Introduction

Now seeing how the bacteria behaves in the lab environment, we go on to see how Xanthomonas infects crops in fields and how our biocontrol agent changes those infection patterns. We will use a stochastic SIR model to simulate the situation and modulate the infection and recovery rates as a function of the biocontrol agent.

Objective

Predict how the biocontrol agent affects the infection and recovery rates and from that make a prescription for the appropriate Biocontrol application time in the field.

Model

SIR Modelling

The spreading of any epidemic disease is modeled by a very common model- The SIR Model is given by the three equations $${ \begin{align*} \frac{dS}{dt} &= -\beta S(t)I(t)\\ \frac{dI}{dt} &= \beta S(t)I(t) -\gamma I(t)\\ \frac{dR}{dt} &= \gamma I(t) \end{align*} } $$ Where, \(S(t), I(t), R(t)\) susceptible, infectious and recovered populations. And \(\beta, \gamma\) are the infection rate and recovery rate respectively.

Parameters

Fig 1: Graph of No. of Susceptible, No. of Infected, No. of Recovered plants with time

Implementing Biocontrol

• Now when we apply the Biocontrol agent, the plants start to recover faster and less is infected. This is implemented in the model by taking a time dependent gamma value which will be high for time t > application time of the biocontrol and low before that.

Assumptions

The assumptions we make for the model:-
1. One infected plant can infect only its nearest and 2nd nearest neighbors. (initial assumption which can change once one introduce randomness to the graph)
2. A plant, once recovered from disease, will not be infected later. The infection rate is greater than the recovery rate naturally, but when the biocontrol agent is introduced, it becomes comparable.
3. The graph is taken to be a \(N\times N\) square grid.

Results and plots

Fig 2: Graph of No. of Susceptible, No. of Infected, No. of Recovered plants with time for a \(N\times N\) square network.

Fig 3: Simulation of spreading of disease with time

Conclusions

1. We see from Fig-2 that once the biocontrol agent is implemented, the recovered population increases very rapidly. From the simulation of figure 3 we also observe a similar trend, noting that the entire grid is not corrupted with the biocontrol agent present in the stipulated time.
2. If we have some idea about what would be the change in the recovery rates with our biocontrol agent from some survey of lab experiments, our model can give an exact prescription of when one should apply the biocontrol in the field in order to stop the entire field from getting infected.

Future scopes

As a natural extension to this model, one can introduce randomness in the interactions and implement something similar to a Watts-Strogatz Random graph to incorporate more real life scenarios.