Introduction to the time-varying reproduction number

Nowcasting and forecasting of infectious disease dynamics

Independent infections model

functions {
  #include "functions/convolve_with_delay.stan"
}

data {
  int n;            // number of time days
  array[n] int obs; // observed onsets
  int<lower = 1> ip_max; // max incubation period
  // probability mass function of incubation period distribution (first index zero)
  array[ip_max + 1] real ip_pmf;
}

parameters {
  array[n] real<lower = 0> infections;
}

transformed parameters {
  array[n] real onsets = convolve_with_delay(infections, ip_pmf);
}

model {
  // priors
  infections ~ normal(0, 10) T[0, ];
  obs ~ poisson(onsets);
}

Prior for infections at time \(t\) is independent from infections at all other time points. Is this reasonable?

What if we convolve infections with infections?

We can use the concept of a convolution to link infections over time:

• Key insight: infections → infections (via generation time)
• This captures the fundamental transmission process in epidemics

Generation time: infection to infection

Here rather than the time from infection (person A) to symptoms (person B, infected by A) we have the time from infection (person A) to infection (person B, infected by A).

This is known as the generation time distribution (\(g(t)\)).

Generation time: notation

We can write the convolution as:

\[ I_t = \mathrm{scaling} \times \sum_{t' < t} I_t' g(t - t') \]

However, unlike the infection to symptoms case here we don’t assume that the two time series have the same magnitude so need to introduce a scaling.

What is this scaling?

The renewal equation as a convolution

The scaling factor: reproduction number

Let’s assume we have \(I_0\) infections at time 0, and the scaling doesn’t change in time.

How many people will they go on to infect?

\[ I = \mathrm{scaling} \times \sum_{t=0}^\infty I_0 g(t) = \mathrm{scaling} * I_0 \]

The scaling can be interpreted as the reproduction number \(R_0\) (assuming a susceptible population).

The renewal equation

If \(R_t\) can change over time, it can be interpreted as the (“instantaneous”) reproduction number:

\[ I_t = R_t \times \sum_{t' < t} I_t' g(t - t') \]

We can estimate \(R_t\) from a time series of infections using the renewal equation.

Your Turn

• Simulate infections using the renewal equation
• Estimate reproduction numbers using a time series of infections
• Combine with delay distributions to jointly infer infections and R from a time series of outcomes

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