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Santa Fe Institute Working Paper 14-XX-XXX
Disease Spreading on Ecological Multiplex Networks
Cecilia S. Andreazzi,1, 2, ∗ Alberto Antonioni,3, † Alireza
Goudarzi,4, ‡ Sanja Selakovic,5, § and Massimo Stella6, ¶
1Departamento de Ecologia, Universidade de Sao Paulo, Sao Paulo, SP 05508-900, Brazil
2Fundacao Oswaldo Cruz, Rio de Janeiro, RJ, 22713-375, Brazil
3Information Systems Department, Faculty of Business and Economics,
University of Lausanne, Lausanne, Switzerland
4Department of Computer Science, University of New Mexico, Albuquerque, NM 87131-0001
5Faculty of Veterinary Medicine, Utrecht University, Utrecht, The Netherlands
6Institute for Complex Systems Simulation,
University of Southampton, Southampton, UK
(Dated: September 16, 2014)
1
Abstract
Processes in which diseases spread and sustain inside of different types of populations were always
interesting subjects for epidemiologists. In recent years, ecologists draw the attention to the fact
that not only hosts but also the other species in the community can affect the process of disease
spread and in that way influence the result of the infection. In nature, we find many parasites
and infectious agents with complex life cycles and which can be transmitted in the ecological
communities through contact interaction or through feeding interaction. Multiplex networks are
layered networks that can represent different types of interactions between agents. In this networks
the agents are part of all the layers, but the structure of their interaction is distinct in each layer.
We recognise multiplex networks approach as a way in which we can question the importance of
the different forms of transmission for the disease spread in the ecosystems. Our results show that
for Trypanosoma cruzi, a parasite that can be transmitted through arthropod vectors or through
feeding on infected prey, the infection is more widespread when considering both layers in the
process. When considering the multiplex structure, species that are not connected through the
food web can be affected because of the inclusion of the vectorial layer. The frequency of vectors in
the community also influenced the infection spread, increasing the speed of infection in the hosts.
We conclude that the multiplex approach is extremely powerful when dealing with different types
of interactions and that non-trivial results can be found when the multilayered structure of the
process is considered.
∗ [email protected]† [email protected]‡ [email protected]§ [email protected]¶ [email protected]
2
I. INTRODUCTION
Organisms differ greatly in how they feed on other species. Trophic interactions, such
as grazing, predation, and parasitism, differ fundamentally on how consumers attack their
victims. This includes whether they kill their victims, how long they remain to feed on a
single victim before killing it or leaving it, and how many victims they feed upon during
their lifetimes [23, 41]. Food webs are maps of the trophic interactions between species
and they are usually represented as networks of species and the energy links between them
[24, 29, 42]. They have been used to identify key linkages between species and processes that
govern ecosystem stability, functioning, and biological diversity [8, 15]. However, food web
studies have historically not included micro-species or parasite interactions. Recently, there
have been efforts to incorporate all the trophic links in food web analysis [8, 21, 22]. These
include how parasites change descriptive parameters of food webs, such as connectivity,
looping, and linkage density [16, 21]. More importantly, parasites appear to play unique
topological roles in food webs. As resources, they have close physical intimacy with their
hosts and thus they are concomitant resources for the same predators. As consumers, they
can have complex life cycles and inverse consumerresource body-size ratios, in contrast to
many free-living consumers [16].
One of the main challenges in community ecology is to include important elements of the
complex life cycles of parasites and understand their effect in the structure and dynamics of
food webs [21, 39]. During their life cycles, parasites can have several free-living and parasitic
stages. They may infect multiple hosts from diverse trophic levels and use intermediary hosts
or vectors to be transmitted to other destination hosts. Several studies have reported that
free-living stages of parasites or ectoparasites may be subject to predation and that predators
can feed on parasites together with their infected prey [19]. Predation also represents a key
transmission pathway, and predators often acquire trophically transmitted parasites from
their prey [19]. Until now accommodating such diverse types of links in the same food web
remained an open question [21].
Here, we propose a new approach to study complex life cycles and transmission modes in
food webs. In this approach, each interaction type between species gives rise to a different
layer in a multi-layered complex network structure called multiplex [9]. Multiplex or multi-
relational networks are multi-layered graphs where nodes appear on all the layers but are
connected according to different topologies on each level [3, 9]. Studies of disease spreading
over multiplex structures have appeared only very recently within the scientific literature [7].
However, to the best of our knowledge, multiplexity has never been applied to the analysis of
disease spreading in real ecological scenarios. Nonetheless, this multiplex approach would be
particularly useful when dealing with multihost parasites that can be transmitted to their
hosts via multiple routes. For example, many protozoa and helminths use intermediary
hosts or vectors to be transmitted from one host to the other, but can also be transmitted
trophically when a predator feeds on an infected prey or vector.
In this dynamical model we are particularly interested in understanding the spatial epi-
demiology of multihost parasites that are trophically transmitted throughout the food web
and also vector-borne via blood-sucking invertebrates. Important human diseases such as
3
Chagas disease, Leishmaniosis, and Malaria are caused by parasites that have such a complex
life cycle [38].
