.. DO NOT EDIT.
.. THIS FILE WAS AUTOMATICALLY GENERATED BY SPHINX-GALLERY.
.. TO MAKE CHANGES, EDIT THE SOURCE PYTHON FILE:
.. "api/gallery/ode_dynamics/epidemics/plot_01_sir_model.py"
.. LINE NUMBERS ARE GIVEN BELOW.

.. only:: html

    .. note::
        :class: sphx-glr-download-link-note

        :ref:`Go to the end <sphx_glr_download_api_gallery_ode_dynamics_epidemics_plot_01_sir_model.py>`
        to download the full example code.

.. rst-class:: sphx-glr-example-title

.. _sphx_glr_api_gallery_ode_dynamics_epidemics_plot_01_sir_model.py:


The Kermack-McKendrick SIR model
======================================

An epidemic grows only if :math:`R_0 S_0 > 1`, peaks when the
susceptible fraction falls to :math:`1/R_0`, and burns out before
infecting everyone. The simulated peak and final size agree with the
closed-form Kermack-McKendrick relations.

.. GENERATED FROM PYTHON SOURCE LINES 12-18

.. code-block:: Python

    import matplotlib.pyplot as plt

    from mathematicskit.ode_dynamics.systems.epidemics import SIRModel, sir_final_size, sir_peak_infected

    s0, i0 = 0.999, 0.001








.. GENERATED FROM PYTHON SOURCE LINES 19-21

Epidemic curves for several reproduction numbers
------------------------------------------------------

.. GENERATED FROM PYTHON SOURCE LINES 21-38

.. code-block:: Python


    fig, ax = plt.subplots(figsize=(7, 4.5))
    for beta in (0.15, 0.3, 0.5):
        system = SIRModel([s0, i0, 0.0], beta=beta, gamma=0.1)
        traj = system.integrate((0.0, 300.0), dt=0.05, method="rk4")
        ax.plot(traj.t, traj.y[:, 1], label=f"R0={system.r0:.1f}")
        print(
            f"R0={system.r0:.1f}: peak I {traj.y[:, 1].max():.4f} (theory {sir_peak_infected(system.r0, s0, i0):.4f}), "
            f"final S {traj.y[-1, 0]:.4f} (theory {sir_final_size(system.r0, s0, i0):.4f})"
        )
    ax.set_xlabel("t")
    ax.set_ylabel("infected fraction I(t)")
    ax.set_title("SIR epidemics: larger R0, earlier and higher peak")
    ax.legend()
    fig.tight_layout()

    plt.show()



.. image-sg:: /api/gallery/ode_dynamics/epidemics/images/sphx_glr_plot_01_sir_model_001.png
   :alt: SIR epidemics: larger R0, earlier and higher peak
   :srcset: /api/gallery/ode_dynamics/epidemics/images/sphx_glr_plot_01_sir_model_001.png
   :class: sphx-glr-single-img


.. rst-class:: sphx-glr-script-out

 .. code-block:: none

    R0=1.5: peak I 0.0637 (theory 0.0637), final S 0.4164 (theory 0.4161)
    R0=3.0: peak I 0.3008 (theory 0.3008), final S 0.0594 (theory 0.0594)
    R0=5.0: peak I 0.4783 (theory 0.4783), final S 0.0070 (theory 0.0070)





.. rst-class:: sphx-glr-timing

   **Total running time of the script:** (0 minutes 0.702 seconds)


.. _sphx_glr_download_api_gallery_ode_dynamics_epidemics_plot_01_sir_model.py:

.. only:: html

  .. container:: sphx-glr-footer sphx-glr-footer-example

    .. container:: sphx-glr-download sphx-glr-download-jupyter

      :download:`Download Jupyter notebook: plot_01_sir_model.ipynb <plot_01_sir_model.ipynb>`

    .. container:: sphx-glr-download sphx-glr-download-python

      :download:`Download Python source code: plot_01_sir_model.py <plot_01_sir_model.py>`

    .. container:: sphx-glr-download sphx-glr-download-zip

      :download:`Download zipped: plot_01_sir_model.zip <plot_01_sir_model.zip>`


.. only:: html

 .. rst-class:: sphx-glr-signature

    `Gallery generated by Sphinx-Gallery <https://sphinx-gallery.github.io>`_