This section about understanding infectious disease dynamics through mathematical models.
Introduction
For iGEM 2024, we are proud participants in the newly introduced Infectious Diseases Village, which gathers teams dedicated to solving some of the most pressing global health challenges. Infectious diseases remain one of the leading causes of morbidity and mortality worldwide, affecting millions of lives annually and contributing to significant social and economic burdens. These diseases can spread rapidly through populations, creating public health crises that require innovative solutions.
One of the promising tools in the fight against infectious diseases is Synthetic Biology (SynBio). By harnessing the power of engineered biological systems, SynBio enables us to develop new therapies, diagnostics, and preventive measures. In our project, SkinBAIT, we are leveraging SynBio to create genetically modified bacteria that produce toxins specifically targeting parasitic infections, offering a safe and effective alternative to current treatments.
Modeling is crucial in understanding infectious disease dynamics and the potential impact of new treatments like SkinBAIT. Through mathematical models, we can simulate the spread and evolution of diseases under different conditions, predict outcomes, and assess the effectiveness of interventions. Models also allow us to explore scenarios that are difficult to test in the lab or real-world settings.
On this page, we will guide you through our modeling approach, including:
The different models we developed to describe the spread of infectious diseases (with and without SkinBAIT's intervention!).
Assumptions we made in each model.
Stability analyses and conditions for different outcomes.
How these models help us predict the effectiveness of SkinBAIT in treating scabies.
How these models contribute to establishing goals in the experimental.
The results from our models will play a crucial role in shaping the goals we set for SkinBAIT's effectiveness. By simulating the disease dynamics and incorporating different treatment application scenarios, we can identify key factors that influence the success of our product. The models will help us determine:
Minimum effectiveness thresholds required for SkinBAIT to significantly reduce scabies transmission rates.
Optimal usage patterns for preventive versus treatment applications.
Population-level impact, allowing us to predict how SkinBAIT can contribute to broader disease control efforts, including the potential for reducing outbreaks.
These insights will guide our product development objectives, ensuring that we not only create a solution that works at an individual level but also one that makes a meaningful difference in public health outcomes. Ultimately, the models allow us to set realistic and data-driven performance goals for SkinBAIT, aligned with both clinical needs and broader epidemiological impact.
Through these models, we aim to provide a robust framework for understanding not only how SkinBAIT can combat scabies, but also how synthetic biology innovations can transform the treatment of infectious diseases globally.
The SIS Model Approach
The SIS (Susceptible-Infected-Susceptible) model is widely used in epidemiology to simulate infectious disease dynamics. Individuals in a population are categorized as either susceptible (\(S\)), meaning they can contract the disease, or infected (\(I\)), meaning they are currently infected and can transmit the disease. A distinctive feature of the SIS model is that individuals can be reinfected after recovery, as opposed to the SIR model where individuals gain immunity post-infection. This makes the SIS model particularly relevant to diseases such as certain viral infections and parasitic diseases, including scabies, where individuals remain susceptible after recovering from an infection.
The SIS model is governed by parameters like the infection rate (\(\beta\)) and the recovery rate (\(\gamma\)), which describe how the disease spreads and recedes. However, it's important to note that the SIS model can be represented in several ways, including differential equations, stochastic processes, or agent-based models. The most common representation, a system of Ordinary Differential Equations (ODEs), assumes deterministic outcomes with a unique solution given a specific set of initial conditions—this uniqueness is a consequence of the Picard-Lindelöf theorem for ODEs (which accounts for Existence-Uniqueness). This solution yields the average behaviour of the system, in contrast, stochastic methods provide a distribution of possible trajectories as a solution. Other representations, such as Stochastic Differential Equations (SDEs) or Markov models, may not guarantee uniqueness: in stochastic systems, the idea of "unique solutions" depends on the type of solution. strong solutions fix the probability space and noise, yielding a unique solution adapted to that noise, like in some ODEs. Weak solutions don't fix the probability space or noise, so instead of a unique outcome, you get a distribution of possible solutions. Therefore, in many cases, it doesn't make sense to talk about a "unique solution" without specifying the probability space and noise. From this we conclude that the choice of model formulation depends on the specific context of the study.
One of the key advantages of the SIS model is its ability to help determine critical thresholds, such as the basic reproduction number (R₀). They are essential in understanding whether an infection will spread within a population or die out. By evaluating the model's parameters, we can identify conditions under which a disease becomes endemic or if eradication is feasible.
In this section, we will explore three different approaches to modeling infectious diseases: deterministic (ODE-based), stochastic, and agent-based. By examining the dynamics of each model, we aim to gain unique insights into disease spread and control, while comparing the strengths and limitations of each approach.
1.1. ODE-Based SIS Model Analysis
As explained above, the SIS model is a simple epidemiological model used to describe the spread of an infectious disease where individuals can become susceptible again after recovery. The model consists of two population compartments:
Susceptible (\(S\)): Individuals who can contract the disease.
Infected (\(I\)): Individuals who have the disease and can spread it by being in contact with Susceptible individuals.
It must be noted that both compartments add up to the Total Population (\(N\)) of the model. We can define \(S\) and \(I\) as the fraction of the Total Population corresponding to their respective compartment, assuming \(N=1\). This approach yields the following expression:
$$
S + I = N
$$
This statement is not trivial, as it comes with the assumption that the Total Population remains constant over the time scale of interest. This assumption is often valid because we are focusing on short-term disease dynamics, where changes in population size are negligible.
The dynamics of the model consist on two different interacions between the different popualtions compartments. First, Susceptible individuals, when in contact with Infected individuals, can contract the disease:
$$
S + I \xrightarrow{\beta} 2I
$$
where \(\beta\) is the transmission rate (given a pairwise interaction).
On the other hand, Infected individuals can recover from the disease and become Susceptible again. In our case, the infected individual can take the current treatments and become disease-free again:
$$
I \xrightarrow{\gamma} S
$$
where \(\gamma\) represents the recovery rate when an infected person takes one of the available treatments.
However, this approach assumes that treatment is always being given to those that are infected. Another option would be to define \(\gamma\) as the recovery rate multiplied by the proportion of infected individuals receiving treatment, and assume that without treatment, recovery doesn't happen. But that would make the model more complex, and we would rather keep it simple for this approach.
From these interactions, we can use mass-action principles to derive the differential equations governing the SIS model. In mass-action, we assume that the rate of infection is proportional to the product of the susceptible and infected populations, as each infected individual has a chance of interacting with a susceptible one. Similarly, the rate of recovery depends on the number of infected individuals. Using these assumptions, the differential equations for the SIS model can be written as follows:
$$
\frac{dS}{dt} = -\beta S I + \gamma I
$$
$$
\frac{dI}{dt} = \beta S I - \gamma I
$$
Now, can we predict the outcome of the spread of the infectious disease? To do so, we need to analyze the system when it reaches equilibrium, where the populations of susceptible and infected individuals no longer change over time. This is known as the steady state of the system, which refers to the situation where the disease dynamics stabilize, and the number of newly infected individuals equals the number of individuals recovering, even though infections and recoveries are still occurring.
The differential equations described above define the rate of change of the population compartments. In order to find the steady state, we set the rates of change equal to zero. This step is crucial because, at steady state, we expect no further change in the number of susceptible and infected individuals over time. In other words, we are looking for a situation where the number of new infections matches the number of recoveries, which results in a constant infected population. Formally, this corresponds to solving the system as \(t\rightarrow \infty\).
Below, the mathematical analysis can be found.
First, both equations are set equal to 0, indicating no rate of change in the population compartments:
$$
\frac{dS}{dt} = -\beta S I + \gamma I = 0
$$
$$
\frac{dI}{dt} = \beta S I - \gamma I = 0
$$
Now, given previous assumptions about the population size being constant, we can apply dimensionality reduction to simplify the system, resulting in a single equation governing the infected population:
$$
S + I = 1 \Longleftrightarrow S = 1 - I
$$
Substituting this into the equation for \(\frac{dI}{dt}\):
$$
\frac{dI}{dt} = \beta (1 - I) I - \gamma I = 0
$$
To find the steady-state solutions, we solve for the values of \(I\) (denoted \(I^*\)) where the rate of change is zero.
$$
\frac{dI}{dt} = \beta (1 - I^*) I^* - \gamma I^* = 0
$$
Rearranging the equation:
$$
\frac{dI}{dt} = I^* (\beta (1 - I^*) - \gamma) = 0
$$
From the equations, we notice there are two fixed points in this model (\(I^*_0\) and \(I^*_1\)):
$$
I^*_0 = 0
$$
$$
\beta (1 - I^*_1) - \gamma = 0 \implies I^*_1 = 1 - \frac{\gamma}{\beta}
$$
From the mathematical analysis, we can conclude that there are two potential outcomes for the disease spread:
Disease eradication: If \(I^*_0 = 0\), the infection disappears entirely, and no individuals remain infected.
Endemic equilibrium: If \(I^*_1 = 1 - \frac{\gamma}{\beta}\), a fraction of the population remains infected indefinitely. This non-zero steady state suggests that, although the infection continues to spread, the rate of new infections matches the rate of recoveries, leading to a stable infected population.
It's important to note that the outcome depends on the values of the transmission rate \(\beta\) and the recovery rate \(\gamma\).
