# Tackling Sleeping Sickness Using a Mathematical Model

When you think of the Trojan Horse, what comes to your mind? Perhaps not the most positive associations: In the tale about the Trojan War, the Greek soldiers left it behind to deceive and eventually eradicate the Trojans. Nowadays, „trojan horse“ labels a specific type of malware program that infect our computers and open backdoors for hackers while pretending they do something benign. Let's face it: The Trojan Horse's reputation could be better.

And yet last week I came across an interesting paper, that shows how a Trojan Horse-technique can be harnessed in a positive way to trick the pathogen of sleeping sickness, scientifically known as Human African Trypanosomiasis (HAT). It is caused by the protozoan parasite Trypanosoma brucei and transmitted by the tsetse fly - to both humans and animals. Sleeping sickness is fatal if untreated. The WHO estimates there are currently 30,000 people infected with HAT.

In his paper published in PLoS Neglected Tropical Diseases, Jan Medlock of the Oregon State University and his team of researchers present a mathematical model that simulates the behavior of the T. brucei and the population genetics of the tsetse fly. It furthermore tests an approach of colonizing tsetse flies with another, genetically modified parasite that kills T. brucei.

This approach is called paratransgenesis, and it works pretty much like the Trojan Horse did.

Paratransgenesis is a technique where a symbiont of a disease-carrier is genetically modified (GMO) in a way that it can eradicate the disease from its host. In the case of HAT, the symbiont is the bacteria Sodalis (its GMO version, respectively) that sneaks inside the tsetse fly's gut with the help of another parasitic bacteria present in the flies' guts, Wolbachia.

Let's check again if our analogy with Greek mythology still holds: The Trojan Horse (Wolbachia) cloaks the Greek soldiers (Sodalis) so they can sneak inside Troy (tsetse fly) and kill its inhabitants (T. brucei). Yup, works. Being a non-geneticist but computer scientist, I would call this a bugfixed tsetse fly.

So where does math come into play? The researchers formulated the way tsetse flies reproduce and the spread of the disease within the flies mathematically. For that purpose, they used differential equations which are commonly used to express laws of nature. They then implemented the model computationally and let it simulate how the „cured“ version of the tsetse fly would spread and how fast the disease could be eliminated.

Medlock's model simulates how effectively the approach works and how long it would take until the bugfixed tsetse fly could have completely replaced its infectious version if released in the wild. The model shows that, after releasing 20% paratransgenic tsetse flies into an infected area, it will take six years until the cured ones gain the upper hand. It shows also that in the same time the spread of HAT could be reduced to zero. That is pretty impressive.

Mathematical models are like the crystal balls of science, just their reputable version. In simple words, creating a mathematical model means mapping a real-life system into mathematical formulas and letting simulations run that either help better understand the system or predict its future behavior. These models are extensively used in physics, engineering, biology, sociology and also in computer science.

Medlock explained to me via email: „Mathematical models, when best used, force us to explicitly identify what we think are the key features of a biological system, allow us to describe these features in an unambiguous way, and help us to understand the implications of these features on the behavior of the system.“

There are more of these mathematical models in our daily lives than you might think. A special form of them, the deterministic finite automata (DFAs) are implemented in traffic lights, vending machines, video games and Apple's speech recognition software Siri - there they'll run every time you issue a voice command. DFAs were the first mathematical models I got to know as a freshman.

So the OSU team proved that the paratransgenic method is actually feasible. Now what? Can it be implemented right away? Can we start cure sleeping sickness within the next six years?

The answer is no. First and foremost, the GMO version of Sodalis capable of fighting T. brucei does not exist yet. Moreover, mathematical models are based on assumptions. Medlock's team based its model on these:

- Mortality of Wolbachia is low in the tsetse fly
- The (to-be-developed) GMO-Sodalis is effective in fighting the parasites

For the simulations to produce meaningful results, the OSU team introduced empirical parameters that incorporate a priori knowledge into the calculations. This is called a white box mathematical model. In the paper, these parameters include biting rates, human and animal population size, HAT incubation time and the reproductivity rate of Wolbachia in the tsetse fly, as well as the tsetse fly's life cycle.

The problem with such models' assumptions is that they might not be actually true in the real life. The empirical data is certainly correct, but it means also the model's results do hold only for a limited set of circumstances.

Of course, sometimes assumptions are necessary. The Greek successfully assumed the Trojans would interpret the Trojan Horse as a gift and bring it into their city. Would they have burnt it down instead, the Greek invasion would have failed.

Speaking of the Trojan Horse: As you will have noticed, I have silently sneaked into Scilogs.com - not with the intention to burn it down (promised!) but to tell you more about the secret math and programs that run our world and either aid science or are inspired by it.

Welcome to #Algoworld!

**References**:

Medlock J, Atkins KE, Thomas DN, Aksoy S, Galvani AP (2013). Evaluating Paratransgenesis as a Potential Control Strategy for African Trypanosomiasis. PLoS Negl Trop Dis 7(8): e2374. doi:10.1371/journal.pntd.0002374

WHO factsheet about Human African Trypanosomiasis: http://www.who.int/mediacentre/factsheets/fs259/en/