• within a few days / weeks of index case
• limited data available
• no or limited intervention
• no depletion of susceptibles
• urgent assessment needed to inform response

• dates of symptom onset

• contact data: exposure (who infected you?) and contact tracing (who could you have infected?)

• dates of exposure / infection

• dates of outcome: death / recovery

• metadata on patients: age, gender, location, occupation, etc.

• data from past outbreaks

Disease-dependent, but generally includes:

• How fast is it growing?
• What is driving the epidemic growth?
• What is the case fatality ratio?
• Who is most severely affected?
• How many cases should we expect in the next days / weeks?
• …

• population: set of all possible observations of a given process/entity
  example: all possible cases of cholera in location xxx

• sample: subset of the population
  example: all cases of cholera in xxx reported last week

• a statistic: quantity used to describe sample / population
  example: % of fatalities in cholera cases in xxx last week

• inference: statement about population(s) from sample(s)
  example: % of fatalities in cholera cases in xxx is greater than in yyy

• when presenting estimates, show the associated uncertainty

• always show the data when possible (not just a model)

• account for the variability in the data

Definition: time interval between the date on infection and the date of symptom onset

Definition: time interval between onset of symptoms in primary and secondary cases.

Definition: time interval between date of infections in primary and secondary cases.

From empirical distribution (data) to estimated distribution.

• choose type of distribution (e.g. normal, Poisson, Gamma)

• find \(\theta_x\) which maximise \(p(x)\), i.e. the likelihood

• visually: best fit between bars (data) and curve (distribution)

Using continuous distributions to model discrete variables:

• flexible distribution (many shapes possibles)

• 2 parameters: shape, scale

• alternatively: mean, coefficient of variation

• typical choice for delay distributions

• needs to be discretised

Definition: the proportion of cases who die of the infection.

• CFR: 60%, N = 10: \[ s_{CFR} = \sqrt{\frac{0.6 \times 0.4}{10}} = 0.155 \]

\[ CI_{95\%} = 0.6 \pm 1.96 \times 0.158 = [0.30 ; 0.90] \]

• CFR: 60%, N = 100: \[ s_{CFR} = \sqrt{\frac{0.6 \times 0.4}{100}} = 0.05 \]

\[ CI_{95\%} = 0.6 \pm 1.96 \times 0.05 = [0.50 ; 0.70] \]

• "case fatality rate": this is a proportion, not a rate

• computation using wrong denominator, i.e. including unknown outcome:

\[ \frac{D}{D + R + U} \]

(leads to underestimating the CFR)

• not accounting for uncertainty, e.g. comparing CFR across groups without statistical tests

Definition: the incidence is the number of new cases on a given time period.

• relies on dates, typically of onset of symptoms

• only daily incidence is non-ambiguous

• other definitions (e.g. weekly) rely on a starting date

• prone to reporting delays

\(log(y) = r \times t + b + \epsilon\:\:\) so that \(\:\:\hat{y} = e^{r \times t + b}\)

with:

• \(r\): growth rate
• b: intercept
• \(\epsilon \sim \mathcal{N}(0, \hat{\sigma_{\epsilon}})\)

Let \(T\) be the time taken by the incidence to double, given a daily growth rate \(r\).

\[ y_2 / y_1 = 2 \:\: \Leftrightarrow e^{rt_2 + b} / e^{rt_1 + b} = 2 \]

\[ \Leftrightarrow e^{r(t_2 - t_1)} = 2 \Leftrightarrow T = log(2) / r \]

Pros:

• fast and simple
• predictions possible
• doubling / halving time readily available
• possible extensions to estimate \(R_0\) from \(r\)

Cons:

• zero incidence problematic
• non mechanistic
• no inclusion of other information (e.g. serial interval)

Serial interval: time interval between onset of symptoms of primary and secondary cases.

\[ \lambda_t = R_0 \times \sum_i w(t - t_i) \]

with: \(\lambda_t\): global force of infection; \(w()\): serial interval distribution; \(t_i\): date of symptom onset

Treat incidence \(y_t\) on day \(t\) as a Poisson distribution of rate \(\lambda_t\):

\[ p(y_t | R_0, y_1, ..., y_{t-1}) = e^{-\lambda_t} \frac{\lambda_t^{y_t}}{(!y_t)} \] with (slight rewriting): \(\lambda_t = R_0 \times \sum_{s = 1}^{t-1} y_s w(t - s)\)

Pros:

• still fast, reasonably simple (1 parameter)
• accommodates zero incidence
• predictions possible: forward simulation (Poisson)
• integrates information on serial interval
• extension to time-varying \(R\) (EpiEstim)

Cons:

• needs information on serial interval
• no spatial processes
• assumes constant reporting
• likelihood / Bayesian methods harder to communicate