As a case study, we modelled the trophic and vectorial transmission of Trypanosoma cruzi,
the etiological agent of Chagas disease. We used data on T. cruzi sylvatic hosts prevalence
in two different regions in Brazil [18, 33]. T. cruzi is a multi-host parasite found in more
than 100 mammalian species [27]. Human infections are generally associated with infected
triatomine bugs (biological vectors), but most recent outbreaks in Brazil were caused by
oral transmission, which is probably the most ancient route of infection among wild animals
[27, 40]. We investigated the effect of each mode of transmission on the epidemiology of
T. cruzi and the relationship between the structural characteristics of host species in the
multiplex and its epidemiological importance. The effect of different community structures,
such as the relative frequency of vectors in the environment was also investigated. We were
able to identify key species for disease spreading and the importance of each transmission
mode in different community structure scenarios.
II. DATASET
Epidemiological studies of T. cruzi infection in wild hosts in two different areas [18, 33]
were used to estimate the probabilities of species interaction and disease transmission in the
model. For the trophic layer, we first built the qualitative food webs based on the diet of
the animals in the areas [1, 6, 11, 12, 30–32]. As there were no species-level classification
of the biological vectors present in the areas, we grouped all the species in a compartment
called ”vectors”. In this procedure, we classified the interaction with “1” if that item was
present in the diet of the animal, and with a “0” if not. We used this binary food web to
constrain the possible interactions between species, i.e., if the interaction is equal to zero in
the binary network it is also zero in the probabilistic network. To estimate the probabilities
of interaction between species in the food webs expected by their body sizes we used the
probabilistic niche model [46]. The species mean body weights were obtained from the
literature [18, 33, 34, 36]. We used the [17] method based on linear model fitting to estimate
the species niche parameters: niche position ni, feeding niche optimum ci and range ri of
suitable preys around that optimum. We then calculated each interaction probability with
the niche model [46].
We estimated the probabilities of interaction in the vectorial layer using the prevalence of
species T. cruzi [18, 33]. We used the proportion of individuals with positive serology for T.
cruzi as a proxy for the probability of interaction between host species and vectors. Because
the trophic route infection tends to be more severe than the vectorial route infection[47],
we assumed that vectors infect animals with positive serology for T. cruzi. We are not
aware of any data on these interaction probabilities. In addition, we used the proportion
of individuals with positive parasitological diagnostics as a proxy for the probability of
disease transmission from the host to the vector, since only the individuals with positive
parasitaemia are able to transmit the parasite.
4
III. MULTIPLEX NETWORKS
A. An Introduction to Multiplexity
Network theory investigates the structural and dynamical properties of a set of entities,
each one represented as a node or vertex; the nodes take part in pairwise interactions,
represented as links or edges. In the last twenty years, the network framework has been
successful in modelling many complex systems, throughout the physical, biological, social,
and technological sciences [26]. In fact, it allowed for an interpretation of many features of
real-world networked systems, such as the emergence of hubs, the “small-world” property
and the organisation in hierarchical structures [2, 26, 45]. Recently, the increasing amount of
available data brought the scientific community to generalise the complex network formalism,
for instance by taking into account also the heterogeneity of real-world interactions [9].
Examples include multi-modal transportation networks in metropolitan areas [25], where
each location can be connected to another by different modes of transport. Analogously,
countries interrelate via various political and economical channels or proteins interact with
each other according to different regulatory mechanics [9, 14]. Also social systems exhibit
different kinds of ties, organised and structured on different levels of social environments,
as have been observed by sociologists since the 1930s [3, 14]. It is in the context of social
network analysis that such representations emerged as multi-relational networks or multiplex
networks [44], i.e. multi-layered graphs where nodes appear on all the layers, but they can
be connected according to different topologies on each layer. Each multiplex layer contains
edges of a given type, e.g. co-location, kinship, commercial, or collaboration relationships
between individuals in a social environment. A visual example of a multiplex is depicted in
Figure 1.
FIG. 1. Example of a multiplex with two layers only (i.e. a duplex). All the nodes are replicated
on all the layers but they are connected differently on each layer.
5
B. Analysis of the Trophic Layer
In our study we analysed two different kinds of interactions that are related to the trans-
mission of T. cruzi. This parasite can be transmitted through both trophic and vectorial
routes. We considered each of those transmission route as a layer in our multiplex. Therefore
we consider only duplexes: a multiplex network consisting of two layers.
In the Canastra dataset we have 20 species, each of which is represented by a node. The
nodes engage in directed trophic interactions and parasitic interactions. A connection is
directed from species A to species B if ”A is eaten by B”. In the trophic layer, the two
nodes having higher in-degree (representing top predators) are the Leopardus pardalis and
the Chrysocyon brachyurus, predating 15 and 14 species, respectively. The trophic height
of a species is the average trophic position of a species in all food chains it belongs to and
it indicates its role as top predator in the food web [13]. In Canastra, the species with the
highest trophic height were also L. pardalis and C. brachyurus. In addition, the species
with higher out-degree (representing the most preyed species) are the Calomys and the
Oligoryzomys, each being predated by 8 species. Furthermore, together with the vectors,
these last two species also have the highest radiality centrality of the network, in other
words they play a locally prominent role in the food web (at a scale comparable to the web
diameter itself). The radiality centrality [20] of node i in a network with n nodes is defined
as
ri :=1
n− 1
∑j
(D − dij + 1), (1)
where di,j is the geodesic distance between nodes i and j (i.e. the minimum number
of hops necessary to go from i to j on the network) while D is the network diameter (i.e.
the maximum geodesic distance between two network nodes). Intuitively, in the context
of food webs, with directed trophic interactions, the radiality centrality can be thought of
as an indicator of the importance of preys for the food web itself (Pearson’s coefficient:
0.99, p-Value≤ 10−7). This correspondence between an ecological property of species and a
network theoretic measure is further evidence for the appropriateness of network science in
the study of ecological systems. Another species that plays an important role in the trophic
layer is the Didelphis albiventris, which has a betweenness centrality [20] of 1. Betweenness
centrality is the ratio of paths that go through a given node over the total number of paths
in a given networks. Therefore, the D. albiventris is present on every trophic pathway in the
food web, and it may therefore play a prominent role in the spreading of Chagas disease.
