Pneumonia dynamics

in bighorn sheep

Kezia Manlove

Dissertation defense

September 14, 2016

Resource challenges in wildlife disease

Low public concern

Mortalities in captivity

Sub-optimal diagnostics

Rare longitudinal datasets

Rare post-mortems

Introduction

Bighorn sheep pneumonia

Manlove et al. (in review)

+ \beta_{1} \sum X_{\text{Ewes with lambs}}

+ \beta_{1} \sum X_{\text{Ewes with lambs}}

+ \beta_{2}\sum X_{\text{Ewes without lambs}}

+ \beta_{2}\sum X_{\text{Ewes without lambs}}

+ \beta_{3} \sum X_{Yearlings}

+ \beta_{3} \sum X_{Yearlings}

logit(P(\text{Lamb survival})) =

logit(P(\text{Lamb survival})) =

\beta_{0}

\beta_{0}

Animal movement histories have three components:

Individual identifier

Spatial coordinates

Timestamp

(I,S,T)

(I,S,T)

I \in {1,...,N}

I \in {1,...,N}

S \in D_{S}

S \in D_{S}

T \in D_{T}

T \in D_{T}

Network constructions rely on intersections of marginalizations of

Nodes = levels of variable between which intersections are calculated

Edges = relative frequency of intersection occurrence

(I,S,T)

(I,S,T)

Other metrics can proxy for social contacts if I, S, T are redundant

A little formalism: Animal movements

Animal movement histories have three components:

Individual identifier

Spatial coordinates

Timestamp

(I,S,T)

(I,S,T)

I \in {1,...,N}

I \in {1,...,N}

S \in D_{S}

S \in D_{S}

T \in D_{T}

T \in D_{T}

(All?) models in population biology rely on some aspect of

(I,S,T)

(I,S,T)

Analysis	Variable aggregated out	Marginal used
Individual home-range
Population-level movements
Individual survival

f_{I,S} = \int_{t \in T}f_{(I,S,T)}dt

f_{I,S} = \int_{t \in T}f_{(I,S,T)}dt

f_{S,T} = \sum_{i \in I}f_{(I,S,T)}

f_{S,T} = \sum_{i \in I}f_{(I,S,T)}

f_{I,T} = \int_{s \in S}f_{(I,S,T)}ds

f_{I,T} = \int_{s \in S}f_{(I,S,T)}ds

T

T

I

I

S

S

Case 1: Social contact network

First claim

All network constructions rely on intersections of some marginal distribution of

(I,S,T)

(I,S,T)

Nodes = levels of variable between which intersections are calculated

Edges = relative frequency of intersection occurrence

Two individuals, and

I_{1}

I_{1}

I_{2}

I_{2}

, with movement paths and

X_{1} = (I = I_{1}, S, T)

X_{1} = (I = I_{1}, S, T)

X_{2} = (I = I_{2}, S, T)

X_{2} = (I = I_{2}, S, T)

X_{1} \cap X_{2}

X_{1} \cap X_{2}

, contains all points

where and co-occur in

I_{1}

I_{1}

I_{2}

I_{2}

(S,T)

(S,T)

(aka their "contacts")

Case 2: Spatial networks

Two sites, and

S_{1}

S_{1}

, with visit histories and

X_{S1} = (I, S = S_{1}, T)

X_{S1} = (I, S = S_{1}, T)

X_{S2} = (I, S = S_{2}, T)

X_{S2} = (I, S = S_{2}, T)

X_{S1} \cap X_{S2}

X_{S1} \cap X_{S2}

The set of path intersections,

is empty.

