# Dropping mortality by increasing connectivity in plant epidemics

Ignacio Taguas<EMAIL_ADDRESS>José A. Capitán  Complex Systems Group
<EMAIL_ADDRESS>Juan C. Nuño<EMAIL_ADDRESS>Department of Applied
Mathematics. Universidad Politécnica de Madrid.
Av. Juan de Herrera, 6, E-28040, Madrid.

###### Abstract

Pathogen introduction in plant communities can cause serious impact and
biodiversity losses that may take long time to manage and restore. Effective
control of epidemic spreading in the wild is a problem of paramount
importance, because of its implications in conservation and potential economic
losses. Understanding the mechanisms that hinder pathogen’s propagation is,
therefore, crucial. Usual modelization approaches in epidemic spreading are
based in compartmentalized models, without keeping track of pathogen’s
concentrations during spreading. In this contribution we present and fully
analyze a dynamical model for plant epidemic spreading based on pathogen’s
abundances. The model, which is defined on top of network substrates, is
amenable to a deep mathematical analysis in the absence of a limit in the
amount of pathogen a plant can tolerate before dying. In the presence of such
death threshold, we observe that the fraction of dead plants peaks at
intermediate values of network’s connectivity, and mortality decreases for
large average degrees. We discuss the implications of our results as
mechanisms to halt infection propagation.

Plant epidemic spreading, deterministic dynamics, stability, random matrices.

##  I Introduction

Introductions of new plant pathogens into previously uncolonized areas is a
major problem, since the feasible lack of defenses of the individuals in the
area might cause fatal losses [1]. One example is the sudden oak death, caused
by the broad host range oomycete Phytophthora ramorum, which has caused
devastating impacts on some North American and European forests [2].
Especially, it has killed millions of oak and tanoak in California since its
first detection in 1995. Another noteworthy example of the impact of plant
pathogen introduction is the massive economic damage caused by Xylella
fastidiosa, a bacterium that affects 563 plant species from the Americas,
Europe, the Middle East and Asia. For example, an exhaustive study of the
impact of this pathogen in olive trees can be found in reference [3].

During an epidemic outbreak, control decisions must be taken to lessen
pathogen impact [4]. However, the restricted amount of time in which actions
must be taken and the lack of information early in the epidemic often make the
choice of action difficult [5, 6] and the consequences can be extremely
detrimental. The previously mentioned sudden oak death is a clear example: as
pointed out by Cunniffe  _et al._ [1], insufficient measures taken to
eradicate the disease in California has led to a point where statewide action
to even slow the spread of P. ramorum is no longer feasible; the pathogen has
spread far enough that the only possible solution at present is local
containment [1].

Thus, developing tools that allow us to better understand the behavior of new
epidemics, and that help us predict what the outcome of different control
measures might be, is a question of paramount importance. The main aim of this
work is the analysis of the dynamics and stability, under different
conditions, of a model for the spread of infections in wild plant populations,
so that it can later be applied to real epidemics.

Most epidemiological models are based on a compartmentalization of individuals
according to their disease status [7, 8]. This is a notable simplification,
for many details of the epidemic are neglected, for example, differences in
response between individuals. The study of infection propagation on networks
provides a way to include in the simulations different parameters related to
the shape of the field in which the epidemic occurs, such as the spatial
location of the individuals and the connectivity between different
individuals. It also allows a better mechanistic understanding of the spread
of the epidemic [9]. Traditionally, networks have been successfully used in
human and animal epidemiology; however, not much work has been done for plant
epidemics [10].

Compartmentalized models like SIS, SIR, and further extensions of these,
generally do not track pathogen concentrations over time. Instead, those
models provide temporal variation of the fraction of individuals belonging to
each class (infected, susceptible, recovered, etc.) during infection
propagation over networked substrates [11, 10]. Clustering individuals, e.g.
plants or trees, into different classes makes infection dynamics more
tractable. Instead, in this contribution we focus on pathogen densities across
infected individuals and their variation over time, and we introduce and fully
analyze a dynamical model for infection spreading defined by density
independent, per-capita fluxes of pathogen between individuals. In addition,
we extract meaningful information when individuals are classified according to
their response to infection. In particular, we assume that an individual is
dead when pathogen’s abundance increases above a certain threshold. This means
that the individual is effectively removed from the network, and does not
contribute to pathogen’s propagation to other neighbouring plants. Individuals
can tolerate moderate epidemic charges (below the death threshold). It is
precisely the presence of this threshold what enriches infection dynamics
—which otherwise is amenable to analytical treatment in the absence of such a
death threshold.

The effect of the substrate (mean field, network, lattice, etc.) on which
epidemics take place has been extensively studied [12, 13, 14, 15, 16, 17].
However, the influence of network average connectivity on the fractions of
healthy, infected or dead individuals has been overlooked, surprisingly. Our
main result is related to the number of dead individuals in the presence of a
finite death threshold. We find that the fraction of dead plants peaks at
intermediate network connectivity: for small connectivity, networks are
basically disconnected and infection comes to a halt with a small number of
dead plants. When networks are connected, increasing average degree favors
channeling the pathogen across the network, the larger the mean connectivity
the smaller the fraction of dead individuals. Therefore, an effective
mechanism to lessen the epidemiological impact in plant communities can be the
facilitation of pathogen spreading over the network —for instance, by planting
new pathogen’s hosts. Such mechanism can be regarded as alternative to
obstructing or limiting propagation by isolating infected individuals.

##  II Epidemic spreading dynamics

Our approach is based on a deterministic epidemic dynamics, that unfolds on
top of a network substrate, for which model parameters are random variables
drawn from specified distributions. Across the network, each node represents
an individual plant. Consider a plant pathogen infecting a network formed by
$n$ plants or trees belonging to the same species. Let $x_{i}$ be the density
of pathogen in plant individual $i$. When isolated (i.e., pathogen is not
transported among individuals) we assume a logistic growth with rate $r>0$ for
pathogen abundance in each plant. This implicitly means that there is a
carrying capacity $K$ within each individual that limits pathogen abundance.

Plant individuals are accessible to host pathogen particles coming from
adjacent plants. Pathogen’s transport is bi-directional: infected individuals
can release pathogen’s particles to neighbors, and can also receive additional
particles from neighboring plants. Such flux of pathogen occurs among
connected individuals $i$ and $j$, for $i\neq j$. Networks are defined in a
way such that two plants are connected if there is a non-zero probability of
contagion between them. This probability depends on the dispersal ability of
the pathogen, which is usually correlated to the distance between plant pairs.
Therefore, we can assume that plants are disconnected if this probability is
small —in other words, only pairs of individuals with probability of contagion
above a threshold will be connected.

