From Seed Dispersal to Superspreaders

An Extreme Value Theory Approach to Wildlife Disease Transmission

Francisca Javiera Rudolph, PhD

University of Florida George A. Smathers Libraries

One Health Center of Excellence

Averages Hide the Story

• Bolnick and colleagues looked across 93 species and found that individual specialization is widespread across taxa - different individuals are doing different things
• Focusing on variation among individuals, not species averages, reveals the mechanisms behind seed dispersal outcomes.

Bolnick et al. The American Naturalist 2003

Zwolak. Biological Reviews 2018

Individual variation in seed dispersers (personality, age, sex, specialization) shapes where seeds go and whether they survive. The link between individual variation and seed fate runs almost entirely through movement ecology. Focusing on variation among individuals, not species averages, reveals the mechanisms behind seed dispersal outcomes. Convex relationship: a population with variation disperses more seeds than predicted from the mean. Concave relationship: a population with variation disperses fewer seeds than predicted from the mean.

The Origin: Aracari & Seed Dispersal

• Primary dispersers of Virola trees in Ecuadorian rainforest
• Most seeds land close to parent — but rare long-distance events are not that rare when we account for individual variation in movement
• Establishing the link with movement and LDD

Holbrook 2011 Biotropica

Many-banded Aracari photo (Pteroglossus pluricinctus) © Diego Mosquera

My dissertation. These birds are the primary dispersal vector for tropical trees. The biological question: where do seeds go? The statistical question: how do you characterize a rare long-distance event that you may have only seen once or twice?

Central Limit Theorem (CLT) & Extreme Value Theory (EVT)

The special rules in statistics

• CLT = gather data -> take average -> repeat many times
  • averages are normally distributed

The Central Limit theorem states that the distribution of sample means approaches a normal distribution as the sample size increases, regardless of the population’s original distribution.

Extreme Value Theory states that regardless of the underlying data-generating process, the behavior of extreme observations — the rare, large events in the tail — converges to a Generalized Extreme Value distribution. The tail has its own universal structure.

WHAT IS EXTREME VALUE THEORY?

The Central Limit Theorem says if you calculate averages, they converge to a Gaussian — regardless of the underlying distribution. That’s why it’s so useful.

EVT is the parallel theorem for maxima. The Fisher-Tippett-Gnedenko theorem (1928, fully proven 1943) says maxima converge to the GEV — also regardless of the underlying distribution.

Ecology has known it needed this tool since 1993. It’s taken us this long to start using it seriously.

EVT

García and Borda-de-Água showed that if you treat long-distance dispersal events as exactly what they are — extremes — you can estimate probabilities of dispersal distances you never observed in your study. That’s the power of EVT. And that’s exactly what I did with my aracari data.

When I allow individual variation in frugivore movement, I get more long-distance dispersal events in my simulated seed shadows. I fit a Generalized Pareto to the tail of those simulated kernels. The shape parameter ξ > 0 — fat tail — and the conditional probability of a seed reaching beyond 500m, 1km, is meaningfully higher under individual variation than under the pooled model.

Consequences of individual variation in animal movement — in frugivores, it translates to an increase in the number of long-distance seed dispersal events.

We know there is a link from mechanism to outcome. Now we need a framework.

Standard kernel fitting is still trying to describe the whole distribution. EVT says: stop trying to fit the whole thing. Focus only on the tail, where it matters. Let the data above a threshold speak for themselves.

From Pattern to Process

Individual variation in movement shapes the tail of the dispersal kernel —

and we can use ξ (shape) to see this

The process
Individual frugivores differ in how far they move. Some are homebodies. Some are explorers. That variation is not just noise, it is the mechanism.

The pattern
Pooled models underestimate long-distance dispersal. When individual variation is modeled explicitly, the tail of the seed shadow is fatter — and ξ > 0 confirms it.

The link
ξ is not a fitting artifact. It is a fingerprint of the movement process. The shape of the seed dispersal tail traces back to how individuals move and how we capture variation across individuals.

EVT has offered ecologists a framework for ‘rare’ events like long-distance dispersal since the 1990s — the tools were there before the field was ready to use them.

Gaines& Denny (1993). The largest, smallest, highest, lowest, longest, and shortest: extremes in ecology. Ecology.
Katz et.al. (2005). Statistics of extremes: modeling ecological disturbances. Ecology
Beisel et.al (2007). Testing the extreme value domain of attraction for distributions of beneficial fitness effects. Genetics

This is the pivot. Everything before — EVT, individual variation, the aracari system — was building to this point.

The argument is simple: ξ is not arbitrary. It reflects something real about how individuals move. When you allow individuals to differ, the resulting seed shadow has a fat tail, and the shape parameter you recover from fitting EVT to that tail traces back to the biological mechanism.

This is what justifies the next move — taking that same logic into disease systems. If ξ encodes movement process in seed dispersal, it encodes transmission process in pathogen dispersal. Same mathematics, new biological context.

When I allow individual variation in frugivore movement, I get more long-distance dispersal events in my simulated seed shadows. I fit a Generalized Pareto to the tail of those simulated kernels. The shape parameter ξ > 0 — fat tail — and the conditional probability of a seed reaching beyond 500m, 1km, is meaningfully higher under individual variation than under the pooled model.

Individual Variation in Disease Transmission

“Population-level analyses often use average quantities to describe heterogeneous systems…”
— Lloyd-Smith et al. Nature 2005

Sound familiar?

