg+bN4ZprwhNciNkj6Iwx5VZWVhZvvvkm/fv3p3HeHhTnzYONG2HiREhICE+AIWZn6MaYcmvy5Mls3bqVW2+9Nf/MKVNcu/N+/Uo/sBJiCd0YU26NHj2aJk2aMGDAgPwzp0yBc84B/0fQlXGW0I0x5dLcuXOZNWsWf/rTn4jOewfo1q2waJHrjKscsYRujCmXRo8eTfXq1Rk2bFj+mV9+6V7zPrGojLOEbowpd7Zt28akSZMYPnw41apVy19gwgRo2BAC3WhUhlkrF2NMuZOYmMjChQtJTEzMP3P3bvjmG3j44XJxM5E/S+jGmHJFVRER2rdvH7jAmjXutUOH0guqlFiVizGmXHniiScYMmQIR48ezT/zt9+ON1MsB3235GUJ3RhTrqgqqkpMTIAKiCefdFUul15aLhO6uOdTlL5OnTrpggULwrJtY0z55qt2yWXrVmja1N3m/9Zb4QksBERkoap2CjTPztCNMeXC/v37mTp1auBkDjBmDBw5Ag8+WPrBlRJL6MaYcuH555+nb9++LF26NP/MnBz46CPo2hWaNSv94EqJtXIxxpR5O3bs4JlnnmHgwIGkpKTkLzBmDCxbBu+8U+qxlSY7QzfGlHlPPPEEBw4c4Mknn8w/c8sWuO8++N3vYMiQ0g+uFFlCN8aUaRs2bOCVV15h2LBhtGrVKvdMVRg50tWdv/56ubuRKC+rcjHGlGl/+ctfiI6O5uGHH84/86234L//hWeegTPOKPXYSpudoRtjyqzvv/+eiRMncs8991C/fv3cM2fMgD/+Efr0gbvvDkt8pS2ohC4iF4rIahFJE5H7A8zvKSJ7RGSJNzwU+lCNMea4o0ePMnLkSBo3bswDDzyQe+bs2a5r3CZN4L33IKpinLsWWeUiItHAK8AFwGZgvohMVtWVeYr+oKoXl0CMxhiTz7hx41i2bBkff/wxVapUOT5j5UoYMADq13cPgT7ttPAFWcqCqUPvAqSp6noAEXkfuBTIm9CNMabUXHfddVSqVImBAwcenzh5MtxyC8TFwVdfVahkDsFVudQHNvm93+xNy6ubiCwVkS9EpE2gFYnIcBFZICILtm/ffgLhGmMqOlXl4MGDxMfHM2TIEHdX6IEDrjXLpZdCvXowbZq7zb+CCSahB2rnk7cDmEVAY1VtD7wEfBJoRao6RlU7qWqnOnXqFCtQY4wBeP/992nVqhXr1693E2bOhPbt4dVX4f/9P5gzB9q2DW+QYRJMQt8MNPR73wDY4l9AVfeq6n5vfAoQKyK1QxalMcZ4mjRpwvnnn0/jxES45x44/3x3a/+MGfDcc1CpUrhDDJtg6tDnA8ki0gT4BbgauMa/gIicDvymqioiXXD/KDJCHawxxnTt2pWuUVHQuTP89BPcfrtrZ56QEO7Qwq7IhK6qR0VkJPAVEA2MVdUVInKbN/914ArgdhE5ChwCrtZw9ctrjCmXXnzxRVYvW8bzVasS+/LLrq7866/hggvCHVrEsP7QjTERb9GPP9KtRw/6Rkcz+fBh5NZb4emn4ZRTwh1aqbP+0I0xZU9ODsybx+7hwxl03nnUOXqUt1u0QObOdf2yVMBkXhTry8UYEzl27IDp02HqVPjsM7K3bmWwCOnAt88/T+0776wwd32eCEvoxpjSceQI7NyZe9i0CdasgdWr3euGDa5stWrQrx/3HzjAl198wRtvvMF5w4eHN/4ywBK6MebEZGfDL7/A+vWQnu7OrnfuhIyM/Il7507Yvz/wehISoHlz9zShm2+G3r2hUyfeHj+eZ4cNY8SIEQy3ZB4US+jGmILt2uUS9oYN7tV//OefISsrd/mYGKhVC0491Q0NG7qbfk49Nfd033D66a61Sp5+yj/99FNuueUWLrjgAkaPHl2KO1y2WSsXYyq6Xbtg7drjw5o17jUtDXbvzl22Vi13S33Tpq4nQ99rkyaQmOjOtk/yIRI5OTl07tyZ6OhovvnmG6pVq3ZS6ytvCmvlYmfoxlQEmZkuSa9adTxh+14z/O4BFIHGjSE5Ga65xj1Q2T9xV69e4qFGRUXx9ddfk5OTY8m8mCyhG1NWZGa6M+Z9+1x99L59ucf374e9e48PGRnuouOmTfDbb+5xbD4NGrikffnlrv46OdkNTZuG7db5mTNn8sYbbzB27Fhq1aoVlhjKOkvoxkSSnBx3gXHlSnc2nZZ2vPpj06YiFwegalV3Jl2zpqvDbtcOGjWCFi2gZUuXuP37D48Qy5YtY8GCBezatYvExMRwh1MmWR26MeGQleUuKq5aBStWuMGXxA8dOl6udm33LEzfkJjomvQlJLjXQOMxZes8bdOmTTRs6Pr/O3ToEJUrVw5zRJHN6tCNKW2ZmbBxozvb/vnn/K9btrizcZ8GDaB1a7j1VmjTxo23auXOssupnJwc/vrXv/LCCy8wf/58Wrdubcn8JFlCN6Y4jh6F6GjXMmTjxvyDL2H/+mvu5aKjXfVH48bwu9+516QkVwXSunWFu419x44dDBkyhC+//JLhw4eTnJwc7pDKBUvoxvgcPuwuJO7Y4QZfixDfhcWNG2HbNvd4syNHci8bF+cSdlKSezhxUtLxpN24sXu+ZRmrCikpP/74I4MGDWLbtm28+uqr3Hbbbe6pQ+ak2RFmyidVdxa9ZQts3348SfsnbP/xHTsC38mYkOAuKDZsCCkprg47M9O9b9jQzWvUCOrUsT5GinD48GEeffRRnn76aRo3bszs2bPp2LFjuMMqVyyhm7LrwAF3Fr18ubuguGmTuxX9l19cIs/MDLxctWruYqNvaNny+HitWsfHmzRxydrOHk/a3LlzGTZsGCtXrmTo0KE8//zznFLBqplKgyV0U3aowuLF8Pnnbpg373jb6kqV3Blz/fpw9tnutV49NyQm5k7YcXHh3Y8KZt++ffTt25fq1aszZcoU+vfvH+6Qyi1L6CbybdgAr7wCH37o6rFFoEsX+NvfXDVI27buhpjo6HBHajz79u3jww8/5MYbb6RatWp88sknnHXWWVQvhTtNKzJL6CYyqcKsWTB6NHzyiauf7t8fHn4YBgxwZ90mYn3yySfcdNNNNGvWjB49etCrV69wh1QhWEI3kWf6dLjvPpg/37XDvu8+GDHCVaOYiLRu3Tpee+01kpKSGDlyJIMHD6ZVq1Z06hTw/hdTQuyyvIkcq1e7vkV693atTl591V3ofPJJS+YRKDMzk48++ogBAwaQnJzMCy+8wPr16wGIiYmxZB4GdoZuwi8tDf7v/+Bf/4LKleGRR2DUKDduIkpWVhbff/89EydOZNKkSezZs4e6devy0EMPMXz4cOrVqxfuECs0S+im9O3ZAzNnwvffw7RpsGiRu+nm9tvdhU6rH49Ib775JnfffTd79+6lWrVqXHbZZVx77bX07t2baLsgHREsoZuSdeAALF3qkvaiRbBwoWs3npMDsbGutco//wlXX+36MzFhpapkZ2cTExPD4sWLufbaaxk7dixdu3alWbNmXHXVVQwYMIC+fftSJQJ7bKzoLKGb0MjKgnXrXK+By5a5pL1smev61ddWvE4dOOssGDgQevRwz5C0pBA2WVlZpKWlsWTJklzDqFGjuOeee6hfvz5NmjQ5Vr5Xr17WWiXCWUI3RcvJOf7ABN+wfbtL1v79dvueLyniunpt2xYGD4aOHV0iD/DsSFOydu/ezZEjR0hMTOTIkSOMGjWKtWvXsnbtWtLT0zl69CgAsbGxtG3blgEDBtC6dWsAEhMT+fzzz8MZvimmoPpDF5ELgReAaOBNVX0qz3zx5l8EHASGquqiwtZp/aGXMFWXYA8fdrfA+wb/94cPu763d+1yrUp8fZ5s3358PCPDzc/Ozr+N6Gj3iLJWrY539+p7tTPvEnHo0CEyMjLIyMhg586dx8Z973/99