fastbeta: Estimate a Time-Varying Infectious Disease Transmission Rate

In davidearn/fastbeta: Fast Estimation of Time-Varying Infectious Disease Transmission Rates

Estimate a Time-Varying Infectious Disease Transmission Rate

Description

Generates a discrete time estimate of a transmission rate \beta(t) from an equally spaced incidence time series and other data.

Usage

fastbeta(series, constants, ...)

Arguments

series	a “multiple time series” object, inheriting from class mts, with three columns storing (“parallel”, equally spaced) time series of incidence, births, and the per capita natural mortality rate, in that order.
constants	a numeric vector of the form c(S0, I0, R0, gamma, delta), specifying an initial state and rates of removal and loss of immunity, in that order.
...	optional arguments passed to deconvolve, if the first column of series represents reported incidence or mortality rather than incidence.

Details

fastbeta works by discretizing the system of ordinary differential equations

\begin{aligned} \frac{\text{d} S}{\text{d} t} &= \nu(t) - \beta(t) S I + \delta R - \mu(t) S \\ \frac{\text{d} I}{\text{d} t} &= \beta(t) S I - \gamma I - \mu(t) I \\ \frac{\text{d} R}{\text{d} t} &= \gamma I - \delta R - \mu(t) R \end{aligned}

where t is understood to be a unitless measure of time relative to the duration of an observation interval, then computing the iteration

\begin{aligned} I_{t + 1} &= \frac{(1 - \frac{1}{2} (\gamma + \mu_{t})) I_{t} + Z_{t + 1}}{ 1 + \frac{1}{2} (\gamma + \mu_{t + 1})} \\ R_{t + 1} &= \frac{(1 - \frac{1}{2} (\delta + \mu_{t})) R_{t} + \frac{1}{2} \gamma (I_{t} + I_{t + 1})}{ 1 + \frac{1}{2} (\delta + \mu_{t + 1})} \\ S_{t + 1} &= \frac{(1 - \frac{1}{2} \mu_{t}) S_{t} - Z_{t + 1} + B_{t + 1} + \frac{1}{2} \delta (R_{t} + R_{t + 1})}{ 1 + \frac{1}{2} \mu_{t + 1}} \\ \beta_{t} &= \frac{Z_{t} + Z_{t + 1}}{2 S_{t} I_{t}} \end{aligned}

where

\begin{aligned} X_{t} &\sim X(t) \quad [X = \beta, \mu, S, I, R] \\ Z_{t} &\sim \int_{t - 1}^{t} \beta(s) S(s) I(s) \, \text{d} s \\ B_{t} &\sim \int_{t - 1}^{t} \nu(s) \, \text{d} s \end{aligned}

and it is understood that indexing starts at t = 0. Z_{t}, B_{t}, and \mu_{t} denote incidence and births between times t - 1 and t and the per capita natural mortality rate at time t; they are supplied together as argument series.

Value

A “multiple time series” object, inheriting from class mts, with four columns (named S, I, R, and beta) storing the result of the iteration described in ‘Details’. It is completely parallel to argument series, having the same tsp attribute.

References

Jagan, M., deJonge, M. S., Krylova, O., & Earn, D. J. D. (2020). Fast estimation of time-varying infectious disease transmission rates. PLOS Computational Biology, 16(9), Article e1008124, 1-39. \Sexpr[results=rd]{tools:::Rd_expr_doi("10.1371/journal.pcbi.1008124")}

Examples

data(sir.e02, package = "fastbeta")
a <- attributes(sir.e02)
str(sir.e02)
plot(sir.e02)

## We suppose that we have perfect knowledge of incidence,
## births, and the data-generating parameters
series <- cbind(sir.e02[, c("Z", "B")], mu = a[["mu"]](0))
colnames(series) <- c("Z", "B", "mu") # FIXME: stats:::cbind.ts mangles dimnames
constants <- c(S0 = sir.e02[[1L, "S"]],
               I0 = sir.e02[[1L, "I"]],
               R0 = sir.e02[[1L, "R"]],
               gamma = a[["gamma"]],
               delta = a[["delta"]])

X <- fastbeta(series, constants)
str(X)
plot(X)

plot(X[, "beta"], ylab = "Transmission rate")
lines(a[["beta"]](time(X)), col = "red") # the "truth"

davidearn/fastbeta documentation built on April 9, 2024, 10:30 a.m.