Figure 3 gives a visual representation of the trophic and vectorial layers in the Canastra
dataset.
In the Pantanal dataset we have 19 species. L. pardalis, also present in this food web,
is the species with the highest trophic height, followed by Cerdocyon thous and Nasua
nasua. Similar to the Canastra dataset, the vectors in Pantanal are the most predated
species (predated by 15 other species). The other species are predated by less than 5
species. Furthermore, the most predated species are those with higher radiality centrality
in the network. A betweenness centrality analysis does not reveal any interesting pattern,
6
compared with the Canastra dataset.
FIG. 2. Visual representation of the multiplex structure, comprising the trophic layer (left) and
the vectorial layer (right) for the Canastra dataset. Note that all the species are present in both
layers but they are connected differently.
C. Analysis of the Multiplex Structure
Following [3], let us define a multiplex M as a formal collection of a finite number m
of single-layered networks. Let us denote with Aα the adjacency matrix of the network in
the α-th layer. Each entry aαij is equal to 1 if nodes i and j are connected on layer α or 0
otherwise. Let us define the adjacency matrix A of the multiplex M as the set Aαα.
Furthermore, we define the edge overlap oij of edge (i, j) between layers α and β as aαij+aβij.
In other words, the edge overlap quantifies the number of times that a given connection is
present in the multiplex structure, across different layers and therefore ≤ oij ≤ m. In our
case, for the Canastra dataset only three edges have an edge overlap of 2 in the whole duplex
(i.e. a multiplex with m = 2 layers only) while there are 14 repeated connections in the
Pantanal dataset. Given the very small numbers in play, we cannot quantify correlations
that explain this difference in overlap in a statistically significant way.
As the simplest characterisation of the multiplex structure, let us introduce also the
aggregated adjacency matrix P , whose entries satisfy pij = 1 if nodes i and j are connected
7
in at least one of the m layers:
pij :=∏α
aαij. (2)
The aggregated matrix P represents the adjacency matrix of the single-layer network
obtained by “projecting” all the layers onto a single network. This matrix cannot encapsulate
the whole information contained in the multiplex structure [3, 4], nonetheless it represents
a simple approximation when analysing multiplexity.
In the Canastra dataset, overlaying together the trophic layer and the vectorial one does
not alter the network hubs, with the Leopardus pardalis still having an in-degree of 15. As
expected, in the aggregated network for both Canastra and Pantanal datasets, obtained by
joining the trophic layer with the oriented edges of the vectorial layer, a boost in radiality
is registered for the vector. This is a preliminary indication that the multiplex structure
will play a predominant role in the spreading of T. cruzi infection mediated by the vectors.
No significant changes in the betweenness centrality measure have been observed in the
aggregated networks.
A more genuine measure of node importance in the multiplex structure is given by the
interdependency factor [3]. The interdependency λi for node i in a multiplex is defined as:
λi :=∑j 6=i
ψijσij
, (3)
where σij is the total number of shortest paths from node i to node j on the whole
multiplex and ψij is the number of shortest paths from node i to node j that make use of
links in two or more layers. Note that a shortest path on a multiplex can jump between
layers at any given node. For instance, the shortest path going from i to k on a duplex
might use the link (i, j) on layer 1 and then the link (j, k) on layer 2 and therefore be
{[(i, j), 1], [(j, k), 2]}. The higher the interdependency λi the easier it is for node i to reach
other nodes in the multiplex by using connections on different layers. Both the two datasets
exhibit a non-negligible average interdependency, with the Canastra dataset having a mean
interdependency of λcan = 0.24 and the Pantanal dataset having a lower value of interdepen-
dency λpan = 0.15. As reported in Figure 3, both in the Canastra and Pantanal datasets,
the vectors have the highest interdependency value. In other words, the multiplex structure,
with its combination of vectorial and trophic interactions, allows the vectors to be more
intensively connected with all the other species. Furthermore, the vectors are not the only
species having a non-zero interdependency. Additionally, the species interdependency was
significantly related to species trophic height (β = -0.14, p < 0.0001, Figure 4). Hence,
these facts are a strong indication that the multiplex structure will play a prominent role in
boosting the effectiveness of the disease spreading.
8
FIG. 3. Interdependency ratio for the species in the Canastra (orange squares) and in the Pantanal
(blue squares) datasets. In both the datasets, the species are ordered based on their body sizes,
from the heaviest to the lighter. The vector is always the last species.
1.0 1.5 2.0 2.5 3.0 3.5 4.0
0.0
0.2
0.4
0.6
0.8
Species trophic height
Spe
cies
inte
rdep
ende
ncy
vectors
L. pardalis
FIG. 4. Relationship between species trophic height and interdependency in Canastra (orange
dots) and Pantanal (blue dots). The position of the vectors and of the top predator Leopardus
pardalis is highlighted.
