fastbeta: Estimate a Time-Varying Infectious Disease Transmission Rate

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

Estimate a Time-Varying Infectious Disease Transmission Rate

Description

Generates a discrete approximation of a time-varying infectious disease transmission rate from an equally spaced disease incidence time series and other data.

Usage

fastbeta(series, sigma = 1, gamma = 1, delta = 0,
         m = 1L, n = 1L, init, ...)

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.
sigma, gamma, delta	non-negative numbers. m*sigma, n*gamma, and delta are the rates of removal from each latent, infectious, and recovered compartment.
m	a non-negative integer indicating a number of latent stages.
n	a positive integer indicating a number of infectious stages.
init	a numeric vector of length 1+m+n+1 giving an initial state with compartments ordered as (S, E, I, R).
...	optional arguments passed to deconvolve, if the first column of series represents observed incidence rather than actual or estimated incidence.

Details

The algorithm implemented by fastbeta is based on an SEIR model with

• m latent stages (E^{i}, i = 1,\ldots,m);

• n infectious stages (I^{j}, j = 1,\ldots,n);

• time-varying rates \beta, \nu, and \mu of transmission, birth, and natural death; and

• constant rates m \sigma, n \gamma, and \delta of removal from each latent, infectious, and recovered compartment, where removal from the recovered compartment implies return to the susceptible compartment (loss of immunity).

It is derived by linearizing of the system of ordinary differential equations

\begin{alignedat}{10} \text{d} & S &{} / \text{d} t &{} = {}& \delta &R &{} - ( && \lambda(t) &{} + \mu(t)) S &{} + \nu(t) \\ \text{d} & E^{ 1} &{} / \text{d} t &{} = {}& \lambda(t) &S &{} - ( && m \sigma &{} + \mu(t)) E^{ 1} &{} \\ \text{d} & E^{i + 1} &{} / \text{d} t &{} = {}& m \sigma &E^{i} &{} - ( && m \sigma &{} + \mu(t)) E^{i + 1} &{} \\ \text{d} & I^{ 1} &{} / \text{d} t &{} = {}& m \sigma &E^{m} &{} - ( && n \gamma &{} + \mu(t)) I^{ 1} &{} \\ \text{d} & I^{j + 1} &{} / \text{d} t &{} = {}& n \gamma &I^{j} &{} - ( && n \gamma &{} + \mu(t)) I^{j + 1} &{} \\ \text{d} & R &{} / \text{d} t &{} = {}& n \gamma &I^{n} &{} - ( && \delta &{} + \mu(t)) R &{} \end{alignedat} \\ \lambda(t) = \beta(t) \sum_{j} I^{j}

and substituting actual or estimated incidence and births for definite integrals of \lambda S and \nu. This procedure yields a system of linear difference equations from which one recovers a discrete approximation of \beta:

\begin{alignedat}{17} &E_{t + 1}^{ 1} &{} = {}& [(1 - \tfrac{1}{2} ( & m \sigma + \mu_{t})) & E_{t}^{ 1} & & & & & & & &{} + Z_{t + 1} & ] &{} / [1 + \tfrac{1}{2} ( & m \sigma + \mu_{t + 1})] \\ &E_{t + 1}^{i + 1} &{} = {}& [(1 - \tfrac{1}{2} ( & m \sigma + \mu_{t})) & E_{t}^{i + 1} &{} + {}& \tfrac{1}{2} & m \sigma ( & E_{t}^{i} &{} + {}& E_{t + 1}^{i} & ) & & ] &{} / [1 + \tfrac{1}{2} ( & m \sigma + \mu_{t + 1})] \\ &I_{t + 1}^{ 1} &{} = {}& [(1 - \tfrac{1}{2} ( & n \gamma + \mu_{t})) & I_{t}^{ 1} &{} + {}& \tfrac{1}{2} & m \sigma ( & E_{t}^{m} &{} + {}& E_{t + 1}^{m} & ) & & ] &{} / [1 + \tfrac{1}{2} ( & n \gamma + \mu_{t + 1})] \\ &I_{t + 1}^{j + 1} &{} = {}& [(1 - \tfrac{1}{2} ( & n \gamma + \mu_{t})) & I_{t}^{j + 1} &{} + {}& \tfrac{1}{2} & n \gamma ( & I_{t}^{j} &{} + {}& I_{t + 1}^{j} & ) & & ] &{} / [1 + \tfrac{1}{2} ( & n \gamma + \mu_{t + 1})] \\ &R_{t + 1} &{} = {}& [(1 - \tfrac{1}{2} ( & \delta + \mu_{t})) & R_{t} &{} + {}& \tfrac{1}{2} & n \gamma ( & I_{t}^{n} &{} + {}& I_{t + 1}^{n} & ) & & ] &{} / [1 + \tfrac{1}{2} ( & \delta + \mu_{t + 1})] \\ &S_{t + 1} &{} = {}& [(1 - \tfrac{1}{2} ( & \mu_{t})) & S_{t} &{} + {}& \tfrac{1}{2} & \delta ( & R_{t} &{} + {}& R_{t + 1} & ) &{} - Z_{t + 1} &{} + B_{t + 1}] &{} / [1 + \tfrac{1}{2} ( & \mu_{t + 1})] \end{alignedat} \\ \beta_{t} = (Z_{t} + Z_{t + 1}) / (2 S_{t} \sum_{j} I_{t}^{j})

where we use the notation

X_{t} \sim X(t) : X = S, E^{i}, I^{j}, R, Z, B, \mu, \beta \\ \begin{aligned} Z(t) &= \int_{t - 1}^{t} \lambda(s) S(s) \, \text{d} s \\ B(t) &= \int_{t - 1}^{t} \nu(s) \, \text{d} s \end{aligned}

and it is understood that the independent variable t is a unitless measure of time relative to the spacing of the substituted time series of incidence and births.

Value

A “multiple time series” object, inheriting from class mts, with 1+m+n+1+1 columns (named S, E, 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

if (requireNamespace("adaptivetau")) withAutoprint({

data(seir.ts02, package = "fastbeta")
a <- attributes(seir.ts02)
str(seir.ts02)
plot(seir.ts02)

## We suppose that we have perfect knowledge of incidence,
## births, and the data-generating parameters
series <- cbind.ts(seir.ts02[, c("Z", "B")], mu = a[["mu"]](0))

args <- c(list(series = series),
          a[c("sigma", "gamma", "delta", "m", "n", "init")])
str(args)

X <- do.call(fastbeta, args)
str(X)
plot(X)

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

})

davidearn/fastbeta documentation built on July 4, 2025, 6:28 p.m.