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Getting Started Example

This vignette walks through a first example of setting up a multi-group epidemic model and calculating useful quantities for analysis using functions in this R package. After installing the package (see instructions in README), load it:

library(multigroup.vaccine)

In our example model, we consider a population of size 10,000 individuals and with the following parameters:

• Group population sizes: two sub-populations, one with size 80,000 (A) and the other with size 20,000 (B)

popsize <- c(80000, 20000)

• Group-to-group contact rates: to model an outbreak spreading among this population, we require quantification of contact among and between members of each group. We ultimately need to create the square matrix contactmatrix that describes the relative amount of contact that members of one group (row) has with each other group (column). The contact matrix could be quantified directly, but we recommend using one of the package functions to create it.

In this example we use a model for “proportionate contacts with preferential mixing”, making use of our contactMatrixPropPref() function, which requires the following components. This contact model can allows for different overall contact rates of members of each group. Here we assume the relative contact rates are 1 and 1.7 for groups A and B. This means that an individual in Group B makes 1.7 contacts for every 1 made by an individual in group A.

relcontact <- c(1, 1.7)

This contact model also can also assume preferential mixing within one’s own group. We assume that the fraction of contacts that are exclusively within-group is 0.2 for group A, and 0.5 for group B. For group A, this means that 20% of their contacts are exclusively with other members of group A, and the remaining 80% of contacts are split between both groups A and B at rates proportional to the population sizes and overall contact rates of each group. The latter element ensures that the overall number of A-to-B and B-to-A contacts are equal, for contact symmetry.

incontact <- c(0.2, 0.5)

With the above elements, we calculate the contact matrix as follows

contactmatrix <- contactMatrixPropPref(popsize, relcontact, incontact)

• Relative susceptibility to infection. Now we are getting into modeling transmission of particular disease within the population described above. Not every contact between an infectious and a susceptible individual necessarily leads to a transmission. The two groups may differ in average susceptibility to infection per contact. With relative susceptibility 1 and 1.05, members of group B are 5% more susceptible to acquiring infection from an infectious contact than group A.

relsusc <- c(1, 1.05)

• Relative transmissibility of infection: The two groups may also differ in transmissibility per contact when infectious. With relative transmissibility 1 and 1.01, infectious members of group B are 1% more likely transmit infection to a susceptible contact compared to infectious members of group A.

reltransm <- c(1, 1.01)

• Overall transmissibility of disease. All the components above are enough to describe the relative transmission rates from one group to another, but we still need an absolute measure of how transmissible the disease is. This is often quantified by a basic reproduction number (\(R_0\)), defined as the average number of transmissions from a typical infectious individual if the rest of the population is susceptible. Here we set \(R_0\) to 1.75.

R0 <- 1.75

• Initial conditions and vaccination: we assume no one is immune due to prior infection (compartment R), and one individual from group B is an initial infectious case (compartment I). We also assume that 30% of group A and 20% of group B are initially immune due to vaccination (compartment V).

initR <- c(0, 0)
initI <- c(0, 1)
initV <- c(0.3, 0.2) * popsize

• Final outbreak size (ODE method): The finalsize() function computes the final number of infected people in each group at the end of the outbreak, by default using a deterministic ordinary differential equation (ODE) model and numerically solving the equations.

finalsize(popsize, R0, contactmatrix, relsusc, reltransm, initR, initI, initV)
#> [1] 14139.571  8749.608

Over one third of the infected individuals are from population B, despite the fact that they comprise only one fifth of the population.

• Final outbreak size (analytic method): We also provide a method to solve an analytic equation for the final outbreak size, which avoids the need to numerically solve for the differential equation trajectory:

finalsize(popsize, R0, contactmatrix, relsusc, reltransm, initR, initI, initV, method = "analytic")
#> [1] 14139.571  8749.609

While the analytic method might seem preferable, it is not necessarily faster or more stable than the ODE numerical solution method, as the analytic method still involves numerical estimation methods required to solve the transcendental equations. We recommend using the default ODE method, especially for models with a large number of groups.

• Final outbreak size (stochastic method): We also provide a method to simulate stochastic (random) outbreaks until no infectious individuals remain, producing a distribution of final outbreak sizes. We can specify the numbers of simulations to run with the nsims argument, which defaults to 1 if unspecified.

finalsize(popsize, R0, contactmatrix, relsusc, reltransm, initR, initI, initV, method = "stochastic",
          nsims = 10)
#>        [,1] [,2]
#>  [1,]     2    3
#>  [2,]     4    9
#>  [3,]     0    3
#>  [4,]     0    1
#>  [5,]     0    1
#>  [6,] 14927 9068
#>  [7,]     0    1
#>  [8,]     3    3
#>  [9,]     5    1
#> [10,] 14577 8763

Each row of the output has the results of one of the stochastic simulations for the final outbreak size of each group (including the initial one case in group B). We see that some simulations end after no or a small number of transmissions due to random luck, while others grow to a size close to the final size result from the deterministic results above.

• Final outbreak size (hybrid method). The simulations for the stochastic method can be slow when the simulated outbreak grows to a large size. When \(R_0 > 1\), we typically see the “bimodal” distribution of outcomes in stochastic simulations in which one set of simulated outbreaks have small final size while another set “escape” to a size that is quite close to the size predicted by the deterministic ODE solution. We created a “hybrid” simulation method that starts by simulating the outbreak stochastically as in the stochastic method. If the simulation reaches a state at which it is extremely unlikely to end in the next transmission generation, the simulation is halted and it is assumed that the final size will be the result from the ODE method.

finalsize(popsize, R0, contactmatrix, relsusc, reltransm, initR, initI, initV, method = "hybrid",
          nsims = 10)
#>        [,1] [,2]
#>  [1,]     0    1
#>  [2,] 14140 8750
#>  [3,]     0    3
#>  [4,] 14140 8750
#>  [5,]     0    1
#>  [6,]     1    2
#>  [7,]     0    1
#>  [8,]     0    3
#>  [9,]     0    1
#> [10,]     0    1

These binaries (installable software) and packages are in development.
They may not be fully stable and should be used with caution. We make no claims about them.
Health stats visible at Monitor.