9
IV. MODEL
A. Multiplex network construction
In this model we consider S different species, each with its corresponding relative fre-
quency fi (see Appendix section for details). Note that the number of considered species S
in the Canastra dataset is 20, while it is 19 in the ecosystem of Pantanal. We applied the
metabolic theory of ecology [5] to estimate species frequency such that f = m−3/4, where m
is species body size. We then normalised the frequencies, i.e.,∑S
i=1 fi = 1. This estimation
has been done following body size datasets in [18, 33].
A node in the network represents an individual of a given species. The attributes of a
node i are: the given species si, its spatial position (xi, yi), and its current state vi that may
be susceptible or infected.
The spatial environment is represented as a square of size L = 1 and the metric we consider
is the Euclidean distance in the two-dimensional space.
The process we have used to obtain ecological multiplex network of about N nodes can be
summarised as follows:
1. A number dNfie of individuals nodes is randomly distributed in the squared space per
each species 1 ≤ si ≤ S.
2. (vectorial layer step) A link between nodes i and j is created in the vectorial layer
with the following probability:
pV(i,j) = V (si, sj)e−kV d(i,j),
where V is the interaction matrix in the vectorial layer (see Appendix section for de-
tails), d(i, j) is the physical distance between nodes i and j, and kV is a free parameter
that controls the importance of the space in the vectorial layer. This parameter is fixed
in order to obtained fairly connected networks on this layer. This creation link process
is repeated for all pairs of nodes (i, j) in the network. The trophic layer is undirected
and we do not consider links where corresponding nodes lie at distance greater than
dmax = 0.2.
3. (trophic layer step) A link between nodes i and j is created in the trophic layer with
the following probability:
pT(i,j) = T (si, sj)e−kT d(i,j),
where T is the interaction matrix in the trophic layer (see Appendix section for details),
d(i, j) is the physical distance between nodes i and j, and kT is a free parameter that
controls the importance of the space in the trophic layer. This parameter is fixed in
order to obtained fairly connected networks on this layer. This creation link process is
repeated for all pairs of nodes (i, j) in the network. Moreover, since the trophic layer
network is to be considered as directed, the link (i, j) can exist while the link (j, i)
is not present and vice versa, T (si, sj) 6= T (sj, si). Again, we do not consider links
where nodes are at distance more than dmax = 0.2.
10
B. Disease diffusion dynamics
Here we describe the disease diffusion dynamics on ecological multiplex networks that are
defined in the previous section. A node, or an individual of a given species, in the network
can be infected or susceptible and its state must be the same in both layers.
We start the simulation infecting all the individuals of species number S (the last species
is the vector species in both datasets) within the circle area of radius R0 = 0.05 and centre
x0 = 0.5, y0 = 0.5, that is, the center of the squared space. Subsequently, the disease
diffusion process evolves as it follows:
1. the vectorial layer is chosen to be considered in the disease diffusion for this step with
probability pvector. Step 2 is then performed when the vectorial layer is considered,
Step 3 otherwise.
2. (vectorial layer disease diffusion) A random node i and a random neighbour j in the
vectorial layer are selected. If the node i is infected also the node j becomes infected
with probability Vd(si, sj).
3. (trophic layer disease diffusion) A random node i and a random neighbour j in the
trophic layer are selected. If the node i is infected also the node j becomes infected
with probability Td(si, sj).
4. Steps 1, 2 and 3 are repeated until the maximum number of time steps is reached.
The process in stages 1, 2 or 3 is repeated N , the size of the populations, times for
each time step. This also means that an individual in the population is chosen once in the
diffusion process at each time step, on average. The two matrices of interactions, V and T ,
and the diffusion matrix Vd, have been estimated from the two datasets in [18, 33]. On the
other hand, we assumed that the diffusion matrix Td is equal to one in each of its values.
C. Model output analysis
To investigate the effect of each layer on the dynamics of the infection, we simulated the
model for diffusion probabilities in the vectorial layer pvector varying from 0 (only the trophic
layer) to 1 (only the vectorial layer).
To compare the dynamics of each species per diffusion probabilities situations, we fitted
a logistic regression to the average proportions of infected individuals per time. The coeffi-
cients of a logistic regression are the steepness parameter of the curves. We then compared
the species coefficients per each diffusion probability on different community compositions
(with 10%, 20% or 80% of vectors). The relaxation time of the average proportion of total
infected individuals was used as a proxy for the velocity of infection spread in the different
diffusion probabilities situations.
11
V. RESULTS
The dynamics of the T. cruzi infection in the Canastra and Pantanal multiplexes are
summarised in Figures 5 and 6. It is possible to observe that in both multiplexes the pro-
portion of infected individuals of all species was low when we had only the trophic layer
(v0). In this situation, there were no infected vectors because vectors are dependent on the
vectorial layer to get infected. The top predators were the species that had a higher propor-
tion of infected individuals. Comparing the communities with different relative proportion
of vectors (columns), there was a slightly increase in the proportion of infected individu-
als when there was a higher relative frequency of vectors in the community (a, d and g).
