SIR

The SIR model was introduced in 1927 by Kermack [1].

In this model, during the course of an epidemics, a node is allowed to change its status from Susceptible (S) to Infected (I), then to Removed (R).

The model is instantiated on a graph having a non-empty set of infected nodes.

SIR assumes that if, during a generic iteration, a susceptible node comes into contact with an infected one, it becomes infected with probability beta, than it can be switch to removed with probability gamma (the only transition allowed are S→I→R).

The dSIR implementation assumes that the process occurs on a directed/undirected dynamic network; this model was introduced by Milli et al. in 2018 [2].

Statuses

During the simulation a node can experience the following statuses:

Name Code
Susceptible 0
Infected 1
Removed 2

Parameters

Name Type Value Type Default Mandatory Description
beta Model float in [0, 1]   True Infection probability
gamma Model float in [0, 1]   True Removal probability

The initial infection status can be defined via:

  • percentage_infected: Model Parameter, float in [0, 1]
  • Infected: Status Parameter, set of nodes

The two options are mutually exclusive and the latter takes precedence over the former.

Methods

The following class methods are made available to configure, describe and execute the simulation:

Configure

class ndlib.models.dynamic.DynSIRModel.DynSIRModel(graph)

Model Parameters to be specified via ModelConfig

Parameters:
  • beta – The infection rate (float value in [0,1])
  • gamma – The recovery rate (float value in [0,1])
DynSIRModel.__init__(graph)

Model Constructor

Parameters:graph – A networkx graph object
DynSIRModel.set_initial_status(self, configuration)

Set the initial model configuration

Parameters:configuration – a `ndlib.models.ModelConfig.Configuration` object
DynSIRModel.reset(self)

Reset the simulation setting the actual status to the initial configuration.

Describe

DynSIRModel.get_info(self)

Describes the current model parameters (nodes, edges, status)

Returns:a dictionary containing for each parameter class the values specified during model configuration
DynSIRModel.get_status_map(self)

Specify the statuses allowed by the model and their numeric code

Returns:a dictionary (status->code)

Execute Simulation

DynSIRModel.iteration(self)

Execute a single model iteration

Returns:Iteration_id, Incremental node status (dictionary node->status)
DynSIRModel.execute_snapshots(bunch_size, node_status)
DynSIRModel.execute_iterations(node_status)

Example

In the code below is shown an example of instantiation and execution of an DynSIR simulation on a dynamic random graph: we set the initial set of infected nodes as 5% of the overall population, a probability of infection of 1%, and a removal probability of 1%.

import networkx as nx
import dynetx as dn
import ndlib.models.ModelConfig as mc
import ndlib.models.dynamic.DynSIRModel as sir

# Dynamic Network topology
dg = dn.DynGraph()

for t in past.builtins.xrange(0, 3):
    g = nx.erdos_renyi_graph(200, 0.05)
    dg.add_interactions_from(g.edges(), t)

# Model selection
model = sir.DynSIRModel(dg)

# Model Configuration
config = mc.Configuration()
config.add_model_parameter('beta', 0.01)
config.add_model_parameter('gamma', 0.01)
config.add_model_parameter("percentage_infected", 0.1)
model.set_initial_status(config)

# Simulate snapshot based execution
iterations = model.execute_snapshots()

# Simulation interaction graph based execution
iterations = model.execute_iterations()
[1]
    1. Kermack and A. McKendrick, “A Contribution to the Mathematical Theory of Epidemics,” Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character, vol. 115, no. 772, pp. 700–721, Aug. 1927
[2]Letizia Milli, Giulio Rossetti, Fosca Giannotti, Dino Pedreschi. “Diffusive Phenomena in Dynamic Networks: a data-driven study”. Accepted to International Conference on Complex Networks (CompleNet), 2018, Boston.