Now, we can move on to determine the local stability of the fixed points. However, it's important to understand what local stability means in the context of a system like this. When we talk about the local stability of a steady state or fixed point, we are essentially asking: if the system starts near this fixed point, will it stay near it, or will it move away? In other words, stability helps us predict the system's behavior in the long term. A stable fixed point means that if the population sizes (susceptible or infected) are perturbed slightly, they will return to their equilibrium values over time. This is like a ball resting at the bottom of a valley: if you push the ball a little to the side, it rolls back to the bottom. On the other hand, an unstable fixed point is like a ball balanced at the top of a hill: even a small nudge will cause it to roll away from that point. In this case, small deviations from the fixed point will grow over time, and the system will not return to equilibrium.
To determine whether a fixed point is stable or unstable, we use a method called linear stability analysis, which examines how the system behaves when slightly perturbed. This will tell us whether small deviations shrink (indicating stability) or grow (indicating instability).
However, the SIS model is described by a non-linear differential equation. Directly solving or analyzing non-linear systems can be complex. Instead, we can simplify the problem by examining how small deviations from a fixed point behave. This process is called linearization. Linearization involves approximating the non-linear system by a linear one near the fixed points. The idea is that close to a fixed point, the non-linear terms in the equations are small and can be approximated by their first-order (linear) terms. This approximation allows us to use tools from linear algebra to analyze the stability of the fixed points. Below, the mathematical analysis can be found.
First, we introduce a small perturbation (\(\epsilon\)) around a fixed point (\(I^*\)) to see how the system reacts. Mathematically, we assume:
$$
I(t) = I^* + \epsilon(t)
$$
where \(I^*\) is the fixed point, and \(\epsilon(t)\) is a small deviation from that fixed point.
By substituting this into the differential equation, we can derive an equation for \(\epsilon(t)\) that tells us how the perturbation evolves over time. The key idea is that the behavior of \(\epsilon(t)\) near \(I^*\) will determine whether the system returns to the fixed point (if \(\epsilon(t)\) decreases) or moves away from it (if \(\epsilon(t)\) increases).
Starting with the differential equation for the infected population:
$$
\frac{dI}{dt} = \beta (1 - I) I - \gamma I
$$
We substitute \(I(t) = I^* + \epsilon(t)\) and expand the equation:
$$
\frac{d(I^* + \epsilon(t))}{dt} = \beta (1 - (I^* + \epsilon(t)))(I^* + \epsilon(t)) - \gamma (I^* + \epsilon(t))
$$
Expanding the right-hand side of the equation by distributing and simplifying terms we get:
The full expansion of the right-hand side becomes:
$$
\frac{d(I^* + \epsilon(t))}{dt} = \beta (1 - I^*) I^* - \gamma I^* + \left[\beta (1 - I^*) - \beta I^* - \gamma \right] \epsilon(t) - \beta \epsilon(t)^2
$$
Since \(I^*\) is a fixed point, we know:
$$
\frac{dI^*}{dt} = \beta (1 - I^*) I^* - \gamma I^* = 0
$$
This allows us to cancel out the term \(beta (1 - I^*) I^* - \gamma I^*\), leaving:
$$
\frac{d(I^* + \epsilon(t))}{dt} = \left[\beta (1 - I^*) - \beta I^* - \gamma \right] \epsilon(t) - \beta \epsilon(t)^2
$$
Since we assume that \(\epsilon(t)\) is small, the \(\epsilon(t)^2\) term can be neglected, leading to the linearized equation:
$$
\frac{d\epsilon(t)}{dt} \approx \left[ \beta (1 - 2I^*) - \gamma \right] \epsilon(t) = \lambda \epsilon(t)
$$
The term \(\lambda = \beta (1 - 2I^*) - \gamma\) is the coefficient that multiplies \(\epsilon(t)\) in the differential equation. It determines the growth or decay rate of the perturbation \(\epsilon(t)\):
If \(\lambda < 0\): The perturbation \(\epsilon(t)\) will decay exponentially over time (\(\epsilon(t) \to 0\)). This means that the system returns to the fixed point, so the fixed point is stable.
If \(\lambda > 0\): The perturbation \(\epsilon(t)\) will grow exponentially over time (\(\epsilon(t) \to \infty\)). This means that the system moves away from the fixed point, so the fixed point is unstable.
In summary, \(\lambda\) tells us whether small deviations from the fixed point will die out (indicating stability) or grow (indicating instability). therefore, this linearization process allows us to understand the stability of each fixed point and predict the long-term behavior of the system based on the values of the parameters \(\beta\) and \(\gamma\).
Now, we can move on to determine the stability of both fixed points of the SIS model:
Stability of \(I^*_0 = 0\):
Substituting \(I^* = 0\) into \(\lambda\):
$$
\lambda_0 = \beta (1 - 2 \times 0) - \gamma = \beta - \gamma
$$
- If \(\beta < \gamma\), then \(\lambda_0 < 0\), and \(I^*_0 = 0\) is stable.
- If \(\beta > \gamma\), then \(\lambda_0 > 0\), and \(I^*_0 = 0\) is unstable.
This means that if the transmission rate \(\beta\) is less than the recovery rate \(\gamma\), the disease will be eradicated, leading to a disease-free population (\(I^*_0 = 0\)).
Stability of \(I^*_1 = 1 - \frac{\gamma}{\beta}\):
Substituting \(I^* = 1 - \frac{\gamma}{\beta}\) into \(\lambda\):
$$
\lambda_1 = \beta \left(1 - 2\left(1 - \frac{\gamma}{\beta}\right)\right) - \gamma = \beta \left(2\frac{\gamma}{\beta} - 1\right) - \gamma = 2\gamma - \beta - \gamma = -\beta +\gamma
$$
- If \(\beta > \gamma\), then \(\lambda_1 < 0\), and \(I^*_1\) is stable.
- If \(\beta < \gamma\), then \(\lambda_1 > 0\), and \(I^*_1\) is unstable.
From this analysis, we can conclude that there is a certain threshold in the parameters that separates the two possible outcomes, that is, whether the disease will spread or die out. As explained above, that threshold can be found when both \(\beta\) and \(\gamma\) have the same value. Hence, we can establish a critical value for the parameters of the model: \(\beta_c = \gamma\) (or, alternatively, \(\gamma_c = \beta\)).
In summary:
If \(\beta < \gamma\): The disease-free equilibrium \(I^*_0 = 0\) is stable, leading to the eradication of the disease.
If \(\beta > \gamma\): The disease persists in the population at the equilibrium \(I^*_1 = 1 - \frac{\gamma}{\beta}\).
These results indicate that the ratio \(\frac{\beta}{\gamma}\), often referred to as the basic reproduction number \(R_0\), is crucial in determining the long-term dynamics of the disease in the population:
If \(R_0 > 1\), the disease will spread. This means that each infected individual, on average, transmits the disease to more than one person. As a result, the number of infections will increase, leading to an outbreak.
If \(R_0 < 1\), the disease will die out. In this scenario, each infected individual transmits the disease to less than one person on average, resulting in a decline in the number of infections over time until the disease is no longer present in the population.
The basic reproduction number \(R_0\) is a key epidemiological metric that helps public health officials understand the potential for disease spread within a community. It is influenced by various factors, including the rate of contact between susceptible and infected individuals (represented by \(\beta\)) and the rate of recovery or death of infected individuals (represented by \(\gamma\)). Thus, \(R_0\) can inform strategies for disease control, such as vaccination or social distancing measures, aimed at reducing transmission rates and ultimately lowering \(R_0\) to below one.
Since the mathematical results can be challenging to interpret at a glance, we've incorporated interactive graphs to help visualize the dynamics. One graph shows the evolution of susceptible and infected individuals over time, while two bifurcation diagrams illustrate how small parameter changes—like the transmission rate—can lead to dramatic shifts in outbreak severity. These tools make it easier to explore critical thresholds and better understand how interventions like SkinBAIT could impact scabies transmission.
To change the parameter values, click on the button for the desired parameter and use the slider that will appear to adjust it.
Below are the key conclusions and insights from the SIS model dynamics, applied to our project. These findings highlight critical factors in controlling the spread of scabies and how SkinBAIT can play a role in preventing outbreaks.
Transmission Rate (\(\beta\))
What it does: Determines how quickly a disease spreads. A higher (\(\beta\)) means more infections.
In Scabies: Factors like population density, close contact, and hygiene can increase (\(\beta\)), making outbreaks more likely. SkinBAIT could help reduce this rate.
Recovery Rate (\(\gamma\))
What it does: Controls how fast infected individuals recover and become susceptible again.
In Scabies: A higher (\(\gamma\)) means faster recovery, but reinfection is possible, stressing the need for ongoing prevention.
Initial Infected Population (\(I_0\))
What it does: The higher the initial number of cases, the faster the disease spreads.
In Scabies: This could represent early outbreak numbers, especially in high-risk areas like close-knit communities.
Bifurcation Analysis
By analyzing how small changes in certain parameters affect disease spread, we can pinpoint critical thresholds where outbreaks escalate. This helps identify when interventions like SkinBAIT are most effective in preventing large-scale outbreaks.
