ODE-based SECIR-type model with COVID-19 variants and vaccinations

This model extends the basic ODE-SECIR model. by adding vaccinations and different immunity levels. Also, this model allows the implicit modeling of a newly arriving variant that takes hold.

Vaccinations are modeled by adding compartments for partially and fully vaccinated persons. Partially and fully vaccinated is to be understood in this context as the person having received a first and second vaccine shot as in 2021. Persons that have recovered from the disease are treated as fully vaccinated from that time forward. Vaccinated persons are added on every day of simulation, see parameters DailyPartialVaccinations and DailyFullVaccinations. All groups can get an infection or get reinfected. Vaccinated persons are less likely to develop symptoms. For example, the probability to develop symptoms when carrying the virus is the base probability from the ODE-SECIR model multiplied with the ReducInfectedSymptomsPartialImmunity parameter.

The ratio of two variants can change over time, which affects the average transmissibility of the disease. Infectiousness of different variants can be set in the parameters.

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

Infection States

The model extends the basic ODE-SECIR model by dividing the compartments based on immunity levels. It contains the following list of InfectionStates:

// Naive immunity
`SusceptibleNaive`
`ExposedNaive`
`InfectedNoSymptomsNaive`
`InfectedNoSymptomsNaiveConfirmed`
`InfectedSymptomsNaive`
`InfectedSymptomsNaiveConfirmed`
`InfectedSevereNaive`
`InfectedCriticalNaive`
`DeadNaive`

// Partial immunity
`SusceptiblePartialImmunity`
`ExposedPartialImmunity`
`InfectedNoSymptomsPartialImmunity`
`InfectedNoSymptomsPartialImmunityConfirmed`
`InfectedSymptomsPartialImmunity`
`InfectedSymptomsPartialImmunityConfirmed`
`InfectedSeverePartialImmunity`
`InfectedCriticalPartialImmunity`
`DeadPartialImmunity`

// Improved immunity
`SusceptibleImprovedImmunity`
`ExposedImprovedImmunity`
`InfectedNoSymptomsImprovedImmunity`
`InfectedNoSymptomsImprovedImmunityConfirmed`
`InfectedSymptomsImprovedImmunity`
`InfectedSymptomsImprovedImmunityConfirmed`
`InfectedSevereImprovedImmunity`
`InfectedCriticalImprovedImmunity`
`DeadImprovedImmunity`

All compartments with the same base state (e.g., ExposedNaive, ExposedPartialImmunity, ExposedImprovedImmunity) represent the same disease stage but with different immunity levels, which affects the later course of infection.

Infection State Transitions

The ODE-SECIRVVS model is implemented as a FlowModel, which defines the derivatives of each flow between compartments. The model follows the same flow pattern as the basic ODE-SECIR model but with three parallel sets of compartments representing different immunity levels.

The key characteristic of this model is that recovered individuals always end up in the improved immunity level, regardless of their starting immunity level. This represents the immunity gained after infection.

For each immunity level (Naive, PartialImmunity, ImprovedImmunity), the following transitions are defined:

`Susceptible*,                 Exposed*`
`Exposed*,                     InfectedNoSymptoms*`
`InfectedNoSymptoms*,          InfectedSymptoms*`
`InfectedNoSymptoms*,          SusceptibleImprovedImmunity` (recovery always leads to improved immunity)
`InfectedNoSymptomsConfirmed*, InfectedSymptomsConfirmed*`
`InfectedNoSymptomsConfirmed*, SusceptibleImprovedImmunity`
`InfectedSymptoms*,            InfectedSevere*`
`InfectedSymptoms*,            SusceptibleImprovedImmunity`
`InfectedSymptomsConfirmed*,   InfectedSevere*`
`InfectedSymptomsConfirmed*,   SusceptibleImprovedImmunity`
`InfectedSevere*,              InfectedCritical*`
`InfectedSevere*,              SusceptibleImprovedImmunity`
`InfectedSevere*,              Dead*`
`InfectedCritical*,            Dead*`
`InfectedCritical*,            SusceptibleImprovedImmunity`

Where * stands for the immunity level suffix (Naive, PartialImmunity, or ImprovedImmunity).

Important: Vaccinations are not implemented as flows between compartments but are handled discretely by the simulation. At the beginning of each simulated day, susceptible individuals are moved between immunity levels according to the specified daily vaccination parameters. This discrete process is separate from the ODE system and is managed by the apply_vaccination function in the model specific Simulation class.

Sociodemographic Stratification

Like the basic ODE-SECIR model, the ODE-SECIRVVS model can be stratified by one sociodemographic dimension, typically age groups. This stratification is important for modeling different vaccination rates, symptom severities, and mortality risks across age groups. The dimension is denoted AgeGroup but can also be used for other interpretations. For stratifications with two or more dimensions, see Model Creation.

