CONCURRENCY TUTORIALS

Exercise 4: Thinking numerically about concurrency at the population level (Code)

```source("concurrency_exercise_4.R")
```

[or whatever you named the file when you saved it).

Note that when you run the full set of code, it may take 10-15 seconds at the end before your output appears; simulation takes time.

The model as currently set will simulate a population in which concurrency is not allowed for either sex; relationships last a mean of 10 months; the average person is in 0.8 relationships at any time; and initial prevalence is 2%. We suggest:

• First run the model as is, and note the levels of prevalence that emerge.
• Change the two lines at the top about enforcing monogamy from T to F, and rerun.
• Try changing one line to F and one line to T and rerun. Try the reverse.
• Play around with other parameters, like the mean degree, or the initial number of infected people, or the mean relational duration. This allows one to see how general the patterns initially observed are.
• At some point, try stepping through the code line by line, so as to really get a sense of what is happening at each point. Feel free to query different objects as you go to see what they equal, in order to increase your intuition. Because of the time loop in the code, you may want to replace the loop with a single time step so you can run each line in the loop independently.

An additional option is to use a Web Interface that hides the code in the background, and simplifies the process of exploring multiple scenarios. One may use this in addition to, or instead of, working with the code directly. This also allows the scenarios to begin with lower prevalence (1%) than the code does by default, and demonstrate that the equilibrium prevalence is still the same. A third option is to explore the code using the R package concurrency.sim.

```####################################
# Parameters
####################################

force.feml.monog <- F                                                         # Should we enforce a rule of female monogamy?
force.male.monog <- F                                                         # Should we enforce a rule of male monogamy?

n.femls <- 5000                                                               # Number of females
n.males <- 5000                                                               # Number of females
expected.meandeg <- 0.8                                                       # Mean degree (= average # of relationships a person is at any moment in time
avg.duration <- 10                                                            # Mean duration of a relationship (in months)
numtimesteps <- 1200                                                          # Length of simulations (in months)
init.number.of.infected.femls <- 100                                          # How many females are infect at the outset of the simulation?
init.number.of.infected.males <- 100                                          # How many males are infect at the outset of the simulation?

beta.by.time.since.inf <- c(rep(0.2055,3),                                    # Probability of transmission per month for active relationship
rep(0.0088,100),rep(0.0614,9),rep(0,10))                                # From Hollingsworth et al. 2008

####################################
# Basic calculations and book-keeping
####################################

n.pop <- n.femls + n.males                                                    # Total pop size
expected.edges <- round(expected.meandeg*(n.pop)/2)                           # Expected # of relationships ("edges") in the population at any time
prob.dissolution <- 1/avg.duration                                            # Daily prob of dissolution for an existing relationship
time.of.aids.death <- length(beta.by.time.since.inf)                          # Length of time from HIV infection until death (in months)
cum.num.feml.aids.deaths <- 0                                                 # Cumulative # of female AIDS deaths
cum.num.male.aids.deaths <- 0                                                 # Cumulative # of female AIDS deaths

feml.prev <- vector()                                                         # Sets up a vector to store female prevalence at each time step
male.prev <- vector()                                                         # Sets up a vector to store male prevalence at each time step

####################################
# Creating the female data frame
####################################

femls <- data.frame(row.names=1:n.femls)                                      # Creates a data frame to store info about the females.  One row per female.
femls\$hiv.status <- rep(0,n.femls)                                            # Create a variable in the female data frame called "hiv.status".  Give all females the value 0 (for the moment).
femls\$inf.time <- NA                                                          # Create a variable in the female data frame called "inf.time" (infection time). Give all females an NA (for the moment).
if (init.number.of.infected.femls>0) {                                        # If there are any initially infected females,
