# Asymmetrical over-infection as a process of plant virus emergence

Frédéric Fabre, Joël Chadœuf, Caroline Costa, Hervé Lecoq, Cecile Desbiez

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Frédéric Fabre, Joël Chadœuf, Caroline Costa, Hervé Lecoq, Cecile Desbiez. Asymmetrical over-infection as a process of plant virus emergence. Journal of Theoretical Biology, 2010, 265 (3), pp.377. 10.1016/j.jtbi.2010.04.027. hal-00608417

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Asymmetrical over-infection as a process of plant
virus emergence

Frédéric Fabre, Joël Chadœuf, Caroline Costa, Hervé Lecoq, Cécile Desbiez

PII: S0022-5193(10)00218-3

DOI: doi:10.1016/j.jtbi.2010.04.027

Reference: YJTBI5976

To appear in: *Journal of Theoretical Biology*

Received date: 23 December 2009

Revised date: 26 April 2010

Accepted date: 26 April 2010

Cite this article as: Frédéric Fabre, Joël Chadœuf, Caroline Costa, Hervé Lecoq and Cécile Desbiez, Asymmetrical over-infection as a process of plant virus emergence, *Journal of Theoretical Biology*, doi:10.1016/j.jtbi.2010.04.027

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1 Asymmetrical over-infection as a process of plant virus emergence.

2

3 Frédéric Fabre<sup>a*</sup>, Joël Chadœuf<sup>b</sup>, Caroline Costa<sup>a</sup>, Hervé Lecoq<sup>a</sup>, Cécile Desbiez<sup>a</sup>

4

5 <sup>a</sup>INRA, UR 407 Unité Pathologie Végétale, F-84140 Avignon, France

6 <sup>b</sup>INRA, UR 546 Biostatistique et Processus Spatiaux, F-84914 Avignon, France

7

8 <sup>*</sup>Author for correspondence:

9 Tel.: +33 4 32 72 28 47

10 Fax: +33 4 32 72 28 42

11 <em>E-mail address:</em> frederic.fabre@avignon.inra.fr

12 <em>Postal address:</em> INRA, UR 407 Unité Pathologie Végétale, F-84140 Avignon, France

13
14 **Abstract**

15

16 Disentangling the role of epidemiological factors in plant pathogen emergences is a
17 prerequisite to identify the most likely future invaders. An example of emergence was
18 recently observed in France: in 10 years, “classic” (CL) strains of *Watermelon mosaic virus*
19 (WMV) were displaced at a regional scale by newly introduced “emerging” (EM) strains.
20 Here we analyse a 3 years dataset describing the co-dynamics of CL and EM strains at field
21 scale using state-space models estimating jointly (i) probabilities of primary and secondary
22 infection and (ii) probabilities of over-infecting with a CL [EM] strain a plant already infected
23 with an EM [CL] strain. Results especially indicate that it is more than 3 times less probable
24 for a CL strain to over-infect an EM infected plant than for an EM strain to over-infect a CL
25 infected plant. To investigate if these asymmetric interactions can explain the CL/EM shift
26 observed at regional scale, an exploratory model describing WMV epidemiology over several
27 years in a landscape composed of a reservoir and a cultivated compartment is introduced. In
28 most simulations a shift is observed and both strains do coexist in the landscape, reaching an
29 equilibrium that depends on the probabilities of over-infection.
30

31

32

33 **Key words:** Biological invasion, Epidemiology, Landscape, State-space model, *Watermelon mosaic virus*.

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## 1. Introduction

Emergences of plant diseases often have detrimental consequences for food production and food security (Strange and Scott, 2005). Alone, plant virus emergences represent 47% of the cases of emerging plant infectious disease (Anderson et al., 2004) and are reported to be increasingly frequent (Jones 2009; Rojas and Gilbertson, 2008; Varma and Malathi, 2003).

Factors driving the emergence of plant viruses are numerous. Anderson et al. (2004) reveal that, for viruses, pathogen introduction (mainly mediated by anthropogenic activities) and change in vector populations were the major factors of emergence. Many other factors including the intrinsic high capacity of viruses to evolve, changes in agricultural practices and altered pathosystem biology have been documented (Anderson et al., 2004; Elena et al., 2009; Jones, 2009; Rojas and Gilbertson, 2008; Woolhouse et al., 2005). A complete pathogen emergence is a 3-step process (Woolhouse et al., 2005): (i) the exposure of a new host species to a pathogen, (ii) the ability of the pathogen to infect an individual of this new host species (i.e. the pathogen and the host must be compatible) and (iii) the subsequent spread of the pathogen in its new host population. However, the term “emergence” often encompasses several phenomena: (i) newly recognised or newly evolved viruses that cause damaging epidemics, (ii) viruses that increase their geographical distribution and (iii) viruses that change their pathogenicity (host range, virulence) (Jones, 2009; Pulliam, 2008). The need to find new ways to restrict virus emergence requires additional insight into the ecological and evolutionary factors involved in these events (Jones 2009; Pulliam, 2008). Identifying these factors and understanding their relative roles (Anderson et al., 2004; Holmes and Drummond, 2007) is needed to predict which pathogens are most likely to become successful invaders in a given environment.
In France, *Watermelon mosaic virus* (WMV, genus *Potyvirus*) is currently undergoing a change in populations revealing a rapid replacement of local strains by “invasive”, recently introduced ones. WMV has a worldwide distribution, mostly in temperate and Mediterranean regions (Lecoq and Desbiez, 2008). It infects more than 170 crops and weed species, causes agronomic problems mostly in cucurbits and, like other potyviruses, it is non-persistently transmitted by at least 35 aphid species (Lecoq and Desbiez, 2008). At the world level, three molecular groups of WMV have been defined (Desbiez et al., 2007), one of which is widespread in France. WMV, severe in melon, only induced very mild symptoms on zucchini squash until the end of the 90’s. However, severe symptoms in zucchini squash leaves and fruits have been observed since 1999 in South-eastern (SE) France. Their appearance correlates with the introduction of new, “emerging” (EM) isolates distant at the molecular level from the “classic” (CL) isolates that have been present for more than 30 years. A survey performed between 2004 and 2007 at the regional scale showed that EM isolates did not spread over long distances, but rapidly displaced the pre-existing CL isolates in all sites where both groups occurred (Desbiez et al., 2009). This case constitutes a well documented example of viral emergence; not of the complete 3-step emergence processes, but rather a case of appearance in a new geographic region of a new strain of a pathogen. Indeed, before their introduction in France through a yet unknown route, EM WMV strains had already been reported to infect cucurbit crops in Eastern Asia. Moreover, as CL strains were pre-existing in France, investigating this case more deeply should provide interesting information on the factor involved in the “paradox of invasion” that is why an emerging viral species can displace a native viral species that should *a priori* be better adapted to the local conditions (Sax and Brown, 2000).

In addition to the regional survey, a field scale study of WMV epidemiology was
performed from 2002 to 2004 in an experimental plot located in SE France. These data,
acquired in natural conditions, provided detailed spatio-temporal maps of the respective dynamics of CL and EM strains of WMV. In this paper, we will first examine the data collected at the field scale using models specifically built in order to elucidate the processes involved in CL strain displacement but hidden from direct observation. In a second step, using an exploratory model, we will investigate whether the processes identified at the field scale can explain the emergence of EM strains at the landscape scale where the whole annual cycle of WMV occurs.

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## 2. Data description

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A plot of 160 zucchini squash F1 Diamant consisting of 8 rows of 20 plants was planted each year from 2002 to 2004 at Montfavet, France. Each plant was surveyed weekly for virus infection until 100% of the plants became infected (week 3 in 2002 and 2004, week 4 in 2003) and a last survey done 4 or 5 weeks later (week 7 in 2002, week 8 in 2003 and 2004). Plants were classified into 4 compartments, (i) H (healthy), (ii) CL (infected by a classic strain), (iii) EM (infected by an emerging strain) and (iv) D (doubly infected by both a classic and an emerging strain), as follows: a young leaf was collected at each sampling date from each plant and tested by DAS-ELISA for WMV infection. If the sample is negative, the plant is classified H. For each positive sample, total RNA was extracted using TRI-Reagent (Molecular Research Center, Inc., Cincinnati, OH) and submitted to one-strep RT-PCR according to the protocol used in the laboratory using primers WMV-5' (5'-GGCTTCTGAGCAAAGATG-3') and WMV-3' (5'-CCCAYCAACTGTYGGAAG-3') (Desbiez et al., 2007). PCR samples were sent to Cogenics (Grenoble, France) for direct sequencing with primer WMV-5'. Sequences were aligned with ClustalW included in DAMBE (Xia, 2000), adding reference sequences belonging to the main molecular groups
defined for WMV (Desbiez et al., 2007, 2009). Neighbour-joining and maximum-parsimony trees were built with MEGA 3.1 (Kumar et al., 2004) and bootstrap analysis performed to assess the robustness of the trees. According to the sequence data, the plant is classified as CL, EM or D.

Moreover, an epidemiological survey (Desbiez et al., 2009) was performed from 2004 to 2008 in the 3 neighbouring “départments” (administrative subdivision, 4-6000 km² each) Gard, Vaucluse and Bouches du Rhône in SE France where EM strains were first detected. Samples of cucurbit crops, received from farmers, farm advisers and seed companies, were tested as described above.