Parameters

The model includes all parameters from the basic ODE-SECIR model plus additional parameters specific to vaccination and variant modeling.

Mathematical variable	C++ variable name	Description
\(\phi\)	`ContactPatterns`	Matrix of daily contact rates between different age groups.
\(\rho\)	`TransmissionProbabilityOnContact`	Transmission risk when susceptible and infected individuals make contact.
\(\xi_{I_{NS}}\)	`RelativeTransmissionNoSymptoms`	Proportion of non-symptomatically infected people who are not isolated.
\(\xi_{I_{Sy}}\)	`RiskOfInfectionFromSymptomatic`	Proportion of infected people with symptoms who are not isolated.
\(\mu_{I_{NS}}^{I_{Sy}}\)	`1 - RecoveredPerInfectedNoSymptoms`	Probability of transition from InfectedNoSymptoms to InfectedSymptoms.
\(\mu_{I_{Sy}}^{I_{Sev}}\)	`SeverePerInfectedSymptoms`	Probability of transition from InfectedSymptoms to InfectedSevere.
\(\mu_{I_{Sev}}^{I_{Cr}}\)	`CriticalPerSevere`	Probability of transition from InfectedSevere to InfectedCritical.
\(\mu_{I_{Cr}}^{D}\)	`DeathsPerCritical`	Probability of dying when in InfectedCritical.
\(T_{E}\)	`TimeExposed`	Time in days an individual stays in the Exposed compartment.
\(T_{I_{NS}}\)	`TimeInfectedNoSymptoms`	Time in days an individual stays in the InfectedNoSymptoms compartment.
\(T_{I_{Sy}}\)	`TimeInfectedSymptoms`	Time in days an individual stays in the InfectedSymptoms compartment.
\(T_{I_{Sev}}\)	`TimeInfectedSevere`	Time in days an individual stays in the InfectedSevere compartment.
\(T_{I_{Cr}}\)	`TimeInfectedCritical`	Time in days an individual stays in the InfectedCritical compartment.
\(s\)	`Seasonality`	Seasonal variation factor affecting transmission.
\(ICU_{cap}\)	`ICUCapacity`	Maximum ICU capacity in the modeled region.
\(TTC_{cap}\)	`TestAndTraceCapacity`	Capacity to test and trace contacts of infected per day.
\(TTC_{maxNoSym}\)	`TestAndTraceCapacityMaxRiskNoSymptoms`	Multiplier for test and trace capacity for cases without symptoms.
\(TTC_{maxSym}\)	`TestAndTraceCapacityMaxRiskSymptoms`	Multiplier for test and trace capacity for symptomatic cases.
\(T_{dyndelay}\)	`DynamicNPIsImplementationDelay`	Delay in days for implementing dynamic NPIs after threshold exceedance.
\(\lambda_{N,i}\)	`ext_inf_force_dummy`	Force of infection for susceptibles with naive immunity.
\(\lambda_{PI,i}\)	`reducExposedPartialImmunity * ext_inf_force_dummy`	Force of infection for susceptibles with partial immunity.
\(\lambda_{II,i}\)	`reducExposedImprovedImmunity * ext_inf_force_dummy`	Force of infection for susceptibles with improved immunity.
\(DV_{PI,i}\)	`DailyPartialVaccinations`	Daily number of first-dose vaccinations per age group.
\(DV_{II,i}\)	`DailyFullVaccinations`	Daily number of second-dose vaccinations per age group.
\(T_{vaccGap}\)	`VaccinationGap`	Time in days between first and second vaccine dose.
\(T_{V1}\)	`DaysUntilEffectivePartialImmunity`	Time in days until first vaccine dose takes full effect.
\(T_{V2}\)	`DaysUntilEffectiveImprovedImmunity`	Time in days until second vaccine dose takes full effect.
\(\delta_{E,PI}\)	`ReducExposedPartialImmunity`	Factor to reduce infection risk for persons with partial immunity.
\(\delta_{E,II}\)	`ReducExposedImprovedImmunity`	Factor to reduce infection risk for persons with improved immunity.
\(\delta_{Sy,PI}\)	`ReducInfectedSymptomsPartialImmunity`	Factor to reduce risk of developing symptoms for persons with partial immunity.
\(\delta_{Sy,II}\)	`ReducInfectedSymptomsImprovedImmunity`	Factor to reduce risk of developing symptoms for persons with improved immunity.
\(\delta_{Sev,PI}\)	`ReducInfectedSevereCriticalDeadPartialImmunity`	Factor to reduce hospitalization/mortality risk for partial immunity.
\(\delta_{Sev,II}\)	`ReducInfectedSevereCriticalDeadImprovedImmunity`	Factor to reduce hospitalization/mortality risk for improved immunity.
\(\kappa\)	`ReducTimeInfectedMild`	Reduction factor for infectious periods with partial or improved immunity.
\(\nu\)	`InfectiousnessNewVariant`	Relative infectiousness of a new variant compared to the original variant.
\(t_{newVar}\)	`StartDayNewVariant`	Day on which the new variant is introduced in the simulation.