In the context of scabies and other infectious diseases, it's important to recognize that the recovery rate \(\gamma\) may decline over time as parasites or infection vectors develop resistance to existing treatments. This decline could lead us to cross the threshold from a state of no infection to endemic equilibrium, making it imperative to implement effective interventions to prevent such a shift. In terms of transmission dynamics, an appropriate value for the transmission rate \(\beta\) in scabies can vary based on factors like population density and social behavior. The selection of \(\beta\) must take into account local conditions, as higher contact rates in densely populated settings could necessitate higher values to accurately reflect the risk of transmission. A critical concern arises if \(\gamma\) becomes less than \(\beta\), as this would indicate that the rate of new infections exceeds the rate of recovery, leading to an increase in the infected population and the potential for sustained outbreaks. The need for innovative solutions becomes clear, and SkinBAIT emerges as a promising candidate to address this challenge. By effectively targeting the scabies mites, SkinBAIT aims to maintain a lower infection rate and enhance recovery, ensuring we do not reach an undesirable endemic state. The implications of this approach will be further explored in the next model we present.
1.2. ODE-Based SIS Model Analysis - SkinBAIT's version
Up until now, we have discussed disease dynamics where only current treatments are considered. Now, in this modified ODE-Based SIS model, we incorporate the potential effects of SkinBAIT, a novel product designed to reduce the spread of parasitic infections by both introducing a new recovery rate and reducing susceptibility for those that take the treatment preventively. Unlike the SIS model discussed previously, which considers only two compartments —Susceptible (\(S\)) and Infected (\(I\))— this extended version introduces two extra compartments: \(K\), representing individuals who use SkinBAIT and are thus less susceptible to infection, and \(J\), representing those individuals who are infected and are using SkinBAIT to recover. The key interactions between these compartments include:
Standard infection transmission between susceptible and infected individuals (where \(\beta\) is the transmission rate):
$$
S + I \xrightarrow{\beta} 2I
$$
Infection transmission between susceptible and infected individuals using SkinBAIT (where \(\beta\) is the transmission rate):
$$
S + J \xrightarrow{\beta} I + J
$$
Recovery of infected individuals through traditional treatments (where \(\gamma\) is the recovery rate):
$$
I \xrightarrow{\gamma} S
$$
Susceptible individuals starting to use SkinBAIT (where \(\alpha\) is the switching rate):
$$
S \xrightarrow{\alpha} K
$$
Individuals discontinuing SkinBAIT use and returning to full susceptibility (where \(\delta\) is the discontinuation rate):
$$
K \xrightarrow{\delta} S
$$
Infection transmission between individuals using SkinBAIT and infected individual (where \(\beta'\) is a lower transmission rate):
$$
K + I \xrightarrow{\beta'} J + I
$$
Infection transmission between individuals using SkinBAIT and infected individuals also using SkinBAIT (where \(\beta'\) is a lower transmission rate):
$$
K + J \xrightarrow{\beta'} 2J
$$
Recovery of infected individuals through the SkinBAIT treatment (where \(\gamma'\) is the recovery rate through SkinBAIT):
$$
J \xrightarrow{\gamma'} K
$$
This model also comes with a set of assumptions:
The total population also remains constant through time, as over our time scale of interest there is no significant variations in population size.
The rate of infection depends on whether the susceptible individuals are using SkinBAIT or not. This is due to the preventive nature of the SkinBAIT product: individuals using it are less susceptible to become infected. hence, their rate of infection is lower (that is, \(\beta > \beta'\)).
Individuals stick to their treatments, in other words, infected individuals using SkinBAIT will not stop using it to use other treatments. Therefore, there is no direct flow between the \(I\) and \(J\) compartments.
These additional dynamics aim to model the combined effects of both traditional treatments and preventive measures on disease persistence and control, offering insights into how SkinBAIT can alter the spread and long-term prevalence of infections in a population.
To capture the dynamics of this extended SIS model, we can use the principle of mass-action kinetics to formulate a system of differential equations (as done with the pevious model). In our case, the interactions between Susceptible (\(S\)), Infected (\(I\)), SkinBAIT Users (\(K\)), and Recovering Users (\(J\)) can be can expressed by the following ODEs:
$$
\frac{dS}{dt} = -\beta S I + \gamma I - \beta S J + \delta K - \alpha S
$$
$$
\frac{dI}{dt} = \beta S I + \beta S J - \gamma I
$$
$$
\frac{dK}{dt} = \alpha S - \beta' K I - \beta' K J + \gamma' J - \delta K
$$
$$
\frac{dJ}{dt} = \beta' K I + \beta' K J - \gamma' J
$$
Note: In this model, we assume that the infection transmission rates are independent of whether individuals come into contact with individuals who are using SkinBAIT or not. Specifically, susceptible individuals (S) are always infected at a rate \(\beta\), and individuals using SkinBAIT (K) are infected at a rate \(\beta’\). We do not consider any reduction in transmission risk from those using SkinBAIT (J) to susceptible individuals (S), meaning there's no differential effect on transmission based on treatment. This differs from typical viral infection models where such dynamics might lead to herd immunity, but it is an assumption made in this model to focus on other aspects of the intervention.
These equations represent the continuous flow of individuals between the four compartments (\(S\), \(I\), \(K\), and \(J\)) over time, governed by the rates of infection, recovery, and SkinBAIT usage. By solving these equations, we can better understand how SkinBAIT affects the transmission dynamics of scabies and similar parasitic diseases. As explained in the previous model, in order to find the solution to the equations when \(t \rightarrow \infty\) we equal all four equations to 0 to give the equilibrium conditions:
$$
\frac{dS}{dt} = -\beta S I + \gamma I - \beta S J + \delta K - \alpha S = 0
$$
$$
\frac{dI}{dt} = \beta S I + \beta S J - \gamma I = 0
$$
$$
\frac{dK}{dt} = \alpha S - \beta' K I - \beta' K J + \gamma' J - \delta K = 0
$$
$$
\frac{dJ}{dt} = \beta' K I + \beta' K J - \gamma' J = 0
$$
One of the key steady states we are interested in is disease-free state, where the infection has been eradicated from the population. This corresponds to both infected compartments, \(I\) (infected individuals not using SkinBAIT) and \(J\) (infected individuals using SkinBAIT), being zero. Other steady state occurs when the infection persists, i.e., when \(I\) and \(J\) are non-zero. This is known as endemic equilibrium, where the disease continues to circulate in the population.
Once we have identified the steady states of our model, the next step is to determine the stability of these states. In simple terms, stability refers to whether the system will return to the steady state after a small disturbance or if it will move away from it. To assess stability, we use a mathematical tool called the Jacobian matrix. This matrix helps us understand how small changes in the population of susceptible, infected, and SkinBAIT-using individuals will affect the overall behavior of the system.
Here's how it works:
Linearization: We take our system of equations and focus on how they behave near the steady states we found earlier. This involves creating a simplified version of the equations that captures the essence of the system's dynamics around those points.
Jacobian Matrix: We build the Jacobian matrix using the partial derivatives of our equations with respect to each variable. In simpler terms, this means we look at how a tiny change in one variable (like the number of susceptible individuals) affects the others (like the number of infected individuals). For our specific case, this is how the Jacobian matrix would look like:
$$
J = \begin{bmatrix}
\frac{\partial f_1}{\partial S} & \frac{\partial f_1}{\partial I} & \frac{\partial f_1}{\partial K} & \frac{\partial f_1}{\partial J} \\
\frac{\partial f_2}{\partial S} & \frac{\partial f_2}{\partial I} & \frac{\partial f_2}{\partial K} & \frac{\partial f_2}{\partial J} \\
\frac{\partial f_3}{\partial S} & \frac{\partial f_3}{\partial I} & \frac{\partial f_3}{\partial K} & \frac{\partial f_3}{\partial J} \\
\frac{\partial f_4}{\partial S} & \frac{\partial f_4}{\partial I} & \frac{\partial f_4}{\partial K} & \frac{\partial f_4}{\partial J}
\end{bmatrix}
$$
Where:
$$
\begin{aligned}
f_1(S, I, K, J) & = -\beta S I + \gamma I - \beta S J + \delta K - \alpha S \\
f_2(S, I, K, J) & = \beta S I + \beta S J - \gamma I \\
f_3(S, I, K, J) & = \alpha S - \beta' K I - \beta' K J + \gamma' J - \delta K \\
f_4(S, I, K, J) & = \beta' K I + \beta' K J - \gamma' J
\end{aligned}
$$
Eigenvalues: The next step is to calculate what are called the eigenvalues of the Jacobian matrix. These eigenvalues provide valuable insights:
- If all eigenvalues have negative real parts, it indicates that any small disturbance will eventually settle back to the steady state. This means the steady state is stable, and the system can recover from small fluctuations.
- If at least one eigenvalue has a positive real part, it means that small disturbances will lead to larger deviations from the steady state, causing the system to move away from it. In this case, the steady state is unstable.
By analyzing the stability of our steady states through the Jacobian, we can better understand how our model will behave in real-world scenarios. This information is crucial for determining how effective interventions like SkinBAIT will be in controlling the spread of scabies and preventing outbreaks.