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### 3. Models description

#### 3.1. A model for the co-dynamics of viral strains at field scale

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This section describes the model skeleton and underlying assumptions. The whole set of equations and additional mathematical details are provided in appendix A and B. In the model, $N(i,t)=[N_H(i,t), N_{CL}(i,t), N_{EM}(i,t), N_D(i,t)]$ is the number of plants in each compartments for a given year $i$ and date $t$ of observation. The model describes the dynamic of $N(i,t)$ at a time interval $\Delta t = 1$ week. Initially, all plants are healthy: $N(i,0)=[N_{tot}, 0, 0, 0]$ ($N_{tot}=160$). Plants can be infected through 8 categories of events resulting from the combination of 2 virus strains (CL and EM), 2 plant categories (when infected by a given virus strain a plant is either healthy or already infected by the other strain) and 2 infection processes (primary and secondary infection) (Fig. 1). Primary infection corresponds to the infection of a plant by aphids having acquired the virus from sources outside the considered field whereas sources are within this field in the case of secondary infection.

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3.1.1. Probabilities of infection of an H plant from *t*-1 to *t*

An H plant at time *t*-1 can become CL, EM or D or remain H at time *t* depending on the probabilities of 4 events (Fig. 1a): (i,ii) *p*<sub>*i*,t*</sub><sup>PI</sup> (CL|H) [resp. *p*<sub>*i*,t*</sub><sup>PI</sup> (EM|H)], probability that “an H plant is infected by primary infection with a CL [resp. EM] strain during Δ*t*” and (iii,iv) *p*<sub>*i*,t*</sub><sup>SI</sup> (CL|H) [resp. *p*<sub>*i*,t*</sub><sup>SI</sup> (EM|H)], probability that “an H plant is infected by secondary infection of a CL [resp. EM] strain during Δ*t*”.

Regarding primary infection, $p_{i,t}^{PI}(CL|H)$ is modelled as $\text{logit}[p_{i,t}^{PI}(CL|H)] = \alpha_{i,CL} + \gamma_{i,t}$ where (i) the parameter $\alpha_{i,CL}$ is, in the logit scale, the probability that an H plant is infected by primary infection with a CL strain during $\Delta t$ in year $i$ if no variation between weeks exists and (ii) $\gamma_{i,t}$ is a set of random effects mutually independent and identically normally distributed with unknown variance $\sigma^2$ accounting for the variability between weeks of the primary infection process. $p_{i,t}^{PI}(EM|H)$ is defined similarly (appendix A – equation A2).

Regarding secondary infection, deriving $p_{i,t}^{SI}(CL|H)$ required to consider the probability $q_i(SI_{CL}|H)$ of the elementary event “a plant infected in the field with a CL strain in year $i$ infects an H plant by secondary infection during $\Delta t$. For a given year $i$, $q_i(SI_{CL}|H)$ was assumed constant and modelled as a pure fixed effect $\beta_{i,CL}$ as $\text{logit}[q_i(SI_{CL}|H)] = \beta_{i,CL}$. Assuming that secondary infection does not depend on the distance between sources of CL strains and H plants, it comes that $p_{i,t}^{SI}(CL|H) = 1 - [1 - q_i(SI_{CL}|H)]^{N_{CL}(i,t-1)+(1-\pi)N_D(i,t-1)}$ where (i) $N_{CL}(i,t-1) + (1-\pi)N_D(i,t-1)$ is the number of potential sources of CL strains in the field at time $t-1$ and (ii) $\pi$ is a parameter describing the mean proportion of EM strains in D plants (1-$\pi$ is the proportion of CL strains). This equation means that an H plant at time $t-1$ become CL at $t$ by secondary infection if at least one of the plants infected in the field with a CL strain at
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date *t*-1 (either CL or D plants) was a source for this strain. Details on its derivation are
provided in appendix B and in appendix A (equations A4, A6) for *p*<sub>*i*,t*</sub><sup>SI</sup> (*EM* | *H*).

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Finally, assuming that during Δt primary and secondary infection events occur independently,

$P_{i,t}^M = (p_{i,t}(H_t | H_{t-1}), p_{i,t}(CL_t | H_{t-1}), p_{i,t}(EM_t | H_{t-1}), p_{i,t}(D_t | H_{t-1}))$, the vector of probabilities

that, in year i, a plant is H [resp. CL, EM and D] at time t given that this plant was H at time t-

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**3.1.2. Probabilities of infection of a singly infected plant from t-1 to t**

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3.1.2. Probabilities of infection of a singly infected plant from *t*-1 to *t*

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A CL plant at time *t*-1 can become D or remain CL at time *t* depending on the probabilities of 2 events (Fig. 1b): (i) *p*<sub>*i*,t*</sub><sup>PI</sup>(*D* | *CL*), probability that “a CL plant is infected by primary infection with a EM strain during Δ*t” and (ii) *p*<sub>*i*,t*</sub><sup>SI</sup>(*D* | *CL*), probability that “a CL plant is infected by secondary infection of an EM strain during Δ*t”. As previously, deriving this probability required to consider the probability *q*<sub>*i*</sub> (*SI*<sub>EM</sub> | *CL*) of “a plant infected in the field with an EM strain is source of infection of an CL plant during Δ*t”.”

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$p_{i,t}^{PI}(D|CL)$ and $q_i(SI_{EM}|CL)$ are modelled with a parameter, $\gamma_{CL}$, defined as the odd ratio between the probability of infecting an H plant with a EM strain and the probability of infecting an CL plant with a EM strain. Values of $\gamma_{CL} >1$ indicate that over-infecting an EM plant with a CL strain is less likely than infecting an H plant with a CL strain. The higher $\gamma_{CL}$, the less likely this event is. Values of $\gamma_{CL} <1$ indicate that over-infecting is more likely. No difference arises when $\gamma_{CL}=1$. For interpretation, odds ratios approximate relative risks for rare events ($p<0.1$). $\gamma_{CL}$ applies indifferently whatever the infection process leading to over-
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infection (primary or secondary infection) and does not depends on the year and on the time lag between the first and the second infection events.

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Finally assuming independence between primary and secondary infection leads to a simple expression of $p_{i,t}(D_t|CL_{t-1})$, the probability that, in year $i$, a plant becomes doubly infected (D) at time $t$ given that this plant was CL at time $t-1$. All the equations necessary to derive $p_{i,t}(D_t|CL_{t-1})$ as well as their analogues for the over-infection of an EM plant with a CL strain (Fig. 1c) are presented in Appendix A (equations A11 to A18).

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### 3.1.3. Dynamics of $N(i,t)$ during an epidemic

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The dynamic of $N(i,t)$ from $t=0$ to $t_{end}$ is modelled as a sequential 3 step stochastic process. During year $i$ from $t-1$ to $t$, (i) the number of H plants remaining H or becoming CL, EM or D is the result of the multinomial process $\sim \text{Multinomial}(N_H(i,t-1), p_{i,t}^M)$, (ii) the number of CL plants becoming D is the result of the binomial process $\sim \text{Binomial}(N_{CL}(i,t-1), p_{i,t}(D_t|CL_{t-1}))$ and (iii) the number of EM plants becoming D is the result of the binomial process $\sim \text{Binomial}(N_{EM}(i,t-1), p_{i,t}(D_t|EM_{t-1}))$. Lastly, the number of plants in each compartment is updated from date $t-1$ to $t$ (Appendix A – equation A22).

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These stochastic processes assume that: (i) all plants in a given compartment at $t-1$ have the same probability to be in another compartment at time $t$, (ii) changes between compartments are independent from one plant to another.

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### 3.1.4. Model inferences

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The vector of model parameters is $\theta=[\alpha_{CL,i}, \alpha_{EM,i}, \beta_{CL,i}, \beta_{EM,i}, \gamma_{EM}, \gamma_{CL}, \pi, \sigma]_{i=1..3}$. Statistical inferences were performed via Bayesian Monte Carlo Markov Chain (MCMC) methods

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(Gelman et al. 2004) with OpenBUGS® 3.0.3 using uninformative prior probability density functions for θ. Consistency between the model and the data was checked using a χ² discrepancy test (Gelman et al. 2004) as well as linear regression of predicted versus observed values of N<sub>CL</sub>(..), N<sub>EM</sub>(..) and N<sub>D</sub>(..). Finally, to investigate more precisely the role of over-infection parameters (γ<sub>EM</sub> and γ<sub>CL</sub>), three models were compared using Bayes Factor (Kass and Raftery, 1995): (i) model M<sub>0</sub>, the model described above, (ii) model M<sub>1</sub> where γ<sub>EM</sub>=γ<sub>CL</sub> (i.e. the probability of over-infection does not depend on the first strain inoculated) and (iii) model M<sub>2</sub> where γ<sub>EM</sub>=γ<sub>CL</sub>=1 (i.e. the probability of over-infection are equal to the probability of infecting an H plant). Further details on model inferences are provided in appendix C.

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213 **3.2. A model of viral epidemiology at landscape scale.**

214

215 **3.2.1. Model description**

216

217 A model of WMV epidemiology over several years in a landscape is described. It simulates
218 $p_{EM}(i)_{\{i=0..y_{end}\}}$, the proportion of EM strains during $y_{end}$ years found in an idealized landscape
219 made of 2 compartments.

