Spatial Spread Model
Theoretical models can help us investigate situations that might otherwise be unsafe or impossible to
execute. In our project design, we were left with a lot of questions regarding the effectiveness of the
Xylencer phages and how to apply them. To answers these questions we made a spatial model that can predict
the spread and effectiveness of the Xylencer phages over time.
To make this theoretical model we first made a theoretical model for the interaction between
Xylella
fastidiosa and the phages. Next, we looked at existing models for the spatial spread of
X.
fastidiosa and recreated a theoretical model to gain insights into how the pathogen spreads itself.
At last,
we were able to combine these theoretical models and generate a model for the spread of the
Xylencer phages and their effectiveness at combatting X. fastidiosa. This model showed us that the
Xylencer
phages are very effective at removing X. fastidiosa from an infected area.
Introduction
Enabling the Xylencer phages to spread themselves is an important part of the Xylencer solution. The sheer number of trees in a vineyard or olive groove would make it impossible to treat all of them individually without it being extremely laborious. On top of that, the asymptomatic plants surrounding the trees will still be infected with X. fastidiosa. This means there will be a continuous source of reintroduction of X. fastidiosa making it impossible to eradicate this pathogen from an infected area.
However, this leaves us with a lot of questions regarding the efficiency of this solution that we designed. Field testing of the Xylencer phages on this scale requires many smallscale experiments in controlled environments before a large scale test would be feasible, if even allowed by current safety regulations. However, it is essential to gain insight into the application before this time, and therefore we developed a theoretical model that can help us find answers on the spreading behavior of the Xylencer phages. Since the discovery of X. fastidiosain Europe in 2013 the construction of spatial spread models has started to predict the effectiveness of control measures to be implemented. Particularly how far the phages would spread is of interest to our project. How big of a range can the phages cover once applied? And for what period will the trees be protected? Also not every farmer might want to apply the Xylencer phages and it is necessary to know if they might accidentally spread to these farms (Read more about this on our Ethics page). To predict how far the phages would spread a better understanding is needed of their spatialspread behavior.
In this subproject, a theoretical model was created based on a model constructed by the EFSA[1] on the spread of X. fastidiosa in the Apulia region. In this model spatial spread model the Xylencer phages had to be incorporated. This way their spreading behavior can be predicted and analyzed to help find the best way of application.
General Approach
To obtain a spatial spread model for the Xylencer phages, two models were combined: a theoretical model on the interaction of the Xylencer phages and their host X. fastidiosa and a theoretical model on the spread of X. fastidiosa. To model the interaction between phages and their respective hosts a LotkaVolterra approach can be used. This was also applied to X. fastidiosa and its phages using the data form a previous characterization of these phages [2]. Furthermore, a model based on the EFSA model is created to predict the spreading of X. fastidiosa. These two models can be combined to create a theoretical model for the spread of the Xylencer phages. In the following sections, these models will be discussed in more detail, outlining how they were built, where the parameters came from and what assumptions have been made.
Interaction of host and phages
To model the interaction of the phages and X. fastidiosa we made use of one of the most established models for ecological systems, the LotkaVolterra model [3]. This model is typically used to describe predatorprey interactions and thus also applicable to phages and bacteria where the bacteria are the prey and phages the predator. Below we describe how we came to the equations and parameters of this model, as well as the assumptions made.

\(\frac{dX}{dt}= \mu \cdot X \cdot (1X)  k_{1} \cdot X \cdot P \)\(\frac{dP}{dt}= k_{2} \cdot P + k_{3} \cdot k_{1} \cdot X \cdot P\)
Table 1: Overview of parameters of the LotkaVolterra model. Parameter Meaning Value Unit Justification \(\mu\) growth rate of phages 0.02 day ^{1} Based on the growth rate used in the EFSA model [1] \(k_{1}\) infection rate of phages 6.2352 E9 ml cell^{1} day ^{1} As characterized by Ahern et al. [2] \(k_{2}\) decay rate of the phages 0.0165 day ^{1} In plants the phages showed a halftime of 42 days [4], this was used to calculate the decay rate in the plant. \(k_{3}\) burst size of the phages 100 PFU cell ^{1} As characterized by Ahern et al. [2]

