The {superspreading} R package
provides a set of functions for understanding individual-level
transmission dynamics, and thus whether there is evidence of
superspreading or superspreading events (SSE). Individual-level
transmission is important for understanding the growth or decline in
cases of an infectious disease accounting for transmission
heterogeneity, this heterogeneity is not accounted for by the
population-level reproduction number (*R*).

Superspreading describes individual heterogeneity in disease transmission, such that some individuals transmit to many infectees while other infectors infect few or zero individuals (Lloyd-Smith et al. 2005).

The {epiparameter} R package is loaded to provide a library of epidemiological parameters. These parameters can be used by the {superspreading} functions. See the Empirical superspreading section of this vignette for its usage.

As an example, offspring distributions are stored in the
{epiparameter} library which contain estimated parameters, such as the
reproduction number (*R*), and
in the case of a negative binomial model, the dispersion parameter
(*k*).

The offspring distribution is the distribution of the number of infectees (secondary case or offspring) that each infector (primary case) produces.

The probability that a novel disease will cause a epidemic (i.e. sustained transmission in the population) is determined by the nature of that diseases’ transmission heterogeneity. This variability may be an intrinsic property of the disease, or a product of human behaviour and social mixing patterns.

For a given value of *R*, if
the variability is high, the probability that the outbreak will cause
epidemic is lower as the superspreading events are rare. Whereas for
lower variability the probability is higher as more individuals are
closer to the mean (*R*).

Here we use *R* to represent
the reproduction number (number of secondary cases caused by a typical
case). Depending on the situation, this may be equivalent to the basic
reproduction number (*R*_{0}, representing
transmission in a fully susceptible population) or the effective
reproduction number at a given point in time (*R*_{t}, representing
the extent of transmission at time *t*). Either can be input into the
functions provided by {superspreading}

The `probability_epidemic()`

function in {superspreading}
can calculate this probability. *k* is the dispersion parameter of a
negative binomial distribution and controls the variability of
individual-level transmission.

```
probability_epidemic(R = 1.5, k = 1, num_init_infect = 1)
#> [1] 0.3333333
probability_epidemic(R = 1.5, k = 0.5, num_init_infect = 1)
#> [1] 0.2324081
probability_epidemic(R = 1.5, k = 0.1, num_init_infect = 1)
#> [1] 0.06765766
```

In the above code, *k* values
above one represent low heterogeneity (in the case *k* → ∞ it is a poisson distribution),
and as *k* decreases,
heterogeneity increases. When *k* equals 1, the distribution is
geometric. Values of *k* less
than one indicate overdispersion of disease transmission, a signature of
superspreading.

When the value of *R*
increases, this causes the probability of an epidemic to increase, if
*k* remains the same.

```
probability_epidemic(R = 0.5, k = 1, num_init_infect = 1)
#> [1] 0
probability_epidemic(R = 1.0, k = 1, num_init_infect = 1)
#> [1] 0
probability_epidemic(R = 1.5, k = 1, num_init_infect = 1)
#> [1] 0.3333333
probability_epidemic(R = 5, k = 1, num_init_infect = 1)
#> [1] 0.8
```

Any value of *R* less than or
equal to one will have zero probability of causing a sustained
epidemic.

Finally, the probability that a new infection will cause a large
epidemic is influenced by the number of initial infections seeding the
outbreak. We define `a`

to be this number of initial
infections.

Given `probability_epidemic()`

it is possible to determine
the probability of an epidemic for diseases for which parameters of an
offspring distribution have been estimated. An offspring distribution is
simply the distribution of the number of secondary infections caused by
a primary infection. It is the distribution of *R*, with the mean of the distribution
given as *R*.

Here we can use {epiparameter} to load in offspring distributions for multiple diseases and evaluate how likely they are to cause epidemics.

```
sars <- epiparameter_db(
disease = "SARS",
epi_name = "offspring distribution",
single_epiparameter = TRUE
)
#> Using Lloyd-Smith J, Schreiber S, Kopp P, Getz W (2005). "Superspreading and
#> the effect of individual variation on disease emergence." _Nature_.
#> doi:10.1038/nature04153 <https://doi.org/10.1038/nature04153>..
#> To retrieve the citation use the 'get_citation' function
evd <- epiparameter_db(
disease = "Ebola Virus Disease",
epi_name = "offspring distribution",
single_epiparameter = TRUE
)
#> Using Lloyd-Smith J, Schreiber S, Kopp P, Getz W (2005). "Superspreading and
#> the effect of individual variation on disease emergence." _Nature_.
#> doi:10.1038/nature04153 <https://doi.org/10.1038/nature04153>..
#> To retrieve the citation use the 'get_citation' function
```

The parameters of each distribution can be extracted:

```
sars_params <- get_parameters(sars)
sars_params
#> mean dispersion
#> 1.63 0.16
evd_params <- get_parameters(evd)
evd_params
#> mean dispersion
#> 1.5 5.1
```

```
family(sars)
#> [1] "nbinom"
probability_epidemic(
R = sars_params[["mean"]],
k = sars_params[["dispersion"]],
num_init_infect = 1
)
#> [1] 0.1198705
family(evd)
#> [1] "nbinom"
probability_epidemic(
R = evd_params[["mean"]],
k = evd_params[["dispersion"]],
num_init_infect = 1
)
#> [1] 0.5092324
```

In the above example we assume the initial pool of infectors is one
(`num_init_infect = 1`

) but this can easily be adjusted in
the case there is evidence for a larger initial seeding of infections,
whether from animal-to-human spillover or imported cases from outside
the area of interest.

We can see that the probability of an epidemic given the estimates of
Lloyd-Smith et al. (2005)
is greater for Ebola than SARS. This is due to the offspring
distribution of Ebola having a larger dispersion (dispersion *k* = 5.1), compared to SARS, which
has a relatively small dispersion (*k* = 0.16).

Lloyd-Smith, J. O., S. J. Schreiber, P. E. Kopp, and W. M. Getz. 2005.
“Superspreading and the Effect of Individual Variation on Disease
Emergence.” *Nature* 438 (7066): 355–59. https://doi.org/10.1038/nature04153.