The Model

The vectors 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, etc. 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 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 ^ be a regular 2D spatial domain. Let t be the time and let x denote the position of some point in We will use the following notation for the state variables.

As in the case of a SEIR model, the total density N of sites susceptible to host a colony of fungus at (x, t) is subdivided into healthy H, latent L, sporulating I and removed (postinfectious) R.

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: a 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; the short range dispersal spore density S1 and the longer range one S2. They are produced by a sporulating colony with rate rp > 0 and may disperse at short range with a constant probability F e [0, 1] and at longer range with probability (1 — F).

We assume that the spores disperse according to a diffusion process with Fickian diffusion coefficient D1 > 0 (short range) or D2 > D1 > 0 (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 ¿1 > 0or ¿2 > 0; we will set ¿1 = ¿2 in the 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:

^(x, t) = V. (D1VS1 (x, t)) — ¿1S1 (x, t) + rpFI(x, t)

(x, t) = V. (D2VS2(x, t)) — ¿2S2(x, t) + rp(1 — F)I(x, 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 Q with vine rows located at the center and surrounded by a region with no vines. Then, if Q is large enough with respect to diffusion coefficients, spores do not reach the boundary and their densities at these points should be equal to 0. Thus, we impose Dirichlet conditions on the boundary

S1 (x, t) = S2(x, t) = 0 for x e 9Q and t > 0. (2.2)

We also set nonnegative initial conditions

S1 (x, 0) = S0(x) > 0, S2(x, 0) = S0(x) > 0 for x e Q. (2.3)

Let Qr c Q denote the area covered by the vine rows. We devise our model in such a way that for all t > 0and x e Q, N(x, t) equals 0 if x g Qr.

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

where r > 0 is the growth rate and K > 0 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 e Qr (inside a row) obeys the classical SEIR model, whereas we set N(x, t) = L(x, t) = I(x, t) = R(x, t) = 0 for t > 0if x e Qr .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 succeed in creating a new colony upon a site). Taking into account (2.4), this yields the second set of equations of our model for x e Qr:

' fr(x, t) = -E(SiSi(x, t) + 52S2(x, t))fgg + rN(x, t) (l - ^jA)

dL(x, t) = +E(SiSi (x, t) + S2S2(x, t))H0, - pL(x, t) < N (x,) P (2.5)

^ dR(x, t) = +i/(x, t) supplemented with nonnegative initial conditions

H(x, 0) = H0(x) > 0, L (x, 0) = L0(x) > 0, / (x, 0) = /0(x) > 0, R (x, 0) = R0(x) > 0 for x e Qr (2.6)

The contact term in (2.5) 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 Vander-plank [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 nonspatial delay equations can be found in [2].

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