# Mixing models

#### George Vega Yon

#### 2025-11-13

## Introduction

This vignette shows how to create a mixing model using the `epiworldR` package. Mixing models feature multiple populations. Each group, called `Entities`, has a subset of the agents. The agents can interact with each other within the same group and with agents from other groups. A contact matrix defines the interaction between agents.

## An SEIR model with mixing

For this example, we will simulate an outbreak featuring three populations. The contact matrix is defined as follows:

\[ \left[% \begin{array}{ccc} 0.9 & 0.05 & 0.05 \\ 0.1 & 0.8 & 0.1 \\ 0.1 & 0.2 & 0.7 \\ \end{array}% \right] \]

This matrix represents the probability of an agent from population \(i\) interacting with an agent from population \(j\). The matrix is row-stochastic, meaning that the sum of each row is equal to 1.

We will build this model using the `entity` class in epiworld. The following code chunk instantiates three entities and the contact matrix.

    library(epiworldR)

    e1 <- entity("Population 1", 3e3, as_proportion = FALSE)
    e2 <- entity("Population 2", 3e3, as_proportion = FALSE)
    e3 <- entity("Population 3", 3e3, as_proportion = FALSE)

    # Row-stochastic matrix (rowsums 1)
    cmatrix <- c(
      c(0.9, 0.05, 0.05),
      c(0.1, 0.8, 0.1),
      c(0.1, 0.2, 0.7)
    ) |> matrix(byrow = TRUE, nrow = 3)

With these in hand, we can proceed to create a mixing model. The following code chunk creates a model, an SEIR with mixing, and adds the entities to the model:

    N <- 9e3

    flu_model <- ModelSEIRMixing(
      name              = "Flu",
      n                 = N,
      prevalence        = 1 / N,
      contact_rate      = 20,
      transmission_rate = 0.1,
      recovery_rate     = 1 / 7,
      incubation_days   = 7,
      contact_matrix    = cmatrix
    )

    # Adding the entities
    flu_model |>
      add_entity(e1) |>
      add_entity(e2) |>
      add_entity(e3)

The function `add_entity` adds the corresponding entity. The default behavior randomly assigns agents to the entities at the beginning of the simulation. Agents may be assigned to more than one entity. The following code-chunk simulates the model for 100 days, summarizes the results, and plots the incidence curve:

    set.seed(331)
    run(flu_model, ndays = 100)
    #> _________________________________________________________________________
    #> Running the model...
    #> ||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||||| done.
    summary(flu_model)
    #> ________________________________________________________________________________
    #> ________________________________________________________________________________
    #> SIMULATION STUDY
    #>
    #> Name of the model   : Susceptible-Exposed-Infected-Removed (SEIR) with Mixing
    #> Population size     : 9000
    #> Agents' data        : (none)
    #> Number of entities  : 3
    #> Days (duration)     : 100 (of 100)
    #> Number of viruses   : 1
    #> Last run elapsed t  : 53.00ms
    #> Last run speed      : 16.96 million agents x day / second
    #> Rewiring            : off
    #>
    #> Global events:
    #>  - Update infected individuals (runs daily)
    #>
    #> Virus(es):
    #>  - Flu
    #>
    #> Tool(s):
    #>  (none)
    #>
    #> Model parameters:
    #>  - Avg. Incubation days : 7.0000
    #>  - Contact rate         : 20.0000
    #>  - Prob. Recovery       : 0.1429
    #>  - Prob. Transmission   : 0.1000
    #>
    #> Distribution of the population at time 100:
    #>   - (0) Susceptible : 8999 -> 1
    #>   - (1) Exposed     :    1 -> 0
    #>   - (2) Infected    :    0 -> 2
    #>   - (3) Recovered   :    0 -> 8997
    #>
    #> Transition Probabilities:
    #>  - Susceptible  0.96  0.04     -     -
    #>  - Exposed         -  0.85  0.15     -
    #>  - Infected        -     -  0.86  0.14
    #>  - Recovered       -     -     -  1.00
    plot_incidence(flu_model)

## Investigating the history

After running the simulation, a possible question is: how many agents from each entity were infected each day? The following code chunk retrieves the agents from each entity and the transmissions that occurred during the simulation:

    library(data.table)

    agents_entities <- lapply(get_entities(flu_model), \(e) {
      entity_get_agents(e)
    }) |> rbindlist()

    head(agents_entities)
    #>    agent entity
    #>    <int>  <int>
    #> 1:   507      0
    #> 2:  1710      0
    #> 3:  8984      0
    #> 4:  7844      0
    #> 5:   760      0
    #> 6:  4379      0

We can retrieve the daily incidence for each entity by merging the transmissions with the agents’ entities. The following code chunk accomplishes this:

    # Retrieving the transmissions
    transmissions <- get_transmissions(flu_model) |>
      data.table()

    # We only need the date and the source
    transmissions <- subset(
      transmissions,
      select = c("date", "source")
    )

    # Attaching the entity to the source
    transmissions <- merge(
      transmissions,
      agents_entities,
      by.x = "source", by.y = "agent"
    )

    # Aggregating by date x entity (counts)
    transmissions <- transmissions[, .N, by = .(date, entity)]

    # Taking a look at the data
    head(transmissions)
    #>     date entity     N
    #>    <int>  <int> <int>
    #> 1:    25      2   170
    #> 2:    26      1   245
    #> 3:    29      1   241
    #> 4:    31      1   223
    #> 5:    25      0   382
    #> 6:    26      0   338

With this information, we can now visualize the daily incidence for each entity. The following code chunk plots the daily incidence for each entity:

    setorder(transmissions, date, entity)

    ran <- range(transmissions$N)
    transmissions[entity == 0, plot(
      x = date, y = N, type = "l", col = "black", ylim = ran)]
    #> NULL
    transmissions[entity == 1, lines(x = date, y = N, col = "red")]
    #> NULL
    transmissions[entity == 2, lines(x = date, y = N, col = "blue")]
    #> NULL

    legend(
      "topright",
      legend = c("Population 1", "Population 2", "Population 3"),
      col = c("black", "red", "blue"),
      lty = 1
    )