How did we come to these equations? arrow_downward
The LotkaVolterra model makes use of the following equations where X is the prey (X. fastidiosa) and P is the predator (the phages).
\(\frac{dX}{dt}= \mu \cdot X  k_{1} \cdot X \cdot P\)
\(\frac{dP}{dt}= k_{2} \cdot P + k_{3} \cdot k_{1} \cdot X \cdot P\)One of the assumptions this model makes is that the growth of prey is unbounded, which is not the case. Therefore, the model had to be adapted to incorporate a carrying capacity for X. fastidiosa To do this, the growth of X. fastidiosa is multiplied by one minus itself divided by the maximum bacterial load to incorporate the maximum carrying capacity of X. fastidiosa in the model. Applying this to the previous equations gives us the final differential equations for the model:
\(\frac{dX}{dt}= \mu \cdot X \cdot (1 \frac{X}{max X})  k_{1} \cdot X \cdot P\)
\(\frac{dP}{dt}= k_{2} \cdot P + k_{3} \cdot k_{1} \cdot X \cdot P\) 
What assumptions did we make? arrow_downward
 The infection rate of the phages is the term by which the growth rate of X. fastidiosa is reduced in proportion to the number of phages and X. fastidiosapresent. This is based on the adsorption rate constant of the phages. This is represented in the \(k_{1} \cdot X \cdot P\) term.
 In the absence of X. fastidiosa the phages undergo exponential decay, this is expressed in the decay rate of phages (\(k_{2}\)).
 The growth rate of the phages is proportional to the amount of X. fastidiosa and phages present. It can be expressed as the adsorption rate constant times the burst size multiplied by the number of phage and X. fastidiosa.
Results
The following graph shows the interaction of X. fastidiosa and its associated phages.
As the graph above shows the phages will be able to reduce the X. fastidiosa bacterial load successfully in just a few days. Also, a high concentration of phages will be left behind to give lasting protection to the plant.
Spatial spread of X. fastidiosa
To predict the spread of X. fastidiosa we created a spatial spread model based on a model made by EFSA[1]. A spatial spread model shows the spreading behavior over an area and is often used to predict the spreading of infections. Since the introduction of X. fastidiosa in Europe, several models have been made to predict the way X. fastidiosa would spread and whether the control measures applied would be effective enough [5], [6]. These measures are mainly focused on killing the insect vector and removal of host plants.
The model makes use of an evenly spaced grid in which each point represents a tree in a simulated olive grove. We assume a distance between trees of 10m and a total area of the grid of 10km x 10km. For each point we define V (x, y, t) as the abundance of vectors colonized by X. fastidiosa in a point (x, y) over time (0, T) and the bacterial load of X. fastidiosa in the plant X (x, y, t) between 0 and 1, for each point (x, y) over time (0, T).
The following system of partial differential equations is used to describe the V(abundance of vectors colonized by X. fastidiosa) and X(the bacterial load in the plant). The ΔV is a term that symbolizes the dispersal of the insect vector. How we came to these equations is explained in the box below.

\(\frac{dV}{dt}= \Delta V  M \cdot V + b \cdot (k(t)V) \cdot \frac {X}{max X}\)\(\frac{dX}{dt}= (s \cdot I \cdot V + \mu \cdot X) \cdot (1X)\)
Table 2: Overview of parameters of the spatial spread of X. fastidiosa model. Parameter Meaning Value M mortality rate of adult insect vectors 0.015 day ^{1} b bacteria acquisition rate 0.0995 k(t) carrying capacity of adult vectors variable over time, see figure 4 below s plant susceptibility 0.014 I bacterial load inoculated 142.86 \mu growth rate of x. fastidiosa μ = 0.02 between the 9th of May and the 14th of October otherwise μ = 0 max X carrying capacity of bacterial load in plant 10 ^{7} max P carrying capacity of phages 10 ^{9}