Consider a focal plant $i$. If there is a link in the network between $i$ and
$j$, the per-capita (per unit of pathogen abundance of the source individual)
in-going flux of pathogen from individual $j$ to $i$ is denoted as $a_{ij}>0$,
and the per-capita out-going flux of pathogen from $i$ to $j$ as $b_{ji}>0$.
Otherwise, if $i$ and $j$ are disconnected, then $a_{ij}=a_{ji}=0$ as well as
$b_{ij}=b_{ji}=0$. In addition, we set $a_{ii}=b_{ii}=0$ for $i=1,\dots,n$.
Then, the amount of pathogen transported from $j$ to $i$ is equal to
$a_{ij}x_{j}$, and the abundance transferred from $i$ to $j$ equals to
$b_{ji}x_{i}$. Consequently, the coupled dynamics of pathogen concentrations
are driven by a system of $n$ coupled differential equations,

$\frac{dx_{i}}{dt}=rx_{i}\left(1-\frac{x_{i}}{K}\right)+\sum_{j=1}^{n}a_{ij}x_{j}-x_{i}\sum_{j=1}^{n}b_{ji},$
(1)

for $i=1,\dots,n$. Observe that sums run over the set of neighbors of node
$i$, given the restrictions imposed above for $a_{ij}$ and $b_{ij}$. The
connectivity of the plant community defines two matrices, $A=(a_{ij})$ and
$B=(b_{ij})$. These per-capita in- and out-going fluxes will be considered as
random variables. The parameters that define pathogen’s growth in isolation
($r$ and $K$) will be varied throughout this study. Graphs are non-directed,
meaning that if there is a flux of pathogen from individual $j$ to individual
$i$, there is also pathogen transfer from $i$ to $j$. These fluxes, however,
do not have to be balanced (this depends on the values of $a_{ij}$ and
$b_{ji}$).

Observe that, if no restrictions are imposed to matrices $A$ and $B$, the
amount of pathogen transferred from a node is not necessarily equal to the
overall pathogen’s amount that all its neighbors receive. In that case, the
system is open and the overall amount of pathogen in the plant community can
increase (in the presence of an external source) or decrease (if there are
additional mechanisms that channel pathogen particles out of the system).
Otherwise, if the amount of pathogen that is transported out of node $i$ is
exactly equal to the overall amount that all of its neighbors receive, then
there are no losses of pathogen during infection spreading. This condition can
be easily formulated, because the total amount of pathogen that comes out from
$i$ is equal to $x_{i}\sum_{j}b_{ji}$, according to previous definitions. On
the other hand, a neighbor $j$ receives from node $i$ an amount of pathogen
equal to $a_{ji}x_{i}$, so the total quantity received by neighbors is
$x_{i}\sum_{j}a_{ji}$, just by summing over the neighbors of node $i$. Thus,
the conditions for no pathogen losses during transport is

$\sum_{j=1}^{n}a_{ji}=\sum_{j=1}^{n}b_{ji}$ (2)

for $i=1,\dots,n$. If these conditions are satisfied, we say that the model is
“conservative”, because there are no losses of pathogen during transfers
between individuals. It is important to observe that the system is still open
and the total pathogen’s abundance in the system can increase from its level
at time $t=0$ until reaching the steady state —this is due to pathogen’s
reproduction inside each individual.

For our model, we have defined three plant states (node compartments)
depending on host’s epidemic concentrations $x_{i}$: individuals can be
healthy (pathogen concentration equal to zero), infected (pathogen abundance
above zero and below a certain death threshold $\delta$), and dead (pathogen
concentration above the death threshold). Therefore, the death threshold can
be interpreted as the maximum pathogen amount an individual can hold to remain
alive. When the death threshold is exceeded, then the plant dies and
disappears from the network. At this moment, it is assumed that the complete
concentration of pathogen vanishes. Thus, plant death modifies immediately
network topology as well as pathogen dynamics.

##  III Qualitative analysis without mortality

Before reporting simulation results obtained for different network
architectures, here we briefly summarize the qualitative analysis of Eq. (1)
in the absence of plant mortality (i.e, in the limit $\delta\to\infty$) in two
scenarios besides the conservative case defined above:

  * (i)

A _per-capita flux balance_ condition is satisfied. In this case, the overall
in-going per-capita flux of a node is balanced by the overall out-going per-
capita flux associated to the same node. This condition reduces, for an
arbitrary node $i$, to

$\sum_{j=1}^{n}a_{ij}=\sum_{j=1}^{n}b_{ji},\qquad i=1,\dots,n.$ (3)

Notice the difference between (2) and (3).

  * (ii)

_No restriction_ is imposed in per-capita flux matrices $A$ and $B$.

We do not make here explicit assumptions about network structure, which is
implicitly contained in matrices $A$ and $B$.

Case (i) has been contemplated because it is amenable to a complete stability
analysis. The proofs of the following results are provided in Appendix A. It
can be shown that, if the network is connected, our model always exhibis two
equilibrium points: $\bm{x}^{\star}=\bm{0}:=(0,\dots,0)$ and
$\bm{x}^{\star}=K\bm{1}=(K,\dots,K)$, for $\bm{1}:=(1,\dots,1)$. Moreover, the
first equilibrium point, which corresponds to pathogen’s clearance, is
unstable. The second one, associated to a full infection situation (all
individuals are infected), is globally asymptotically stable, though.
Therefore, in the absence of a death threshold, it is expected that all
individuals within a connected component of the network will end up infected.

The stability analysis yields comparable results for the general case of
arbitrary (unrestricted) per-capita pathogen fluxes between individuals. In
(ii) it is not possible to compute explicit expressions for the equilibrium
points. However, if matrices $A$ and $B$ are random, it is _almost sure_ that,
for connected networks (i.e., there is a single giant component), the only
equilibria observed for Eq. (1) correspond to either pathogen’s clearance,
$\bm{x}^{\star}=\bm{0}$, or full infection, $\bm{x}^{\star}=(x_{i}^{\star})$,
with $x_{i}^{\star}>0$ for every node, $x_{i}^{\star}$ not necessarily equal.
Two situations can occur, depending on network’s connectivity: (a) a single
equilibrium point emerges, $\bm{x}^{\star}=\bm{0}$, which is stable; and (b)
both equilibrium points arise, being unstable the one associated to pathogen’s
clearance, and stable the one relative to pathogen’s full infection. Note that
the full infection equilibrium emerges and is stable when the pathogen’s
clearance one becomes unstable.

Although it is difficult to analytically show that the full infection
equilibrium exists (case b), we numerically found that this is precisely the
most probable case, especially for sparse random networks above (but close to)
the percolation threshold. For larger network average connectivity, case (a)
becomes more frequent (not dominant, though), and model realizations in which
infection disappears can be observed in simulations (see Fig. 6 in Appendix
B). Observe that the qualitative analysis summarized here for the general case
(ii) applies as well for the conservative scenario defined by Eq. (2).

Therefore, in the absence of a death threshold, it is expected that the
pathogen spreads throughout the entire network with high probability if the
network is connected. Considering a maximum value in pathogen’s concentration
above which an individual is regarded as dead, however, makes very hard to
predict which plant is to become extinct and which one will survive (although
infected) starting from arbitrary initial conditions, unless resorting to the
numerical integration of the system of differential equations.

