Ecology of infectious diseases:
How climate & community-level interactions
shape host-parasite dynamics.
Paul J. Hurtado
Mathematical Biosciences Institute
Aquatic Ecology Laboratory
Department of Evolution, Ecology & Organismal Biology,
The Ohio State University
About Paul…
About Paul…
About Paul…
Mathematical Biosciences Institute
The Ohio State University | Columbus, OH
http://mbi.osu.edu/
Aquatic Ecology Laboratory (AEL)
Kinnear Rd, West Campus
http://ael.osu.edu/
Part I
Disease Dynamics in Consumer Populations:
Consequences of Resource-Mediated
Transmission and Infectiousness
Acknowledgements
The Daphnia Crew
Spencer Hall (IU, Bloomington),
Megan Duffy (GA Tech), and their labs.
The Ellner-Hairston lab at Cornell
Stephen Ellner, Nelson Hairston Jr.,
Joe Simonis, Mike Cortez, and others.
Mathematical Biosciences Institute (MBI)
Libby Marschall, Joe Tien,
Marisa Eisenberg, Suzanne Robertson,
Yunjiao Wang, Rebecca Tien, others.
Others:
Frank M. Hilker (Univ. of Bath)
Chris Schepper (Cornell, CAM)
Background
Algae (n)
Daphnia/host (x+y)
Spores (z)
Goals
Typical predator-prey models driven by population size.
Typical infectious disease models driven by population size.
Models with both often tacitly assume only indirect crosstalk
between processes, mediated by population size.
The data say direct interactions exist, i.e., some disease
parameters depend directly on the predator-prey interaction.
Q1: How do these dependencies affect
prey-predator-disease dynamics?
Q2: How does density-dependent transmission
give rise to instances of bistability?
The Model
n
x
y
The (Simplified) Model
n
x
y
Model Comparisons
Expected Dynamics?
• SI (SIZ) Dynamics?
• Host-Resource Dynamics?
Ross’s SI Model
•Ross, R. 1908. Report on the prevention of malaria in Mauritius.
London. http://archive.org/details/onpreventi00rossreportrich
•Ross, R. 1916. "An Application of the Theory of Probabilities to the
Study of a priori Pathometry. Part I". Proceedings of the Royal Society A:
Mathematical, Physical and Engineering Sciences 92 (638): 204–226.
doi:10.1098/rspa.1916.0007.
•Ross, R.; Hudson, H. P. 1917. "An Application of the Theory of
Probabilities to the Study of a priori Pathometry. Part II". Proceedings of
the Royal Society A: Mathematical, Physical and Engineering Sciences 93
(650): 212. doi:10.1098/rspa.1917.0014.
•Ross, R.; Hudson, H. P. 1917. "An Application of the Theory of
Probabilities to the Study of a Priori Pathometry.--Part III". Proceedings
of the Royal Society B: Biological Sciences 89 (621): 507.
doi:10.1098/rspb.1917.0008.
Mandal et al. Malaria Journal 2011, 10:202; Fig 2.
Key Insight: Epidemic Thresholds
• Thresholds are criteria (mathematical inequalities) used to
predict qualitative outcomes. (“Dynamical systems” terms)
• Ex. Water can be a solid, liquid or a gas at sea level:
Solid if T < 0⁰C, Liquid if 0⁰C < T < 100 ⁰C, Gas if T > 100 ⁰C
• Ex. Allee effects for endangered species: Initial size too low,
populations die out. Above threshold, populations grow.
VERY common threshold in biology:
Population growth or extinction determined by the average
lifetime reproductive output of individuals. Greater than 1,
growth, less than 1, decline.
“SI” Model of Disease Transmission
Assume:
1) Births = Deaths
2) This implies N=S+I constant
3) Infection is life long
dS/dt = m(S+I) – βSI – mS
dI/dt = βSI – mI
N constant, so no need for dS/dt!
dI/dt = β(N-I)I – mI
dI/dt = β(N – m/β – I)I
Equilibrium number infected?
dI/dt = 0 “no change over time”
I=0 or I = N – m/β
Growth vs. decline in I?
dI/dt positive or negative?
