Notation for pandemic potential pathogens

Basic set-up

Let \(k \in \{\text{nat}, \text{acc}, \text{del}\}\) be the possible types of release event, each with rate of occurance \(\lambda_k\) per year such that:

• \(\lambda_\text{nat} \sim \pi(\lambda_\text{nat})\) is the rate that the pathogen naturally spills over into humans,
• \(\lambda_\text{acc} \sim \pi(\lambda_\text{nat})\) is the rate that the pathogen accidentally spills over into humans,
• and \(\lambda_\text{del} \sim \pi(\lambda_\text{del})\) is the rate that the pathogen is deliberately released in humans.

We model the release events as being independent, each taking a Poisson distribution, such that the number of releases \(r_k\) of each type in a given year is \[ r_k \sim \text{Poisson}(\lambda_k). \]

Given a release of type \(k\), we suppose that there is probability \(p_k\) that a pandemic would occur, such that \[ X_k \sim \text{Binomial}(r_k, p_k), \] where \(X_k\) is the number of pandemics of type \(k\) that happen. We imagine the harm coming about from any pandemic as being \(h \sim \pi(h)\), and the total yearly harm from this pathogen as \(H = \sum_k H_k = \sum_k hx_k\). The expected yearly harm is \[ \mathbb{E}[H] = \sum_k \mathbb{E}[H_k] = \mathbb{E}[h] \sum_k \mathbb{E}[X_k], \] that is, just the expected harm per pandemic, multiplied by the expected number of pandemics. In this set-up, because the expected harm from each type of pandemic is assumed to be the same, it is sufficient only to consider the expected number of pandemics.

Let’s try simulating one draw from the number of yearly pathogens:

#' You could give these priors and draw them, but I'm just using a fixed value
lambda_nat <- 0.05
lambda_acc <- 0.01
lambda_del <- 0.005

(r_nat <- rpois(1, lambda_nat))

## [1] 1

(r_acc <- rpois(1, lambda_acc))

## [1] 0

(r_del <- rpois(1, lambda_del))

## [1] 0

#' You could give these priors and draw them, but I'm just using a fixed value
p_nat <- 0.05
p_acc <- 0.1
p_del <- 0.2

(x_nat <- rbinom(n = 1, size = r_nat, prob = p_nat))

## [1] 0

(x_acc <- rbinom(n = 1, size = r_acc, prob = p_acc))

## [1] 0

(x_del <- rbinom(n = 1, size = r_del, prob = p_del))

## [1] 0

#' The total number of pandemics in the year was
(x_tot <- x_nat + x_acc + x_del)

## [1] 0

It’s was 0 – good news I hope. What about the expected number of yearly pandemics under this scenario? We can simulate 5000 years and see what the average is:

set.seed(1)

yearly_pathogens <- function(lambda, p) {
  K <- length(lambda)
  r <- rpois(K, lambda)
  x <- rbinom(K, size = r, prob = p)
  return(x)
}

lambda_values <- c(lambda_nat, lambda_acc, lambda_del)
p_values <- c(p_nat, p_acc, p_del)

yearly_pathogens(lambda_values, p_values)

## [1] 0 0 0

N <- 5000
x_wide <- replicate(N, yearly_pathogens(lambda = lambda_values, p = p_values))
colSums(t(x_wide))

## [1] 8 7 6

sum(colSums(t(x_wide)))

## [1] 21

So, under these settings, there were an average of 0.0042 pandemics per year. These were comprised of 8 natural outbreaks, 7 accidental releases and 6 deliberate releases in 5000 years.

Adding pandemic prevention research

Pandemic prediction research primarily has the effect of:

• Decreasing \(\lambda_{\text{nat}}\)
• Increasing \(\lambda_{\text{acc}}\)
• Increasing \(\lambda_{\text{del}}\)
• Reducing the size of \(h\)

Let \(S \in \{0, 1\}\) be the scenario in which pandemic prediction research takes place \(S = 1\) or does not \(S = 0\). We are interested in determining \(\mathbb{E}[H \, | \, S = 0]\) and \(\mathbb{E}[H \, | \, S = 1]\).