##  IV Plant mortality and network dynamics

In the presence of a (finite) death threshold $\delta$, every infected plant
whose epidemic charge exceeds the mortality threshold will die. When mortality
occurs, automatically, the corresponding node is disconnected from the
network, as well as the associated in- and out-going fluxes (which are set
equal to zero). Importantly, network topology is temporally coupled with the
infection propagation dynamics. Initial networks initially can change its size
and topological properties due to the demise of some of the individuals during
the epidemic. As we will show below, the introduction of a mortality threshold
in pathogen’s concentrations changes drastically the scenario portrayed in the
qualitative analysis of the model without such an upper bound on tolerable
epidemic charges.

In simulations, the entries of matrices $A$ and $B$ were independently and
randomly drawn from a uniform distribution $U(0,1)$. The dynamical model (1)
was integrated numerically until convergence to an equilibrium steady state.
As the dynamics unfolds, a number of nodes can go extinct until no
concentration exceeds the threshold across the system. Nodes that remain alive
will reach the corresponding equilibrium state.

Dynamics was integrated using two network architectures: random graphs drawn
from the Erdös-Renyi (ER) [18], and random geometric (RG) [19] models. In the
absence of any particular knowledge about plant locations, we assume that
links are drawn at random. Specifically, it is assumed that there is a uniform
probability of linking two nodes, as defined by the classical ER model. The
outcome of the ER model is a graph $G(n,p)$ in which $n$ is the number of
nodes and $p$ is the probability that two nodes are linked —i.e., links are
placed randomly and independently between distinct pairs of nodes with
probability $p$.

In reality, however, the probability of contagion is straightforwardly related
to distance. As a consequence, it is natural to assume that closer nodes have
a larger probability of being connected. This assumption is applied to build
RG networks. A random geometric network is the simplest kind of spatial
network. Nodes are embedded in a metric space, and two nodes could be
connected if and only if the euclidean distance between them is in a given
range —for instance, smaller than a certain neighborhood radius, $R$ [20].
Therefore, two nodes that are within the same area of influence are randomly
connected as in the ER model. Nodes that are out of this area of influence are
not connected.

Nodes in our RG networks are drawn uniformly on the unit square
$[0,1]\times[0,1]$. We assume that connection probability explicitly depends
on the distance between nodes. We used the following continuous function,

$p_{ij}=\begin{cases}1-\left(\frac{d_{ij}}{R}\right)^{2},&d_{ij}\leq R,\\\
0,&d_{ij}>R,\end{cases}$ (4)

which means that when the distance between $i$ and $j$ is smaller than $R$,
there is a non-zero probability of connection dependent on that distance (the
closer the nodes are, the more chances they are connected). When the distance
between pairs of nodes is above the radius, they are never connected. Samples
of the RG network model with size $n$ and radius $R$ will be denoted by
$G(n,R)$. Our results are not dependent of the specific functional form of
connection probability given by Eq. (4).

Figure 1: Number of healthy, infected and dead plants. Upper (lower) panels
correspond to ER (RG) graphs. A, upper panel: example of an ER network drawn
from $G(100,0.03)$ with a connection probability $p=0.03$; lower panel:
example of a RG network with $n=100$ and $R=0.15$ on the unit square. B,
number of healthy (green) and infected (red) nodes in simulations carried out
without a death threshold ($\delta\to\infty$) as functions of mean degrees of
networks. C, number of healthy (green), infected (red), and dead (black) nodes
in the presence of the death threshold $(\delta=0.9K)$ as functions of initial
network’s mean degree. In B and C panels, the horizontal axis is calculated as
the expected mean degree of the initial network —i.e, the mean averaged degree
over realized networks, which is equal to $\text{E}[k]=(n-1)p$ for ER sampled
graphs. The shading areas are defined by the $10$-th and $90$-th percentiles.
Here $r=0.5$ and $K=1$. A maximum in the number of dead nodes arises in both
cases; in ER networks this happens at a mean degree of around $3$, and in RG
networks at a mean degree of around $5$. Each panel was calculated by
averaging over $1200$ initial conditions, networks and matrix realizations.

Simulations were conducted using networks formed by $100$ nodes. Unless the
contrary is specified, we used a growth rate value $r=0.5$ and a carrying
capacity value $K=1$. As for the death threshold, we set it as $\delta=0.9K$.
As initial condition, we picked up randomly a node as infected with an initial
pathogen’s load randomly drawn from the uniform distribution $U(0,\delta)$.

##  V Simulation results

In Figure 1 we report results for the infection process operating on networks
with increasing connectivity. Panel A shows two samples of ER and RG network
models with about $L=150$ links overall. Clearly, for that number of links, ER
graphs are closer to the percolation threshold —estimated as $p_{c}\approx
1/n$, [21]—, which is evidenced by a large connected component. For $L\approx
150$, RG networks are comprised of more isolated nodes and smaller components.
The radius above which a giant component arises in RG graphs is estimated as
$R_{c}\approx\sqrt{\log n/(n\pi)}$ [20]. Panel B shows how the number of
healthy and infected nodes vary as functions of the mean degree when plant
mortality is not considered (i.e., when the death threshold goes to infinity),
whereas panel C reports the same results for the number of healthy, infected,
and dead nodes versus mean degree in the presence of a finite death threshold.
We observe in both panels that ER and RG curves are similar, but displaced to
the right for RG graphs.

As shown in Appendix B, when the network is connected, we expect that every
node is infected (at least in the range of average degrees reported in Fig.
1), in the absence of plant mortality. For mean degrees well above the
percolation threshold, this is what we observe in panel B. On the other hand,
if the network is not connected, it is apparent that pathogen’s spreading will
not progress in those components that were not initially infected. It is only
close to the percolation threshold of these networks that the curve of healthy
individuals starts declining. This is consistent with the expected degree
$\text{E}[k]$ at the percolation threshold ($k_{i}$ stands for the degree of
node $i$). For an ER graph $G(n,p)$, $\text{E}[k]\approx(n-1)p$ [21], which is
of the order of unity at $p=p_{c}\approx 1/n$. For a RG graph $G(n,R)$,
$\text{E}[k]\approx n\pi R^{2}$ [20], which reduces to $\text{E}[k]\approx\log
n$ at $R=R_{c}\approx\sqrt{\log n/(n\pi)}$. Therefore, for $n=100$ we expect
the transition at mean degrees about $4.6$ for RG networks and about $1$ for
ER networks. The decline of the number of healthy plants in both cases is
observed at mean degrees consistent with these estimations (Fig. 1). This
analysis suggest that network connectivity is an important driver of the
overall outcome of infection spreading.

Plant mortality arises for finite values of the death threshold $\delta$.
According to Fig. 1, the number of dead nodes peaks at a mean degree value
around $\text{E}[k]\approx 3$ for ER graphs and around $\text{E}[k]\approx 5$
for RG networks. We observe this maximum in mortality independently of the
values taken by the growth rate, carrying capacity, and death threshold. These
parameters values do affect the location of the maximum, though. The same
phenomenon occurs in the conservative scenario for ER graphs, see Figure 2.
Conservation of transported pathogen in RG networks yields similar results.

