SPARSE-MOD Key Features

Time windows

One of the benefits of the SPARSEMODr design is that the user can specify how the values of certain model parameters might change over time. In this particular example, we show how the time-varying transmission rate, β_t might change in a stepwise fashion due to ‘interventions’ and ‘release of interventions’. We assume that when a parameter value changes between two time windows, there is a linear change over the number of days in that window. In other words, the user specifies the value of the parameter acheived on the last day of the time window. Note, however, that the user is instead allowed to supply daily parameter values to avoid this linear-change assumption. Here we show an example of a pattern of time-varying transmission rate that a user might specify, and how the C++ code is interpreting these values on the back-end.

# Set up the dates of change. 5 time windows
n_windows = 5
# Window intervals
start_dates = c(mdy("1-1-20"),  mdy("2-1-20"),  mdy("2-16-20"), mdy("3-11-20"),  mdy("3-22-20"))
end_dates =   c(mdy("1-31-20"), mdy("2-15-20"), mdy("3-10-20"), mdy("3-21-20"), mdy("5-1-20"))

# Date sequence
date_seq = seq.Date(start_dates[1], end_dates[n_windows], by = "1 day")

# Time-varying beta
changing_beta = c(0.3,            0.1,            0.1,            0.15,            0.15)

#beta sequence
beta_seq = NULL

beta_seq[1:(yday(end_dates[1]) - yday(start_dates[1]) + 1)] =
  changing_beta[1]

for(i in 2:n_windows){

  beta_temp_seq = NULL
  beta_temp = NULL

  if(changing_beta[i] != changing_beta[i-1]){

    beta_diff = changing_beta[i-1] - changing_beta[i]
    n_days = yday(end_dates[i]) - yday(start_dates[i]) + 1
    beta_slope = - beta_diff / n_days

    for(j in 1:n_days){
      beta_temp_seq[j] = changing_beta[i-1] + beta_slope*j
    }

  }else{
    n_days = yday(end_dates[i]) - yday(start_dates[i]) + 1
    beta_temp_seq = rep(changing_beta[i], times = n_days)
  }

  beta_seq = c(beta_seq, beta_temp_seq)

}

beta_seq_df = data.frame(beta_seq, date_seq)
date_breaks = seq(range(date_seq)[1],
                  range(date_seq)[2],
                  by = "1 month")


ggplot(beta_seq_df) +
  geom_path(aes(x = date_seq, y = beta_seq)) +
  scale_x_date(breaks = date_breaks, date_labels = "%b") +
  labs(x="", y=expression("Time-varying "*beta*", ("*beta[t]*")")) +
  # THEME
  theme_classic()+
  theme(
    axis.text = element_text(size = 10, color = "black"),
    axis.title = element_text(size = 12, color = "black"),
    axis.text.x = element_text(angle = 45, vjust = 0.5)
  )

Dispersal kernel

As we discuss in the documentation (see ?SPARSEMODr::Movement), we allow migration between populations in the meta-population to affect local and regional transmission dynamics. For now, migration is determined by a simple dispersal kernel, although we are working on adding more customizable gravity kernels. The user can control the shape of this kernel with the dist_phi option, as follows: $$ p_{i,j} = \frac{1}{\text{exp}(d_{i,j} / \phi)}, $$ where p_i, j is the probability of moving from population j to population i and d_i, j is the euclidean distance between the two populations.

We can see how ϕ (i.e., dist_phi in the model input) controls the probability below. In general, larger values of dist_phi make it more likely for hosts to travel farther distances.

#> Warning: A numeric `legend.position` argument in `theme()` was deprecated in ggplot2
#> 3.5.0.
#> ℹ Please use the `legend.position.inside` argument of `theme()` instead.
#> This warning is displayed once every 8 hours.
#> Call `lifecycle::last_lifecycle_warnings()` to see where this warning was
#> generated.

Transmission Types

As we describe in the documentation (e.g., see ?SPARSEMODr::model_interface), we allow the user to implement frequency-dependent (FD) or density-dependent (DD) transmission in the SPARSEMODr models.

For FD transmission, we divide the user-specified value of transmission rate by the total number of hosts within a given sub-population. For example, in the classic SEIR model, where S, E, I, and R, are the numbers (i.e., integers) of susceptible, exposed, infectious, and recovered hosts, respectively, we would have the following expression describing mass-action transmission per sub-population, i: $$ \beta_i \frac{S_i}{N_i} I_i, $$ where β_i is the user-specified transmission rate for sub-population i, and N_i is the total host population size for sub-population i. Therefore, with FD transmission, the effect of a single infectious host on the risk of infection is modulated by the fraction of the host population that is still susceptible, irrespective of the host population density (i.e., number of hosts per unit area) within that sub-population.

Alternatively, for DD transmission, the user can specify a (non-)linear Monod equation that describes the relationship between host population density and the transmission rate β via the model’s (optional) parameter, dd_trans_monod_k. The Monod equation is: $$ \beta_{\text{realized}} = \beta_{\text{max}} \frac{\text{Dens}}{K + \text{Dens}}, $$ where β_max is the maximum possible transmission rate across all densities, Dens is the density of the focal host population (i.e., number of hosts per unit area), and K is a constant that controls the effect of density on the transmission rate and is user-controlled by specifying dd_trans_monod_k. More specifically, K is the half-velocity constant at which point β_realized/β_max = 0.5.

We can see how dd_trans_monod_k controls the transmission rate below. In general, larger values of dd_trans_monod_k mean that transmission rate is more strongly limited by population density.