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STS506 L. Leticia R. et al.
Using the (forward) Chapman-Kolmogorov equation we have the
(Chemical) Master equation:

(,) () = ( + 1)( − 1) (+1,−1) + ( + 1) − [ + ] (1)
(,+1) (,)

The reproductive number, R corresponds to the expected total of new
cases originated by a typical infected case. Based on model (1) it is equal
to /. This number is paramount for determining if the outbreak will
infect few individuals or it if can develop into an epidemic.
Understanding the transmission process and its parameters can help
us to identify outbreak interventions that can reduce R below the
threshold value. In this work, we assume the SIR compartmental model
and evolving in a network of contacts that is fully known. The objective
is estimating the infectious agent SIR parameters based on surveillance-
alike information. That is, we only observe the time–aggregated counts
of new cases entering status .
We also propose the introduction of surrogate epidemic models that
can reduce the computational burden of the original ABC. This is
compared to the regular ABC using simulated data on random networks.

2. Methodology
2.1 Epidemics in Networks
In the considered SIR epidemic model in a simple network G(, ),
the infectious agent is independently transmitted between pairs of
connected individuals. This transmission event occurs from an infective
to a connected susceptible individual with rate > 0. On the other
hand, infective individuals recover independently after a time that
is exponentially distributed with parameter > 0.
In this model, the reproductive number is R net = ( ), where is
1
1
the excess degree (Newman, 2002) and ( ) = ( )/() − 1 under
2
1
the configuration model, where is the degree of a randomly selected
vertex. The parameter corresponds to the probability of transmission
between a susceptible and an infective individual, throughout the
infectious period of the latter. That is
∞
= ∫ (1 − − ) (),
1
0
where (⋅) is the infectious period. Then
Rnet= ( ).
+ 1
(2)
2.2 ABC-MCMC
The ABC (Del Moral, et, al. 2012) is a class of methods that provide
an alternative to the likelihood computation. It is a rejection sampler for

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