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\begin{document}

\title{African swine fever endemic persistence in wild boar populations: key
mechanisms explored through modelling}



\author[1]{Vincenzo gervasi}%
\author[1]{Vittorio Guberti}%
\affil[1]{Istituto Superiore per la Protezione e la Ricerca Ambientale}%


\vspace{-1em}



  
  \date{May 25, 2024}


\begingroup
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\selectlanguage{english}
\begin{abstract}
African swine fever (ASF), is a serious global concern from an
ecological and economic point of view. While it is well established that
its main transmission routes comprise contact between infected and
susceptible animals and transmission through contaminated carcasses, the
specific mechanism leading to its long-term persistence is still not
clear. Among others, a proposed mechanism involves the potential role of
convalescent individuals, which would be able to shed the virus after
the end of the acute infection. Using a spatially explicit, stochastic,
individual-based model, we tested: 1) if ASF can persist when
transmission occurs only through infected wild boars and infected
carcasses; 2) if the animals that survive ASF can play a relevant role
in increasing ASF persistence chances; 3) how hunting pressure can
affect the ASF probability to persist. The scenario in which only direct
and carcass-mediated transmission were contemplated had 52\% probability
of virus persistence 10 years after the initial outbreak. The inclusion
of survivor-mediated transmission corresponded to slightly higher
persistence probabilities (57\%). ASF prevalence during the endemic
phase was generally low, ranging 0.1-0.2\%. The proportion of
seropositive individuals gradually decreased with time and ranged 4.5 --
6.6\%. Our results indicate that direct and carcass-mediated infection
routes are sufficient to explain and justify the long-term persistence
of ASF at low wild boar density and the ongoing geographic expansion of
the disease front in the European continent. During the initial years of
an ASF outbreak, hunting should be carefully evaluated as a management
tool, in terms of potential benefits and negative side-effects, and
combined with an intensive effort for the detection and removal of wild
boar carcasses. During the endemic phase, further increasing hunting
effort should not be considered as an effective strategy. Additional
effort should be dedicated to finding and removing as many wild boar
carcasses as possible.%
\end{abstract}%



\sloppy


\textbf{African swine fever endemic persistence in wild boar
populations:}

\textbf{key mechanisms explored through modelling}

Vincenzo Gervasi\textsuperscript{1}, Vittorio Guberti\textsuperscript{1}

\textsuperscript{1} \textbf{Istituto Superiore per la Protezione e la
Ricerca Ambientale --} Ozzano Emilia (BO), Italy

\textbf{Corresponding author:}

Vincenzo Gervasi

Istituto Superiore per la Protezione e la Ricerca Ambientale

Ozzano Emilia (BO), Italy

Ph: +39 064403315

\textbf{Email:}

\emph{vincent.gervasi@gmail.com}

\emph{} \textbf{Running title:} African swine fever endemic persistence
mechanisms.\emph{} \textbf{Summary:} African swine fever (ASF), is a
serious global concern from an ecological and economic point of view.
While it is well established that its main transmission routes comprise
contact between infected and susceptible animals and transmission
through contaminated carcasses, the specific mechanism leading to its
long-term persistence is still not clear. Among others, a proposed
mechanism involves the potential role of convalescent individuals, which
would be able to shed the virus after the end of the acute infection.
Using a spatially explicit, stochastic, individual-based model, we
tested: 1) if ASF can persist when transmission occurs only through
infected wild boars and infected carcasses; 2) if the animals that
survive ASF can play a relevant role in increasing ASF persistence
chances; 3) how hunting pressure can affect the ASF probability to
persist. The scenario in which only direct and carcass-mediated
transmission were contemplated had 52\% probability of virus persistence
10 years after the initial outbreak. The inclusion of survivor-mediated
transmission corresponded to slightly higher persistence probabilities
(57\%). ASF prevalence during the endemic phase was generally low,
ranging 0.1-0.2\%. The proportion of seropositive individuals gradually
decreased with time and ranged 4.5 -- 6.6\%. Our results indicate that
direct and carcass-mediated infection routes are sufficient to explain
and justify the long-term persistence of ASF at low wild boar density
and the ongoing geographic expansion of the disease front in the
European continent. During the initial years of an ASF outbreak, hunting
should be carefully evaluated as a management tool, in terms of
potential benefits and negative side-effects, and combined with an
intensive effort for the detection and removal of wild boar carcasses.
During the endemic phase, further increasing hunting effort should not
be considered as an effective strategy. Additional effort should be
dedicated to finding and removing as many wild boar carcasses as
possible.

\textbf{Keywords} : ASF transmission, convalescent, chronic carrier,
individual-based model, \emph{Sus scrofa,} virus persistence.

\textbf{INTRODUCTION}

African swine fever (ASF) and its ongoing spread in several European and
Asian countries (Depner et al., 2017; Lu et al., 2020; Penrith, 2020) is
a serious global concern from an ecological and economic point of view
(Guberti et al., 2019; Pitts and Whitnall, 2019). The disease is caused
by a highly virulent virus of the \emph{Asfaviridae} family, which
affects wild boar (\emph{Sus scrofa} ), domestic pigs and African wild
suids (Blome et al., 2013; de Carvalho Ferreira et al., 2013), leading
to almost 100\% lethality of infected individuals (Blome et al., 2013).
After its first Eurasian appearance in Georgia in 2007, the disease has
spread to the Russian Federation through trans-Caucasian countries, to
Belarus and Ukraine, then to Belarus and Ukraine, Lithuania, Estonia,
Latvia, Poland, Czech Republic, Hungary, Romania, Bulgaria, Belgium,
Slovakia, Greece, Serbia and Germany (Depner et al., 2017; Blome et al.,
2020). A similar eastward spreading is occurring in Asia, currently
affecting China, Hong Kong, North Korea, South Korea, Laos, Vietnam,
Myanmar, Cambodia, Indonesia, Philippines, Timor-Leste, Papua New
Guinea, and India (Blome et al., 2020; Penrith, 2020).

At the arrival of the virus in the EU, ASF was expected to exhibit the
typical epidemic pattern of highly virulent acute infections, which
often generate self-limiting localized epidemic waves with a high
probability of a rapid fade-out. Such expectation was justified by the
fact that the wild boar was the sole infected host, and by the absence
of a competent arthropod vector (Gabriel et al., 2011; Chenais et al.,
2019; O'Neill et al., 2020). After few years of field experiences,
however, it is now evident that the virus is able to persist in low
density wild boar populations several years after the first epidemic
outbreak, with an endemic prevalence usually around 1\%, although some
local variation has been observed (Nurmoja, Schulz et al., 2017;
Pautienius et al., 2018). The virus endemic persistence at low wild boar
density is enhanced by its stability at a wide range of environmental
conditions (Mazur-Panasiuk et al., 2019). ASF virus has been shown to
persist in frozen meat for several months, and may persist in carcasses,
forest soil and water for several weeks (Mebus et al., 1997) allowing an
efficient indirect transmission through both contaminated carcasses and
the environment (De Carvalho Ferreira et al., 2014; Probst et al., 2017;
Carlson et al., 2020; Fischer et al., 2020). However, the inner
mechanism leading to the long-term persistence at both low wild boar
density and prevalence is still not clear, as virus persistence in the
environment and in wild boar carcasses is highly variable and mediated
by wild boar behaviour, while the endemic persistence of the virus is
invariably observed (Blome et al., 2020). Several alternative hypotheses
have been advanced and are currently being explored, but no one has been
confirmed and validated so far.

