#' Fitting 2 simple viral infection models to influenza data
#'
#' @description This function runs a simulation of a compartment model
#' using a set of ordinary differential equations.
#' The model describes a simple viral infection system in the presence of drug treatment.
#' The user provides initial conditions and parameter values for the system.
#' The function simulates the ODE using an ODE solver from the deSolve package.
#' The function returns a matrix containing time-series of each variable and time.
#'
#' @param U : initial number of uninfected target cells : numeric
#' @param I : initial number of infected target cells : numeric
#' @param V : initial number of infectious virions : numeric
#' @param X : initial level of immune response : numeric
#' @param dI : rate at which infected cells die : numeric
#' @param dV : rate at which infectious virus is cleared : numeric
#' @param p : rate at which infected cells produce virus : numeric
#' @param g : unit conversion factor : numeric
#' @param k : rate of killing of infected cells by T-cells (model 1) or virus by Ab (model 2) : numeric
#' @param a : activation of T-cells (model 1) or growth of antibodies (model 2) : numeric
#' @param alow : lower bound for activation rate : numeric
#' @param ahigh : upper bound for activation rate : numeric
#' @param b : rate at which virus infects cells : numeric
#' @param blow : lower bound for infection rate : numeric
#' @param bhigh : upper bound for infection rate : numeric
#' @param r : rate of T-cell expansion (model 1) : numeric
#' @param rlow : lower bound for expansion rate : numeric
#' @param rhigh : upper bound for expansion rate : numeric
#' @param dX : rate at which antibodies decay (model 2) : numeric
#' @param dXlow : lower bound for decay rate : numeric
#' @param dXhigh : upper bound for decay rate : numeric
#' @param fitmodel : fitting model 1 or 2 : numeric
#' @param iter : max number of steps to be taken by optimizer : numeric
#' @return The function returns a list containing the best fit timeseries,
#' the best fit parameters, the data and the AICc for the model.
#' @details Two simple compartmental ODE models mimicking acute viral infection
#' with T-cells (model 1) or antibodies (model 2) are fitted to data.
#' @section Warning: This function does not perform any error checking. So if
#' you try to do something nonsensical (e.g. specify negative parameter or starting values),
#' the code will likely abort with an error message.
#' @examples
#' # To run the code with default parameters just call the function:
#' \dontrun{result <- simulate_modelcomparison_fit()}
#' # To apply different settings, provide them to the simulator function, like such:
#' result <- simulate_modelcomparison_fit(iter = 5, fitmodel = 1)
#' @seealso See the Shiny app documentation corresponding to this
#' function for more details on this model.
#' @author Andreas Handel
#' @importFrom utils read.csv
#' @importFrom dplyr filter rename select
#' @importFrom nloptr nloptr
#' @export
simulate_modelcomparison_fit <- function(U = 1e6, I = 0, V = 1, X = 1,
dI = 2, dV = 4, p = 0.1, g = 0, k = 0.001,
a = 0.001, alow = 1e-5, ahigh = 10,
b = 0.001, blow = 1e-6, bhigh = 10,
r = 0.1, rlow = 1e-3, rhigh = 10,
dX = 1, dXlow = 0.1, dXhigh = 10, fitmodel = 1, iter = 1)
{
##all sub-functions are specified first
#########################################
#ode equations for model 1
model1ode <- function(t, y, parms)
{
with(
as.list(c(y,parms)), #lets us access variables and parameters stored in y and parms by name
{
dUdt = -b*V*U
dIdt = b*V*U - dI*I - k*X*I
dVdt = p*I - dV*V - g*b*V*U
dXdt = a*V + r*X
list(c(dUdt, dIdt, dVdt,dXdt))
}
) #close with statement
} #end function specifying the ODEs
#########################################
#ode equations for model 2
model2ode <- function(t, y, parms)
{
with(
as.list(c(y,parms)), #lets us access variables and parameters stored in y and parms by name
{
dUdt = -b*V*U
dIdt = b*V*U - dI*I
dVdt = p*I - dV*V - k*X*V - g*b*V*U
dXdt = a*V*X - dX*X
list(c(dUdt, dIdt, dVdt, dXdt))
}
) #close with statement
} #end function specifying the ODEs
###################################################################
#function that fits the ODE model to data
###################################################################
modelcompfitfunction <- function(params, fitdata, Y0, xvals, fitmodel, fixedpars, fitparnames, LOD)
{
names(params) = fitparnames #for some reason nloptr strips names from parameters
modelpars = c(params,fixedpars)
#call ode-solver lsoda to integrate ODEs
if (fitmodel == 1)
{
odeout <- try(deSolve::ode(y = Y0, times = xvals, func = model1ode, parms=modelpars, atol=1e-8, rtol=1e-8));
}
if (fitmodel == 2)
{
odeout <- try(deSolve::ode(y = Y0, times = xvals, func = model2ode, parms=modelpars, atol=1e-8, rtol=1e-8));
}
#extract values for virus load at time points where data is available
modelpred = odeout[match(fitdata$xvals,odeout[,"time"]),"V"];
#since the ODE returns values on the original scale, we need to transform it into log10 units for the fitting procedure
#due to numerical issues in the ODE model, virus might become negative, leading to problems when log-transforming.
