Abstract
This thesis focuses on the spread of organisms in both ecological and epidemiological
contexts. In most of the studies presented, displacement is modeled with a spatial
kernel function, which is characterized by scale and shape. These are measured by the
net squared displacement (or kernel variance) and kurtosis, respectively. If organisms
disperse by the assumptions of a random walk or correlated random walk, a Gaussian
shaped kernel is expected. Empirical studies often report deviations from this, and
commonly leptokurtic distributions are found, often as a result of heterogeneity in the
dispersal process.
In two of the included papers, the importance of the kernel shape is tested, by using a
family of kernels where the shape and scale can be separated effectively. Both studies
utilize spectral density approaches for modeling the spatial environment. It is
concluded that the shape is not important when studying the population distribution in
a habitat/matrix context. The shape is however important when looking at the
invasion of organisms in a patchy environment, when the arrangement of patches
deviates from randomly distributed. The introduced method for generating patch
distribution is also compared to empirical distributions of patches (farms and old
trees) and it is concluded that the assumptions used for modeling of the spatial
environment are consistent with the observed patterns. These assumptions include
fractal properties such that the same aggregational patterns are found at different
scales.
In a series of papers, movements of animals are considered as vectors for betweenherd disease spread. The studies are based on data found in databases kept by the
Swedish Board of Agricultural (SJV), consisting of reported movements, as well as
farm location and characteristics. The first study focuses on the distance related
probability of contacts between herds. In the following papers, the analysis is
expanded to include production type and herd size. Movement data of pigs (and cattle
in Paper I) are analyzed with Bayesian analysis, implemented with Markov Chain
Monte Carlo (MCMC). This is a flexible approach that allows for parameter
estimations of complex models, and at the same time includes parameter uncertainty.
In Paper IV, the effects of the included factors are investigated. It is shown that all
three factors (herd size, production type structure and distance related probability of
contacts) are expected to influence disease spread dynamics, and the production type
i
structure is found to be the most important factor. This emphasizes the value of
keeping such information in central databases. The models presented can be used as
support for risk analysis and disease tracing. However, reliable data is always a
problem, and implementation may be improved with better quality data.
The thesis also shows that utilizing spatial kernels for description of the spatial spread
of organisms is an appropriate approach. However, these kernels must be flexible and
flawed assumptions about the shape may lead to erroneous conclusions. Hence, the
joint distribution of kernel shape and scale should be estimated. The flexibility of
Bayesian analysis, implemented with MCMC techniques, is a good approach for this,
and further allows for implementation of more complex models where other factors
may be included.
ii
Populärvetenskaplig sammanfattning
Alla organismer måste kunna sprida sig. Att kunna uppskatta denna spridning är en av
grundstenarna för att förstå både ekologiska och epidemiologiska processer. Denna
avhandling innehåller två delar. Dels generella frågeställningar om betydelsen av
spridningskaraktärer inom ekologiska och epidemiologiska system, och dels
tillämpade studier med målet att modellera spridning av smittsamma djursjukdomar
mellan gårdar, med fokus på djurtransporter.
Det är vanligt att man representerar spridningen med en spridningsfunktion, dvs. en
funktion som beskriver spridning beroende på avståndet. Spridningsfunktionen kan
karakteriseras med två mått; dels en skalfaktor, som talar om ifall spridning generellt
sker över långa avstånd eller begränsat till kortare distanser, och dels en
spridningsfunktionens form, som avgör ifall det är stora skillnader i
spridningsavstånden. Att matematisk kunna särskilja dessa aspekter gör att man kan
studera när och hur dessa karaktärer är viktiga att ta hänsyn till när man studerar
spridning av organismer. I min avhandling visar jag bland annat att formen är viktig
att ta hänsyn till när spridningen sker mellan enheter som är aggregerade i rummet.
Det är alltså viktigt att utvärdera karaktärer på landskapet som spridningen sker i.
Detta resultat har konsekvenser för studier av invasion av främmande arter, arters
möjlighet att nyttja de resurser som finns i landskapet och smittspridning.
Det finns tydliga paralleller mellan ekologiska och epidemiologiska studier. Där man
inom t.ex. bevarandebiologin är ute efter att optimera överlevnaden hos en hos en art,
är man inom epidemiologin ute efter att få smittan att dö ut. Avhandlingen innehåller
även en serie studier som analyserar hur gårdar i Sverige är i kontakt med varandra
via djurtransporter. Detta analyseras ur ett smittskyddsperspektiv, och med syfte att
öka förståelsen för smittspridning. Alla djurtransporter av gris och nöt måste
rapporteras till jordbruksverket, och utifrån detta har jag försökt beskriva
(matematiskt) sannolikheten för att en gård skicka djur till en annan. Grisnäringen är
väldigt strukturerad i Sverige. Olika besättningar fokuserar på att t.ex. avel,
slaktgrisproduktion eller uppfödning av gyltor (suggor kallas gyltor fram till och med
att de fött kultingar en gång). En mer detaljerad analys har därför utförts på
gristransporter. Sannolikheten för kontakt mellan gårdarna har uppskattats beroende
på avstånd mellan gårdarna, besättningsstorlek och produktionstyp. Alla faktorerna är
viktiga att ta hänsyn till, men produktionsinriktningsstrukturen är den faktor som mest
påverka smittspridningsdynamiken. Detta är en del av ett långsiktigt projekt vars mål
iii
är att kunna simulera smittspridning i Sverige av smittsamma sjukdomar, som t.ex.
mul- och klövsjuka.
iv
Table of Contents
Abstract ........................................................................................................................... i
Populärvetenskaplig sammanfattning ........................................................................... iii
Publications included in thesis ..................................................................................... vii
Publications not included in thesis .............................................................................. viii
Abbreviations ................................................................................................................ ix
1 Introduction ............................................................................................................. 1
1.1 Aims ................................................................................................................. 3
2 The spatial kernel .................................................................................................... 5
2.1 Movement, dispersal and the spatial kernel ..................................................... 6
2.2 Kernel kurtosis and disease spread ................................................................. 11
2.2.1 Kurtosis as measurement of heterogeneity in the spatial process ........... 11
2.2.2 The spatial kernel in Papers I and III ....................................................... 13
2.2.3 Kurtosis and stochastic disease transmission .......................................... 14
2.3 Relative and absolute distance dependence.................................................... 14
3 The spatial context ................................................................................................ 19
3.1 Segmented landscapes .................................................................................... 20
3.2 Point pattern landscapes ................................................................................. 21
4 Modeling of between-herd contacts ...................................................................... 24
4.1 Data................................................................................................................. 25
4.2 Overview of Papers I – IV .............................................................................. 26
4.3 Use of the model in risk assessment ............................................................... 30
5 Bayesian analysis and Markov Chain Monte Carlo .............................................. 33
5.1 A small bibliographic study ........................................................................... 35
5.2 Bayesian and frequentistic statistics ............................................................... 37
6 Conclusions and recommendations for further research ....................................... 41
6.1 The spatial kernel ........................................................................................... 41
6.2 Neutral point pattern landscapes .................................................................... 42
6.3 Disease spread modeling ................................................................................ 42
6.3.1 Further development of the contact model .............................................. 43
6.3.2 Application to specific diseases ............................................................... 44
Acknowledgements ...................................................................................................... 45
References .................................................................................................................... 47
v
Publications included in thesis
This thesis is based on the following seven papers, which will be referred to in the
text by their Roman numerals:
I.
Lindström, T., Sisson, S.A., Nöremark, M., Jonsson, A., Wennergren, U. 2009.
Estimation of distance related probability of animal movements between
holdings and implications for disease spread modeling. PREVENTIVE
VETERINARY MEDICINE. 91, 85–94.
II.
Lindström, T., Sisson, S.A., Stenberg Lewerin, S., Wennergren, U. 2010.
Estimating animal movement contacts between holdings of different
production types. PREVENTIVE VETERINARY MEDICINE. In press.
doi:10.1016/j.prevetmed.2010.03.002
III.
Lindström, T., Sisson, S.A., Stenberg Lewerin, S., Wennergren, U. Bayesian
analysis of animal movements related to factors at herd and between herd
levels: Implications for disease spread modeling. Submitted manuscript.
IV.
Lindström, T., Stenberg Lewerin, S., Wennergren, U. Expected effect of herd
size, production type and between herd distances on the dynamic of between
herd disease spread. Manuscript.
V.
Lindström, T., Håkansson, N., Westerberg, L., Wennergren, U. 2008. Splitting
the tail of the displacement kernel shows the unimportance of kurtosis.
ECOLOGY. 89, 1784–1790.
VI.
Lindström, T., Håkansson, N., Wennergren, U. The shape of the spatial kernel
and its implications for biological invasions in patchy environments.
Manuscript.
VII.
Westerberg, L., Lindström, T., Nilsson, E., Wennergren, U. 2008. The effect
on dispersal from complex correlations in small-scale movement.
ECOLOGICAL MODELLING. 213, 263–272.
Papers I, II, V and VI are reprinted with permission from the publishers.
vii
Publications not included in thesis
1. Håkansson, N., Jonsson, A., Lennartsson, J., Lindström, T., Wennergren, U.
Generating structure specific networks. ADVANCES IN COMPLEX
SYSTEMS. In press.
2. Nöremark, M., Håkansson, N., Lindström, T. Wennergren, U., Sternberg
Lewerin, S., 2009. Spatial and temporal investigations of reported movements,
births and deaths of cattle and pigs in Sweden. ACTA VETERINARIA
SCANDINAVICA. 51:37.
viii
Abbreviations
CRW
Correlated random walk
FMD
Foot and mouth disease
LDD
Long distance dispersal
MAM
Mass action mixing
MCMC
Markov Chain Monte Carlo
NPPL
Neutral point pattern landscape
RW
Random walk
SJV
Swedish board of agriculture
ix
1 Introduction
This thesis focuses on the spread of organisms and in the context of both ecological
and epidemiological studies. At first glance, these may seem like diverse topics, but
largely they are based on very similar concepts. Epidemiology is often regarded to
have been founded by Hippocrates of Cos (460 BC to 377 BC) who introduced the
word epidemic and was the first to consider the distribution and spread of disease in a
rational manner (Merrill and Timmreck 2006). A modern definition of epidemiology
can be found in Smith (2006, p.1), where it is described as the study of “distribution
and determinants of disease in populations”. The word epidemiology is for some
authors reserved for human diseases (Kleinbaum et al. 2007), but Dohoo et al. (1994)
points out that there are no linguistic or scientific reasons to make such distinction.
