SI

The SI 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 only from Susceptible (S) to Infected (I).

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

SI assumes that if, during a generic iteration, a susceptible node comes into contact with an infected one, it becomes infected with probability β: once a node becomes infected, it stays infected (the only transition allowed is S→I).

The dSI 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

Parameters

Name Type Value Type Default Mandatory Description
beta Model float in [0, 1]   True Infection 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.DynSIModel.DynSIModel(graph)

Model Parameters to be specified via ModelConfig

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

Model Constructor

Parameters:graph – A dynetx graph object
DynSIModel.set_initial_status(self, configuration)

Set the initial model configuration

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

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

Describe

DynSIModel.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
DynSIModel.get_status_map(self)

Specify the statuses allowed by the model and their numeric code

Returns:a dictionary (status->code)

Execute Simulation

DynSIModel.iteration(self)

Execute a single model iteration

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

Example

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

import networkx as nx
import dynetx as dn
import ndlib.models.ModelConfig as mc
import ndlib.models.dynamic.DynSIModel as si

# 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 = si.DynSIModel(dg)

# Model Configuration
config = mc.Configuration()
config.add_model_parameter('beta', 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.