ODE-based SIR-type model

The ODE-based SIR-type module models and simulates the epidemic using an ODE-based SIR-type model approach. The model is particularly suited for simple simulations of infectious diseases in a population and getting started with the MEmilio library. The model assumes infectiousness upon transmission and does not allow for pre-symptomatic infectious or noninfectious phases. The model assumes perfect immunity after recoveryIt is thus only suited for epidemic use cases and, mostly, early epidemic phases.

A generalization of the model that includes a pre-symptomatic noninfectious stage is the ODE-SEIR model.
A generalization of the model that includes pre-symptomatic noninfectious and infectious stages is the ODE-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.

In the following, we present the model in detail.

The infection states and the transitions are visualized in the following image.

Infection States

The model contains the following list of InfectionStates:

`Susceptible`
`Infected`
`Recovered`

Infection State transitions

The ODE-SIR model is implemented as a CompartmentalModel, which defines the derivative of the aggregated compartment values in time.

Sociodemographic Stratification

In the ODE-SIR model, the population can be stratified by one sociodemographic dimension. This dimension is denoted AgeGroup but can also be used for other interpretations. For stratification with two or more dimensions, see Model Creation.

The number of age groups is specified in the model constructor and the model can be initialized with:

mio::oseir::Model model(nb_groups)

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 the Susceptible compartment.
\(N\)	`populations.get_total()`	Total population.
\(T_{I}\)	`TimeInfected`	Time in days an individual stays in the Infected compartment.

Initial conditions

The initial conditions of the model are defined 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::osir::InfectionState::Infected}] = nb_inf_t0;
model.populations[{mio::AgeGroup(0), mio::osir::InfectionState::Recovered}] = nb_rec_t0;

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

For age-resolved simulations, 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::osir::InfectionState::Infected}] = 1/nb_groups * nb_inf_t0;
   model.populations[{i, mio::osir::InfectionState::Recovered}] = 1/nb_groups * nb_rec_t0;

   model.populations.set_difference_from_group_total<mio::AgeGroup>(
       {i, mio::osir::InfectionState::Susceptible}, 1/nb_groups * nb_total_t0);
}

Nonpharmaceutical Interventions

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

Basic dampings can be added to the ContactPatterns as follows:

contact_matrix[0] = mio::ContactMatrix(Eigen::MatrixXd::Constant(nb_groups, nb_groups, 1/nb_groups * cont_freq));

// Add a uniform damping across all age groups
contact_matrix.add_damping(Eigen::MatrixXd::Constant(nb_groups, nb_groups, 0.7), mio::SimulationTime(30.));

Simulation

The ODE-SIR model can be simulated using the simulate function, which tracks the compartment sizes over time:

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

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

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::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>(result.get_num_time_points());

// Access data at 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();
double last_time = result.get_last_time();

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
std::vector<std::string> labels = {"S", "I", "R"};
result.print_table(labels);

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

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

Visualization

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

Examples

An example can be found at examples/ode_sir.cpp.

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