Figure 2: Conservative model. Here we reproduce panels B and C of Fig. 1 for
the dependence of the number of individuals within compartments (healthy,
infected, or dead) with ER network’s mean degree, when conservation of
pathogen —Eq. (2)— is imposed in the dynamics.

These maxima in the number of dead plants seem counter-intuitive. One might
expect that the higher the connectivity of the network, the lower the number
of dead plants. However, there is an intermediate network connectivity with
the highest death rate. To test whether these maxima were due to the networks
being disconnected for lower mean degrees, we repeated these simulations for
small-world networks (Appendix C). We used the Watts-Strogatz (WS) small-world
model [22], that introduces link rewiring of regular networks yielding graphs
formed by a single connected component. In this case, we do not observe a
maximum in the number of dead nodes for any mean degree; plant mortality
decreases monotonically as the mean degree increases. The reason is that, in
small-world networks, no matter how low the mean degree is, the network is
always connected. Infection can spread throughout the network, and for larger
degrees the epidemic load can be channeled out through a higher number of
ways, thus reducing the asymptotic pathogen’s abundance of each node and,
consequently, not exceeding the death threshold.

Once the network is connected, or has just a few isolated nodes, if nodes are
connected on average with a higher number of neighbors, pathogen particles can
be transported more efficiently between plants and the overall pathogen’s
charge is distributed across the network, leading to more individuals with
concentrations below the threshold $\delta$. It is difficult for the pathogen
to reach the death threshold in a given individual, for there are many out-
going fluxes to many other individuals, swiftly distributing the pathogen.

This hypothesis can be confirmed in Figure 3, which shows the relationship
between network connectivity and pathogen’s temporal dynamics. We have
considered ER graphs with three connectivity values: (i) below the maximum in
the number of death nodes ($\text{E}[k]=1$), (ii) at the maximum
($\text{E}[k]=3$), and (iii) well above the maximum ($\text{E}[k]=8$). For
these cases we have obtained the distributions of the size of the connected
component which the initially infected node belongs to, as well as the size
distribution of the largest connected component that ends up fully infected
—note that, according to our analytical results, once the process has relaxed
and every survivor has abundance below the death threshold in an infected
component, we expect that the full component will remain infected until
reaching the steady state (this is illustrated in Fig. 3, lower panels). As
the initially infected node is in a larger component, we observe that the
temporal dynamics leads to increasing death events. However, for mean degrees
leading to fully connected networks, the average pathogen’s load per
individual is smaller, because pathogen particles are more evenly distributed
due to network’s higher connectivity. Indeed, for low mean degree
($\text{E}[k]=1$), the size of the initially infected component is small, and
the network is initially broken into pieces, so infection is able to progress
only within the initial group and the expected number of deaths is low. For
intermediate mean degree ($\text{E}[k]=3$), the initially infected component
is large, but the epidemic process breaks the network and, as a consequence,
the size of the largest component that ends up fully infected lowers. For
large connectivity ($\text{E}[k]=8$), almost every plant belongs to the
initially infected component. Such connectivity allows for an effective
distribution of the overall pathogen’s load across nodes, many of them
remaining below the death threshold. The initial component’s size in this case
decreases because dead nodes are removed from the initial cluster. Hence, the
maxima in mortality are explained: once a network is connected, higher
connectivity implies lower mortality rates, due to the distribution of
pathogen over a larger number of nodes as the epidemic spreads.

Figure 3: Random network connectivity and pathogen’s temporal dynamics. Panels
A, B, and C correspond to ER networks with mean degrees $\text{E}[k]=1$,
$\text{E}[k]=3$, and $\text{E}[k]=8$, respectively. Upper panels depict the
endstate of sampled networks after the disease has spread. Healthy (green),
infected (red), and dead (black) nodes are represented. Blue links remain
after reaching the endstate (links connecting dead nodes are colored in grey).
In all cases, in the initial state of the network only one node is infected.
As for the endstate, in A the fractions $(H,I,D)$ of healthy, infected, and
dead plants are $(H,I,D)=(0.95,0.03,0.02)$. In B and C we end up with
fractions $(H,I,D)=(0.04,0.66,0.3)$ and $(H,I,D)=(0,0.89,0.11)$, respectively.
The left middle panel shows the distribution of the size of the connected
component which the initially infected node belongs to ($1200$ realizations
for each mean degree). Blue, purple, and orange histograms correspond to
$\text{E}[k]=1$, $\text{E}[k]=3$, and $\text{E}[k]=8$, respectively. The right
middle panel depicts the size distribution of the largest completely infected
component at the end of simulations, showing that networks break up once the
dynamics unfolds. Lower panels show the temporal variation of each node
pathogen’s concentration. The dashed line indicates the death threshold value.
The location of the distributions, which correlates to existence of a giant
component, is put into correspondence with temporal dynamics, leading to a
small fraction of infections and deaths (A), a maximum number of dead nodes
(B), and the distribution of pathogen’s loads across the network, leading to
fewer deaths (C).

To see how remaining model parameters affect plant mortality, we set the
carrying capacity to $K=1$ and varied the death threshold ($\delta\in[0.1,1]$)
and the growth rate ($r\in[0.1,5]$). The percentage of dead nodes as function
of the growth rate and the death threshold are shown in the heat map
represented in Figure 4.

Figure 4: Fraction of dead nodes for variable death threshold and growth rate.
For fixed growth rates, mortality decreases as the death threshold increases.
When the death threshold is constant, the fraction of dead nodes augments for
increasing growth rates until reaching a rather constant plateau. This can
occur sharply or smoothly depending on the values of the death threshold.

For fixed values of the growth rate, the number of dead nodes decreases as the
death threshold increases. This is an intuitive result, since a higher death
threshold implies a lesser fraction of killed plants. When the death threshold
is equal to the carrying capacity ($K=1$), the number of death nodes for any
growth rate is significantly lower than with any other death threshold.

If the death threshold is fixed, mortality increases as the growth rate
augments until reaching an approximately constant value that depends on
$\delta$. Such increase can be sharper (smoother) for smaller (larger) values
of $\delta$. This is due to the fact that, when the ratio between growth rate
and death threshold is large enough, nodes die too fast, disconnecting the
network and hindering the spread of the disease. In summary, mortality becomes
more pronounced for large pathogen’s growth rates and small values of the
maximum pathogen’s concentration that an individual can tolerate.

##  VI Discussion

In this work we have presented and fully analyzed a mathematical model of pest
dynamics, acting on plant communities, that keeps track of pathogen’s
concentration across individuals over time. Our approach considers a death
threshold, above which plants die. We have shown that the introduction of this
threshold modifies substantially the infection dynamics. Contrary to the case
where no threshold exists, individual deaths modify network topology over
time, because individuals whose pathogenic charge exceeds the threshold are no
longer connected. A counter-intuitive result turns out: networks with higher
connectivity yield a larger fraction of individuals that survive. We can
explain the phenomenon using a channeled flux analog: if the percolation
threshold has been crossed over and there exists a giant component, for equal
pathogen growth rate and death threshold values, a higher connectivity enables
a better drainage of pathogen particles across the network, which precludes a
larger number of individuals from reaching their death threshold. What remains
is a network formed by infected individuals that are resistant to the pathogen
in the long run.