The definitions of ecology also differs somewhat between authors, but most
definitions are based on that given by Ernst Haeckel 1866, here with translation given
in Worster (1994, p.192), ”the science of the relations of living organisms to the
external world, their habitat, customs, energies, parasites, etc.”. This definition is in
many ways similar to the more recent (as well as more poetic) definition given by
Ehrlich and Roughgarden (1987, p.vii): “Ecology is about the living beings of earth
and how they interact with one another and with their nonliving environments”.
By the definitions above, it is clear that the two fields are concerned with similar
topics, and an alternative (and perhaps somewhat more provoking) definition of
epidemiology given by Smith (2006, p.1) is that “Epidemiology is nothing more than
ecology with a medical and mathematical flavor”. That aside, many concepts
developed in ecology have found their way into epidemiological research of
infectious diseases. One example is given by Heesterbeek (2002), who outlines the
history of the basic reproductive number, 𝑅0 , defined as the number of offspring
produced by one individual (or sometimes female). Other examples are the
relationship between the SIR (Susceptible-Infected-Recovered) models and Levins
metapopulations (Grenfell and Keeling 2007) as well as cyclic behavior of hostpathogen fluctuations (Bjørnstad et al. 2002) and traveling waves in biological
invasions (Mollison 1977).
Given the definitions of both research fields, it should also be easily realized that the
spatial factor is a major issue within both ecological and epidemiological research.
Any organism (human, animal, pathogen or other) interacts with both their biotic and
abiotic environment in a spatial context. As this environment is typically both
1
spatially and temporally variable, all organisms must rely on the ability to disperse
and colonize new habitat or hosts, and features of these dispersal abilities are
important factors of the fitness of the organism (McPeek and Holt 1992, Hovestadt et
al. 2001, Achter and Webb 2006, Phillips et al. 2008). In the field of landscape
ecology, this spatial context is often referred to as a landscape (Keitt 2000) and, since
the landscape can have diverse characteristics; we may expect different aspects of the
dispersal ability to be important in different landscapes.
While epidemiology and ecology share many features, studies of the former must
(among others) include other pathways for dispersal of pathogens. Four papers
included in this thesis focus on veterinary epidemiology and more specifically on
transmission of diseases between livestock herds. Pathways such as wind dispersal
(e.g. Crauwels et al. 2003, Garner et al. 2006) and insect vectors (e.g. Erasmus 1975,
Murray 1995) are pathways with clear resemblance to ecological studies, but
transmission between herds is (of course) also influenced by human activities. Human
induced contacts between herds pose a great risk for transmission of many livestock
diseases, and while these contact cannot be understood on the premises of ecological
research, they commonly still have a spatial element (Robinson and Christley 2007,
Velthuis and Mourits 2007, Ersbøll and Nielsen 2008, Dubé et al. 2009, Nöremark et
al. 2009a).
One type of contact that is of particular interest is the movement of animals between
herds, and this is the focus of Papers I – IV. While different diseases can be
transmitted by different pathways, the introduction of an infected animal to a new
herd is always a major risk factor for infectious diseases (Févre et al. 2006, OrtizPelaez et al. 2006, Rweyemamu et al. 2008). The importance of animal movements
for disease transmission has long been recognized, and Fischer (1980) states that a
changed attitude towards animal movements contributed to a decrease in transmission
of cattle disease at the end of the 19th century. Special attention to animal movements
between herds has received particular interest after the UK 2001 outbreak of foot and
mouth disease (FMD), where animal movements contributed to both large number of
transmissions between herds as well as many long distance transmissions (Ferguson et
al. 2001, Keeling et al. 2001, Kiss et al. 2006, Dubé et al. 2009).
2
1.1 Aims
The unifying factor in this thesis is the analysis of (spatial) spread of organisms in
both ecological and epidemiological contexts. The general aim is to develop models
and methods to increase our knowledge regarding spread of organisms in spatially
explicit settings. While the papers included deal with similar concepts, the specific
aims are somewhat diverse and vary between the different studies. I will therefore list
the aims here and in bracket state which papers are concerned with each aim.
 To develop models that describe between-herd contact patterns via live animal
movements. This forms part of a larger research project with the long term aim
of creating predictive models for disease spread between Swedish livestock
herds, which can be used for risk assessment (Papers I, II, III, IV).
 To contribute to theory regarding the importance of the shape of the spatial
kernel (as described by kurtosis) used for modeling spread of organisms in
spatially explicit systems. The aim is more specifically to investigate when
kurtosis is expected to be important to include when modeling organisms in
spatially explicit systems (Papers V, VI).
 To develop analytical tools for translating observed data into relevant
parameters that can be used in the modeling organisms spread (Papers I, II, III,
VI, VII).
3
2 The spatial kernel
Papers I, III, IV, V and VI utilize a spatial kernel in the modeling of spread of
organisms. The term “spatial kernel”, which is more commonly used in
epidemiological studies (e.g. Keeling et al. 2001, Boender et al. 2007) is used in this
thesis. I use the term as it may apply to both ecological and epidemiological studies. It
is however known by many names, and the terminology in ecology includes dispersal
kernel (Mollison 1991, Clark 1998, Skarpaas and Shea 2007), redistribution kernels
(Kot et al. 1996, Lockwood et al. 2002, Westerberg and Wennergren 2003), dispersal
curves (Nathan et al. 2003) and displacement kernels (Santamaría et al. 2007, Preece
and Mao 2009). In epidemiological studies it is sometimes also referred to as the
contact kernel (Mollison 1977, Ferguson et al. 2001).
The spatial kernel used in this thesis describes the density at some distance 𝐷 from the
source by the function
𝐹 (𝐷, 𝑎, 𝑏) =
𝐷 𝑏
−( )
𝑒 𝑎
(1)
𝑆
where 𝐷 is the distance, 𝑎 and 𝑏 are parameters determining the scale and shape of the
kernel and 𝑆 is used to normalize the distribution. For the one dimensional case it is
normalized by 𝑏⁄(2𝑎𝛤 (1⁄𝑏)) and in two dimensions by 2𝜋𝑎2 𝛤 (2⁄𝑏)⁄𝑏 . The
function has been labeled a generalized normal distribution (Nadarajah 2005) and
power exponential (Klein et al. 2006). The reason why this function is used is that it
contains some well known distributions as special cases. For parameter 𝑏 = 1 it is a
exponentially decreasing function and for 𝑏 = 2, the distribution may be rewritten as
a Gaussian distribution. These are distributions often used in both ecological and
epidemiological studies.
Rather than just describing the spatial kernel by the parameters 𝑎 and 𝑏, the kernel
can be characterized by two more relevant measures; the scale and the shape. The
scale is in this thesis measured as a two dimensional variance and calculated as the
second moment around zero (see Paper V). This measure is chosen because it, in
some instances, can be estimated as the expected net squared displacement by
analysis of small scale movement patterns of individual organisms. The shape is
measured by kurtosis, which is calculated as the fourth moment divided by the square
of the second moment (as provided in Paper V).
5
2.1 Movement, dispersal and the spatial kernel
In the beginning of the 20th century, Albert Einstein studied diffusion of molecules by
random walk movements and later, ecologists found that similar approaches could be
used to study the dispersal of living organisms (Vogel 2005). By representing the path
as a series of consecutive steps, estimation of displacement distances at larger scales
may be inferred from analysis of the small scale patterns of the steps. Ideally, these
steps should have an ecological or behavioral relevance (Turchin 1998), such as a
rhizome branching for clonal growth of plants (Cain 1990) or the flight of a butterfly
between host plants (Kareiva and Shigesada 1983). The movement path of
continuously moving organisms may also be represented by discrete steps, usually
defined by some time interval. Turchin (1998) labels these “artificial moves” and
points out that choosing an appropriate duration could be problematic, and both overand under-sampling poses problems to estimation of displacement distances. Yet,
analysis of small-scale movement patterns remains an important tool for analysis of
movement and dispersal.
θ1
l1
θ2
l2
Figure 1. Consecutive steps of a path represented by discrete moves, each with a step length, li, and
turning angle, θi. To satisfy the assumptions of a correlated random walk, no correlations are allowed
between parameters θ and l and autocorrelations are also prohibited. The distribution of θ may however
deviate from uniform between –π and π.
To satisfy the assumptions of a random walk (RW), no correlations or
autocorrelations of turning angles and step length should be present, and a move is
equally probable to occur in any direction. Deviations from RW can be implemented
in specialized RW-models. A correlated random walk (CRW) allows for non uniform
6
turning angles, making some turning angles more probable. Turning angles are
measured relative to the previous direction and the assumptions of the CRW are
violated if turning angles are distributed such that they are more probable in a
geographical direction (see figure 1). Turning angles are measured relative to previous
direction and are commonly centered around zero, making moves continuing in the
similar direction more probable. A CRW allows for relative correlations but is
violated if absolute, for example geographical, bias exists.
If a path is described by a CRW, the expected net squared displacement of the
organism after 𝑛 steps, E(𝑅𝑛2 ), can be approximated by (Kareiva and Shigesada 1983)
E(𝑅𝑛2 ) ≈ 𝑛 (E(𝑙2 ) + 2E(𝑙)2
𝜓
)
1−𝜓
(2)
where E(𝑙) is the mean move length, E(𝑙2 ) is the mean squared move length and 𝜓 is
the average cosine of the turning angle. The approximation holds for 𝑛 ≫ 1 and
symmetric distribution of turning angle, meaning that left turns and right turns are
equally probable. It is noteworthy that the expected net squared displacement
increases linearly with 𝑛.
For both RW and CRW, the spatial probability density of an organism converges to a
Gaussian distribution as 𝑛 → ∞ (Okubo 1980). The resulting probability distribution
of an organism’s spatial location relative to its start point (assuming movement on a
two dimensional surface) is then
𝑃(𝑥, 𝑦|𝜎 2 ) =
1
2
2
2
𝑒 −(𝑥 +𝑦 )/(2𝜎 )
2
2𝜋𝜎
(3)
where 𝑥, 𝑦 are the spatial coordinates. Following for example Cain (1991), 𝜎 2 = 2𝐷𝑛,
where 𝐷 is the diffusion constant used in diffusion models, which may be given from
E(𝑅𝑛2 ) via 𝐷 = E(𝑅𝑛2 )⁄(4𝑛).
Living organisms are however not molecules and therefore they may not necessarily
be modeled accurately by a random walk assumptions. The distribution of observed
dispersal distances are commonly leptokurtic, and this is found for a wide range of
taxa within both plants (Johnson and Webb 1989, Clark 1998, Clark et al. 1999,
Nathan et al. 2003, Yamamura 2004, Skarpaas and Shea 2007), lichens (Marshall
1996, Walser 2004) and animals (Inoue 1978, Fraser et al. 2001, Schweiger et al.