Given the summer peaks observed in several countries, one hypothesis is
that the ASF virus might have found an alternative competent arthropod
vector species that could replace the \emph{Ornithodoros} ticks, absent
in Eurasia. Soft and hard ticks, different species of flies, tabanids
and mosquitoes have been proposed (Bonnet et al., 2020), but arthropods
do not influence the spread of the virus in wild boar populations (Herm
et al., 2021). Another hypothesis relies on the possibility that during
its spreading in recent years, the ASF virus might have gone through a
process of attenuation, thus reducing its virulence and lethality
(Gallardo et al., 2017; Nurmoja, Petrov et al., 2017). Although
different wild boar mortality rates have been observed during the ASF
spreading in different parts of Estonia and in Latvia (Zani et al.,
2018; Gallardo et al., 2019), suggesting the possibility that moderately
virulent variants of the ASF virus might be present in the population,
no confirmation has been provided so far that a low virulent ASF virus
might play a role in its persistence (Blome et al., 2020). Moreover,
modelling work shows that, even if the two variants were both present in
a wild boar population, the attenuated ASF virus would rapidly go
extinct at the expense of the highly virulent one (Nielsen et al.,
2021).

A third proposed mechanism involves the potential role of infectious
survivors, which would still be able to shed the virus after the end of
the acute infection, thus favouring the long-term persistence of the
disease. Current knowledge (S\selectlanguage{ngerman}ánchez-Vizcaíno et al., 2015; Ståhl et al.,
2019) suggests that ASF survivors (hereafter called convalescents) may
still carry and transmit the virus after the acute disease phase, but
then fully recover from the infection and become immune for life (i.e.
category 2 according to Stahl et al., 2019). The possibility that some
of them might develop a persistent infection, accompanied by a subacute,
chronic disease, has also been proposed (Arias and Sánchez-Vizcaíno,
2002; Category 1 according to Sthal et al., 2019).

The role of infectious survivors is therefore still discussed. It is
known that the virus can still be present in survivors for roughly
60--70 days and up to 91days (Petrov et al., 2018). However, lab
experiments suggest a very low probability of infection between
convalescent and susceptible individuals, and no virus survival beyond
100 days (Nurmoja, Petrov et al., 2017; Petrov et al., 2018; Ståhl et
al., 2019). Other experiments on domestic pigs in a controlled
environment, though, seem to indicate that transmission of the ASF virus
via infectious survivors does occur at least in a 55 days post-exposure
window (Eblé et al., 2019). Lacking a conclusive evidence, though, the
possibility that surviving wild boars carrying both virus and antibodies
could shed and transmit the virus, even though at a very low rate,
cannot be disregarded (Blome et al., 2020).

The lack of a clear and verified persistence mechanism for ASF also has
consequences on the available management options for its containment and
eradication. Currently, carcass removal and wild boar culling are the
two main available strategies implemented in the affected areas (Lange
et al., 2018), but their effectiveness strongly depends on the relative
importance of the different ASF transmission routes. Wild boar culling,
which aims at a reduction in wild boar density, mainly affects virus
transmission rates between live individuals; therefore, the
effectiveness of culling as a control measure depends on how relevant
infected and convalescent individuals are in ASF persistence. Carcass
removal, on the other hand, only affects the ASF transmission route
which involves dead wild boar; its effectiveness, therefore, strongly
depends on the importance of carcass-mediated transmission in ASF
persistence. Accordingly, it is of paramount importance that all the ASF
persistence mechanisms be determined and ranked.

In this paper, we explored the relative role of different ecological and
epidemiological factors in the long- term persistence of ASF in wild
boar populations. We first assessed the likelihood of the disease to
persist through direct and carcass-mediated infection, but without the
contribution of any survivor-mediated transmission; in a second scenario
we explored the potential for ASF convalescents to play a role in
disease persistence. To this aim, we ran and analysed a spatially
explicit, stochastic, individual-based model, which mimicked the
demography and spatial dynamics of a wild boar population, the
epidemiology of the ASF virus through the different proposed
transmission routes, and population management through wild boar
harvest. Finally, we assessed the sensitivity of ASF persistence to
changes in all the main ecological, epidemiological, and
management-related parameters, including hunting rate, thus ranking them
in order of importance as determinants of ASF persistence probability.
We discuss the implications of our results for the disease surveillance
and control in the affected countries, and for the ongoing effort to
limit its spread in new, still unaffected areas.

\textbf{METHODS}

\textbf{2.1 Model structure}

The model we used resembled the structure of a SEIR
(susceptible-exposed-infected-recovered) epidemiological model (Anderson
and May, 1992), with the inclusion of a spatially-explicit, stochastic,
individual-based structure. It mimics the structure of the model
presented in Lange et al. (2018), with a particular focus on the
mechanisms of virus transmission. We built it and ran it using the
software Netlogo 6.1.1 (Wilensky, 1999).

All processes took place in a grid of 120x120 km (area = 14,400
km\textsuperscript{2}). We divided the grid into 1,600 3x3 km cells,
which represented the smaller simulated spatial unit. Such units,
covering an area of 9 km\textsuperscript{2}, corresponded to a
reasonable estimate of a wild boar's core home range (Leaper et al.,
1999). Each spatial unit was characterized by its local wild boar
density, defined as the number of individuals having their home range
centred in each cell. This state variable was then used as an input
parameter for the reproduction and dispersal processes.

Each wild boar was characterized by a series of state variables, which
defined its role and behaviour in the model. First, a wild boar was
assigned a sex and classified into one of the three age classes:
juveniles (0-1-year-old), yearlings (1-2 years old); adults (older than
2 years). Additional individual state variables were the reproductive
state (only for females) and the dispersal state (only for yearlings).
Finally, each individual could be classified in one of the eight model
compartments: susceptible, exposed, infected, convalescent, immune,
infectious carcass, non-infectious carcass, hunted. The ``convalescent''
compartment included the individuals which survived the acute phase of
the disease and were passing through the recovery process. In such phase
they were still able to transmit the infection for a limited amount of
time, until total recovery. The duration of the infectious period in
convalescents was controlled by parameter \selectlanguage{greek}\emph{χ,} \selectlanguage{english}whose value was
determined through a numerical optimization process (see below for
details). The ``infectious carcass'' compartment included the
individuals which succumbed ASF and whose decomposing bodies could still
transmit the virus. Once a carcass lost its potential infectiousness it
was transferred to the absorbing ``non-infectious carcass'' state. The
duration of a carcass infectious period was controlled by
parameter\selectlanguage{greek}\emph{Ι,} \selectlanguage{english}also derived from the optimization process. Such
period was by default 50\% shorter in summer than in winter.