As for the robustness of our main result, we have used different network
structures (ER, RG) and implemented different conditions referred to potential
pathogen’s losses or gains during transportation (balanced fluxes,
conservative approach, unrestricted fluxes). The maximum in mortality arises
as long as the network changes from being disconnected to connected —we do not
find the maximum in model networks that have no percolation transition, such
as in the WS model. Therefore, the incidence of the epidemic across a
population is determined by network structure. As far as we know, fatality
rates in plant epidemics have not been previously connected with network
average connectivity. In addition, we varied model parameters (pathogen’s
growth rate and death threshold) and —not surprisingly— we found that
restrictive death thresholds and highly reproductive pathogens yield the
highest fractions of dead nodes.

Finally, we have used random values for per-capita fluxes of pathogen between
individuals. Random matrices represent an appropriate framework to model large
systems for which it is difficult to infer actual interaction strengths, as
well as to provide consistent predictions about diversity and stability [23,
24].

Except for the position of the maximum, which in RG networks takes place at
larger mean degrees, we did not find remarkable differences with the dynamics
on ER networks regarding fatality and infectivity. The RG model is more
realistic because nodes are embedded in a plane, as plants in communities, and
neighboring relations between them are better defined depending on distance.
However, these features are not determinant when it comes to quantify
mortality rates during epidemic spreading.

Temporal dynamics shown in Figure 3 illustrate the appearance of the maximum
in mortality. When a giant component starts forming, most of the nodes get
infected, resulting in the propagation of the disease. However, since the mean
degree is not very high, it takes longer to reach all the nodes (see Fig. 3B,
lower panel). Pathogen grows inside infected hosts and take longer to spread,
so a higher number of plants can effectively cross over the death threshold,
increasing mortality numbers. Larger connectivity (above the maximum),
however, implies that dead nodes reach the threshold in a more synchronous way
(Fig. 3C, lower panel). Because connectivity is large, the network remains
connected and the qualitative analysis provided in Section III ensures that a
full infection equilibrium state (for the plants that remain alive) will be
reached with high probability.

Countless studies based on compartmentalized models have focused on finding
thresholds in effective spreading rates above which infection progresses and
reaches every node (see [25] and references therein). For SIS models on
heterogeneous networks, the mean field epidemic threshold is known to be equal
to $\langle k\rangle/\langle k^{2}\rangle$, where $\langle\cdot\rangle$ stands
for an average value [26]. Such thresholds do not arise in our model because
pathogen can be effectively spread across the giant component. For that
reason, our results have to be recast in terms of how pathogen distributes
depending on network connectivity.

Our work helps infer effective mechanisms that can hinder, or even halt,
epidemic spreading across plant communities. For communities in which plants
are scattered, it is hard that the infection propagates among disconnected
network components, at least, in short time scales. Obviously, long-range
dispersal can occur and uncolonized components can become infected in the long
run. However, in diverse communities, packed individuals are accessible to
pathogen transfers. Probably local containment to refrain propagation is not
effective in highly connected systems. Our results suggest that a plausible
mechanism to alleviate pathogen charges across communities is precisely
increasing the connectivity by planting new individuals. For example, Viburnum
acerifolium [27] is a shrub that coexists with the Californian oak, and both
are infected by P. ramorum [28]. Shrub individuals grow fast compared to trees
and could alleviate oak’s pathogen levels shortly after being planted, if
oak’s epidemic spreading could be effectively approximated by the mechanisms
that drive our model dynamics. New individuals will “drain” pathogen particles
from the species to be protected. Although infection may pervade the whole
system, larger connectivity values will lower pathogen’s loads, which will be
more tolerable for plants targeted by conservation strategies. Thanks to
pathogen reduction measures like this, infected individuals, ultimately, will
survive until recovery protocols are available.

##  Appendix A Stability results for per-capita flux balance

In this section we provide proofs for the results stated in Section III for
the per-capita flux balanced dynamics. We first calculate the attractors of
the dynamics.

######  Theorem 1.

Assume that the graph $M$ associated to matrices $A$ and $B$ is connected, and
that per-capita fluxes are balanced, i.e., Eq. (3) holds. Then the only
equilibrium points of (1) are either $x^{\star}_{1}=\dots=x^{\star}_{n}=0$ or
$x^{\star}_{1}=\dots=x^{\star}_{n}=K$.

###### Proof.

We want to solve the non-linear system of equations

$rx^{\star}_{i}\left(1-\frac{x^{\star}_{i}}{K}\right)+\sum_{j=1}^{n}a_{ij}x^{\star}_{j}-x^{\star}_{i}\sum_{j=1}^{n}b_{ji}=0,$
(5)

$i=1,\dots,n$. Completing squares, we can write the system as

$\left(x^{\star}_{i}-\frac{K}{2}\right)^{2}=\frac{K^{2}}{4}+\frac{K}{r}\bigg{(}\sum_{j=1}^{n}a_{ij}x^{\star}_{j}-x^{\star}_{i}\sum_{j=1}^{n}b_{ji}\bigg{)}=0.$
(6)

Define the new variables $y_{i}:=x^{\star}_{i}-\frac{K}{2}$. With these new
variables, the system reduces to

$y_{i}^{2}=\frac{K^{2}}{4}+\frac{K}{r}\bigg{(}\sum_{j=1}^{n}a_{ij}y_{j}-y_{i}\sum_{j=1}^{n}b_{ji}\bigg{)}=0.$
(7)

Without loss of generality, we can sort the entries of vector $\bm{y}$ by
relabelling equations and write $-\frac{K}{2}\leq y_{1}\leq y_{2}\leq\dots\leq
y_{n}$. Then (7) for $i=1$ yields

$\displaystyle y_{1}^{2}$
$\displaystyle=\frac{K^{2}}{4}+\frac{K}{r}\bigg{(}\sum_{j=1}^{n}a_{1j}y_{j}-y_{1}\sum_{j=1}^{n}b_{j1}\bigg{)}$
(8)
$\displaystyle\geq\frac{K^{2}}{4}+\frac{K}{r}\bigg{(}\sum_{j=1}^{n}a_{1j}-\sum_{j=1}^{n}b_{j1}\bigg{)}y_{1}=\frac{K^{2}}{4},$

where we have used the assumption that $y_{j}\geq y_{1}$ for $j=2,\dots,n$ and
the equality of in- and out-going fluxes, cf. Eq. (3). From (8) we have that
either $y_{1}=-\frac{K}{2}$ or $y_{1}\geq\frac{K}{2}$. We consider these two
cases separately.

