ODE-based SECIR-type model

The ODE-SECIR module models and simulates an epidemic using an ODE-based SECIR-type model approach. The model is particularly suited for pathogens with pre- or asymptomatic infection states and when severe or critical symptoms are possible. The model assumes perfect immunity after recovery. It is thus only suited for epidemic use cases and, mostly, early epidemic phases.

• A generalization of the model that allows for Gamma or Erlang distributed stay times is the LCT-SECIR model.

• A generalization of the model that allows for arbitrary distributed stay times is the IDE-SECIR model.

• A generalization of the model that includes three immunity layers and vaccination is the ODE-SECIRVVS model.

• A generalization of the model that includes three immunity layers, vaccination, and waning immunity is the ODE-SECIRTS model.

The infection states and the transitions (also see next two sections) are visualized in the following graph.

Infection States

The model contains the following list of InfectionStates:

`Susceptible`
`Exposed`
`InfectedNoSymptoms`
`InfectedNoSymptomsConfirmed`
`InfectedSymptoms`
`InfectedSymptomsConfirmed`
`InfectedSevere`
`InfectedCritical`
`Recovered`
`Dead`

While the states InfectedNoSymptomsConfirmed and InfectedSymptomsConfirmed are available, they are not used in the current implementation and detection is only modeled implicitly through detection rates based on user-defined criteria.

Infection State Transitions

The ODE-SECIR model is implemented as a FlowModel, which defines the derivatives of each flow between compartments. This allows for explicit computation of new transmissions, infections, and hospitalizations. Additionally, the aggregated compartment values can be computed with minimal overhead. The defined transitions FromState, ToState are:

`Susceptible,                 Exposed`
`Exposed,                     InfectedNoSymptoms`
`InfectedNoSymptoms,          InfectedSymptoms`
`InfectedNoSymptoms,          Recovered`
`InfectedNoSymptomsConfirmed, InfectedSymptomsConfirmed`
`InfectedNoSymptomsConfirmed, Recovered`
`InfectedSymptoms,            InfectedSevere`
`InfectedSymptoms,            Recovered`
`InfectedSymptomsConfirmed,   InfectedSevere`
`InfectedSymptomsConfirmed,   Recovered`
`InfectedSevere,              InfectedCritical`
`InfectedSevere,              Recovered`
`InfectedSevere,              Dead`
`InfectedCritical,            Dead`
`InfectedCritical,            Recovered`

Sociodemographic Stratification

In the ODE-SECIR model, the population can be stratified by one sociodemographic dimension. This 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 implements the following parameters:

Mathematical variable	C++ variable name	Description
\(\phi\)	ContactPatterns	Matrix of daily contact rates / number of daily contacts between different age groups.
\(\rho\)	TransmissionProbabilityOnContact	Transmission risk for people located in one of the susceptible compartments.
\(\xi_{I_{NS}}\)	RelativeTransmissionNoSymptoms	Proportion of nonsymptomatically infected people who are not isolated.
\(\xi_{I_{Sy}}\)	RiskOfInfectionFromSymptomatic	Proportion of infected people with symptoms who are not isolated (time-dependent if TestAndTraceCapacity used).
\(N_j\)	Nj	Total population of age group \(j\).
\(D_i\)	Di	Number of deaths in age group \(i\).
\(T_{E}\)	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.
\(\mu_{I_{NS}}^{I_{Sy}}\)	1 - RecoveredPerInfectedNoSymptoms	Probability of transition from compartment InfectedNoSymptoms to InfectedSymptoms.
\(\mu_{I_{Sy}}^{I_{Sev}}\)	SeverePerInfectedSymptoms	Probability of transition from compartment InfectedSymptoms to InfectedSevere.
\(\mu_{I_{Sev}}^{I_{Cr}}\)	CriticalPerSevere	Probability of transition from compartment InfectedSevere to InfectedCritical.
\(\mu_{I_{Cr}}^{D}\)	DeathsPerCritical	Probability of dying when in compartment InfectedCritical.