Initial conditions

The initial conditions of the model are represented by the class Populations which defines the number of individuals in each sociodemographic group and InfectionState. Before running a simulation, you should set the initial values for each compartment across all immunity levels.

Below is an example showing how to initialize all compartments for the ODE-SECIRVVS model:

for (mio::AgeGroup i = 0; i < model.parameters.get_num_groups(); i++) {
    model.populations[{i, mio::osecirvvs::InfectionState::ExposedNaive}]                                = 10;
    model.populations[{i, mio::osecirvvs::InfectionState::ExposedImprovedImmunity}]                     = 11;
    model.populations[{i, mio::osecirvvs::InfectionState::ExposedPartialImmunity}]                      = 12;
    model.populations[{i, mio::osecirvvs::InfectionState::InfectedNoSymptomsNaive}]                     = 13;
    model.populations[{i, mio::osecirvvs::InfectionState::InfectedNoSymptomsNaiveConfirmed}]            = 13;
    model.populations[{i, mio::osecirvvs::InfectionState::InfectedNoSymptomsPartialImmunity}]           = 14;
    model.populations[{i, mio::osecirvvs::InfectionState::InfectedNoSymptomsPartialImmunityConfirmed}]  = 14;
    model.populations[{i, mio::osecirvvs::InfectionState::InfectedNoSymptomsImprovedImmunity}]          = 15;
    model.populations[{i, mio::osecirvvs::InfectionState::InfectedNoSymptomsImprovedImmunityConfirmed}] = 15;
    model.populations[{i, mio::osecirvvs::InfectionState::InfectedSymptomsNaive}]                       = 5;
    model.populations[{i, mio::osecirvvs::InfectionState::InfectedSymptomsNaiveConfirmed}]              = 5;
    model.populations[{i, mio::osecirvvs::InfectionState::InfectedSymptomsPartialImmunity}]             = 6;
    model.populations[{i, mio::osecirvvs::InfectionState::InfectedSymptomsPartialImmunityConfirmed}]    = 6;
    model.populations[{i, mio::osecirvvs::InfectionState::InfectedSymptomsImprovedImmunity}]            = 7;
    model.populations[{i, mio::osecirvvs::InfectionState::InfectedSymptomsImprovedImmunityConfirmed}]   = 7;
    model.populations[{i, mio::osecirvvs::InfectionState::InfectedSevereNaive}]                         = 8;
    model.populations[{i, mio::osecirvvs::InfectionState::InfectedSevereImprovedImmunity}]              = 1;
    model.populations[{i, mio::osecirvvs::InfectionState::InfectedSeverePartialImmunity}]               = 2;
    model.populations[{i, mio::osecirvvs::InfectionState::InfectedCriticalNaive}]                       = 3;
    model.populations[{i, mio::osecirvvs::InfectionState::InfectedCriticalPartialImmunity}]             = 4;
    model.populations[{i, mio::osecirvvs::InfectionState::InfectedCriticalImprovedImmunity}]            = 5;
    model.populations[{i, mio::osecirvvs::InfectionState::SusceptibleImprovedImmunity}]                 = 6;
    model.populations[{i, mio::osecirvvs::InfectionState::SusceptiblePartialImmunity}]                  = 7;
    model.populations[{i, mio::osecirvvs::InfectionState::DeadNaive}]                                   = 0;
    model.populations[{i, mio::osecirvvs::InfectionState::DeadPartialImmunity}]                         = 0;
    model.populations[{i, mio::osecirvvs::InfectionState::DeadImprovedImmunity}]                        = 0;

    // Set the SusceptibleNaive compartment as the difference from the total population
    model.populations.set_difference_from_group_total<mio::AgeGroup>(
        {i, mio::osecirvvs::InfectionState::SusceptibleNaive}, 1000);
}

After setting the initial populations, you also need to set the vaccination parameters. Vaccinations are realized discretely by moving susceptible individuals to the partially and fully vaccinated immunity levels at the beginning of each day of the simulation. For example, to set daily vaccinations that increase over time:

// Prepare and resize vaccinations parameter for the entire simulation period
const size_t daily_vaccinations = 10;
model.parameters.get<mio::osecirvvs::DailyPartialVaccinations<double>>().resize(
    mio::SimulationDay((size_t)tmax + 1));
model.parameters.get<mio::osecirvvs::DailyFullVaccinations<double>>().resize(
    mio::SimulationDay((size_t)tmax + 1));

// Set increasing number of vaccination over time
for (size_t i = 0; i < tmax + 1; ++i) {
    auto num_vaccinations = static_cast<double>(i * daily_vaccinations);
    model.parameters
        .get<mio::osecirvvs::DailyPartialVaccinations<double>>()[{(mio::AgeGroup)0, mio::SimulationDay(i)}] =
        num_vaccinations;
    model.parameters
        .get<mio::osecirvvs::DailyFullVaccinations<double>>()[{(mio::AgeGroup)0, mio::SimulationDay(i)}] =
        num_vaccinations;
}

Nonpharmaceutical Interventions

The ODE-SECIRVVS model supports nonpharmaceutical interventions (NPIs) through dampings to the contact matrix. These dampings reduce the contact rates between different groups to simulate interventions like lockdowns.

