Challenge Team Interim Report

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    Team Number: 044

    School Name: Highland HS

    Area of Science: Epidemiology

    Project Title: Deadly Viral Outbreak Simulations

Final Report

We propose to simulate the spread of a deadly, airborne virus through a given section of city, under given environmental factors. In our simulation we will include mutation rates for the virus, symptom discovery rates, mortality rates for victims of the virus, and the rate at which the virus would spread through the air. In short, using a given source of infection, we will simulate the spread of infection.

Our program will provide key insight into the areas and people most at risk during an outbreak. When it is known where a disease will travel, how it will react to certain stimuli, and how it will act in certain situations, then measures may be taken to stop the spread of the disease faster, more effectively, and with greater efficiency. Our project is a possible supercomputing project because of the many factors and equations involved in an epidemiological simulation such as what we purpose. The project could be expanded to cover a greater area and include a greater population. Diseases are prevalent in our society. An epidemic is a large, short outbreak of a disease. Some of the factors that are involved in the spread of an infectious disease are infectious agents such as: mode of transmission, latent period, susceptibility and resistance. There are many ways in which diseases are transmitted, but in our project we will focus on diseases that are transmitted person to person. Some diseases that are transmitted this way are Measles, Chickenpox, Mumps, Rubella, Smallpox, Influenza, Poliomyelitis, Herpes, Strep throat, and Syphilis. In our project we plan to make a epidemiologic model of the Influenza virus.

We plan to model an epidemic of the influenza virus by using the S-I-R method. This method divides a given population into three different groups. A person in our model is first Susceptible, then becomes Infected, and then becomes Removed. Removed means simply immune to the disease or dead. This is an S-I-R simulation. S-I-R simulations are designed to simulate other epidemics like: Measles, Chicken pox, and Rubella.

This method also uses some assumptions: 1. The population size, in our equations N, is constant. This is acceptable because we are modeling the flu epidemic, which runs its course in less than a year so we don't have to worry about births or account for massive changes in population. 2. The population size, N, is homogeneously mixed in our sample environment so we can use a constant rate of infection, which in our equations is r or the daily contact rate. 3. Individuals recover and are removed from the infected class at a constant rate proportional to the number of invectives, In our equations a or the daily removal rate. 4. The latent period, the time between the moment of exposure to the time when one can infect the Susceptible class, is zero. Thus:

(NS(t)) = -rSNI
(NI(t)) = rSNI - aNI
(NR(t)) = aNI
NS(0) = NSo > 0, NI(0) = NIo > 0, NR(0) = NRo => 0
NS(t) + NI(t) + NR(t) = N
And if you divide out N:
S(t) = -rSI
I(t) = rSI - aI
S(0) = So > 0 , I(0) = Io >0
R(t) = 1 - S(t) - I(t)

Another key element of our simulation is given in a theorem that gives us a way to tell if an epidemic, an exponential growth of a disease, is to occur. This theorem is: Let (S(t), I(t)) be the solutions of the previous rate equations. If the average number of contacts of an infective, r / a, times the number of susceptibles at t = 0 is less than or equal to one, then I(t) decreases to zero as t approaches infinity and we have no epidemic. If r / a multiplied by the number of susceptibles at t = 0 is greater than one, then I(t) first increases to the maximum number of infected which is equal to: 1 - Ro - ( a / r ) - [ln( S(infinity) / So )] / ( r / a) and then decreases to zero as t approaches infinity. The susceptible fraction S(t) is a decreasing function and the limiting value S(infinity) is the unique root in (0, ( a / r)) of the equation.

What all this means is that we now have a way to map out the exponential growth of an epidemic and we can now get real data to compare with our model. We must digress and say that this model does not account for age and other factors. However we are currently researching a way to adapt this model account for some of these factors. We are also researching how to represent this graphically to see the spread of the disease with diffusion equations.

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