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library(bbmle)
library(deSolve)
data1 = c(106, 121, 136, 156, 183, 248, 279, 276, 239, 185, 160, 187, 150, 193, 153, 153, 138, 143, 119)
t1 = c(0, 1, 2, 3, 4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17, 18)
base1 = c(105.1240, 106.3563, 107.5804, 108.7790, 109.9351, 111.0325, 112.0557, 112.9902, 113.8230, 114.5424, 115.1383, 115.6027, 115.9291, 116.1133, 116.1530, 116.0481, 115.8005, 115.4142, 114.8952)
model_SIR = function(t, theta) {
SIR = function(Time, State, Pars) { with(as.list(c(State, Pars)), { DecreaseS = -1 * beta * S * I / N DecreaseI = -1 * alpha * I dS = DecreaseS dI = -1 * DecreaseS + DecreaseI return(list(c(dS, dI))) }) } with(as.list(theta), { state1 = c(S = N - I1, I = I1) pars1 = c(beta = beta1, alpha = alpha1, N = N) state2 = c(S = N - I2, I = I2) pars2 = c(beta = beta2, alpha = alpha1, N = N) out1 = ode(y = state1, times = t, func = SIR, parms = pars1, method = 'ode45') out2 = ode(y = state2, times = t, func = SIR, parms = pars2, method = 'ode45') return(cbind(-diff(out1[,"S"]), -diff(out2[,"S"]))) })
}
loglik_negbin = function(theta, data, mean) {
# for negative bimonial, mean = r p /(1-p) r = theta[["r"]]; # if r < 0, return a tiny likelihood if (r < 0) return(-1e10) sum(dnbinom(x=data, size=r, mu=mean, log=TRUE))
}
loglik_poisson = function(theta, data, mean) {
sum(dpois(x=data, lambda=mean, log=TRUE))
}
SIR_fit_given_logL = function(times, data, base, theta0, model, logL) { # construct a general negative log-likelyhood function
L <- function() { # reconstruct the vector of named pairs of # parameters from the list of arguments l=length(theta0) theta=c() vars=names(theta0) for (i in 1:l) { item=c(x=get(vars[i])) names(item) <- vars[i] theta <-c(theta, item) } l = model(times, theta) #"base" is the vector of baseline values mean = l[,1] + l[,2] + base # compute the negative log-likelihood L = -logL(theta, data, mean) # if not a number, return a very small likelihood if (is.na(L)) return(1e10) return(L) }
# replace the input arguments of L by the list of parameters formals(L)←as.list(theta0) # mle2 # sometimes confint may find a better solution. # In this case we need to refit the model with the better # solution as a starting paoint, because the parameter # names in the returned better fit can be wrong #fit = mle2(L, method = “BFGS”, start = as.list(theta0), control = list(maxit = 1e6*length(theta0))) fit = mle2(L, method = “Nelder-Mead”, start = as.list(theta0), control = list(maxit = 1e6*length(theta0), ndeps = 1e-4, reltol = 1e-10)) # if there is only one parameter to be fitted # the returned confidence interval is a vector # change it to a matrix return(fit) }
SIR_fit = function(times, data, base, theta0, model) {
# we start with negative binomial # if theta0 contains no r, if (is.na(theta0['r'])) theta0 = c(theta0, r = 10000) while (TRUE) { no_r = is.na(theta0['r']); if (!no_r) { p = SIR_fit_given_logL(times, data, base, theta0, model, loglik_negbin) print(p) } # if the fitted shape parameter r is too large (> 1000) # then it is approximately poisson # in this case we use poisson r1 = coef(p)["r"] if (no_r || r1[["r"]] > 1000) { # we need to remove r from theta if (!no_r) { old_theta0 = coef(p) theta0 = c() for (i in 1:length(old_theta0)) if (names(old_theta0[i]) != 'r') theta0 = c(theta0, old_theta0[i]) } p = SIR_fit_given_logL(times, data, base, theta0, model, loglik_poisson) print(p) } conf = confint(p) if(is.vector(conf)) { conf = matrix(conf, nrow = 1) } if(is.matrix(conf)) { break } theta0 = coef(conf) names(theta0) = names(coef(p)) print(theta0) if (is.na(theta0['r'])) theta0 = c(theta0, r = 10000) } return(list(fit=p, conf = conf, r=coef(p)["r"]))
}
c = SIR_fit(c(0:19), data1, base1, c(beta1 = 0.3, alpha1 = 0.2, N = 1377, I1 = 1, beta2 = 0.3, I2 = 1), model_SIR)
print©
aicc = AICc(c"fit", nobs = length(data1))
print(aicc)