\(R_0\) is an average. It hides the same individual variation we saw in dispersal kernels:

• Most individuals infect zero or one
• A few infect many — superspreaders

Individual reproductive number, \(\nu\), as a random variable representing the expected number of secondary cases caused by a particular infected individual.

• Same \(R_0\), six different epidemiological worlds.
• More heterogeneity means more outbreaks die out, but the ones that don’t are explosive.
• k describes the shape of that individual variation

The opening quote of Lloyd-Smith is the same argument I’ve been making about seed dispersal for the last five slides. Population averages hide the mechanism. R₀ hides individual variation in infectiousness exactly the way a pooled dispersal kernel hides individual variation in movement.

Lloyd-Smith’s solution was statistical — fit a distribution, estimate R0 and k. That works when you have contact tracing data, which you have for SARS, measles, smallpox. You do not have that for wildlife pathogens. You have GPS collars.

The branching process is the entire model — there is no SIR, no compartments, no population dynamics. It is a purely probabilistic model of individual-level transmission heterogeneity.

The Mechanistic Foundation

Ponciano & Capistrán (2011) derive the incidence rate from first principles:

• An infected individual has realized disease dispersion \(a\)
• Number of successful transmission encounters \(X(a)\) follows a pure birth process
• Probability of at least one infection:

\[P(X(a) \geq 1) = 1 - e^{-a \cdot b \cdot h(I)}\]

When dispersion effort \(\Lambda\) is exponentially distributed across individuals:

\[P(X(a) \geq 1) = \int_0^\infty \left(1 - e^{-\lambda h(I)}\right) \alpha e^{-\alpha \lambda} \, d\lambda = \frac{h(I)}{h(I) + \alpha}\]

Ponciano & Capistrán. PLoS Computational Biology 2011

Individual variation matters here:
The exponential distribution for \(\Lambda\) already allows individuals to differ in how much they disperse pathogen.
But the exponential has a thin tail — it does not generate superspreaders.

What if dispersion effort follows a heavier-tailed distribution?
What if — like seed dispersers — some infected individuals are true long-distance movers?

Ponciano and Capistran gave us the scaffolding. The pure birth process is elegant: the probability that you produce at least one infection grows with your realized dispersion effort a. The exponential model for Lambda averages over individual heterogeneity and gives you the LHD incidence rate. But the exponential is thin-tailed — it cannot generate the kind of extreme dispersion events that produce superspreaders. We need to go one step further.

From Movement to Superspreaders: The Lomax Emerges

What if dispersion effort \(a\) is itself heterogeneous?

Allow dispersion effort \(a\), to be distributed as an exponential distribution, whose parameter is sampled from a gamma distribu- tion:

Let \(A \sim \text{Gamma}(\theta, \tau)\) — individual variation in transmission potential — and \(Y \mid A \sim \text{Exp}(A)\) — realized dispersion given that potential.

Then the marginal distribution of \(Y\) is:

\[f_Y(y) = \int_0^\infty a e^{-ay} \cdot \frac{\theta^\tau}{\Gamma(\tau)} a^{\tau-1} e^{-\theta a} \, da = \frac{\tau \theta^\tau}{(y + \theta)^{\tau+1}}\]

This is the Lomax distribution — special case of a GPD - a heavy-tailed distribution with support on \([0, \infty)\).
The tail emerges mechanistically from the compound structure.

And the probability of transmission under Lomax-distributed dispersion:

\[P(X(a) \geq 1) = \int_0^\infty \left(1 - e^{-abE}\right) \frac{\tau \theta^\tau}{(a+\theta)^{\tau+1}} \, da\]

The Gamma is the movement

\(\theta\) and \(\tau\) are not abstract parameters — they characterize how individuals in a population move. Some animals are homebodies, some are explorers.

That variation across individuals, captured by the Gamma, is what drives the heavy tail.

Allow the movement to vary, and EVT drops out the other end.

Rudolph & Ponciano (2025) Biorxiv | Rudolph (in.prep)

Ponciano-Capistrán give you a first-principles framework where transmission probability depends on a dispersion effort parameter a, and they model the heterogeneity in a across individuals using an exponential distribution.

what if we let that exponential rate itself vary? Instead of a fixed rate, you draw it from a Gamma distribution — and that Gamma is the movement. The parameters θ and τ characterize how individuals in a population move. The Gamma is biologically interpretable as the movement kernel.

when you integrate out that Gamma-distributed rate, the marginal distribution of dispersion effort is Lomax. And the Lomax is a special case of the Generalized Pareto — which is exactly the EVT family

here is a natural extension of an existing framework, and when you make this one biologically motivated move, EVT drops out the other end.

Simulation

Where the Framework Goes: American Mink in Chile

Invasive semiaquatic carnivore in Chilean Patagonia

• Disperses along river networks
• Documented hosts: canine distemper, parvovirus, Mycobacterium bovis
• Potential bridge host to native species: coypu, pudú, otters

Santibañez et.al. (2025). Frontiers in Veterinary Science
Hernandez et.al. (2024) Acta Tropica

How can we use mink movement along river corridors predict pathogen spread potential and spillover risk to native species?

Lack of data and time asks for a simulation framework

Crego et.al. (2018). PLoS One

American mink in Chilean waterways — invasive, semiaquatic, dispersing along river networks. This is a tractable one-dimensional dispersal problem. They carry pathogens and are positioned to act as bridge hosts. The question is whether the tail of their movement kernel predicts pathogen spread potential before outbreak data exist.

Thank you

Questions?

Francisca Javiera Rudolph, PhD javiera.av@ufl.edu

• Global Fellows UFIC
• F. Hernandez Univ Austral Chile
• JM Ponciano