When both layers had the same importance for the diffusion of the infection (v0.5), we had
a higher proportion of infected individuals in all species. Increasing the relative frequency
of vectors in the community decreased the proportion of infected vectors. Once again, the
top predators were the species with the higher proportion of infected individuals. When the
transmission was only the vectorial layer (v1), the two multiplexes differed in the infection
dynamics. In Canastra, only three species (sp2 is L. pardalis, sp11 is Marmorsops incanus
and sp20 is the vector) became infected, independent of the relative frequency of vectors.
However, the higher the frequency of vectors in the community, the slower was the infection
in the vectors.In the Pantanal multiplex, more species became infected with transmission
occurring only in the vectorial layer. More interesting, there was no relationship between
the trophic height of the species and the proportion of infected individuals. Increasing the
relative frequency of vectors in the community accelerated the infection in all species except
for the vectors.
The logistic regression coefficients indicated how steep was the increase in infection for
each species. In Canastra, (Figure 7), it was possible to identify different behaviours de-
pending on the species trophic height. In general, top predators had a steeper increase in the
proportion of infection compared to lower trophic levels. They were also more affected by
the probability of disease diffusion in the vectorial layer pvector, that is, the higher the disease
diffusion in the vectorial layer, the steeper is the infection in time. However, in very high
pvector only sp2 (L. pardalis) and sp11 (M. incanus) had infection, which is in accordance
with the results found in Figure 5. Vectors were differently affected by the pvector then other
species. Because their interaction pattern, infection in vectors is dependent on adjacent host
getting infected first. This creates a delay in the disease diffusion dynamics.
In Pantanal multiplex (Figure 8), the top predators was also the species that had, in
general, steeper increase in infection. However, the relationship between the logistic regres-
sion coefficients and the probability of disease diffusion in the vectorial layer (pvector) had a
bell shape in Pantanal, while in Canastra it was skewed to the right. Vectors were also not
affected by the probability of disease diffusion in the vectorial layer (pvector).
As reported in Figure 9, our simulation showed that the spreading of the disease on the
multiplex depends in a non-monotic way on the importance of the vector layer. More in de-
tails, for different values of the relative frequency of vectors (IV), increasing the probability
of diffusion in the vectorial layer initially makes the spread of the disease faster, in other
12
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
FIG. 5. Average proportion of infected individuals per time for the initial 500 time steps in
Canastra multiplex. Species colours are related to species trophic height in the trophic layer, such
as that hot-colored species are the ones with a higher trophic heights (top predators). The vector
is sp20. From a to c are the results for a community with 10% of vectors relative frequency, d to
f for a community with 20% of vectors relative frequency and g to i for a community with 80% of
vectors relative frequency. v0 is for pvector = 0, v0.5 is pvector = 0.5 and v1 is pvector = 1.
13
(a) (b) (c)
(d) (e) (f)
(g) (h) (i)
FIG. 6. Average proportion of infected individuals per time for the initial 500 time steps in Pantanal
multiplex. Species colours are related to species trophic height in the trophic layer, such as that
hot-colored species are the ones with a higher trophic heights (top predators). The vector is sp19.
From a to c are the results for a community with 10% of vectors relative frequency, d to f for a
community with 20% of vectors relative frequency and g to i for a community with 80% of vectors
relative frequency. v0 is for pvector = 0, v0.5 is pvector = 0.5 and v1 is pvector = 1.
14
(a) (b) (c)
FIG. 7. Logistic regression coefficients for the average proportion of infected individuals per time
in Canastra multiplex for the different diffusion probabilities in the vectorial layer pvector. Species
colours are related to species trophic height in the trophic layer, such as that hot-colored species
are the ones with a higher trophic heights (top predators). The vector is sp20.
(a) (b) (c)
FIG. 8. Logistic regression coefficients for the average proportion of infected individuals per time
in Pantanal multiplex for the different diffusion probabilities in the vectorial layer pvector. Species
colours are related to species trophic height in the trophic layer, such as that hot-colored species
are the ones with a higher trophic heights (top predators). The vector is sp19.
words, the time step (i.e. relaxation time) at which 90% of the population gets infected
decreases. However, in general, the relaxation time increases when the importance of the
vectorial layer is higher than a given threshold, as reported also in V. This non-monotonic
behaviour is not evident for the Canastra dataset, for IV = 80%, at least at our discrete
observation scale for the vectorial layer importance. Nonetheless, this phenomenon is intrin-
sically related to the multiplex structure: when the vectorial layer becomes preponderant
the infection diffuses mainly by vector interactions, which involve only a relatively small
fraction of nodes, so that the other species on the trophic layer get infected by predator-
prey interactions after some characteristic diffusion time. This behaviour is compatible with
our results coming from the logistic regression coefficients.
15
FIG. 9. Relaxation time for the Canastra (left) and the Pantanal (right) multiplexes on a log-log
scale, for different values IV of initial vector abundance. The relaxation time represents the first
time step at which at least 90% of the total population in the multiplex is inflected. The presence
of local minima and non-monotonic behavior is evident from the plots, symptom of the different
role played by the trophic and the vectorial layer.
(Min. Relaxation Time, Vector Layer Importance)Relative frequency of vectors (IV) Pantanal Canastra
0.1 (112, 0.35) (143, 0.55)0.2 (77, 0.50) (115, 0.7)0.4 (75, 0.65) (135, 0.80)0.8 (118, 0.85) (317, 0.95)
TABLE I. Table of the minimum relaxation time, and relative vector layer importance, versus the
relative frequency of vectors for the Pantanal and Canastra datasets, respectively. The minimum
relaxation time is a measure of how fast the disease spreads, in terms of simulation time steps, over
at least 90% of the total population. In the multiplex, different importances of the vector layer
bring to different minimum relaxation times, depending on the initial vector abundance.