S_{2}

S_{2}

BUT, not empty if we aggregate over (= marginalize out) time

X_{S1} = (I, S = S_{1})

X_{S1} = (I, S = S_{1})

X_{S2} = (I, S = S_{2})

X_{S2} = (I, S = S_{2})

and

X_{S1} \cap X_{S2}

X_{S1} \cap X_{S2}

= frequency with which and share animals

S_{1}

S_{1}

S_{2}

S_{2}

A little formalism: Animal movements

Animal movement histories have three components:

Individual identifier

Spatial coordinates

Timestamp

(I,S,T)

(I,S,T)

I \in {1,...,N}

I \in {1,...,N}

S \in D_{S}

S \in D_{S}

T \in D_{T}

T \in D_{T}

(All?) models in population biology rely on some aspect of

(I,S,T)

(I,S,T)

Analysis	Variable aggregated out	Marginal used
Individual home-range
Population-level movements
Individual survival

f_{I,S} = \int_{t \in T}f_{(I,S,T)}dt

f_{I,S} = \int_{t \in T}f_{(I,S,T)}dt

f_{S,T} = \sum_{i \in I}f_{(I,S,T)}

f_{S,T} = \sum_{i \in I}f_{(I,S,T)}

f_{I,T} = \int_{s \in S}f_{(I,S,T)}ds

f_{I,T} = \int_{s \in S}f_{(I,S,T)}ds

T

T

I

I

S

S

Case 1: Social contact network

First claim

All network constructions rely on intersections of some marginal distribution of

(I,S,T)

(I,S,T)

Nodes = levels of variable between which intersections are calculated

Edges = relative frequency of intersection occurrence

Two individuals, and

I_{1}

I_{1}

I_{2}

I_{2}

, with movement paths and

X_{1} = (I = I_{1}, S, T)

X_{1} = (I = I_{1}, S, T)

X_{2} = (I = I_{2}, S, T)

X_{2} = (I = I_{2}, S, T)

X_{1} \cap X_{2}

X_{1} \cap X_{2}

, contains all points

where and co-occur in

I_{1}

I_{1}

I_{2}

I_{2}

(S,T)

(S,T)

(aka their "contacts")

Case 2: Spatial networks

Two sites, and

S_{1}

S_{1}

, with visit histories and

X_{S1} = (I, S = S_{1}, T)

X_{S1} = (I, S = S_{1}, T)

X_{S2} = (I, S = S_{2}, T)

X_{S2} = (I, S = S_{2}, T)

X_{S1} \cap X_{S2}

X_{S1} \cap X_{S2}

The set of path intersections,

is empty.

S_{2}

S_{2}

BUT, not empty if we aggregate over (= marginalize out) time

X_{S1} = (I, S = S_{1})

X_{S1} = (I, S = S_{1})

X_{S2} = (I, S = S_{2})

X_{S2} = (I, S = S_{2})

and

X_{S1} \cap X_{S2}

X_{S1} \cap X_{S2}

= frequency with which and share animals

S_{1}

S_{1}

S_{2}

S_{2}

Second claim

Network constructions can proxy for one another under appropriate correlations of

I, S, \text{ and } T

I, S, \text{ and } T

First claim

All network constructions rely on intersections on some marginal distribution of

(I,S,T)

(I,S,T)

Nodes = levels of variable between which intersections are calculated

Edges = relative frequency of intersection occurrence

Third claim

Correlations in can be predicted from animal behaviors

I, S, \text{ and } T

I, S, \text{ and } T

Do these claims hold (in silico, or in vivo)?

In silico

Simulated movement trajectories for

N = 50 animals over

T = 100 timesteps on an

S = 10x10 grid

Three governing movement parameters (varied systematically on log-scale)

Social affinity (desire to stay with current associates)

Spatial affinity (desire to stay within a particular territory)

Habitat heterogeneity (desirability of particular sites for all animals)

Recorded

Social contact network metrics (degree, modularity, transitivity, average distance)

Spatial network metrics (using animal-visits-multiple-sites as edges)

Home-range and defensibility (= day-range:home-range)

Group size distributions (fit with a Poisson model; summarize with )

\lambda

\lambda

\gamma =

\gamma =

\delta =

\delta =

\rho =

\rho =

Pneumonia dynamics

in bighorn sheep

Introduction

Bighorn sheep pneumonia

Bighorn sheep status

Hells Canyon

Early analyses

1. How do die-offs and persistent disease compare?

2. Is transmission frequency- or density-dependent?

3. Do all infected animals pose similar risks to lambs?

When do we need social contact networks?

Main findings

Main Findings

"One Health" or Three?

Is disease modeling becoming more cross-disciplinary?

Main points

Objectives

Why does population structure matter?

Question

A framework relating network constructions

Motivating question

Acknowledgements