Vfq16/P008/DUCTJk3o3r077733HqpKw4YNqVOnDsnJySQnJ9OiRQtSUlJo2bIlcXYHbZlwUv2hi0g08ApwAbAZmC8ik1V1pV+x/kCyN5wNvOa9RoacHJe4DhyAgwfdcOhQ7kSXmekS4NGjxRsKWkbVbVc1NOM5OS6p+m/D/33eRJ2ZmfuRY8GIjna3x9ep44Yzz3TvfU9szzs0bhy2x5WVhJycHHJycsjOziYuLg4RITMzk0OHDlGjRg1EhN27d7Nv3z6ys7OPlc07fuTIEbp06QJAamoqv/zyy7Hb3b/88kvS0tI4cuQIhw8fPvaamZnJgQMH2LdvH/Hx8YwdOxaA4cOH8+uvvzJ58mQAunXrxtKlSwPGX6VKFU477TTOP//8Y9OefPJJTj/9dABEhE2bNlnPhuVYMFUuXYA0VV0PICLvA5cC/gn9UuBddaf7c0SkhojUVdWtIY/4yy95bMgQJu7dezxhqaLe6zGqpCYkEHvkCI9mZvIFMNubNQKYWsRmagJzvfE7gE3Ap977y4ElvoIiIOK27/tDEaFlbCxT6tYFES779VeqR0czrm5diIrinPR0tng/dQHUf12eHtWq8e4ZZ0BUFN2WLePcU07hn96jw5rMmsVR3776b9u3fHQ0g7p04f8uuQTi40l68klu79mT+/7wBzKOHOHMUaNyxy7i7sT0Lb9jB3+85hoeeOABMjIyaNOmDf/4xz+48YYbWLFiBb179871WQX6lffUU08xbNgwli9fTq9evRg3bhwDBgzgm2++YdCgQUUu/+677x4rf+WVV/LVV1/RuXNn3nvvPe68884il586dSqdO3dm/PjxjBw5kuXLl9OwYUP++c9/8thjjwVMxv42btx4rPxDDz1EVlYWMTExPPjgg7z22mv5tpeXr/xrr73GRx99xLZt2wB4/fXX+d///perbGxsLJUqVSIhIYGEhIRcTf9atGiR63Fsd999N4cOHeLUU0+lVq1auYb4+Ph8cQwePDjXe0vm5VswCb0+Lp/5bCb/2XegMvWBXAldRIYDwwEaNWpU3FidU06hbuPGtN22LVcCRcQdrH6D9O8PlSrRYM0a2v76K1x/PVSpQrPp09n1888uicXEuNfo6ONJLSrKPW382WchJobkMWOovmOHu8ElJoYzH3uMKt4NFHn279h4w4YN4YknAGj/yCOuRYCXSLv++c/s3LmzwGUB2rZtC3ffDcA5d99Ny9at4aabAOhzyy3k+D/tJtDy3bvDjTcC0G/NGpL79YPLLiNu3z4unjOn0NgBWrVqBUBcXBwDBw48dnHslFNO4bLLLity+WbNmgFQo0YNrrrqKup7NwbVrVuXq6++usjlG3gtXurWrct1111HnTp1ADjjjDO4/vrri1zeV7558+YMGzaMhIQEAFJSUrj55psREaKjo48NUVFRucZ9fY7069eP6tWrH1v/tddeS8eOHfMt4/++UqVKx8rff//93HHHHcfiGjt2LNnZ2VSqVIm4uDji4uKIKqTb3bu9Y8BnyJAhBZY1psg6dBG5Euinqjd774cAXVT1Tr8ynwP/UNWZ3vtvgHtVdWFB67U6dGOMKb7C6tCDufV/M9DQ730DYMsJlDHGGFOCgkno84FkEWkiInHA1cDkPGUmA9eL0xXYUyL158YYYwpUZB26qh4VkZHAV7hmi2NVdYWI3ObNfx2YgmuymIZrtnhjyYVsjDEmkKBuLFLVKbik7T/tdb9xxTUeMcYYEybWfa4xxpQTltCNMaacsIRujDHlhCV0Y4wpJ4LqnKtENiyyHfj5BBevDewIYTglwWIMDYsxNCzGkxcp8TVW1TqBZoQtoZ8MEVlQ0J1SkcJiDA2LMTQsxpMX6fGBVbkYY0y5YQndGGPKibKa0MeEO4AgWIyhYTGGhsV48iI9vrJZh26MMSa/snqGbowxJg9L6MYYU06UuYQuIheKyGoRSROR+8MYx1gR2SYiy/2mnSoiU0Vkrfda02/eA17Mq0WkXynE11BEpovIKhFZISJ/isAY40Vknogs9WJ8JNJi9NtutIgsFpHPIjFGEUkXkWUiskREFkRojDVEZJKI/OQdl90iKUYRaeF9fr5hr4jcFUkxFklVy8yA6753HdAUiAOWAq3DFMv5QEdgud+0Z4D7vfH7gae98dZerJWAJt4+RJdwfHWBjt54NWCNF0ckxShAgjcei3uMa9dIitEv1j8D/wY+i7Tv2ttuOlA7z7RIi/Ed4GZvPA6oEWkx+sUaDfwKNI7UGAPGHc6Nn8CH3A34yu/9A8ADYYwnidwJfTVQ1xuvC6wOFCeub/lupRzr/4ALIjVGoAqwCPe82oiKEfcErm+A3n4JPdJiDJTQIyZGoDqwAa8hRiTGmCeuvsCsSI4x0FDWqlwKehh1pDhNvSc1ea++x7WHNW4RSQI64M6AIypGrypjCbANmKqqERcj8DxwL+D/ZO5Ii1GBr0VkobiHsUdajE2B7cDbXtXVmyJSNcJi9Hc1MNEbj9QY8ylrCV0CTCsL7S7DFreIJAAfAXep6t7CigaYVuIxqmq2qqbgzoK7iEjbQoqXeowicjGwTQt54HneRQJMK43vuruqdgT6AyNE5PxCyoYjxhhcFeVrqtoBOICrvihIOP9m4oDfAx8WVTTAtLDmo7KW0CP9YdS/iUhdAO91mzc9LHGLSCwumU9Q1Y8jMUYfVd0NzAAujLAYuwO/F5F04H2gt4i8F2ExoqpbvNdtwH+BLhEW42Zgs/cLDGASLsFHUow+/YFFqvqb9z4SYwyorCX0YB5YHU6TgRu88Rtw9da+6VeLSCURaQIkA/NKMhAREeAtYJWqPhehMdYRkRreeGWgD/BTJMWoqg+oagNVTcIdb9+q6nWRFKOIVBWRar5xXP3v8kiKUVV/BTaJSAtv0u+AlZEUo5/BHK9u8cUSaTEGFs4K/BO8WHERrsXGOuAvYYxjIrAVyML9p74JqIW7eLbWez3Vr/xfvJhXA/1LIb5zcT//UoEl3nBRhMXYDljsxbgceMibHjEx5om3J8cvikZMjLj66aXesML3dxFJMXrbTAEWeN/3J0DNCIyxCpABnOI3LaJiLGywW/+NMaacKGtVLsYYYwpgCd0YY8oJS+jGGFNOWEI3xphywhK6McaUE5bQKxARyfZ6kVvh9XD4ZxEp1WNARB4VkT4luP6W3j4uFpFmIvJjMZe/S0SqBFGup6/nxULKpIjIRcXZ/onEUty4TPllCb1iOaSqKaraBtdR10XA30szAFV9SFWnleAmBgL/U9UOqrpOVc/JW0BEogtZ/i5cW+RQSMF9xifqLkIXi6kALKFXUOpuER8OjBQnSUR+EJFF3nAOgIiMF5FLfcuJyAQR+b2ItBHXl/kSEUkVkWT/9Xudbo0TkeXi+un+f970cSJyhTeeLiKPeNtbJiItvekJIvK2Ny1VRC73pvcVkdle+Q+9fmr8t3kRLgneLCLTvWn7vdee4vqH/zewzLu78nPvl8pyERkkIn8E6gHTfcvnWf+F4vrynglc5je9qrj+8ed7vwwu9e5kfhQY5H1GgwKV8/usnvXb3zsDxVLQ/hcUV4DvI9c2vOkPefEsF5ExIiLe9D+KyEqv7PsF7WfBR5gJi3Df2WRD6Q3A/gDTdgGn4c4E471pycACb7wH8Ik3fgquC9QY4CXgWm96HFA5z3rPwvWe6Htfw3sdB1zhjacDd3rjdwBveuNPA8/7LVsTqA18D1T1pt2Hd2dpnu0+DNyTd59xd3keAJp47y8H/uVX7hS/mGoHWG88rme9ZFynTP/h+F2jTwLX+fYTdydzVWAo8LLfOgoqdzuuz50Yb96peWMpaP8LiytP/AVtw/+ux/HAJd74FqBSnu8uYPzhPq5tOD7YGbrx9RgXC/xLRJbheplrDaCq3wFniEgiro+Lj1T1KDAbeFBE7gMaq+qhPOtdDzQVkZdE5EKgoJ4efZ2GLcT1Lw+uT5dXfAVUdRfuwRetgVniutu9AffwgeKYp6obvPFlQB8ReVpEzlPVPUUs2xLYoKpr1WW09/zm9QXu9+KagUuyjQKso6ByfYDXvc8VVd0ZYNmC9r+wuPwVtI1eIjLX+957A2286anABBG5DjhazP00YRIT7gBM+IhIUyAb13vc34HfgPa4qrhMv6LjgWtxnVMNA1DVf4vIXGAA8JWI3Kyq3/oWUNVdItIe6AeMAK7yLZvHYe81m+PHo5C/G1LBnfEPPrG9BdwZui++NSJyFq6O+x8i8rWqPlrE8gX1kyHA5aq6OtdEkbODLBdofwNtI9/+i0hKEMv6ls9VTkTigVeBTqq6SUQexiVpcN/r+bhuZP8mIm0Kit9EDjtDr6BEpA7wOq5KQHHVKVtVNQcYgnsEl884XN00qrrCW74psF5VX8T1Otcuz/prA1Gq+hHwN1xXqcH6Ghjpt66awBygu4ic4U2rIiLNi7HOXESkHnBQVd8DnvWLbx/ukX15/QQ0EZFm3nv/xPoVcKdf/XOHAtZVULmvgdtEJMabfmqA5Qva/8Li8hdoG77kvcOrj/dd24gCGqrqdNyDPWoACYXEbyKEJfSKpbJ3gW4FMA33R/6IN+9V4AYRmQM0J/fZ7G/AKuBtv3UNApZ7P79bAu/m2VZ9YIY3fxzucV3Behyo6V2oWwr0UtXtuDrpiSKSiktwLYuxzrzOBOZ58f3F2ybAGOCLvBdFVTUTdxH5c+/i489+sx/DVVmlinto+GPe9OlAa99F0ULKvQls9KYvBa7JG0tB+19EXP7ybUNdH/T/wlU/fYLrnhrcP/P3vGqYxcBor2xB8ZsIYb0tmiKJawu9DPfQ6aLqmo0xYWJn6KZQ4m4C+gl4yZK5MZHNztCNMaacsDN0Y4wpJyyhG2NMOWEJ3RhjyglL6MYYU05YQjfGmHLi/wOUTzXGQWlhlgAAAABJRU5ErkJggg==\n",
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"S, I, R, C = sir_model(N, i_0, beta=popt[0], gamma=popt[1], delta=0, steps=len(y))\n",
"times = list(range(len(y)))\n",
"fig, ax = plt.subplots(nrows=1, ncols=1)\n",
"ax.plot(times, y, linestyle='-', color='r', label='Data')\n",
"ax.plot(times, C, linestyle='-.', color='k', label='SIR Model')\n",
"ax.legend(loc='best')\n",
"plt.xlabel('Days since first detected case')\n",
"plt.title('\"Confirmed\" COVID-19 Infections in Denmark')\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"What happens if we run the model into the future?"
]
},
{
"cell_type": "code",
"execution_count": 17,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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GOf7gKk9RatSdgP7GmAPALKC7MUZHA1eqFBkyBLZsgR0Ovss057nnqOXlRccOHRx7YHWRQhO1iDwhIsEiUh8YAiwXkTudHplSymFuu83qrufIWnXasWMs/u03Bl57LR6eno47sLqE9qNWqgyoVQs6d3bsU4pfT5xIOhBz772OO6jKV7EStYisEJFoZwWjlHKegQPhp5/AUQ8PzvnsM6oZQ+fRox1zQHVZWqNWqowYMMB6/+KLqz/WmRMnWHTwIAObN9feHiWgxEbPO3fuHIcOHSIjI6OkTlmm+fr6EhwcjLe3t92hKBdRty5ERFjNH48/fnXHKlepEkvffpsqdeo4JjhVoBJL1IcOHaJixYrUr18fY0xJnbZMEhGSk5M5dOgQDXQWaHWBgQPhX/+CX3+1ZoK5Uh5eXnQcOdJxgakClVjTR0ZGBlWrVtUkXQKMMVStWlX/elGXGDjQep8//8qPcfaPP/hnmzYkLljgkJhU4Uq0jVqTdMnRa63y06wZtGx5db0/tn/8MW9v3sy++HjHBaYKVKZuJlYowsheq1evpmXLloSFhXHmzJliHX/+/PkkJiY6JS6lHGXgQFi1Cq50pIc2u3aRXK4cvR9+2LGBqcsqU4m6KGbOnMkjjzxCQkICfn5+xdr3ShO1UiVp4EBrdvIrarkQgYUL8e/ZE+/KlR0em8pfmUzUK1asoGvXrsTExNC8eXPuuOMORISpU6fy2WefMWHCBO644w4AJk6cSGRkJCEhIYwfPz7vGDNmzCAkJITQ0FDuuusuvv/+exYsWMCjjz5KWFgYe/fuZe/evdx44420bduWzp07s3PnTgD2799Phw4diIyM5KmnnrLlGqiyKzQUGjSAuXOLv2/iokVE7N9PfEiI4wNTl2Xf5LZdu1667rbbYPRoSE+Hm2669PvYWOt1/DjExFz83YoVxTr95s2b2b59O7Vr16ZTp06sXbuW4cOHs2bNGqKjo4mJiSEuLo7du3ezYcMGRIT+/fuzatUqqlatygsvvMDatWsJCgrixIkTBAYG0r9//7x9AXr06MHbb79NkyZNWL9+PaNHj2b58uWMHTuWv/3tb9x9991MmTKlWHErdbWMsfpUv/EGpKZas5UX1aJ589gE1DzfKVuViDJZowZo164dwcHBeHh4EBYWxoEDBy7ZJi4ujri4OMLDw2nTpg07d+5k9+7dLF++nJiYGIKCggAIDAy8ZN/Tp0/z/fffM2jQIMLCwhg5ciRHcudDWrt2LbfffjsAd911l/MKqdRl9O8PmZkQF1e8/Rbt3Ut4eDjBkZHOCUzly74adUE1YH//gr8PCip2DfqvypUrl/fZ09OTrKysS7YREZ544glG/qW/6OTJkwvtVZGTk0NAQAAJCQn5fq+9MpSdOnWCKlVg4UK49dai7XMiKYm1a9fy5JNPOjc4dYkyW6Muit69ezNt2jROnz4NwG+//caxY8fo0aMHn332GcnJyQCcOHECgIoVK5KamgpApUqVaNCgAZ9//jlgJf0tW7YA0KlTJ2blDmM2c+bMEi2TUgBeXtCnD3z1FWRnF22fJePHk5OTQ7TWpkucJuoC3HDDDQwdOpQOHTrQunVrYmJiSE1NpWXLljz55JNERUURGhrKQw89BMCQIUOYOHEi4eHh7N27l5kzZ/L+++8TGhpKy5Yt+fLLLwF4/fXXmTJlCpGRkZw6dcrOIqoyrF8/63bP+vVF237RV19R3Rgi+vRxbmDqEkZEHH7QiIgIif9LZ/gdO3Zw7bXXOvxc6vL0mquCpKRAtWrw8MPw4osFb3suPZ3qFSowoHFjpjlq+D11EWPMJhGJyO87rVErVUYFBECXLlY7dWG+f+89UkSI1t4ettBErVQZ1q8fJCbCvn0Fb7fo44/xAXqNHVsicamLaaJWqgzr1896L6xWfcOtt/L8TTdRsXZt5welLqGJWqkyrFEjaNGi8ETda9w4Hv3qq5IJSl1CE7VSZVy/frByJVyuA1L8jBlsz+2xpOyhiVqpMq5fP8jKgq+/zv/7cQ8+yG2DB5dsUOoiZSpRv/DCC7Rs2ZKQkBDCwsJYX9QOpA6WkJDA4sWL85YXLFjAi7n9o2JjY5kzZ84l