- 1 Set up
- 2 Data
- 3 Parameters
- 4 Running the model
- 4.1 Default options
- 4.2 Reducing the accuracy of the approximate Gaussian Process
- 4.3 Adjusting for future susceptible depletion
- 4.4 Adjusting for truncation of the most recent data
- 4.5 Projecting the reproduction number with the Gaussian Process
- 4.6 Fixed reproduction number
- 4.7 Breakpoints
- 4.8 Weekly random walk
- 4.9 No delays
- 4.10 Non-parametric infection model

The `estimate_infections()`

function encodes a range of different model options. In this vignette we apply some of these to the example data provided with the *EpiNow2* package, highlighting differences in inference results and run times. It is not meant as a comprehensive exploration of all the functionality in the package, but intended to give users a flavour of the kind of model options that exist for reproduction number estimation and forecasting within the package, and the differences in computational speed between them. For mathematical detail on the model please consult the model definition vignette, and for a more general description of the use of the function, the estimate_infections workflow vignette.

We first load the *EpiNow2* package and also the *rstan* package that we will use later to show the differences in run times between different model options.

```
library("EpiNow2")
library("rstan")
#> Loading required package: StanHeaders
#>
#> rstan version 2.32.6 (Stan version 2.32.2)
#> For execution on a local, multicore CPU with excess RAM we recommend calling
#> options(mc.cores = parallel::detectCores()).
#> To avoid recompilation of unchanged Stan programs, we recommend calling
#> rstan_options(auto_write = TRUE)
#> For within-chain threading using `reduce_sum()` or `map_rect()` Stan functions,
#> change `threads_per_chain` option:
#> rstan_options(threads_per_chain = 1)
```

In this examples we set the number of cores to use to 4 but the optimal value here will depend on the computing resources available.

`options(mc.cores = 4)`

We will use an example data set that is included in the package, representing an outbreak of COVID-19 with an initial rapid increase followed by decreasing incidence.

```
library("ggplot2")
example_confirmed[1:60]
reported_cases <-ggplot(reported_cases, aes(x = date, y = confirm)) +
geom_col() +
theme_minimal() +
xlab("Date") +
ylab("Cases")
```

Before running the model we need to decide on some parameter values, in particular any delays, the generation time, and a prior on the initial reproduction number.

Delays reflect the time that passes between infection and reporting, if these exist. In this example, we assume two delays, an *incubation period* (i.e. delay from infection to symptom onset) and a *reporting delay* (i.e. the delay from symptom onset to being recorded as a symptomatic case). These delays are usually not the same for everyone and are instead characterised by a distribution. For the incubation period we use an example from the literature that is included in the package.

```
example_incubation_period#> - lognormal distribution (max: 14):
#> meanlog:
#> - normal distribution:
#> mean:
#> 1.6
#> sd:
#> 0.064
#> sdlog:
#> - normal distribution:
#> mean:
#> 0.42
#> sd:
#> 0.069
```

For the reporting delay, we use a lognormal distribution with mean of 2 days and standard deviation of 1 day. Note that the mean and standard deviation must be converted to the log scale, which can be done using the `convert_log_logmean()`

function.

```
LogNormal(mean = 2, sd = 1, max = 10)
reporting_delay <-
reporting_delay#> - lognormal distribution (max: 10):
#> meanlog:
#> 0.58
#> sdlog:
#> 0.47
```

*EpiNow2* provides a feature that allows us to combine these delays into one by summing them up

```
example_incubation_period + reporting_delay
delay <-
delay#> Composite distribution:
#> - lognormal distribution (max: 14):
#> meanlog:
#> - normal distribution:
#> mean:
#> 1.6
#> sd:
#> 0.064
#> sdlog:
#> - normal distribution:
#> mean:
#> 0.42
#> sd:
#> 0.069
#> - lognormal distribution (max: 10):
#> meanlog:
#> 0.58
#> sdlog:
#> 0.47
```

If we want to estimate the reproduction number we need to provide a distribution of generation times. Here again we use an example from the literature that is included with the package.

```
example_generation_time#> - gamma distribution (max: 14):
#> shape:
#> - normal distribution:
#> mean:
#> 1.4
#> sd:
#> 0.48
#> rate:
#> - normal distribution:
#> mean:
#> 0.38
#> sd:
#> 0.25
```

Lastly we need to choose a prior for the initial value of the reproduction number. This is assumed by the model to be normally distributed and we can set the mean and the standard deviation. We decide to set the mean to 2 and the standard deviation to 1.

