This model is based on integro-differential equations.

The eight compartments

- `Susceptible` ($S$), may become Exposed at any time

- `Exposed` ($E$), becomes InfectedNoSymptoms after some time

- `InfectedNoSymptoms` ($I_{NS}$), becomes InfectedSymptoms or Recovered after some time

- `InfectedSymptoms` ($I_{Sy}$), becomes InfectedSevere or Recovered after some time

- `InfectedSevere` ($I_{Sev}$), becomes InfectedCritical or Recovered after some time

- `InfectedCritical` ($I_{Cr}$), becomes Recovered or Dead after some time

- `Recovered` ($R$)

- `Dead` ($D$)

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

The variables $\sigma_{z_1}^{z_2}$ refer to a transition from a compartment $z_1$ to a compartment $z_2$.


The model parameters used are the following:

| Mathematical variable | C++ variable name | Description |

|---------------------------- | --------------- | -------------------------------------------------------------------------------------------------- |

| $\phi$ | `ContactPatterns` | Average number of contacts of a person per day. |

| $k$ | `Seasonality` | The influence of the seasons is taken into account with the seasonality parameter. |

| $\rho$ | `TransmissionProbabilityOnContact` | Transmission risk for people located in the Susceptible compartment. |

| $\xi_{I_{NS}}$ | `RelativeTransmissionNoSymptoms` | Proportion infected people with no symptoms who are not isolated. |

| $\xi_{I_{Sy}}$ | `RiskOfInfectionFromSymptomatic` | Proportion of infected persons with symptoms who are not isolated. |

| $N$ | `m_N` | Total population. |

| $D$ | Entry of `m_populations` | Number of dead people. |

| $\mu_{z_1}^{z_2}$ | `TransitionProbabilities` | Probability of transitioning from compartment $z_1$ to compartment $z_2$. |

| $\gamma_{z_1}^{z_2}(\tau)$ | `TransitionDistributions` | Expected proportion of people who are still in compartment $z_1$ $\tau$ days after entering this compartment and who will move to compartment $z_2$ later in the course of the disease. |

The simulation runs in discrete time steps using a non-standard numerical scheme. This approach is based on the paper ["A non-standard numerical scheme for an age-of infection epidemic model" by Messina et al., Journal of Computational Dynamics, 2022](https://doi.org/10.3934/jcd.2021029).

- [IDE minimal example](../../examples/ide_secir.cpp)

- The file [parameters_io](parameters_io.h) provides functionality to compute initial data for the IDE-SECIR model based on real data. An example for this initialization method can be found at [IDE initialization example](../../examples/ide_initialization.cpp).

- There are various options for initializing a fictional scenario. Regardless of the approach, you must provide a history of values for the transitions and additional information to compute the initial distribution of the population in the compartments. This information must be of the following type:

- You can state the number of total confirmed cases `total_confirmed_cases` at time $t_0$. The number of recovered people is set accordingly and the remaining values are derived in the model before starting the simulation.

- You can set the number of people in the `Susceptible` compartment at time $t_0$ via `m_populations`. Initial values of the other compartments are derived in the model before starting the simulation.

- You can set the number of people in the `Recovered` compartment at time $t_0$ via `m_populations`. Initial values of the other compartments are derived in the model before starting the simulation.

- If none of the above is used, the force of infection formula and the values for the initial transitions are used consistently with the numerical scheme proposed in [Messina et al (2022)](https://doi.org/10.3934/jcd.2021029) to set the `Susceptible`s.