+6xYsYLoaJ1XWDle+/bWw775Nn/88Qcf/vEH0wYNKvG41J/se4S8hK1bt45Fixbx448/Uq5cOY4fP05mZqYtsSQkJBAfH89NuQNP9e/fn/79+9sSi1KentC3rzXsaVaW9dRinrg46mRnU2fECNviU2WoRn3kyBGCgoLyxvgICgqidu3a1K9fn+PHjwMQHx9P19xR/VauXElYWBhhYWGEh4fnPRr+3//+l9atWxMaGsq4ceMALjucaWxsLKNGjaJz5840bdqURYsWkZmZydNPP83s2bMJCwtj9uzZfPDBBzzwwAN5sX777bcX7fNXaWlp3HvvvURGRhIeHp73xKNSV6pfPzh5Etatu3j95IkT+aR8eejQwZ7AFGBTjfrBB+EyYxVdsbAwmDTp8t/fcMMNTJgwgaZNm9KzZ08GDx5MVFTUZbd/+eWXmTJlCp06deL06dP4+vqyZMkS5s+fz/r16/H3988b42PEiBH5DmcKcODAAVauXMnevXvp1q0be/bsYcKECcTHx/PGG28A8MEHH1x07vz2udALL7xA9+7dmTZtGikpKbRr146ePXtSvnz5Yl83pQB69QJvb1i0CDp3ttZlZ2XxXHw8vevWZahXmfnj2yWVmRp1hQoV2LRpE++++y7VqlVj8ODBlyTIC3Xq1ImHHnqIyZMnk5KSgpeXF99++y333HMP/v7+gDW8aUHDmQLcdttteHh40KRJExo2bJhX2y5IYfvExcXx4osvEhYWRteuXcnIyODgwYNXdmGUAipVgqgoK1Gft2HjRo7n5BD9yCP2BaaAItSojTG+wCqgXO72c0RkfMF7Faygmq8zeXp60rVrV7p27Urr1q358MMP8fLyIicnB+CiWbvHjRtH3759Wbx4Me3bt+fbb79FRC4ZnrS4w5kWZXjTwvYREebOnUuzZs0KPZZSRRUdbf21u28fNGwIixYtwtPTk95Dh9odWplXlBr1WaC7iIQCYcCNxpj2To3KCXbt2sXu3bvzlhMSEqhXrx7169dn06ZNAMy9YG6ivXv30rp1ax5//HEiIiLYuXMnN9xwA9OmTSM9PR2whjctaDhTgM8//5ycnBz27t3Lvn37aNas2UXDoeYnv30u1Lt3b/73v/9xfkCtzZs3X+XVUcpK1PBnrXrRW2/RuVkzqlSpYl9QCihCohbL6dxF79yX44fcc7LTp08zbNgwWrRoQUhICImJiTzzzDOMHz+esWPH0rlzZzw9PfO2nzRpEq1atSI0NBQ/Pz/69OnDjTfeSP/+/YmIiCAsLIyXX34Z4LLDmQI0a9aMqKgo+vTpw9tvv42vry/dunUjMTEx72biX+W3z4Weeuopzp07R0hICK1atdJ5F5VDNGoEzZtbifqX779n68mTRDdsaHdYCqwaYGEvwBNIAE4DLxW2fdu2beWvEhMTL1nn7oYNGyaff/65becvi9dcXZ1HHhHx9hZ55dahAsjOxYvtDqnMAOLlMjm1SDcTRSRbRMKAYKCdMabVX7cxxowwxsQbY+KTkpIc9L8RpVRJ6tcPzp2Dj7/ZSGNvb5r27m13SIpids8TkRRjzArgRuCnv3z3LvAuWBMHOCrA0qygXiVKuaKOHaFy5dNsPbWfMeGtMR5lpmOYSyv0X8EYU80YE5D72Q/oCRTex0wpVep4eUGXiAN4mOvpd+fddoejchWlRl0L+NAY44mV2D8TkUsfl1NKuYUh97Zi4bLvKN/J7kjUeYUmahHZCoSXQCxKKZtJTg6Rjfbg4dGURYvguuvsjkhBGXoyUSlVuM2ffkrz9s1oHvwJ+Qwzo2xSphK1p6cnYWFhtGrVin79+pGSkmJ3SMX2zDPP5PXfVsrRqm3axFNAzKAOJCTAoUN2R6SgjCVqPz8/EhIS+OmnnwgMDGTKlCl2hwRYfdnPP8aulJ2uWb2aZzp0YMh9DQDQ2bdcQ5lK1Bfq0KEDv/32G3D5YUqPHj3KgAEDCA0NJTQ0lO+//x6AV199lVatWtGqVSsm5Q5c8vjjj/Pmm2/mHf+ZZ57hlVdeAWDixIlERkYSEhLC+PHWMCkHDhzg2muvZfTo0bRp04Zff/013+3AGi2vWbNm9OzZk127djn92qiy6fhPP7EoPp4zvXvTvLk13kdhcymqEnK5J2Gu5lWUJxOjpkfJ9M3TRUQkMytToqZHyUdbPhIRkbTMNImaHiWzts0SEZGUMykSNT1K5ibOFRGRpLQkiZoeJQt2LhARkSOpR4r05E/58uVFRCQrK0tiYmJkyZIlIiLSvXt3+fnnn0VE5IcffpBu3bqJiMhtt90mr732Wt4+KSkpEh8fL61atZLTp09LamqqtGjRQn788Uf58ccfpUuXLnnnuvbaa+WXX36RpUuXyv333y85OTmSnZ0tffv2lZUrV8r+/fvFGCPr1q0TEbnsdufPl5aWJqdOnZJGjRrJxIkTi1RefTJRFcfUYcMEkK1z5oiIyD/+IeLrK5KWZnNgZQQFPJlYpgaZPXPmDGFhYRw4cIC2bdvSq1evi4YpPe/s2bMALF++nBkzZgBW+3blypVZs2YNAwYMyBv7eeDAgaxevZp//OMfHDt2jMOHD5OUlESVKlWoW7cukydPJi4ujvBwq+PM6dOn2b17N3Xr1qVevXq0b2+NbxUXF5fvdqmpqQwYMCBvaFWdCUY5y6Lff6duYCCtBgwArEGaJk+G5cv/HLBJ2cO2RL0idkXeZ29P74uW/b39L1qu7Fv5ouUg/6CLlmtWqFmkc55voz516hTR0dFMmTKF2NjYAocp/SuRyz90GRMTw5w5c/j9998ZMmRI3vZPPPEEI0eOvGjbAwcOXDTQ/+W2mzRpUpGGRlXqamRkZPDNmjUMGzYs72nELl2gQgVrkCZN1PYqk23UlStXZvLkybz88sv4+flddpjSHj168NZbbwGQnZ3NH3/8QZcuXZg/fz7p6emkpaXxxRdf0Dl3SowhQ4Ywa9Ys5syZQ0xMDGANSTpt2jROn7YGIPztt984duzYJTFdbrsuXbrwxRdfcObMGVJTU1mojYbKCVa+/z5paWlE9+qVt65cOejd20rUBdRPVAkok4kaIDw8nNDQUGbNmnXZYUpff/11vvvuO1q3bk3btm3Zvn07bdq0ITY2lnbt2nHdddcxfPjwvOaKli1bkpqaSp06dahVqxZgTQE2dOhQOnToQOvWrYmJicl3LOrLbdemTRsGDx5MWFgYt956a97/FJRypIXvvIM/0C13ztDzoqPht98cP3WeKh5T0J/yVyoiIkLi4+MvWrdjxw6uvfZah59LXZ5ec1UUkpNDfR8fwqpV48sLppEDOHoUatWCZ58FHfbcuYwxm0QkIr/vymyNWill2TZvHgezs+mfz5CmNWpAu3boU4o200StVBm38J13AOj70EP5fh8dDRs2WLVrZQ9N1EqVcYm7d9OufHlqhoTk+/35Hh+LF5dgUOoimqiVKuNmHjjAsj17Lvt9aCgEB+tTinbSRK2UokLNyz+LYIxVq46Lg9xnwVQJ00StVBl2V8OGjA0LK3S76GhIS4OVK50fk7pUmUjUycnJhIWFERYWRs2aNalTp07ecmZm5kXbTpo0ifT09HyPM3z4cBITEws8V1JSEtdddx3h4eGsXr26WHEmJCSwWBsCVUk5c4aggwep4uNT6Kbdu4Ofn/b+sEuZSNRVq1YlISGBhIQERo0axT//+c+8ZZ+//EgLStRTp06lRYsWBZ5r2bJlNG/enM2bNxf74RRN1KpELV/Oa9nZPPP884Vu6ucHPXroU4p2KROJOj/Lli0jPDyc1q1bc++993L27FkmT57M4cOH6datG926dbtkn65du3L+QZ4KFSrw5JNPEhoaSvv27Tl69CgJCQk89thjLF68mLCwMM6cOUNcXBwdOnSgTZs