` list(mean = 2, sd = 0.1) rt_prior <-`

We are now ready to run the model and will in the following show a number of possible options for doing so.

By default the model uses a renewal equation for infections and a Gaussian Process prior for the reproduction number. Putting all the data and parameters together and tweaking the Gaussian Process to have a shorter length scale prior than the default we run.

```
estimate_infections(reported_cases,
def <-generation_time = generation_time_opts(example_generation_time),
delays = delay_opts(delay),
rt = rt_opts(prior = rt_prior)
)#> Warning: There were 11 divergent transitions after warmup. See
#> https://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
#> to find out why this is a problem and how to eliminate them.
#> Warning: Examine the pairs() plot to diagnose sampling problems
# summarise results
summary(def)
#> measure estimate
#> <char> <char>
#> 1: New infections per day 2237 (956 -- 4859)
#> 2: Expected change in daily reports Likely decreasing
#> 3: Effective reproduction no. 0.88 (0.58 -- 1.3)
#> 4: Rate of growth -0.034 (-0.16 -- 0.085)
#> 5: Doubling/halving time (days) -21 (8.2 -- -4.4)
# elapsed time (in seconds)
get_elapsed_time(def$fit)
#> warmup sample
#> chain:1 22.546 17.871
#> chain:2 43.610 35.665
#> chain:3 21.929 18.197
#> chain:4 18.185 17.063
# summary plot
plot(def)
```

To speed up the calculation of the Gaussian Process we could decrease its accuracy, e.g. decrease the proportion of time points to use as basis functions from the default of 0.2 to 0.1.

```
estimate_infections(reported_cases,
agp <-generation_time = generation_time_opts(example_generation_time),
delays = delay_opts(delay),
rt = rt_opts(prior = rt_prior),
gp = gp_opts(basis_prop = 0.1)
)#> Warning: There were 19 divergent transitions after warmup. See
#> https://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
#> to find out why this is a problem and how to eliminate them.
#> Warning: Examine the pairs() plot to diagnose sampling problems
# summarise results
summary(agp)
#> measure estimate
#> <char> <char>
#> 1: New infections per day 2229 (1060 -- 4985)
#> 2: Expected change in daily reports Likely decreasing
#> 3: Effective reproduction no. 0.88 (0.61 -- 1.2)
#> 4: Rate of growth -0.036 (-0.15 -- 0.076)
#> 5: Doubling/halving time (days) -19 (9.1 -- -4.8)
# elapsed time (in seconds)
get_elapsed_time(agp$fit)
#> warmup sample
#> chain:1 14.264 17.067
#> chain:2 14.501 16.719
#> chain:3 17.512 24.243
#> chain:4 16.937 17.436
# summary plot
plot(agp)
```

We might want to adjust for future susceptible depletion. Here, we do so by setting the population to 1000000 and projecting the reproduction number from the latest estimate (rather than the default, which fixes the reproduction number to an earlier time point based on the given reporting delays). Note that this only affects the forecasts and is done using a crude adjustment (see the model definition).

```
estimate_infections(reported_cases,
dep <-generation_time = generation_time_opts(example_generation_time),
delays = delay_opts(delay),
rt = rt_opts(
prior = rt_prior,
pop = 1000000, future = "latest"
)
)#> Warning: There were 9 divergent transitions after warmup. See
#> https://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
#> to find out why this is a problem and how to eliminate them.
#> Warning: Examine the pairs() plot to diagnose sampling problems
# summarise results
summary(dep)
#> measure estimate
#> <char> <char>
#> 1: New infections per day 2219 (934 -- 4735)
#> 2: Expected change in daily reports Likely decreasing
#> 3: Effective reproduction no. 0.88 (0.57 -- 1.2)
#> 4: Rate of growth -0.034 (-0.16 -- 0.076)
#> 5: Doubling/halving time (days) -20 (9.1 -- -4.3)
# elapsed time (in seconds)
get_elapsed_time(dep$fit)
#> warmup sample
#> chain:1 24.086 18.080
#> chain:2 25.069 31.131
#> chain:3 24.617 25.153
#> chain:4 21.874 18.766
# summary plot
plot(dep)
```