VI. DISCUSSION
We here introduced a promising approach for the study of parasites that have complex life
cycles and multiple hosts. The use of multiplex networks to study epidemiological dynam-
ics is incipient and revealed non-expected results when different types of interactions were
considered in the infection transmission [37, 43]. Our study showed that the epidemiology
of parasites that can be spread via multiple routes of infection have non-trivial dynamics.
For the specific case of T.cruzi infection, we found that the epidemiological dynamics is
largely influenced by the multiplex structure, and that when both layers are considered a
much higher proportion of infected individuals is expected. The structural roles played by
each species in the multiplex was related to its importance for the infection transmission,
which enable us to identify key species for this process.
In random multiplexes it was found that the behaviour of the epidemic process depends
on the strength and nature of the coupling between the networks [37]. This previous study
found that the global epidemic threshold of the process turned out to be smaller than the
epidemic thresholds of the two layers separately. This implies that an endemic state may
arise even if the epidemics is not endemic in any of the two layers separately [37]. In our study
16
we found that the number of species affected and the proportion of infected individuals was
much higher when both layers were considered in the transmission. This result indicates
that even in multiplexes with highly structured layers that differ in their topologies, the
spreading of a disease via multiple routes may have a higher efficiency then when it happens
in only one layer. The multilayered transmission, which is observed in many parasites with
complex life cycles, seems to be a very efficient strategy for the diffusion on multiple hosts
that differ on their probability of infection.
Ecoepidemiological studies of T.cruzi transmission in wild hosts have already pointed that
the trophic interactions among hosts are important for the disease dynamics [18, 33, 34].
The epidemiological importance of top predators as bioaccumulators of trophic transmitted
parasites was observed in the model results as the proportion of infected individuals was
related to the trophic height of the species. Other interesting result is that D. albiventris
was the species with the highest betweenness in the multiplex. Marsupials and specifically
Didelphis species are recognised as one of the most important T.cruzi reservoirs [35]. Its
topological importance in the T.cruzi transmission multiplex indicates that the structural
role played by each species may be related to its importance for the transmission of the
parasite. We recognise that the model we simulated is tremendously simplistic face the
complexity of the biology and epidemiology of T.cruzi. However, even using the most simple
SI structure we were able to catch a qualitative dynamics that is consistent with the T.cruzi
behaviour. Thus, the qualitative predictions for the importance of each layer is still valid.
Many zoonoses, that is, diseases and infections that are naturally transmitted between
vertebrate animals and humans, have multiple hosts and complex life cycles. Examples of
zoonoses that are transmitted to humans by moderately to highly generalised arthropod
vectors include Malaria, Leishmaniasis, Chagas disease, West Nile virus, plague, and Lyme
disease [38]. All of them are characterised by at least one layer of transmission among its
hosts, which may have complex structures. The multiplex approach presented here could
improve our understanding of the epidemiology and the evolution of these parasites and
help to elaborate more efficient control strategies for the diseases they cause. More layers as
for example the network of human interactions and its socio-ecological characteristic could
be included to make the model more realistic.
VII. CONCLUSIONS AND FUTURE WORK
The multiplex approach of our specific system can be further developed and it can ques-
tion not only the importance of the transmission way of certain diseases in the ecosystem
but also to understand many other general processes that are important for the structure,
functioning and stability of the ecosystem. Differentiating the susceptible, infectious and
recovered/immune individuals could help us to understand the importance of the epidemi-
ological state for the dynamics of the disease through the ecosystem. Futher, weighting the
nodes and links of the multiplex can lead us to understanding of bioenergetic flow in the
ecosystem and change of the interaction strength between species influenced by the parasite
T.cruzi. Also, inserting into the model further spatial elements would be critical for the
17
analysis of the real-world ecosystems.
A first point for future direction would be applying the multiplex analysis to bigger
datasets, possibly encompassing more species and more types of interactions.
The multiplex approach may be powerful tool when dealing with different type of inter-
actions in complex systems that can be represented by complex networks. The important
steps considered in the dealing with these kind of systems are temporal and spatial scales
involved in the each class of interactions which form each layer of the multiplex [28]. This
idea should be further developed.
ACKLOWLEDGEMENT
All the authors would like to thank the whole SFI CSSS2014 organising team for the
greatly rewarding experience of the Santa Fe Summer School 2014. M.S. acknowledges the
EPSRC and the DTC in Complex Systems Simulation. A.G. would like to acknowledge the
support from the National Science Foundation of the United States of America under grant
CDI-1028238 and CCF-1318833. C.S.A is supported by the CNPq, Brazil. S.S would like
to acknowledge the support from the Complexity Program of The Netherlands Organisation
for Scientific Research (NWO).