220 The reservoir compartment, available all year long, includes potential wild reservoir hosts of
221 WMV. It is characterized by $p_{EM}(i)$, the proportion of EM strains among the reservoir hosts
222 infected by WMV in year $i$ ($1-p_{EM}(i)$ is the proportion of CL strains).

223 The cultivated compartment, available during $T_{max}$ weeks a year, is composed of $N_{field}$ fields
224 similar to our experimental plots (with $N_{tot}$ plants). Using the field model, the dynamics of
225 $N(i,f,t|\Omega_i)=[N_H(i,f,t), N_{CL}(i,f,t), N_{EM}(i,f,t), N_D(i,f,t)]$ are simulated, $N(i,f,t|\Omega_i)$ being the number
226 of H, CL, EM and D plants during year $i$ ($1 \le i \le y_{end}$), in the field $f$ ($1 \le f \le N_{field}$) at date $t$ ($1 \le
227 $ $t \le T_{max}$) simulated for a given model parameter $\Omega_i$. $\Omega_i=[p_i^s(PI_{CL}|H), p_i^s(PI_{EM}|H), q_i^s(SP_{CL}|H),$
228 $q^s(SP_{EM}|H)$, $\gamma^s_{EM}$, $\gamma^s_{CL}$, $\pi^s$] (the upper script S is added to the notation defined in the “field scale” model to indicate that these parameters are those of the “landscape scale” model) is defined as follow:

231 (i) For the sake of simplicity, the probabilities of primary infection rates of CL [resp. EM] strains $p^s_i(PI_{CL}|H)$ [resp. $p^s_i(PI_{EM}|H)$] depend only on the year $i$ and not on weeks. Primary infection events are inoculations to cultivated plants of viruses originating from the reservoir compartment (no between fields infections are considered). If $p^s$ denotes the probability of having a primary infection event, and given that the relative proportion of EM strains in the reservoir compartment in the previous winter is $p_{EM}(i-1)$, it is assumed that $p^s_i(PI_{EM}|H)=p^s.p_{EM}(i-1)$ and $p^s_i(PI_{CL}|H)=p^s.(1-p_{EM}(i-1))$: the probability of primary infection by a given strain is proportional to the frequency of this strain in the reservoir compartment during the previous winter (i.e. there is an annual “reset” of WMV populations in the reservoir compartment).

241 (ii) The probabilities of secondary infection, $q^s_i(SP_{CL}|H)$ and $q^s_i(SP_{EM}|H)$, are equal to $q^s$.

242 (iii) The probabilities of over-infection are assessed as previously assuming that $\gamma^s_{EM}=\gamma^s_{CL}.\delta$ where $\delta>0$.

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245 Initially ($i=1, t=1$), only CL strains are in the reservoir hosts ($p_{EM}(i=0)=0$). Thus only CL strains can be introduced in the cultivated compartment through primary infection events.
246
247 However during the first year a single plant infected with an EM strain is introduced at time $t_{intro}$ in $N_{intro}$ out of $N_{field}$. In these field(s), EM strains can only spread by secondary infection
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249 during this first year (i.e. between-fields events of primary infection are not modelled).

250
251 The dynamic of $p_{EM}(i)$ is simulated by iterating 2 steps for $i$ in $1..y_{end}$:
252 (i) Step 1. Given the values of $p^S$, $q^S$, $\gamma^S_{CL}$, $\pi^S$, $\delta$ and $p_{EM}(i-1)$, the dynamics of $N(i,f,t|\Omega_i)$ are assessed. If $i=1$, a single plant infected with an EM strain is introduced at time $t_{intro}$ in $N_{intro}$ out of $N_{field}$.

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256 (ii) Step 2. $p_{EM}(i)$ is the ratio between $S_{EM}(i)$ and $S_{Tot}(i)$, respectively defined as:

$$257 S_{EM}(i) = \sum_{f=1}^{N_{field}} \int_{t=1}^{T_{max}} (N_{EM}(i, f, t) + \pi^S N_{D}(i, f, t)).dt$$

$$258 S_{Tot}(i) = \sum_{f=1}^{N_{field}} \int_{t=1}^{T_{max}} (N_{CL}(i, f, t) + N_{EM}(i, f, t) + N_{D}(i, f, t)).dt$$

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As such, $p_{EM}(i)$ is proportional to the cumulated dynamic of EM plants with respect to the cumulated dynamic of CL, EM and D infected plants over all the fields of the landscape. D plants are supposed to be source of EM strains proportionally to $\pi^S$.

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## 3.2.2. Landscape model analysis

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Analyses were focused on the effect of $\gamma^S_{EM}$ and $\gamma^S_{CL}$. First, the effect of $\delta$ ($\delta=\gamma^S_{EM}/\gamma^S_{CL}$) on the temporal dynamics of $p_{EM}(i)$ for $i \in \{1.. y_{end}\}$ was studied. $\gamma^S_{CL}$ was set to 2.95 and $p_{EM}(i)$ was assessed for values of $\delta$ ranging from 1 ($\gamma^S_{EM}=\gamma^S_{CL}$) to 40 ($\gamma^S_{EM}=40$. $\gamma^S_{CL}$). Secondly, the final state reached after 40 years by $p_{EM}(y_{end})$ was studied in a ($\gamma^S_{EM}, \gamma^S_{CL}$) plane with $1 \le \gamma^S_{EM} \le \gamma^S_{CL} \le 1000$. In practise, a log-regular grid with 900 points in this ($\gamma^S_{EM}, \gamma^S_{CL}$) plane was considered. Three final states were possible: (i) EM strains replaced CL strains (it exists $i \in \{2.. y_{end}\}$ such that $p_{EM}(j \ge i)=1$), (ii) EM strains disappeared following their introduction (it exists $i \in \{2.. y_{end}\}$ such that $p_{EM}(j \ge i)=0$), and (iii) EM and CL continued to coexist at $y_{end}$ (for
all $i \in \{2.. y_{end}\}$, $0<p_{EM}(i)<1$). For each points of the grid in the plane $(y_{EM}^S, y_{CL}^S)$, the probabilities of the 3 possible final states were estimated by their proportions using 1000 independent replicates. In all simulations, primary and secondary infection probabilities were set to their overall mean from data analysis ($p^S=0.11, q^S=0.035$) as well as $\pi^S$ set to 0.56. $N_{field}$ was set to 25, $t_{intro}$ and $N_{intro}$ to 1, $T_{max}$ to 10 weeks and $y_{end}$ to 40.

## 4. Results

### 4.1. Epidemiology of CL and EM strains of WMV at the field scale

#### 4.1.1. Experimental results at the field scale

The set of spatio-temporal maps of the disease dynamics observed are presented in online supplementary material (Fig. S1). During the 3 years, CL and EM strains were always first detected in the same plot as early as the second week after planting. Only one, EM1, of the four subgroups of EM strains observed in SE France (Desbiez et al., 2009), was found repeatedly in the experimental plot (data not shown). Complete infection of the plot was observed after 3 or 4 weeks and, at week 7 or 8, the number of D plants was close to 100%, indicating that cross protection between CL and EM strains is not fully efficient (Fig. S1).

#### 4.1.2. Goodness of fit of the model

The baseline model ($M_0$) was consistent with the data. A good fit between observed and predicted values of the numbers of CL, EM and D plants ($N_{CL}(i,t), N_{EM}(i,t), N_{D}(i,t)$ respectively) was observed for the 3 years (Fig. 2). The overall $r^2$ value between observed and predicted values of $N_{CL}(i,t), N_{EM}(i,t), N_{D}(i,t)$ was 0.98 and 95 % of observed values were included in a 90% credibility interval ($n=74$). Moreover, the Bayesian p-value of the $\chi^2$ discrepancy did not reveal any significant inconsistency between the model and the data: the
p-values were 0.87, 0.69 and 0.9 for N<sub>CL</sub>, N<sub>EM</sub> and N<sub>D</sub> respectively. The statistical summary of the parameters of the model M<sub>0</sub> are provided in Table 1.

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4.1.3. Mean annual probabilities of primary infection

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Mean probabilities refer here to the probability of primary infection of an H plant averaged
over the duration of the epidemic. Their values ranged from 0.05 to 0.31 depending on the
year and the strain (Fig. 3). The overall mean was 0.11. From 2002 to 2004, the primary
infection rate by EM strains was significantly higher than the one of CL strains in 2003 and
2004 (Bayesian p-value of 0.047 and 0 respectively) but not significantly different in 2002
(Bayesian p-value of 0.66). Finally, the hypotheses of a decrease or an increase through years
of the mean probabilities of primary infection for CL and EM strains were tested. No
significant trend was detected in mean probabilities of primary infection by a CL strain but
the mean probability of primary infection by an EM strain was significantly higher in 2004
than in 2002 or 2003 (Bayesian p-value of 0.01 and 0.02 respectively).