The analytical framework included two scenarios, one in which disease
transmission occurred only through direct contact between susceptible
and infected individuals, or between susceptible individuals and
infected carcasses, another in which we added a third possible
transmission route, which involved the role of ASF convalescents, while
keeping the other two transmission mechanisms in place. Transmission
routes are shown in Fig. 1.

\textbf{2.2 Virus transmission and disease course}

For each of the three transmission pathways, we considered two different
contact probabilities, depending on the spatial location of the virus
source and of the susceptible individual: 1) a first potential
transmission occurred with probability P\emph{\textsubscript{i,i}} ,
between individuals whose home range centres fell both in cell
\emph{i.}This parameter referred mainly to those wild boars which were
likely to belong to the same social group; 2) a second potential
transmission occurred with probability P\emph{\textsubscript{i,j}} ,
between individuals whose home range fell in the two neighbouring cells
\emph{i}and \emph{j} , respectively. This parameter referred to the wild
boars which were likely to belong to different but neighbouring social
groups.

After infection, individuals were moved into the ``exposed'' state to
allow for a 3-day period of incubation and latency (Blome et al., 2013),
then transferred into the ``infected'' compartment. There, an infected
wild boar had a \selectlanguage{greek}\emph{γ} \selectlanguage{english}probability to die because of the ASF acute
infection within a 5-day period, whose value was optimized before model
running. The individuals which did not survive the disease were
transferred each day into the ``infectious carcass'' compartment. In the
first scenario, survivors were directly moved to the ``immune'' state
and could neither be re-infected, nor transmit the disease to others
(Fig. 1). In the second scenario, ASF survivors we moved into the
``convalescent'' state at 5 dpi, became seropositive at 15 dpi and
remained in that compartment until full recovery, according to
parameter\selectlanguage{greek}\emph{χ} . \selectlanguage{english}Then, if still alive, they were transferred into the
``immune'' state (Fig. 1).

\textbf{2.3 Mortality, reproduction, dispersal}

Besides lethality, we considered also natural and hunting mortality
rates. For natural mortality we applied a 0.18 annual rate to juveniles
and 0.12 to the other two age classes (To\selectlanguage{ngerman}ïgo et al., 2008). Individuals
who died of natural causes were transferred directly to the
``non-infectious carcass'' compartment of the model. Hunting rate in the
post-recruited population was initially set at 0.3 for all age classes.
Then, to test for the effect of an increased hunting pressure on virus
persistence probability, we also ran additional scenarios in which
hunting pressure ranged from 0.4 to 0.6.

At the beginning of each year, each female wild boar was assigned a
reproductive state, controlled by an age and density-dependent
reproduction probability. All reproductive females were assigned a
delivery reproduction day, ranging from 1 to 60, to allow for a uniform
distribution of births during the birth season. Litter size was set to 4
for juveniles, 5 for yearlings, 6 for adults (Bieber and Ruf, 2005).
Then, in the assigned day, they gave birth to the age-specific number of
piglets with a 50:50 sex ratio. The piglets coordinates initially
corresponded to those of their mother.

At the beginning of the dispersal season, all yearling individuals were
assigned a dispersal state, generated through a sex-specific dispersal
probability (0.7 for males, 0.4 for females; Truvé et al., 2004), a
dispersal starting day, ranging from 1 to 40, and a dispersal duration
ranging from 1 to 14 days for males, from 1 to 7 days for females (Truvé
et al. 2004). Then, each day and for the entire duration of the
dispersal season, each dispersing individual moved from its current cell
to the neighbouring cell with the lowest wild boar density, thus
mimicking the effort by dispersing animals to avoid intra-specific
competition for resources. All model parameters are summarized in Tab.
1.

\textbf{2.4 Model initialization}

We initialized the model with a starting population of 43,200 wild boar,
corresponding to a density of 3.0 individuals / km\textsuperscript{2}.
The initial coordinates of all individuals were randomly generated
within the grid limits, thus producing a homogeneous spatial
distribution of wild boar density. We set sex ratio at 50:50, whereas
age classes were attributed according to the stable age distribution
estimated by Bieber and Ruf (2005): 60\% juveniles, 20\% yearlings, 20\%
adults. All individuals except one (randomly picked) were initially
assigned to the ``susceptible'' state. The remaining individual was
defined as ``infected'' and placed in the centre of the simulated grid.
The model proceeded in daily time-steps for a period of 10 years.
Reproduction took place each year for a period of 60 days during the
months of April and May. Natal dispersal occurred for a period of 40
days and started on June 1\textsuperscript{st}. The hunting season
lasted for 150 days between October and February. Each day, model
processes were performed according to the following order: density
update, disease incubation, virus transmission, disease-related
mortality, recovery, carcass decomposition, hunting, reproduction,
dispersal, aging.

\textbf{2.5 Parameter optimization}

For several of the main parameters, known to play a role in ASF
persistence, there was no reliable estimate obtained in the field from
the affected wild boar populations. In particular, virus transmission
rates mediated by infected or convalescent individuals, or by wild boar
carcasses, are substantially unknown. A few experimental estimates exist
but with different virus genotypes (de Carvalho Ferreira et al., 2013;
Eblé et al., 2019) and obtained mainly on pigs and in controlled
conditions. The applicability of such estimates to the complex
eco-epidemiological conditions of a natural wild boar population are
therefore dubious. Similar considerations can be done for other crucial
parameters, such as the duration of carcass infectivity in the field or
the disease lethality rate in the field.

To overcome this limitation and provide the model with appropriate
values of the missing parameters, we used an optimization process based
on a series of demographic and epidemiological criteria regarding the
evolution of the disease in a wild boar population. In particular, we
optimized model parameters to mimic the evolution of the ASF spread in
Latvia during the period 2014-2019, as described in O\selectlanguage{polish}ļš\selectlanguage{english}evskis et al.
(2020). First, we selected all parameters for which we had no
field-based reliable estimate. They were the three ASF infection
probabilities (direct, carcass, convalescent), the duration of a carcass
infectious period, the duration of a convalescent infectious period,
disease lethality rate and the ratio between the infection rate within
and between wild boar social groups. Then, we ran 1,000 iterations of
our model, each time with a randomly selected value for each of the
parameters to be optimized. For each iteration, we reported the number
of days of disease persistence, the maximum virus prevalence and
seroprevalence during the 10-year period and the minimum observed
population density. Finally, we picked the iterations which satisfied
all of the following criteria, derived from the Latvian study case
(O\selectlanguage{polish}ļš\selectlanguage{english}evskis et al., 2020):

\begin{enumerate}
\tightlist
\item
  The virus was still circulating in the population at the end of the
  simulated study period
\item
  The highest observed virus prevalence ranged 2-5\%
\item
  The highest observed seroprevalence ranged 5-10\%
\item
  Population density decreased by at least 70\% during the epidemic
\end{enumerate}

The parameter values which maximized the probability of all the four
criteria to be met were picked as input values for our model, thus
assuring the highest ASF persistence chances and a realistic
epidemiological and demographic evolution of the study system. The
values for each parameter, resulting from the optimization process, are
shown in Tab. 1, whereas the density probability functions associated to
each parameter are available in Figs. S1 and S2 in the Online Supporting
Information.

