Modeling of the invasion of a fungal disease over a vineyard

Transcription

1 1 Modeling of the invasion of a fungal disease over a vineyard J-B. Burie 1, A. Calonnec 2, and M. Langlais 1 1 UMR CNRS 5466, Mathématiques Appliquées de Bordeaux INRIA futurs, Equipe Anubis case 26, Université Victor Segalen Bordeaux rue Léo Saignat, 3376 Bordeaux Cedex, France burie@sm.u-bordeaux2.fr 2 INRA-CR de Bordeaux UMR INRA-ENITA en Santé végétale BP Villenave d Ornon calonnec@bordeaux.inra.fr Summary. The spatiotemporal spreading of a fungal disease over a vineyard is investigated using a SEIR-type model coupled with a set of partial differential equations describing the dispersal of the spores. The model takes into account both short and long range dispersal of spores and growth of foliar surface. Results of numerical simulations are presented. A mathematical result for the asymptotic behavior of the solutions is given as well. Key words: SEIR model, dispersal, diffusion, large time behavior 1.1 Introduction Integrated pest management offers an attractive alternative to routine chemical application by treating only in response to disease risk indicators. Powdery mildew, caused by the fungus Uncinula necator, is the most economically important and wide spread disease of grapevine. For this disease, the main factor of risk is a timing of the attack early in the season combined to the phenological stage of the host. The leaves are the first to be infected, and there is a spatial relationship between maps of frequency of leaves diseased early in the season with maps of frequency of bunches with high severity [8, 4]. A better knowledge of the mechanisms of the disease propagation could help to improve its control at the plot scale by tailoring treatments to local specific needs, or at the estate scale by treating only specific plots. We aim at building a mathematical model of this fungal epidemic with a particular stress upon the dispersal mechanism of the spores produced by the

2 2 J-B. Burie et al. colonies of fungus. Already a lot of work has been done upon the subject of dispersal for various kinds of dispersers such as animals, seeds, spores... (see e.g. [1, 11, 6] and references therein). In particular, we want to investigate the role of a dual dispersal mechanism in which the spores produced may either disperse inside the vine stock and germinate near the colony (short range dispersal) or may be lifted up above the vine rows and fall far from the colony (long range dispersal). Our goal is to build a model which is a simpler version, and as a consequence easier to analyse, of a much more elaborate one: [5]. This latter model couples a mechanistic model for the growth of each vine stock in the vineyard with a dispersal model using ray tracing like techniques at the vine stock scale and a distribution law at the vineyard scale for the spores escaping the vine stock. In [13], the authors considered a 2D spatial model based upon such a dual dispersal mechanism using diffusion theory coupled with a Vanderplank equation [12]. Using this Vanderplank equation leads to delay equations that complexify the mathematical analysis of the model. Instead, in this paper, we will use a SEIR compartmental model like in classical epidemiology (see e.g. [1, 3]) to take into account the local extension of the disease. In the non spatial case, a comparison between these two approaches can be found in [9]. During an epidemic lasting a whole season from bud break until grapes maturation, the growth of the host cannot be neglected. We include a description of the host growth in our model. We also take into consideration the specific spatial organisation of vineyards that are made of several separate rows. This paper is organised as follows. After having described the model, we perform a mathematical analysis and present numerical simulations. 1.2 The model The vector of a fungal disease are the spores produced by the colonies of fungus that lie on the vegetal tissue which may be leaves, buds, fruits... We assume for simplicity that the time variation of the surface of a colony can be neglected. Then as in [9] we consider the unit of disease to be a colony and for the host to be a site, that is the surface occupied by a colony. The cycle of the epidemic is as follows: when spores fall upon the vegetal tissue, they may create a new colony which will produce spores after some latency period and during some sporulating period. Let Ω be a regular 2D spatial domain. Let t be the time and x denote the position of some point in Ω. We will use the following notations for the state variables. Like in the case of a SEIR model, the total density N of sites suceptible to host a colony of fungus at (x, t) is subdivided into healthy H, latent L, sporulating I and removed (postinfectious) R.