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315 **4.1.4. Annual probabilities of secondary infection**

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The probabilities that a plant infected with a CL or an EM strain was a source of infection for
an H plant during a week ranged from 0.002 to 0.07 depending on the year and the strain
(Table 1). The overall mean was 0.036. The mean probabilities of secondary infection did not
exhibit any clear annual trend. For a given year, the mean probabilities of secondary infection
of a CL strain were significantly higher than the one of an EM strain in 2003 and 2004
(Bayesian p-values of 1 and 0.99 in 2003 and 2004 respectively) but not in 2002 (Bayesian p-
values 0.89). The role of D plants in secondary infection events as sources of CL and EM

318

319

320

321

322

323
strains was modelled by the parameter $\pi$ which is the mean frequency of EM strains in these
plants (see appendix B for details). The mean value of $\pi$ was estimated as 0.56 with a 90%
Bayesian posterior credibility interval ranging from 0.34 to 0.79. Accordingly, within plant
competition between CL and EM strains in D plants did not significantly favour the spread of
one or another strain.

329

330 **4.1.5. Odd ratios of over-infection**

331

The mean value of the odd ratio for over-infecting an EM plant ($\gamma_{EM}$) and a CL plant ($\gamma_{CL}$) were respectively 8.92 and 2.95 (Table 1). Accordingly, over-infecting a CL plant with an EM strain is about 3 times less probable than infecting an H plant but over-infecting an EM plant with a CL strain is about 9 times less probable than infecting an H plant. Overall, it is 3 times less probable for a CL strain to infect an EM plant than for an EM strain to infect a CL plant.
Moreover, when considering the 95% Bayesian posterior credibility interval and Bayesian p-values, it appears that $\gamma_{EM}$ and $\gamma_{CL}$ are significantly greater than one. Thus over-infecting a previously infected plant is significantly less probable than infecting a healthy plant. $\gamma_{EM}$ is also significantly greater than $\gamma_{CL}$. The comparison of model $M_0$ (baseline model with $\gamma_{EM} \neq \gamma_{CL}$ i.e. the probability of over-infection depends on the first strain inoculated), $M_1$ (model with $\gamma_{EM} = \gamma_{CL}$ i.e. the probability of over-infecting does not depend on the first strain inoculated) and $M_2$ (model with $\gamma_{EM} = \gamma_{CL} = 1$ i.e. the probability of over-infection are equal to the probability of infecting an H plant) leads to the same conclusion with Bayes factors of respectively 4 for $2 \cdot \ln(BF_{1,0})$ (support for $M_0$) and 157 for $2 \cdot \ln(BF_{2,0})$ (strong evidence for $M_0$).

Back to the meaning of odd ratios, and taking the probabilities of secondary infection in 2003
as example, the value of the odd ratios for over-infection imply that, although the mean

348
probability of secondary infection of CL strains on a healthy plant was higher than the one of EM strains ($q_2(SI_{CL} | H) = 0.033$ and $q_2(SI_{EM} | H) = 0.011$), this is no longer the case when considering the infection by secondary spread of a previously infected plant ($q_2(SI_{CL} | EM) = 0.0038$ and $q_2(SI_{EM} | CL) = 0.0037$ according to equations A12 and A16 of Appendix A).

## 4.2. Landscape scale model

The landscape model simulates the dynamic of the proportion of EM strains in a landscape during 40 years ($p_{EM}(i)$ with $i \in \{1..40\}$) following the accidental introduction of a single cucurbit seedling ($N_{intro}=1$ out of $N_{field}=25$ fields with 160 plants) infected with an EM strain. The effect of the parameter, $\delta = \gamma_{EM}^S / \gamma_{CL}^S$, measuring how many times it is less likely for a CL strain to infect an EM plant than for an EM strain to infect a CL plant, on the dynamic of the proportion of EM strains in a landscape was particularly investigated. A value of $\delta > 1$ indicates that it is $\delta$ times more probable for an EM strain to infect a CL plant than for a CL strain to infect an EM plant. Inversely, a value of $\delta < 1$ indicates that it is $1/\delta$ times more probable for a CL strain to infect an EM plant than for an EM strain to infect a CL plant. In Fig. 4a, the odd ratio for over-infecting an CL plant, $\gamma_{CL}^S$, was set to its reference value 2.95 and the dynamic of the proportion of EM strains in the landscape was plotted for 4 values of $\delta$ (corresponding to 4 values of the odd ratio for over-infecting an CL plant, $\gamma_{EM}^S$) [$\delta=1$ ($\gamma_{EM}^S=2.95$), $\delta=3$ ($\gamma_{EM}^S \approx 8.8$), $\delta=10$ ($\gamma_{EM}^S \approx 29$) and $\delta=40$ ($\gamma_{EM}^S=118$)]. For most simulations, the proportion of EM strains in the landscape rapidly increased during the first 15 years and then reached a noisy steady state where both CL and EM strains of WMV coexisted in the landscape. In simulations where $\delta$ ranged between 0.001 and 1000, coexistence between EM
and CL strains almost always occurred for $0.05 \le \delta \le 10$ (Fig. 5a). When coexistence occurred, the level of the steady state between CL and EM strains mostly depended on $\delta$ (Fig. 5b). For $\delta=1$, the proportion reached by EM strains in the landscape, 0.57, was close to their proportion in D plants, $\pi^S$. However, this correspondence was not observed for higher values of $\pi^S$ (not illustrated).

Coexistence of EM and CL strains was not always observed. The probability that EM strains disappeared following their introduction is higher than 0.25 for values of $\delta \le 0.05$ and even higher than 0.5 for values of $\delta \le 0.01$ (Fig. 5c). Similarly, EM strains could replace CL strains (Fig. 5d). However this event was infrequent in the range of the plane ($\gamma^S_{EM}, \gamma^S_{CL}$) explored and only occurred with probabilities higher than 0.5 for values of $\delta \ge 50$. Finally, for a given value of $\delta$ (e.g. $\gamma^S_{EM}=100, \gamma^S_{CL} - \delta =100$) it was somewhat more likely for EM strains to replace CL strains than for EM strains to go extinct in the inverse case ($\gamma^S_{CL}=100, \gamma^S_{EM} - \delta =0.01$). This asymmetry resulted from a trade-off between the endemic and native origins of CL and EM strains respectively and their relative fitness in competition in doubly infected plant.

## 5. Discussion

The shift between CL and EM strains of WMV observed in France since 2000 is a striking example of the invasion paradox, locally adapted strains being rapidly displaced by invading ones originating possibly from Asia (Desbiez et al., 2009). A similar observation was recently reported by Gómez et al. (2009) in Spain where, after their recent introduction, new isolates of *Pepino mosaic virus* (PepMV-CH2) have spread in a few years in an epidemic fashion into a niche previously occupied by PepMV-EU isolates. As a first step in this study, we developed stochastic models to disentangle the field co-dynamics of CL and EM strains with regard to several epidemiological mechanisms (primary and secondary infection and a
wide range of host-virus interactions in co-infected host). These models also explicitly deal
with the uncertainty in the observation process. They are examples of state-space model, a
class of model now widely used in various fields of ecology (e.g. Clark, 2005; Cressie et al.,
2009; Soubeyrand et al., 2009) notably due to its propensity to reconcile the point of view of
statisticians and mechanism based modellers.

The main originality of the model was to define probabilities of over-infection that can
handle the different kinds of plant-virus interaction arising in mixed infections. A virus may
stimulate the replication and/or the movement of another virus (Atabekov and Dorokhov,
1984) (Karyieja et al., 2000). A plant resistant to a virus can even become infected with the
help of another virus (Dodds and Hamilton, 1972) (Wang et al., 2004) or another strain of the
same virus (Gomez et al., 2009). Diverse mechanisms are involved in these synergistic
interactions, including efficient inhibition of posttranscriptional gene silencing by at least one
of the viruses (for a review see Latham and Wilson, 2008; Cuellar et al., 2009). However, one
virus can also prevent the infection of a second, a phenomenon known as cross-protection
(Fraser, 1998) and used to control some major virus diseases (Lecoq, 1998). To our
knowledge, very few papers deal with the mathematical modelling of interactions between
plant viruses (Zhang et al., 2000; Zhang and Holt, 2001). Only Zhang and Holt (2001)
modelled the effect of cross protection on plant-virus epidemiology assuming that infection
by the protective virus precludes completely infection by the severe one, but not vice-versa.
More generally for plant pathogens, many model-based studies used the concept of the basic
reproduction number ($R_0$) to analyse disease invasion and persistence from the microscopic to
the regional scale. However, most of these studies rely on the assumption that the invading
pathogen is not competing with resident pathotypes (see Gilligan and van den Bosh (2008) for
a review and references for exceptions to this trend).
In our case, inferences on the odd ratio for over-infecting an EM plant ($\gamma_{EM}$) and a CL plant ($\gamma_{CL}$) firstly reveal that over-infecting an already infected plant is less probable than infecting a healthy plant but nevertheless occurs as revealed by both experimental and field observations. Normally, over-infection between strains of the same potyvirus is supposed to be suppressed by the mechanisms of cross-protection, possibly related to RNA silencing (Ratcliff et al., 1999); cross-protection was once considered as a taxonomic criterion for species/strain demarcation (Ward and Shukla, 1991). The efficiency of cross-protection among potyviruses depends on the molecular relatedness between strains, highly divergent isolates being poorly cross-protective (Nakazono-Nagaoka et al., 2009; Valkonen et al., 2002; Wang et al., 1991; You et al., 2005). CL and EM isolates have 4 to 6% divergence in their coat protein (CP) nucleotide sequence (Desbiez et al., 2009). At this level of divergence, cross-protection between potyvirus isolates is usually effective (Nakazono-Nagaoka et al., 2009; Lecoq, 1998). However some exceptions were already observed and the relation between CP sequence relatedness and protection efficiency has been questioned (Valkonen et al., 2002).