How did we come to these equations? arrow_downward
These equations are adaptions from the model made by the EFSA [1], to compute the bacterial load instead of a disease level of the plants.
\(\Delta V\)
This term symbolizes the dispersal of the insect. To compute Δ V we used the convolve function of the scipy.ndimage python package to apply a dispersal kernel. To obtain the dispersal kernel for the spread of the vector we used data from the EFSA model [1] (table A.5). This gave us the 95% range for the short distance dispersal kernel of the vector fitted to the spread in Apulia. This interval was fitted to a normal distribution which gave a mean of 237 m and a standard deviation of 99 m.
\( M \cdot V\)
This term symbolizes the natural mortality rate of the insect.\(b \cdot (k(t)  V) \cdot \frac{X}{max X}\)
This term describes the chance of a vector acquiring X. fastidiosa\( s \cdot I \cdot V\)
This term describes the new infections of X. fastidiosa in plants form the insect vector.\( \mu \cdot X\)
The growth rate of X. fastidiosa under favorable conditions. This means μ = 0.02 between the 9th of May and the 14th of October otherwise μ = 0.\( 1  \frac{X}{max X}\)
This is incorporated because X. fastidiosa has a maximum carrying capacity so the bacterial load can never become higher than max X. 
What assumptions did we make? arrow_downward
 Once an adult vector feeds on an infected plant it becomes infectious immediately and can transfer the X. fastidiosa to other infected (increasing the disease level) or healthy plants. The bacterial titer in the plant influences the possibility of X. fastidiosa being taken up by the insect, and once an insect in infected its status will remain infectious . Only the feeding behavior and dispersal of the vector will influence the propagation of X. fastidiosa.
 A transmission function is used to describe the chance of successful transmission of X. fastidiosa. This function considers (i) the susceptibility of the host plant, (ii) vector feeding behavior and preference (iii) the vector's capacity to transmit X. fastidiosa.
 A plant can be successfully inoculated several times, and this contributes to the overall bacterial growth. This bacterial population growth comes from the propagation of X. fastidiosa in the plant. Propagation of X. fastidiosa only takes place between the 9^{th} of May and the 14th of October due to the favorable weather conditions. [1]
 Vectors can only transmit the disease during their adult stage, the transition from preimaginal state to adult takes place in the spring (May 10^{th}) and from adult to preimaginal in autumn (October 27^{th}). The function of the vector population also contains a mortality rate describing the survival pattern of the vector.
 It is assumed the average dispersal of the insect per day is 237 m (standard deviation 99m) in a random direction. This was approximated by applying a convolution using a dispersal kernel based on a normal distribution.
Results
This model was used to create a visualization of the the spread of X. fastidiosa over 6 years, using the conditions described below.

Initial Conditions arrow_downward
The following initial conditions were applied:
\(V(x,y,0) = 0\)
\(X(x,y,0) = I(x,y)\)
\(x,y \in [0,i],[0,j]\)
I(x,y) is made under the assumption the disease was brought in by a propagule population of colonized insect vectors. Therefore, as initial conditions a small region of 9 x 9 trees are infected and have an initial bacterial load of 2500000.
The olive grove was approximated by the following series of mesh points:
\(x = i \cdot \Delta x\)
\(y = j \cdot \Delta y\)
Time steps of a day were made numbering the days of the year 1 to 365 starting on the first of January, with repetitions of this cycle for the number of years the model was run.
Reflective boundary conditions were applied based on the assumption the insects would fly back onto the field around the borders. As the borders of the field would also mean an end to the vegetation in which they live. This was implemented using the mirror option of the python function convolve.
Spatial spread of Xylencer phages
After successfully predicting the spread of X. fastidiosa we are ready to expand the model with our phages. We use the model on the interaction of the phages and X. fastidiosa to adapt the equations in the spatial spread model. Also, two new differential equations were introduced, one on the number of phages (P (x, y, t)) and the other on the abundance of insect vectors carrying the phage(Vp (x, y, t)) for each mesh point.

\(\frac{dV}{dt}= \Delta V  M \cdot V + b \cdot (k(t)V) \cdot \frac{X}{max X}\)\(\frac{dX}{dt}= (s \cdot I \cdot V + \mu \cdot X) \cdot (1X)  k_{1} \cdot X \cdot P \)\(\frac{dVp}{dt}= \Delta Vp  M \cdot Vp + b \cdot (k(t)Vp) \cdot \frac{P}{max P}\)\(\frac{dP}{dt}= s \cdot I \cdot Vp  k_{2} \cdot P + k_{3} \cdot k_{1} \cdot X \cdot P\)
Table 3: Overview of parameters of the spatial model for the spread of the Xylencer phages. Parameter Meaning Value M mortality rate of adult insect vectors 0.015 day ^{1} b bacteria acquisition rate 0.0995 k(t) carrying capacity of adult vectors variable over time, see figure 4 below s plant susceptibility 0.014 I bacterial load inoculated 142.86 μ growth rate of x. fastidiosa μ = 0.02 between the 9th of May and the 14th of October otherwise μ = 0 max X carrying capacity of bacterial load in plant 10 ^{7} max P carrying capacity of phages 10 ^{9} k_{1} infection rate of phages 6.2352 E9 ml cell^{1} day ^{1} k_{2} decay rate of the phages 0.0165 day ^{1} k_{3} burst size of the phages 100 PFU cell ^{1}

