Introduction

We decided to model the deployment of our bed bug trap in a realistic environment, to understand how it was likely to work and what parameters might be important. This task involved a number of different modeling tasks, that posed different problems: modeling the diffusion of pheromones in the room coming both from a natural bed bug nest and the traps that we plan to deploy; modeling the movement of bed bugs influenced both by the pheromone field and their nest; and modeling the fungal epidemic that we plan to induce in the bed bug population.

Though the model is complex and includes many parameters, it has already allowed us to draw several conclusions, and as the model is improved and the parameters are refined other conclusions will follow. For instance, using realistic diffusion parameters it is clear that the pheromone field is relatively rapidly established and so the precise nature and concentration of pheromones is probably not critical. In contrast, the delay between infection and death of bed bugs is critical for ensuring the eradication of the nest. Our modeling thus helps understand critical aspects of the proposed design.

Why modeling? Descriptions of our goals

As our project was growing, we understood the importance of modeling our solution in order to beter to understand how it would work. Indeed, even if we had the opportunity to do laboratory experiments to test our traps, it would be very time consuming without the aid of modeling. A model run on a computer would allow us to better understand our results and refine the trap design and field trial protocol faster. The field and laboratory trails would also help us to parameterize our model and thus refine it. Modeling our project quickly became one of our top priorities.

When we began, we wanted to test various parameters:

• The effect of furniture
Our trap will most likely be used in a room with furniture (like a bedroom for example). Even if we have the formula to model the diffusion of gases (the Fick's laws of diffusion), this model doesn't take into account the effects of obstacles, which may have an impact on diffusion, and so on the efficiency of our traps.
• The effect of air flows
Since we will put our traps in an inhabited room, it may have openings to another room or even outside. These can cause drafts, which could have an impact on pheromone diffusion, and so on the efficiency of our traps
• The effect of the trap design and organization of the pheromones and fungus
During the development of our project, we had the opportunity to choose between two types of trap for the pheromones and the fungus. One consisted of an opened box with free circulation and the other was a closed trap with small openings. Since the two designs are different, this may have had an impact on the efficiency of our traps.

This list is of course not complete and many other aspects will affect our traps efficiency and we hope to test them using our model.

Unfortunately, among the three parameters we initially wanted to investigate, only one of them has been successfully incorporated into the program, because of the complexity of the mathematical models and their solutions

How did we model?

Our program had to model the diffusion of pheromones through a room containing furniture. It needed to include the behavior of bed bugs influenced by the pheromones and to describe their deaths (caused by the fungus). This model will have to manage a lot of parameters, from the diffusion of pheromones to the bedbugs movement and the interactions between all these variables. Because of the complexity of the phenomena, building the program by describing every action occurring in our model would be too difficult. Fortunately for us, we could use the language NetLogo, which allowed us to use agent-based modeling, a type of modeling particularly efficient for complex system modeling.

Agent-based modeling allows us to define the behavior of the agents (the diffusing molecules, the bedbugs,...) by defining the interactions between those agents. With the NetLogo language, those agents are patches, representing the environment where other agents are interacting and the turtles, mobile agents who can move through the world made by patches and interact with the other agents (patches or turtles). Those agents allowed us to build our current program. Thanks to this, modeling systems with multiples agents are interacting with every other agent is simpler.

What did we Model ?

Modeling of Diffusion

When we begin the development of our project, we first searched on the iGEM's database, in order to find if any teams had already worked on pheromone diffusion. After a few research among the previous iGEM's teams, we found out that the Valencia team of 2014 had also worked on pheromone diffusion and had modeled it using NetLogo. Because of the similarity between their work and ours, we inspired ourselves with the design of their modeling of diffusion when we model this aspect.

Fick's law of diffusion, mathematical model

As the 2014 Valencia team already found out, there is many ways to model diffusion. Here, we will use the Fick's law of diffusion to model it because of its simplicity and the amount of documentation about it we have access to. But there could be an impact of diffusion within a fluid in motion, the effect of air flows on diffusion, this can be modeled by the Navier-Stokes equations. Here we choose to not use it and hypothesize a room without drafts, but since it is the subject of another aspect of the program, we will develop it later.

How the formula is used in our model

The most well known form of the Fick's law of diffusion is described as it follows :

It is a partial differential equation with D being the diffusion coefficient, c being the pheromone concentration and \(\nabla\) being the gradient operator.
Since we consider our diffusion coefficient as constant, the equation can be set as:

With \(\Delta\) being the laplacian operator.
This equation is called the Heat equation and is well known to modelise various transfert problems. In our case, this equation model the diffusion of pheromones in a case without air flows. In a 2D environment, the equation would be :

\begin{equation} \frac{\partial c}{\partial t} = D\times\Delta c = D\times(\frac{\partial^2 c}{\partial x^2} + \frac{\partial^2 c}{\partial y^2}) \end{equation}

Adapt the formula to our model, discretisation

Even though we have our mathematical model, we still have to adapt it to our program. In order to do it, we use the finite difference method to approximate the solution of our equation. This method allows the program to compute on every position the approximate solution of our equation. By this, we are able to monitor the diffusion of the phermones at each point in time and space.