3 1 Modeling of the invasion of a fungal disease over a vineyard 3 We want to devise a model that takes into account multiple ranges of dispersal for the spores in order to investigate their different roles for the spreading of the epidemic. Spores may disperse separately or as infection units (packages of spores). For simplicity, we only take into account two ranges for dispersal: short range (spores disperse inside the vine stock where they come from), and a longer range (spores disperse at the vineyard scale). Let S(x, t) denote the density of spores produced by the colonies. The spores total density S is subdivided acccording to the range of dispersal, short range dispersal spores density S 1 and longer range one S 2. They are produced by a sporulating colony with rate r p > and may disperse at short range with a constant probability F [, 1] and at longer range with probability (1 F ). We assume that the spores disperse according to a diffusion process with Fickian diffusion coefficient D 1 > (short range) or D 2 > D 1 > (longer range) as in [13]. Using Fickian diffusion for long range dispersal may seem unrealistic at first. But the spores are not necessarily taken away along dominating wind directions. The dispersal is also due to turbulence that provides the energy to tear off the spores from the leaves. Spores fall upon the vineyard with some deposition rate δ 1 > or δ 2 >, we will set δ 1 = δ 2 in numerical simulations. We thus find the first set of equations of our model that describes the production of spores by the colonies and their dispersal: S 1 t (x, t) =. (D 1 S 1 (x, t)) δ 1 S 1 (x, t) + r p F I(x, t) S 2 t (x, t) =. (D 2 S 2 (x, t)) δ 2 S 2 (x, t) + r p (1 F )I(x, t) (1.1) for x Ω and t >. Moreover, we assume that no spores come from outside the vineyard. The spores produced by the fungus colonies should freely escape from the vineyard. To simulate this, we choose a computing domain Ω with vine rows located at the center and surrounded with by a region with no vines. Then, if Ω is large enough with respect to diffusion coefficients, spores do not reach the boundary and their densities at these points should be equal to. Thus, we impose Dirichlet conditions on the boundary S 1 (x, t) = S 2 (x, t) = for x Ω and t > (1.2) We also set nonnegative initial conditions S 1 (x, ) = S 1(x), S 2 (x, ) = S 2(x) for x Ω (1.3) Let Ω r Ω denote the area covered by the vine rows. We devise our model in such a way that for all t > and x Ω, N(x, t) equals if x Ω r. The powdery mildew epidemic has no impact upon the growth of the host. This growth brings new sites available for colonization. We study the epidemic during one single season, then we assume that the time variation of the total number of colony sites inside the rows obeys a logistic law

4 4 J-B. Burie et al. N (x, t) = rn(x, t) t ( 1 ) N(x, t), for x Ω r (1.4) K where r > is the growth rate and K > the carrying capacity. Although r and K are constant for simplicity, we could introduce spatial heterogeneities for the host growth assuming r and K depend on x. Provided r, K are bounded, our results can be easily extended to handle this. Next the local evolution of the disease at some point x Ω r (inside a row) obeys the classical SEIR model, whereas we set N(x, t) = L(x, t) = I(x, t) = R(x, t) = for t if x Ω r. Let p and i denote the mean duration of the latency and infectious period respectively. Let E be the inoculum effectiveness (probability for the spores to succed into creating a new colony upon a site). Taking into account (1.4), this yields the second set of equations of our model for x Ω r : ( ) H t (x, t) = E(δ 1S 1 (x, t) + δ 2 S 2 (x, t)) H(x,t) N(x,t) + rn(x, t) 1 N(x,t) K L t (x, t) = +E(δ 1S 1 (x, t) + δ 2 S 2 (x, t)) H(x,t) N(x,t) 1 pl(x, t) I t (x, t) = + 1 p L(x, t) 1 i I(x, t) R t (x, t) = + 1 i I(x, t) (1.5) supplemented with nonnegative initial conditions H(x, ) = H (x), L(x, ) = L (x), I(x, ) = I (x), R(x, ) = R (x) for x Ω r (1.6) The contact term in (1.5) is based upon a proportionate mixing assumption. Though our model includes host growth, this assumption is in agreement with the underlying hypothesis of classical epidemiologic models in phytopathology (see Vanderplank [12]) that states that the rate of increase of diseased tissue is proportional to the amount of spores multiplied by the probability that these spores fall upon healthy tissues. A similar approach for including host growth in a model of phytopathology but with non spatial delay equations can be found in [2]. 1.3 Theoretical results We have the following existence result for our model. Theorem 1. The system (1.1),(1.5) is well posed: let H, L, I,R be in L (Ω) and S 1, S 2 be in L 2 (Ω), the system posseses a unique componentwise nonnegative solution that exists globally in time.

