MGDrivE2: One Node Epidemiological Dynamics

Table of Contents

1. Parameterization
2. Initialization of the Petri Net
3. Equilibrium Conditions and Hazard Functions
4. Simulation of Fully Specified SPN Model
   1. Deterministic: ODE Solutions
   2. Stochastic: CLE Solutions
5. References

# simulation functions
library(MGDrivE2)
# inheritance patterns
library(MGDrivE)
# plotting
library(ggplot2)

# basic inheritance pattern
cube <- MGDrivE::cubeMendelian()

Parameterization

Several parameters are necessary to setup the structural properties of the Petri Net, as well as calculate the population distribution at equilibrium, setup initial conditions, and calculate hazards. Again, we specify all entomological parameters as for the mosquito-only simulation (see “MGDrivE2: One Node Lifecycle Dynamics”) as well as additional parameters for the SEI-SIS dynamics. Like the aquatic stages, \(\frac{1}{q_{\mathit{EIP}}}\) will give the mean dwell time for incubating mosquitoes, and variance by \(\frac{1}{n_{\mathit{EIP}} \cdot q_{\mathit{EIP}}^{2}}\). The model requires muH, mortality rate in humans, because equilibrium dynamics are simulated (that is, human populations follow an “open cohort” with equal birth and death rates). A table of (case-sensitive) epidemiological parameters the user needs to specify is given below. Note that all parameters must be specified as a rate per day. For a detailed discussion of these parameters in the context of malaria models, see Smith and McKenzie (2004).

Please note that f and Q must be specified; this is because future versions of MGDrivE2 will include additional vector control methods such as IRS (indoor residual spraying) and ITN (insecticide treated nets). In the presence of ITNs/IRS f will vary independently as a function of time depending on intervention coverage.

Additionally, we specify a total simulation length of 300 days, with output stored daily.

# entomological and epidemiological parameters
theta <- list(
  # lifecycle parameters
  qE = 1/4,
  nE = 2,
  qL = 1/3,
  nL = 3,
  qP = 1/6,
  nP = 2,
  muE = 0.05,
  muL = 0.15,
  muP = 0.05,
  muF = 0.09,
  muM = 0.09,
  beta = 16,
  nu = 1/(4/24),
  # epidemiological parameters
  NH = 1000,
  X = 0.25,
  f = 1/3,
  Q = 0.9,
  b = 0.55,
  c = 0.15,
  r = 1/200,
  muH = 1/(62*365),
  qEIP = 1/11,
  nEIP = 6
)

# simulation parameters
tmax <- 250
dt <- 1

We also need to augment the cube with genotype specific transmission efficiencies; this allows simulations of gene drive systems that confer pathogen-refractory characteristics to mosquitoes depending on genotype. The specific parameters we want to attach to the cube are b and c, the mosquito to human and human to mosquito transmission efficiencies. We assume that transmission from human to mosquito is not impacted in modified mosquitoes, but mosquito to human transmission is significantly reduced in modified mosquitoes. For detailed descriptions of these parameters for modelling malaria transmission, see Smith & McKenzie (2004) for extensive discussion.

# augment the cube with RM transmission parameters
cube$c <- setNames(object = rep(x = theta$c, times = cube$genotypesN), nm = cube$genotypesID)
cube$b <- c("AA" = theta$b, "Aa" = 0.35, "aa" = 0)

Initialization of the Petri Net

The SEI-SIS disease transmission model sits “on top” of the existing MGDrivE2 structure, using the default aquatic and male “places”, but expanding adult female “places” to follow an Erlang-distributed pathogen incubation period (called the extrinsic incubation period, EIP) and adding SIS human states as well. Information on how to choose the proper EIP distribution can be found in the help file for ?makeQ_SEI().

The transitions between states is also expanded, providing transitions for females to progress in infection status, adding human dynamics, and allowing interaction between mosquito and human states. All of these additions are handled internally by spn_T_epiSIS_node().

# Places and transitions
SPN_P <- spn_P_epiSIS_node(params = theta, cube = cube)
SPN_T <- spn_T_epiSIS_node(spn_P = SPN_P, params = theta, cube = cube)

# Stoichiometry matrix
S <- spn_S(spn_P = SPN_P, spn_T = SPN_T)

Equilibrium Conditions and Hazard Functions

Now that we have set up the structural properties of the Petri Net, we need to calculate the population distribution at equilibrium and define the initial conditions for the simulation.