2004, Walters et al. 2006, Coombs and Rodríguez 2007). Leptokurtic means that the
distribution has a higher kurtosis than a Gaussian distribution. Making this
7
ecologically relevant, Clark et al. (2001, p.537) state that “…the leptokurtosis in the
distribution also produces greater distinction between short- and long-range dispersal
events”. The opposite, platykurtic distributions, are rarely found, but Holmes (1993)
showed that such distributions may be expected by the mere fact that animals have a
finite speed when moving. Various explanations have been proposed for the observed
leptokurtic distributions. It is most commonly considered a result of a heterogeneous
dispersal process (Hawkes et al. 2009) which may be due to behavioral differences
(Dobzhansky and Wright 1943, Shigesada et al. 1995, Fraser et al. 2001) or a result of
different external factors influencing the dispersal (Johnson and Webb 1989,
Shigesada et al. 1995, Clark 1998, Clark et al. 2001). Other reasons include loss of
individuals during dispersal (Schneider 1999) and temporal variation in the diffusion
process (Yamamura et al. 2007).
Figure 2. Kurtosis, 𝜿, vs. movement step using the results from resampling method 4 in Paper VII. Here,
𝟐
𝜿 was calculated as 𝜿 = 𝐄(𝑹𝟒𝒏 )/(𝐄(𝑹𝟐𝒏 )) , where 𝐄(𝑹𝟒𝒏 ) is the mean fourth powered displacement and
𝐄(𝑹𝟐𝒏 ) is the mean squared displacement. The kurtosis tends to 𝜿 = 𝟐, which is the kurtosis of a two
dimensional Gaussian distribution following the definition by raw moments in Paper V.
In Paper VII we analyzed the movement pattern of the springtail Protaphorura
armata and how the pattern changed in response to food and conspecifics. We used
expected net squared displacement as a measurement of the animals tendency to stay
in the area. Note that this is the same measure as the variance of the spatial kernel.
8
The shape of the spatial kernel was however not within the scope of this research and
therefore not included. Figure 2 shows estimations of kurtosis of resampling method 4
and how it changes with time (one step has a fixed duration of one second). As 𝑛
becomes large, kurtosis tends to 𝜅 = 2, which is the kurtosis expected of a two
dimensional Gaussian distribution (Clark et al. 1998, Paper V). While no formal proof
is provided, this indicate that inclusion of higher order correlations in movement
parameters in the resampling method change the prediction on displacement scale
(measured as E(𝑅𝑛2 )) but not the shape (measured as 𝜅).
Kurtosis is generally considered to be a measure of “shape”. More specifically, it is
defined as the fourth moment divided by the square of the second moment, the latter
being the variance. Symmetrical kernels (that is assuming equal probability of
dispersal in every direction) are centered at the origin of the dispersal event for both
one- and two-dimensional cases. Hence it makes sense to evaluate this property as
“raw moment”, meaning that the moments are evaluated around the center rather than
the mean. This distinction makes no differences in the one-dimensional case, but it
aids the interpretation for the two-dimensional case. Clark et al. (1999) points out that
there is no convention for calculating kurtosis for two-dimensional kernels, but states
that if the kernel is symmetrical, bivariate measures of kurtosis are undesirably
complex. They further argue that the moments should be defined by the distance from
zero (rather than Cartesian x-y-coordinates), which is also consistent with the
dispersal distance data often collected in field studies, and kurtosis calculated by such
moments capture the desired measure of shape.
It should be stressed that kurtosis is dimensionless. Any unit 𝑢 cancels out in the
division of the fourth moment, with unit 𝑢4 , by the square of the second moment, with
unit 𝑢2 . Thereby, the shape is not confused with the scale. In my opinion this makes
the analysis of the spatial kernel by its moments superior to more subjective studies of
long distance dispersal (LDD), where LDD events are defined by some fixed distance
or some (arbitrary) percentages of the observed dispersal distances (Nathan 2006).
Many studies have focused on the kernel characteristics, in particular in relation to
invasion speed (e.g. Mollison 1977, van den Bosch et al. 1990, Shigesada et al. 1995,
Kot et al. 1996). Kot (1996) summarizes that there are three types of spatial kernels
that needs to be considered for studies of biological invasion:
1) Kernels with exponentially bounded tails. Invasions with such dispersal show a
constant invasion rate where the range of the population expands linearly (or,
equivalently, the square root of the area increases linearly). The kernels used in
9
Papers I, III, IV, V, VI and VII falls under this category for parameter 𝑏 ≥ 1
1
(i.e. 𝜅 ≤ 3 ).
3
2) Kernels without exponentially bounded tails but with finite moments. Such
kernels give rise to accelerating invasions. The kernels used in Papers I, III, IV,
1
V and VI falls under this category for parameter 𝑏 < 1 (i.e. 𝜅 < 3 )
3
3) Kernels without finite moments. These are extremely fat tailed kernels, and
analytic approximations of invasion speed is only possible for special cases.
The kernel used in the thesis cannot produce such behavior.
For the latter, it may seem unrealistic that the dispersal distances are characterized by
non finite moments, but Levy flights, which fall under this category, have shown good
fit with observed data (Viswanathan et al. 1996, Schick et al. 2008). However,
Skarpaas and Shea (2007, p.421) points out that “all realized dispersal distributions
consist of a finite number of dispersal events and therefore have exponentially
bounded tails”. Also, accelerating invasions may be the result of other factors than the
lack of exponentially bounded dispersal distances. For example, recent studies of cane
toad invasion in Australia conclude that the accelerating invasion speed is a result of
heterogenic environments and different evolutionary pressures at the front of the
invasion (Urban et al. 2008).
Spatial kernels (though most often not referred to as such) have also been
implemented in metapopulation studies. In an effort to introduce spatial realism,
Hanski (1994) introduced the incidence function. In Hanski’s formulation,
colonization between patches depending on distance is assumed to follow a negative
exponential distribution (i.e. 𝑏 = 1 in the spatial kernel used in this thesis). Hawkes
(2009) points out that while some authors, such as Heinz et al. (2006), have
incorporated different functions, metapoulation studies have generally continued to
utilize the negative exponential function. Hawkes also points out that the empirical
evidence clearly shows that deviations from this should be expected, and one should
not use the negative exponential function by default.
10
2.2 Kernel kurtosis and disease spread
Many epidemiological studies of infectious diseases have implemented spatial kernels
(Mollison 1977, van den Bosch et al. 1990, Mollison et al. 1994, Keeling et al. 2001,
Ferguson et al. 2001, Boender et al. 2007, Boender et al. 2008). Mollison et al. (1994,
p.120) states that “Key questions for spatial disease dynamics concern long distance
contacts…” and they conclude that analysis of transport pattern is a promising
approach. It is from this insight that the analysis in Papers I and III estimates the
spatial kernel of between-herd contact probabilities via animal movements. In
ecological studies, a Gaussian shape may (theoretically) be assumed, but
mechanistically based assumptions regarding kurtosis of the kernels used for human
induced contacts are difficult to obtain. There is no underlying random walk process,
and therefore no reason to assume a Gaussian distribution.
2.2.1 Kurtosis as measurement of heterogeneity in the spatial process
There are also no mechanistic reasons for assuming a (discrete or continuous) mixture
of Gaussian distributions, as is often done in ecological studies (Hawkes 2009) when
explaining leptokurtosis. In Paper III we however state that kurtosis can be interpreted
as a measure of heterogeneity of the spatial process (such as dispersal, colonization or
infection), even though we do not assume that there is a mixture of Gaussian
processes. Intuitively this makes sense, and here I present a further analysis of the
subject.
Assume a mixture of processes, where process 𝑖 is described by a spatial kernel 𝑓𝑖 (𝑥)
(where 𝑥 is the distance) with kurtosis 𝜅̂ , meaning that all processes have the same
kurtosis. Kurtosis is defined as the fourth moment, 𝜇4 , divided by the square of the
second moment, 𝑉 2 . Therefore, for each process 𝑖
∞
∫0 𝑥 4 𝑓𝑖 (𝑥)𝑑𝑥
𝜇4𝑖
=
= 𝜅̂
𝑉𝑖2 (∫∞ 𝑥 2 𝑓 (𝑥)𝑑𝑥 )2
𝑖
0
(4)
Hence, 𝜇4𝑖 = 𝜅̂ 𝑉𝑖2 . The mixture process can then be summarized by a resulting
mixture distribution 𝐹, consisting of a fraction 𝑤𝑖 (where ∑𝑖 𝑤𝑖 = 1) of every mixture
component 𝑖. Then the fourth moment of 𝐹, 𝜇4𝐹 , is
11
∞
𝜇4𝐹 = ∫ 𝑥 4 ∑(𝑤𝑖 𝑓𝑖 (𝑥)) 𝑑𝑥 = ∑ 𝑤𝑖 𝜇4𝑖
0
𝑖
(5)
𝑖
and the variance, 𝑉̅ , is given by
∞
𝑉̅ = ∫ 𝑥 2 ∑(𝑤𝑖 𝑓𝑖 (𝑥)) 𝑑𝑥 = ∑ 𝑤𝑖 𝑉𝑖 .
0
𝑖
(6)
𝑖
The kurtosis of 𝐹 is then given as
∑𝑖 𝑤𝑖 𝜇4𝑖
∑𝑖 𝑤𝑖 𝜅̂ 𝑉𝑖2
∑𝑖 𝑤𝑖 𝑉𝑖2
𝜇4𝐹
𝜅𝐹 = 2 =
=
= 𝜅̂
(∑𝑖 𝑤𝑖 𝑉𝑖 )2 (∑𝑖 𝑤𝑖 𝑉𝑖 )2
(∑𝑖 𝑤𝑖 𝑉𝑖 )2
𝑉̅
(7)
Hence, we may conclude that 𝜅𝐹 > 𝜅̂ if
2
∑ 𝑤𝑖 𝑉𝑖2
𝑖
> (∑ 𝑤𝑖 𝑉𝑖 )
(8)
𝑖
(both 𝜅𝐹 and 𝜅̂ are positive).
