# 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

### Preface

In this vignette, we show how to set up and run MGDrivE2 simulations that utilize the SEI (Susceptible-Exposed-Infectious) mosquito model of epidemiological dynamics coupled to the SIS (Susceptible-Infectious-Susceptible) human model of epidemiological dynamics. The SEI mosquito model uses an Erlang-distributed incubation (E) period to approximate general non-exponential dwell times in the E state. The SIS human model on the other hand, is a deliberate simplifying caricature of infection dynamics in the human population; despite this, it is widely used and provides a useful base case due to analytically tractable dynamics as well as a foundation from which more complex models can be built. This vignette shows examples of transmission dynamics interacting with population genetic dynamics in a single well-mixed node.

We start by loading the MGDrivE2 package, as well as the MGDrivE package for access to inheritance cubes and ggplot2 for graphical analysis. We will use the basic cube to simulate Mendelian inheritance for this example.

    # 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).

Parameter | Description
--- | ---
`NH` | total human population size
`X` | equilibrium prevalence of disease in humans
`f` | mosquito feeding rate
`Q` | proportion of blood meals taken on humans (human blood index in field literature)
`b` | mosquito to human transmission efficiency
`c` | human to mosquito transmission efficiency
`r` | rate of recovery in humans
`muH` | mortality rate in humans
`qEIP` | inverse of mean duration of EIP
`nEIP` | shape parameter of Erlang-distributed EIP

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)

### Deterministic: ODE Solutions

As in the “MGDrivE2: One Node Lifecycle Dynamics” vignette, we will first numerically simulate the mean-field ODE approximation to the stochastic trajectory, using the approximate hazards suitable for continuous-state approximation (see `?spn_hazards()`). Internally, MGDrivE2 uses the high quality numerical solvers in from `deSolve` to integrate a mean-field approximation to the stochastic model. We also plot the adult mosquito and human dynamics to see the impact on disease incidence in humans as our released allele spreads through the mosquitoes.

    # 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")

The most obvious point of these plots is that males never exhibit latent or infectious stages. As males do not bite humans, and vertical transmission of most diseases is exceedingly rare or unknown, male mosquitoes do not participate in epidemiological dynamics. Looking at the females, we see our releases, starting at day 50, in the susceptible compartment. Notice that it takes a very low proportion of latent/infected female mosquitoes to contribute to disease incidence in humans. However, it is clear that disease incidence is reduced after releases of modified, and thus non-transmitting, mosquitoes.

Further, we can observe the disease dynamics in female mosquitoes and compare that to what is happening in the human population.

    # 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")

We see the initial equilibrium of SEI mosquitoes and SIS humans. Then, as releases are performed, we see disease incidence fall quickly in mosquitoes and that slowly impact incidence in humans.

### Stochastic: Chemical Langevin Equation Solutions

As a further example, we run a single stochastic realization of the same simulation, using the `cle` sampler with \(\Delta t = 0.1\), approximating 10 jumps per day. As the adult male mosquitoes do not contribute to infection dynamics, we will only view the adult female mosquito and human dynamics here.

    # 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")

Analyzing one stochastic realization of this system, we see some similarities and some striking differences. The releases are clearly visible, lower left-hand plot, and we see that the initial dynamics are similar to the ODE dynamics. However, it is quickly apparent that the releases are not reducing transmission adequately, that in fact, disease incidence is increasing rapidly in human and female mosquitoes. There are two main possibilities for this: first, that the stochastic simulation just happens to drift like this, a visual reminder that there can be significant differences when the well-mixed, mean-field assumptions are relaxed, or that the step size (\(\Delta t\)) is too large, and the stochastic simulation is a poor approximation of the ODE solution. Further tests, with \(\Delta t = 0.05\) and \(\Delta t = 0.15\), returned similar results, indicating that this is an accurate approximation but still highlighting the importance of testing several values of \(\Delta t\) for consistency.

### References

- Smith, D. L., & McKenzie, F. E. (2004). Statics and dynamics of malaria infection in Anopheles mosquitoes. Malaria journal, 3(1), 13.