Inferences on the odd ratio for over-infecting an EM plant ($\gamma_{EM}$) and a CL plant ($\gamma_{CL}$) also indicate that it is 3 times less probable for a CL strain to infect an EM plant than for an EM strain to infect a CL plant. Examples of such asymmetrical efficiency are quite rare in the literature (e.g. between strains of *Sugarcane mosaic virus* (Krstic, 1995), *Potato virus A* (Valkonen et al., 2002), *Cucumber mosaic virus* (Tian et al., 2009) or between *Oilseed rape mosaic virus* and *Tobacco mild green mosaic virus* (Aguilar et al., 2000)). This asymmetry might be related to a more efficient accumulation, migration and/or silencing suppression in EM strains, thus overcoming the cross-protection mechanisms. Whereas strains of some potyviruses rarely infect the same cell (Dietrich and Maiss 2003; Takahashi et al., 2007), the high frequency of natural recombinants in WMV (Desbiez and Lecoq, 2008) suggests that cell
447 co-infection is frequent in this virus; the asymmetry between EM and CL strains might
448 happen at this level. These hypotheses remain to be investigated. Asymmetrical interactions
449 have also been recently revealed in animal pathology. Wolfe et al. (2007) showed that mice
450 previously infected with *Bordetella pertussis* were not protected against a later infection with
451 *B. parapertussis* (the main agents of whooping cough), while primary infection with *B.
452 parapertussis* conferred cross-protection. Much attention has been given to the mathematical
453 modelling of cross-protection in animal pathology and especially virology as it is a key factor
454 of vaccination success (e.g. human influenza A (Ferguson et al., 2003)). However, with a few
455 exception (Restif et al., 2008 and references therein), asymmetric cross-immunity has also
456 been overlooked in most mathematical models.

457 The model goodness of fit was satisfactory. However, improvements in several ways
458 are possible by taking advantage of the flexibility of the state-space models. Modelling the
459 spatiotemporal dynamics of CL and EM strains on a daily time step is particularly appealing.
460 Such space-time data analysis often improves our understanding of the mechanisms
461 governing both animal and plant epidemics (e.g. Keeling et al., 2001; Otten et al., 2003;
462 Soubeyrand et al., 2009). Here it will provide inferences about the distance of dispersion of
463 secondary infection events for example, a process seldom estimated *in natura* for non-
464 persistently transmitted viruses. Bayesian inferences on spatiotemporal process are of special
465 interest as they account coherently for parameter variability (Cressie et al., 2009). If classical
466 Bayesian inferences are difficult to handle, statistical methods like approximate Bayesian
467 calculations (Beaumont et al., 2002) could be an alternative.

468

469 The second part of this work investigated the consequences, at the landscape scale, of
470 the asymmetrical probabilities of over-infection revealed at the field scale during the course of
471 the annual epidemiological cycle. An explanatory model is proposed in order to fit the scales
at which hosts and aphid-borne viruses naturally interact. Simulations addressed the case of an accidental introduction of a cucurbit seedling infected with an EM strain. The results indicated that $\delta$, the parameter measuring how many times it is less likely for a CL strain to infect an EM plant than for an EM strain to infect a CL plant, deeply impacted the equilibrium between CL and EM strains in the landscape throughout the following mechanism. Assuming for example $\delta > 1$ implies that plants remained solely EM infected longer than solely CL infected. In other words, the higher $\delta$ is, the longer is the lag time necessary for an EM infected plant to become D infected relatively to the lag time necessary for a CL plant to become D. In turn, this tends to increase the cumulated dynamics of plants solely infected with EM strains in the fields of the landscape and thus the proportion of reservoir hosts infected with EM strains. A consequence of this mechanism is that, for some value of $\delta$, CL strains mostly persist in the environment in D plants. The persistence of viral isolates in mixed infected plants was proposed by Gómez et al. (2009) to explain PepMV-EU isolates persistence in Spain after PepMV-CH2 spread.

In the simulation with the parameters closest to the field observation (Fig. 4a, $\delta=3$ - black curve), CL and EM strains almost always coexist in the landscape at a ratio of about 3:7. Although this is not a formal validation, these figures should be compared to the data of the epidemiological survey in the Gard and Bouches du Rhône regions where the proportion of EM strains seems to reach an equilibrium near 70-80% (Fig. 4b) in less than 10 years after their introduction. In Vaucluse, the situation is different as the proportion of CL strains decreased steadily and reached only 1% in 2008. Although it is tempting to bring nearer these data, the epidemiological situation of WMV is much more complex than the simulation scenario with in particular 4 subgroups of EM isolates probably originating from as many introduction events. Only one subgroup of isolates was detected in the field data (EM1). The value of $\delta$ derived thus characterizes the interactions between CL and EM1 strain in D plants.
The landscape model describes a disease with discontinuous crop-host availability typical of the annual crops in temperate climates. During the crop-free season, viruses and vectors are hosted by "reservoirs", wild hosts playing then a major role. Up to date, viruses in the wild compartment have received little attention. Consequently, in the model, the reservoir compartment is simply seen as a mirror reflecting, from year *n* to year *n*+1, the proportion of EM strain according to its evolution in the sole cultivated compartment. Despite this simplification, one should notice that the trend detected in the mean probability of primary infection by CL and EM strains (Fig. 3) corroborates this hypothesis. Nevertheless, further research on virus-plant pathosystems at the interface between cultivated and natural vegetation is needed (Jones, 2009). Firstly, this need arises from the importance of studying epidemics at the scale where they naturally occur as well as the need to match this scale with the one at which control strategy must be deployed (Dybiec et al. 2004; Gilligan 2008). In this regard, the recent focus on the landscape epidemiology of plant disease (Plantegenest et al. 2007) is a promising field of research. A global sensitivity analysis of the landscape model aiming to rank the relative role of model parameters on the probabilities of extinction and coexistence as well as on the proportion of EM strains at equilibrium could help to point out the major processes that should be studied in priority at the landscape scale. Secondly this need arises because alternation between hosts in a heterogeneous landscape tends to select for generalist viruses which in turn are more likely to jump over the species boundary (Elena et al., 2009), and are thus more likely to successfully emerge.

517

518 Acknowledgments

519

We are grateful to Catherine Wipf-Scheibel and Charlotte Chandeysson for their excellent technical assistance. We also thank Amandine Lê Van for her enthusiasm with early
522 modelling work on this dataset and Denis Fargette and Cindy E. Morris for their comments

523 and help with the English.

524
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**Table 1.** Statistical summary of the inference on parameters of model M₀. Main statistics (mean, q-5%: percentile 5%, q-25%: quartile 25%, median, q-75%: quartile 75%, q-95: percentile 95%) of the marginal posterior distribution are derived from a MCMC sample of size 10⁵. For the secondary infection process, the values provided for the β's parameters (transformed in the logit⁻¹ scale), estimate directly the probability that a plant infected with a CL or an EM strain is the source of infection of an H plant by secondary infection during a Δt time step ($q_i(SI_{CL}|H)$ and $q_i(SI_{EM}|H)$ with $1 \le i \le 3$).

722

723

<table><thead><tr><th>Process</th><th>Parameter</th><th>mean</th><th>q-5%</th><th>q-25%</th><th>median</th><th>q-75%</th><th>q-95%</th></tr></thead><tbody><tr><td rowspan="7">Primary infection</td><td>α<sub>1,CL</sub></td><td>-3.49</td><td>-7.20</td><td>-4.90</td><td>-3.41</td><td>-2.09</td><td>-0.10</td></tr><tr><td>α<sub>2,CL</sub></td><td>-4.85</td><td>-7.99</td><td>-6.32</td><td>-4.99</td><td>-3.58</td><td>-0.79</td></tr><tr><td>α<sub>3,CL</sub></td><td>-4.54</td><td>-7.59</td><td>-5.56</td><td>-4.40</td><td>-3.47</td><td>-1.87</td></tr><tr><td>α<sub>1,EM</sub></td><td>-3.69</td><td>-7.32</td><td>-5.04</td><td>-3.62</td><td>-2.34</td><td>-0.51</td></tr><tr><td>α<sub>2,EM</sub></td><td>-4.36</td><td>-7.52</td><td>-5.82</td><td>-4.50</td><td>-3.06</td><td>-0.27</td></tr><tr><td>α<sub>3,EM</sub></td><td>-2.36</td><td>-5.40</td><td>-3.35</td><td>-2.25</td><td>-1.32</td><td>0.33</td></tr><tr><td>σ</td><td>3.28</td><td>1.80</td><td>2.41</td><td>3.01</td><td>3.84</td><td>5.64</td></tr><tr><td rowspan="7">Secondary infection</td><td>logit<sup>-1</sup>(β<sub>1,CL</sub>)</td><td>0.050</td><td>0.031</td><td>0.046</td><td>0.051</td><td>0.057</td><td>0.064</td></tr><tr><td>logit<sup>-1</sup>(β<sub>2,CL</sub>)</td><td>0.033</td><td>0.025</td><td>0.030</td><td>0.033</td><td>0.037</td><td>0.042</td></tr><tr><td>logit<sup>-1</sup>(β<sub>3,CL</sub>)</td><td>0.071</td><td>0.039</td><td>0.054</td><td>0.067</td><td>0.084</td><td>0.119</td></tr><tr><td>logit<sup>-1</sup>(β<sub>1,EM</sub>)</td><td>0.039</td><td>0.022</td><td>0.034</td><td>0.040</td><td>0.044</td><td>0.051</td></tr><tr><td>logit<sup>-1</sup>(β<sub>2,EM</sub>)</td><td>0.011</td><td>0.007</td><td>0.009</td><td>0.011</td><td>0.013</td><td>0.015</td></tr><tr><td>logit<sup>-1</sup>(β<sub>3,EM</sub>)</td><td>0.009</td><td>10<sup>-8</sup></td><td>10<sup>-5</sup></td><td>0.001</td><td>0.013</td><td>0.042</td></tr><tr><td>π</td><td>0.56</td><td>0.34</td><td>0.46</td><td>0.56</td><td>0.66</td><td>0.79</td></tr><tr><td rowspan="2">Over Infection</td><td>γ<sub>CL</sub></td><td>2.95</td><td>2.08</td><td>2.54</td><td>2.90</td><td>3.31</td><td>3.98</td></tr><tr><td>γ<sub>EM</sub></td><td>8.92</td><td>6.09</td><td>7.51</td><td>8.70</td><td>10.09</td><td>12.54</td></tr></tbody></table>