From equation 6 we know that 𝑉̅ is real (both 𝑤𝑖 and 𝑉𝑖 are real and positive for all 𝑖)
and therefore
∑ 𝑤𝑖 (𝑉𝑖 − 𝑉̅ )2 ≥ 0
(9)
𝑖
Expanding equation 9 we obtain
∑ 𝑤𝑖 (𝑉𝑖2 − 2𝑉𝑖 𝑉̅ + 𝑉̅ 2 ) ≥ 0
(10)
𝑖
and if this is rewritten as three separate summations, then
∑ 𝑤𝑖 𝑉𝑖2 − ∑ 2𝑤𝑖 (𝑉𝑖 𝑉̅) + ∑ 𝑤𝑖 𝑉̅ 2 ≥ 0.
𝑖
𝑖
(11)
𝑖
where ∑𝑖 2𝑤𝑖 (𝑉𝑖 𝑉̅ ) = 2𝑉̅ ∑𝑖 𝑤𝑖 𝑉𝑖 = 2𝑉̅ 2 and
∑𝑖 𝑤𝑖 = 1). Equation 11 can then be rewritten as
∑𝑖 𝑤𝑖 𝑉̅ 2 = 𝑉̅ 2 ∑𝑖 𝑤𝑖 = 𝑉̅ 2
∑ 𝑤𝑖 𝑉𝑖2 − 2𝑉̅ 2 + 𝑉̅ 2 = ∑ 𝑤𝑖 𝑉𝑖2 − 𝑉̅ 2 ≥ 0.
𝑖
𝑖
12
(12)
(since
By the definition of 𝑉̅ in equation 6, this is equal to
2
2
∑ 𝑤𝑖 𝑉𝑖2 − (∑ 𝑤𝑖 𝑉𝑖 ) ≥ 0 ⇔ ∑ 𝑤𝑖 𝑉𝑖2 ≥ (∑ 𝑤𝑖 𝑉𝑖 ) . (13)
𝑖
𝑖
𝑖
𝑖
This is the essential relationship from equation 8 and we may conclude that any
mixture distributions of components with the same arbitrary kurtosis 𝜅̂ will have
kurtosis larger than 𝜅̂ unless 𝑤𝑖 = 1 for exactly one component (meaning that there is
only one component) or 𝑉𝑖 = 𝑉̅ for all 𝑖 (meaning that all mixture components are
identical). This supports the interpretation of kernel kurtosis in Paper IV, that kurtosis
may be seen as a measure of the heterogeneity of the spatial processes.
2.2.2 The spatial kernel in Papers I and III
Papers I and III analyses the spatial kernel of between-herd contacts, with the aim of
making inference about disease spread modeling. In Paper I all farms are assumed to
have the same characteristics and only one kernel is fitted to describe distance
dependence. However, it is concluded that the data is better described as a mixture of
distance dependent contacts (described by the spatial kernel) and non distance
dependent contacts, the latter following mass action mixing (MAM) assumptions.
This gives a better fit for both long and short distance contacts. In Paper III the
contacts are described using a separate kernel for every combination of production
types of the start and end herd. In this study, the MAM component is not included,
mainly for simplicity. While exclusion of the MAM component in some sense is a
simplification, many more parameters are used to describe the distance dependence.
There are eight production types included in the study, and each kernel is defined by
two parameters ( 𝑉 and 𝜅 ) adding up to 128 parameters (though not completely
independent since hierarchical priors are used in the estimation). The spatial kernels
are found to differ between production types, both in terms of the scale and shape
aspects. Further, a good fit is found (by visual comparison with the data) when each
kernel is fitted to describe the contacts between herds of two production types, rather
than treating all herds as equal (as is done in Paper I). When the kurtosis of the
kernels fitted in Paper III are compared to the kurtosis of a single kernel fitted in
Paper I (from analysis excluding the MAM component), it is shown that most kernels
fitted in Paper III have a lower kurtosis. Hence, a lower kernel kurtosis is found for
contacts between herds of two specific production types, which can be explained by
13
this being a more homogenous process. This is well in line with the interpretation of
kurtosis as a measure of heterogeneity of the spatial process.
2.2.3 Kurtosis and stochastic disease transmission
High kurtosis may also cause stochastic disease transmission dynamics (Molison
1977). If there is a strong spatial component such that infections are more likely to
occur at shorter distances, local susceptibles will rapidly become depleted. Long
distant transmissions, described by the fat tail of the leptokurtic kernel, may then
transmit the disease to distant, virgin areas. When this happens, there will be a rapid
increase of infected units in the new area and Keeling et al. (2001) states that such
dynamic was observed in the UK 2001 outbreak of FMD. The kernel shape is
therefore an important feature for disease spread modeling, but it may be difficult to
make assumptions about this characteristic. Hence, rather than assuming a fixed
shape, the joint distribution of both shape and scale should be estimated, as is done in
Papers I and III.
2.3 Relative and absolute distance dependence
It is of course possible to study the spatial aspect of for instance between-herd
contacts without implementation of a spatial kernel. For instance, in Nöremark et al.
(2009a) we concluded that the large amount of long distance animal movements
between herds support an initial total standstill (meaning no animal movements
allowed) in all of Sweden, rather than a regionalization, in the case of an outbreak of
for instance FMD. However, it is not straightforward to implement observed distances
directly for modeling purposes. This is particularly the case if the herds are not
randomly distributed, which is found for both cattle and pig herds in Sweden. These
are mainly located in the south and along the coast. Further, in Paper VI it is
concluded that the distributions of herds of both species show a non random
distribution over multiple scales. Figure 3 shows the distribution of between-herd
distances of Swedish pig herds (data from Paper III). This would be the expected
distribution of movement distances if there was in fact no distance dependence. Thus,
merely analyzing the observed distances may give the false impression that there is a
strong spatial component, when in fact the movements only reflect the distribution of
the herds.
14
Figure 3. The distribution of distances between Swedish pig herds.
The actual distance dependence in the probability of contacts between herds clearly
needs to be separated from the spatial distribution of herds. This reasoning also holds
for ecological studies, and the observed non random distribution of trees presented in
Paper VI stresses the importance of separating this distribution from the dispersal
distances when modeling dispersal in this system.
In Paper VI spatial kernels are implemented in two different ways, corresponding to
different assumptions on the spatial processes. These are referred to as relative and
absolute distance dependence, respectively. In the latter, the distribution of other
patches (I here use the term patch as in Paper VI to refer to both habitats in ecological
settings and infective units in epidemiological settings, and I will use the term
landscape to refer to patch distribution) does not influence the probability of infective
contacts or dispersal events (I will refer to either of these as colonization) between
two patches. The probability of colonization from one patch to another depends only
on the distance between the two patches. In relative distance dependence, the
colonization probability is also dependent on other patches and the total colonization
probability from a patch is distributed over all other patches. This approach is used in
the modeling of animal movement contacts (Papers I, III and IV). If absolute distance
dependence is used instead, it would correspond to the assumption that more animal
movements are shipped from herds in areas with many other herds. In my opinion, the
15
assumption of the relative distance dependence is better fitted to epidemiological
studies focusing on disease transmission via animal movements.
The assumptions of absolute distance dependence better agrees with passive dispersal
in an ecological setting (and also with wind dispersal of pathogens). For instance,
seed dispersed by wind will land in some location independent of the distribution of
suitable habitat patches. The probability of dispersal decreases with distance, but the
parent plant cannot control weather the seed lands in a suitable habitat or in a hostile
matrix where it is unable germinate. Hence, the distribution of other patches does not
influence the probability of colonization from one patch to another and a higher
colonization rate should be expected in areas with high density of patches.
Animals with active dispersal can be argued to follow both assumptions. As they may
continue to move until they encounter a suitable habitat patch they could be assumed
to follow the suppositions of the relative distance dependence. However, unlike the
trucks moving animals between herds, there is a probability that the animal does not
find a suitable patch, in particular if there are great distances involved. Hence
mortality during dispersal needs to be taken into account. While general models, such
as those used in Papers V and VI, may be used to provide valuable information about
population processes, any model that is applied to a specific system needs to be
modified to incorporate relevant factors for that particular system.
It is also noteworthy that the distinction between absolute and relative distance
dependence may have small effect if patches are randomly distributed, but as
mentioned above this is not the case in any of the distributions analyzed in Paper VI.
Figure 4 shows how absolute and relative distance dependence may differ in their
predictions of colonization depending on the patch distribution. This can be related to
the results in Paper VI, where it is shown that in random landscapes the kernel
kurtosis is not important for modeling colonization processes, and this result also
holds for non random landscapes when using the assumptions of relative distance
dependence. However, if absolute distance dependence is assumed, the kernel kurtosis
becomes highly important in non random landscapes. Further, if kurtosis is interpreted
as a measure of heterogeneity in dispersal as shown above, this means that individual
differences within a population may be highly important in heterogeneous landscapes,
when dispersal follows the assumptions of absolute distance dependence.
16
Figure 4. Demonstrating the differences between colonization modeled with absolute and relative
distance dependence. Colonization was simulated from the encircled patch (left column) for 10000 steps,
using kernels with 𝛋 = 𝟑. 𝟑𝟑 and 𝑽 = 𝟎. 𝟎𝟎𝟐𝟓(relative to the unit square). Kernels were normalized such
that the expected probability of colonization in a random patch distribution was 10% (in the relative
distance dependence it was exactly 10%) when all other patches were unoccupied and 2000 patches were
used. The right column shows the frequency of colonization distances simulated with absolute (black
bars) and relative (white bars) distance dependence in the corresponding patch distribution in the left
column. In the random distribution (top row) there is little difference between the relative and absolute
distance dependence. In the non random distributions (second and third row) the simulated distances
differ such that the relative distance dependence leads to a higher number and more long-distance
colonization when simulated from an isolated patch (second row), and the reversed is found if
colonization is simulated from a patch in a dense area (third row).
17
Figure 5. The marginal joint posterior probability density of variance and kurtosis of the kernel estimated
in Paper III for movements from Piglet producers to Fattening herds.
One might interpret this such that the kurtosis did not need to be included in the
analysis of Paper III. There are however three important factors why this needs to be
included. First, the stochastic aspect is not included in Paper VI. As mentioned in
Paper IV, we should expect more stochastic dynamic when the kernel is leptokurtic.
Secondly, the information can be useful when the specific contact probabilities of
individual herds are of interest (see section 5.3). Thirdly, the estimation of the
variance and kurtosis is not independent. As demonstrated in figure 5, there can be a
clear correlation in the posterior probability of these measures. If the kernel shape is
assumed beforehand, one will likely obtain an erroneous estimate of the variance. A
joint probability of the scale and shape should therefore be estimated.