#Therefore, we enforce a minimum value of 1e-10 for virus load before log-transforming
logvirus=c(log10(pmax(1e-10,modelpred)));
#since the data is censored,
#set model prediction to LOD if it is below LOD
#this means we do not penalize model predictions below LOD
logvirus[(fitdata$outcome<=LOD & (fitdata$outcome-logvirus)>0)] = LOD
#return the objective function, the sum of squares,
#which is being minimized by the optimizer
return(sum((logvirus-fitdata$outcome)^2))
} #end function that fits the ODE model to the data
############################################################
#the main function, which calls the fit function
############################################################
#some settings for ode solver and optimizer
#those are hardcoded here, could in principle be rewritten to allow user to pass it into function
atolv=1e-8; rtolv=1e-8; #accuracy settings for the ODE solver routine
maxsteps = iter #number of steps/iterations for algorithm
#load data
#This data is from Hayden et al 1996 JAMA
#We only use the data for the no-drug condition here
LOD = hayden96flu$LOD[1] #limit of detection, log scale
fitdata = subset(hayden96flu, txtime == 200, select=c("HoursPI", "LogVirusLoad")) #only fit some of the data
colnames(fitdata) = c("xvals",'outcome')
#convert to days
fitdata$xvals = fitdata$xvals / 24
Y0 = c(U = U, I = I, V = V, X = X); #combine initial conditions into a vector
xvals = seq(0, max(fitdata$xvals), 0.1); #vector of times for which solution is returned (not that internal timestep of the integrator is different)
#combining fixed parameters into a parameter vector
fixedpars = c(dI=dI,dV=dV,p=p,k=k, g=g);
if (fitmodel == 1)
{
par_ini = as.numeric(c(a=a, r=r, b=b))
lb = as.numeric(c(alow, rlow, blow))
ub = as.numeric(c(ahigh, rhigh, bhigh))
fitparnames = c('a','r','b')
}
if (fitmodel == 2)
{
par_ini = as.numeric(c(a=a, dX=dX, b=b))
lb = as.numeric(c(alow, dXlow, blow))
ub = as.numeric(c(ahigh, dXhigh, bhigh))
fitparnames = c('a','dX','b')
}
#this line runs the simulation, i.e. integrates the differential equations describing the infection process
#the result is saved in the odeoutput matrix, with the 1st column the time, all other column the model variables
#in the order they are passed into Y0 (which needs to agree with the order in virusode)
bestfit = nloptr::nloptr(x0=par_ini, eval_f=modelcompfitfunction,lb=lb,ub=ub,opts=list("algorithm"="NLOPT_LN_NELDERMEAD",xtol_rel=1e-10,maxeval=maxsteps,print_level=0), fitdata=fitdata, Y0 = Y0, xvals = xvals, fitmodel=fitmodel, fixedpars=fixedpars,fitparnames=fitparnames,LOD=LOD)
#extract best fit parameter values and from the result returned by the optimizer
params = bestfit$solution
names(params) = fitparnames #for some reason nloptr strips names from parameters
modelpars = c(params,fixedpars)
#time-series for best fit model
if (fitmodel == 1)
{
odeout <- try(deSolve::ode(y = Y0, times = xvals, func = model1ode, parms=modelpars, atol=1e-8, rtol=1e-8));
}
if (fitmodel == 2)
{
odeout <- try(deSolve::ode(y = Y0, times = xvals, func = model2ode, parms=modelpars, atol=1e-8, rtol=1e-8));
}
#compute SSR for final fit. See comments inside of fitting function for explanations.
modelpred = odeout[match(fitdata$xvals,odeout[,"time"]),"V"];
logvirus=c(log10(pmax(1e-10,modelpred)));
logvirus[(fitdata$outcome<=LOD & (fitdata$outcome-logvirus)>0)] = LOD
ssrfinal=(sum((logvirus-fitdata$outcome)^2))
#compute AICc
N=length(fitdata$outcome) #number of datapoints
K=length(par_ini); #fitted parameters for model
AICc= N * log(ssrfinal/N) + 2*(K+1)+(2*(K+1)*(K+2))/(N-K)
#list structure that contains all output
result = list()
result$ts = odeout
result$bestpars = params
result$AICc = AICc
result$SSR = ssrfinal
#return the data not on a log scale for consistency
fitdata$outcome = 10^fitdata$outcome
fitdata$varnames = 'V_data'
colnames(fitdata) = c("xvals",'yvals','varnames')
result$data = fitdata
#The output produced by the fitting routine
return(result)
}
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