724

725
726 **Figure legends**

727 **Fig. 1.** Possible transitions from time *t*−1 to time *t* in year *i* between the 4 plant compartments (H: healthy, CL: infected by a classic strain, EM: infected by an emerging strain and D: infected by both a CL and an EM strain). **a)** The possible transitions of an H plant depend on the probabilities of 4 events: (i,ii) *p*<sub>*i*,t*</sub><sup>PI</sup> (CL|H) [resp. *p*<sub>*i*,t*</sub><sup>PI</sup> (EM|H)], probability that “an H plant is infected by primary infection with a CL [resp. EM] strain during *Δt*” and (iii,iv) *p*<sub>*i*,t*</sub><sup>SI</sup> (CL|H) [resp. *p*<sub>*i*,t*</sub><sup>SI</sup> (EM|H)], probability that “an H plant is infected by secondary infection of a CL [resp. EM] strain during *Δt*”. **b,c)** The possible transitions of a CL [resp. EM] plant depend on the probabilities of 2 events: (i) *p*<sub>*i*,t*</sub><sup>PI</sup> (*D*|CL) [resp. *p*<sub>*i*,t*</sub><sup>PI</sup> (*D*|EM)], probability that “a CL [resp. EM] plant is infected by primary infection with a EM [resp. CL] strain during *Δt*” and (ii) *p*<sub>*i*,t*</sub><sup>SI</sup> (*D*|CL) [resp. *p*<sub>*i*,t*</sub><sup>SI</sup> (*D*|EM)] probability that “a CL [resp. EM] plant is infected by secondary infection of an EM [resp. CL] strain during *Δt*”.

738

739 **Fig. 2.** Observed and predicted temporal dynamics of WMV epidemics in cucurbit plots from 740 2002 to 2004. The field contained 160 curcurbit plants. Blue, red and green dots indicate the 741 observation of the number of plants infected with a classical strain (CL), infected with an 742 emerging strain (EM) and infected with both strains (D) respectively. Lines indicate the mean 743 number of CL, EM and D plants respectively as predicted by the model. For each prediction, 744 the extent of a 90% posterior confidence interval is indicated.

745

746 **Fig. 3.** Mean annual probabilities of primary infection by CL and EM strains from 2002 to 747 2004. Boxes indicate the lower (25%) and upper (75%) quartiles, and symbols indicate the 748 mean values (stars for CL strain, dots for EM strain). Dotted lines extending from each end of 749 the box show the extent of the 90% posterior confidence interval.
750

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766

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774

**Fig. 4.** Simulated and observed dynamics of the proportion of EM strains of WMV following its introduction in a landscape. **a)** Dynamics of the proportion of EM strains of WMV during the 40 first years ($p_{EM}(i)$ with $i \in \{1..40\}$) after their introduction in a landscape composed of 25 fields, each field having 160 cucurbit plants. The introduction consists in putting, at the onset ($i=1, t_{intro}=1$) and in 1 field, a single plant infected with an EM strain. The dynamics of $p_{EM}(i)$ is indicated for 4 values of $\delta$, the ratio between $\gamma^S_{EM}$ and $\gamma^S_{CL}$ ($\delta=1$ in red, $\delta=3$ in black, $\delta=10$ in green and $\delta=40$ in blue). $\delta$ measures how many times it is less likely for a CL strain to infect an EM plant than for an EM strain to infect a CL plant. For each value of $\delta$, a sample of 20 independent realizations of the model are plotted (dotted lines), and the mean of $p_{EM}(i)$ indicated in solid line (the mean is assessed with the 1000 simulation realized). Others model parameters were set to the values detailed in the model description section. **b)** Annual dynamics of the proportion of EM strains detected in the sample collected during an epidemiological survey performed from 2004 to 2008 in the 3 neighbouring regions where EM strains were first detected in France.

**Fig. 5.** Effect of $\gamma^S_{EM}$, $\gamma^S_{CL}$ and their ratio $\delta$ on the final state of the landscape model 40 years after the introduction in 1 field out of 25 a single plant infected with an EM strain of WMV. $\gamma^S_{EM}$ [resp. $\gamma^S_{CL}$] measures how many times over-infecting an EM [resp. CL] plant with a CL [resp. EM] strain is less likely than infecting an H plant with a CL [resp. EM] strain. Their ratio $\delta= \gamma^S_{EM} / \gamma^S_{CL}$ is then the differential ability for over-infection between EM and CL strains. **a)** Probabilities that EM and CL still coexist in the landscape 40 years after the initial introduction as a function of $\gamma^S_{EM}$ and $\gamma^S_{CL}$. **b)** When EM and CL strains coexist at $y_{end}=40$, the mean value of $p_{EM}(y_{end})$ is plotted as a function of $\gamma^S_{EM}$ and $\gamma^S_{CL}$. **c)** Probabilities that EM strains disappeared in the landscape as a function of $\gamma^S_{EM}$ and $\gamma^S_{CL}$. **d)** Probabilities that EM
strains replaced CL strains in the landscape as a function of $\gamma^{s_{EM}}$ and $\gamma^{s_{CL}}$. For all graphs, model parameters other than $\gamma^{s_{EM}}$ and $\gamma^{s_{CL}}$ were set to the values detailed in the model description section. Probabilities were estimated based on 1000 independent realizations of the model. In graphs a) and b), the black dot is positioned at the value of $\gamma^{s_{EM}}$ and $\gamma^{s_{CL}}$ estimated from the field data set ($\gamma^{s_{EM}} = 9.19$ and $\gamma^{s_{CL}} = 2.89$).

780
781 **Appendix A. The model describing the co-dynamics of viral strains at field scale**

782

783 In the model, plants were classified into 4 compartments: (i) H (healthy), (ii) CL (infected by a classic strain), (iii) EM (infected by an emerging strain) and (iv) D (doubly infected by both a CL and an EM strain). For a given year *i* and date *t* of observation, $N(i,t)=[N_H(i,t), N_{CL}(i,t), N_{EM}(i,t), N_D(i,t)]$ is the vector of the number of plants in each compartment. The model describes the dynamic of $N(i,t)$ at a $\Delta t$ time interval equal to 1 week given that initially all plants are healthy: $N(i,0)=[N_{tot,0}, 0, 0]$ ($N_{tot}=160$).

789

790 **Probabilities of infection of an H plant from t-1 to t**

791 From time t-1 to time t, an H plant can remain H or become CL, EM or D with probabilities $p_{i,t}^M = (p_{i,t}(H_t | H_{t-1}), p_{i,t}(CL_t | H_{t-1}), p_{i,t}(EM_t | H_{t-1}), p_{i,t}(D_t | H_{t-1}))$. These transitions between compartments depend on the probabilities of 4 events (Fig. 1a).

793

794 $p_{i,t}^{PI}(CL|H)$ [resp. $p_{i,t}^{PI}(EM|H)$], are the probabilities that “an H plant is infected by primary infection with a CL [resp. EM] strain during $\Delta t$”. They are modelled as:

795

796 $\text{logit}[p_{i,t}^{PI}(CL|H)] = \alpha_{i,CL} + \gamma_{i,t} \quad (A1)$

797 $\text{logit}[p_{i,t}^{PI}(EM|H)] = \alpha_{i,EM} + \gamma_{i,t} \quad (A2)$

798 where (i) $\alpha_{i,CL}$ [resp. $\alpha_{i,EM}$] is, in the logit scale, a parameter describing the probability that an H plant is infected by primary infection with a CL [resp. EM] strain during $\Delta t$ in year $i$ if no variation between weeks exists and (ii) $\gamma_{i,t}$ is a set of random effects mutually independent and identically normally distributed with unknown variance $\sigma^2$.

