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| { | ||
| "cells": [ | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "# Exploring Wavefront Speed with an SEIR Rabies Model\n", | ||
| "\n", | ||
| "[A Simple Model for the Spatial Spread and Control of Rabies - Källén, Arcuri, and Murray - 1985](https://doi.org/10.1016/S0022-5193(85)80276-9)" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "A simple model for the spatial spread of rabies models the dynamics of teh fron of an epizootic wave.\n", | ||
| "\n", | ||
| "Populations will be **S**, susceptible, **E**, exposed/incubating, **I**, infectious/rabid, and (untracked) _R_, removed/deceased.\n", | ||
| "\n", | ||
| "$$\n", | ||
| "\\delta S / \\delta t = -KIS \\\\\n", | ||
| "\n", | ||
| "\\delta I / \\delta t = D \\delta^2 I / \\delta x^2 + KIS - \\mu I\n", | ||
| "$$ (1)\n", | ||
| "\n", | ||
| "where $K$ is the transmission coefficient and $1 / \\mu$ is the life expectancy of an infective fox.\n", | ||
| "\n", | ||
| "$$\n", | ||
| "D = kA\n", | ||
| "$$ (2)\n", | ||
| "\n", | ||
| "where $D$ is the diffusion coefficient, $k$ is the rate at which infective foxes leave their territories which have average area A. $1 / k$ is the average time until a fox leaves its territory. Foxes remain in their territory until affected by rabies which leads to random migratory behavior.\n", | ||
| "\n", | ||
| "We will assume a static population and ignore birthrate for the moment expecting that we will simulate sufficiently short time scales.\n", | ||
| "\n", | ||
| "Some non-dimensions quantities:\n", | ||
| "\n", | ||
| "$$\n", | ||
| "u = I / S_0 \\\\\n", | ||
| "\n", | ||
| "v = S / S_0 \\\\\n", | ||
| "\n", | ||
| "x^* = (K S_0 / D)^{1/2} x \\\\\n", | ||
| "\n", | ||
| "t^* = K S_0 t \\\\\n", | ||
| "\n", | ||
| "r = \\mu / K S_0\n", | ||
| "$$ (3)\n", | ||
| "\n", | ||
| "Dropping the asterisks gives the following model equations:\n", | ||
| "\n", | ||
| "$$\n", | ||
| "\\delta u / \\delta t = \\delta^2 u / \\delta x^2 + u(v - r) \\\\\n", | ||
| "\n", | ||
| "\\delta v / \\delta t = - u v\n", | ||
| "$$ (4)\n", | ||
| "\n", | ||
| "where $u \\ge 0$ and $0 \\lt v \\le 1$.\n", | ||
| "\n", | ||
| "$R = 1 / r$ is the basic reproduction rate of the disease, $R_0$\n", | ||
| "\n", | ||
| "There is a typical critical minimum susceptible density, $S_c$ below which the infection cannot persist:\n", | ||
| "\n", | ||
| "$$\n", | ||
| "S_c = \\mu / K = r S_0\n", | ||
| "$$ (5)\n", | ||
| "\n", | ||
| "The proportion, $a$ of susceptibles which remain after the infective wave has passed satisfies\n", | ||
| "\n", | ||
| "$$\n", | ||
| "0 \\lt a \\lt r \\lt 1 \\\\\n", | ||
| "\n", | ||
| "a - r {log} (a) = 1\n", | ||
| "$$ (6)\n", | ||
| "\n", | ||
| "The waveforms for the two species travel together at the same speed, $c$, which is bounded below by\n", | ||
| "\n", | ||
| "$$\n", | ||
| "c = 2 \\sqrt {1 - r}\n", | ||
| "$$ (7a)\n", | ||
| "\n", | ||
| "$$\n", | ||
| "c = 2[D(KS_0 - \\mu)]^{1/2} = 2[D\\mu(1/r-1)]^{1/2}\n", | ||
| "$$ (7b)" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "## Estimates for the Epidemiological Parameters\n", | ||
| "\n", | ||
| "$$\n", | ||
| "r = 0.5\n", | ||
| "$$\n", | ||
| "\n", | ||
| "> An infective fox first goes through an incubation period that can vary from 12 to 110 days, and then a rabid state lasting from 3 to 10 days. A life expectancy of about 35 days gives\n", | ||
| "\n", | ||
| "$$\n", | ||
| "\\mu = 10 yr^{-1}\n", | ||
| "$$\n", | ||
| "\n", | ||
| "> If we assume an average territory to be about $5 km^2$ and that an infective fox leaves its territory about the time it becomes rabid, that is about a month, then\n", | ||
| "\n", | ||
| "$$\n", | ||
| "k \\approx 12 yr^{-1} \\\\\n", | ||
| "\n", | ||
| "D \\approx 60 km^2 yr^{-1}\n", | ||
| "$$\n", | ||
| "\n", | ||
| "Estimated wave-speed $\\approx 50 km / year$" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "## Fox Population Density from Data\n", | ||
| "\n", | ||
| "\n", | ||
| "\n", | ||
| "Fig. 1 Fluctuations in fox population density as a function of the passage of a rabies epizootic after data from the Centre National d'Etudes sur la Rage, 1977. $S_0$ is the population density in the absence of rabies and $S_c$ the critical population density below which the infection dies out. (Redrawn from MacDonald (1980), page 19).\n" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "## Travelling Wave Shape for Simplified Model\n", | ||
