Team:Warwick/Modelling/Ebola

Introduction
Due to the recent Ebola outbreak in western Africa, we began to wonder: could our system be adapted to treat Ebola? You can imagine our surprise upon discovering that one of the leading methods currently being researched is siRNA targeting. As a result, this would mean that with the specific sequence, we could insert this into our system, inject an individual and the replicon would take care of the rest. Upon full treatment, we would then administer theophylline, thereby stopping the replicon.

Naturally we cannot work with Ebola in our lab, so we resorted to the computer. We created an epidemiological model using partial differential equations to create animations to show the spread of Ebola, and to begin to look at a vaccination policy, i.e. how many individuals would we need to vaccinate to eradicate the infection? These kinds of systems form the foundations of modelling vaccination procedures, and are used by many to predict outcomes.

The Equations
[1] We began contruction of our model by first going to the literature. Most of our work is based upon that of Li and Zou (link above), who created a mathematical model for spatial spread of infectious diseases within a spatially continuous domain. Their focus was on diseases with a fixed latent period. This is not a problem for Ebola as the average latent period is between 8 and 10 days. Li and Zou made assumptions in the making of their model, these are: the disease has full immunity following "recovery", the population inhabits a space that is spatially heterogeneous as well as continuous (the model initially focuses on just one spatial dimension) and a few mathematical simplifications such as assuming that the rates are constant.

The model that we adopted is as follows: \[\begin{cases} \frac{\partial S(t,x)}{\partial t}= \mu + D_S \frac{\partial ^2 S(t,x)}{\partial x^2} -kS(t,x) -rI(t,x)S(t,x) \\ \frac{\partial L(t,x)}{\partial t}= D_L \frac{\partial ^2 L(t,x)}{\partial x^2} +rI(t,x)S(t,x) - \bar{d}L(t,x) -\epsilon\int_{-\infty}^{\infty}rI(t-\tau,y)S(t-\tau,y)f_\alpha(x-y)dy \\ \frac{\partial I(t,x)}{\partial t}= D_I \frac{\partial ^2 I(t,x)}{\partial x^2}- (\sigma +\gamma+k) +\epsilon\int_{-\infty}^{\infty}rI(t-\tau,y)S(t-\tau,y)f_\alpha(x-y)dy \\ \frac{\partial R(t,x)}{\partial t}= D_R \frac{\partial ^2 R(t,x)}{\partial x^2} +\gamma_L L(t,x)+\gamma I(t,x)- kR(t,x) \end{cases} \] where we have that:
$S$ represents the susceptible class of individuals (i.e. those that can be infected)
$L$ represents the latent class (i.e. those who have been exposed and infected but not capable of infecting others)
$I$ represents the infectious class (i.e. those who can infect others)
$R$ represents and the removed class (i.e. those who have been removed from the chain of transmission e.g. through death or recovery) respectively
$\sigma$ is the disease-induced mortality rate
$\gamma$ is the recovery rate
$k$ is the natural death rate
$D$ is the diffusion rate, with the subscript noting the different diffusion rates of the classes of individuals
$\mu$ is the recruitment of susceptible individuals
$\bar{d}$ is a parameter incorporating several constants
Note also that the system incorporates time delays, because, as is the case with any infection, it takes time for it to affect an individual.

Mathematically it is very difficult to deal with partial differential equations directly, even in a numerical sense. As a result, we further went on to simplify the situation by introducing a few more mathematical assumptions to create a visual model in MATLAB illustrating how the disease would spread if we had a group of people arranged in a rectangular array. [2]

To do this we adapted the MATLAB code taken from the reference (above). We changed the system previously mentioned to now incorporate a stochastic element of disease propagation, thereby illustrating in each run of the code, a potentially different outcome. The animation below illustrates a single run of this experiment. The green represents those individuals who are susceptible, red is those who are infected with the shade of red being proportional to the stage of infection (darker means earlier on in the infectious cycle) and grey represents those individuals who are "recovered" which also includes those who have died.The conditions used are as follows:
- We fix the dimensions of the array to be 10x10.
-The latent period of infection is 10 days.
-The maximum number of days to model is 30.
- We started with a single infected individual.


Below we have included a figure illustrating the effect of running the experiment several times and taking the average result to determine the number of people infected. The figure displays an area plot of the SIR percentages over time.

Conclusion

While our program still needs developing, we hope it could be a useful tool for other IGEM teams, and scientists in modelling disease propagation. Please feel free to download the software and develop it more/adapt it for your own purposes. Possible adaptations could include options for applying this model to different population densities, different probabilities of infection from different cultures, different densities of vectors carrying the disease. We hope that due to the open source nature of IGEM, that this program can be continuously updated by IGEM teams throughout the years. This data could be useful in predicting which diseases will become dangerous, and how quickly, by fitting parameters in the above equations to real time data for such diseases. With population densities taken into account, this model could be used to predict which areas are most at risk from infection, and a range of dates for when diseases will reach certain places. Being able to accurately and quickly predict the scale of diseases will allow for better methods to curb epidemics such as the ebola epidemic.

It is clear to see that Ebola is an epidemic of enormous danger to millions of people, and it is our hope that our replicon can be used to help in the fight against the spread of Ebola, and other diseases. Our replicon could vastly improve the effectiveness of siRNA targeting, in theory vastly improving current Ebola treatments, and potentially saving lives.