18
3 The spatial context
The studies in Papers I, III, IV, V and VI concern modeling of organisms in a
spatially explicit context, meaning that the location of every object of interest (in
these studies habitats and/or farms) has a specified location in a heterogeneous
landscape (Dunning et al. 1995). Spatially implicit models may be appropriate for
studying some population processes, mainly when the key interest is the effect of
subdivided populations or communities. These models have been used to study for
instance source/sink dynamics (Pulliam and Danielson 1995), population synchrony
(Ruokolainen et al. 2009), food web stability (Holt 2002) and drug resistance at
hospitals (Smith et al. 2004). However, spatially implicit models suffer from lack of
realism and are unable to incorporate important factors such as distance dependence
in the process studied (Dunning et al. 1995, Débarre et al. 2009).
Furthermore, one of the key features of spatially explicit models is the ability to
include the spatial arrangement of a landscape. This is a central issue in Papers V and
VI, where the interplay between landscape features and kernel properties are
considered. Spatial processes are modeled with neutral landscapes, defined by Keitt
(2000) as landscapes where the value at any point in the landscape can be considered
random. Keitt also pointed out that this does not exclude models with spatial
autocorrelation. In the studies presented in both Papers V and VI, spatially explicit
landscapes are generated with a spectral density approach. This is given as a two
dimensional extension of spectral time series analysis. Any time series can be
represented by a set of cosine terms and a random time series is made up of terms
where there is no relationship between the amplitude and the frequency, and such time
series is referred to as white noise. If instead there is a relationship such that low
frequencies have higher amplitude, a smoother time series is obtained and this is
referred to as red noise. Figure 6 shows schematically the relationship between the
cosine terms and the resulting time series obtained by summation of the terms. A
measurement of the noise color can be obtained by plotting log(frequency) vs.
log(amplitude) and calculating 𝛾 as the negative slope of a line fitted to the plot.
Hence, a large 𝛾 means that the time series is autocorrelated (red) and if 𝛾 = 0 the
time series is random (white). A negative 𝛾 (referred to as blue noise) means that
there the high frequencies are dominant and a negative autocorrelation is obtained.
Analysis of the spectral density by 𝛾 assumes a linear relationship between the
log(frequency) and log(amplitude) and is usually referred to as 1/f noise.
19
Figure 6. Describing how different relationships between amplitude and frequency produce different
time series characteristics. The left columns show sine waves where there is no relationship between
frequency and amplitude of the waves (top row) and waves where high frequency waves have a lower
amplitude (bottom row). The summation of the waves are shown in the right column and in the lower
row the low frequency waves dominate the pattern and yield a smother time series.
3.1 Segmented landscapes
The spatially explicit studies presented in this thesis all focus on two dimensional
space, and Keitt (2000) show how the same approach may be used to analyze and
generate two dimensional grid data. In Paper V, the continuous values of the
generated grid are digitalized to obtain a segmented representation of the landscape,
such that it consists of cells that are either habitat or matrix. The value of 𝛾 then
determines the arrangement of the habitat cells, such that they are randomly
20
distributed or show a clustered pattern. A different way of generating such pattern is
the algorithm developed by Hiebler (2000), which rearranges habitat patches such that
a grid landscape is obtained, where every cell has a specified probability of
neighboring cells having the same property (that is habitat or matrix). Conceptually
(as well as methodologically) these methods differ in that Hieblers nearest neighbor
method considers the spatial autocorrelation at a specific scale, while the spectral
density approach considers the overall structure of habitat arrangement.
3.2 Point pattern landscapes
For some studies, it may be more convenient to represent the landscape by the
coordinates of the patches as a point pattern. In Paper VI, a spectral density approach
is used to generate neutral point pattern landscapes (NPPL) and also to analyze
empirical distributions. Figure 7 show the procedure of generating the NPPL. First, a
grid surface consisting of randomly distributed random numbers is generated.
Secondly, by considering this grid in the spectral domain, the spectral density function
of the surface is filtered by introducing a required 𝛾 (introducing a relationship
between log(frequency) and log(amplitude)), then transforming the spectral domain
into a new grid. This step introduces the spatial autocorrelation measured as
continuity.
In the third step, the contrast of the NPPL is introduced. This is to be interpreted as a
measure of how different dense and sparse areas are, while the continuity measures
how areas with similar density are located relative to each other. Landscapes with
high continuity are characterized by large areas of similar patch density. The values of
the grid are transformed by a method based on spectral mimicry, as introduced for
time series by Cohen et al. (1999). This involves replacing the values in the grid by
values from a wanted distribution, thereby controlling the mean and variance, but at
the same time keeping the autocorrelation structure as introduced by 𝛾. Cohen et al.
substitute the values of the grid, which are normally distributed, with random values
from another a normal distribution of. In Paper VI, the method is instead used to
substitute the values of the grid with random values from a gamma distribution. A
normal distribution has negative values, and this is not desirable if the surface is to be
used as a probability density for patch distribution. By transforming the grid values to
obtain gamma distributed values, it is possible to get a grid with wanted coefficient of
variation, and at the same time avoiding negative values. The reason for introducing
the third step is that it was discovered that the use of a normal distribution could not
21
produce NPPL with sufficiently high contrast values as found in the observed
landscapes (particular in the analyzed distributions of trees).
In the final step, the patches are distributed according to the surface obtained by step
three. Based on a method introduced by Mugglestone and Renshaw (1996), the
contrast and continuity values may be calculated on the point pattern distribution. The
values differ from the input values of 𝛾 and coefficient of variation, and the
relationship between input and output values is found iteratively. Thereby it is
possible to obtain NPPL with comparable characteristics to empirically observed
point patterns.
The assumed linear relationship between the amplitude and frequencies implies a self
similarity over different scales, and therefore the pattern may be defined as a fractal.
This means that the same pattern found at smaller scales is also found at larger scale.
The analyzed data of both farms and tree distributions show a good fit with the linear
assumptions (figure VI-6). Halley et al. (2004) list several processes that may lead to
fractal spatial patterns. Two of these are relevant to consider for the analyzed
distributions. First, power law dispersal (such as Levy flights) may cause such
patterns. If the considered tree species have such highly leptokurtic dispersal, it could
explain the fractal distribution. Secondly, Halley et al. state that observed fractal
distributions may just reflect some other underlying fractal distribution. If suitable
farmland has a fractal distribution, it may explain a fractal distribution of farms.
Likewise, fractal distributions of the considered tree species may be explained by a
fractal distribution of suitable habitats. However, both the observed tree species and
farm distributions are the results of several different processes, including geological,
ecological (mainly for the tree species) and multiple anthropogenic factors. It may not
be expected that these factors together influence the distribution in similar fashion on
several scales. As further pointed out by Halley et al., a log-log plot may give a faulty
impression of linearity, and one should be careful when drawing conclusions about
inherent fractal properties of observed patterns. We do however in Paper VI conclude
that the linear assumption produce a good fit. In instances where this is not found, one
may use a different function to estimate the relationship between the amplitude and
frequency. This does however require additional parameters, and may make the
interpretation of these parameters more difficult.
22
Higher Continuit y
f iltering
Low er Continuity
f iltering
Low Contrast
transformation
High Contrast
transformation
High Contrast
transformation
Low Contrast
transformation
(1)
(2)
(3)
(4)
Contrast: 2.2
Continuity: 1.7
Contrast: 4.9
Continuity: 2.1
Contrast: 1.5
Continuity: 1.1
Contrast: 4.2
Continuity: 1.0
Figure 7. Illustrating the method used to create landscapes with different patch arrangement.
1. Generating a surface of random values (normal distributed).
2. Filtering to obtain surfaces of different Continuity. This measure the tendency of nearby areas to have similar patch density.
3. Transforming into gamma distributed values to incorporate Contrast and avoid negative values. Contrast is calculated as the coefficient of
variation of patch density and is a measurement of difference between dense and sparse areas.
4. Distributing habitats in proportion to the surface generated in step 3.
23
Ripley (2004) gives two other basic methods for generating non random point pattern
landscapes. Firstly, the cluster process involves initially distributing a given amount
of clusters at random locations, and subsequently distributing patches around these
points. This method may be difficult to fit to observed patterns, in particular when the
observed pattern does not show a distribution of clearly distinct clusters, as is the case
with the analyzed patch distributions in Paper VI. Secondly, a point pattern may be
generated as a Poisson process with varying intensity. Usually this is done to generate
landscapes with a trend in a specific direction, and this approach may be well fitted
for situations where such gradient is an important part of the study.
One limitation of the analysis method presented in Paper VI is that it does not provide
any information about the underlying process causing the observed pattern. This is
however also a benefit. As the method for generating NPPL relies on few assumptions
(except for the assumption of linearity in the frequency domain, which may be relaxed
if required), the method is well suited for testing the effect of observed landscape
characteristic by replicating observed distributions. While some iteration is required
to get the relationship between the input parameters and the analyzed parameters of
the generated NPPL, there is also a clear relation between these parameters.
Further, continuity and contrast may be understood as empirically relevant features of
a landscape, both relating to habitat fragmentation. This is a major issue in
conservation biology, and usually refers to the process of habitat loss resulting in
remaining habitats that show a fragmented pattern (Fahrig 1997). By looking at the
generated NPPL in figure 7 it can easily be seen that a single measure of the
fragmented state of a point pattern landscape is not sufficient. Both continuity and
contrast provide important information about the distribution of habitat patches.
Another benefit of the spectral density approach is that it may be extended to include
anisotropy, meaning that spatial relations of patches (or continuous habitats for
situation where this is a more relevant description) are not independent of the
direction. One might expect to find such a pattern if the observed distribution is
caused by some underlying process with different forces working in for example a
north-south direction than an east-west direction.
24
4 Modeling of between-herd contacts
Four of the papers included in this thesis focus on disease transmission between
livestock herds via live animal movements. Different pathogens may be transmitted
between herds via different routes, such as other farming related contacts, wild animal
vectors and wind. Relocation of infected animals is however often considered a major
risk factor (Févre et al. 2006, Ortiz-Pelaez et al. 2006, Rweyemamu et al. 2008) and a
good description of such contacts is essential for modeling of most livestock diseases.
4.1 Data
The data utilized in the studies presented in Papers I – IV are based on reports to SJV.
EU legislations state that member states must keep databases on all livestock holdings
and register movements of cattle and pigs. Many countries also include more
information than the minimum requirements. In Sweden movement data is reported
by the farmer, and for cattle movement the data is reported by the farmer at both the
receiving and sending holding, while pig movements are only required to be reported
by the farmer receiving the animal shipment. Cattle movements are reported at the
individual level, using a specific identity number for each animal, whereas pigs are
reported at the movement level.