In our case, we decided to use numerical solutions to analyze the dynamics of our modified SIS model. These methods allow us to approximate the solutions of our system of ordinary differential equations (ODEs), particularly when the equations are too complex for exact analytical solutions. While analytical solutions provide exact answers in the form of equations, numerical solutions offer approximate answers using computational techniques. This is especially useful in cases where:
The equations are too complicated to solve algebraically.
The system behavior is complex and requires simulation over time.
Numerical methods work by breaking down the problem into smaller, manageable parts. For our SIS model, we discretize time into small intervals and calculate the values of our compartments at each time step based on the rates of change defined by our ODEs.
In our analysis, we used the `odeint` function from the `scipy.integrate` library in Python to implement this numerical solution. This function takes our system of equations and initial conditions and iteratively calculates the populations of \(S\), \(I\), \(K\), and \(J\) at each time step. `odeint` is based on the LSODA algorithm, which automatically selects between two integration methods: the Adams method for smooth problems and the Backward Differentiation Formula (BDF) for stiff problems. This adaptability allows `odeint` to efficiently handle a wide range of ODE systems, including those that may present numerical difficulties. By running this simulation over a specified period, we can observe how the dynamics of the system evolve, allowing us to analyze the effectiveness of SkinBAIT and its impact on disease spread and recovery.
Since this can be challenging to interpret at a glance, we have also incorporated interactive graphs to help visualize the dynamics of the populations over time for different parameters. These tools make it easier to explore critical thresholds and better understand how interventions like SkinBAIT could impact scabies transmission.
To change the parameter values, click on the button for the desired parameter and use the slider that will appear to adjust it.
The results from our ODE-based SIS model with SkinBAIT offer exciting insights into how our synthetic biology solution can effectively control and eradicate scabies infections (amongst others). Here’s what we’ve learned:
Impact of SkinBAIT: Introducing SkinBAIT into the population dramatically reduces the spread of scabies. As the treatment becomes more widespread, the number of infected individuals (both those using SkinBAIT and those not yet using it) significantly drops. SkinBAIT’s ability to lower the transmission rate directly curbs the spread of scabies (and othe parasitic infestations), making it a powerful tool in the fight against parasitic infestations.
Interplay of Key Parameters: The success of SkinBAIT isn’t just about introducing it—it’s about how different factors work together. For example:
- The lower the transmission rate among those using SkinBAIT, the faster we see a decline in infections.
- Recovery rates are also vital. SkinBAIT boosts recovery, so people infected can return to a healthy state much quicker.
- Rapid adoption of SkinBAIT and sustained usage (keeping the discontinuation rate low) are critical. The more people use it, and the longer they stick with it, the better the chances of stopping the disease entirely.
Path to Eradication: To eliminate scabies, we need to optimize several key parameters:
- High effectiveness: SkinBAIT must significantly reduce transmission, lowering the risk of spread.
- Widespread use: SkinBAIT must be rapidly adopted across the population. The faster and more consistently it’s used, the more we can tip the balance toward disease eradication.
- Sustained action: Reducing the number of people who stop using SkinBAIT is crucial. Keeping the treatment in play over long periods ensures that scabies has less chance of making a comeback.
Crossing the Threshold: Our model shows that there are critical thresholds where small changes in SkinBAIT usage or effectiveness can make a big difference. Once we pass a certain point, the disease starts to disappear from the population, moving us toward eradication. This is the sweet spot we’re aiming for—where SkinBAIT can tip the balance and drive scabies to extinction.The model indicates that SkinBAIT is most effective when it can significantly lower the transmission rate (\(\beta')\) and accelerate recovery (\(\gamma')\) while maintaining high adoption rates (\(\alpha)\) and low cessation rates (\(\delta)\). These factors combine to push the system toward disease eradication, especially in scenarios where natural transmission (\(\beta)\) remains high. Controlling for environmental factors that influence \(\beta\) and maximizing SkinBAIT’s impact are thus essential strategies for successful eradication.
Below are six figures, each depicting the final infected population (including both individuals using and not using SkinBAIT) for various parameter combinations. When a parameter is not part of the sweep, the following values are used: \(\beta = 0.7\), \(\gamma = 0.3\), \(\beta' = 0.5\), \(\gamma' = 0.3\), \(\alpha = 0.3\), \(\delta = 0.4\), and \(I_0 = 0.3\).
In conclusion, SkinBAIT’s true power lies in its ability to break the cycle of infection and reinfection. By lowering transmission, speeding up recovery, and ensuring widespread and sustained use, we can push the scabies infection rates down to zero. With the right combination of these factors, we can not only control scabies outbreaks but ultimately eliminate them, providing a lasting solution to a persistent problem.
1.3. SSA-Based SIS Model Analysis
In our project, we implemented a Stochastic Simulation Algorithm (SSA) version of the SIS model to enhance our understanding of the dynamics of infectious diseases, particularly scabies. The decision to use an SSA approach stems from the need to accurately capture the unpredictable nature of disease transmission, which is often influenced by random interactions between individuals. Unlike Ordinary Differential Equations (ODEs), which offer a deterministic framework assuming predictable trajectories based on fixed parameters, SSAs incorporate randomness, acknowledging that biological systems are inherently stochastic.
In an ODE model, we would observe how susceptible individuals become infected and how infected individuals recover at consistent rates, resulting in smooth, predictable curves. However, real-world disease outbreaks are rarely that straightforward. Factors such as population density, social behavior, and random contacts can lead to significant variations in infection rates. By employing an SSA model, we can explore the probabilistic nature of disease spread, providing insights into how likely outbreaks are to occur under various conditions.
For instance, in a community with a low initial infection rate, a few random infections might lead to a sudden spike in cases, while the same initial conditions could result in a slow decline in infections in another scenario. This variability is crucial for public health interventions, as understanding the likelihood of outbreaks can inform strategies such as vaccination campaigns and public awareness initiatives.
To implement our stochastic model, we used the Gillespie algorithm, a well-established method for simulating Continuous-Time Markov Chains (CTMC). The algorithm models the occurrence of events in continuous time, with the time between events determined stochastically. Each simulation generates a statistically accurate trajectory of the system, providing one possible outcome for a stochastic process. While Gillespie works with discrete events and states, as the population size grows, these discrete jumps become infinitesimally small, and the system’s behavior tends to approximate the solutions of Stochastic Differential Equations (SDEs).
What is significant about this type of simulation is that it allows for a more nuanced analysis compared to deterministic models. Instead of just obtaining the expected trajectory, we generate a distribution of possible outcomes. This enables us to use statistical methods to analyze the system's behavior, such as identifying worst-case scenarios or estimating the likelihood of rare events, providing deeper insight into the range of possible dynamics.
In our case, each susceptible individual has a chance of coming into contact with an infected individual, leading to transmission if the conditions are right. This transmission occurs at a rate determined by the parameter \(\beta\), which reflects how infectious the disease is. On the other hand, infected individuals recover and transition back to the susceptible state at a rate defined by \(\gamma\). By incorporating these interactions, we can model how the infection spreads through the population over time, capturing the essence of disease dynamics in a way that mirrors real-life scenarios.
Transitioning from an ODE framework to an SSA model requires several adaptations. First, instead of continuous equations describing changes in populations, we define discrete events that occur at specific rates. We must also specify the initial sizes of susceptible and infected populations carefully, as the inherent randomness of the model means that small variations in initial conditions can lead to significantly different outcomes. Moreover, there is a strong dependency on noise. Additionally, we conduct multiple simulations—typically 1000 runs in our case—to account for the stochastic nature of the model. This allows us to average the results and quantify the variability in infection and recovery rates.
The significance of using the SSA model is profound. It enables us to delve deeper into how randomness influences disease dynamics, particularly for infectious diseases like scabies, where interaction patterns among individuals can lead to unpredictable transmission dynamics. The insights obtained from our SSA model are crucial for developing effective public health strategies. Understanding the likelihood of outbreaks under various conditions can inform decisions about intervention timing and resource allocation. Furthermore, by modeling the stochastic nature of infection spread, we can assess the potential effectiveness of our innovative treatment, SkinBAIT, in real-world scenarios.
In summary, the SSA model complements our deterministic analyses by providing a probabilistic framework that reflects the complexities of infectious disease transmission. This approach not only enhances our predictions but also facilitates informed decision-making in managing outbreaks, ultimately contributing to our goal of effectively addressing scabies and similar infectious diseases. Below, we incorporated an interactive graph that allows us to simulate different conditions, making our model more understandable and accessible for analysis.
To change the parameter values, click on the button for the desired parameter and use the slider that will appear to adjust it.
The insights gained from our SSA model have significantly enriched our understanding of infectious disease dynamics, particularly regarding the interplay between various parameters and the stochastic nature of disease transmission. One of the most striking observations is how the total population size influences the model's behavior. In scenarios with larger populations, the outcomes appear more deterministic, resembling those predicted by traditional ODE models. This phenomenon can be attributed to the law of large numbers: as the population increases, the impact of random fluctuations diminishes, leading to more predictable aggregate behavior. Essentially, with more individuals interacting, the average rates of infection and recovery stabilize, making it seem as if the system operates under fixed rules rather than random events. This is closely related to the convergence towards continuous models.
Conversely, in smaller populations, the effects of stochasticity become more pronounced. Here, we witness significant variability in infection rates and recovery outcomes, as random interactions can lead to abrupt spikes in cases or sudden declines in infection. This variability underscores the importance of considering population size when modeling infectious diseases, as it can fundamentally alter the dynamics of disease spread.
