Metapopulation Networks and the Invasion Threshold

The Levins metapopulation tracks the fraction of patches occupied with a mean-field equation, and random-graph models track who contacts whom, but neither describes a pathogen jumping between whole subpopulations linked by travel. A disease can spread perfectly well within a city yet still fail to reach the next city if too few infected travelers make the trip. That gives two separate thresholds: a local one that decides whether an outbreak grows inside a subpopulation, and a global invasion threshold that decides whether it spreads between them.

Reaction–diffusion metapopulations

Model a set of subpopulations sitting on the nodes of a mobility network with degree distribution $P(k)$. Inside each node, individuals run a local reaction — an SIR or SIS epidemic — while individuals diffuse along the edges to neighboring nodes at some per-capita mobility rate $p$ (Colizza & Vespignani 2007, Nat. Phys.). The state is the number of susceptible, infectious, and recovered individuals in every node, and the dynamics couple local transmission to between-node transport. This is the theory behind global spread models: the reaction–diffusion picture of continuous space becomes a discrete network when space is a set of cities joined by flights.

Two thresholds, not one

Local growth is governed by the familiar basic reproduction number: a subpopulation sustains an outbreak only if $R_0>1$. Between-subpopulation spread is governed by a separate quantity, the global invasion threshold $R_*$, defined as the expected number of subpopulations seeded by one already-infected subpopulation (Colizza & Vespignani 2007, Phys. Rev. Lett.). The metapopulation is invaded only when

$$R_* > 1.$$

A disease can clear the local bar and fail the global one: it burns through the seed city but sends too few infected travelers onward to ignite a second city. Raising $R_0$ is not enough; the pathogen also has to move.

What sets R*

During a local outbreak of size proportional to $(R_0-1)$, a node exports infected individuals to each neighbor at a rate set by the mobility $p$ and the number of residents. Summing the seedings a node delivers over the course of its outbreak gives a threshold of the schematic form

$$R_* \;\propto\; p\,\bar{N}\,\frac{(R_0-1)^2}{R_0^2}\,\frac{\langle k^2\rangle}{\langle k\rangle},$$

where $\bar N$ is the mean subpopulation size and $\langle k^2\rangle/\langle k\rangle$ measures degree heterogeneity. Three levers raise $R_*$: faster mobility $p$, larger subpopulations $\bar N$ (bigger outbreaks export more), and more heterogeneous mobility. In a scale-free network the second moment $\langle k^2\rangle$ is large, so hubs make the metapopulation far easier to invade — heterogeneity lowers the mobility needed to cross $R_*=1$.

Gravity and radiation coupling

The mobility rates are not uniform. Empirically, the flux of travelers between two places follows a gravity model,

$$w_{ij} \;\propto\; \frac{N_i^{\,\alpha}\,N_j^{\,\beta}}{d_{ij}^{\,\gamma}},$$

with flux growing in the product of the two population sizes and decaying with distance $d_{ij}$. Large, close cities exchange the most travelers and seed each other first, which is why epidemics tend to reach big hubs early regardless of geographic proximity. The radiation model is a parameter-free alternative that predicts commuting flux from the population lying within the radius between origin and destination.

A worked example

Take a scale-free mobility network with local $R_0=3$, so every subpopulation would sustain an outbreak on its own. At a mobility rate of $p=5\times10^{-5}$ per individual per step, the outbreak stays trapped in the seed subpopulation: the expected number of onward seedings is below one, so $R_*<1$. Raise mobility to $p=10^{-3}$ and each infected subpopulation now exports enough travelers to seed several neighbors before its own outbreak fades, so $R_*>1$ and the epidemic sweeps the whole network. The local reproduction number never changed; only the mobility crossed the global invasion threshold.

In code

We build a small metapopulation on a scale-free graph, run a discrete SIR reaction with between-node mobility seeded in one subpopulation, and count how many subpopulations experience a real outbreak below and above the invasion threshold.

