For a recent talk in my department I talked a little bit about agent based modeling and in the process I came across the simple but quite interesting SIR model in epidemiology. The inspiration for this post was Simon Dobson's post on Epidemic spreading processes, which will provide a much more detailed scientific background and take you through some of the code step by step. However as a brief introduction

I've made some minor tweaks to the model by adding vaccinated and dead states. I've also unified the function based approach into a single Parameterized class, which takes care of initializing, running and visualizing the network.

In this blog post I'll primarily look at how we can quickly create complex visualization about this model using HoloViews. In the process I'll look at some predictions this model can make about herd immunity but won't be giving it any rigorous scientific treatment.

The Code

Here's the code for the model relying only on numpy, networkx, holoviews and matplotlib in the background.

In [1]:
import collections
import itertools
import math

import numpy as np
import numpy.random as rnd
import networkx as nx

import param
import holoviews as hv

DEAD = 'D'

class SRI_Model(param.Parameterized):
    Implementation of the SRI epidemiology model
    using NetworkX and HoloViews for visualization.
    This code has been adapted from Simon Dobson's
    code here:
    In addition to his basic parameters I've added
    additional states to the model, a node may be
    in one of the following states:
      * Susceptible: Can catch the disease from a connected node.
      * Vaccinated: Immune to infection.
      * Infected: Has the disease and may pass it on to any connected node.
      * Recovered: Immune to infection.
      * Dead: Edges are removed from graph.

    network = param.ClassSelector(class_=nx.Graph, default=None, doc="""
        A custom NetworkX graph, instead of the default Erdos-Renyi graph.""")
    visualize = param.Boolean(default=True, doc="""
        Whether to compute layout of network for visualization.""")
    # Initial parameters
    N = param.Integer(default=1000, doc="""
        Number of nodes to simulate.""")
    mean_connections = param.Number(default=10, doc="""
        Mean number of connections to make to other nodes.""")
    pSick = param.Number(default=0.01, doc="""
        Probability of a node to be initialized in sick state.""", bounds=(0, 1))

    pVaccinated = param.Number(default=0.1, bounds=(0, 1), doc="""
        Probability of a node to be initialized in vaccinated state.""")
    # Simulation parameters
    pInfect = param.Number(default=0.3, doc="""
        Probability of infection on each time step.""", bounds=(0, 1))
    pRecover = param.Number(default=0.05, doc="""
        Probability of recovering if infected on each timestep.""", bounds=(0, 1))
    pDeath = param.Number(default=0.1, doc="""
        Probability of death if infected on each timestep.""", bounds=(0, 1))
    def __init__(self, **params):
        super(SRI_Model, self).__init__(**params)
        if not
            self.g = nx.erdos_renyi_graph(self.N, float(self.mean_connections)/self.N)
            self.g =
        self.vaccinated, self.infected = self.spreading_init()
        self.model = self.spreading_make_sir_model()
        self.color_mapping = [SPREADING_SUSCEPTIBLE,
                              SPREADING_RECOVERED, DEAD]
        if self.visualize:
            self.pos = nx.spring_layout(self.g, iterations = 50,
                                        k = 2/(math.sqrt(self.g.order())))

    def spreading_init(self):
        """Initialise the network with vaccinated, susceptible and infected states."""
        vaccinated, infected = 0, []
        for i in self.g.node.keys():
            self.g.node[i]['transmissions'] = 0
            if(rnd.random() <= self.pVaccinated): 
                self.g.node[i]['state'] = SPREADING_VACCINATED
                vaccinated += 1
            elif(rnd.random() <= self.pSick):
                self.g.node[i]['state'] = SPREADING_INFECTED
                self.g.node[i]['state'] = SPREADING_SUSCEPTIBLE
        return vaccinated, infected

    def spreading_make_sir_model(self):
        """Return an SIR model function for given infection and recovery probabilities."""
        # model (local rule) function
        def model( g, i ):
            if g.node[i]['state'] == SPREADING_INFECTED:
                # infect susceptible neighbours with probability pInfect
                for m in g.neighbors(i):
                    if g.node[m]['state'] == SPREADING_SUSCEPTIBLE:
                        if rnd.random() <= self.pInfect:
                            g.node[m]['state'] = SPREADING_INFECTED
                            g.node[i]['transmissions'] += 1

                # recover with probability pRecover
                if rnd.random() <= self.pRecover:
                    g.node[i]['state'] = SPREADING_RECOVERED
                elif rnd.random() <= self.pDeath:
                    edges = [edge for edge in self.g.edges() if i in edge] 
                    g.node[i]['state'] = DEAD

        return model

    def step(self):
        """Run a single step of the model over the graph."""
        for i in self.g.node.keys():
            self.model(self.g, i)

    def run(self, steps):
        Run the network for the specified number of time steps
        for i in range(steps):

    def network_data(self):
        Return the network edge paths and node positions,
        requires visualize parameter to be enabled.
        if not self.visualize:
            raise Exception("Enable visualize option to get network data.")