5 1 Modeling of the invasion of a fungal disease over a vineyard 5 Proof. The proof of this theorem follows standard arguments (see e.g. [7]) and will not be detailed here. The large time behavior of the solutions can be described as follows. Theorem 2. If the hypothesis of the previous existence theorem are satisfied, then as t goes to infinity, S 1 (x, t) and S 2 (x, t) converge to in the L 2 (Ω) and H 1 (Ω) norms. And there are nonnegative functions H and R such that for all x Ω r, H (x) + R (x) = K and lim H(x, t) = H (x) t + lim t + L(x, t) = lim lim R(x, t) = R (x) t + I(x, t) = t + This result must be carefully interpreted since our model is valid for only one single season. It means that the epidemic finally dies out at the end of the season when the growth of the host is achieved. Proof. It is easy to show that for all x, t, N, H, L, I, R are nonnegative and bounded by K. As N t = rn(1 N/K) we have lim t + N(x, t) = K. From (1.5) we have R I, hence for all x, R(x, t) converges to t = 1 i some limit R (x) as t goes to infinity. Moreover I(x, t) L 1 (, + ) for all x. Next, as I t = + 1 p L 1 I i I, we also have t (x, t) L (, + ) for all x. This classicaly proves that for all x, lim t + I(x, t) =. By integrating over Ω the last two equations in (1.5) we similarly prove that lim t + Ω I(x, t) =. And as I(t, x) is bounded by K uniformly in x, t, for any integer p 1, we also have lim t + Ω Ip (x, t) =. Let. 2 denote the L 2 (Ω) norm. Multiplying the first equation in system (1.1) by S 1, integrating over Ω and using oung s inequality, we find d S dt + D 1 S δ 1 S c I 2 where c is some constant. Since lim t + Ω I2 (x, t) = and δ 1 >, it is easy to prove that lim t + S 1 2 (t) =. Similarly, by multiplying the first equation in system (1.1) by 2 S 1, we have lim t + S 1 2 (t) = and the same results hold true for S 2. Adding the last three equations of (1.5), since S 1 and S 2 are nonnegative, D = L + I + R is increasing with respect to t and converges to some limit. We thus find that L(x, t) converges to some limit L (x) as t goes to infinity. And as N = H + L + I + R converges to K, H(x, t) converges to some limit H (x) as well.

6 6 J-B. Burie et al. It remains to prove that L (x) =. Since rn(1 N/K) = N adding the first two equations of (1.5) and integrating with respect to t yields: H(x, t) + L(x, t) + 1 p t L(x, s) ds + N(x, ) = H(x, ) + L(x, ) + N(x, t) Hence, L(x, t) L 1 (, + ) for all x, hence for all x, L (x) =. We also give a threshold condition for successful establishment of the disease but only in the non spatial case. We thus investigate the linear stability of the non trivial equilibrium point (S 1, S 2, H, L, I, R) = (,, K,,, ) of the set of differential equations (1.1) (with. (D i S i (x, t)) = ) coupled with equations (1.5). Thanks to the Routh-Hurwitz criterium, we find the following basic reproductive rate of the disease: R = Er p i, and the threshold condition is R > 1. The biological interpretation is straightforward, R equals the number of spores r p produced by a single colony during the duration of the infectious period i multiplied by the probability that these spores succeed into creating a new colony. 1.4 Numerical experiments An example of field data is available (Calonnec et al., personnal communication) of a powdery mildew epidemic over a 5 rows vineyard. It shows that without fungicide treatment the disease invades all the vineyard within 3 months. We make a simulation of this particular vineyard. Each row is 66 m long and.5 m wide, and the distance between two rows is 1.5 m. We choose a rectangular computing domain Ω such that the 5 rows are located at the center of the domain and that Ω is 3 times larger than the vineyard. As mentioned before, doing so the Dirichlet conditions at the boundary of Ω describe the fact that the spores may freely disperse out of Ω. Parameters of the model as well as roughly realistic values are listed in Table 1.1. With these parameters, the basic reproductive rate of the disease in the homogeneous case is R = 1. We now explain how the values of the dispersal parameters δ a = δ 2 and D i were estimated. All spores lifted up in the atmosphere fall within half an hour so the deposition rates δ are more or less equal to 5 day 1. To estimate the diffusion coefficients D 1 and D 2, we focus on the spores dispersal mechanism alone. Let D a diffusion coefficient and δ a deposition rate, the density S of spores dispersed in the atmosphere and produced by a single source obeys the following equation: { S t (x, t) =. (D S(x, t)) δs(x, t), (x, t) R2 R + S(x, ) = Dirac(x), x R 2