Another important insight relates to the parameter ratio of \(\beta\) (the infection rate) to \(\gamma\) (the recovery rate). In deterministic models, this ratio often serves as a clear threshold for disease outcomes. For instance, a ratio greater than one typically indicates that the infection will spread, while a ratio less than one suggests that the disease will die out. However, in our SSA framework, this threshold becomes less clear due to the inherent randomness in individual interactions. With stochasticity, even a ratio that suggests the potential for an outbreak might not guarantee one; small fluctuations can lead to unexpected results, including scenarios where an outbreak does not occur despite a seemingly favorable \(\frac{\beta}{\gamma}\) ratio. (You can try this on our interactive graph!!)
Additionally, we observed that variations in the infection and recovery rates significantly impact the overall dynamics of the disease. For example, a slight increase in the infection rate \(\beta\) can lead to dramatically different outcomes, particularly in smaller populations where stochastic events can easily tip the balance between infection and recovery. Conversely, in larger populations, the same increase might not yield as significant an impact, reinforcing the need to consider the population context when interpreting model results.
The insights gained from our SSA model also highlight the importance of tailoring public health strategies to account for the stochastic nature of disease dynamics. For instance, interventions aimed at reducing the transmission rate \(\beta\) should consider not only the average behavior of the population but also the potential for sudden outbreaks driven by random interactions. Similarly, understanding that recovery dynamics can vary widely in different contexts reinforces the need for flexible and adaptive strategies in response to emerging outbreaks.
In conclusion, the SSA model has revealed a complex landscape of infectious disease dynamics, emphasizing the significance of parameter interactions, population size, and the unpredictable nature of disease transmission. By embracing these insights, we can better inform public health policies and interventions, ultimately enhancing our ability to manage and mitigate the spread of infectious diseases like scabies and other parasitic infestations.
Building on these findings, our next model will introduce SkinBAIT into the SSA framework. By incorporating this innovative treatment into our stochastic model, we aim to explore how the application of SkinBAIT can influence the dynamics of infection spread. This transition will allow us to investigate not only the efficacy of the treatment but also how it interacts with the inherent randomness of disease transmission, providing deeper insights into potential intervention strategies for managing scabies outbreaks.
1.4. SSA-Based SIS Model Analysis - SkinBAIT's version
In our ongoing effort to combat scabies outbreaks, we have developed a stochastic SIS model that incorporates the innovative treatment SkinBAIT. This model aims to explore the interaction between the traditional disease dynamics and the therapeutic effects of SkinBAIT, providing valuable insights into its potential efficacy and impact on infection spread.
The model extends our previous work by integrating SkinBAIT as a new intervention strategy. SkinBAIT is designed to be a targeted treatment that utilizes genetically engineered Cutibacterium acnes bacteria to produce toxins lethal to scabies mites while remaining harmless to humans. By introducing this treatment into our stochastic framework, we can investigate how it modifies the dynamics of infection within a population characterized by inherent randomness.
Similar to our earlier modeling efforts, we have employed the Gillespie algorithm to simulate the stochastic dynamics of the system. This algorithm enables us to efficiently model the random events that drive changes in population states, capturing the variability in infection spread more accurately than deterministic approaches. The new model retains the core components of the SIS framework, with added complexity to account for the interactions between susceptible individuals, infected individuals, and those using SkinBAIT. Specifically, we have defined additional species: those using SkinBAIT (\(K\)) and infected individuals using SkinBAIT (\(J\)). The model tracks the transitions between these states through several defined reactions, including the initiation and discontinuation of SkinBAIT usage, as well as the infection dynamics associated with both traditional and SkinBAIT-treated individuals.
One of the key features of this model is its ability to simulate multiple stochastic runs, allowing us to assess the variability in outcomes and the impact of different parameters on the dynamics of disease transmission. By running numerous simulations, we can capture a range of potential scenarios, enabling us to better understand the behavior of the system under varying conditions.
Through this new model, we aim to answer critical questions about the role of SkinBAIT in managing scabies outbreaks. Specifically, we will examine how varying parameters such as transmission rates (\(\beta\) and \(\beta'\)), recovery rates (\(\gamma\) and \(\gamma'\)), and the initiation and discontinuation rates of SkinBAIT usage (\(\alpha\) and \(\delta\)) influence the overall dynamics of infection. Furthermore, we seek to understand how these factors interact with population size and the inherent stochasticity of disease spread, leading to a more comprehensive understanding of the treatment's potential effectiveness.
Ultimately, the insights gained from this stochastic SIS model with SkinBAIT will inform our experimental pipeline and public health strategies, enhancing our capacity to address scabies outbreaks effectively. By linking the therapeutic potential of SkinBAIT with the dynamic nature of infectious disease transmission, we hope to contribute significantly to the development of innovative solutions for managing parasitic infections in the future.
Below, we incorporated an interactive graph that allows us to simulate different conditions, making our model more understandable and accessible for analysis.
To change the parameter values, click on the button for the desired parameter and use the slider that will appear to adjust it.
Through our exploration of the stochastic SIS model with SkinBAIT, we gained several critical insights into the dynamics of infectious disease spread. One of the most notable observations is that the thresholds separating endemic outcomes from disease eradication are not as distinct as we initially thought when comparing this model with the ODE-based approach. This realization underscores the complexities introduced by stochasticity in disease dynamics and emphasizes the need for our solution to be robust enough to surpass these thresholds with a substantial margin to be effective and marketable.
We recognized that the stochastic nature of the model causes variability in outcomes, suggesting that factors such as population size can significantly influence the predictability of disease dynamics. In larger populations, the deterministic characteristics became more pronounced, while smaller populations displayed more erratic behavior. This variability highlights the importance of considering population size and the inherent uncertainty in disease spread when developing our product.
These insights underline the necessity of a well-rounded strategy in the development of SkinBAIT, ensuring that it not only meets the established thresholds for disease control but also provides a buffer against the inherent uncertainties of disease dynamics. This understanding will inform our next steps as we continue to refine our models and approach to infectious disease management.
1.5. Network of Agents Based SIS Model Analysis
A network of agents refers to a collection of individual entities, or "agents", that interact with each other based on a set of rules or behaviors. In the context of infectious diseases, these agents can represent people in a population, each with their own unique characteristics and states. The connections between agents can represent social relationships, such as friendships, family ties, or other interactions that can influence the spread of a disease. By modeling agents in a network, we can better understand how diseases spread through communities and how individual actions impact overall disease dynamics.
The agent-based SIS (Susceptible-Infected-Susceptible) model is a computational simulation that provides insights into the dynamics of infectious diseases on a network of interconnected individuals, or agents. This model allows us to explore how disease spreads through a population and how different factors, such as individual interactions and network structure, influence infection rates and recovery.
In this model, each individual in the population can be in one of two states: Susceptible (\(S\)) or Infected (\(I\)). At the start of the simulation, a certain percentage of individuals are randomly infected, while the rest remain susceptible. The model operates on a small-world network, which is a type of graph that simulates social connections among individuals. In this network, each individual is connected to a few nearest neighbors, and a probability factor allows for some randomness in connections, creating a mix of local and long-range interactions.
The dynamics of infection in this model are based on two key processes:
Infection: An infected individual can transmit the disease to susceptible neighbors based on a predefined infection rate (\(\beta\)). The probability of transmission depends on the number of infected neighbors and the connections each individual has.
Recovery: Infected individuals have a chance of recovering and becoming susceptible again after a certain period, governed by a recovery rate (\(\gamma\)). This cyclic nature of the SIS model represents diseases where individuals can be re-infected after recovery.
We specifically chose to model our agent-based SIS framework on a small-world network because it closely mimics the social structure found in real-world populations. Small-world networks possess unique characteristics that distinguish them from other types of networks, such as regular lattices and random networks:
Short Path Lengths: In a small-world network, any two individuals can be connected through a relatively small number of steps. This trait allows for efficient transmission of information or disease, as individuals can interact with others outside their immediate social circle. This aspect is particularly relevant for infectious diseases, which often spread through social contacts that may not be geographically proximate.
High Clustering: Small-world networks exhibit high clustering, meaning that individuals are more likely to be connected to the same neighbors. This property allows for localized outbreaks, as an infected individual can quickly spread the disease to closely connected friends and family. This characteristic is essential for understanding how infections can persist and spread within communities.
In contrast, regular lattices have a structured layout where each individual is connected in a predictable pattern, leading to long distances between nodes and potentially slower disease spread. On the other hand, random networks have connections that are entirely random, which can lead to unrealistic scenarios where distant individuals are just as likely to infect each other as neighbors. While they can capture some aspects of disease transmission, they may not reflect the nuanced social structures and interaction patterns present in actual populations.
The agent-based SIS model is particularly relevant for studying infectious diseases because it accounts for the complexity and variability of individual interactions within a population. Unlike traditional models that assume a homogeneous population, the agent-based approach allows us to consider how social networks and individual behaviors impact disease spread. This is essential for understanding diseases that are transmitted through close contact, such as scabies and other parasitic infestations, as it reflects real-world scenarios more accurately.