\textbf{2.6 Analysis of model results}

After the parameter optimization process, we performed the actual model
runs and compared the disease course under the sole direct and
carcass-mediated transmission (scenario 1), and under the action of all
three mechanisms (direct, carcass-mediated, convalescents; scenario 2).
For each scenario, we ran the model 1,000 times. In each iteration and
at each time step, we recorded the number of infected individuals
(ASF\textsuperscript{+}), seropositive individuals
(Ab\textsuperscript{+}), and the number of both virus and seropositive
wild boar (ASF\textsuperscript{+} \& Ab\textsuperscript{+}). We also
kept track of the number of infected carcasses and of total population
size. We stopped model running as soon as the virus disappeared from the
population and had no more chances to be transmitted through the
pathways considered in that specific scenario. In such a case we also
recorded the year and day of virus extinction. For each simulated day,
we also recorded the number of ASF infections occurring through each of
the three transmission routes, thus obtaining an estimate of how
relevant each source of infection was in virus transmission and
persistence.

\textbf{2.7 Sensitivity analysis}

To more fully explore the relative importance of each model parameter in
affecting ASF persistence, we also performed a regression-based global
sensitivity analysis (Saltelli et al., 2008; but see Nsoesie et al.,
2012 for an application on individual-based epidemiological models).
Once more, we ran 1,000 iterations of the model, this time randomly
selecting each parameter from a uniform distribution ranging from the
optimized parameter value to its 150\% values. For each iteration, we
recorded the number of days of virus persistence in the population.
Then, we visually checked the linearity of the relationship between the
input parameter and the resulting virus persistence. Following, we
standardized all input parameter values using the z-score method
(Kreyszig, 1979), and performed a generalized linear regression using
virus persistence as the response variable and the standardized model
parameters as predictors. To account for the overdispersion in the data
we used a quasi-Poisson distribution for the response variable. The
regression coefficients of each predictor provided an estimate of the
sensitivity of ASF persistence to changes in that parameter (Saltelli et
al., 2008). The z-score standardization made all regression coefficients
comparable, although the initial model parameters were measured on
different scales.

\textbf{RESULTS}

\textbf{3.1 ASF dynamic}

In both simulated scenarios, ASF exhibited an infection dynamic which
could be characterized in four distinct phases (Guberti et al., 2019):
invasion (years 1-2), first epidemic wave (years 3-4), endemic phase
(years 5-7), second epidemic wave (years 8-10). During the initial
invasion phase, the disease remained rather localized with a low virus
prevalence, in average 0.01\%, and a mild reduction in wild boar density
(Tab.2 and Fig. 2). This was also the phase during which the ASF virus
had the highest chances to disappear from the wild boar population: when
simulating only transmission through direct and carcass contacts, the
virus failed in reaching the endemic status in 40\% of the model runs
after two years; when adding the survivor-mediated infection, the
extinction probability after the first two years was slightly reduced to
34\% (Fig. 3).

During the first epidemic wave (years 3-4), the ASF virus exhibited a
rapid geographic spread and a progressive increase in both prevalence
and seroprevalence. In scenario 1 (two transmission pathways) the ASF
prevalence and seroprevalence were in average 1-2\% (Tab. 2), with peaks
of 2.87\% in virus prevalence and 6.52\% in seroprevalence (Fig. 2).
When accounting for role of convalescents in disease transmission
(scenario 2), the average virus and seroprevalence slightly increased,
with peaks of 4.52\% and 6.61\%, respectively. During this phase, the
virus faded out in 1\% of the iterations in both scenarios,
corresponding to a 59\% persistence probability and 65\% in scenario 2
(Fig. 3).

After spreading across the whole study area, ASF entered its endemic
phase in years 5-7, during which virus prevalence was in average lower
than 0.5\% in both scenarios (Tab. 2). Seroprevalence progressively
decreased during the endemic years, averaging 1.79\% and 2.11\% in the
two scenarios, respectively (Fig. 2). This period was also the one
during which the population reached its lowest density, in average about
0.8-0.9 individuals / km\textsuperscript{2}, depending on the scenario
(Tab. 2).

Despite the low virus prevalence, ASF had low probabilities to disappear
from the population during the endemic phase: extinction rate during
this period was 1\% in scenario 1 and 2\% in scenario 2, so that at the
end of seventh year the ASF virus was still present in 58\% of the
iterations in scenario 1 (only direct and carcass-mediated transmission)
and in 63\% of cases in scenario 2 (all three transmission pathways
included).

The last phase of the ASF dynamic occurred in years 8-10, during which a
second lower epidemic wave emerged (Fig. 2), with average prevalence and
seroprevalence ranging 1-2\% and 2-3\%, respectively. At the end of the
10-year simulated period, the ASF virus was still present in the wild
boar population in 52\% of the iterations in scenario 1 and in 57\% in
scenario 2 (Tab. 2, Fig. 3).

Overall, about 58\% of the virus transmissions occurred directly between
an infected and a susceptible wild boar, 38\% by mean of an infected
carcass, 4\% due to a convalescent wild boar (Fig. 4a). These
proportions, though, were not constant over the 10 simulated years.
Direct transmission was relatively more frequent in the initial years of
the epidemic, when about 65\% of the infections occurred using this
pathway; carcass-mediated transmission, instead, was relatively less
frequent at the beginning of the simulated period, when it accounted for
only about 20\% of the infections, but became progressively more
important, especially during the endemic phase. In those years, in fact,
it represented 40\% of all ASF transmissions. Moreover, carcass-mediated
virus transmission was strongly correlated to wild boar density. As
shown in Fig, 5, this infection pathway accounted for only about 20\% of
all virus transmission when wild boar density was around 3 individuals /
km\textsuperscript{2}, but it increased to 60\% when density decreased
to 1.0 / km\textsuperscript{2} or lower values. The proportion of
survivor-mediated virus transmission remained rather constant during the
simulated period, ranging 2-4\% of all infections. Finally, the
proportion of infections occurring within and between social groups was
also rather constant during the study period, with about 55\% of the
virus transmission taking place within the same 3x3 km cell, the
remaining 45\% between two neighbouring cells (Fig. 4b).