7 1 Modeling of the invasion of a fungal disease over a vineyard 7 where Dirac(x) is the Dirac function. Then the total amount of fallen spores upon the vineyard at some point x R 2 is d(x) = + δs(x, t) dt, d(x) is the the probability density of fallen spores. It can be explicitly computed and its variance is σ = D/δ. The values of D 1 and D 2 in table 1.1 have been chosen so that σ =1 m for the short range dispersal, and σ =2 m for the long range dispersal. parameter description Table 1.1. Model parameters value δ 1 short range deposition rate 5 day 1 δ 2 long range deposition rate 5 day 1 D 1 short range diffusion coefficient 5 m 2 day 1 D 2 long range diffusion coefficient 2 m 2 day 1 r p spores production 1 4 spores day 1 colony site 1 F short range vs. long range dispersion.8 E inoculum effectiveness.1% p latency period duration 1 days i infectious period duration 1 days K carrying capacity of the colony sites 4 m 2 colony sites r growth rate of the colony sites.1 day 1 We start the infection at t = with one latent colony at the center of the vineyard over one vine stock. For simplicity we take an initial uniform site density for all the vine stocks. Hence the initial conditions are H (x) = 4 m 2 colony sites and L (x) = I (x) = R (x) = for x in the rows except for x [ 1/4; 1/4] 2 where H (x) = I (x) = R (x) = and L (x) = 4 m 2 colony sites density. We also set S1 (x) = S 2 (x) = for all x. The duration of the simulation is 9 days. Results of the simulation for these parameters are displayed on Fig They show the proportion of diseased colony sites with respect to spatial location P (x) = D(x)/N(x) = (L(x)+I(x)+R(x))/N(x)) 3, 6 and 9 days after the beginning of the infection (Fig ). The epidemic first invades the central row of the vineyard (day 3) then it reaches the other rows until almost all the vineyard has been contaminated at day 9. We also display short and long range spores density with respect to the spatial location at day 9 on Fig. 1.4 and 1.5. Short range spores mostly stay over the row where they are produced whereas the distribution of long range spores is more uniform. The lower spore density in the central row is due to the fact that the corresponding colonies have attained the postsporulating phase.

8 8 J-B. Burie et al. proportion of diseased surface at day Fig Proportion of diseased colony sites in the vineyard at day 3 proportion of diseased surface at day Fig Proportion of diseased colony sites in the vineyard at day 6

9 1 Modeling of the invasion of a fungal disease over a vineyard 9 proportion of diseased surface at day Fig Proportion of diseased colony sites in the vineyard at day 9 density of spores S1 (short range) at day Fig Density S 1 of short range dispersal spores over the vineyard at day 9

