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928 Add a flow chart for the lct model (#943)
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‎cpp/models/lct_secir/README.md‎

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# LCT SECIR model
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This model is based on the Linear Chain Trick.
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The Linear Chain Trick provides the option to use Erlang-distributed sojourn times in the compartments through the use of subcompartments.
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The normal ODE models have (possibly unrealistic) exponentially distributed sojourn times.
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The LCT model can still be described by an ordinary differential equation system.
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For the concept see
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- Lena Plötzke, "Der Linear Chain Trick in der epidemiologischen Modellierung als Kompromiss zwischen gewöhnlichen und Integro-Differentialgleichungen", 2023. (https://elib.dlr.de/200381/, German only)
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- P. J. Hurtado und A. S. Kirosingh, "Generalizations of the ‘Linear Chain Trick’: incorporating more flexible dwell time distributions into mean field ODE models“, 2019. (https://doi.org/10.1007/s00285-019-01412-w)
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The eight compartments
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- Susceptible, may become exposed at any time
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- Exposed, becomes infected after some time
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- InfectedNoSymptoms, becomes InfectedSymptoms or Recovered after some time
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- InfectedSymptoms, becomes InfectedSevere or Recovered after some time
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- InfectedSevere, becomes InfectedCritical or Recovered after some time
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- InfectedCritical, becomes Recovered or Dead after some time
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- Recovered
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- Dead
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- `Susceptible` ($S$), may become exposed at any time
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- `Exposed` ($E$), becomes infected after some time
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- `InfectedNoSymptoms` ($I_{NS}$), becomes InfectedSymptoms or Recovered after some time
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- `InfectedSymptoms` ($I_{Sy}$), becomes InfectedSevere or Recovered after some time
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- `InfectedSevere` ($I_{Sev}$), becomes InfectedCritical or Recovered after some time
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- `InfectedCritical` ($I_{Cr}$), becomes Recovered or Dead after some time
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- `Recovered` ($R$)
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- `Dead` ($D$)
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are used to simulate the spread of the disease.
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It is possible to include subcompartments for the five compartments Exposed, InfectedNoSymptoms, InfectedSymptoms, InfectedSevere and InfectedCritical.
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A simple example can be found at: examples/lct_secir.cpp.
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Below is an overview of the model architecture and its compartments.
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![tikzLCTSECIR](https://github.com/SciCompMod/memilio/assets/70579874/6a5d5a95-20f9-4176-8894-c091bd48bfb7)
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| Mathematical variable | C++ variable name | Description |
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|---------------------------- | --------------- | -------------------------------------------------------------------------------------------------- |
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| $\phi$ | `ContactPatterns` | Average number of contacts of a person per day. |
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| $\rho$ | `TransmissionProbabilityOnContact` | Transmission risk for people located in the susceptible compartments. |
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| $\xi_{I_{NS}}$ | `RelativeTransmissionNoSymptoms` | Proportion of nonsymptomatically infected people who are not isolated. |
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| $\xi_{I_{Sy}}$ | `RiskOfInfectionFromSymptomatic` | Proportion of infected people with symptomps who are not isolated. |
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| $N$ | `m_N0` | Total population. |
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| $D$ | `D` | Number of death people. |
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| $n_E$ | Defined via `InfectionState` | Number of subcompartments of the Exposed compartment. |
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| $n_{NS}$ | Defined via `InfectionState` | Number of subcompartments of the InfectedNoSymptoms compartment. |
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| $n_{Sy}$ | Defined via `InfectionState` | Number of subcompartments of the InfectedSymptoms compartment. |
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| $n_{Sev}$ | Defined via `InfectionState` | Number of subcompartments of the InfectedSevere compartment.|
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| $n_{Cr}$ | Defined via `InfectionState` | Number of subcompartments of the InfectedCritical compartment. |
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| $T_E$ | `TimeExposed` | Average time in days an individual stays in the Exposed compartment. |
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| $T_{I_{NS}}$ | `TimeInfectedNoSymptoms` | Average time in days an individual stays in the InfectedNoSymptoms compartment. |
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| $T_{I_{Sy}}$ | `TimeInfectedSymptoms` | Average time in days an individual stays in the InfectedSymptoms compartment. |
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| $T_{I_{Sev}}$ | `TimeInfectedSevere` | Average time in days an individual stays in the InfectedSevere compartment. |
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| $T_{I_{Cr}}$ | `TimeInfectedCritical` | Average time in days an individual stays in the InfectedCritical compartment. |
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| $\mu_{I_{NS}}^{R}$ | `RecoveredPerInfectedNoSymptoms` | Probability of transition from compartment InfectedNoSymptoms to Recovered. |
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| $\mu_{I_{Sy}}^{I_{Sev}}$ | `SeverePerInfectedSymptoms` | Probability of transition from compartment InfectedSymptoms to InfectedSevere. |
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| $\mu_{I_{Sev}}^{I_{Cr}}$ | `CriticalPerSevere` | Probability of transition from compartment InfectedSevere to InfectedCritical. |
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| $\mu_{I_{Cr}}^{D}$ | `DeathsPerCritical` | Probability of dying when in compartment InfectedCritical. |
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The notation of the compartments with indices here stands for subcompartments and not for age groups or similar, as in some ODE models. Accordingly, $I_{NS,n_{NS}}$, for example, stands for the number of people in the $n_{NS}$-th subcompartment of the InfectedNoSymptoms compartment.
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## Examples
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A simple example can be found at [LCT minimal example](../../examples/lct_secir.cpp).
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‎cpp/models/ode_secir/README.md‎