During the 10 simulated years, the ASF force of infection exhibited a
seasonal cycle, with a peak in spring and summer at about 1.05, after
newly born piglets entered the susceptible compartment. The force
progressively decreased during the following seasons, exhibiting
negative values (0.95 -- 1.00) throughout winter, starting a new cycle
at the onset of the successive reproductive season. The temporal trend
in the force of infection is shown in Fig. S3 in the Online Supporting
Information.

\textbf{3.2 Sensitivity analysis}

After visually inspecting the relationship between the input parameters
and the resulting ASF persistence, all of them appeared as clearly
linear, except for the annual hunting rate, whose effect was not. As
shown in Fig. 6, the effect of hunting, expressed as the resulting
lowest wild boar density observed during the 10-year period, could be
broken down into two segments of different slopes. For this reason, we
estimated two different sensitivity values for the hunting rate
parameter, one for the rates corresponding to a minimum wild boar
density \textless{} 0.75 individuals / km\textsuperscript{2}, one for
the rates corresponding to a wild boar density \textgreater{} 0.75
individuals / km\textsuperscript{2}.

The global sensitivity analysis revealed than not all the input
parameters had a significant effect on ASF persistence. Of the infection
probabilities related to each of the three transmission pathways, direct
and carcass-mediated transmission exhibited significantly different from
zero sensitivity values (Tab. 3), whereas the sensitivity of
survivor-mediated transmission was not significant. Similarly,
increasing the duration of a carcass infectivity period significantly
increased the ASF persistence, whereas increasing the duration of a
convalescent wild boar infectivity period did not (Tab. 3). ASF
lethality exhibited a significant but negative sensitivity value (Tab.
3), implying that an increase in the proportion of fatal disease
outcomes produced a reduction in virus persistence, because of the
reduced time available for infected wild boars to transmit the disease.
The proportion of reproducing females in the population, on the
contrary, exhibited a positive and significantly different from zero
sensitivity value (Tab. 3), suggesting that an increased reproductive
performance at the population level corresponded to an increased
probability of disease persistence over time.

When analysing the effect of hunting rate in the two different density
segments, the results of the sensitivity analysis exhibited rather
different relationships. The sensitivity of ASF persistence to changes
in hunting rate when wild boar density was higher than 0.75 /
km\textsuperscript{2} was the highest among all tested parameters,
whereas the same parameter did not exhibit any significant effect on ASF
persistence when wild boar density was lower than 0.75 /
km\textsuperscript{2} (Tab. 3 and Fig. 6). At 1.5 wild boars /
km\textsuperscript{2} ASF was expected to persist in the population at
least 10 years, but a reduction of wild boar density to its half
corresponded to an expected persistence of about three years (Fig. 6),
which resulted in a disease fade-out at the end of the first epidemic
wave. On the contrary, further increasing hunting effort to reduce wild
boar density to even lower values did not result in any further
reduction in the expected duration of the epidemic (Fig. 6).

\textbf{DISCUSSION}

Our model captured well both the epidemiological and the demographic
dynamics observed in the affected areas during the first years of the
epidemic, in terms of population size reduction, average prevalence and
seroprevalence, and long-term persistence of the disease at low wild
boar density during the endemic phase. Although we used the
epidemiological data reported for Latvia (O\selectlanguage{polish}ļš\selectlanguage{english}evskis et al., 2020) as a
reference for model parameterization, the dynamics emerging from our
study were typical for ASF in wild boar in most of the surveillance data
reported for northern and eastern Europe since the ASF initial outbreak:
ASF reduced infected populations by 70-80\% during the first 4-5 years
of the epidemic, as reported in most of the Baltic countries and in
Poland (Depner et al., 2017; O\selectlanguage{polish}ļš\selectlanguage{english}evskis et al., 2020; Nielsen et al.,
2021); peaks in the ASF virus prevalence were usually around 5\%, with
the average prevalence during the whole period ranging 1-2 \% (Depner et
al., 2017; Nurmoja, Schulz et al., 2017); seroprevalence peaked at
values around 10\% and then progressively decreased during the endemic
phase, as recently reported for Estonia and Latvia (Nielsen et al.,
2021). Moreover, the additional parameters selected through the
optimization procedure were all in the range of values obtained from
field and laboratory data during these years, even though no a-priori
information was used to select them (Tab. 1, Figs. S1 and S2). The model
exhibited a rather slow dynamic, especially during the initial period
after virus release (years 1-2), when prevalence remained well below 1\%
and raised slowly towards a clear first epidemic wave. Besides from
being an intrinsic property of the system, such pattern was determined
by the large area used for simulation, which caused a dilution effect of
the epidemiological parameters during the initial years. Data estimated
exclusively on the initially infected area would have shown higher
prevalence and faster spread of the virus. This should be taken into
account when comparing model dynamics with surveillance data reported
from small affected areas, shortly after the initial virus detection.

In terms of mechanistic disease dynamics, our model results indicate
that the two transmission pathways so far considered as the main
infection routes, namely direct and carcass-mediated, are sufficient to
explain and justify the long-term survival of the ASF virus at low wild
boar density and the ongoing geographic expansion of the disease front
in the European continent. The addition of a third transmission
mechanism, mediated by ASF survivors during their convalescent phase,
did not change drastically the disease dynamics, nor substantially
increased the ASF virus persistence probabilities. Three specific
results clearly indicate that survivors play a minor role in virus
persistence: 1) the temporal trend in the main epidemiological
parameters (prevalence and seroprevalence) was similar in the scenarios
with and without the inclusion of survivor-mediated transmission (Tab.
2); 2) persistence probabilities at five and ten years were
substantially the same for the two scenarios (Tab. 2); 3) the
sensitivity values of all the parameters involved in the
survivor-mediated infection were not significantly different from zero
(Tab. 3).

The role of different transmission mechanisms in ASF persistence,
though, is far from being clarified, and several parallel approaches are
being developed to explore the issue. In a recent work, Lange et al.
(2021) proposed a comparison of different alternative ASF persistence
mechanisms, based on the Estonian case study. They estimated a less than
20\% persistence probability after 10 years for a scenario involving
only direct and carcass-mediated virus transmission. They also found
that the inclusion of convalescents with up to 4 weeks of transient
infectivity did not increase ASF persistence rates, unless it was
combined with a reduction of disease lethality from 95\% to 80\% (Lange
et al., 2021). Instead, they reported that a small proportion (0.1 --
1.0 \%) of life-long infectious carriers would drastically increase ASF
long-term persistence probabilities. Alternative mechanisms, such as a
shortened protection by maternal antibodies and the possibility of
immunity loss after recovery were not related to an increase in ASF
persistence in their model (Lange et al., 2021). Using a similar
modelling approach and surveillance data for Eastern Poland, Pepin et
al. (2020) obtained results which are more in agreement with our
findings: they estimated 50-60\% ASF persistence rates running an
individual-based model which comprised only direct and carcass-mediated
infection, but estimated such persistence on a time horizon of only 2
years, which makes the comparison with our study not optimal. Finally,
O'Neill et al. (2020) presented a different modelling approach to the
study of ASF persistence in wild boar, which made use of a
deterministic, population-based, compartmental model (Keeling and
Rohani, 2008). They reported that the observed epidemiological patterns
of ASF could not be matched when accounting only for infected and
carcass-mediated transmission, and that the inclusion of a re-infection
probability for ASF survivors allowed to obtain long-term disease
persistence and the same epidemiological trends reported in the affected
countries (O'Neill et al., 2020). The apparently contrasting results of
these different modelling exercises confirms the complexity of the
ecological and epidemiological mechanisms on which ASF persistence
relies. In such complexity, our results suggest that the main infection
routes through which ASF can persist at low wild boar density might have
been already unveiled. Although identifying alternative or additional
mechanisms is relevant and needed, the main focus should be kept on the
role of infectious live wild boar and infectious carcasses, which are
likely to explain a large part of the observed dynamics in the affected
countries.