  * (a)

Let $y_{1}=-\frac{K}{2}$. Since the network is connected, exists a shortest-
path over the network such that node $1$ is connected to node $n$, either
directly through an existing link or by a finite series of steps using
intermediate nodes. Let $\\{i_{1},i_{2},\dots,i_{s}\\}$ denote the index
sequence for the path connecting node $1$ and node $n$. Then $a_{1,i_{1}}>0$
and assume that $y_{i_{1}}>-\frac{K}{2}$. Because
$\sum_{j}a_{1j}y_{j}>-\frac{K}{2}\sum_{j}a_{1j}$ (notice that the strict
inequality is due the variables being sorted, which implies that
$y_{k}>-\frac{K}{2}$ for $k>i_{1}$), Eq. (7) for $i=1$ reduces to

$\displaystyle 0$
$\displaystyle=\sum_{j=1}^{n}a_{1j}y_{j}+\frac{K}{2}\sum_{j=1}^{n}b_{j1}$ (9)
$\displaystyle>\frac{K}{2}\bigg{(}-\sum_{j=1}^{n}a_{1j}+\sum_{j=1}^{n}b_{j1}\bigg{)}=0,$

which is obviously a contradiction. Therefore $y_{i_{1}}=-\frac{K}{2}$, which
implies that $y_{1}=\dots=y_{i_{1}}=-\frac{K}{2}$.

This argument can be iterated until reaching node $n$: now consider node
$i_{1}$, which in the path is connected to $i_{2}$. If $i_{2}>i_{1}$ then we
apply the same argument (using that $a_{i_{1},i_{2}}>0$ in Eq. (7)
particularized for $i=i_{1}$) to prove that
$y_{1}=\dots=y_{i_{2}}=-\frac{K}{2}$. If $i_{2}<i_{1}$, then we iterate the
procedure for the following node in the sequence, $i_{3}$. At the end of the
path we will reach node $i_{s}$, connected to the endpoint at node $n$. Then
Eq. (7) for $i=i_{s}$, together with the fact that $a_{i_{s},n}>0$, implies
that $y_{1}=y_{2}=\dots=y_{n}=-\frac{K}{2}$ in this case. This solution
reduces to $x^{\star}_{1}=\dots=x^{\star}_{n}=0$.

  * (b)

Assume now that $y_{1}\geq\frac{K}{2}$. Now particularize (7) for $i=n$. Using
that $y_{j}\leq y_{n}$, we can find the following upper bound for $y_{n}^{2}$:

$\displaystyle y_{n}^{2}$
$\displaystyle=\frac{K^{2}}{4}+\frac{K}{r}\bigg{(}\sum_{j=1}^{n}a_{nj}y_{j}-y_{n}\sum_{j=1}^{n}b_{jn}\bigg{)}$
(10)
$\displaystyle\leq\frac{K^{2}}{4}+\frac{K}{r}\bigg{(}\sum_{j=1}^{n}a_{nj}-\sum_{j=1}^{n}b_{jn}\bigg{)}y_{n}=\frac{K^{2}}{4},$

which implies that $-\frac{K}{2}\leq y_{n}\leq\frac{K}{2}$. Since $y_{1}\leq
y_{n}\leq\frac{K}{2}$ and $y_{1}\geq\frac{K}{2}$ by hypothesis, we get
$y_{1}=y_{n}=\frac{K}{2}$. This obviously leads to the solution
$x^{\star}_{1}=\dots=x^{\star}_{n}=K$.

This completes the proof of the theorem. ∎

Observe that this result is general for connected graphs, irrespective of the
specific network structure yielded by matrix $M$. In other words, we have not
made any assumption on the distribution of the links, some of which can be
absent (some $a_{ij}=0$ or $b_{ij}=0$), to prove that the only equilibrium
points are those associated to full infection or to the absence of the
infection.

We now focus on the stability of these equilibrium points. It can be shown
that the equilibrium point associated to pathogen coexistence across
individuals, $\bm{x}^{\star}=K\bm{1}$, is _globally asymptotically stable_. We
proceed as follows: (i) first we show that $\bm{x}^{\star}=K\bm{1}$ is
asymptotically stable; (ii) then we show that the other equilibrium point,
$\bm{x}^{\star}=\bm{0}$, is unstable; (ii) as the origin is unstable, global
stability of the coexistence equilibrium arises because the state space of
feasible solutions is invariant to the dynamical system, which implies that
every trajectory with positive initial conditions will converge to
$\bm{x}^{\star}=K\bm{1}$.

The Jacobian matrix can be expressed as $J=D+A$, where $D$ is a diagonal
matrix $D=(d_{ii})$ whose diagonal entries are given by

$d_{ii}=r-\frac{2rx_{i}^{\star}}{K}-\sum_{j=1}^{n}b_{ji}.$ (11)

The stability of the coexistence equilibrium point follows as a corollary of
the Gershgorin’s circle theorem, which we reproduce here for the sake of
completeness:

######  Theorem 2.

Let $S$ be a complex $n\times n$ matrix with entries $s_{ij}$. Let
$R_{i}=\sum_{j\neq i}|s_{ij}|$ be the sum of the absolute values of the non-
diagonal entries in the $i$-th row. Let $D(s_{ii},R_{i})\subseteq\mathbb{C}$
be a closed disc centered at $s_{ii}$ with radius $R_{i}$ (such a disc is
called a Gershgorin disc). Then every eigenvalue of $S$ lies within at least
one of the Gershgorin discs $D(m_{ii},R_{i})$.

Then, stability follows directly:

######  Corollary 1.

If condition (3) is satisfied, the equilibrium point of (1) associated to full
infection, $x^{\star}_{1}=\dots=x^{\star}_{n}=K$, is asymptotically stable.

###### Proof.

The Jacobian matrix, in this case, reduces to $J=D+A$ with
$d_{ii}=-r-\sum_{j}b_{ji}$. The Gershgorin’s circle theorem helps show that
all the eigenvalues of $J$ have strictly negative real part. Indeed, for each
row of $J$, the radius of the $i$-th disc is $R_{i}=\sum_{j}a_{ij}$ because
$a_{ij}\geq 0$. Therefore each disc
$D\big{(}-r-\sum_{j}b_{ji},\sum_{j}a_{ij}\big{)}$ is centered on the real
axis, and each one is contained in the complex semi-plane
$\text{Re}\,z\leq-r$, because the right-most point of the disc is the real
number $-r-\sum_{j}b_{ji}+\sum_{j}a_{ij}=-r$. Thus, any eigenvalue satisfies
$\text{Re}\,\lambda\leq-r$ and has strictly negative real part. Hence the
equilibrium point is asymptotically stable. ∎

Again, this result is independent of whether matrices $A$ and $B$ have an
adjacency matrix $M$ superimposed or not. It is general as long as the
condition of per-capita flux balance, Eq. (3), holds.

