ODE-based SEAIR-type model
===========================

This model is an extended ODE-type model. The six compartments

- **Susceptible** (:math:`S`): may become Exposed at any time.
- **Exposed** (:math:`E`): becomes Asymptomatic after some time.
- **Asymptomatic** (:math:`A`): becomes Infected or Recovered after some time. People which are not included in the count of confirmed cases, either because they are asymptomatic or because of insufficient testing.
- **Infected** (:math:`I`): becomes Recovered or Dead after some time. It is assumed to be equal to confirmed cases here.
- **Recovered** (:math:`R`)
- **Dead** (:math:`D`)

are used to simulate the spread of the disease.

A detailed description of the model can be found in the publication
`Tsay et al. (2020), Modeling, state estimation, and optimal control for the US COVID-19 outbreak <https://doi.org/10.1038/s41598-020-67459-8>`_.

Below is an overview of the model architecture and its compartments.

.. image:: https://github.com/SciCompMod/memilio/assets/69154294/9d1bd9d1-9c6d-484f-aac1-6aead0b34185
   :alt: secir_model

.. list-table::
   :header-rows: 1
   :widths: 20 20 60

   * - Mathematical variable
     - C++ variable name
     - Description
   * - :math:`\alpha_{a}`
     - ``SocialDistancing``
     - Infectivity of the Asymptomatic compartment. This includes ContactPatterns, the TransmissionProbabilityOnContact and the influence of social distancing.
   * - :math:`\alpha_{i}`
     - ``Quarantined``
     - Infectivity of the Infected compartment. This includes ContactPatterns, the TransmissionProbabilityOnContact and the influence of quarantine.
   * - :math:`N`
     - ``pop_total``
     - Represents the total population.
   * - :math:`T_{E}`
     - ``TimeExposed``
     - Time in days an individual stays in the Exposed compartment.
   * - :math:`\frac{\mu_A^R}{T_A}`
     - ``RecoveryRateFromAsymptomatic``
     - The rate at which people recover from the Asymptomatic compartment. This includes the probability of not getting a positive test and the time spent in the Asymptomatic compartment before recovery.
   * - :math:`\frac{1-\mu_A^R}{T_A}`
     - ``TestingRate``
     - The rate at which people transit from the Asymptomatic to the Infected compartment. This includes the probability of getting a positive test and the time spent in the Asymptomatic compartment before testing positive.
   * - :math:`\frac{\mu_I^R}{T_I}`
     - ``RecoveryRate``
     - The rate at which people recover. This includes the probability of not dying and the time spent in the Infected compartment before recovery.
   * - :math:`\frac{1-\mu_I^R}{T_I}`
     - ``DeathRate``
     - The rate at which people die. This includes the probability of dying and the time spent in the Infected compartment before dying.
   * - :math:`\frac{1}{T_R}`
     - ``TimeRecoveredInv``
     - Inverse time in days an individual stays in the Recovered compartment before becoming Susceptible again.

- An example can be found in the
  `examples/ode_seair.cpp <https://github.com/SciCompMod/memilio/blob/main/cpp/examples/ode_seair.cpp>`_.
- For an example of using this model to define an optimization problem, see the
  `examples/ode_seair_optimization.cpp <https://github.com/SciCompMod/memilio/blob/main/cpp/examples/ode_seair_optimization.cpp>`_.


The code documentation for the model can be found at :CPP-API:`mio::oseair` .