802 $p_{i,t}^{SI}(CL|H)$ [resp. $p_{i,t}^{SI}(EM|H)$] are probabilities that “an H plant is infected by secondary infection of a CL [resp. EM] strain during $\Delta t$”. They depend on the probabilities $q_i(SI_{CL}|H)$

803
804 [resp. $q_i(SI_{EM} | H)$] that “a plant infected in the field with a CL [resp. EM] strain in year $i$ infects an H plant by secondary infection during $\Delta t$”. In the model, it is assumed that

$$806 \quad \operatorname{logit}[q_i(SI_{CL} | H)] = \beta_{i,CL} \quad (A3)$$

$$807 \quad \operatorname{logit}[q_i(SI_{EM} | H)] = \beta_{i,EM} \quad (A4)$$

808 where $\beta_{i,CL}$ [resp. $\beta_{i,EM}$] is, in the logit scale, a parameter modelling the probability that a plant infected with a CL [resp. EM] strain is a source of infection for an H plant by secondary spread during $\Delta t$ in year $i$.

811 Following the calculations detailed in appendix B, it comes that

$$812 \quad p_{i,t}^{SI}(CL | H) = 1 - [1 - q_i(SI_{CL} | H)]^{N_{CL}(i,t-1) + (1-\pi)N_D(i,t-1)} \quad (A5)$$

$$813 \quad p_{i,t}^{SI}(EM | H) = 1 - [1 - q_i(SI_{EM} | H)]^{N_{EM}(i,t-1) + \pi N_D(i,t-1)} \quad (A6)$$

814 where $\pi$ is a parameter modelling the mean proportion of EM strains in D plants ($1-\pi$ is the proportion of CL strains).

816 Finally, assuming that during $\Delta t$ primary and secondary infection events occur independently, the probability that, in year $i$, a plant is H [resp. CL, EM and D] at time $t$ given that this plant was H at time $t-1$ can be derived:

$$820 \quad p_{i,t}(H_t | H_{t-1}) = [1 - p_{i,t}^{PI}(CL | H)] \cdot [1 - p_{i,t}^{SI}(CL | H)] \cdot [1 - p_{i,t}^{PI}(EM | H)] \cdot [1 - p_{i,t}^{SI}(EM | H)] \quad (A7)$$

$$821 \quad p_{i,t}(CL_t | H_{t-1}) = [1 - p_{i,t}^{PI}(EM | H)] \cdot [1 - p_{i,t}^{SI}(EM | H)] \cdot [p_{i,t}^{PI}(CL | H) \cdot p_{i,t}^{SI}(CL | H) + p_{i,t}^{PI}(CL | H) \cdot (1 - p_{i,t}^{SI}(CL | H)) + (1 - p_{i,t}^{PI}(CL | H)) \cdot p_{i,t}^{SI}(CL | H)] \quad (A8)$$

$$822 \quad p_{i,t}(EM_t | H_{t-1}) = [1 - p_{i,t}^{PI}(CL | H)] \cdot [1 - p_{i,t}^{SI}(CL | H)] \cdot [p_{i,t}^{PI}(EM | H) \cdot p_{i,t}^{SI}(EM | H) + p_{i,t}^{PI}(EM | H) \cdot (1 - p_{i,t}^{SI}(EM | H)) + (1 - p_{i,t}^{PI}(EM | H)) \cdot p_{i,t}^{SI}(EM | H)] \quad (A9)$$

$$823 \quad p_{i,t}(D_t | H_{t-1}) = 1 - p_{i,t}(H_t | H_{t-1}) - p_{i,t}(CL_t | H_{t-1}) - p_{i,t}(EM_t | H_{t-1}) \quad (A10)$$
824

825

### Probabilities of over-infection of a singly infected plant from *t-1* to *t*

826

From time *t-1* to time *t*, a CL plant can remain CL or become D with probability $p_{i,t}(D_t|CL_{t-1})$ that depends on the probabilities of 2 events (Fig. 1b): (i) $p_{i,t}^{PI}(D|CL)$, the probability that “a CL plant is infected by primary infection with an EM strain during $\Delta t$” and (ii) $p_{i,t}^{SI}(D|CL)$, the probability that “a CL plant is infected by secondary infection of an EM strain during $\Delta t$”. As previously, deriving the latter probability required to introduce the probability $q_i(SI_{EM}|CL)$ that “a given plant infected in the field with an EM strain is the source of infection of a CL plant by secondary infection during $\Delta t$”.

827

828

829

830

831

832

833

Defining the parameter $\gamma_{CL}$ as the odd ratio between the probability of infecting an H plant with an EM strain and the probability of infecting an CL plant with an EM strain implies that

834

835

$$p_{i,t}^{PI}(D|CL) = \frac{\text{odds}[p_{i,t}^{PI}(EM|H)]}{\text{odds}[p_{i,t}^{PI}(EM|H)] + \gamma_{CL}} \quad (A11)$$

836

$$q_i(SI_{EM}|CL) = \frac{\text{odds}[q_i(SI_{EM}|H)]}{\text{odds}[q_i(SI_{EM}|H)] + \gamma_{CL}} \quad (A12)$$

837

As previously (Appendix B), it comes that

838

$$p_{i,t}^{SI}(D|CL) = 1 - [1 - q_i(SI_{EM}|CL)]^{N_{EM}(i,t-1) + \pi N_D(i,t-1)} \quad (A13)$$

839

Finally, assuming independence between primary and secondary infections, leads to

840

$$p_{i,t}(D_t|CL_{t-1}) = 1 - [1 - p_{i,t}^{PI}(D|CL)] \cdot [1 - p_{i,t}^{SI}(D|CL)] \quad (A14)$$

841

842

Similarly, an EM plant at time *t-1* can remain EM or become D at time *t* (Fig. 1c) with probability $p_{i,t}(D_t|EM_{t-1})$. The same developments as previously lead successively to

843

844

$$p_{i,t}^{PI}(D|EM) = \frac{\text{odds}[p_{i,t}^{PI}(CL|H)]}{\text{odds}[p_{i,t}^{PI}(CL|H)] + \gamma_{EM}} \quad (A15)$$
$$845 \qquad q_i(SI_{CL} | EM) = \frac{odds[q_i(SI_{CL} | H)]}{odds[q_i(SI_{CL} | H)] + \gamma_{EM}} \qquad (A16)$$

$$846 \qquad p_{i,t}^{SI}(D | EM) = 1 - [1 - q_i(SI_{CL} | EM)]^{N_{CL}(i,t-1) + (1-\pi)N_D(i,t-1)} \qquad (A17)$$

847 And

$$848 \qquad p_{i,t}(D_t | EM_{t-1}) = 1 - [1 - p_{i,t}^{PI}(D | EM)] \cdot [1 - p_{i,t}^{SI}(D | EM)] \qquad (A18)$$

849

850 **Dynamics of N(i,t) during an epidemic**

851 The dynamic of N(i,t) from t=0 to t_end is modelled as a sequential 3 step stochastic processes:

852 (i) $\Delta H(i,t)=[\Delta H_H(i,t), \Delta H_{CL}(i,t), \Delta H_{EM}(i,t), \Delta H_D(i,t)]$, the vector of the number of H plants in year i at date t-1 remaining H at date t or becoming respectively CL, EM or D at date t, is drawn from a multinomial distribution:

$$855 \qquad \Delta H(i,t) \sim \text{Multinomial}(N_H(i,t-1), p_{i,t}^M) \qquad (A19)$$

856 (ii) $\Delta CL(i,t)$, the number of CL plants in year i at date t-1 becoming D at date t is the result of a binomial process:

$$858 \qquad \Delta CL(i,t) \sim \text{Binomial}(N_{CL}(i,t-1), p_{i,t}(D_t | CL_{t-1})) \qquad (A20)$$

859 (iii) $\Delta EM(i,t)$, the number of EM plants in year i at date t-1 becoming D at date t is the result of a binomial process:

$$861 \qquad \Delta EM(i,t) \sim \text{Binomial}(N_{EM}(i,t-1), p_{i,t}(D_t | EM_{t-1})) \qquad (A21)$$

862 Then N(i,t) is updated from time t-1 to time t,

$$863 \qquad N(i,t) = \begin{pmatrix} N_H(i,t-1) - \Delta H_{CL}(i,t) - \Delta H_{EM}(i,t) - \Delta H_D(i,t) \\ N_{CL}(i,t-1) + \Delta H_{CL}(i,t) - \Delta CL(i,t) \\ N_{EM}(i,t-1) + \Delta H_{EM}(i,t) - \Delta EM(i,t) \\ N_D(i,t-1) + \Delta H_D(i,t) + \Delta CL(i,t) + \Delta EM(i,t) \end{pmatrix} \qquad (A22)$$

864

865
866 Appendix B: derivation of the probabilities of secondary infection

867 Suppose that each infected plant sends on another plant (through aphid population dynamics)
868 a Poisson number of viral particules with mean λ independent of the strain of the viral
869 particules. Suppose also that the frequency of viral particles of EM strain is π in doubly
870 infected plants. Thus a plant infected by the two viral strain sends a Poisson number of viral
871 particles of EM strain with mean π.λ and a Poisson number of particles of CL strain with
872 mean (1 − π).λ.

873 A given plant then receives a Poisson number of viral particles of EM strain with mean
874 Λ = (N<sub>EM</sub> + πN<sub>D</sub>).λ where N<sub>EM</sub> is the number of plants infected with EM strain only at the
875 preceding date and N<sub>D</sub> the number of plants doubly infected at the same date.

876 Let $p(EM,H)$ be the probability that a viral particle of EM strain is transmitted to an healthy
877 plant at a given date.