Basic dampings can be added to the contact matrix as follows:

// Create a contact matrix with baseline contact rates
auto& contacts = model.parameters.get<mio::osecirvvs::ContactPatterns<double>>();
auto& contact_matrix = contacts.get_cont_freq_mat();
contact_matrix[0].get_baseline().setConstant(0.5);
contact_matrix[0].get_baseline().diagonal().setConstant(5.0);

// Add a damping that reduces contacts by 30% starting at day 5
contact_matrix[0].add_damping(0.3, mio::SimulationTime(5.0));

For more complex scenarios, such as real-world , you can implement detailed NPIs with location-specific dampings as in the ODE-SECIR model. The ODE-SECIRVVS model supports the same contact locations (e.g., home, school, work, other) and can apply different dampings to each location.

Example of defining locations and interventions for a detailed scenario:

// Define different contact locations
enum class ContactLocation {
    Home = 0,
    School,
    Work,
    Other,
    Count,
};

// Define intervention types
enum class Intervention {
    Home,
    SchoolClosure,
    HomeOffice,
    GatheringBanFacilitiesClosure,
    PhysicalDistanceAndMasks,
    SeniorAwareness,
};

The model also supports dynamic NPIs based on epidemic thresholds:

// Configure dynamic NPIs
auto& dynamic_npis = params.get<mio::osecirvvs::DynamicNPIsInfectedSymptoms<double>>();
dynamic_npis.set_interval(mio::SimulationTime(3.0));  // Check NPI every 3 days
dynamic_npis.set_duration(mio::SimulationTime(14.0)); // Apply NPI for 14 days
dynamic_npis.set_base_value(100'000);                // Base value to trigger NPI is population of 100,000
dynamic_npis.set_threshold(200.0, dampings);         // Trigger at 200 cases per 100,000

Simulation

The SECIRVVS model offers the same simulation functions as the SECIR model:

simulate: Standard simulation that tracks the compartment sizes over time
simulate_flows: Extended simulation that additionally tracks the flows between compartments

Basic simulation:

double t0 = 0;       // Start time
double tmax = 30;    // End time
double dt = 0.1;     // Time step

// Run a standard simulation
mio::TimeSeries<double> result = mio::osecirvvs::simulate<double>(t0, tmax, dt, model);

During simulation, the model handles several special processes:

Daily Vaccinations: At the beginning of each simulated day, the apply_vaccination function updates the immunity level of susceptible individuals based on the vaccination parameters.
Variant Evolution: If configured, the apply_variant function updates the transmission probability based on the existance of a new variant over time.

For both simulation types, you can also specify a custom integrator:

auto integrator = std::make_unique<mio::RKIntegratorCore>();
integrator->set_dt_min(0.3);
integrator->set_dt_max(1.0);
integrator->set_rel_tolerance(1e-4);
integrator->set_abs_tolerance(1e-1);

mio::TimeSeries<double> result = mio::osecirvvs::simulate(t0, tmax, dt, model, std::move(integrator));

Output

The output of the simulation is a mio::TimeSeries object containing the sizes of each compartment at each time point. For a standard simulation, you can access the results as follows:

// Get the number of time points
auto num_points = static_cast<size_t>(result.get_num_time_points());

// Access data at a specific time point
Eigen::VectorXd value_at_time_i = result.get_value(i);
double time_i = result.get_time(i);

// Access the last time point
Eigen::VectorXd last_value = result.get_last_value();

You can print the simulation results as a formatted table:

// Print results to console with default formatting
result.print_table();

// Print with custom column labels (you'll need a lot for this model!)
std::vector<std::string> labels = {"S_naive", ... };
result.print_table(std::cout, labels);

The order of the compartments is as defined in the InfectionState enum.

Additionally, you can export the results to a CSV file for further analysis or visualization:

// Export results to CSV
result.export_csv("simulation_results.csv", labels);

Visualization

To visualize the results of a simulation, you can use the Python package m-plot and its documentation.

Examples

The extended model is used in the 2021_vaccination_sarscov2_delta_germany simulation. An easier example can be found in the examples/ode_secirvvs.cpp.

Examples of the basic ODE-SECIR model can be found at:

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