2DBo0KO8R8Y0bN9KxY0dCQ0Np164dp06d4umnn2b27NmEhYUxe/bsEr0equw59MknSPnyEBVVpO2jo2H/ftixw8mBqUvYNihT1788qpqf6OhoHnnkkbztY2NjiY2N5fjx43ljaZy3YsWKIp87IyOD2NhYli1bRtOmTbn77rt56623ePDBB3n11Vf57rvvCAoKKvAYaWlptG/fnhdeeIHHHnuM9957j//7v/9jwoQJxMfH88Ybb3D8+HGef/55vv32W8qXL89LL73Eq6++yrhx4xg8eDCzZ88mMjKSP/74A39//4v2VcqZJCeH62bNok/dukwtV65I+/Tta70vXAiF/GGpHKxM1qizs7Np0KABTZs2BWDYsGGsWrWqWMfw8fEhOreDadu2bTlw4MAl2/zwww8kJibSqVMnwsLC+PDDD/nll1/YtWsXtWrVIjIyEoBKlSrh5VWmRpxVNvtx3ToO5+TQ+ZZbirxPcDCEh2s7tR3sG+a0GDXgv24fFBRU7P0vdOHwolfK29s7b/hRT09PsrKyLtlGROjVqxeffvrpReu3bt2qQ5cqWy385huMMdz0r38Va7/oaHjhBUhOhqpVnRScukSZrFFnZGRw4MAB9uR28v/oo4+Iym2nq1ixYr6j212J9u3bs3bt2rzzpKen8/PPP9O8eXMOHz7Mxo0bAUhNTSUrK8uh51aqIF98/DGdOnakWrVqxdovOhpycuDrr50UmMpXmUzUvr6+TJ8+nUGDBtG6dWs8PDwYNWoUACNGjKBPnz753kwsrmrVqvHBBx9w++23ExISQvv27dm5cyc+Pj7Mnj2bMWPGEBoaSq9evcjIyKBbt24kJibm3UyMj49n+PDhVx2HUhfas2wZW/fuZWAxkzRARARUr67NHyVNhzl1Y3rNVX7+27cvjy9ezIE1a6jXqVOx97/3Xpg3D5KSwNvbCQGWUTrMqVIqz7yVK2nr739FSRqs5o9Tp2DtWgcHpi6r0ERtjLnGGPOdMWaHMWa7MWZsSQSmlHK8Qxs3sj4tjYFXmKQBevWypun64gsHBqYKVJQadRbwsIhcC7QH/m6M0V6USpVCi157DYCBDz54xceoWNGaS3HePOvGonK+QhO1iBwRkR9zP6cCO4A6V3IyZ7SHq/zptVb5uf+DD/hh6lSa33TTVR0nJgYOHYLcjkvKyYrVRm2MqQ+EA+vz+W6EMSbeGBOflJR0yb6+vr4kJydrAikBIkJycjK+vr52h6JcjKePD9fdd99VH6dfP+tG4pw5DghKFarIvT6MMRWAlcALIjKvoG3z6/Vx7tw5Dh06REZGxpXGqorB19eX4OBgvPW2vMo199FHWRkXx4vLluFfyBAJRdG3LyQmwr591pjV6uoU1OujSE8mGmO8gbnAzMKS9OV4e3vToEGDK9lVKeUAuxYu5Nt9+/ALDHTI8WJirK56P/4Ibds65JDqMorS68MA7wM7RORV54eklHK4lBT+tX8/20aPxng4plfuzTeDl5c2f5SEovyLdQLuArobYxJyX1d3J0IpVaLOfv45ZGbiOXSow44ZGGhNKDBnjo5R7WxF6fWxRkSMiISISFjuS0e3V6oUufmJJ7itfHnIHbHRUWJiYM8e2LrVoYdVf6FPJirl5pKOHePbEydoHBnp8Lt+t9wCHh7a/OFsmqiVcnNz5s4lW4Qhr7/u8GNXqwZdu8Ls2dr84UyaqJVyc7OmT6dFixa0bt3aKce//XbYvRs2bXLK4RWaqJVya4c2bmT1xo0MqVfPaZNVxMSAjw/MnOmUwys0USvl1j57/nkEGPLPfzrtHAEB1sMvs2ZBdrbTTlOmaaJWyo3N/OYb2vr706RXL6ee54474PffYflyp56mzNJErZSb2jZ3Lj+eOcOwPn2cfq6+faFyZW3+cBZN1Eq5qZkTJ+IN3P7CC04/l68v3HqrNfTpmTNOP12Zo4laKTc1fvFivp08maBmzUrkfHfcAampsHBhiZyuTNFErZSb8gsMpMuYMSV2vqgoqFMHPvigxE5ZZmiiVsoNPRoZydsOHNejKDw94Z57YOlS+PXXEj2129NErZSbyUlKYtOmTezaubPEz33PPdb0XFqrdixN1Eq5GY9Zs1guwsvvvVfi527Y0BpRb9o0nU/RkTRRK+VGJCeH42+8AZGReNo0mv/w4XDgAHz3nS2nd0uaqJVyI6unTKH2zz+zols322IYMACqVIGpU20Lwe1oolbKjby1cCHlvb1p9+ijtsXg62t11Zs3D5KTbQvDrWiiVspNHDt2jLkrVjBs9GiHTF57NYYPh8xMvanoKJqolXIT055+mnPnzjFq1Ci7QyE0FDp3hilTdKAmR9BErZQbyMrI4O2pU+kaEEDz5s3tDgeAMWNg/3746iu7Iyn9NFEr5QbmPfEEv2RnM9YFatPn3XILBAfD//5ndySlnyZqpUo5ycnhlffeo7G3N/2efdbucPJ4e8Pf/gbffguJiXZHU7ppolaqlFv79ttsSEvjoVtvxdPHx+5wLnL//VCunNaqr5YmaqVKuQ1Ll1LDw4NhLpgNq1WDu+6yen8cPWp3NKWXJmqlSrmHvvySvUeO2N4l73IefRTOnoVJk+yOpPTSRK1UKfbr+vUAlK9e3eZILq9pU2sC3DffhFOn7I6mdCo0URtjphljjhljfiqJgJRSRbNn2TIatm/P1GHD7A6lUE88AX/8YSVrVXxFqVF/ANzo5DiUUsVUbdo0nvLyoq8TZxh3lPBw6N0bXnsN0tPtjqb0KTRRi8gq4EQJxKKUKqo9e6g8ezZPP/AAtcLC7I6mSJ58EpKSrKcVVfE4rI3aGDPCGBNvjIlPSkpy1GGVUvl4btAgvvTwgMceszuUIuvc2apVv/iitlUXl8MStYi8KyIRIhJRrVo1Rx1WKfUXW9esYXxCAqtDQqBWLbvDKZZ//xtOnICXX7Y7ktJFe30oVco8+txzBAQE8OTcuXaHUmxt2sCgQVZbtfarLjpN1EqVIks//ZS4uDieHj+eKvXq2R3OFXnuOcjIgAkT7I6k9ChK97xPgXVAM2PMIWPMfc4PSyn1V+fS03n4nntoVKECo0ePtjucK9asGYwaBW+/DVu22B1N6VCUXh+3i0gtEfEWkWAReb8kAlNKXWzSoEFsP3uWV8aMwcfFxvQorgkTIDDQGgpVxO5oXJ82fShVChxct45nFi+mf82a3Pzvf9sdzlULDIT//AdWr4ZPP7U7GteniVopFyc5Ofxj4EAAJs+ZY3M0jnPvvRAZCY88Aikpdkfj2jRRK+XiknfvZtvx4zxz003U69TJ7nAcxsMD3noLjh2DBx+0OxrX5mV3AEqpggU1a8aWX3/FLzDQ7lAcrm1baxyQ55+HW2+Ffv3sjsg1aY1aKRclOTm8PXQoZ06coELNmi43KYCjPPUUhITAiBHWwzDqUpqolXJRq0eP5m+ffsqcf/3L7lCcysfHmljg+HGr3Vp7gVxKE7VSrmjlSrq89x7re/TgzjIwNmh4OEycCF9+Ca+8Ync0rkcTtVIu5vCPP7J2wABo3Jh28+ZhPMrGf6Zjx1rt1OPGWd321J/Kxi9AqVLi9OnTRHftSvTJk6RMnw6VKtkdUokxBqZNg4YNrfFADhywOyLXoYlaKReRlZXFkCFD2JKWxsznnyegY0e7QypxlSrB/PnWHIt9+ujNxfM0USvlAnKyshjesydfffUVU6ZM4aYnn7Q7JNu0aGEl63374JZbrAGcyjpN1ErZLCcrixEtW/LhypU8c9ddjBo1yu6QbBcVBR9+aLVVDxigyVoTtVI2ys7MZGSrVrz/88881bkz42fMsDsklzFkCLz3HixdCv37w5kzdkdkH03UStkk/fhxBtarx9Rdu3iyUyeeXbHC7pBczvDh8P778O23ZbvNWhO