In particular, the temporal trend in the proportion of ASF infections
occurring with each of the two mechanisms (Fig. 4) shows that direct and
carcass---mediated transmissions are likely to play different roles in
different phases of the ASF epidemic. During the initial invasion phase,
which in our model roughly corresponded to the first two years after
virus invasion, almost 70\% of the infections occurred directly between
infected and susceptible individuals (Fig. 4a), and in particular within
the same social group (Fig. 4b). This quantification is substantially
different from what reported by Pepin et al. (2020), who estimated that
53-66\% of all virus transmission would be due to a contact between a
susceptible wild boar and an infectious carcass. It should be noted,
though, that those quantifications were based on an initial wild boar
density ranging 0.5-2.0 individuals / km\textsuperscript{2}, as opposed
to the 3.0 / km\textsuperscript{2} used in our model. Accordingly, we
also observed that carcass-mediated ASF transmission became relatively
more frequent and even predominant for decreasing wild boar density
values (Fig. 5), suggesting that carcasses are likely to be the most
important infection route during the endemic phase, when the wild boar
population density has been reduced by 70-80\% after the first epidemic
wave. After entering its endemic phase, ASF seems to be maintained
essentially by infected carcasses, which act as a reservoir for the
virus in small pockets, until the wild boar population bounces back to
density levels which re-allow an effective virus transmission through
direct boar-to-boar contacts.

Such prolonged period of endemicity, which some of the affected
countries in north and eastern Europe are experiencing in these years,
is likely to be challenging both for disease surveillance and for the
efforts of its eradication. One of the most challenging results of our
study is the evidence that a long-term disease persistence was
compatible with a very low endemic prevalence, which ranged in average
from 0.2 to 0.3\% (Tab. 2). This means that at any given time during the
endemic phase, only 2-3 wild boars out of 1000 in the population were
infected. Moreover, our model reported an average of about 40 infected
carcasses in the whole study area during the endemic phase,
corresponding to a density of about one carcass / 300
km\textsuperscript{2}. In such conditions, the evidence of ASF presence
in a given area can remain substantially invisible to surveillance. In
this phase, both passive and active surveillance are likely to be poorly
effective in detecting the disease, because the likelihood of hunting an
ASF infected wild boar and that of retrieving an infected carcass in the
forest are both rather low. On the other hand, seropositive individuals
represented about 6\% of the wild boar population at the beginning of
the endemic period, decreasing to about 1\% after three years (Fig. 2),
making much more likely to detect seropositive than virus positive
animals during the endemic phase. Accordingly, in most of the affected
countries the number of virus positive wild boar in hunting bags and the
number of infected carcasses detected in the forest rapidly dropped to
zero after the end of the first epidemic wave, whereas the number of ASF
seropositive cases reported through hunted individuals progressively
increased in subsequent years (Boklund et al., 2018; Nielsen et al.,
2021). In most of the cases, seropositive animals are the sole reported
cases for long periods of time during the endemic phase. Such
epidemiological landscape, in which the probability to detect the virus
in dead wild boar is extremely low, makes the infection status of the
involved wild boar population uncertain.

Our results confirm the possibility for ASF to persist for long times
with a very low endemic prevalence, which ranged in average from 0.2 to
0.3\%, and at very low wild boar density (Tab. 2). This means that at
any given time during the endemic phase, only 2-3 wild boars out of 1000
in the population were infected. Moreover, our model reported an average
of about 40 infected carcasses in the whole study area during the
endemic phase, corresponding to a density of about one carcass / 300
km\textsuperscript{2}. In such conditions, the evidence of ASF presence
in a given area can remain substantially invisible to surveillance. In
this phase, both passive and active surveillance are likely to be poorly
effective in detecting the disease, because the likelihood of hunting an
ASF infected wild boar and that of retrieving an infected carcass in the
forest are both rather low. On the other hand, seropositive individuals
represented about 6\% of the wild boar population at the beginning of
the endemic period, decreasing to about 1\% after three years (Fig. 2),
making much more likely to detect seropositive than virus positive
animals during the endemic phase. Accordingly, in most of the affected
countries the number of virus positive wild boar in hunting bags and the
number of infected carcasses detected in the forest rapidly dropped to
zero after the end of the first epidemic wave, whereas the number of ASF
seropositive cases reported through hunted individuals progressively
increased in subsequent years (Boklund et al., 2018; Nielsen et al.,
2021). In most of the cases, seropositive animals are the sole reported
cases for long periods of time during the endemic phase. Such
epidemiological landscape, in which the probability to detect the virus
in dead wild boar is extremely low, makes the infection status of the
involved wild boar population uncertain.

In terms of wild boar population management, our model results confirm
that the effort of eradicating or just controlling ASF is a hard
challenge, but they also indicate that some options are more likely to
be effective than others. In particular, the sensitivity analysis
revealed that the effectiveness of wild boar hunting is limited. Hunting
effects are more apparent during the initial invasion and epidemic
phases, when wild boar density is still at relatively high values (Fig.
6). However, the main effect of hunting is just to shorten the
transition from the epidemic to the endemic condition, through an
initial reduction in population density. High hunting pressure might
also generate unwanted effects, inducing compensatory population growth
rate and accelerated generation time, higher juvenile female
contribution to the reproductive set and earlier reproduction (Morelle
et al., 2020). Moreover, the potentially limited benefits of increased
hunting are likely to be counteracted by several of its side effects,
such as increased wild boar movements, virus contamination risks and
potential human-related long-distance transport of the ASF virus
(Guberti et al., 2019).

Afterwards, when ASF enters its endemic phase, hunting has a negligible
role in increasing the overall probability of virus fade out, because
during that period ASF is mainly transmitted and sustained through
infected carcasses. (Fig. 5). In our modelling conditions, the density
threshold marking such loss of hunting effectiveness was estimated at
0.75 wild boar/km\textsuperscript{2}, but such threshold is likely to be
context dependent and difficult to be estimated with the sole hunting
data. Additionally, it should be noted that modelling hunting as a fixed
proportion of population size, as we did in our model, might reduce
model realism at very low population densities. Hunters' effectiveness,
in fact, is expected to decrease when a wild boar population is sparser,
making it hard to accomplish the same hunting goals achieved at higher
population densities.