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Appendix A: Model Parameters
In this section we give the model parameters for the datasets of Canastra and Pantanal.
frequencyChrysocyon-brachyurus-(CHR) 0.08Leopardus-pardalis-(LEO) 0.14Cerdocyon-thous-(CER) 0.20Lycalopex-vetulus-(LYC) 0.35Conepatus-semistriatus-(CON) 0.38Didelphis-albiventris-(DID) 0.61Lutreolina-crassicaudata-(LUT) 1.47Caluromys-philander-(CAL) 2.50Nectomys-squamipes-(NES) 2.64Monodelphis-sp-(MON) 4.43Marmorsops-incanus-(MAR) 5.82Oxymycterus-delator-(OXY) 5.91Cerradomys-subflavus-(CER) 6.23Necromys-lasiurus-(NEL) 7.22Akodon-montensis-(AKM) 9.09Akodon-sp-(AKO) 10.58Gracilinanus-agilis-(GRA) 13.32Oligoryzomys-spp-(OLI) 14.13Calomys-sp-(CAL) 14.91Vector-(VEC) f_vector
FIG. 10. Frequencies for species in the dataset of Canastra. The frequency of vector species is
a free parameter of the model and all the following frequencies have been re-normalised in the
numerical simulations as a function of fvector.
CHR LEO CER LYC CON DID LUT CAL NES MON MAR OXY CER NEL AKM AKO GRA OLI CAL VECChrysocyon;brachyurus;(CHR) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Leopardus;pardalis;(LEO) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Cerdocyon;thous;(CER) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Lycalopex;vetulus;(LYC) 0.32 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Conepatus;semistriatus;(CON) 0.32 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Didelphis;albiventris;(DID) 0.34 0.30 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Lutreolina;crassicaudata;(LUT) 0.37 0.34 0.32 0.28 0.27 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Caluromys;philander;(CAL) 0.00 0.37 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Nectomys;squamipes;(NES) 0.40 0.37 0.35 0.31 0.30 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Monodelphis;sp;(MON) 0.42 0.40 0.38 0.34 0.33 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Marmorsops;incanus;(MAR) 0.44 0.41 0.40 0.36 0.35 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Oxymycterus;delator;(OXY) 0.44 0.42 0.40 0.36 0.35 0.31 0.22 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Cerradomys;subflavus;(CER) 0.44 0.42 0.40 0.36 0.36 0.32 0.22 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Necromys;lasiurus;(NEL) 0.45 0.43 0.41 0.37 0.37 0.33 0.23 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Akodon;montensis;(AKM) 0.46 0.44 0.42 0.39 0.38 0.34 0.25 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Akodon;sp;(AKO) 0.47 0.45 0.43 0.40 0.39 0.36 0.26 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Gracilinanus;agilis;(GRA) 0.00 0.46 0.45 0.00 0.00 0.37 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Oligoryzomys;spp;(OLI) 0.49 0.47 0.45 0.42 0.41 0.38 0.29 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Calomys;sp;(CAL) 0.49 0.47 0.46 0.42 0.42 0.38 0.29 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Vector;(VEC) 0.00 0.00 0.78 0.81 0.81 0.84 0.89 0.91 0.91 0.92 0.90 0.90 0.90 0.88 0.84 0.79 0.71 0.68 0.66 0.00
FIG. 11. Trophic matrix for the species interactions in the dataset of Canastra, T . The diffusion
matrix, Td, for the trophic layer is equal to one in each of its values.
22
VECChrysocyon+brachyurus+(CHR) 0.06Leopardus+pardalis+(LEO) 0.31Cerdocyon+thous+(CER) 0.15Lycalopex+vetulus+(LYC) 0.00Conepatus+semistriatus+(CON) 0.00Didelphis+albiventris+(DID) 0.00Lutreolina+crassicaudata+(LUT) 0.00Caluromys+philander+(CAL) 0.00Nectomys+squamipes+(NES) 0.00Monodelphis+sp+(MON) 0.12Marmorsops+incanus+(MAR) 0.31Oxymycterus+delator+(OXY) 0.04Cerradomys+subflavus+(CER) 0.00Necromys+lasiurus+(NEL) 0.00Akodon+montensis+(AKM) 0.00Akodon+sp+(AKO) 0.00Gracilinanus+agilis+(GRA) 0.00Oligoryzomys+spp+(OLI) 0.00Calomys+sp+(CAL) 0.00Vector+(VEC) 0.00
VECChrysocyon+brachyurus+(CHR) 0.00Leopardus+pardalis+(LEO) 0.31Cerdocyon+thous+(CER) 0.00Lycalopex+vetulus+(LYC) 0.00Conepatus+semistriatus+(CON) 0.00Didelphis+albiventris+(DID) 0.00Lutreolina+crassicaudata+(LUT) 0.00Caluromys+philander+(CAL) 0.31Nectomys+squamipes+(NES) 0.00Monodelphis+sp+(MON) 0.00Marmorsops+incanus+(MAR) 0.19Oxymycterus+delator+(OXY) 0.00Cerradomys+subflavus+(CER) 0.08Necromys+lasiurus+(NEL) 0.00Akodon+montensis+(AKM) 0.01Akodon+sp+(AKO) 0.02Gracilinanus+agilis+(GRA) 0.00Oligoryzomys+spp+(OLI) 0.00Calomys+sp+(CAL) 0.07Vector+(VEC) 0.00
FIG. 12. Left: Vector matrix for the species interactions in the dataset of Canastra, V . In this
layer only vector-host interactions are allowed, that is, all other matrix values are set to be zero.
