Models description
==================

Examples used in this documentation rely on several models.

Following sections show how they works.

The cantilever beam model
-------------------------

.. image:: /_static/beam.svg
  :scale: 100 %
  :alt: alternate text
  :align: center

We consider a cantilever beam defined by its Young’s modulus ``E``, its length ``L`` and its section modulus ``I``.
One end of the cantilever beam is fixed in a wall and we apply a concentrated
bending load ``F`` at the other end of the beam, resulting in a deviation ``Y``.
The mechanical equation ruling the deviation is


.. raw:: html

   <div style="margin-left:20px;width:300px;height:120px;">
.. math::

   Y = \frac{FL^3}{3EI}

.. raw:: html

   </div>


This model is implemented in Modelica language:

.. code::

   model deviation
     output Real y;
     input Real E (start=3.0e7);
     input Real F (start=3.0e4);
     input Real L (start=250);
     input Real I (start=400);
   equation
     y=(F*L^3)/(3*E*I);
   end deviation;




The epidemiological model
--------------------------

This model describes epidemics which propagate through human contact.
An isolated population is considered, whose total number is constant.
The people are divided in three categories:

* the Susceptibles (who are not sick yet),
* the Infected (who are currently sick),
* the Removed (who are either dead or immune).

The disease can only be propagated from Infected to Susceptibles.
This happens at a rate called ``infection rate`` :math:`\beta`.
An Infected becomes Removed after an infection duration :math:`\gamma` corresponding to the inverse of the ``healing_rate``.

.. image:: /_static/epid.png
   :scale: 50 %
   :alt: alternate text
   :align: center

The evolution of the number of Susceptibles, Infected and Removed over
time writes:




.. math::

   \begin{aligned}
   \frac{\partial S}{\partial t}(t) &= - \frac{\beta}{N} S(t) I(t) \\
   \frac{\partial I}{\partial t}(t) &= \frac{\beta}{N} S(t) I(t) - \gamma I(t) \\
   \frac{\partial R}{\partial t}(t) &= \gamma I(t)
   \end{aligned}




This model is implemented in Modelica language. The default simulation time is 50 units of time (days for instance).

.. code::

   model epid

   parameter Real total_pop = 763;
   parameter Real infection_rate = 2.0;
   parameter Real healing_rate = 0.5;

   Real infected;
   Real susceptible;
   Real removed;

   initial equation
   infected = 1;
   removed = 0;
   total_pop = infected + susceptible + removed;

   equation
   der(susceptible) = - infection_rate * infected * susceptible / total_pop;
   der(infected) = infection_rate * infected * susceptible / total_pop -
                   healing_rate * infected;
   der(removed) = healing_rate * infected;

   annotation(
       experiment(StartTime = 0.0, StopTime = 200.0, Tolerance = 1e-6, Interval = 0.1));
   end epid;

We focus on the effect of the ``infection_rate`` and ``healing_rate`` on the evolution of the ``infected`` category.
Hence **two input variables** and **one time-dependent output**.

References
~~~~~~~~~~
.. [anonymous1978] *Influenza in a boarding school*. In: British Medical Journal 587.1,
   `pdf <https://pmc.ncbi.nlm.nih.gov/articles/PMC1603269/pdf/brmedj00115-0064.pdf>`__