On the other hand, we can show that the equilibrium point associated to
pathogen’s clearance, $\bm{x}^{\star}=\bm{0}$, is unstable. We state this as a
proposition:

######  Proposition 1.

If condition (3) holds, the equilibrium point of (1) associated to full
pathogen extinction, $x^{\star}_{1}=\dots=x^{\star}_{n}=0$, is unstable.

###### Proof.

The Jacobian matrix has diagonal entries given by $d_{ii}=r-\sum_{j}b_{ji}$.
Then trivially the vector $\bm{1}^{T}$ is an eigenvector of $J$ with
eigenvalue $r$, because
$\sum_{j}J_{ij}=d_{ii}+\sum_{j}a_{ij}=r+\sum_{j}a_{ij}-\sum_{j}b_{ji}=r$. As
$J$ has constant row-sums, then $\bm{1}$ is eigenvector with eigenvalue equal
to each row sum. Because there is an eigenvalue with strictly positive real
part, this point is unstable, as stated. ∎

To finish with the qualitative analysis, we need to rule out the possibility
that trajectories do not cross the boundaries of the space of feasible
solutions, $x_{i}=0$, $i=1,\dots,n$. But it is easy to check that any
trajectory starting with initial condition $\bm{x}(0)$ in the interior of the
space $\mathbb{R}^{n}_{+}=\\{\bm{x}\in\mathbb{R}^{n}|x_{i}\geq
0,i=1,\dots,n\\}$ remains in that space, i.e., the space of feasible solutions
is invariant. We state this as a proposition:

######  Proposition 2.

The state space of feasible solutions associated to Eq. (1),
$\mathbb{R}^{n}_{+}$, is invariant.

###### Proof.

First consider the initial condition $\bm{x}(0)=\bm{0}$. Since $\bm{x}=\bm{0}$
is an equilibrium point, the trajectory remains in $\mathbb{R}^{n}_{+}$.

Now consider an initial condition such that $x_{i}(0)\geq 0$ for all
$i=1,\dots,n$ and some of the initial values verify $x_{j}(0)>0$. Then it is
easy to see that $x_{i}(t)\geq 0$ for all $t>0$ and all $i=1,\dots,n$. Assume
that some variable $x_{k}(t_{a})=0$ vanishes at $t=t_{a}>0$. At that time it
holds that

$\left.\frac{dx_{k}}{dt}\right|_{t=t_{a}}=\sum_{s=1}^{n}a_{ks}x_{s}(t_{a})\geq
0.$ (12)

Because the derivative is non-negative, the flux of the ODE system does not
allow the trajectory to cross the axis $x_{k}=0$. The same holds for the
remaining variables. Hence any initial condition
$\bm{x}(0)\in\mathbb{R}^{n}_{+}$ yields a trajectory contained in the state
space of feasible solutions. ∎

This proposition, together with corollary 1 and proposition 1 yields the
following corollary:

######  Corollary 2.

If condition (3) is satisfied, the equilibrium point of (1) associated to full
infection, $x^{\star}_{1}=\dots=x^{\star}_{n}=K$, is globally asymptotically
stable.

###### Proof.

This follows trivially because the state space $\mathbb{R}^{n}_{+}$ is
invariant and the unique stable equilibrium point is $\bm{x}^{\star}=K\bm{1}$.
Hence all trajectories will converge to $\bm{x}^{\star}=K\bm{1}$ and its basin
of attraction will be $\mathbb{R}^{n}_{+}-\\{\bm{0}\\}$, i.e., the full state
space except the unstable equilibrium point. ∎

##  Appendix B Stability results for the general case

The behavior described in Appendix A for the flux-balanced model is recovered
_almost surely_ if no restrictions are imposed in matrices $A$ and $B$.
Regarding system’s equilibria, it is always found the one associated to
pathogen’s clearance, $\bm{x}^{\star}=\bm{0}$. According to the results
provided below in this Appendix, it is very likely that an equilibrium point
associated to full infection, $\bm{x}^{\star}$ with $x_{i}^{\star}>0$ for
$i=1,\dots,n$, exists. But, if the network is connected, no equilibria can
arise that combine infected and healthy individuals _almost surely_ :

######  Proposition 3.

Consider the dynamics (1) with unrestricted, random per-capita flux matrices
$A$ and $B$ (with i.i.d. entries). If the adjacency matrix $M$ defines a
connected graph, almost surely no equilibrium points $\bm{x}^{\star}$ exist
such some pathogen abundances are positive and some of them are exactly equal
to zero.

###### Proof.

Assume, without loss of generality, that the first $k$ abundances of
$\bm{x}^{\star}$ are equal to zero, $x_{1}^{\star}=\dots=x_{k}^{\star}=0$, and
the remaining ones are positive, $x_{i}^{\star}>0$ for $i=k+1,\dots,n$. Then
(5) reduces, for $i=1,\dots,k$, to

$\sum_{j=k+1}^{n}a_{ij}x_{j}^{\star}=0.$ (13)

Let $A_{1}$ be the submatrix of $A$ formed by the columns from $j=k+1$ to
$j=n$ and the rows from $i=1$ to $i=k$. Then if Eq. (13) was true, this would
imply that the random matrix $A_{1}$ is not full rank. But this is a
contradiction because any (connected) random matrix is full rank —see
Corollary 1.2 in [29].

Observe that, if the graph were disconnected, then we could find equilibria
with non-zero entries within one or more connected components and zero entries
in other components. Eq. (13) would not impose any restriction because matrix
elements $a_{ij}=0$ between disconnected components. For example, if infection
spreading starts in a node within a component, at the end this component will
reach a fully infected state, whereas the remaining components will have
healthy individuals. ∎

Therefore, we can expect _almost surely_ that the only equilibrium points of
(1) with unrestricted, random per-capita flux matrices $A$ and $B$, are either
$\bm{x}^{\star}=\bm{0}$ or $\bm{x}^{\star}$ with all entries $x_{i}^{\star}>0$
for $i=1,\dots,n$, if the latter is found as solution of (5). This is similar
to what we found analytically in Appendix A for the balanced case. In
principle, more than one single equilibrium associated to full infection with
all $x_{i}^{\star}>0$ could arise. In practice, the majority of realizations
should exhibit only the two aforementioned equilibria, as in the balanced
case. Observe that this applies to the conservative case, for it being a
particular case of the unrestricted flux situation.

Can we say something about the stability of these two equilibria? The
following proposition holds:

######  Proposition 4.

Let $\bm{x}^{\star}=(x_{i}^{\star})$ be a solution of (5), and

$J(\bm{x}^{\star})=\emph{diag}\biggl{(}r-\frac{2rx_{i}^{\star}}{K}-\sum_{j=1}^{n}b_{ji}\biggr{)}+A$
(14)

be the Jacobian matrix evaluated at that equilibrium point
—$\emph{diag}(\bm{u})$ stands for a diagonal matrix defined by vector
$\bm{u}$. Then, if the full infection equilibrium $\bm{x}^{\star}$
($x_{i}^{\star}>0$) exists, it holds that

$(J(\bm{x}^{\star})\bm{x}^{\star})_{i}=-\frac{r{x_{i}^{\star}}^{2}}{K},$ (15)

for $i=1,\dots,n$. Moreover, if the full infection equilibrium arises, at the
full pathogen’s clearance equilibrium we find that

$(J(\bm{0})\bm{x}^{\star})_{i}=\frac{r{x_{i}^{\star}}^{2}}{K},$ (16)

for $i=1,\dots,n$ and $\bm{x}^{\star}$ the full pathogen infection equilibrium
point.