Right: Vector diffusion matrix, Vd, for the dataset of Canastra. Again, all other matrix values are
set to be zero.
frequencyLeopardus/pardalis/(LEO) 0.24Cerdocyon/thous/(CER) 0.35Nasua/nasua/(NAS) 0.55Sus/scrofa/(feral)/(SUS) 0.20Pecari/tajacu/(PEC) 0.28Hydrochaeris/hydrochaeris/(HYD) 0.28Tayassu/pecari/(TAY) 0.30Euphractus/sexcinctus/(EUP) 0.63Philander/frenatus/(PHI) 4.02Thrichomys/pachyurus/(THR) 4.14Clyomys/laticeps/(CLY) 5.09Holochilus/brasiliensis/(HOL) 5.91Cerradomys/scotti/(CES) 8.93Monodelphis/domestica/(MON) 10.22Oecomys/mamorae/(OEC) 11.16Thylamys/macrurus/(THY) 12.64Calomys/callosus/(CAL) 15.88Gracilinanus/agilis/(GRA) 19.17Vector/(VEC) f_vector
FIG. 13. Frequencies for species in the dataset of Pantanal. The frequency of vector species is
a free parameter of the model and all the following frequencies have been re-normalised in the
numerical simulations as a function of fvector.
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LEO CER NAS SUS PEC HYD TAY EUP PHI THR CLY HOL CES MON OEC THY CAL GRA VECLeopardus;pardalis;(LEO) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Cerdocyon;thous;(CER) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Nasua;nasua;(NAS) 0.32 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Sus;scrofa;(feral);(SUS) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Pecari;tajacu;(PEC) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Hydrochaeris;hydrochaeris;(HYD) 0.30 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Tayassu;pecari;(TAY) 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Euphractus;sexcinctus;(EUP) 0.32 0.30 0.27 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Philander;frenatus;(PHI) 0.38 0.37 0.34 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Thrichomys;pachyurus;(THR) 0.38 0.37 0.34 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Clyomys;laticeps;(CLY) 0.39 0.38 0.35 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Holochilus;brasiliensis;(HOL) 0.40 0.38 0.36 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Cerradomys;scotti;(CES) 0.41 0.40 0.38 0.00 0.00 0.00 0.00 0.00 0.21 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Monodelphis;domestica;(MON) 0.42 0.41 0.39 0.00 0.00 0.00 0.00 0.00 0.21 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Oecomys;mamorae;(OEC) 0.42 0.41 0.39 0.00 0.00 0.00 0.00 0.00 0.22 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Thylamys;macrurus;(THY) 0.43 0.42 0.40 0.00 0.00 0.00 0.00 0.00 0.23 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Calomys;callosus;(CAL) 0.44 0.43 0.41 0.00 0.00 0.00 0.00 0.00 0.24 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Gracilinanus;agilis;(GRA) 0.45 0.44 0.42 0.00 0.00 0.00 0.00 0.00 0.25 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00 0.00Vector;(VEC) 0.00 0.66 0.68 0.65 0.66 0.00 0.66 0.68 0.74 0.74 0.74 0.00 0.74 0.73 0.73 0.72 0.69 0.66 0.00
FIG. 14. Trophic matrix for the species interactions in the dataset of Pantanal, T . The diffusion
matrix, Td, for the trophic layer is equal to one in each of its values.
VECLeopardus-pardalis-(LEO) 0.33Cerdocyon-thous-(CER) 0.38Nasua-nasua-(NAS) 0.63Sus-scrofa-(feral)-(SUS) 0.57Pecari-tajacu-(PEC) 0.57Hydrochaeris-hydrochaeris-(HYD) 0Tayassu-pecari-(TAY) 0.25Euphractus-sexcinctus-(EUP) 0Philander-frenatus-(PHI) 0.5Thrichomys-pachyurus-(THR) 0.07Clyomys-laticeps-(CLY) 0.2Holochilus-brasiliensis-(HOL) 0.26Cerradomys-scotti-(CES) 0.18Monodelphis-domestica-(MON) 0.75Oecomys-mamorae-(OEC) 0.29Thylamys-macrurus-(THY) 0.28Calomys-callosus-(CAL) 0.25Gracilinanus-agilis-(GRA) 0.36Vector-(VEC) 0
VECLeopardus-pardalis-(LEO) 0.00Cerdocyon-thous-(CER) 0.00Nasua-nasua-(NAS) 0.20Sus-scrofa-(feral)-(SUS) 0.05Pecari-tajacu-(PEC) 0.00Hydrochaeris-hydrochaeris-(HYD) 0.00Tayassu-pecari-(TAY) 0.00Euphractus-sexcinctus-(EUP) 0.05Philander-frenatus-(PHI) 0.25Thrichomys-pachyurus-(THR) 0.02Clyomys-laticeps-(CLY) 0.03Holochilus-brasiliensis-(HOL) 0.00Cerradomys-scotti-(CES) 0.00Monodelphis-domestica-(MON) 0.17Oecomys-mamorae-(OEC) 0.05Thylamys-macrurus-(THY) 0.00Calomys-callosus-(CAL) 0.00Gracilinanus-agilis-(GRA) 0.29Vector-(VEC) 0.00
FIG. 15. Left: Vector matrix for the species interactions in the dataset of Pantanal, V . In this
layer only vector-host interactions are allowed, that is, all other matrix values are set to be zero.
Right: Vector diffusion matrix, Vd, for the dataset of Canastra. Again, all other matrix values are
set to be zero.
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