We might further want to adjust for right-truncation of recent data estimated using the estimate_truncation model. Here, instead of doing so we assume that we know about truncation with mean of 1/2 day, sd 1/2 day, following a lognormal distribution and with a maximum of three days.

```
LogNormal(
trunc_dist <-mean = Normal(0.5, 0.1),
sd = Normal(0.5, 0.1),
max = 3
)#> Warning in new_dist_spec(params, "lognormal"): Uncertain lognormal distribution
#> specified in terms of parameters that are not the "natural" parameters of the
#> distribution (meanlog, sdlog). Converting using a crude and very approximate
#> method that is likely to produce biased results. If possible, it is preferable
#> to specify the distribution directly in terms of the natural parameters.
trunc_dist#> - lognormal distribution (max: 3):
#> meanlog:
#> - normal distribution:
#> mean:
#> -1
#> sd:
#> 0.14
#> sdlog:
#> - normal distribution:
#> mean:
#> 0.83
#> sd:
#> 0.13
```

We can then use this in the `esimtate_infections()`

function using the `truncation`

option.

```
estimate_infections(reported_cases,
trunc <-generation_time = generation_time_opts(example_generation_time),
delays = delay_opts(delay),
truncation = trunc_opts(trunc_dist),
rt = rt_opts(prior = rt_prior)
)#> Error in vapply(delays[parametric], attr, "weight_prior", FUN.VALUE = logical(1)): values must be length 1,
#> but FUN(X[[3]]) result is length 0
# summarise results
summary(trunc)
#> Error in object[[i]]: object of type 'builtin' is not subsettable
# elapsed time (in seconds)
get_elapsed_time(trunc$fit)
#> Error in (function (cond) : error in evaluating the argument 'object' in selecting a method for function 'get_elapsed_time': object of type 'builtin' is not subsettable
# summary plot
plot(trunc)
```

Instead of keeping the reproduction number fixed from a certain time point we might want to extrapolate the Gaussian Process into the future. This will lead to wider uncertainty, and the researcher should check whether this or fixing the reproduction number from an earlier is desirable.

```
estimate_infections(reported_cases,
project_rt <-generation_time = generation_time_opts(example_generation_time),
delays = delay_opts(delay),
rt = rt_opts(
prior = rt_prior, future = "project"
)
)#> Warning: There were 8 divergent transitions after warmup. See
#> https://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
#> to find out why this is a problem and how to eliminate them.
#> Warning: Examine the pairs() plot to diagnose sampling problems
# summarise results
summary(project_rt)
#> measure estimate
#> <char> <char>
#> 1: New infections per day 2200 (1010 -- 4815)
#> 2: Expected change in daily reports Likely decreasing
#> 3: Effective reproduction no. 0.87 (0.59 -- 1.2)
#> 4: Rate of growth -0.036 (-0.15 -- 0.08)
#> 5: Doubling/halving time (days) -19 (8.6 -- -4.5)
# elapsed time (in seconds)
get_elapsed_time(project_rt$fit)
#> warmup sample
#> chain:1 23.881 34.602
#> chain:2 18.612 18.302
#> chain:3 26.161 28.837
#> chain:4 21.156 29.987
# summary plot
plot(project_rt)
```

We might want to estimate a fixed reproduction number, i.e. assume that it does not change.

```
estimate_infections(reported_cases,
fixed <-generation_time = generation_time_opts(example_generation_time),
delays = delay_opts(delay),
gp = NULL
)# summarise results
summary(fixed)
#> measure estimate
#> <char> <char>
#> 1: New infections per day 16780 (9407 -- 30551)
#> 2: Expected change in daily reports Increasing
#> 3: Effective reproduction no. 1.2 (1.1 -- 1.3)
#> 4: Rate of growth 0.05 (0.035 -- 0.065)
#> 5: Doubling/halving time (days) 14 (11 -- 20)
# elapsed time (in seconds)
get_elapsed_time(fixed$fit)
#> warmup sample
#> chain:1 1.270 0.951
#> chain:2 1.109 0.804
#> chain:3 1.151 0.893
#> chain:4 1.242 0.926
# summary plot
plot(fixed)
```

Instead of assuming the reproduction number varies freely or is fixed, we can assume that it is fixed but with breakpoints. This can be done by adding a `breakpoint`

column to the reported case data set. e.g. if we think that the reproduction number was constant but would like to allow it to change on the 16th of March 2020 we would define a new case data set using

```
data.table::copy(reported_cases)
bp_cases <- bp_cases[,
bp_cases <-:= ifelse(date == as.Date("2020-03-16"), 1, 0)
breakpoint ]
```