In the context of our project, the agent-based SIS model enhances our understanding of how SkinBAIT —our innovative treatment for scabies— can influence disease dynamics. By simulating various scenarios, we can assess how effectively SkinBAIT can reduce infection rates and prevent outbreaks. This model complements our previous ODE- and SSA-based approaches by providing a more nuanced view of disease transmission, especially under different population structures and interaction patterns.
Below, we incorporated an interactive graph that allows us to simulate different conditions, making our model more understandable and accessible for analysis.
To change the parameter values, click on the button for the desired parameter and use the slider that will appear to adjust it.
The exploration of the agent-based SIS model has yielded profound insights that have the potential to reshape our understanding of infectious disease dynamics. One of the most significant revelations is the intricate interplay between network structures and disease transmission. By modeling individual agents within a small-world network, we discovered that the connectivity patterns of a population greatly influence the spread of infection. This realization highlights the importance of social structures in epidemiological modeling and underscores the need for tailored intervention strategies that consider not only biological factors but also the complex web of interactions that define communities. This fits with the feedback we received during our interview with Dr. Omar López López, who gave us insights on the differences of disease spreding between Spain and Malawi (where tight-knit communities favor this spreading) and how he takes this into account in his job.
Moreover, the agent-based approach has highlighted the stochastic nature of disease dynamics, challenging traditional views of predictability in outbreaks. Our findings show that small changes in behavior or contact patterns can lead to significantly different outcomes, underscoring the need for public health strategies that are flexible and adaptive. This shift emphasizes the importance of designing interventions that account for the uncertainties and variations inherent in human behavior.
The model also captures the nuanced responses of individual agents to infection and recovery, offering insights into the societal factors influencing disease spread. By simulating diverse reactions within a population, we recognized that disease eradication is not solely a biological challenge but also a sociocultural one. This perspective encourages us to engage with communities and tailor health messaging and resources to their unique dynamics.
Ultimately, the insights gained from our agent-based SIS model enhance our understanding of disease dynamics and inspire a paradigm shift in public health approaches. By embracing the complexities of social networks and the stochastic nature of infection, we can develop targeted strategies that aim not only to eradicate diseases but also to empower communities to build resilience against future outbreaks. This journey toward a more integrated understanding of infectious diseases is a transformative vision for the future of public health.
As we transition to our next model, which incorporates the dynamics of SkinBAIT, these insights will be crucial. Understanding how social and individual behaviors influence infection rates will guide our modeling of SkinBAIT's impact. By integrating these dynamics within the agent-based framework, we aim to assess both the efficacy of our solution and its optimal deployment in real-world settings, further bridging the gap between scientific research and practical applications in combating scabies and other infectious diseases.
1.6. Network of Agents Based SIS Model Analysis - SkinBAIT's version
In this section, we present an analysis of a modified agent-based SIS model using a network of agents to simulate the dynamics of scabies infection and the application of the SkinBAIT treatment. This model is particularly relevant for understanding how the spread of infectious diseases can be influenced by network structures and individual interactions, offering insights that extend beyond traditional compartmental models.
The agent-based SIS model operates on a Strogatz-Watts small-world network, which captures the characteristics of real-world social networks through a combination of local connectivity and long-range connections. Each agent (or node) in the network represents an individual within a population that can exist in one of four states:
Susceptible (\(S\)): An individual not currently infected.
Infected (\(I\)): An individual currently infected.
Susceptible to SkinBAIT (\(K\)): An individual not currently infected using SkinBAIT.
Infected with SkinBAIT (\(J\)): An individual currently infected who is being treated with SkinBAIT.
The model is built on several key assumptions that dictate its dynamics:
Transmission Mechanism: The infection spreads through interactions between agents in the network. The probability of infection depends on the number of infected neighbors and is influenced by two parameters:
Beta (β): The infection rate for the standard infection process.
Beta Prime (β'): The decreased infection rate for individuals using SkinBAIT.
Recovery Rates: Each infected individual has a chance to recover and return to the susceptible state, characterized by:
Gamma (γ): The recovery rate from standard infection.
Gamma Prime (γ'): The recovery rate from infection for individuals treated with SkinBAIT.
Transition States: The model incorporates additional dynamics:
Alpha (α): The probability that a susceptible individual transitions to using SkinBAIT.
Delta (δ): The probability that an individual using SkinBAIT reverts back to the susceptible state.
Network Structure: The network is characterized by:
Population Size: The total number of individuals in the network.
Connected Neighbors: The average number of direct connections each agent has.
Rewiring Probability: The likelihood of connections being rewired to create shortcuts, which can enhance the spread of the infection.
The model employs a deterministic simulation approach, running multiple iterations to gather statistical data on the dynamics of the different states over time. The simulation tracks the fraction of the population in each state (S, I, K, and J) across a specified time horizon. This agent-based SIS model is highly relevant for understanding the transmission dynamics of scabies, especially in the context of the SkinBAIT treatment. By considering the network structure, the model reveals how individual interactions and treatment uptake can significantly influence the overall disease dynamics. This approach offers valuable insights for public health interventions, highlighting the importance of targeted treatments and community-level strategies in controlling infectious diseases.
Below, you can find an interactive graph that allows you to explore the outcome of different system conditions.
To change the parameter values, click on the button for the desired parameter and use the slider that will appear to adjust it.
The analysis of our network of agents-based SIS model has yielded several significant insights that deepen our understanding of scabies transmission dynamics and the effectiveness of the SkinBAIT treatment. These findings build upon the foundational knowledge we established in previous sections, particularly regarding the interplay between infection rates, recovery rates, and the role of SkinBAIT in managing scabies outbreaks.
One of the most striking observations from our simulations is the tendency for a remaining population of individuals using SkinBAIT to persist even in the absence of active infection. This phenomenon aligns with our earlier findings on the dynamics of treatment uptake and its effects on susceptibility. Despite a decline in the scabies infection rate to zero, individuals remain in the susceptible-to-SkinBAIT (\(K\)) state due to the alpha (\(\alpha\)) parameter, which governs the transition from susceptible to the SkinBAIT treatment. However, it is evident that, once the disease is completely erradicated, this parameter should decrease to 0, as the product would be no longer needed.
Moreover, introducing SkinBAIT into the model significantly influences the overall disease dynamics compared to the previous model, particularly in its potential to push the system toward the threshold of no disease and facilitate eradication. The treatment acts as a critical intervention that alters the natural course of the infection. By enhancing the recovery rates and reducing the effective infection rates, SkinBAIT contributes to lowering the prevalence of scabies. This proactive approach is vital, as it not only aids in managing current outbreaks but also helps in steering the population toward a stable state free from infection.
On another note, our model has highlighted how various parameters influence the overall dynamics and outcomes of the scabies epidemic:
Infection and Recovery Rates (\(\beta\), \(\beta'\), \(\gamma\), \(\gamma'\)): The balance between infection rates and recovery rates is crucial. Lower values of beta (\(\beta\)) and beta prime (\(\beta'\)) lead to fewer infections, ultimately resulting in a quicker return to equilibrium. Conversely, higher values of these parameters significantly increase the infection prevalence, leading to a sustained population of infected individuals, even when SkinBAIT is applied.
Treatment Transition Rates (\(\alpha\), \(\delta\)): The parameters governing the transition to and from SkinBAIT play a pivotal role in the model. An increase in the transition rate to SkinBAIT (\(\alpha\)) can enhance the treatment's reach, reducing the infected population more rapidly. However, if the rate of reverting to the susceptible state (\(\delta\)) is also high, it may counteract these efforts, leading to persistent populations of individuals still requiring treatment.
Network Connectivity: The structural properties of the network, including connected neighbors and rewiring probability, have a significant impact on infection spread. Higher connectivity facilitates quicker transmission of the infection, whereas a lower rewiring probability can lead to more isolated clusters of infection, allowing for localized outbreaks. This underscores the importance of community engagement and targeted interventions in densely connected networks.
When comparing our agent-based model findings to those from our deterministic SIS model, we observe notable differences in outcomes. While the deterministic model typically offers a clear threshold for the transition between endemic states and disease eradication, the agent-based approach reveals a more nuanced picture. In particular, the stochasticity of individual interactions within the network can lead to unpredictable fluctuations in infection rates, demonstrating how local dynamics can influence global outcomes.
These insights are critical as they highlight the multifaceted nature of infectious disease transmission and the efficacy of SkinBAIT. Understanding the interplay between treatment uptake and infection dynamics not only informs the development of effective public health strategies but also emphasizes the need for continuous adaptation in response to changing conditions within the population.
1.7. Conclusions of the SIS Model Approach
The conclusions from our modeling work offer valuable insights into the dynamics of scabies transmission and control, providing a strong foundation for our experimental pipeline. Through a combination of different modeling approaches, we have been able to explore the intricate balance between infection spread, recovery rates, and the impact of our SkinBAIT treatment.
Key Findings
Across all models, the introduction of SkinBAIT has consistently shown its potential to significantly reduce and even eradicate scabies infections. Our models confirm that the success of SkinBAIT relies not just on its individual effectiveness, but on how widely and consistently it is adopted in a population. Furthermore, we’ve observed critical thresholds where small improvements in SkinBAIT’s effectiveness or adoption rates can tip the balance from endemic infection to complete disease eradication. These findings guide our focus on maximizing the efficacy and reach of our product.