1UjbZOHMmX//+O28MGsTza9aUmW54xXXPPTBzJqxbBx06wJ49dkdU8vSXoVQJ+2XtWgCixo5lzw8/8PfPPrM5Itd3++1WrTo5Gdq3h6+/tjuikqWJWqmSkpnJ13fcQePrr+e7V18F4JrrrrM5qNKjc2f44QeoXdtqBnnsMcjMtDuqkqGJWqkScHbdOoiMpMsnn/BgixZEDh1qd0ilUuPGsH49/O1v1iPn110HGzfaHZXzaaJWyomOHj3KqPBwQjt2JP3YMfy//JKJ27dToWZNu0Mrtfz84M034YsvrJnM27e3Hj935xuNmqiVcoKUAwd47plnaNy4Me9v3Urv0FCy1q+3+pkph7jlFtixw5oo93//sx49/89/ID3d7sgcTxO1Ug706/r1PBoZSd0GDXj62Wfp1asX23fs4PWEBCrVrWt3eG6ncmWYMgW2boUuXeBf/4L69eGZZ6yZY9yFJmqlrpLk5DDvsce4qXp16rdvz6vx8fSrV4+EOXOYN28eTZs2tTtEt9eqFSxYAGvWWO3Wzz4LdevCnXfCN99AdrbdEV4dI04YpTsiIkLi4+MdflylXEXasWNsnT+fDiNGIDk5hJUvT3JmJvd07Mh9//kP9a+/3u4Qy7Rdu+D1160ZzlNSIDgYYmKsqb46dwZvb7sjvJQxZpOIROT7nSZqpQqXfvw462fM4PoTJ/BetYp/rl3LOzk5HE9Kwj8oiIPr1lGnbVu3nS6rtMrIgIULYcYMq2Z99qw1Ql+vXlZTSefO1jRgnp52R6qJWqkiy8nK4tDGjexes4Yta9awuVw5fty+nZ07dpAjwg/GcF3btmxr2ZLjISF0eeABTc6lRFqa9dDMokUQFwcHD1rrK1a0JtkNCfnz1aIFlC9fsvFddaI2xtwIvA54AlNF5MWCttdErVyR5OSQnZyMV1ISJ3ftYuGSJXS+804adOnC6jfeYNQjj7D37FnOXrBPnaAg2nToQHjdukQGBtJ15Egq1KljWxmU4xw8aLVpr1kDmzfDtm1WMj+vZk1o0MDqTdKwodXmXaOG9apZ03ovV85x8RSUqL2KsLMnMAXoBRwCNhpjFohIouNCVGVKTg5kZ5Nx+jTG05NylSohOTkcS0wkOzOTrLNn896z/P3JDgggMz2dM5s2Uc3Pj6ZVq5KdlsbHy5YR2rMnYYMHc3L/fl4bNoz09HROpaaSkpbGyfR0UgIDSQFSkpNJSUnhJeBh4DgwDPgQaNClC4HXXEPTwED6BgfTuGlTmrRpQ6s+fah27bU2XijlTHXrwtCh1gusn+X+/VYPkh07rM/79sHatVZbd07OpccICICqVaFKFesVHGzNUuNwIlLgC+gALL1g+QngiYL2adu2rVyJrXPmiH8VI7XLe0lzHx9p5uMj/gFGatcKlGbNmknTuteIX4CRB0MCRJo2lbWNgsWvspFnx8eKiMicF54Wv8pGavt5SmNvb2ng42Ut1wqSRo0aSb0a1cS3kpGJLQNFGjWSGQ2ri28lI9OnPS8iIhNHDBXfikZql/OUBl5eUsfHU3wrGqlVu7rUr19fagdWFt+KyJzmQSL16smE+pWlXAVk5dovRUTkgb6dpFwFpJa3h1zj6SnVfTykXHmkZq0aEhwcLNUq+Uu58siPDauJ1Kkj99W1lo+kHhERkVsiG4uPP1LD00NqeXhIFW9jLdesITVr1pQAfx/x8UfS6lQXqVFDbqztIxWCPCQzK1NERNo3rirefkh1Y6SaMVLR24i3n5GgoCAJCgqSCr6eUqEKIkFBIlWrSnh1TwkK9s67/g2r+4qPDxJojFQxRvy8EJ9yRgICAiQgIEB8vY3UqG5EAgJEAgKkbhUjDZqVz9u/SgUP8fZCKoFUBPHxQrx9PKRChQri6+srHgZpVB0Ra6Jp8fNBQkOriohI1tmzAhT6iqxl7ZuVu9yza0MREfl1wwYBxNsDqe3hIS3KlZMqNT2lRatgueOOO+Tv994rDa8PkAdHdBH59FPJXLZMOjwbJlNWTMqLP2p6lEzfPF1ERDKzMiVqepR8tOUjERFJy0yTqOlRMmvbLBERSTmTIlHTo2Ru4lwREUlKS5Ko6VGyYOcCERE5knpEoqZHyZLdS0RE5GDKQYmaHiXf7P1GRET2ntgrUdOjZMX+FSIisjNpp0RNj5K1B9eKiMi2o9skanqUbDi0QURENh/ZLFHTo2Tzkc0iIrLh0AaJmh4l245uExGRtQfXStT0KNmZtFNERFbsXyFR06Nk74m9IiLyzd5vJGp6lBxMOSgiIkt2L5Go6VF5v70FOxdI1PQoSUpLEhGRuYlzJWp6lKScSRERkVnbZknU9ChJy0wTEZGPtnwkUdOj8n570zdPl6jpUXnX8t34d6XHhz3ylqdsmCI3fnxj3vKkdZOk3yf98pYnrp0oA2cPzFv+z+r/yODPB+ctT1gxQe6Ye0fe8lPLn5LY+bF5y+O+GSf3L7g/b/nhpQ/L6EWj85bHLhkrY5eMzVsevWi0PLz04bzl+xfcL+O+GZe3HDs/Vp5a/lTe8u2f3yEPL5ggGzaILFwoEvnSYOk14T/ywAMit98uUmPMQKk7ZKJ07SpXDIiXy+TUQmvUQB3g1wuWDwGXDFBgjBkBjACoe4X9Rf0CAihf3Yc6mf7UP1ceMXDCP4lrAmtS75pW5KSc5OTBY1QIrg0VW+OffpIKu5PwDwwAoHKN6lSs7UPjzErUyfInw2Sz1i+JprUbUrtGY84cPcTaI+up0KQRlG9KpVO/UmnfKXwDrHnpgq6pQ+W65Qg5G0i1HF9SPM6yoVwy4Q1bE1SlDid+2cnG5K34tQqFcsEEHd9JwC9b8Pb3A6B24/pUOfgjHc5Wp7L48LtHOpt9TtC5VScq+gVw+OdNJPzxMz5trgfPQGof3kSVw7vyyl+/RUOqnvuNnhl18MWL/Z6pbPc+Se+2ffDx9GHvT6tIPHMQ2kUD3tQ7sIrtqYfy9m/UqgG/VDzDzWfrA7DDO4U93qn0jxyEMYaffvyaQx7HIew2a/ufl5JhUvP2bx5WD5KPcFNmPTCGTV7HOF4umxsjhmCMYcP6eWSWPwutbgegydYvOBP4Z/vsteG1+SPjDD2zrH//1V6/keVfju5tYvDy8uK7dTOpGuQLYXeDlxdtv59KjSbBAHh4edHp5oZU9PIjpmIkXt7evJ+2iqDAOgyN+hvenp5MXvYsra5pDD3G4ennR9+vxhDRzPop1mnblgGfDqBj3Y480vERAPp/2p8eDXowtv1YAPrMPEyTpv0gcgjegP+hqnhX8L+i36oqWzw8oHIliIy0lj/OgLCaMC63c8+tn0GHYHjESbOnFdpGbYwZBPQWkeG5y3cB7URkzOX20TZqpZQqnoLaqIvywMsh4JoLloOBw44ITCmlVOGKkqg3Ak2MMQ2MMT7AEGCBc8NSSil1XqFt1CKSZYx5AFiK1T1vmohsd3pkSimlgCIkagARWQwsdnIsSiml8qGDMimllIvTRK2UUi5OE7VSSrk4TdRKKeXinDJ6njEmCfjlCncPwhqKoSzRMpcNWmb3dzXlrSci1fL7wimJ+moYY+Iv93SOu9Iylw1aZvfnrPJq04dSSrk4TdRKKeXiXDFRv2t3ADbQMpcNWmb355TyulwbtVJKqYu5Yo1aKaXUBTRRK6WUi3OZRG2MudEYs8sYs8cYM87ueBzFGHONMeY7Y8wOY8x2Y8zY3PWBxphvjDG7c9+rXLDPE7nXYZcxprd90V8dY4ynMWazMWZR7rJbl9kYE2CMmWOM2Zn7792hDJT5n7m/65+MMZ8aY3zdrczGmGnGmGPGmJ8uWFfsMhpj2hpjtuV+N9kYY4ocxOXm6CrJF9bwqXuBhoAPsAVoYXdcDipbLaBN7ueKwM9AC+C/wLjc9eOAl3I/t8gtfzmgQe518bS7HFdY9oeAT4BFuctuXWasuXKH5372AQLcucxY0/TtB/xylz8DYt2tzEAXoA3w0wXril1GYAPWHLQGWAL0KWoMrlKjbgfsEZF9IpIJzAJutjkmhxCRIyLyY+7nVGAH1g/8Zqz/sMl9vyX3883ALBE5KyL7gT1Y16dUMcYEA32BqResdtsyG2MqYf0H/T6AiGSKSApuXOZcXoCfMcYL8Mea/cmtyiwiq4ATf1ldrDIaY2oBlURknVhZe8YF+xTKVRJ1fhPo1rEpFqcxxtQHwoH1QA0ROQJWMgeq527mLtdiEvAYkHPBOncuc0MgCZie29wz1RhTHjcus4j8BrwMHASOAKdEJA43LvMFilvGOrmf/7q+SFwlUefXVuNW/QaNMRWAucCDIvJHQZvms65UXQtjTDRwTEQ2FXWXfNaVqjJj1SzbAG+JSDiQhvUn8eWU+jLntsvejPUnfm2gvDHmzoJ2yWddqSpzEVyujFdVdldJ1G49ga4xxhsrSc8UkXm5q4/m/jlE7vux3PXucC06Af2NMQewmrG6G2M+xr3LfAg4JCLrc5fnYCVudy5zT2C/iCSJyDlgHtAR9y7zecUt46Hcz39dXySukqjddgLd3Du77wM7ROTVC75aAAzL/TwM+PKC9UOMMeWMMQ2AJlg3IUoNEXlCRIJFpD7Wv+VyEbkT9y7z78Cvxphmuat6AIm4cZmxmjzaG2P8c3/nPbDuwbhzmc8rVhlzm0dSjTHtc6/V3RfsUzi776hecBf1JqweEXuBJ+2Ox4Hluh7rT5ytQELu6yagKrAM2J37HnjBPk/mXoddFOPOsCu+gK782evDrcsMhAHxuf/W84EqZaDMzwI7gZ+Aj7B6O7hVmYFPsdrgz2HVjO+7kjICEbnXaS/wBrlPhhflpY+QK6WUi3OVpg+llFKXoYlaKaVcnCZqpZRycZqolVLKxWmiVkopF6eJWimlXJwmaqWUcnH/DxmpNWhWXxCzAAAAAElFTkSuQmCC\n",
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"S, I, R, C = plot_sir(N, i_0, beta=popt[0], gamma=popt[1], delta=0, steps=1000, verbose=True)"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"The total number of infections stabilizes, and the pandemic abates. The effective reproductive number $R_0$ is less than 1.\n",
"\n",
"Let's do a quick sanity check.\n",
"Recall that $R_0 = \\beta/\\gamma$ for the SIR model _initially_, when the number infected, recovered, or dead are negligible compared to the total population.\n",
"Once some of the population has recovered, some of the people an infectious person encounters wiil be people who have recovered and not susceptible to reinfection (according to the model).\n",
"To first order, $\\beta$ is effectively reduced by $S/N$."
]
},
{
"cell_type": "code",
"execution_count": 18,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"51261262636618.664"
]
},
"execution_count": 18,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# R_0 at the time the first infection is detected\n",
"popt[0]/popt[1] # greater than 1. According to the model, the infection will spread if R_0 > 1."
]
},
{
"cell_type": "code",
"execution_count": 19,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"219047240451.27252"
]
},
"execution_count": 19,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# After 1000 timesteps, R_0 is different \n",
"(S[-1]/N)*popt[0]/popt[1] # less than 1: According to the model, the infection will abate."
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"# Fitting curves to the cumulative incidence function \n",
"\n",
"The SIR model and its variants attempt to capture the dynamics of transmission and recovery,\n",
"For some parameter choices, that leads to a sigmoidal shape for the cumulative incidence of infections.\n",
"\n",
"Some pandemic predictions simply fit a sigmoidal function to the cumulative number of infections, an approach\n",
"that dates back to William Farr in the 1800s. \n",
"King et al. (2015) showed that this approach does not work well for AIDS or Ebola.\n",
"Nonetheless, this is the approach the IHME takes: they fit a scaled and shifted Gaussian cdf to \n",
"cumulative incidence.\n",
"\n"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"## Sigmoidal Models\n",
"\n",
"\n",
"Common sigmoidal models include CDFs of unimodal distributions (such as the Gaussian cdf $\\Phi(x)$) and the sigmoid function $1/(e^{-x} + 1)$, the inverse of the logistic function.\n",
"\n",
"To allow this function to be shifted and scaled, we can introduce the family of curves\n",
"\n",
"\\begin{equation}\n",
"\\sigma(x; a, b, c) \\equiv \\frac{c}{e^{-b(x-a)}+1}.\n",
"\\end{equation}\n",
"\n",
"The parameter $a$ controls where the function crosses $c/2$; $b$ controls how rapidly it increases, and $c$ is its asymptotic limit.\n",
"\n",
"Let's fit this to the Danish data."