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Below is an overview of the model architecture and its compartments.
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![secir_model](https://github.com/SciCompMod/memilio/assets/69154294/d99b257b-1cc2-447d-8e32-ba3aa8329c0c)
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![secir_model](https://github.com/SciCompMod/memilio/assets/70579874/46b09e8a-d083-4ef9-8328-21975890b60f)
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| Mathematical variable | C++ variable name | Description |
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|---------------------------- | --------------- | -------------------------------------------------------------------------------------------------- |
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| $\phi$ | `ContactPatterns` | Matrix of daily contact rates / number of daily contacts between different age groups. |
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| $\xi_{I_{Sy}}$ | `riskFromInfectedSymptomatic` | Proportion of infected people with symptomps who are not isolated (time-dependent if `TestAndTraceCapacity` used). |
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| $N_j$ | `Nj` | Total population of age group $j$. |
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| $D_i$ | `Di` | Number of death people of age group $i$. |
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| $\frac{1}{T_{E}}$ | `rateE` | Time in days an individual stays in the Exposed compartment (Computed from `SerialInterval` and `IncubationPeriod`). |
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| $\frac{1}{T_{I_{NS}}}$ | `rateINS` | Time in days an individual stays in the InfectedNoSymptoms compartment (Computed from `SerialInterval` and `IncubationPeriod`). |
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| $\frac{1}{T_{E}}$ | `rateE` | Reciprocal of the time in days an individual stays in the Exposed compartment (Computed from `SerialInterval` and `IncubationPeriod`). |
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| $\frac{1}{T_{I_{NS}}}$ | `rateINS` | Reciprocal of the time in days an individual stays in the InfectedNoSymptoms compartment (Computed from `SerialInterval` and `IncubationPeriod`). |
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| $T_{I_{Sy}}$ | `TimeInfectedSymptoms` | Time in days an individual stays in the InfectedSymptoms compartment. |
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| $T_{I_{Sev}}$ | `TimeInfectedSevere` | Time in days an individual stays in the InfectedSevere compartment. |
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| $T_{I_{Cr}}$ | `TimeInfectedCritical` | Time in days an individual stays in the InfectedCritical compartment. |

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