R

library(igraph)

set.seed(1834)
G <- sample_pa(40, m = 2, directed = FALSE)
nb <- adjacent_vertices(G, V(G))
n <- 40; M <- 3000; beta <- 0.9; gamma <- 0.3

run <- function(p, steps = 120) {
  S <- rep(M, n); I <- rep(0, n); R <- rep(0, n)
  S[1] <- M - 20; I[1] <- 20
  for (t in seq_len(steps)) {
    ni <- rbinom(n, S, 1 - exp(-beta * I / M))
    nr <- rbinom(n, I, gamma)
    S <- S - ni; I <- I + ni - nr; R <- R + nr
    for (comp in c("S", "I", "R")) {
      x <- get(comp); leave <- rbinom(n, x, p); x <- x - leave
      for (i in seq_len(n)) if (leave[i] > 0) {
        k <- nb[[i]]
        x[k] <- x[k] + rmultinom(1, leave[i], rep(1 / length(k), length(k)))
      }
      assign(comp, x)
    }
  }
  sum(R > 0.02 * M)
}
sapply(c(5e-5, 1e-3), run)   # reached below vs above invasion

Python

import numpy as np
import networkx as nx
import polars as pl

G = nx.barabasi_albert_graph(40, 2, seed=1834)   # scale-free mobility net
nbrs = [list(G[i]) for i in G.nodes()]
n, M = G.number_of_nodes(), 3000                 # subpops, individuals each
beta, gamma = 0.9, 0.3                            # local R0 = beta/gamma = 3


def run(p, steps=120):
    rng = np.random.default_rng(1834)
    S = np.full(n, M); I = np.zeros(n, int); R = np.zeros(n, int)
    S[0], I[0] = M - 20, 20                       # seed one subpopulation
    for _ in range(steps):
        newinf = rng.binomial(S, 1 - np.exp(-beta * I / M))
        newrec = rng.binomial(I, gamma)
        S -= newinf; I += newinf - newrec; R += newrec
        for comp in (S, I, R):                    # mobility: leave, then split
            leave = rng.binomial(comp, p)
            comp -= leave
            for i in range(n):
                if leave[i] and nbrs[i]:
                    share = rng.multinomial(leave[i], [1 / len(nbrs[i])] * len(nbrs[i]))
                    for j, c in zip(nbrs[i], share):
                        comp[j] += c
    return int((R > 0.02 * M).sum())              # subpops with a real outbreak


rows = [{"mobility_p": p, "subpops_reached": run(p)} for p in (5e-5, 1e-3)]
print(pl.DataFrame(rows))

shape: (2, 2)
┌────────────┬─────────────────┐
│ mobility_p ┆ subpops_reached │
│ ---        ┆ ---             │
│ f64        ┆ i64             │
╞════════════╪═════════════════╡
│ 0.00005    ┆ 1               │
│ 0.001      ┆ 40              │
└────────────┴─────────────────┘

Julia

using Graphs, Distributions, Random

function run(p; steps = 120)
    rng = MersenneTwister(1834)
    G = barabasi_albert(40, 2; rng = rng)
    n, M, beta, gamma = 40, 3000, 0.9, 0.3
    S = fill(M, n); I = zeros(Int, n); R = zeros(Int, n)
    S[1] = M - 20; I[1] = 20
    for _ in 1:steps
        ni = [rand(rng, Binomial(S[i], 1 - exp(-beta * I[i] / M))) for i in 1:n]
        nr = [rand(rng, Binomial(I[i], gamma)) for i in 1:n]
        S .-= ni; I .+= ni .- nr; R .+= nr
        for comp in (S, I, R)
            for i in 1:n
                leave = rand(rng, Binomial(comp[i], p))
                comp[i] -= leave
                k = neighbors(G, i)
                isempty(k) && continue
                share = rand(rng, Multinomial(leave, fill(1 / length(k), length(k))))
                comp[k] .+= share
            end
        end
    end
    count(R .> 0.02 * M)
end

[run(p) for p in (5e-5, 1e-3)]   # reached below vs above invasion

Why it matters

The invasion threshold explains why local control is not the whole story: a pathogen with $R_0>1$ everywhere can still be stopped from going global if travel is slow or evenly spread, and can be tipped over $R_*=1$ by air travel that concentrates flux through a few hubs. It reframes containment around mobility — travel restrictions, border screening, cordons — as acting on a different threshold than vaccination, which acts on local $R_0$. The heterogeneity term is why real epidemics seed large, well-connected cities first and why scale-free mobility makes global spread so hard to prevent.

Related

• Metapopulations and the Levins Model
• Random-Graph Models
• Networks in Ecology and Epidemiology
• Reaction–Diffusion and Spatial Spread
• Compartmental Models (SIR)
• Quantitative Methods