The function equilibrium_SEI_SIS() calculates the equilibrium distribution of female mosquitoes across SEI stages, based on human populations and force-of-infection, then calculates all other equilibria. We set the logistic form for larval density-dependence in these examples by specify log_dd = TRUE.

# SEI mosquitoes and SIS humans equilibrium
#  outputs required parameters in the named list "params"
#  outputs initial equilibrium for adv users, "init
#  outputs properly filled initial markings, "M0"
initialCons <- equilibrium_SEI_SIS(params = theta, phi = 0.5, log_dd = TRUE,
                                   spn_P = SPN_P, cube = cube)

With the equilibrium conditions calculated (see ?equilibrium_SEI_SIS()), and the list of possible transitions provided by spn_T_epiSIS_node(), we can now calculate the rates of those transitions between states.

# approximate hazards for continuous approximation
approx_hazards <- spn_hazards(spn_P = SPN_P, spn_T = SPN_T, cube = cube,
                              params = initialCons$params, type = "SIS",
                              log_dd = TRUE, exact = FALSE, tol = 1e-8,
                              verbose = FALSE)

Simulation of Fully Specified SPN Model

Similar to previous simulations, we will release 50 adult females with homozygous recessive alleles 5 times, every 10 days, but starting at day 20. Remember, it is critically important that the event names match a place name in the simulation. The simulation function checks this and will throw an error if the event name does not exist as a place in the simulation. This format is used in MGDrivE2 for consistency with solvers in deSolve.

# releases
r_times <- seq(from = 20, length.out = 5, by = 10)
r_size <- 50
events <- data.frame("var" = paste0("F_", cube$releaseType, "_", cube$wildType, "_S"),
                     "time" = r_times,
                     "value" = r_size,
                     "method" = "add",
                     stringsAsFactors = FALSE)

# run deterministic simulation
ODE_out <- sim_trajectory_R(x0 = initialCons$M0, tmax = tmax, dt = dt, S = S,
                            hazards = approx_hazards, sampler = "ode", method = "lsoda",
                            events = events, verbose = FALSE)

# summarize females/males by genotype
ODE_female <- summarize_females_epi(out = ODE_out$state, spn_P = SPN_P)
ODE_male <- summarize_males(out = ODE_out$state)

# add sex for plotting
ODE_female$sex <- "Female"
ODE_male$sex <- "Male"
ODE_male$inf <- "S"

# plot
ggplot(data = rbind(ODE_female, ODE_male)) +
  geom_line(aes(x = time, y = value, color = genotype)) +
  facet_grid(inf ~ sex, scales = "fixed") +
  theme_bw() +
  ggtitle("SPN: ODE solution - Mosquitoes")

# summarize females/humans by genotype
ODE_female <- summarize_females_epi(out = ODE_out$state, spn_P = SPN_P)
ODE_humans <- summarize_humans_epiSIS(out = ODE_out$state)

# add species for plotting
ODE_female$species <- "Mosquitoes"
ODE_humans$species <- "Humans"

# plot
ggplot(data = rbind(ODE_female,ODE_humans) ) +
  geom_line(aes(x = time, y = value, color = genotype, linetype = inf)) +
  facet_wrap(. ~ species, scales = "free_y") +
  theme_bw() +
  ggtitle("SPN: ODE solution")

# delta t
dt_stoch <- 0.1

# run CLE simulation
CLE_out <- sim_trajectory_R(x0 = initialCons$M0, tmax = tmax, dt = dt,
                            dt_stoch = dt_stoch, S = S, hazards = approx_hazards,
                            sampler = "cle", events = events, verbose = FALSE)

# summarize females/humans by genotype
CLE_female <- summarize_females_epi(out = CLE_out$state, spn_P = SPN_P)
CLE_humans <- summarize_humans_epiSIS(out = CLE_out$state)

# plot
ggplot(data = rbind(CLE_female,CLE_humans) ) +
  geom_line(aes(x = time, y = value, color = inf)) +
  facet_wrap(~ genotype, scales = "free_y") +
  theme_bw() +
  ggtitle("SPN: CLE Approximation")