When pig holdings are registered in the database, the farmer is required to submit a
map showing the location of the holding, as well as the production type and the
maximum capacity of fattening pigs and sows, respectively, (Anonymous 2010). No
such requirements exist for registering of cattle holdings. The coordinates found in the
database are instead approximated from a register of land use (Nöremark et al.
2009a).
The entries in the databases are derived from reports by the farmers and some degree
of erroneous and missing reports are expected. The data was therefore edited, as
described in Nöremark et al. (2009a), and cattle movements with inconsistent reports
were removed (for instance an animal movement reported by the farmer sending the
animals could not be matched with a movement reported by the receiving farmer).
Such data editing could not be performed on the pig movement data as this is based
on single reports. Also, inactive holdings are supposed to be removed from the
database, but since this is not always done, holdings that had not reported any
25
movement of animals (including movements to slaughter houses) or births or deaths
for a one year period were removed.
The production type and maximum capacities are reported for a pig holding when it is
first registered, and the information is not necessarily updated if changed. Hence, the
information registered for a holding may no longer be accurate. While databases may
provide important information, data quality will always be a major issue of concern.
This needs to be accounted for when conclusions are drawn from the analysis and
used for risk assessment. The database could be used more efficiently in risk
assessment if data quality was improved. Mainly, updates of data entries would be
particularly useful, as well as more detailed instructions to the farmers on how to
report information production types. For instance, Nöremark et al. (2009b) showed
that different farmers had somewhat different interpretations of the production types.
4.2 Overview of Papers I – IV
Papers I – IV presents the development of the contact model. More details are added
throughout this series of papers, and I here give an overview of how the model
progresses. Paper I focuses exclusively on the distance dependent aspect of betweenherd animal movements. Previous studies have reported that contacts between
holdings occur more frequently between nearby holdings (Boender et al. 2007,
Robinson and Christley 2007) and this is expected to influence the disease
transmission dynamic (Tildesley and Keeling 2009). In Paper I, two models are used
and compared in their ability to describe the probability of movements of cattle and
pigs between herds. The first model is based on the spatial kernel described in section
2 and highly leptokurtic distributions are estimated for both species. The second
model is a mixture model, where contact probability is modeled as arising from a
mixture of the spatial kernel and a MAM component. By comparing the posterior
probabilities of the model, it is shown that the mixture model is superior in describing
the observed contacts. Also, by simulating contacts with each model and comparing
the obtained structure with a set of network measures, it is shown that the models are
expected to differ in predictions of disease transmission.
Paper II is the only spatially implicit study included in this thesis. Herds are
considered as spatially separate, but their explicit locations are not included. The
paper focuses on the production type structure of pig holdings (production types of
cattle holdings are not included in the database), and the probability of animal
26
movements between holdings are modeled as dependent on this structure. As many
holdings have more than one production type, contact probabilities are modeled by
the assumption that holdings behave as a mixture of the types. Rather than assuming
equal proportions of each type, the dominance of different production types in
determining the contacts of a holding is estimated.
Pig farming in Sweden mainly follows a breeding pyramid structure (Anonymous
2009), as illustrated in figure 8, and animals are expected to mainly be moved from
holdings at the top of the pyramid and downwards. The pyramid includes four of the
seven production types from Paper II. At the top of the pyramid are the Nucleus
herds, where breeding is performed. These herds are then expected to send animals to
Multiplying herds, which produce gilts. These are mainly expected to be moved to
Piglet producers, and the piglets are then sent to Fattening herds.
Nucleus Herds
Multiplying herds
Piglet producers
Fattening herds
Figure 8. The breeding pyramid of Swedish pig farming. Animals are expected to mainly be moved
downwards in the pyramid
Farrow-to-finish herds are not included in the pyramid as the whole chain of
production is integrated within the herd. Often, however, holdings with this type are
concurrently operating as Piglet producers (Anonymous 2009). The Sow pool system
is also not included in the pyramid. A Sow pool consists of a center, where sows are
covered or inseminated, and a number of satellites. The sow is moved to the satellite
for farrowing and subsequently moved back to the center. See Paper II for more
details.
27
In general, the results presented in Paper II are in agreement with the expectations
from the structure of the pig farming industry. The predicted contact pattern generally
follows the expectations of the breeding pyramid and the Sow pool system, and also
the analysis showed that Farrow-to-finish herds are estimated to have few contacts.
However, there are unexpectedly many movements predicted between Sow pool
centers, and it is concluded that this result may be a result of erroneous reports in
database.
Paper III combines and builds on the analysis of Papers I and II, and the analysis is
further expanded to include herd sizes. The production type structure is modeled as in
Paper II and the probability of contacts depending on between-herd distances and herd
sizes is modeled as being dependent on the production types. The MAM component
of the distance dependence is however not included, and this is removed for two
reasons, the first being for simplicity. Given the large amount of parameters used in
the model presented in Paper III, it is indeed valuable to reduce these. Secondly,
describing the distance dependence strictly by the spatial kernel promotes a more
transparent interpretation of the results. One of the aims of Paper III is to compare the
parameters estimated for different production types. The variance and kurtosis
describes the scale and shape, respectively, of the distance dependence of the contact
probabilities. Adding a third quantity (the amount of MAM) makes comparison of
these features more difficult. The importance of having quantitative measures for the
scale and shape of distance dependent contact probabilities is well illustrated in this
paper. These quantities allows for comparison of the spatial component of the disease
transmission via the analyzed contacts. Further, the included simulation study
demonstrates that the scale (measured as 2D-variance) estimated for different
production type combinations also is expected to result in differences in disease
transmission at different distances.
The data is inherently weak for spatial kernels describing the distance-related
probability of movements between holdings of production types that are not
commonly in contact. To improve the estimates of these contacts, hierarchical priors
are implemented. This involves a prior, with a set of hyper parameters that are
estimated by the parameters, rather than specified beforehand. The parameters are
concurrently influenced by the hierarchical prior, and thereby estimates of parameters
are indirectly influenced by each other, a concept known as borrowing strength
(Gelman et al. 2004). Figure 9 illustrates this concept.
This approach is particularly appropriate when one expects that the parameters are not
entirely independent, which makes sense in the case with the spatial kernels used for
28
modeling of between-herd contacts. Parameters based on little (or even no) data will
be highly influenced by the hierarchical prior, while parameters estimated from strong
data is only moderately influenced by this. Without the implementation of hierarchical
priors, rare contacts can be estimated to have a resemblance to MAM, and therefore
an overestimated influence on long distance disease transmission.
Figure 9. Schematic description of hierarchical Bayesian modeling. Posterior distribution of parameters
θ1, θ2,… θn are estimated from the data and hierarchical prior ψ. The latter is estimated by the
parameters θ1, θ2,… θn (and a hyperprior, not shown in picture). Thereby, the estimation of any
parameter θk is influenced by the estimates of other parameters. This concept is referred to as borrowing
strength.
In Paper IV, the importance of the parameters included in the model presented in
Paper III are investigated and related to general theory about disease transmission
dynamics. Simulations are performed where the influence of different parameters are
removed. It is found that the production type structure is the most important factor,
and this reduces the mean epidemic size compared to a model where disease
transmission is simulated by MAM assumptions. Further, the production type
structure is shown to induce a largely stochastic behavior of an outbreak.
While the number of early infections is estimated at lower numbers in the full model
compared to MAM assumptions, it should be stressed that this is under the
assumption that the index herd is of arbitrary production type. If the index herd is of a
29
type with many contacts, for instance high up in the breeding pyramid, there will be a
rapid increase in the number of infected herds. This is illustrated by figure 10.
Figure 10. Examples of individual simulations from Paper IV with Full model (black line) and MAM
model (grey line) the number of infected herds plotted vs time. In this example, the index herd had many
contacts, and there is a rapid increase of the number of infected herd compared to a MAM model.
4.3 Use of the model in risk assessment
The long-term aim of the model developed in Papers I – III is to develop a method for
simulation of disease transmission. Therefore, the general contact pattern is estimated
based on the observed contacts between herds with different characteristics and
locations are in contact via animal movements. Papers III and IV demonstrate how
this model can be used to make general predictions of disease transmission via animal
movements. It is shown that differences depending on the index herd are expected,
both in the number of contacts as well as in the distance at which these contacts occur.
Another possible implementation of the model is to provide more detailed information
on risk assessment of individual herds. Suppose an infectious disease is discovered at
a herd, hereafter referred to as the index herd. In order to control the outbreak, it is of
great importance to map out the contacts of the index herd. The database kept by SJV
can be used to trace any reported movements from a herd, but delayed reports are
common (Nöremark et al. 2009a). The model presented in Paper III is parameterized
30
such that it describes the general contact pattern of a herd, given its location and
characteristics. By simulating animal movements from the index herd, it is possible to
obtain a picture of which herds are most likely to have had contacts with the index
herd, including secondary and higher degree infections. Figure 11 shows the result of
such simulations. Note the different patterns arising depending on the production type
of the index herd.
It should be stressed that this is not intended to be a substitute for classic disease
tracing. It may however be used as a complement, for example as prior information on
the risk of herds becoming infected via animal movement. It should also be pointed
out that further developments of the model may improve the ability to predict contacts
between holdings.
31
Figure 11. Results of simulation of disease transmission with the model, parameterization and simulation method presented in Paper IV. Each dot
shows the location of a herd and colors indicate the probability of infection. Each map show the result of 9000 simulations for a time period of 28
days, starting at the encircled, black herd, which is reported with production type (from left to right) Nucleus herd, Multiplying herd and Fattening
herd, respectively. Arrows point out the location of herds with the three highest probability of becoming infected and annotate the estimated
probability.
32
5 Bayesian analysis and Markov Chain Monte Carlo
Papers I, II and III focus largely on parameter estimation. Bayesian inference is
utilized and implemented with Markov Chain Monte Carlo (MCMC). MCMC is a
flexible analytic tool that allows for probabilistic parameter estimation of complex
models in the presence of parameter uncertainty. A Markov Chain is a discrete
random process where the current state of the chain only depends on the previous and
Monte Carlo is a term used for algorithms that repeatedly draw random numbers from
a probability distribution. It may be used for numerical integration based on random
numbers, and this integration is required in order to perform Bayesian inference. The
idea of the MCMC is to simulate (correlated) draws from the posterior distribution of
the parameters. The chain is initialized with initial parameter values (in some
parameter space where the posterior density is larger than 0) and updated repeatedly
within the Markov Chain, with the parameter values of the next state depending only
on the current parameter values. For multi-parameter models it is generally
convenient to perform updates of one parameter at the time within each step of the
chain, yet the mixing may be improved by joint updates if there are strong
correlations between parameters.
