**EpiLPS** (Gressani et al. 2022) is the acronym
for **Epi**demiological modeling (tool) with
**L**aplacian-**P**-**S**plines.
It proposes a new Bayesian methodology for estimating the instantaneous
reproduction number \(\mathcal{R}_t\),
i.e. the average number of secondary cases generated by an infectious
agent at time \(t\); a key metric for
assessing the transmission dynamics of an infectious disease and a
useful reference for guiding interventions of governmental institutions
in a public health crisis. The EpiLPS project builds upon two strong
pillars in the statistical literature, namely Bayesian P-splines and
Laplace approximations, to deliver a fast and flexible methodology for
inference on \(\mathcal{R}_t\). EpiLPS
requires two (external) inputs: (1) a time series of incidence counts
and (2) a (discrete) serial interval distribution.

The underlying model for smoothing incidence counts is based on the negative binomial distribution to account for possible overdispersion in the data. EpiLPS has a two-phase engine for estimating \(\mathcal{R}_t\). First, Bayesian P-splines are used to smooth the epidemic curve and to compute estimates of the mean incidence count of the susceptible population at each day of the outbreak. Second, in the philosophy of Fraser (2007), the renewal equation is used as a bridge to link the estimated mean incidence and the reproduction number. As such, the second phase delivers a closed-form expression of \(\mathcal{R}_t\) as a function of the B-spline coefficients and the serial interval distribution.

Another key strength of EpiLPS is that two different strategies can
be used to estimate \(\mathcal{R}_t\).
The first approach called LPSMAP is completely sampling-free and fully
relies on Laplace approximations and maximum *a posteriori* (MAP)
computation of model hyperparameters for estimation. Routines for
Laplace approximations and B-splines evaluations are typically the ones
that are computationally most intensive and are therefore coded in C++
and integrated in R via the **Rcpp** package, making them
time irrelevant. The second approach is called LPSMALA
(Laplacian-P-splines with a Metropolis-adjusted Langevin algorithm) and
is fully stochastic. It samples the posterior of the model by using
Langevin diffusions in a Metropolis-within-Gibbs framework. Of course,
LPSMAP has a computational advantage over LPSMALA. Thanks to the
lightning fast implementation of Laplace approximations, LPSMAP
typically delivers estimates of \(\mathcal{R}_t\) in a fraction of a
second.