init.inf.f <- sample(1:n.femls, init.number.of.infected.femls)              # Sample from vector of female ids with size of initially infected females
femls\$hiv.status[init.inf.f] <- 1                                           # Give them infection
init.inf.time.f <- sample(0:(-time.of.aids.death+2),
init.number.of.infected.femls, replace=TRUE)              # Sample times backwards from present to length of infection
femls\$inf.time[init.inf.f] <- init.inf.time.f                               # Assign.
}

####################################
# Male data frame
####################################
# All parallel to female data frame above
males <- data.frame(row.names=1:n.males)
males\$hiv.status <- 0
males\$inf.time <- NA
if (init.number.of.infected.males>0) {
init.inf.m <- sample(1:n.males, init.number.of.infected.males)              # Sample from vector of female ids with size of initially infected females
males\$hiv.status[init.inf.m] <- 1                                           # Give them infection
init.inf.time.m <- sample(0:(-time.of.aids.death+2),
init.number.of.infected.males, replace=TRUE)            # Sample times backwards from present to length of infection
males\$inf.time[init.inf.m] <- init.inf.time.m                               # Assign.
}

####################################
# Initial contact network
####################################

edgelist <- data.frame(row.names=1:expected.edges)                            # Create a data frame that will be used to hold the IDs of the relationship pairs
if (force.feml.monog==FALSE) {                                                # If we are *not* enforcing female monogamy,
edgelist\$f <- sample(1:n.femls,expected.edges,replace=T)                #   sample from the female IDs with replacement, and assign them to a column in the data frame called "f"
} else {                                                                      #     otherwise,
edgelist\$f <- sample(1:n.femls,expected.edges,replace=F)                #   sample from the female IDs without replacement, and assign them to a column in the data frame called "f"
}
if (force.male.monog==FALSE) {                                                # Same for males
edgelist\$m <- sample(1:n.males,expected.edges,replace=T)
} else {
edgelist\$m <- sample(1:n.males,expected.edges,replace=F)
}

table(tabulate(edgelist\$f))                                                   # These rather odd lines provide a check to see a table of relationships per person in the population. When monogamy is enforced for
table(tabulate(edgelist\$m))                                                   #   either sex, the corresponding table should be limited to 0's and 1's; when it os not enforced, the table is not limited.

####################################                                          # Everything until now was set-up; now this is the core of the simulation.
# Time loop                                                                   # We will simulate forward in time, considering transmission, vital dynamics, and relational dynamics at each time point.
####################################

for (time in 1:numtimesteps) {

# Transmissions
hiv.status.feml.partners <- femls\$hiv.status[edgelist\$f]                # Create a vector containing the HIV status for the female partner in each relationship
hiv.status.male.partners <- males\$hiv.status[edgelist\$m]                # Create a vector containing the HIV status for the male partner in each relationship

sdpf <- which(hiv.status.male.partners==0 &                             # Make a vector of IDs for the relationships that are serodiscordant with positive female (SDPF)
hiv.status.feml.partners == 1)
f.in.sdpf <- edgelist\$f[sdpf]                                           # Make a vector of IDs for the females in the SDPF relationships
inf.time.sdpf <- time - femls\$inf.time[f.in.sdpf]                       # Make a vector that identifies how long before present these females were infected
prob.trans.sdpf <- beta.by.time.since.inf[inf.time.sdpf]                # Determine probability of transmission in the SDPM relationships, based on time since the male was infected
trans.sdpf <- rbinom(length(sdpf),1,prob.trans.sdpf)                    # Flip a weighted coin for each SDPF relationship to determine if transmission occurs
newly.inf.males <- edgelist\$m[sdpf[trans.sdpf==1]]                      # Double-indexing (an R specialty!)  This lets you get the identities of the newly infected males in one step.
males\$hiv.status[newly.inf.males] <- 1                                  # Assign the newly infected males an hiv.status of 1
males\$inf.time[newly.inf.males] <- time                                 # Assign the newly infected males an infection time of time

sdpm <- which(hiv.status.feml.partners==0 &                             # All parallel for serodiscordant with positive male (SDPM)
hiv.status.male.partners == 1)
m.in.sdpm <- edgelist\$m[sdpm]
inf.time.sdpm <- time - males\$inf.time[m.in.sdpm]
prob.trans.sdpm <- beta.by.time.since.inf[inf.time.sdpm]
trans.sdpm <- rbinom(length(sdpm),1,prob.trans.sdpm)
newly.inf.femls <- edgelist\$f[sdpm[trans.sdpm==1]]