Updates are typically performed by one of the following two methods. If the
distribution of some parameter 𝑥 conditional on the other parameters, 𝜽 , is of a
standard distributional form, call it 𝑓 (𝑥 |𝜽) , then Gibbs sampling may be
implemented. This means that the parameter is updated by a random draw from the
distribution 𝑓(𝑥 |𝜽) . If the conditional distribution of 𝑥 is not of a known form,
updates are commonly performed with Metropolis-Hastings steps. This involves
proposing new values 𝑥̇ given the current value 𝑥𝑡 from some distribution, 𝑞 (𝑥̇ |𝑥𝑡 ),
and subsequently accepting or rejecting it with probability
𝑚𝑖𝑛 (1,
𝑃(𝑥̇ | … )𝑞(𝑥𝑡 |𝑥̇ )
),
𝑃(𝑥𝑡 | … )𝑞 (𝑥̇ |𝑥𝑡 )
(14)
where 𝑃(𝑥| … ) is the posterior probability of 𝑥. Commonly, random walk proposals
are utilized, and 𝑞 (𝑥̇ |𝑥𝑡 ) = 𝑞 (𝑥𝑡 |𝑥̇ ) in the case of symmetrical random walk. If the
proposed value is accepted, then 𝑥𝑡+1 = 𝑥̇ . If 𝑥̇ however is rejected, then 𝑥𝑡+1 = 𝑥𝑡 .
As the MCMC is typically initialized with some arbitrary value, the initial updates,
known as the burn-in, needs to be removed. Only after converging does the MCMC
properly sample from the posterior density. While there are analytic tools to aid the
33
assessment of convergence (e.g. Venna et al. 2003, Gamerman and Lopes 2006),
visual inspection is commonly used. It is however important to try different initial
parameter values to ensure that the MCMC converges to the same region of high
posterior probability, independent of the initial values.
Figure 12. The first 5000 updates of the MCMC estimation of the marginal posterior densities of 𝒄̇ 𝟐 in
Paper III. SPC – Sow Pool Centers, SPS - Sow Pool Satellites, FH – Fattening herds, FF – Farrow to
finish, NH – Nucleus herds, MH – Multiplying herds, PP – Piglet producers, MI – Missing information.
Figure 12 shows the initial updates of 𝑐̇2 in one Markov Chain used in Paper III. In
this example, the simulation was initialized by setting all parameters 𝑐̇2,𝑘 = 0. The
34
chain appears to have stabilized after about 300 steps. To avoid any confusion, it
needs to be pointed out that a much longer chain than shown in figure 12 is used in
Paper III. For the full analysis we simulated 7 parallel chains, each with 500000
iterations, and discarded the first 100000 updates as burn-in. Other parameters
showed poorer mixing and required a longer burn-in.
In both Gibbs sampling and Metropolis-Hastings updates, the current parameters are
given from the previous step, making the random draws from the posterior
distribution inherently autocorrelated. Hence, it is important to simulate a chain long
enough to ensure that the chain explores the whole posterior density of parameters.
High autocorrelation means that there is poor mixing and a longer chain has to be
simulated. Hence, implementation of MCMC techniques may require substantial
effort to improve the mixing. While symmetric random walk proposals, as used in the
original algorithm (Metropolis et al. 1953), are often sufficient, mixing may be
improved by having a fixed proposal distribution, called the independence MCMC
sampler. This should ideally be similar to the posterior distribution of the parameters,
but it needs to span the whole posterior distribution (or it will not sample the whole
posterior). Naturally, this may be difficult to know beforehand. In Paper I, such a
distribution is created by simulating an initial Markov Chain with random walk
Metropolis-Hastings proposals and subsequently fitting a multivariate Gaussian
distribution to the sampled values of 𝑎 and 𝑏 (see Paper I for details on the
parameters). To ensure that the distribution is wide enough, the scale is doubled.
5.1 A small bibliographic study
This section provides a small bibliographic study in order to show the trend of
Bayesian methodology and MCMC techniques in the research fields that are the focus
of this thesis. While the estimations in Papers I – III concerns description of animal
movements in the context of disease spread modeling, this thesis also contains
ecological modeling with partly similar features. An analysis for this field is therefore
also included. The ISI Web of Knowledge (http://apps.isiknowledge.com) allows for
tracking changes in research topics over time by analyzing relevant publications. The
aim in this section is to show the trend of MCMC and Bayesian implementation in the
fields of ecology and disease spread research. For ecology, publications published
between 1975 and 2009 were selected by searching for Topic=(bayes*) and
Topic=(MCMC) OR Topic=(Markov Chain Monte Carlo), respectively. Topic
terms performs search of Title, Abstract and Keywords. To obtain ecological
35
publications only, the results were filtered by Subject Areas=(ECOLOGY) and
selected relevant journals. The following journals were selected: TRENDS IN
ECOLOLOGY AND EVOLUTION, ECOLOGICAL MONOGRAPHS, ECOLOGY
LETTERS,
ECOLOGY,
AMERICAN
NATURALIST,
ECOLOGICAL
APPLICATIONS, JOURNAL OF ECOLOLOGY, JOURNAL APPLIED
ECOLOGY, OECOLOGIA, OIKOS, ECOGRAPHY, JOURNAL OF ANIMAL
ECOLOGY, ECOLOGICAL MODELING, THEORETICAL POPULATION
BIOLOGY, and PROCEEDINGS OF THE ROYAL SOCIETY OF LONDON
SERIES B.
For disease spread studies it is less apparent which relevant journals to select as these
may be found in journals focusing on quite different topics. The publications were
therefore selected based only on topic search as Topic=(bayes* disease spread) OR
Topic=(bayes* infect* disease) OR Topic=(bayes* epidemi*) and Topic=(MCMC
disease spread) OR Topic=(MCMC infect* disease) OR Topic=(MCMC
epidemi*), respectively. While using different search criteria may change the specific
results, it is unlikely to change the general trend (which is the focus of this
bibliographical study).
As shown in figure 13, few ecological or disease spread studies using Bayesian or
MCMC methods were published in either ecological or disease spread literature prior
to the mid 90’s. Much of this may be explained by the development of high
performance computers. Access to fast computers has certainly made the use of
Bayesian inference and in particular MCMC methods possible, leading some authors
(Spiegelhalter 2004) to describe the recent trend in epidemiology as an MCMC
revolution. It is theoretically possible to estimate the parameters of any mathematical
model as long as a likelihood function and prior can be formulated, yet computation
time and convergence may be a problem (O’Neill and Roberts 1999).
While Bayesian inference is used in only a minority of the published papers within
both studies of disease spread and ecology, the trend shows a clear increase. Since
Bayesian statistics has some useful properties (see below) it is likely that this trend
will continue. Implementation with MCMC relies on computational speed, and with
faster computers, a continued increase in the use of MCMC should be expected.
36
Figure 13. Publications per year with (top panel) MCMC or Markov Chain Monte Carlo and (bottom
panel) Bayes* included in title, abstract or keywords. Publications were selected through ISI Web of
Knowledge.
5.2 Bayesian and frequentistic statistics
It is not within the scope of this thesis to penetrate the deep philosophical differences
between Bayesian and frequentistic methods of inference. A profound discussion of
these two frameworks and their relation to the concepts of probability as well as
theory and models may be found in for instance Dawid (2004). Computational
37
advances are however not the only reasons why researchers are increasingly turning to
Bayesian methods. A growing skepticism against the use and relevance of for
example p-values has lead to a still ongoing debate within both ecology and
epidemiology over which method is preferable (Dixon and Ellison 1996, Dennis
1996, Burton et al. 1998, Dunson 2001, Clark 2005, Cripps 2008).
In this debate, Bayesian inference has received much criticism regarding the
subjectiveness of the prior. It is indeed true that a prior needs to be included in order
to obtain a posterior distribution (of parameters or models) as given by Bayes
theorem. However, as pointed out by Dixon and Ellison (1996), not all Bayesians use
subjective probabilities. Rather, by using uninformative priors, the researcher may let
the data inform the belief about the parameter(s) or model(s). While the prior may not
be uniform between −∞ and ∞, it may be uniform on a limited parameter space, or as
stated in Paper III, the prior is “proportional to one on the support of the parameters”.
One clear advantage of the Bayesian methods, and indeed the main reason why this
methodology is used in Papers I – III, is of a pragmatic nature. It is a flexible
framework that allows for parameterization of more complex mathematical models
than is possible with classic frequentistic methods (Dunson 2001, Clark 2005). I am
not aware of any frequentistic method that would allow for parameterization of for
example the model presented in Paper III. Moreover, Bayesian statistics may be
favorable also for simpler models as it, if required, allows for extension of the model
to include further complexity.
Another pragmatic issue is however the possibility to communicate results to other
researchers. While Lee (2004) argued that the interpretation of Bayesian statistics is
more intuitive, most scientists are more familiar with the frequentistic concepts such
as confidence intervals. Clark (2005) pointed out that their Bayesian counterpart, the
credibility intervals, are most often very similar given vague priors and similar
assumptions and for simple analyses little insight might be gained by switching to
Bayesian inference. In addition, while Bayesian methods allow for fitting of more
complex models, implementing these methods may require extensive computational
work (Clark 2005, Lawson and Zhou 2005). It is therefore, I believe, justified to use a
frequentistic method if it answers the question at hand. Dennis (1996, p.1101) argued
that “Bayesian and frequentistic statistics cannot logically coexist”. I would however
argue that such radical statement misses the question of why and how a study is
performed. The choice of epistemological framework should be based on what type of
data is available, and what the purpose of the study is. If predictions and testing
different scenarios are the central issue, as is often the purpose of applied studies of
38
both epidemiology and ecology, then I believe that Bayesian inference is the method
of choice. This is agreed upon by for instance Berger (2003), who concluded that for
epidemiological studies, frequentistic statistics may be preferred in controlled
experiments, while Bayesian methods are superior for creating public policies. Also,
Ellison (2004) argued that the Bayesian iterative process of gaining more information
through several experiments and including the result of previous studies as prior
information has proven successful for adaptive management in ecology.