0< I < N – m/β
which requires βN/m > 1
The Model
n
x
y
Consumer/Host
Parasite/Disease
Basic Reproduction Number
Basic Reproduction Number
Host-Resource Dynamics
Host-Resource Dynamics
Three-species Dynamics
Three-species Dynamics
Q1: Why bistability?
Why Bistability?
Simplify the model to find out!
First, transform the direct transmission (3D) model by
(n, x, y) –> (n, p = x+y, i = y/p)
This gives,
Host-Resource Dynamics
Host-Resource Dynamics
Host-Resource Dynamics
Host-Resource Dynamics
Host “Hydra Effect”
Hydra Effect +
Density Dependent Transmission
Bistability Summary
1. Bistability arises in these models due to
•
•
The presence of a “Hydra Effect” in the hostresource (Rosenzweig-MacArthur) model, and
Transmission is density dependent.
2. Such dynamics are only likely in nature when
disease mortality is strong enough to
“stabilize” consumer-resource cycling.
3. Hydra effect in tri-trophic foodweb models?
Three-species Dynamics
Three-species Dynamics
Constant vs. Resource-Dependent?
Predict by Relative Sensitivity (Slope)
Summary
1. Resource-dependent epidemiological processes
can significantly affect disease dynamics!
2. Qualitatively, “simple” models still perform well.
3. We can (usually) predict whether resourcedependent epidemiological rates will have a
stabilizing or destabilizing effect on the threespecies equilibrium using relative sensitivities.
4. New insights into other three-species dynamics!
5. New math questions!
Part II
Hypoxia in Lake Erie’s Central Basin:
How Annual Variation In Temperature,
Dissolved Oxygen Affect Fishes
Paul Hurtado (OSU), Yuan Lou (OSU), Elizabeth Marschall (OSU),
Kevin Pangle (CMU) & Stuart Ludsin (OSU)
(Work in Progress!)
Based on Hadley Centre HadCM3 climate model
http://www.globalwarmingart.com/wiki/File:Global_Warming_Predictions_Map_jpg
Hypoxia (Low Dissolved O2)
• Worldwide water-quality problem
• Hypoxia can
– reduce fish growth rates and induce stress
– reduce foraging success
– force fish to aggregate in a 1-2m band of the
water column, and
– reduce availability of predation refugia.
• Consequences for fish disease poorly known.
• Warming climate = more severe hypoxia.
Ultimate Goal
Data from Lake Erie: 1987-2005
Model
We model the Central Basin of Lake Erie, ignore horizontal space, dividing
the water column into 24 “patches” each roughly 1 meter deep.
N1
N1
N2
N2
N3
N…
N24
Leave based on
“patch” quality
Redistribute
N3
N…
N24
Model
We model the Central Basin of Lake Erie, ignore horizontal space, dividing
the water column into 24 “patches” each roughly 1 meter deep.
N1
N1
N2
N2
N3
N…
N24
∑mξiNi
mфi(N)
N3
N…
N24
Model
We model the Central Basin of Lake Erie, ignore horizontal space, dividing
the water column into 24 “patches” each roughly 1 meter deep.
Population Dynamics
Population Dynamics
Population Dynamics
Problem: Quality Index ri
• Movement based on r = G/μ can lead to high
tolerance for high mortality rates!
• Solution? Stimulus ≠ Response! r = g(G)f(1/μ)
f(1/μ)
1/μ
1/μ
1/μ
What next?
1. Finalize the movement model and describe the
consequences of hypoxia and variation in
hypoxia events annually from 1987-2005.
2. Incorporate multiple fish species with different
life histories, e.g., temperature tolerances.
3. Incorporate different microparasite disease
models (e.g., SI, SIR, SEI, ???).
Preliminary Results
• Hypoxia induced aggregation can increase
disease risk.
• Stress effects appear to be less significant.
• Model results are sensitive to the choice of
movement rule – need to get it right!
• It remains unclear how T and DO variation
together impact annual population dynamics.
Concluding Remarks
• Simple (and not-so-simple) models are essential
tools for studying complex biological systems.
• Plenty of room to build bridges between existing
theory in ecology, epidemiology, evolution,
physiology, physical processes, climate change, …
Doing it right requires good data!
• Biology is experiencing a massive quantitative
revolution & fueling advances in math, stats,
physics, computing, … all of the other sciences.
Questions?