## Estimation of original replica figures

During the utilizing, we remember methods for calculating the a€?initiala€? copy number, for example. at the start of an outbreak, and also for estimating the a€?time-dependenta€? reproduction quantity whenever you want during an outbreak, also the requisite hypotheses for any means. Proposed extensions and selection implemented into the program will also be introduced.

## Attack rate (AR)

In traditional SIR model of infection indication, the fight rates (AR : the amount in the populace sooner or later infected) is related to the standard replica quantity , by roentgen 0 probeer dit = a?’ wood 1 a?’ AR S 0 AR a?’ 1 a?’ S 0 in which S _{0} could be the initial percentage of susceptible inhabitants. The necessary presumptions were homogeneous mixing, sealed inhabitants, no intervention throughout the episode.

## Great gains (EG)

As described by Wallinga & Lipsitch , the great growth rate through the early state of an episode can be for this original copy proportion. The exponential growth rate, denoted by roentgen, are defined because of the a capita change in few latest situation per unit period. As chance data include integer respected, Poisson regression is suggested to approximate this parameter [6, 10], in the place of linear regression associated with logged frequency. The reproduction numbers try computed as roentgen = 1 M a?’ roentgen where M may be the second creating purpose of the (discretized) generation energy distribution. It is necessary to decide on an interval when you look at the epidemic bend over which progress are great. We propose to make use of the deviance oriented R-squared statistic to steer this preference. No presumption is created on mixing inside the populace.

## Max possibility opinion (ML)

This design, suggested by light & Pagano , relies on the presumption that the many secondary matters caused by a directory situation was Poisson marketed with expected price R. offered observance of (N _{0,} letter _{1}, …, letter _{T} ) experience cases over successive opportunity models, and a generation opportunity submission w, roentgen try projected by making the most of the log-likelihood LL roentgen = a?‘ t = 1 T wood e a?’ I? t I? t N t N t ! in which I? _{t} = roentgen a?‘ _{i = 1} t N _{t a?’ i} w _{i} . Right here once more, the chance need to be computed on a period of great increases, as well as the deviance R-squared assess enable you to find the most useful course. No expectation is manufactured on blending in the inhabitants.

The method thinks the epidemic contour is actually analysed from first circumstances on. Should this be incorrect, the initial replica wide variety are overestimated, as supplementary cases is going to be assigned to too little directory situations: we applied a correction as defined in further document 1: Supplementary content S1. It’s also feasible to account fully for importation of situations throughout the epidemic.

## Sequential bayesian means (SB)

This method, although launched as a€?real-time bayesiana€? by their writers, most precisely enables sequential opinion associated with the initial replica number. It utilizes an approximation to your SIR design, wherein chance at energy t + 1, N(t + 1) is roughly Poisson distributed with mean N(t)e (I?(R a?’ 1)) , in which 1 I? the average time of the transmittable stage. The suggested algorithm, expressed in a Bayesian framework, begins with a non-informative prior from the circulation of the copy numbers R. The distribution was up-to-date as new data is noticed, using P R | N 0 , … , N t + 1 = P N t + 1 | R , N 0 , … , letter t P roentgen | N 0 , … , letter t p-n 0 , … , letter t + 1 ) This means that, the prior distribution for roentgen utilized on each new day could be the rear circulation from the past day. At every opportunity, the function with the posterior might be computed combined with the highest chances occurrence period. As prior to, the technique requires that the crisis is within a period of exponential progress, i.e. it does not be the cause of prone depletion; they implicitly utilizes an exponential submission for all the generation energy; and assumes random mixing into the inhabitants.