A final issue supporting the use of Bayesian statistics for the models in Papers I – III
is of an epistemological kind. Frequentistic statistics is based on the probability of
observing an outcome given the model (and model parameters) and if the outcome is
too improbable, the model is falsified and discarded. Bayesian statistics is instead
concerned with the probability of the model (and model parameters) given the
observed outcome. Models may be compared by for instance the ratio of the posterior
probabilities of the models, as is done in Paper I. The frequentistic approach may be
well suited if the purpose is to find the mechanism of the underlying process causing
the observed outcome. However, for such complex processes as the system of animal
trading in the livestock sector, it is not realistic to expect to capture something even
close to a “true” model. Where and when an animal shipment is sent from one holding
to another depends on a wide range of factors. These include social and economic
aspects, legal issues (such as animal welfare legislations), population growth, logistics
and indeed highly unpredictable human behavior. Hence, we are mainly confined to
phenomenological modeling and should aspire to capture the important features of the
observed pattern. Parameter uncertainty is expected in such modeling and this
uncertainty would be disregarded if parameters are estimated with a maximum
likelihood approach. If the full posterior is estimated, the uncertainty can be
incorporated in simulations or when inference is made directly from the parameters.
39
6 Conclusions and recommendations for further research
This thesis has contributed to both ecological and epidemiological studies by
improving theoretical understanding of spatially explicit spread of organisms. Further,
models and analytical tools were developed that can be used specifically as a basis for
preventative disease spread models in the livestock sector. The following sections
outline specific conclusions and recommendations for further research.
6.1 The spatial kernel
The spatial spread of organisms is often described by a spatial kernel. By efficiently
separating the shape and the scale of the kernel, it is possible to compare the influence
of these features. The scale, in this thesis measured as 2D-variance or expected net
squared displacement, is always an important factor when modeling the spread of
organisms. The shape, measured as the kernel kurtosis, can also be interpreted as a
measure of heterogeneity in the spatial process, which in ecological studies may be
caused by individual differences. This may be important, but depends on what process
is investigated, as well as the spatial context.
Using quantitative measures for characterization of the spatial kernel by its shape and
scale also allows for comparison of different processes. These characteristics provide
insight to the spatial aspect of probabilities of between herd contacts via animal
movements. Herds of production types with kernels generally estimated to have a
large variance are expected to have many long distance contacts and thereby the
possibility to transmit diseases to distant areas. Herds of production types with
estimated kernels of generally high kurtosis are concluded to be heterogeneous in
their contact structure.
The spatial kernels estimated for the animal movements are generally leptokurtic, and
in Paper III it is demonstrated that the kernel kurtosis differs between different
production types. For both ecological and epidemiological studies it is often hard to
estimate the shape of the kernel beforehand. The kurtosis may not always be directly
important, but erroneous assumptions about kurtosis may also influence estimation of
the kernel variance. Therefore, one should be careful about making such assumptions,
and rather aim for estimation of the joint probability of scale and shape. Bayesian
inference is a promising approach for this.
41
Hawkes (2009) argues that the unquestioned use of negative exponential distribution
should be avoided, and instead scientists should use distributions based on
mechanistic reasons. While I do agree with the first part, I believe that mechanistic
deductions of distance dependence may be too complex to obtain, in both ecological
and epidemiological studies. Instead we may often be confined to phenomenological
estimations. In order to implement this, one requires flexible distributions. Further it
is important to compare the predictions of the estimated distribution with observed
data, and thereby make sure that the predictions are sufficient for both short and long
distances.
The resampling method presented in Paper VII can also be used to estimate dispersal
parameters of moving animals. It does however not account for individual differences.
Therefore, it should not be used when this is expected to be an important aspect in the
modeling scenario. If so, a further development of the method using a hierarchical
modeling approach is recommended, as introduced by Morales et al. (2004), where
individual differences can be incorporated.
6.2 Neutral point pattern landscapes
The method presented in Paper VI is a promising approach for modeling point pattern
landscapes. The landscapes are described by two parameters: the contrast, which
describes how different dense and sparse areas are, and the continuity, which
describes if areas of similar density are located nearby each other or scattered in the
landscape. These measures can also be estimated from observed data, and thereby
modeling results can be related to real situations.
Further development of the NPPL methodology should focus on introducing classes
of the patches. In an ecological setting these may represent habitat qualities or age
classes (which is relevant for modeling of organisms associated with trees). In
epidemiological modeling of transmission of livestock disease, such classes may
represent production types or herd sizes.
6.3 Disease spread modeling
The model developed can be used to aid risk assessment as a complement to classical
methods. Contact tracing based on the database of reported movements may in initial
42
stages of an outbreak be difficult due to delayed reports. Modeling contact scenarios
can therefore aid the early phase of disease control.
The model can also be used to provide insight to dynamics of disease transmission via
the analyzed contacts, as well as sort out which factors are most important to consider
in modeling the system. Thereby, the importance of reliable data can be pinpointed. It
is concluded that for modeling of disease spread via pig movements, the production
type structure is the most important factor to consider. Improving data quality of this
can improve the possibility to use the database in risk assessment.
6.3.1 Further development of the contact model
To develop a model, it may sometimes be necessary to “squint” a bit and focus on the
big picture. It is however important to retrospectively open the eyes and see what
details may have been missed. Animal movements between herds are influenced by
numerous factors, and the model presented should not be expected to have captured
all relevant features. The main excluded aspect, perhaps, is the reoccurrence of
contacts between the same herds. In particular, this is expected to be an important
aspect of the contacts between the actors of the Sow pool system. If recurrent contacts
are included in the model, this is expected to reduce the number of infections and this
effect may particularly be important at later stages of an outbreak. It may also be used
to provide better estimates when used for risk assessment. If recurrent movements of
animals between holdings are common, prior reported movements may be used to
improve the estimation of contacts of individual holdings. It should be expected that
recurrent contacts are more common between holdings of some production types than
others (such as Sow pool satellites and centers), and therefore this should be included
as a parameter for each production type combination of the sending and receiving
holding.
While cattle are only considered in Paper I, the cattle farming also involves different
production types. Beef and dairy herds are expected to have different contact
structures, and it should be expected that dairy herds send male calves to beef
producers for fattening. The report forms of herd registrations in Sweden does not
include differentiation of dairy and beef herds, but information of dairy herds may be
obtained from the dairy farmers organization “Svensk mjölk”. From the results of
Nöremark et al. (2009a), it should be expected that seasonal patterns are important to
include in modeling of contacts between cattle herds.
43
6.3.2 Application to specific diseases
The long term aim of the developed model is to use it in modeling of scenarios of
specific diseases. For that purpose, disease specific features will have to be
incorporate. These include incubation time and recovery rates, and it may also be
necessary to consider the intra herd structure. Holdings of different production types
and sizes are further expected to differ in efforts to increase biosecurity (Nöremark
2009b, Anonymous 2009, Nöremark et al. submitted), which is an important factor to
include in modeling scenarios.
Further, other contacts are expected to be important pathways for between-herd
transmissions for most diseases. Farming related contacts (for example milk trucks,
shared equipment, veterinarians etc.) and other visitors can contribute to the spread of
disease. The modeling framework developed here can also be applied to such contact.
Obtaining reliable data for these contacts will however be a major challenge, and
parameters will probably have to be based largely on expert opinions. This may also
be combined with observed data, and Bayesian modeling with expert opinions as
informative prior is a suitable approach.
Some diseases may also be transmitted via wildlife and in these instances the disease
dynamic may not be understood (nor modeled) by the discourses of one single
discipline. For example bluetongue disease, a viral disease infecting ruminants
(mainly sheep), is transmitted via biting midges belonging to the genus Culicoides
(Erasmus 1975). The spread of blue tongue disease needs to be described as both a
local process, resulting from the movement pattern of the Culicoides, and a largescale process with long distance transmission (Gerbier et al. 2008). For example long
distance animal movements may quickly carry the disease further than is expected
from the flight of the Culicoides. A flexible model that can be fitted to both ecological
and human processes can be very useful for modeling of such disease and I argue that
the general approach with a spatial kernel is very useful. This shows that it is not only
possible to use similar models for ecological and epidemiological studies, but also
advantageous.
44
Acknowledgements
First of all I would like to thank my supervisor, Uno Wennergren. I have in particular
appreciated three things about working with Uno. First, he has always been very
positive and encouraging. Secondly, he has allowed me to be very free in my
decisions and allowed me to take any path I have found suitable (both geographically
and scientifically). Thirdly, he has always shown a great understanding in the
importance of a balance between the work and non-work related dimensions of life.
Also, (as a fourth “runner-up”) Uno has never refrained from investing in gadgets,
which has made work both easier and more motivating.
I want to thank Scott Sisson for letting me visit School of Mathematics and Statistics
at UNSW in Sydney (twice!). Special thanks for the patience he has shown in
teaching a biologist about Bayesian inference and Markov Chain Monte Carlo
methods. I am also grateful for Scotts insightful and detailed comments on my work,
as well as for his ability to always find time for this, even with tight deadlines.
I am also grateful to my colleges at the Theoretical Biology group, in particular Lars
Westerberg for all stimulating conversations (both scientific and not). I also want to
thank Lars for his valuable comments on my thesis. I further wish to recognize my
fellow PhD students Frida Lögdberg, Sara Gudmundsson and Nina Håkansson for
creating a positive working atmosphere. A special thank you to Nina, whose
mathematical and computing skills has improved the quality of much of the research
in this thesis. Further thanks to all the people at the Biology department. I have
enjoyed being around people with deep knowledge in all the different fields of
biology.
I further want to thank my colleagues at Skövde University; Annie Jonsson (thank
you for all the comments on my thesis) and Jenny Lennartsson.
Thank you Maria Nöremark and Susanna Stenberg Lewerin at the National Veterinary
Institute (SVA) for all the fruitful discussions over the years. I have learnt much.
Also, thanks to all the nice people in the Physics building, I have had many good
laughs at lunch time.
I am grateful to the Swedish Emergency Management Agency (KBM) for funding and
the Swedish Board of Agriculture for supplying the data of animal movements to my
45
cooperative partners at SVA. Also, I want to thank the Administration Board of
Östergötland for supplying the tree distribution data used in Paper VI.
…………………………………………………………………………………………
I want to thank my partner Dana Cordell for the support during my PhD, for cheering
me up when frustration has struck, and also for raising the standard of what it means
to be a PhD student (“why?” can be a very important question). Also, a big thank you
for helping me make the cover of the thesis.
Tack till mamma och pappa för att ni alltid låtit mig göra mina egna val, och för allt
stöd ni gett mig för att klara av det. Och tack för den bästa systern jag haft, som alltid
har tid att lyssna.
Having an outlet in playing music has been very valuable during my time as a PhD
student and therefore a special thanks to Kristian Rödin and Björn Lorenzoni of the
band(s) Mudfly/Bubbla and Niclas Peterson of Kokong. Finally, thank you to all my
friends (no one mentioned, no one forgotten) who have made life more enjoyable for
the past years.
46
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