Modified SIR model for host aggregation

The standard SIR model has a mass-action transmission term, but how can we improve biological realism by incorporating the social behavior of hosts?

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I’ve been exploring models from parasitology that describe the population dynamics of host-parasite interactions. One of the main challenges that these models address are that parasite loads are unevenly distributed across hosts, typically following a negative binomial distribution (though I’m doing some work on whether it might be a Poisson-Poisson mixture). This reflects real-world observations where a few hosts carry most parasites, while most hosts have very few parasites.

Anderson and May’s work (1978) is particularly relevant. They incorporated parasite aggregation into their ODEs with terms that account for this uneven distribution. The dP/dt model for parasite pop dynamics includes the terms: -alpha * P + -alpha * ((P²)/H) * ((k+1)/k), where these variables represent:

• P: Total number of parasite individuals

• alpha: Parasite-induced mortality rate, aka virulence

• H: Total number of host individuals

• k: Degree of aggregation (small k is very clumped, big k approached no clumping aka a Poisson distribution)

These terms are derived directly from the second moment of the negative binomial, which has the identity: E(X²) = (mu²) * ((k+1)/k) + mu, where mu = P/H, and Anderson and May’s parameterization is just an expansion of -alpha * H * E(X²).

For SIR models, though, things are familiar but look a little different. Following Barlow (2000), we can modify it to swap the mass action transmission process term:

beta * S * I

to account for aggregation of hosts, such that beta * S * I becomes:

k * S * log(1 + (beta * I) / k),

where k is the aggregation parameter in a negative binomial distribution, as in:

X ~ NegBin(mu, k).

sir_NegBin_model <- function(time, state, parameters) {
  with(as.list(c(state, parameters)), {
    # Equations
    dS <- -k * S * log(1 + (beta * I) / k)
    dI <- k * S * log(1 + (beta * I) / k) - gamma * I
    dR <- gamma * I

    # Return the rate of change
    list(c(dS, dI, dR))
  })
}

It would be interesting to consider varying trajectories for pathogen evolution under different host aggregation behaviors. We could use the Price equation to model strain evolution and resulting impacts on the virulence parameter, alpha.

References

Anderson, R. M., & May, R. M. (1978). Regulation and stability of host-parasite population interactions. Journal of animal ecology, 47(1), 219-247.

Barlow, N. D. (2000). Non‐linear transmission and simple models for bovine tuberculosis. Journal of Animal Ecology, 69(4), 703-713.

Kong, L., Wang, J., Han, W., & Cao, Z. (2016). Modeling heterogeneity in direct infectious disease transmission in a compartmental model. International journal of environmental research and public health, 13(3), 253.