Importance of Each Model:
ODE-Based SIS Model:
This model provided a clear, analytical framework to understand the relationship between infection dynamics and intervention with SkinBAIT. By focusing on deterministic outcomes, we gained insight into how different parameter combinations influence long-term disease behavior. This model’s strength lies in its ability to offer straightforward, steady-state predictions and help identify thresholds for eradication. However, it assumes a homogeneous population and does not capture random variations, which may limit its applicability in real-world scenarios.
SSA-Based SIS Model:
The stochastic version introduced random fluctuations, reflecting the unpredictability of real-life disease spread. This model’s strength lies in its ability to simulate smaller populations or situations where randomness plays a key role. However, it lacks the clarity of the ODE model when identifying precise thresholds, making it more difficult to predict outcomes in a controlled environment. Nonetheless, it emphasizes the importance of ensuring robustness in our SkinBAIT design, accounting for random variations in treatment outcomes.
Agent-Based SIS Model:
By simulating individual interactions, the agent-based model helped us explore how network dynamics and individual-level behaviors influence disease transmission. This model is particularly important for understanding the effect of social structures, such as close-knit communities or isolated populations, on the spread of scabies. While its complexity can make it challenging to analyze at a larger scale, its flexibility allows us to simulate diverse scenarios and inform our strategy for targeted interventions.
Impact on Our Experimental Pipeline:
The insights gained from these models are directly informing our experimental design. The identification of key parameters—such as transmission rates (\(\beta\) and (\(\beta'\)), recovery rates (\(\gamma\) and (\(\gamma'\)), and SkinBAIT adoption-discontinuation rates (\(\alpha\) and (\(\delta\))—guides us in setting priorities for lab work and testing. By targeting these variables, we can focus on optimizing SkinBAIT’s effectiveness, ensuring its rapid adoption, and achieving sustained use to push the system toward eradication thresholds.
Ultimately, these models not only help us understand the theoretical dynamics of scabies control but also shape the practical steps we need to take in our project. By highlighting the importance of parameter tuning, widespread adoption, and the need for robust strategies to handle real-world variability, our modeling work lays the foundation for a successful experimental pipeline and impactful final product.
2. The Lotka-Volterra Approach
In previous models, we primarily explored the dynamics of disease transmission and recovery at a population level. These models have been useful for understanding how scabies, and other infectious diseases, can spread and persist within a population, as well as the potential impact of treatments on reducing infection. However, they did not address the crucial question of how effective our product, SkinBAIT, needs to be at an individual level. This is a key aspect because while it is important to control the spread of disease, it is equally vital to determine how potent the treatment must be to eliminate the parasites in an infected host. For this reason, we now shift our focus to understanding the required effectiveness of SkinBAIT in combating scabies and similar parasitic infestations.
To do this, we employ a mathematical approach inspired by the Lotka-Volterra model, which is commonly used to describe predator-prey interactions in nature. The Lotka-Volterra model is a system of differential equations that models the dynamics between two populations: predators and prey. The prey population increases in the absence of predators, but its growth is limited by the predator population, which in turn relies on the prey population for sustenance. In this model, predators reduce the prey population, and the prey supports the predators’ growth until a balance between the two populations is reached. This framework allows us to describe the interactions between two competing species in a simplified yet insightful way.
In the context of our project, we adapted the Lotka-Volterra model to represent scabies as the "prey" and our genetically engineered C. acnes cells, which produce toxins that target and eliminate scabies, as the "predators." This adaptation enables us to model the interaction between scabies and our toxin-producing bacteria, helping us investigate how effective our engineered solution needs to be to achieve satisfactory results. Note that there will be significant differences with the original model, but we were inspired by it nonetheless.
In our model, we focus on the effectiveness of the engineered bacteria in killing scabies mites. However, it is crucial to recognize that the timescale of toxin production is significantly shorter than the dynamics of scabies population growth and spread. Therefore, we assume a quasi-steady-state equilibrium condition, where the production of toxins occurs rapidly relative to the other processes in the model. This allows us to treat the effectiveness of the product as directly proportional to the effectiveness of the protein produced by the bacteria. Thus, our model simplifies the complexity of interactions, focusing on the key relationship between the toxin's effectiveness and its impact on scabies populations, making it a valid approach for assessing the necessary effectiveness of SkinBAIT.
The goal of this model is to predict how effective SkinBAIT needs to be in eradicating scabies infestations at the individual level. By simulating how the parasitic population (scabies) and the treatment population (engineered bacteria) interact over time, we can gain insights into the critical parameters that will influence the success of our product. This model helps answer questions such as:
How many bacteria (or how much toxin) is required to eliminate the scabies population?
How quickly does the scabies population decline once SkinBAIT is applied?
How does the spread of treatment across the population affect scabies eradication?
Understanding the relationship between scabies and SkinBAIT through modeling allows us to predict the product's effectiveness before and during laboratory trials. This reduces experimental costs, informs optimal dosage strategies, and provides an understanding of the dynamics between the parasite and the toxin-producing bacteria over time. Additionally, the model helps us refine the design of our synthetic biology solution, ensuring that SkinBAIT can be as efficient as possible in real-world applications.
The model simulates a network of individuals, where each individual has a concentration of scabies and a concentration of toxin-producing bacteria (predator). These concentrations are represented as two vectors:
s for the current parasitic load in each individual,
c for the concentration of bacteria or the amount of toxin being applied to fight scabies.
Each individual is a "node" in a network, and these nodes are connected based on a graph structure. In this case, a "graph" means a representation of how individuals in a population might be connected or interact with each other. Some individuals may be directly connected (like in a household), while others might be further apart (like in a community).
The interactions between these nodes are governed by ordinary differential equations (ODEs), which describe how the populations of scabies and toxin-producing bacteria change over time.
The model uses the following parameters:
\( \beta\) (Scabies growth rate) - The rate at which the scabies population grows if left untreated.
\( K\) (Carrying capacity) - The maximum population of scabies that can exist in the skin environment.
\( \delta\) (Death rate) - The natural rate at which scabies die off without any intervention.
\(\gamma\) (Effectiveness of the toxin) - The rate at which the scabies population decreases due to the presence of the engineered bacteria.
\( \alpha \) (Diffusion rate) - The rate at which scabies spread or how the toxin spreads through a network of individuals.
\(f(t)\) (External control function) - This represents the application of SkinBAIT (e.g., through "vaccination" or "application" or introduction of engineered bacteria) to the population over time.
The core of this model is the interaction between the scabies (prey) and the toxin-producing bacteria (predator):
The dynamics of the scabies population (prey) over time is described by the equation:
$$
\frac{ds}{dt} = \beta \cdot \left( 1 - \frac{s}{K} \right) \cdot s - \delta \cdot s - \gamma \cdot c \cdot s - \alpha \cdot L \cdot s
$$
This equation can be broken down as follows:
The first term, \(\beta \cdot \left( 1 - \frac{s}{K} \right) \cdot s\), represents the natural growth of scabies, which slows down as the population nears the carrying capacity.
The second term, \(\delta \cdot s\), represents the natural death rate of scabies.
The third term, \(\gamma \cdot c \cdot s\), captures how the bacteria kill scabies. The effectiveness is proportional to the concentration of both the bacteria and the scabies.
The fourth term, \(\alpha \cdot L \cdot s\), represents the diffusion of scabies across individuals in the population, using the Laplacian matrix (\(L\)) to capture how scabies spread through connected individuals.
For an (unweighted) graph \(G\) with \(n\) vertices, the adjacency matrix \(A\) of \(G\) is an \(n \times n\) matrix where: \(A_{i, j} = 1\) if there is an edge from vertex \(i\) ( \(v_i\)) to vertex \(j\) ( \(v_j\)), and \(A_{i, j} = 0\) otherwise.
This matrix fully describes the graph's structure, including all vertices and edges.
As we are modeling pairwise interaction between people, the adjacency matrix \(A\) will be symmetric (undirected graph).
This means \(A_{i, j} = A_{j, i}\) for all pairs of individuals in the network.
$$
A_{i, j} := \begin{cases}
1, & \{v_i, v_j\} \in E, \\
0, & \{v_i, v_j\} \notin E.
\end{cases}
$$
Another useful matrix is the degree matrix \(D\), an \(n \times n\) diagonal matrix where: \(D(i, i)\) is the number of edges incident to the vertex \(i\), and \(D(i, j) = 0\) for \(i \neq j\).
$$
D_{i, i} = \deg (v_i) = \sum_{j = 1}^n A_{i, j} = \sum_{j \, :\, v_j \in \mathcal{N}(v_i)} \, 1.
$$
where \(\mathcal{N}(v_i)\) are the neighbors of the vertex \(v_i\), which consists of all other vertices that are directly connected to \(v_i\) by an edge.
To explain how we can model diffusion in a graph, we take the simplest differential equation describing the dynamics of diffusion of a concentration \(x_i\) on a vertex \(v_i\) accounting for movement to/from all of its neighbors.
We assume that the rate at which \(x_i\) changes due to diffusion is given by the net rate from a vertex to its neighbors.