We then use this instead of `reported_cases`

in the `estimate_infections()`

function:

```
estimate_infections(bp_cases,
bkp <-generation_time = generation_time_opts(example_generation_time),
delays = delay_opts(delay),
rt = rt_opts(prior = rt_prior),
gp = NULL
)# summarise results
summary(bkp)
#> measure estimate
#> <char> <char>
#> 1: New infections per day 2356 (1932 -- 2898)
#> 2: Expected change in daily reports Decreasing
#> 3: Effective reproduction no. 0.89 (0.86 -- 0.92)
#> 4: Rate of growth -0.027 (-0.034 -- -0.02)
#> 5: Doubling/halving time (days) -25 (-35 -- -20)
# elapsed time (in seconds)
get_elapsed_time(bkp$fit)
#> warmup sample
#> chain:1 1.819 2.313
#> chain:2 1.927 2.225
#> chain:3 2.038 3.406
#> chain:4 2.024 2.143
# summary plot
plot(bkp)
```

Instead of a smooth Gaussian Process we might want the reproduction number to change step-wise, e.g. every week. This can be achieved using the `rw`

option which defines the length of the time step in a random walk that the reproduction number is assumed to follow.

```
estimate_infections(reported_cases,
rw <-generation_time = generation_time_opts(example_generation_time),
delays = delay_opts(delay),
rt = rt_opts(prior = rt_prior, rw = 7),
gp = NULL
)# summarise results
summary(rw)
#> measure estimate
#> <char> <char>
#> 1: New infections per day 2136 (1056 -- 4315)
#> 2: Expected change in daily reports Likely decreasing
#> 3: Effective reproduction no. 0.87 (0.63 -- 1.2)
#> 4: Rate of growth -0.035 (-0.11 -- 0.044)
#> 5: Doubling/halving time (days) -20 (16 -- -6.3)
# elapsed time (in seconds)
get_elapsed_time(rw$fit)
#> warmup sample
#> chain:1 4.182 6.196
#> chain:2 4.202 6.631
#> chain:3 4.999 7.652
#> chain:4 4.127 4.635
# summary plot
plot(rw)
```

Whilst *EpiNow2* allows the user to specify delays, it can also run directly on the data as does e.g. the EpiEstim package.

```
estimate_infections(
no_delay <-
reported_cases,generation_time = generation_time_opts(example_generation_time)
)#> Warning: There were 32 divergent transitions after warmup. See
#> https://mc-stan.org/misc/warnings.html#divergent-transitions-after-warmup
#> to find out why this is a problem and how to eliminate them.
#> Warning: Examine the pairs() plot to diagnose sampling problems
#> Warning: Bulk Effective Samples Size (ESS) is too low, indicating posterior means and medians may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#bulk-ess
#> Warning: Tail Effective Samples Size (ESS) is too low, indicating posterior variances and tail quantiles may be unreliable.
#> Running the chains for more iterations may help. See
#> https://mc-stan.org/misc/warnings.html#tail-ess
# summarise results
summary(no_delay)
#> measure estimate
#> <char> <char>
#> 1: New infections per day 2759 (2325 -- 3311)
#> 2: Expected change in daily reports Decreasing
#> 3: Effective reproduction no. 0.87 (0.76 -- 0.98)
#> 4: Rate of growth -0.042 (-0.098 -- 0.0045)
#> 5: Doubling/halving time (days) -17 (150 -- -7.1)
# elapsed time (in seconds)
get_elapsed_time(no_delay$fit)
#> warmup sample
#> chain:1 34.378 41.249
#> chain:2 24.455 27.261
#> chain:3 28.171 43.757
#> chain:4 32.226 33.559
# summary plot
plot(no_delay)
```

The package also includes a non-parametric infection model. This runs much faster but does not use the renewal equation to generate infections. Because of this none of the options defining the behaviour of the reproduction number are available in this case, limiting user choice and model generality. It also means that the model is questionable for forecasting, which is why were here set the predictive horizon to 0.

```
estimate_infections(reported_cases,
non_parametric <-generation_time = generation_time_opts(example_generation_time),
delays = delay_opts(delay),
rt = NULL,
backcalc = backcalc_opts(),
horizon = 0
)# summarise results
summary(non_parametric)
#> measure estimate
#> <char> <char>
#> 1: New infections per day 2539 (2066 -- 3083)
#> 2: Expected change in daily reports Decreasing
#> 3: Effective reproduction no. 0.92 (0.8 -- 0.99)
#> 4: Rate of growth -0.023 (-0.044 -- -0.0015)
#> 5: Doubling/halving time (days) -30 (-460 -- -16)
# elapsed time (in seconds)
get_elapsed_time(non_parametric$fit)
#> warmup sample
#> chain:1 1.631 0.451
#> chain:2 1.862 0.363
#> chain:3 1.969 0.454
#> chain:4 2.113 0.539
# summary plot
plot(non_parametric)
```