Therefore, if transmission and persistence mechanisms are different in
the different stages of an ASF epidemic, also management actions should
be modulated depending on which phase a given affected area is
experiencing. To this aim, our study indicates that during the initial
years, and especially during the first epidemic wave, hunting as a
management tool should be carefully evaluated in terms of potential
benefits and negative side-effects, and combined with an intensive
effort for the detection and removal of wild boar carcasses. During the
endemic phase, when both virus prevalence and wild boar density are low,
further increasing hunting effort should not be considered as an
effective option. Instead, additional effort should be dedicated to
finding and removing as many wild boar carcasses as possible. In
epidemiological terms, this would correspond to shortening a carcass'
infectious period, a parameter which exhibited a high sensitivity value
during all phases of the ASF simulated course (Tab. 3).

Finally, the sensitivity analysis revealed a third relevant ASF
persistence mechanism, which offers an additional management
opportunity: ASF persistence probability was significantly and
positively influenced by spring recruitment, expressed in the model as
the proportion of females of all ages giving birth to piglets (Tab. 3).
Newly born wild boar, in fact, provide each year a new input of
susceptible individuals, potentially suitable for infection and virus
transmission, thus allowing the typical increase in ASF prevalence
during summer, observed in several of the affected European countries
(Boklund et al., 2018). Moreover, an increased reproductive performance
also generates a higher population growth rate during the endemic phase
of the ASF epidemic, allowing wild boar density to recover quickly, and
increasing the chances that a second lower epidemic wave might occur.
Ecological theory has long recognized the link between food availability
and recruitment in large herbivorous mammals such as wild boar (Gaillard
et al., 1998). Therefore, management actions such as winter supplemental
feeding, which are a widespread practice in most of the European
countries currently affected by ASF (Guberti et al., 2019), should be
considered as powerful enhancers of ASF persistence and strongly limited
or banned. Other modelling work (O'Neill et al., 2020) has shown that
ASF is more likely to persist in wild boar populations with increased
reproductive performance and increased carrying capacity, which are the
typical demographic and ecological consequences of widespread
supplemental feeding. Such population-level effects of artificial
feeding are likely to be further magnified by the local spatial and
behavioural effect: feeding sites, in fact, increase wild boar spatial
aggregation, favour direct or indirect contact between neighbouring
social groups, and overall are likely to increase virus transmission
rates in the population.

\textbf{Acknowledgments}

All the Authors were supported by the European Union's Horizon 2020
Research and Innovation Programme under grant agreement No 773701
(DEFEND).

\textbf{Conflict of interest statement}

The authors certify that they have no affiliations with or involvement
in any organization or entity with any financial interest or
non-financial interest in the subject matter or materials discussed in
this manuscript.

\textbf{ETHICAL statement}

The authors confirm that the ethical policies of the journal, as noted
on the journal's author guidelines page, have been adhered to. No
ethical approval was required as this is a modelling article with no
original research data.

\textbf{DATA AVAILABILITY statement}

Data sharing is not applicable to this article as no new data were
created or analysed in this study.

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1--11, DOI: 10.1038/s41598-018-24740-1.\selectlanguage{english}
\begin{longtable}[]{@{}llll@{}}
\toprule
Parameter & Description & Value & Source / Notes\tabularnewline
\midrule
\endhead
\emph{N\textsubscript{0}} & Initial population size & 43,200 &
-\tabularnewline
\emph{D\textsubscript{0}} & Initial density (wild boar /
km\textsuperscript{2}) & 3 & -\tabularnewline
\emph{A\textsubscript{0}} & Initial age-distribution (juveniles,
yearlings, adults) & 0.6, 0.2, 0.2 & Bieber and Ruf 2005\tabularnewline
\emph{P\textsubscript{d}} & Direct transmission probability (infected -
susceptible) & 0.0035 & Numerically optimized with data from O\selectlanguage{polish}ļš\selectlanguage{english}evskis
et al. 2020\tabularnewline
\emph{P\textsubscript{c}} & Carcass transmission probability & 0.00016 &
Numerically optimized with data from O\selectlanguage{polish}ļš\selectlanguage{english}evskis et al.
2020\tabularnewline
\emph{P\textsubscript{s}} & Convalescent transmission probability &
0.00038 & Numerically optimized with data from O\selectlanguage{polish}ļš\selectlanguage{english}evskis et al.
2020\tabularnewline
\selectlanguage{greek}\emph{ε} & Incubation time (days) & 3 & Blome et al. 2013\tabularnewline
\selectlanguage{greek}\emph{γ} & Disease lethality & 0.946 & Numerically optimized with data
from O\selectlanguage{polish}ļš\selectlanguage{english}evskis et al. 2020\tabularnewline
\emph{I} & Carcass infectious period (days) & 85 & Numerically optimized
with data from O\selectlanguage{polish}ļš\selectlanguage{english}evskis et al. 2020\tabularnewline
\selectlanguage{greek}\emph{χ} & Convalescents infectious period (days) & 77 & Numerically
optimized with data from O\selectlanguage{polish}ļš\selectlanguage{english}evskis et al. 2020\tabularnewline
\emph{M} & Natural mortality rate (juveniles, yearlings, adults) & 0.18,
0.12, 0.12 & Toigo et al. 2008\tabularnewline
\emph{h} & Annual hunting rate & 0.3 & -\tabularnewline
\emph{R} & Reproduction probability (juveniles, yearlings, adults) &
0.3, 0.8, 0.9 & Bieber and Ruf 2005\tabularnewline
\emph{L} & Litter size (juveniles, yearlings, adults) & 4, 5, 6 & Bieber
and Ruf 2005\tabularnewline
\emph{d} & Dispersal probability (females, males) & 0.4, 0.7 & Truv\selectlanguage{ngerman}é et
al. 2004\tabularnewline
\bottomrule
\end{longtable}\selectlanguage{ngerman}