878 Suppose that this healthy plant receives $n$ particles of strain EM, and that each particle can be
879 transmitted independently from another particle. The probability that this plant is not infected
880 by these particles is $[1 - p(EM,H)]^n$.

881 A healthy plant receiving a Poisson number of viral particles of EM strain with mean
882 Λ = (N<sub>EM</sub> + πN<sub>D</sub>).λ remains uninfected by an EM strain with probability

$$
\begin{align*}
E\left[(1 - p(EM, H))^n\right] &= \sum_{n=0}^{\infty} \frac{(1 - p(EM, H))^n}{n!} \Lambda^n \exp(-\Lambda) \\
&= \exp(-p(EM, H) \cdot \Lambda) \\
&= \exp[-p(EM, H) \cdot (N_{EM} + \pi \cdot N_D) \cdot \Lambda]
\end{align*}
$$

884
885 Similarly, an healthy plant is infected by a given EM plant (so that N<sub>EM</sub> = 1 and N<sub>D</sub> = 0 in the
886 preceding formula) with probability

887
$$
q(SI_{EM} | H) = 1 - \exp(-p(EM, H) \cdot \lambda)
$$
Thus a healthy plant receiving a Poisson number of viral particles of EM strain with mean

$\Lambda = (N_{EM} + \pi N_{D}).\lambda$ remains uninfected by an EM strain with probability

$\exp[-p(EM, H).(N_{EM} + \pi N_{D})\lambda] = [1 - q(SI_{EM} | H)]^{N_{EM} + \pi N_{D}}$

and becomes EM infected with probability

$1 - [1 - q(SI_{EM} | H)]^{N_{EM} + \pi N_{D}}$
894 Appendix C: Model inferences

895 Link between the observed data and the model

896 During the first weeks until 100% of the plants became infected by WMV (week 3 in 2002 and 2004, and week 4 in 2003), maps of the epidemics were established at a weekly interval.

897 This is precisely the Δt time step adopted in the model. Thus, assuming that the DAS-ELISA and RT-PCR tests used to detect viruses have high levels of sensitivity and specificity, both 898 ΔH(i,t), ΔCL(i,t) and ΔEM(i,t) are directly observed. After this period of weekly data 899 collection, a last map of the epidemic is observed 4 weeks later (week 7 in 2002, week 8 in 900 901 2003 and 2004 denoted thereafter t<sub>end</sub>). Here, only N(i,t<sub>end</sub>) is observed. Given that no map 902 was observed in week t<sub>end</sub> -1 it is impossible to derive ΔH(i,t<sub>end</sub>), ΔCL(i,t<sub>end</sub>) and ΔEM(i,t<sub>end</sub>). 903 However, N(i,t<sub>end</sub>) can be related to the model output N(i,t<sub>end</sub>-1) by noting that 904

$$
905 N(i, t_{\text{end}}) \sim \text{Multinomial} \left( N_{\text{tot}}, \begin{pmatrix} p_i(H_{t_{\text{end}}}) \\ p_i(CL_{t_{\text{end}}}) \\ p_i(EM_{t_{\text{end}}}) \\ p_i(D_{t_{\text{end}}}) \end{pmatrix} \right)
$$

906 where the probabilities of being respectively H, CL, EM and D at t<sub>end</sub> during year i can be
907 derived using classical conditional probability rules given for example that to be H at time t a
908 plant must be H at time t-1 and not be infected from times t-1 to t:

909 $p_i(H_{tend}) = p_i(H_{tend-1}) \cdot p_i(H_{tend} | H_{tend-1})$

910 And, because $p(H_{t_{end}-1}) = \frac{N_H(i, t_{end}-1)}{N_{tot}}$, it comes

911 $p_i(H_{t_{end}}) = \frac{N_H(i, t_{end}-1)}{N_{tot}} \cdot p_i(H_{t_{end}} | H_{t_{end}-1})$

912 Similarly, it results that

913 $p_i(CL_{t_{end}}) = \frac{N_{CL}(i, t_{end}-1)}{N_{tot}} \cdot [1 - p_i(D_{t_{end}} | CL_{t_{end}-1})] + \frac{N_{H}(i, t_{end}-1)}{N_{tot}} \cdot p_i(CL_{t_{end}} | H_{t_{end}-1})$
$$914 \qquad p_i(EM_{t_{end}}) = \frac{N_{EM}(i, t_{end}-1)}{N_{tot}} \cdot [1 - p_i(D_{t_{end}} | EM_{t_{end}-1})] + \frac{N_H(i, t_{end}-1)}{N_{tot}} \cdot p_i(EM_{t_{end}} | H_{t_{end}-1})$$

$$915 \qquad p_i(D_{tend}) = 1 - p_i(H_{tend}) - p_i(CL_{tend}) - p_i(EM_{tend})$$

These equations defined a Markov chain linking dates t—1 to t. The likelihood at $t_{end}$, given the last observation realized in the field, is obtained by numerical integration of all the possible states of $N(i, t_{end}-1)$ by simulation of this Markov chain.

## Bayesian statistical inferences

Statistical inferences were performed via Bayesian Monte Carlo Markov Chain (MCMC) methods (Gelman et al. 2004) with OpenBUGS® 3.0.3 (Bayesian inference Using Gibbs Sampling) software (available from http://www.mrc-bsu.cam.ac.uk/bugs/) (Thomas et al. 2006).

The vector of model parameters is $\theta=[ \alpha_{CL,i}, \alpha_{EM,i}, \beta_{CL,i}, \beta_{EM,i}, \gamma_{EM}, \gamma_{CL}, \pi, \sigma ]_{i=1..3}$. To reflect our initial lack of knowledge we used uninformative prior probability density function (PDF) for $\theta$. Log-uniform[10⁻³, 10³] probability distributions were assigned to $\gamma_{CL}$ and $\gamma_{EM}$. Normal(0,100) probability distributions were assigned to $\alpha$'s and $\beta$'s parameters, a Beta(1,1) probability distribution assigned to $\pi$ and a half positive Normal(0,100)I(0,+∞) assigned to $\sigma$. To check for convergence of the MCMC chain to its ergodic target distribution, we ran 3 chains and used the Gelman-Rubin diagnostics implemented in OpenBUGS® 3.0.3 (Brooks and Gelman 1998). Marginal posterior PDFs were summarised by statistics computed from the MCMC samples: mean, median, 2.5th and 97.5th percentile defining the 95% Bayesian posterior credibility interval. From the joint posterior distribution of model parameter, derived quantities such as the mean probability that an H plant is infected by primary or secondary infection are easy to obtain in OpenBUGS.
To check for consistency between the model and the data, the $\chi^2$ discrepancy test proposed by Gelman (2004, p 175) was used (see also Rivot et al. (2008) for details on its practical implementation). $\chi^2$ discrepancy statistics were computed for the observation of $N_{CL}(i,t)$, $N_{EM}(i,t)$, $N_D(i,t)$ summed over the available observation units i and t. Extreme Bayesian p values (<0.05 or > 0.95) reveal inconsistencies between model predictions and the actual data.

Simple predicted versus observed values of these variables were also regressed and the coefficient of determination ($r^2$), which provides a measure of the proportion of the total variance explained by the model, was estimated.

In order to investigate more precisely the role of the process of over-infection, a model comparison approach was used. Let's denote $M_0$ the previous model parameterisation. Two alternative formulations of $M_0$ were fitted to the dataset. In the model $M_1$, it is assumed that $\gamma_{EM}=\gamma_{CL}$ (i.e. the probability of over-infecting an already infected plant differs from the probability of infecting an H plant but does not depend on the strain (CL or EM) first inoculated). In the model $M_2$, it is assumed that $\gamma_{EM}=\gamma_{CL}=1$ (i.e. the probability of over-infecting an already infected plant is the same as the probability of infecting an H plant).

Models $M_1$, $M_2$ and $M_0$ were compared using Bayes factors. The Bayes factor of 2 alternative models $M_0$ and $M_1$, $BF_{1,0}$, is defined as the ratio of the marginal likelihood of the data under model $M_1$, [$y|M_1$], to that obtained under model $M_0$, [$y|M_0$]. It is a measure of the relative credibility of $M_1$ compared with $M_0$ for given dataset. In practise, [$y|M_i$] can be approximated as the harmonic mean of the likelihood function (Kass and Raftery, 1995). These approximations of Bayes factor are accurate enough to be interpreted in a logarithm scale according to guidelines provided by Kass and Raftery (1995).
<figure><img src="image_3.png" /></figure>

<figure><img src="image_4.png" /></figure>

<figure><img src="image_5.png" /></figure>
Year: 2002

<figure><img src="image_6.png" /></figure>

Year: 2003

<figure><img src="image_7.png" /></figure>

Year: 2004

<figure><img src="image_8.png" /></figure>
<figure><img src="image_9.png" /><figcaption>4. Figure</figcaption></figure>
<figure><img src="image_10.png" /><figcaption>(a)</figcaption></figure>

Years after initial introduction of an EM strain

<figure><img src="image_11.png" /><figcaption>(b)</figcaption></figure>

Year of sample collection
<figure><img src="image_12.png" /><figcaption>4. Figure</figcaption></figure>

<figure><img src="image_13.png" /></figure>

<figure><img src="image_14.png" /></figure>

<figure><img src="image_15.png" /></figure>