        nodeMarkers = []
        overlay = []
        points = np.array([self.pos[v] for v in self.g.nodes_iter()])
        paths = []
        for e in self.g.edges_iter():
            xs = [ self.pos[e[0]][0], self.pos[e[1]][0] ]
            ys = [ self.pos[e[0]][1], self.pos[e[1]][1] ]
            paths.append(np.array(zip(xs, ys)))
        return paths, points

    def stats(self):
        Return an ItemTable with statistics on the network data.
        state_labels = hv.OrderedDict([('S', 'Susceptible'), ('V', 'Vaccinated'), ('I', 'Infected'),
                                    ('R', 'Recovered'), ('D', 'Dead')])
        counts = collections.Counter()
        transmissions = []
        for n in self.g.nodes_iter():
            state = state_labels[self.g.node[n]['state']]
            counts[state] += 1
            if n in self.infected:
        data = hv.OrderedDict([(l, counts[l])
                               for l in state_labels.values()])
        infected = len(set(self.infected))
        unvaccinated = float(self.N-self.vaccinated)
        data['$R_0$'] = np.mean(transmissions) if transmissions else 0
        data['Death rate DR'] = np.divide(float(data['Dead']),self.N)
        data['Infection rate IR'] = np.divide(float(infected), self.N)
        if unvaccinated:
            unvaccinated_dr = data['Dead']/unvaccinated
            unvaccinated_ir = infected/unvaccinated
            unvaccinated_dr = 0
            unvaccinated_ir = 0
        data['Unvaccinated DR'] = unvaccinated_dr
        data['Unvaccinated IR'] = unvaccinated_ir
        return hv.ItemTable(data)

    def animate(self, steps):
        Run the network for the specified number of steps accumulating animations
        of the network nodes and edges changing states and curves tracking the
        spread of the disease.
        if not self.visualize:
            raise Exception("Enable visualize option to get compute network visulizations.")

        # Declare HoloMap for network animation and counts array
        network_hmap = hv.HoloMap(key_dimensions=['Time'])
        sird = np.zeros((steps, 5))
        # Declare dimensions and labels
        spatial_dims = [hv.Dimension('x', range=(-1.1, 1.1)),
                        hv.Dimension('y', range=(-1.1, 1.1))]
        state_labels = ['Susceptible', 'Vaccinated', 'Infected', 'Recovered', 'Dead']

        # Text annotation
        nlabel = hv.Text(0.9, 0.05, 'N=%d' % self.N)

        for i in range(steps):
            # Get path, point, states and count data
            paths, points = self.network_data()
            states = [self.color_mapping.index(self.g.node[n]['state'])
                      for n in self.g.nodes_iter()]
            state_array = np.array(states, ndmin=2).T
            (sird[i, :], _) = np.histogram(state_array, bins=list(range(6)))
            # Create network path and node Elements
            network_paths = hv.Path(paths, key_dimensions=spatial_dims)
            network_nodes = hv.Points(np.hstack([points, state_array]),
            # Create overlay and accumulate in network HoloMap
            network_hmap[i] = (network_paths * network_nodes * nlabel).relabel(group='Network', label='SRI')

        # Create Overlay of Curves
        extents = (-1, -1, steps, np.max(sird)+2)
        curves = hv.NdOverlay({label: hv.Curve(zip(range(steps), sird[:, i]), extents=extents,
                                               key_dimensions=['Time'], value_dimensions=['Count'])
                              for i, label in enumerate(state_labels)},
                              key_dimensions=[hv.Dimension('State', values=state_labels)])
        # Animate VLine on top of Curves
        distribution = hv.HoloMap({i: (curves * hv.VLine(i)).relabel(group='Counts', label='SRI')
                                   for i in range(steps)}, key_dimensions=['Time'])
        return network_hmap + distribution

The style

HoloViews allows use to define various style options in advance on the Store.options object.

In [2]:
# Increase dpi and select the slider widget
%output dpi=120 holomap='widgets'

# Set colors and style options for the Element types
from holoviews import Store, Options
from holoviews.core.options import Palette
opts = Store.options()

opts.Path      = Options('style', linewidth=0.2, color='k')
opts.ItemTable = Options('plot', aspect=1.2, fig_size=150)
opts.Curve     = Options('style', color=Palette('hot_r'))
opts.Histogram = Options('plot', bgcolor='w', show_grid=False)
opts.Overlay   = Options('plot', show_frame=False)
opts.HeatMap   = Options('plot', show_values=False, show_grid=False,
                          aspect=1.5, xrotation=90)

opts.Overlay.Network = Options('plot', xaxis=None, yaxis=None, bgcolor='w')
opts.Overlay.Counts  = Options('plot', aspect=1.2, show_grid=True)

opts.Points    = {'style': Options(cmap='hot_r', s=50, edgecolors='k'),
                  'plot':  Options(color_index=2)}
opts.VLine     = {'style': Options(color='k', linewidth=1),
                  'plot':  Options(show_grid=True)}
HoloViewsJS successfully loaded in this cell.

Next we'll simply enable the Seaborn plot style defaults because they look a bit better than the HoloViews defaults for this kind of data.

In [3]:
import seaborn

Herd Immunity

Experiment 1: Evaluating the effects of a highly infectious and deadly disease in a small population with varying levels of vaccination

Having defined the model and defined the model we can run some real experiments. In particular we can investigate the effect of vaccination on our model.

We'll initialize our model with only 50 inviduals, who will on average make 10 connections to other individuals. Then we will infect a small population ($p=0.1$) so we can track how the disease spreads through the population. To really drive the point home we'll use a very infectious and deadly disease.

In [4]:
experiment1_params = dict(pInfect=0.08, pRecover=0.08, pSick=0.15,
                          N=50, mean_connections=10, pDeath=0.1)

Low vaccination population (10%)

Here we'll investigate the spread of the disease in population with a 10% vaccination rate:

In [5]:
sri_model = SRI_Model(pVaccinated=0.1, **experiment1_params)