Measuring and visualizing the effective reproduction number for various epidemics

When infection is spreading through a population, it is often convenient to work with the effective
reproduction number R, which is defined as the actual average number of secondary cases per primary case. The value
of R reflects the impact of control measures and depletion of susceptible persons during the epidemic. If R exceeds
1, the number of cases will inevitably increase over time, and a large epidemic is possible. To stop an epidemic, R
needs to be persistently reduced to a level below 1 .

We can visualize the epidemic curve for
in
and its respective reproduction number based on
as follows:

It is possible to obtain likelihood-based estimates of R. The relative likelihood p_{ij} that
case i has been infected by case j, given their difference in time of symptom
onset t_i-t_j, can be expressed in terms of the probability distribution for the generation
interval. This distribution for the generation interval is available for many infectious diseases, and we denote it
by w(\tau). The relative likelihood that case i has been infected by case
j is then the likelihood that case i has been infected by case j
, normalized by the likelihood that case i has been
infected by any other case k:
\Large p_{ij}= \frac{w(t_i-t_j)}{\displaystyle\sum_{{i} \ne {k}} w(t_i-t_k)}

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