]
},
{
"cell_type": "code",
"execution_count": 20,
"metadata": {},
"outputs": [],
"source": [
"def g(x, a, b, c):\n",
" '''\n",
" Wrapper for the sigmoid to pass to the curve-fitting function\n",
" ''' \n",
" return c/(np.exp(-b*(x-a))+1)"
]
},
{
"cell_type": "code",
"execution_count": 21,
"metadata": {
"scrolled": true
},
"outputs": [
{
"data": {
"text/plain": [
"[<matplotlib.lines.Line2D at 0x7f98b0dde0a0>]"
]
},
"execution_count": 21,
"metadata": {},
"output_type": "execute_result"
},
{
"data": {
"image/png": "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\n",
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"# test g visually.\n",
"z = np.array(range(-8,15,1))\n",
"plt.plot(z, g(z,3,1/2,5))"
]
},
{
"cell_type": "code",
"execution_count": 22,
"metadata": {},
"outputs": [
{
"data": {
"text/plain": [
"array([5.00000000e+02, 1.11921669e-02, 1.16292975e+06])"
]
},
"execution_count": 22,
"metadata": {},
"output_type": "execute_result"
}
],
"source": [
"# fit the DK data\n",
"popts, pvocs = curve_fit(g, x, y, p0=[len(x)/2, 0.02, np.max(y)/2], bounds = ([0, 0, 0], [500, 500, N]), maxfev=10000)\n",
"popts"
]
},
{
"cell_type": "code",
"execution_count": 23,
"metadata": {},
"outputs": [
{
"data": {
"image/png": 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\n",
"text/plain": [
"<Figure size 432x288 with 1 Axes>"
]
},
"metadata": {
"needs_background": "light"
},
"output_type": "display_data"
}
],
"source": [
"# plot\n",
"Cs = g(x, popts[0], popts[1], popts[2])\n",
"times = list(range(len(y)))\n",
"fig, ax = plt.subplots(nrows=1, ncols=1)\n",
"ax.plot(times, y, linestyle='-', color='r', label='Data')\n",
"ax.plot(times, C[:len(times)], linestyle='-.', color='k', label='SIR Model')\n",
"ax.plot(times, Cs, linestyle=':', color='b', label='Sigmoid Model')\n",
"ax.legend(loc='best')\n",
"plt.xlabel('Days since first detected case')\n",
"plt.title('\"Confirmed\" COVID-19 Infections in Denmark')\n",
"plt.show()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"Both models agree reasonably well with past data. But they have rather different predictions about the future.\n",
"\n",
"Let's look at how the predictions vary over time, as the dataset grows."
]
},
{
"cell_type": "code",
"execution_count": 24,
"metadata": {},
"outputs": [],
"source": [
"def plot_predictions(y, n, points):\n",
" '''\n",
" plot SIR and sigmoid models fitted to the first n data\n",
" \n",
" Parameters\n",
" ----------\n",
" y : list of floats\n",
" number of infections at each time point\n",
" n : int\n",
" fit the models to the first n elements of y\n",
" points : int\n",
" number of time points for which to plot the prediction\n",
" \n",
" Returns\n",
" -------\n",
" no return value\n",
" '''\n",
" # fit the two models\n",
" x = list(range(len(y)))\n",
" popt, pvoc = curve_fit(f, x[:n], y[:n], p0=[1, 0.5], bounds = (0, [500, 1]),\n",
" maxfev=10000)\n",
" popts, pvocs = curve_fit(g, x[:n], y[:n], p0=[len(x)/2, 0.02, np.max(y)/2],\n",
" bounds = ([0, 0, 0], [500, 500, N]), maxfev=10000)\n",
" times = list(range(points))\n",
" S, I, R, C = sir_model(N, i_0, beta=popt[0], gamma=popt[1], delta=0, steps=points)\n",
" Cs = g(times, popts[0], popts[1], popts[2])\n",
" fig, ax = plt.subplots(nrows=1, ncols=1)\n",
" ax.plot(x, y, linestyle='-', color='r', label='Data')\n",
" ax.plot(times, C, linestyle='-.', color='k', label='SIR Model')\n",
" ax.plot(times, Cs, linestyle=':', color='b', label='Sigmoid Model')\n",
" ax.vlines(x[n-1], 0, np.max([C, Cs]), label='truncation time')\n",
" ax.legend(loc='best')\n",
" plt.xlabel('Days since first detected case')\n",
" plt.title('\"Confirmed\" COVID-19 Infections in Denmark')\n",
" plt.show()"
]
},
{
"cell_type": "code",
"execution_count": 25,
"metadata": {},
"outputs": [
{
"data": {
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"The predictions depend on the model and details of the data. \n",
"A few additional days of data can drastically change the predictions. \n",
"For some end dates, the two models have nearly identical predictions; for others, \n",
"they differ by a factor of 20 or more.\n",
"\n",
"Unlike the SIR model, the sigmoid model does not \"know\" how many people are in the population, so even if recovery conveys immunity, the cumulative number of infections in the sigmoid model can exceed the size of the population.\n",
"\n",
"This mindless \"curve fitting\" approach is not a reliable basis for predicting the course of the pandemic, predicting ICU demand, predicting the effect of an intervention, allocating resources, or setting public policy.\n",
"It is a mechanical application of models and algorithms to data, not science.\n",
"\n",
"Unsurprisingly, models of this type do not predict infections well in practice. \n",
"The [most-cited predictions](https://covid19.healthdata.org/global?view=total-deaths&tab=trend) have been those promulgated by the [Institute for Health Metrics and Evaluation](http://www.healthdata.org/) (IHME) at the University of Washington.\n",
"They say, \n",
"\n",
">Our model is designed to be a planning tool for government officials who need to know how different policy decisions can radically alter the trajectory of COVID-19 for better or worse.\n",
"\n",
">Our model aimed at helping hospital administrators and government officials understand when demand on health system resources will be greatest. (http://www.healthdata.org/covid/faqs#differences%20in%20modeling, last visited 23 January 2021)\n",
"\n",
"IHME's model is described [here](https://static-content.springer.com/esm/art%3A10.1038%2Fs41591-020-1132-9/MediaObjects/41591_2020_1132_MOESM1_ESM.pdf).\n",
"Early in the pandemic, IHME predicted that there would be about 60,000 COVID-19 deaths in ths U.S.\n",
"As of 18 February 2021, there have been [more than 490,000](https://coronavirus.jhu.edu/us-map).\n",
"Between March and August, 2020, next-day deaths were within the IHME 95% prediction interval only about \n",
"30% of the time, despite revisions of the model https://arxiv.org/abs/2004.04734.\n",
"(We shall not delve into how IHME produces its prediction intervals.)\n",
"See also https://www.vox.com/future-perfect/2020/5/2/21241261/coronavirus-modeling-us-deaths-ihme-pandemic.\n",
"\n",
"One feature of the sigmoidal models is that they predict rapid declines after the peak (the decline is just as rapid as the rise), a feature that\n",
"some politicians like. \n",
"That feature is built into the sigmoidal curve: it is an _assumption_, not something that comes from the data.\n",
"\n",
"There are also more complicated models of pandemics, including models with more \"compartments,\" models based on other sigmoidal functions (such as the Gaussian CDF), and models that allow various kinds of heterogeneity within the population and simulate interactions between individuals.\n",
"Some have proved more accurate than others.\n",
"Just how poorly some models perform is obscured by the practice of revising the model and projections frequently--even daily. \n",
"\n",
"Sometimes, a moddle with little substantive foundation can nonetheless predict the behavior of a system, at least if the system is not changing quickly. \n",
"Accurate predictions of the effect of interventions (masks, lockdowns, vaccination, and so on) require a close tie between the world and the model: the model needs a substantive basis. \n",
"For causal inferences, the model needs to be a _generative model_ or _response schedule_.\n",
"\n",
"George Box famously wrote, \"all models are wrong, but some are useful.\"\n",
"While that quotation is often invoked as a general license to throw models at data, it should instead \n",
"prompt questions:\n",
"\n",
"* useful for what?\n",
"* how can you tell whether a model is useful for a particular goal?\n",
"\n",
"In the same paper, Box wrote, \"It is inappropriate to be concerned with mice when there are tigers abroad.\"\n",
"Most of the fretting people do about statistical models is at the level of mice (e.g., asymptotics under unrealistic assumptions), while the problems with most \n",
"statistical models are tigers: the models have little or no connection to the actual scientific question."
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"## New variants\n",
"\n",
"As of this writing (10 February 2021), new strains of COVID-19 have been detected in the UK, South Africa, and the US, where cases of the new strain have a doubling time of about 10 days. The UK strain is estimated to be 50% to 70% more transmissible than the original virus. There is mounting evidence that the strain is also more lethal.\n",
"https://www.wsj.com/articles/new-u-k-covid-19-variant-could-be-more-deadly-british-officials-say-11611338370 (last visited 22 January 2021).\n"
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