***
SIS
***

The SIS model was introduced in 1927 by Kermack [#]_.
 
In this model, during the course of an epidemics, a node is allowed to change its status  from **Susceptible** (S) to **Infected** (I).

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

SIS 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 again to susceptible with probability lambda (the only transition allowed are S→I→S).

The dSIS implementation assumes that the process occurs on a directed/undirected dynamic network.

--------
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
lambda  Model  float in [0, 1]           True       Recovery probability
======  =====  ===============  =======  =========  =====================

The initial infection status can be defined via:

    - **fraction_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.


-------
Example
-------

In the code below is shown an example of instantiation and execution of an DynSIS 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 probability of recovery of 1%.

.. code-block:: python

    import networkx as nx
    import dynetx as dn
    import ndlib.models.ModelConfig as mc
    import ndlib.models.dynamic as dm
    from past.builtins import xrange

    # Dynamic Network topology
    dg = dn.DynGraph()

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

    # Model selection
    model = dm.DynSISModel(dg)

    # Model Configuration
    config = mc.Configuration()
    config.add_model_parameter('beta', 0.01)
    config.add_model_parameter('lambda', 0.01)
    config.add_model_parameter("fraction_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()



.. [#] W. O. 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