###### Proof.

It is easy to compute that

$(J(\bm{x}^{\star})\bm{x}^{\star})_{i}=rx_{i}^{\star}-\frac{2r{x_{i}^{\star}}^{2}}{K}-x_{i}^{\star}\sum_{j=1}^{n}b_{ji}+(A\bm{x}^{\star})_{i}\\\
=-\frac{r{x_{i}^{\star}}^{2}}{K}+rx_{i}^{\star}-\frac{r{x_{i}^{\star}}^{2}}{K}-x_{i}^{\star}\sum_{j=1}^{n}b_{ji}+(A\bm{x}^{\star})_{i},$
(17)

but in the last equality all the terms except the first vanish because
$\bm{x}^{\star}$ is a solution of (5). Similarly,

$\displaystyle(J(\bm{0})\bm{x}^{\star})_{i}$
$\displaystyle=rx_{i}^{\star}-x_{i}^{\star}\sum_{j=1}^{n}b_{ji}+(A\bm{x}^{\star})_{i}=\frac{r{x_{i}^{\star}}^{2}}{K}$
(18)
$\displaystyle+rx_{i}^{\star}-\frac{r{x_{i}^{\star}}^{2}}{K}-x_{i}^{\star}\sum_{j=1}^{n}b_{ji}+(A\bm{x}^{\star})_{i},$

and we get (16). ∎

Figure 5: Predicted and observed eigenvalues. Observed rightmost eigenvalue of
the Jacobian matrix (14) at $\bm{x}^{\star}=\bm{0}$, together with predicted
values given by Eq. (16) (green dots). Red dots stand for predicted —cf. Eq.
(15)— _vs._ observed stability eigenvalues at the full infection equilibrium
point, in realizations where both equilibria coexist. A total of $10^{4}$
Erdös-Renyi model networks realizations ($n=100$ nodes) were taken to produce
both panels. A corresponds to $p=0.1$, and B was obtained for $p=0.2$, both
values well above the percolation threshold. As $p$ approaches one, the
agreement between predicted and observed becomes almost perfect.

Expressions (15) and (16) can be used to provide accurate approximations of
the rightmost eigenvalue of the Jacobian matrix for both equilibria. If
$\bm{v}=(v_{i})$ was an eigenvector of $J$, we would have that
$(J(\bm{x}^{\star})\bm{v})_{i}=\lambda v_{i}$. If all entries are real and
$v_{i}>0$, then we could write $\lambda$ as

$\lambda=\frac{1}{n}\sum_{i=1}^{n}\frac{(J(\bm{x}^{\star})\bm{v})_{i}}{v_{i}}.$
(19)

Numerically we find that the eigenvector $\bm{v}$ associated with the
eigenvalue that determines stability can be approximated by $\bm{x}^{\star}$,
if this full infection equilibrium point exists. Then Equations (15) and (16)
can be written as

$\frac{1}{n}\sum_{i=1}^{n}\frac{(J(\bm{x}^{\star})\bm{x}^{\star})_{i}}{x_{i}^{\star}}=-\frac{r}{K}\overline{\bm{x}^{\star}}$
(20)

and

$\frac{1}{n}\sum_{i=1}^{n}\frac{(J(\bm{0})\bm{x}^{\star})_{i}}{x_{i}^{\star}}=\frac{r}{K}\overline{\bm{x}^{\star}},$
(21)

respectively. Here $\overline{\bm{v}}:=\frac{1}{n}\sum_{i=1}^{n}v_{i}$. If the
full infection equilibrium exists and the approximation for the eigenvector
$\bm{v}\approx\bm{x}^{\star}$ is correct, then the right-most eigenvalue can
be approximated by $-\frac{r}{K}\overline{\bm{x}^{\star}}$ for the full
infection equilibrium (which will be stable), and by
$\frac{r}{K}\overline{\bm{x}^{\star}}$ for the full pathogen clearance (which,
as a consequence, will be unstable). Figure 5 shows the goodness of such
approximations for ER networks with different connectivity values. We observe
that the approximation works better for increasing values of the connection
probability $p$, yielding an almost perfect agreement in the limit $p\to 1$.

Figure 6: Probability of stability of the full infection and pathogen’s
clearance equilibria as function of the expected mean degree. Ten thousand ER
model realizations ($n=100$) were calculated for each connection probability
$p$, and we estimated probabilities as observed frequencies for each
equilibria. As $\text{E}[k]=(n-1)p$ increases, the equilibrium
$\bm{x}^{\star}=\bm{0}$ becomes more likely to be asymptotically stable.

We conclude that, in situations where these two equilibrium points appear, the
one associated to full infection is stable, and the one associated to
pathogen’s clearance is not. In summary, the qualitative behavior of critical
points in the general case is very similar to the case of per-capita flux
balance.

To finish the qualitative analysis of the general model, we have studied the
effect of network connectivity in ER graphs on the stability of the two
equilibria that can appear in this case. We provide the results of this
exploration in Figure 6.

We observe that, for small mean degrees (in particular, for those reported in
Fig. 1), the full infection equilibrium is always stable and the pathogen’s
clearance is unstable. However, for largely connected networks, in some cases
the latter equilibrium becomes stable —in which cases the coexistence
equilibrium does not exist. This is more apparent for fully connected
networks, for which about a $5\%$ of the realizations yield pathogen’s
clearance as endstate. It is worth mentioning that the sum of the two
probabilities is numerically equal to one, so we did not find additional
equilibria across all the realizations.

##  Appendix C Small-world networks

In this contribution, we have shown that mortality peaks at intermediate
network connectivity values, and this phenomenon can be ascribed to the
potential fragmentation of the network as infection spreads. However, not
every model of random network formation exhibits a percolation transition due
to the formation of a giant component. According to our results, in this case
the maxima should not appear.

Figure 7: Small-world networks. Here the fraction of healthy, infected, and
dead nodes is represented as function of the mean degree of network samples of
the WS model —notice that the minimum possible average degree in this model is
$\text{E}[k]=2$. Because model networks are connected, we observe no maximum
in the number of dead nodes. We used a rewiring probability $\phi=0.01$ to
generate WS model networks. Parameter values for $r$, $K$, and $\delta$ are
the same as in Fig. 1. We averaged over $1200$ realizations.

Figure 7 summarizes simulation results for our epidemic spreading dynamics on
to of the WS model for small-world network structure (compare these results
with those reported in Fig. 1C). As the WS model generates connected networks,
infection spreads throughout the network for any value of the mean degree.
This explains that no maximum is observed in plant mortality.

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