ODE-based SEAIR-type model

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

• Susceptible (\(S\)): may become Exposed at any time.

• Exposed (\(E\)): becomes Asymptomatic after some time.

• Asymptomatic (\(A\)): 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 (\(I\)): becomes Recovered or Dead after some time. It is assumed to be equal to confirmed cases here.

• Recovered (\(R\))

• Dead (\(D\))

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.

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

Mathematical variable	C++ variable name	Description
\(\alpha_{a}\)	SocialDistancing	Infectivity of the Asymptomatic compartment. This includes ContactPatterns, the TransmissionProbabilityOnContact and the influence of social distancing.
\(\alpha_{i}\)	Quarantined	Infectivity of the Infected compartment. This includes ContactPatterns, the TransmissionProbabilityOnContact and the influence of quarantine.
\(N\)	pop_total	Represents the total population.
\(T_{E}\)	TimeExposed	Time in days an individual stays in the Exposed compartment.
\(\frac{\mu_A^R}{T_A}\)	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.
\(\frac{1-\mu_A^R}{T_A}\)	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.
\(\frac{\mu_I^R}{T_I}\)	RecoveryRate	The rate at which people recover. This includes the probability of not dying and the time spent in the Infected compartment before recovery.
\(\frac{1-\mu_I^R}{T_I}\)	DeathRate	The rate at which people die. This includes the probability of dying and the time spent in the Infected compartment before dying.
\(\frac{1}{T_R}\)	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.

• For an example of using this model to define an optimization problem, see the examples/ode_seair_optimization.cpp.

The code documentation for the model can be found at mio::oseair .