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 need to set the initial values for each compartment:

// Set total population size
model.populations.set_total(nb_total_t0);

// Set values for each InfectionState in the specific age group
model.populations[{mio::AgeGroup(0), mio::osecir::InfectionState::Exposed}] = nb_exp_t0;
model.populations[{mio::AgeGroup(0), mio::osecir::InfectionState::InfectedNoSymptoms}] = nb_car_t0;
model.populations[{mio::AgeGroup(0), mio::osecir::InfectionState::InfectedNoSymptomsConfirmed}] = 0;
model.populations[{mio::AgeGroup(0), mio::osecir::InfectionState::InfectedSymptoms}] = nb_inf_t0;
model.populations[{mio::AgeGroup(0), mio::osecir::InfectionState::InfectedSymptomsConfirmed}] = 0;
model.populations[{mio::AgeGroup(0), mio::osecir::InfectionState::InfectedSevere}] = nb_hosp_t0;
model.populations[{mio::AgeGroup(0), mio::osecir::InfectionState::InfectedCritical}] = nb_icu_t0;
model.populations[{mio::AgeGroup(0), mio::osecir::InfectionState::Recovered}] = nb_rec_t0;
model.populations[{mio::AgeGroup(0), mio::osecir::InfectionState::Dead}] = nb_dead_t0;

// Set the susceptible population as difference to ensure correct total population
model.populations.set_difference_from_total({mio::AgeGroup(0), mio::osecir::InfectionState::Susceptible}, nb_total_t0);

For age-resolved models, you need to set the initial conditions for each age group. Additionally, you can use set_difference_from_group_total to set the susceptible compartment as the difference between the total group size and all other compartments:

for (auto i = mio::AgeGroup(0); i < nb_groups; i++) {
    model.populations[{i, mio::osecir::InfectionState::Exposed}] = fact * nb_exp_t0;
    // ...other states...
    model.populations.set_difference_from_group_total<mio::AgeGroup>(
        {i, mio::osecir::InfectionState::Susceptible}, fact * nb_total_t0);
}

Nonpharmaceutical Interventions

In the SECIR model, nonpharmaceutical interventions (NPIs) are implemented through dampings to the contact matrix. These dampings reduce the contact rates between different groups to simulate interventions.

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

// Create a contact matrix with constant contact rates between all groups
mio::ContactMatrixGroup& contact_matrix = model.parameters.get<mio::osecir::ContactPatterns<double>>();
contact_matrix[0] = mio::ContactMatrix(Eigen::MatrixXd::Constant(1, 1, cont_freq));

// Add a damping that reduces contacts by 70% starting at day 30
contact_matrix[0].add_damping(0.7, mio::SimulationTime(30.));

For age-resolved models, you can apply different dampings to different groups:

contact_matrix[0] = mio::ContactMatrix(Eigen::MatrixXd::Constant((size_t)nb_groups, (size_t)nb_groups, fact * cont_freq));

// Add a damping that reduces contacts within the same age group by 70% starting at day 30
contact_matrix.add_damping(Eigen::VectorX<ScalarType>::Constant((size_t)nb_groups, 0.7).asDiagonal(),
                         mio::SimulationTime(30.));

For more complex scenarios, such as real-world venue closures or lockdown modeling, you can implement detailed NPIs with location-specific dampings. The ODE-SECIR model supports contact matrices for different locations (e.g., home, school, work, other) and can apply different dampings to each location.

Example for defining different contact locations:

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

// Map contact locations to strings for loading data files
const std::map<ContactLocation, std::string> contact_locations = {
    {ContactLocation::Home, "home"},
    {ContactLocation::School, "school_pf_eig"},
    {ContactLocation::Work, "work"},
    {ContactLocation::Other, "other"}
};

You can create intervention types that target specific locations with different intensities:

// Different types of NPI
enum class Intervention
{
    Home,
    SchoolClosure,
    HomeOffice,
    GatheringBanFacilitiesClosure,
    PhysicalDistanceAndMasks,
    SeniorAwareness,
};