Thus,
$$
\frac{dx_i}{dt} = -\alpha \sum_{j\, :\, v_j \in\mathcal{N}(v_i)} (x_i - x_j).
$$
where \(\alpha\) is the diffusion coefficient (diffusivity) and controls how quickly the diffusion process occurs across the graph.
This expression can be rewritten noticing that \(\sum_{j\, :\, v_j \in\mathcal{N}(v_i)} \, \cdot = \sum_{j = 1}^n A_{i, j} \, \cdot\) where \(A\) is the previously defined adjacency matrix:
$$
\frac{dx_i}{dt} = -\alpha \sum_{j = 1}^n A_{i, j} (x_i - x_j) = \alpha \left( \sum_{j = 1}^n A_{i, j} \, x_j - x_i \sum_{j = 1}^n A_{i, j} \right).
$$
Moreover, using the identity \(\deg (v_i) = \sum_{j = 1}^n A_{i, j}\),
$$
\frac{dx_i}{dt} = -\alpha \left( \deg(v_i) \, x_i + \sum_{j = 1}^n A_{i, j} \, x_j \right) = -\alpha \sum_{j = 1}^n \left( \delta_{i, j} \, \deg(v_i) - A_{i, j} \right) x_j
$$
where \(\delta_{i, j}\) is the Kronecker delta function which is equal to 1 when \(i = j\) and zero otherwise.
The last expression can be connected to another widely used graph matrix called Laplacian matrix.
This is a symmetric, \(n \times n\) matrix defined as \(L := D - A\) whose elements are
$$
L_{i, j} := \delta_{ij} \deg(v_i) - A_{i, j} = \begin{cases}
-A_{i, j}, & i \neq j, \\
\deg(v_i), & i = j.
\end{cases}
$$
Thus, by the definition of matrix multiplication, the system in vector form is given by
$$
\frac{dx}{dt} = -\alpha \, L \, x.
$$
The concentration of toxin-producing bacteria remains fixed unless more bacteria are introduced through an external event, such as a "vaccination" or "treatment application." This process is controlled by a function \(f(t)\), which acts like a schedule for administering the bacteria to the population, similar to how a vaccine or treatment might be given at specific times. The function \(f(t)\) determines when and how much of the bacteria is introduced, and it allows for precise control over the treatment.
The bacteria concentration, represented by \(c\), doesn’t naturally grow or spread on its own. Instead, it relies on these external inputs—meaning we decide when and how much to apply. This is different from how scabies reproduce and spread on their own. The equation that captures this is:
$$
\frac{dc}{dt} = f(t)
$$
This means the change in the bacteria population over time depends entirely on \(f(t)\), or how we control their introduction. The bacteria then work to reduce the scabies population, with the effectiveness of this elimination controlled by a parameter \(\gamma\), which tells us how effective the bacteria’s toxin is at killing scabies.
The population and their interactions are represented using a graph, which is a mathematical structure used to model relationships. In this case, the graph \(G = (V, E)\) is made up of:
Nodes (\(V\)): Each node represents an individual in the population who could be infected with scabies.
Edges (\(E\)): These are the connections between individuals, showing possible pathways for scabies to spread. For example, an edge might represent physical contact or close proximity between two people.
The connectivity of the graph—how individuals are connected and how scabies might spread—is captured by a mathematical object called the Laplacian matrix (\(L\)). The Laplacian matrix helps us understand how scabies can move from one individual to another in the network. In simpler terms, it tracks the flow of scabies across the population based on who is connected to whom.
How the Model Works
Initial Conditions: At the start of the model, a small part of the population is infected with scabies, while the engineered bacteria are not yet introduced. This mimics a real-world situation where an infection has begun to spread, but treatment has not yet been applied.
Dynamics of Spread: As time goes on, the scabies population grows and spreads to other individuals in the network. This growth happens according to:
Growth Rate (\(\beta\)): This is the rate at which scabies reproduce and spread to new individuals.
Carrying Capacity (\(K\)): This is the maximum number of scabies that can exist in a given host before the population stops growing due to limitations in resources.
Scabies are also eliminated by:
Natural Death (\(\delta\)): The rate at which scabies die off naturally without intervention.
Toxin Production (\(\gamma c\)): The rate at which the engineered bacteria, once introduced, kill the scabies.
Diffusion Across the Network: This term, involving the Laplacian matrix \(L\), models how scabies move from one individual to another across the network, based on their connections.
Vaccination/Treatment application: The function \(f(t)\) introduces the bacteria into the population at specific times, similar to how a vaccination campaign might target certain parts of the population at different intervals. Once the bacteria are introduced, they begin to act locally, killing the scabies in the areas where they were applied.
Evolution Over Time: As the system evolves, both the scabies and bacteria populations change over time. The model tracks these changes at each node (each individual in the population), showing how the infection spreads or diminishes based on various factors like the effectiveness of the bacteria, the timing of treatment, and the network structure. Over time, the goal is to see how well the bacteria can reduce or eliminate the scabies infestation.
In this way, the model allows us to simulate different scenarios and determine how effective the bacteria need to be, and when and where they should be applied, in order to control or eradicate scabies in a population.
Parameter Sweep for Treatment Effectiveness
To gain insights into the effectiveness of SkinBAIT in various scenarios, we conducted a comprehensive parameter sweep. This approach involved systematically varying key parameters in our model to identify the optimal combinations for effectively controlling scabies infestations.
First, we explored the relationship between the effectiveness of the toxin, represented by the parameter \(\gamma\), and the intervals of vaccination or treatment application. By varying \(\gamma\) against different vaccination intervals, we aimed to understand how varying levels of treatment effectiveness influence the dynamics of scabies populations over time. This exploration was critical because it allowed us to assess the impact of different treatment strengths and their timing on the overall success of the intervention.
Next, we examined the interaction between \(\gamma\) and vaccination size, which refers to the number of nodes (individuals) that receive the treatment during each vaccination event. This second sweep was crucial for evaluating how the scale of vaccination efforts impacts the overall effectiveness of SkinBAIT. By analyzing the relationship between treatment effectiveness and the size of the vaccinated population, we could identify the most effective strategies for application.
Through these parameter sweeps, we aimed to determine the optimal combination of vaccination intervals, treatment sizes, and product effectiveness required to achieve satisfactory results in eradicating scabies. This systematic exploration not only enhances our understanding of the dynamics between scabies and SkinBAIT but also informs future experimental designs and strategies for real-world applications, ensuring that our approach to combating this parasitic infestation is both efficient and effective.
2.1 Conclusions of the Lotka-Volterra Approach
The application of the Lotka-Volterra model to our project has yielded several key insights regarding the dynamics between scabies and our engineered bacteria, SkinBAIT. These insights are crucial for shaping our experimental pipeline and guiding our strategy for introducing this innovative treatment to society.
Understanding Treatment Effectiveness: Our parameter sweeps revealed that the effectiveness of the toxin, denoted by the parameter \(\gamma\), plays a significant role in determining the success of SkinBAIT. Specifically, we found that higher effectiveness is essential for achieving rapid declines in the scabies population, especially when treatment is applied at strategically timed intervals. This highlights the need to focus on optimizing the potency of our engineered bacteria to ensure it can effectively combat scabies infestations.
Optimal Vaccination Strategies: The exploration of vaccination intervals and sizes provided valuable information on how to effectively administer SkinBAIT. We learned that the timing and scale of treatment application are critical factors in controlling scabies populations. By identifying the optimal intervals and sizes for treatment, we can design targeted vaccination campaigns that maximize the impact of SkinBAIT, thereby enhancing its overall effectiveness in real-world scenarios.
Modeling Dynamics Over Time: The ability to simulate the interactions between scabies and SkinBAIT over time allowed us to visualize potential outcomes of different treatment strategies. This dynamic modeling approach helps us anticipate how varying treatment effectiveness, vaccination intervals, and population sizes may influence the eradication of scabies. Such foresight is invaluable for refining our experimental designs, allowing us to conduct trials that are better aligned with our goals.
Informing Experimental Designs: The insights gained from our modeling efforts have direct implications for our experimental pipeline. We can now prioritize the development of SkinBAIT formulations that align with the identified parameters for effective treatment, ensuring that our laboratory trials focus on the most promising strategies. Additionally, understanding the interactions within the population helps us design experiments that mimic realistic conditions, leading to more relevant results.
Preparing for Real-World Application: As we plan to introduce SkinBAIT to society, our findings provide a roadmap for effective implementation. By demonstrating the importance of targeted vaccination strategies and the relationship between treatment effectiveness and population dynamics, we can communicate the value of our solution to stakeholders, healthcare professionals, and the community. This ensures that our approach is not only scientifically sound but also practically applicable in real-world settings.
Future Research Directions: The modeling framework established through the Lotka-Volterra approach opens avenues for further research. We can explore additional factors influencing scabies dynamics, such as environmental conditions, host behavior, and potential resistance development. This ongoing research will enhance our understanding and help us adapt our strategies to evolving circumstances.
In summary, the Lotka-Volterra model has significantly advanced our understanding of the relationship between SkinBAIT and scabies, providing us with crucial insights that will guide our experimental pipeline and inform our approach to introducing this innovative treatment to society. By focusing on optimizing treatment effectiveness and strategic application, we are well-positioned to tackle scabies infestations effectively and make a meaningful impact in the fight against this parasitic disease as a first step to combat infectious diseases.