femls\$hiv.status[newly.inf.femls] <- 1
femls\$inf.time[newly.inf.femls] <- time

# Deaths to AIDS
femls.dying.of.AIDS <- which(time-femls\$inf.time==time.of.aids.death)   # Which females have been HIV+ long enough to die of AIDS?
males.dying.of.AIDS <- which(time-males\$inf.time==time.of.aids.death)   # Which females have been HIV+ long enough to die of AIDS?
cum.num.feml.aids.deaths <-                                             # Increase cumulative # of female AIDS deaths by the newly dying females
cum.num.feml.aids.deaths + length(femls.dying.of.AIDS)
cum.num.male.aids.deaths <-                                             # Increase cumulative # of male AIDS deaths by the newly dying males
cum.num.male.aids.deaths + length(males.dying.of.AIDS)

# End of ties because of death
edges.with.feml.dying.of.aids <-                                        # Determine the IDs of those relationships involving dying women
which(edgelist\$f %in% femls.dying.of.AIDS)
edges.with.male.dying.of.aids <-
which(edgelist\$m %in% males.dying.of.AIDS)                        # Determine the IDs of those relationships involving dying women
edges.with.either.dying.of.aids <-
c(edges.with.feml.dying.of.aids,edges.with.male.dying.of.aids)    # Combine the two
if (length(edges.with.either.dying.of.aids>0)){                         # Remove those edges from the edgelist
edgelist <- edgelist[-edges.with.either.dying.of.aids,]
}

# End of other ties randomly

edges.coinflip <- rbinom(dim(edgelist)[1],1,prob.dissolution)           # Flip a weighted coin for each remaning relationship
edges.to.break <- which(edges.coinflip==1)                              # Make vector of those edges to break
if (length(edges.to.break>0)) {                                         # If there are any edges to break,
edgelist <- edgelist[-edges.to.break,]                            #   then break them.
}

num.ties.to.add <- expected.edges - dim(edgelist)[1]                    # Determine how many edges to add. (We assume same as # broken in this simple model.)

if (force.feml.monog==F) {                                        #   and if we are *not* enforcing female monogamy,
f <- sample(1:n.femls,num.ties.to.add,replace=T)            #     sample from the female IDs with replacement,
} else {                                                          #   else if we are enforcing female monogamy,
femls.with.no.ties <- setdiff(1:n.femls,edgelist\$f)         #     determine which women do not currently have a relationship, and
f <- sample(femls.with.no.ties,num.ties.to.add,replace=F)   #     sample from them without replacement
}
if (force.male.monog==F) {                                        #   and if we are *not* enforcing male monogamy,
m <- sample(1:n.males,num.ties.to.add,replace=T)            #     sample from the male IDs with replacement,
} else {                                                          #   else if we are enforcing male monogamy,
males.with.no.ties <- setdiff(1:n.males,edgelist\$m)         #     determine which men do not currently have a relationship, and
m <- sample(males.with.no.ties,num.ties.to.add,replace=F)   #     sample from them without replacement
}
new.edges <- data.frame(f,m)                                      # Combine the new males and females into pairs in a data frame
edgelist <- data.frame(rbind(edgelist,new.edges))                 # "Bind" that data frame onto the end of our existing esgelist
row.names(edgelist) <- 1:dim(edgelist)[1]                         # Rename the rows of the updated edgelist so they are consecutive numbers again.
}

# Insert new births
new.feml.ids <- femls.dying.of.AIDS                                     # Make a vector of the IDs of the women who just died. We shall "replace" them with a new arrival, by
femls\$hiv.status[new.feml.ids] <- 0                                     #   setting the HIV status of that node back to 0, and
femls\$inf.time[new.feml.ids] <- NA                                      #   setting the infection time for that node back to NA.

new.male.ids <- males.dying.of.AIDS                                     # All parallel with males.
males\$hiv.status[new.male.ids] <- 0
males\$inf.time[new.male.ids] <- NA

# Track prevalence

feml.prev[time] <- mean(femls\$hiv.status)                                     # Calculate current female prevalence and append it onto the female prevalence vector.
male.prev[time] <- mean(males\$hiv.status)                                     # Calculate current male prevalence and append it onto the male prevalence vector.

}

windows(20,10)                                                                # These two lines set up a plotting window
par(mfrow=c(1,2))                                                             #    with two panels.
plot(feml.prev,main= "Female prevalence over time ",ylim=c(0,0.5))             # Plot female prevalence over time
plot(male.prev,main= "Male prevalence over time ",ylim=c(0,0.5))               # Plot male prevalence over time

```

(c) Steven M. Goodreau, Samuel M. Jenness, and Martina Morris 2012. Fair use permitted with citation. Citation info: Goodreau SM and Morris M, 2012. Concurrency Tutorials, http://www.statnet.org/concurrency