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, with the following conditions:
- 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
.......Write up conclusion...........