{
"cells": [
{
"cell_type": "markdown",
"id": "ac19c62d-0304-40ef-9ce2-32f4f340488a",
"metadata": {},
"source": [
"# Sensitivity Analysis\n",
"\n",
"The following tutorial demonstrates how one may perform a sensitivity analysis of a simulation model via `calisim`. We will first import our required dependencies."
]
},
{
"cell_type": "code",
"execution_count": 99,
"id": "3a462176-a0ab-4cd7-93c4-7653db5b3292",
"metadata": {},
"outputs": [],
"source": [
"from calisim.data_model import (\n",
"\tDistributionModel,\n",
"\tParameterDataType,\n",
"\tParameterSpecification,\n",
")\n",
"from calisim.example_models import SirOdesModel\n",
"from calisim.sensitivity import (\n",
"\tSensitivityAnalysisMethod,\n",
"\tSensitivityAnalysisMethodModel,\n",
")\n",
"import numpy as np\n",
"import pandas as pd\n",
"from scipy.integrate import solve_ivp"
]
},
{
"cell_type": "markdown",
"id": "1e8112ea-b09f-4cc9-8ad7-94f8cc927101",
"metadata": {},
"source": [
"## SIR Model Parameters and Initial Conditions\n",
"\n",
"We next define our forward model. We will use an SIR (susceptible, infected, and recovered) compartmental model, combined with SciPy's solver for ordinary differential equations. The SIR model is expressed as a system of ordinary differential equations where:\n",
"\n",
"| Parameter | Value | Description |\n",
"|-----------|-------|-------------|\n",
"| β (beta) | 0.4 | Infection rate: probability of transmission per contact per time unit |\n",
"| γ (gamma) | 0.1 | Recovery rate: fraction of infected recovering per time unit |\n",
"| | | **Average infectious period = 1 / γ = 10 time units** |\n",
"\n",
"With the following compartments:\n",
"\n",
"| Compartment | Symbol | Initial Value | Description |\n",
"|------------|--------|---------------|-------------|\n",
"| Susceptible | S0 | 999 | Individuals who can catch the disease (N - I0 - R0) |\n",
"| Infected | I0 | 1.0 | Individuals currently infected and can spread the disease |\n",
"| Recovered | R0 | 0 | Individuals recovered or removed; no longer infectious |"
]
},
{
"cell_type": "code",
"execution_count": 100,
"id": "adb3cf44-f6dd-4ddb-8152-f710902a07a7",
"metadata": {},
"outputs": [],
"source": [
"def sir_simulate(parameters: dict) -> np.ndarray | pd.DataFrame:\n",
" def dX_dt(_: np.ndarray, X: np.ndarray) -> np.ndarray:\n",
" S, I, _ = X\n",
" dotS = -parameters[\"beta\"] * S * I / parameters[\"N\"]\n",
" dotI = (\n",
" parameters[\"beta\"] * S * I / parameters[\"N\"] - parameters[\"gamma\"] * I\n",
" )\n",
" dotR = parameters[\"gamma\"] * I\n",
" return np.array([dotS, dotI, dotR])\n",
"\n",
" X0 = [parameters[\"S0\"], parameters[\"I0\"], parameters[\"R0\"]]\n",
" t = (parameters[\"t\"].min(), parameters[\"t\"].max())\n",
" x_y = solve_ivp(\n",
" fun=dX_dt, y0=X0, t_span=t, t_eval=parameters[\"t\"].values.flatten()\n",
" ).y\n",
"\n",
" df = pd.DataFrame(dict(dotS=x_y[0, :], dotI=x_y[1, :], dotR=x_y[2, :]))\n",
" return df"
]
},
{
"cell_type": "markdown",
"id": "93a95b93-b79e-45b1-aa0e-760c380b686e",
"metadata": {},
"source": [
"We will perform a simulation study with the following ground-truth parameters:"
]
},
{
"cell_type": "code",
"execution_count": 101,
"id": "95c54892-4ce1-41bc-85c0-62f910ccabdf",
"metadata": {},
"outputs": [
{
"data": {
"text/html": [
"\n",
"\n",
"
\n",
" \n",
" \n",
" | \n",
" beta | \n",
" gamma | \n",
" N | \n",
" I0 | \n",
" R0 | \n",
" S0 | \n",
"
\n",
" \n",
" \n",
" \n",
" | 0 | \n",
" 0.4 | \n",
" 0.1 | \n",
" 1000 | \n",
" 1.0 | \n",
" 0 | \n",
" 999.0 | \n",
"
\n",
" \n",
"
\n",
"
\n",
"\n",
"
\n",
" \n",
" \n",
" | \n",
" dotS | \n",
" dotI | \n",
" dotR | \n",
" day | \n",
"
\n",
" \n",
" \n",
" \n",
" | 0 | \n",
" 999.000000 | \n",
" 1.000000 | \n",
" 0.000000 | \n",
" 0 | \n",
"
\n",
" \n",
" | 1 | \n",
" 998.534208 | \n",
" 1.349201 | \n",
" 0.116592 | \n",
" 1 | \n",
"
\n",
" \n",
" | 2 | \n",
" 997.906105 | \n",
" 1.819995 | \n",
" 0.273899 | \n",
" 2 | \n",
"
\n",
" \n",
" | 3 | \n",
" 997.059813 | \n",
" 2.454180 | \n",
" 0.486007 | \n",
" 3 | \n",
"
\n",
" \n",
" | 4 | \n",
" 995.919926 | \n",
" 3.308098 | \n",
" 0.771976 | \n",
" 4 | \n",
"
\n",
" \n",
" | 5 | \n",
" 994.385263 | \n",
" 4.457212 | \n",
" 1.157524 | \n",
" 5 | \n",
"
\n",
" \n",
"
\n",
"