| "\n", | ||
| "\n", | ||
| "\n", | ||
| "Fig. 2 The shape of the (non-dimensional) travelling wave for $r = 0.5$. Here $r = \\mu / KS_0$ where $1 / \\mu$ is the life expectancy of an infective fox, $K$ the transmission coefficient and $S_0$ the undisturbed fox density.\n" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": null, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [ | ||
| "import matplotlib.pyplot as plt\n", | ||
| "from matplotlib.figure import Figure\n", | ||
| "\n", | ||
| "class Incubation:\n", | ||
| " def __init__(self, model, verbose: bool = False) -> None:\n", | ||
| " return\n", | ||
| "\n", | ||
| " def census(self, model, tick) -> None:\n", | ||
| " return\n", | ||
| "\n", | ||
| " def __call__(self, model, tick) -> None:\n", | ||
| " return\n", | ||
| "\n", | ||
| " def plot(self, fig: Figure = None):\n", | ||
| " fig = plt.figure(figsize=(12, 9), dpi=128) if fig is None else fig\n", | ||
| " yield\n", | ||
| " return" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": null, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [ | ||
| "class Transmission:\n", | ||
| " def __init__(self, model, verbose: bool = False) -> None:\n", | ||
| " return\n", | ||
| "\n", | ||
| " def __call__(self, model, tick) -> None:\n", | ||
| " return\n", | ||
| "\n", | ||
| " def plot(self, fig: Figure = None):\n", | ||
| " fig = plt.figure(figsize=(12, 9), dpi=128) if fig is None else fig\n", | ||
| " yield\n", | ||
| " return" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "code", | ||
| "execution_count": 1, | ||
| "metadata": {}, | ||
| "outputs": [], | ||
| "source": [ | ||
| "from laser_generic import Model\n", | ||
| "from laser_generic import Susceptibility\n", | ||
| "from laser_generic import Infection\n", | ||
| "import numpy as np\n", | ||
| "import pandas as pd\n", | ||
| "\n", | ||
| "# Fox density from 1/km^2 to 4+/km^2 (higher in suburban and urban areas)\n", | ||
| "# Each patch will represent 5km^2\n", | ||
| "scenario = pd.DataFrame(data=np.full(201, 20, dtype=np.int32), columns=[\"population\"])\n", | ||
| "\n", | ||
| "model = Model(scenario)\n", | ||
| "model.components = [\n", | ||
| " Susceptibility,\n", | ||
| " Incubation,\n", | ||
| " Infection,\n", | ||
| " Transmission\n", | ||
| "]" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "## Survival Fraction for Simplified Model\n", | ||
| "\n", | ||
| "\n", | ||
| "\n", | ||
| "Fig 3. The fraction $a (= S/ S_0)$ of susceptible foxes which survive an epidemic wave as a function of $r (= \\mu / K S_0)$: $a$ and $r$ are related by $a - r log(a) = 1$ where $0 \\lt a \\lt r \\lt 1$.\n" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "## Considering Reproduction\n", | ||
| "\n", | ||
| "> If $S_0$ is the carrying capacity in a particular rabies-free habitat, we can add a logistic population growth term to the equation for the susceptible foxes...\n", | ||
| "\n", | ||
| "$$\n", | ||
| "\\delta S / \\delta t = -KIS + \\beta S(1 - S / S_0)\n", | ||
| "$$ (9)\n", | ||
| "\n", | ||
| "Then\n", | ||
| "\n", | ||
| "$$\n", | ||
| "\\delta u / \\delta t = \\delta^2 u / \\delta x^2 + u(v - r) \\\\\n", | ||
| "\n", | ||
| "\\delta v / \\delta t = -uv + bv(1 - v)\n", | ||
| "$$ (10)\n", | ||
| "\n", | ||
| "where $b = \\beta / KS_0 = r \\beta / \\mu$" | ||
| ] | ||
| }, | ||
| { | ||
| "cell_type": "markdown", | ||
| "metadata": {}, | ||
| "source": [ | ||
| "\n", | ||
| "## Travelling Wave Shape with Births\n", | ||
| "\n", | ||
| "\n", | ||
| "\n", | ||
| "Fig 6. The shape of the travelling wave solution when the susceptible foxes have logistic population growth with a net birth rate of 0.5 per year, that is $b = 0.05$ ($b$ is the net birth rate times the life expectancy of an infective fox). Here the model parameter $r = 0.5$." | ||
| ] | ||
| } | ||
| ], | ||
| "metadata": { | ||
| "kernelspec": { | ||
| "display_name": "Python 3", | ||
| "language": "python", | ||
| "name": "python3" | ||
| }, | ||
| "language_info": { | ||
| "codemirror_mode": { | ||
| "name": "ipython", | ||
| "version": 3 | ||
| }, | ||
| "file_extension": ".py", | ||
| "mimetype": "text/x-python", | ||
| "name": "python", | ||
| "nbconvert_exporter": "python", | ||
| "pygments_lexer": "ipython3", | ||
| "version": "3.12.1" | ||
| } | ||
| }, | ||
| "nbformat": 4, | ||
| "nbformat_minor": 2 | ||
| } |
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