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"# hetGPy SIR Example\n",
"\n",
"This document walks through a two-dimenstional SIR example.\n",
"\n",
"### The SIR Simulation Model\n",
"\n",
"A Susceptible-Infected-Recovered (SIR) model is a [canonical epidemiological compartmental model](https://en.wikipedia.org/wiki/Compartmental_models_in_epidemiology). The core idea is that we can model the spread of an infectious disease by dividing the population into _compartments_ of those who have not had the disease (**S**), those who are infected (**I**) and those who have recovered (**R**), subject to infection rates $\\beta$ and recovery rates $\\gamma$:\n",
"\n",
"
\n",
"\n",
"Our specific implementation is a stochastic version of the SIR model via the Gillespie algorithm from [Hu and Ludkovski (2017)](https://epubs.siam.org/doi/10.1137/15M1045168) where the two model inputs are the size of the initial susceptible population $S_0$ and the initial infected population $I_0$ and our output is the number of cumulative infections in the population (note that both the inputs and outputs are scaled to $[0,1]$ to simplify the calculations, but are otherwise not meaningful). \n",
"\n",
"\n",
"`hetGPy` contains an implemention of this stochastic SIR model (which is also available in the original `hetGP` R package). Note that the example simulations take 1-2 minutes to run.\n",
"\n",
"This walkthrough is also essentially a Python replicate of the R examples from [Binois and Gramacy (2018)](https://cran.r-project.org/web/packages/hetGP/vignettes/hetGP_vignette.pdf) and [Gramacy (2020)](https://bookdown.org/rbg/surrogates/chap10.html#chap10varp) Ch. 10.2.1.\n"
]
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{
"cell_type": "code",
"execution_count": 1,
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"source": [
"import numpy as np\n",
"from hetgpy import hetGP\n",
"from hetgpy.test_functions import sirEval\n",
"from scipy.stats.qmc import LatinHypercube\n",
"from scipy.stats import norm\n",
"from scipy.io import loadmat\n",
"\n",
"\n",
"# space filling design of inputs\n",
"seed = 10\n",
"rand = np.random.default_rng(seed)\n",
"lhs = LatinHypercube(d = 2, seed = seed)\n",
"X = lhs.random(200)\n",
"\n",
"# replicate each input location between 1 and max_reps times\n",
"max_reps = 100\n",
"reps = rand.choice(max_reps + 1, size = len(X), replace = True) \n",
"idxs = np.repeat(np.arange(len(X)),reps)\n",
"\n",
"# run SIR simulation\n",
"X = X[idxs,:]\n",
"Y = np.zeros(len(X))\n",
"for i in range(len(X)):\n",
" Y[i] = sirEval(X[[i],:],seed = i).squeeze()\n",
"\n",
"# predictive grid\n",
"xseq = np.linspace(0,1,100)\n",
"xgrid = np.array([(y,x) for x in xseq for y in xseq])"
]
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"cell_type": "code",
"execution_count": 2,
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"source": [
"model = hetGP()\n",
"model.mle(\n",
" X = X,\n",
" Z = Y,\n",
" covtype = \"Matern5_2\",\n",
" lower = np.array([0.05,0.05]),\n",
" upper = np.array([10.0,10.0]),\n",
" maxit = 1e4\n",
")\n",
"preds = model.predict(xgrid)"
]
},
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"cell_type": "markdown",
"metadata": {},
"source": [
"And then we can plot our results:"
]
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{
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"image/svg+xml": [
"\n",
"\n",
"\n"
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""
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],
"source": [
"%config InlineBackend.figure_formats = ['svg']\n",
"from matplotlib import cm\n",
"import matplotlib.pyplot as plt\n",
"f, ax = plt.subplots(nrows=1,ncols=2, figsize = (9,6))\n",
"\n",
"# data\n",
"m = preds['mean'].reshape(100,100)\n",
"v = (preds['sd2'] + preds['nugs']).reshape(100,100)\n",
"\n",
"ax0 = ax[0].imshow(m,origin='lower',extent=[0,1,0,1],cmap='viridis')\n",
"ax1 = ax[1].imshow(v,origin='lower',extent=[0,1,0,1],cmap='magma')\n",
"ax[0].set_title('Predictive Mean')\n",
"ax[1].set_title('Predictive Variance')\n",
"for a in ax:\n",
" a.set_xlabel('S0')\n",
" a.set_ylabel('I0')\n",
"\n",
"kws = dict(fraction=0.046,pad=0.04)\n",
"f.colorbar(ax0, ax = ax[0],**kws)\n",
"f.colorbar(ax1, ax = ax[1],**kws)\n",
"f.tight_layout()"
]
},
{
"cell_type": "markdown",
"metadata": {},
"source": [
"As discussed in the other resources linked at the beginning of this example, the interpretation of this example is that when there are a large amount of initital Susceptibles ($S_0 > ~ 0.8$) and a moderate amount of initial Infecteds ($I_0 > ~ 0.2$) see high variance in that some stochastic realizations result in large amounts of cumulative infections, but others do not. This is due to the high nonlinearity of an SIR model."
]
}
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![\"Diagram](\"https://commons.wikimedia.org/wiki/File:Diagram_of_SIR_epidemic_model_states_and_transition_rates.svg#/media/File:Diagram_of_SIR_epidemic_model_states_and_transition_rates.svg\")