What assumptions did we make? arrow_downward
 All previous assumptions made for the LotkaVolterra model and the spatial spread model of X. fastidiosa are still true.
 The possibility of phages being taken up by the vector is dependent on their titer in the plant. Other parameters for transmission are similar to the transmission of the pathogen.
 A plant can be inoculated with phages by the vector several times. Once an insect takes up the phages they will remain on the vector due to the adhesion protein on their capsid.
 The vectors carrying X.fastidiosa and the vectors carrying the phages are not necessarily the same but they are sampled from the same population and they can be.
 We assume the phages have a maximum carrying capacity that is 100 times bigger than that of the bacterial load of X. fastidiosa because of the burst size.
Results
This model was created to asses the effectiveness of the Xylencer project. To see the effectiveness of the spreading of the phages a scenario will be run with which we can analyze the spreading of the phages by the insect through the field. We will apply the PDB to 3 trees in a field infected with X. fastidiosa

Initial Conditions arrow_downward
The following initial conditions were applied:
\(V(x,y,0) = 0\)
\(Vp(x,y,0) = 0\)
\(X(x,y,0) = 5^{5}\)
\(P(x,y,0) = Ip(x,y)\)
\(x,y \in [0,i],[0,j]\)
As initial condition all trees are assumed to be infected with a bacterial load of 50000 cells ml^{1}. The phage is applied at three points in this region with the PDB. Assuming an initial phage titer equal to 1 ^{4}.
The olive grove was approximated by a grid of 10km x 10 km with trees 10 m apart from one another. This is simulated with the following series of mesh points:
\(x = i \cdot \Delta x\)
\(y = j \cdot \Delta y\)
Time steps of a day were made numbering the days of the year 1 to 365 starting on the first of January, with repetitions of this cycle for the number of years the model was run.
Reflective boundary conditions were applied based on the assumption the insects would fly back onto the field around the borders. As the borders of the field would also mean an end to the vegetation in which they live.
This model shows that the phages can successfully combat X. fastidiosa by spreading themselves over the field. Within 40 days it can clear X. fastidiosa from a large field. This means our approach will work to combat x. fastidiosa in an infected region. It takes a long time for the symptoms to show, and once they do, the disease has already spread far and the disease is in an advanced stage. The Xylencer phages would be able to quickly tackle X. fastidiosa in a large area, saving the infected plants.
However, it also means the control of where the phages will spread to is lacking when they are spreading with such a high speed. This means we have to think carefully about the spreading of the phages as explained on our safety page. Also, a more accurate model could be achieved when more data would be available on the transmission of our enhanced phages by insects. This would be interesting to give further insights in the application of our project.
The script created for this model can be found here.

References arrow_downward
 C. Bragard et al., “Update of the Scientific Opinion on the risks to plant health posed by Xylella fastidiosa in the EU territory,” EFSA J., vol. 17, no. 5, May 2019.
 S. J. Ahern, M. Das, T. S. Bhowmick, R. Young, and C. F. Gonzalez, “Characterization of novel virulent broadhostrange phages of Xylella fastidiosa and Xanthomonas.,” J. Bacteriol., vol. 196, no. 2, pp. 459–71, Jan. 2014.
 J. D. Murray, Ed., Mathematical Biology, vol. 17. New York, NY: Springer New York, 2002.
 M. Das, T. S. Bhowmick, S. J. Ahern, R. Young, and C. F. Gonzalez, “Control of Pierce’s Disease by Phage,” PLoS One, vol. 10, no. 6, p. e0128902, Jun. 2015.
 S. M. White, J. M. Bullock, D. A. P. Hooftman, and D. S. Chapman, “Modelling the spread and control of Xylella fastidiosa in the early stages of invasion in Apulia, Italy,” Biol. Invasions, vol. 19, no. 6, pp. 1825–1837, Jun. 2017.
 L. Bosso, D. Russo, M. Di Febbraro, G. Cristinzio, and A. Zoina, “Potential distribution of Xylella fastidiosa in Italy: A maximum entropy model,” Phytopathol. Mediterr., vol. 55, no. 1, pp. 62–72, Jan. 2016.