// Different levels of NPI
enum class InterventionLevel
{
    Main,
    PhysicalDistanceAndMasks,
    SeniorAwareness,
    Holidays,
};

A complex lockdown scenario with multiple interventions starting on a specific date can be implemented via:

auto start_lockdown_date = mio::Date(2020, 3, 18);
auto start_lockdown = mio::SimulationTime(mio::get_offset_in_days(start_lockdown_date, start_date));

// Apply different dampings for each intervention type
contact_dampings.push_back(contacts_at_home(start_lockdown, 0.6, 0.8));
contact_dampings.push_back(school_closure(start_lockdown, 1.0, 1.0));
contact_dampings.push_back(home_office(start_lockdown, 0.2, 0.3));
contact_dampings.push_back(social_events(start_lockdown, 0.6, 0.8));
contact_dampings.push_back(physical_distancing(start_lockdown, 0.4, 0.6));

A more advanced structure to automatically activate interventions based on threshold criteria is given by DynamicNPIs. Dynamic NPIs can be configured to trigger when the number of symptomatic infected individuals exceeds a certain relative threshold in the population. In contrast to static NPIs which are active as long as no other NPI gets implemented, dynamic NPIs are checked at regular intervals and get activated for a defined duration when the threshold is exceeded. As above, different dampings contact_dampings can be assigned to different contact locations and are then triggered all at once the threshold is exceeded. The following example shows how to set up dynamic NPIs based on the number of 200 symptomatic infected individuals per 100,000 population. It will be active for at least 14 days and checked every 3 days. If the last check after day 14 is negative, the NPI will be deactivated.

// Configure dynamic NPIs with thresholds
auto& dynamic_npis = params.get<mio::osecir::DynamicNPIsInfectedSymptoms<double>>();
dynamic_npis.set_interval(mio::SimulationTime(3.0));  // Check every 3 days
dynamic_npis.set_duration(mio::SimulationTime(14.0)); // Apply for 14 days
dynamic_npis.set_base_value(100'000);                // Per 100,000 population
dynamic_npis.set_threshold(200.0, contact_dampings);         // Trigger at 200 cases per 100,000

Simulation

The SECIR model offers two simulation functions:

1. simulate: Standard simulation that tracks the compartment sizes over time

2. simulate_flows: Extended simulation that additionally tracks the flows between compartments

Standard simulation:

double t0 = 0;       // Start time
double tmax = 50;    // End time
double dt = 0.1;     // Time step

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

Flow simulation for tracking transitions between compartments:

// Run a flow simulation to additionally track transitions between compartments
auto result = mio::osecir::simulate_flows(t0, tmax, dt, model);
// result[0] contains compartment sizes, result[1] contains flows

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> secir = mio::osecir::simulate(t0, tmax, dt, model, std::move(integrator));

Output

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

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

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

// Access the last time point
Eigen::VectorXd last_value = secir.get_last_value();
double last_time = secir.get_last_time();

For flow simulations, the result consists of two mio::TimeSeries objects, one for compartment sizes and one for flows:

auto result = mio::osecir::simulate_flows(t0, tmax, dt, model);

// Access compartment sizes
auto compartments = result[0];

// Access flows between compartments
auto flows = result[1];

You can print the simulation results as a formatted table:

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

// Print with custom column labels
std::vector<std::string> labels = {"S", "E", "C", "C_confirmed", "I", "I_confirmed", "H", "U", "R", "D"};
secir.print_table(labels);

Additionally, you can export the results to a CSV file:

// Export results to CSV with default settings
secir.export_csv("simulation_results.csv");

The ODE-SECIR model also provides utility functions to extract specific measures, such as the reproduction number:

// Calculate R value at a specific time index
auto r_at_index = mio::osecir::get_reproduction_number(time_idx, sim);

// Calculate R values for the entire simulation
Eigen::VectorXd r_values = mio::osecir::get_reproduction_numbers(sim);

Visualization

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

Examples

Different examples can be found at:

• examples/ode_secir.cpp

• examples/ode_secir_ageres.cpp

• examples/ode_secir_parameter_study.cpp

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