Rt Estimation¶

The epimodels.rt module estimates the time-varying reproduction number \(R_t\) directly from incidence data, without assuming an underlying model. It implements the sliding-window approach of Cori et al. (2013) — the method behind EpiEstim — with a gamma-distributed serial interval and a gamma prior/posterior on \(R_t\).

Basic usage¶

from epimodels.rt import estimate_rt

# incidence: daily case counts (regular spacing)
result = estimate_rt(
    incidence,
    window=7,        # sliding window length (time steps)
    si_mean=4.0,     # serial interval mean (time steps)
    si_sd=2.0,       # serial interval standard deviation
)

result.rt_mean       # posterior mean Rt at each window end
result.rt_low        # 95% credible interval
result.rt_high
result.times         # window end times

result.plot()        # Rt curve with credible band and Rt = 1 line

Interpretation¶

\(R_t > 1\): the epidemic is growing at that time;
\(R_t < 1\): it is declining;
When a window contains (almost) no cases, the posterior falls back to the prior (as in EpiEstim) — treat those windows as uninformative.

Tuning¶

Parameter	Effect
`window`	Larger windows smooth Rt but lag changes
`si_mean` / `si_sd`	Serial interval assumptions (in time steps)
`prior_mean`/`prior_sd`	Gamma prior on Rt (default mean 5, sd 5)
`level`	Credible interval width (default 0.95)
`times`	Real timestamps for reporting (e.g. dates)

Validation against a simulated epidemic¶

import numpy as np
from epimodels.continuous import SIR
from epimodels.rt import estimate_rt

model = SIR()
t_eval = np.arange(0, 60)
model([989, 10, 0], [0, 60], 1000, {"beta": 0.9, "gamma": 0.3}, t_eval=t_eval)

incidence = np.maximum(-np.diff(model.traces["S"]), 0)  # daily new cases
result = estimate_rt(incidence, window=7, si_mean=3.0, si_sd=1.5)

# Early Rt should be near R0 = beta/gamma = 3, declining below 1 after
# the susceptible pool is depleted.
result.plot()

Reference:: Cori, A. et al. (2013). A new framework and software to estimate time-varying reproduction numbers during epidemics. American Journal of Epidemiology, 178(9), 1505-1512.