Tab. 1 -- Summary of the main parameters used to build and run the
spatially explicit, stochastic, individual-based model of ASF spread
into a wild boar population.\selectlanguage{english}
\begin{longtable}[]{@{}lll@{}}
\toprule
& Scenario 1 (direct + carcasses) & Scenario 2 (direct + carcasses +
convalescents)\tabularnewline
\midrule
\endhead
Invasion phase (years 1-2) & Invasion phase (years 1-2) & Invasion phase
(years 1-2)\tabularnewline
ASFV\textsuperscript{+} & 0.05\% (0.01 -- 0.13) & 0.07\% (0.01 --
0.17)\tabularnewline
ASFV\textsuperscript{+} \& Ab\textsuperscript{+} & 0.09\% (0.01 -- 0.21)
& 0.04\% (0.01 -- 0.11)\tabularnewline
Ab\textsuperscript{+} & 0.06\% (0.01 -- 0.18) & 0.08\% (0.01 --
0.21)\tabularnewline
Population density (wild boar / km\textsuperscript{2}) & 2.64 (2.58 --
2.69) & 2.62 (2.54 -- 2.70)\tabularnewline
First epidemic wave (years 3-4) & First epidemic wave (years 3-4) &
First epidemic wave (years 3-4)\tabularnewline
ASFV\textsuperscript{+} & 0.67\% (0.31 -- 0.92) & 0.74\% (0.42 --
1.14)\tabularnewline
ASFV\textsuperscript{+} \& Ab\textsuperscript{+} & 0.44\% (0.23 -- 0.68)
& 0.49\% (0.27 -- 0.73)\tabularnewline
Ab\textsuperscript{+} & 1.32\% (1.04 -- 2.39) & 1.90\% (1.19 --
2.55)\tabularnewline
Population density (wild boar / km\textsuperscript{2}) & 1.31 (0.94 --
1.51) & 1.13 (0.87 -- 1.44)\tabularnewline
Endemic phase (years 5-7) & Endemic phase (years 5-7) & Endemic phase
(years 5-7)\tabularnewline
ASFV\textsuperscript{+} & 0.10\% (0.01 -- 0.33) & 0.13\% (0.02 --
0.35)\tabularnewline
ASFV\textsuperscript{+} \& Ab\textsuperscript{+} & 0.11\% (0.02 -- 0.23)
& 0.09\% (0.01 -- 0.22)\tabularnewline
Ab\textsuperscript{+} & 1.64\% (1.40 -- 2.37) & 2.02\% (1.61 --
2.46)\tabularnewline
Population density (wild boar / km\textsuperscript{2}) & 0.92 (0.55 --
1.02) & 0.87 (0.51 -- 0.98)\tabularnewline
Second epidemic wave (years 8-10) & Second epidemic wave (years 8-10) &
Second epidemic wave (years 8-10)\tabularnewline
ASFV\textsuperscript{+} & 0.50\% (0.16 -- 0.66) & 0.55\% (0.17 --
0.60)\tabularnewline
ASFV\textsuperscript{+} \& Ab\textsuperscript{+} & 0.60\% (0.22 -- 0.90)
& 0.36\% (0.10 -- 0.58)\tabularnewline
Ab\textsuperscript{+} & 1.95\% (1.20 -- 2.84) & 2.19\% (1.30 --
2.92)\tabularnewline
Population density (wild boar / km\textsuperscript{2}) & 2.19 (1.91 --
2.59) & 2.03 (1.87 -- 2.54)\tabularnewline
5-year persistence probability & 55\% & 64\%\tabularnewline
10-year persistence probability & 52\% & 57\%\tabularnewline
\bottomrule
\end{longtable}

Tab. 2- Summary average statistics corresponding to the two simulated
scenarios used to test the different long-term persistence mechanisms of
ASF in a wild boar population. 95\% Cis are shown in parentheses.\selectlanguage{english}
\begin{longtable}[]{@{}lllll@{}}
\toprule
Parameter & Parameter & Sensitivity & SE & p-value\tabularnewline
\midrule
\endhead
Symbol & Description & & &\tabularnewline
\emph{P\textsubscript{d}} & Transmission probability from infected wild
boars & \textbf{-0.067} & \textbf{0.010} & \textbf{\textless{}
0.001}\tabularnewline
\emph{P\textsubscript{c}} & Transmission probability from infected
carcasses & \textbf{-0.107} & \textbf{0.012} & \textbf{\textless{}
0.001}\tabularnewline
\emph{P\textsubscript{s}} & Transmission probability from ASF survivors
& -0.001 & 0.011 & 0.88\tabularnewline
\selectlanguage{greek}\emph{χ} & Duration of ASF survivors' infectivity period & 0.016 & 0.017
& 0.11\tabularnewline
\selectlanguage{greek}\emph{Ι} & Duration of carcasses infectivity period & \textbf{0.088} &
\textbf{0.010} & \textbf{\textless{} 0.001}\tabularnewline
\selectlanguage{greek}\emph{γ} & Disease lethality & \textbf{-0.042} & \textbf{0.011} &
\textbf{\textless{} 0.001}\tabularnewline
\emph{h} & Hunting rate (wild boar density \textgreater{} 0.75 /
km\textsuperscript{2}) & \textbf{-0.385} & \textbf{0.088} &
\textbf{0.001}\tabularnewline
\emph{h} & Hunting rate (wild boar density \textless{} 0.75 /
km\textsuperscript{2}) & 0.006 & 0.059 & 0.21\tabularnewline
\emph{R} & Proportion of females reproducing in the population &
\textbf{0.171} & \textbf{0.086} & \textbf{0.04}\tabularnewline
\bottomrule
\end{longtable}

Tab. 3- Sensitivity of ASF persistence to changes in the main
epidemiological and demographic parameters. The values result from a
global regression-based sensitivity analysis based on standardized input
values. Sensitivity values significantly different from zero are
highlighted in bold font.

FIGURE LEGENDS

Fig. 1 -- Epidemiological compartments used to build the spatially
explicit, stochastic, individual-based model of ASF spread into a wild
boar population. In scenario 1 only direct and carcass-mediated
transmissions were considered. In scenario 2 the three transmission
mechanisms were included. The following notations are used for model
parameters: \selectlanguage{greek}\emph{γ} \selectlanguage{english}= disease lethality; \selectlanguage{greek}\emph{χ} \selectlanguage{english}= convalescents
infectious period; \selectlanguage{greek}\emph{Ι} \selectlanguage{english}= Carcass infectious period; \emph{M} =
natural mortality rate; \emph{h} = hunting rate.

Fig. 2 -- Average daily proportions of ASF infected (a) and
sero-prevalent (b) wild boars, resulting from a model in which disease
transmission occurred either directly (infected-susceptible) or through
an infected carcass.

Fig. 3 -- Daily virus persistence probabilities associated to each of
the two simulated scenarios of the spatially explicit, individual-based
model of ASF in wild boar. Scenarios 1 include only direct and
carcass-mediated virus transmission, whereas scenario 2 also includes
survivor-mediated transmission.

Fig. 4 -- Proportion of ASF virus infections occurring with each of the
three simulated transmission pathways (a), and their relative
distribution within and between wild boar social groups (b).

Fig. 5 -- Functional relationship between wild boar population density
and the proportion of ASF infections occurring through infected
carcasses.

Fig. 6 -- Relationship between the minimum observed wild boar density
during the simulated 10-year period and the ASF persistence expressed in
days.

\textbf{Supporting Information}

Additional Supporting Information may be found in the online version of
this article:

\textbf{Fig. S1} -- Results of the optimization process for the three
ASF transmission rates

\textbf{Fig. S2} -- Results of the optimization process for four
additional model parameters.

\textbf{Fig. S3} -- Temporal trend in the ASF force of infection.

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