10 1 J-B. Burie et al. density of spores S2 (long range) at day Fig Density S 2 of long range dispersal spores over the vineyard at day 9 Finally we investigate the influence of the parameter F over the intensity of the epidemic keeping other parameters of the simulation at the same values as above. If F = only long range dispersion takes place. The proportion of diseased colony sites is displayed upon Fig.1.6 at day 9. Compared with Fig.1.3, the disease intensity is very low upon each row. If F = 1 only short range takes place. As shown upon Fig.1.7, the epidemic has attained its maximum intensity but only in the main part of the central row whereas the other rows have not been contaminated. As pointed out in [11, 13], the rate of expansion of the epidemic needs both short and long range dispersal f its vectors to reach an optimal value. This is even more evident in the case of separate rows of vine: without long distance dispersal, the disease hardly reaches the rows where the initial contamination did not take place, while whithout short distance dispersal, local extension of the disease is not strong enough to ensure a high level of contamination. 1.5 Conclusion We designed a mathematical model for a fungal disease of the vine. It takes into account the host growth occuring during the epidemic and a dual dispersal mechanism of the spores together with the spatial organization in rows of vine of vineyards. We were able to give an existence result for the solutions of the model, a description of the long time behavior of the solutions and a threshold theorem in the homogeneous case.

11 1 Modeling of the invasion of a fungal disease over a vineyard 11 proportion of diseased surface at day Fig Proportion of diseased colony sites at day 9 - long range dispersal only proportion of diseased surface at day Fig Proportion of diseased colony sites at day 9 - short range dispersal only

12 12 J-B. Burie et al. Numerical simulations show how short and long range dispersal interact with the row structure of the vineyard to allow the epidemic to reach an optimal rate of infection. In future work, we plan to investigate the existence of traveling waves for this kind of model and, if they exist, to find how the different parameters of the model influence the speed of these waves. We also plan to design a more mechanistic and biologically realistic description of the dispersal mechanism which would allow us to compare our results with field data available and perform a parameter identification. References 1. Bailey, N. T. J.: The mathematical theory of infectious disease and its applications. Second edition, Hafner Press, New ork (1975) 2. Blaise, P., Gessler, C.: An extended Progeny Parent Ratio Model. 1. Theoretical Development, Journal of Phytopathology - Phytopathologische eitschrift, 134, (1992) 3. Brauer, F., Castillo-Chavez, C.: Mathematical Models in Population Biology and Epidemiology, Texts in Applied Mathematics 4. Springer Verlag, Berlin Heidelberg New ork (21) 4. Calonnec, A., Cartolarao, P., Deliere, L., Chadoeuf, J.: Powdery mildew on grapevine: effect of the date of primary contamination on the disease development on leaves and the damages on grape. Proceedings of Organisation for biological and integrated control of noxious animals and plants Workshop, Brescia, Italy, in press. 5. Calonnec, A., Latu, G., Naulin, J., Roman, J., Tessier, G.: Parallel Simulation of the Propagation of Powdery Mildew in a Vineyard. Proceedings of Euro-Par 25, Lecture Notes in Computer Science, 3648, (25) 6. Clark, J. S., Fastie, C., Hurtt, G., et al: Reid s paradox of rapid plant migration, Bioscience, 48, (1998) 7. Henry, D.: Geometric Theory of Semilinear Parabolic Equations, Lecture Notes in Mathematics 84. Springer-Verlag, Berlin Heidelberg New ork (1981) 8. Peyrard, N., Calonnec, A., Bonnot,,F., Chadoeuf, J.: Explorer un jeu de données sur grille par tests de permutation. Revue Statistique Appliqué, LIII, (25) 9. Segarra, J., Seger, M. J., van den Bosch, F.: Epidemic dynamics and patterns of plant diseases, Phytopathology, 91, (21) 1. Shigesada, N., Kawasaki, K.: Biological invasions: Theory and practice. Oxford Series un Ecology and Evolution. Oxford University Press, Oxford (1997). 11. Shigesada, N., Kawasaki, K.: Invasion and the range expansion of species: effects of long-distance dispersal. In: Bullock, J., Kenward, R., Hails, R. (eds) Dispersal Ecology, pp , Blackwell Science, Malden MA (22) 12. Vanderplank, J. E.: Plant Diseases: Epidemics and Control. Academic Press, New ork (1963) 13. awolek, M. W., adoks, J. C.: Studies in focus development: An optimum for the dual dispersal of plant pathogens. Phytopathology, 82, (1992)