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Lab 10 1 Lab 10: Spreading of a Virus Background Information In every field of engineering, it is often necessary to analyze the behavior of extremely complex systems. Examples of such systems include the cooling of a metal rod in a pool of liquid, the transfer of materials throughout a chemical plant, the flow of electricity in a circuit, etc. If a strict mathematical analysis of such systems is performed, a set of high-order differential equations often results. However, it is usually extremely difficult (or even impossible) to solve such systems in a timely manner. This lab demonstrates a way to simplify such problems through the use of a discrete time analysis. The method will ignore the exact solutions to differential equations, and instead form an estimate by modelling the system’s behavior. The example in this lab will consider population dynamics to study the spread of a virus. Ironically, this has applications not only in human viruses but also computer viruses. Just like humans, computers typically spread viruses through contact with others. To begin developing this model, we will look at one individual and see how they can get sick. The easiest way to get sick is to have contact with someone. Take the following vector, P, and imagine you are in the middle, P(2): Healthy, P(1) You at P(2) Sick, P(3) There are 3 types of people around you: Healthy (0), Sick (1) and Immune (2). An Immune person will be one who was Sick and has now recovered from the virus. If we convert the above vector to a numeric, it would be the following (assuming you are healthy): P(1) =0 P(2) = 0 P(3) = 1 The chances of you getting sick will rely on two conditions being met (1) How many sick people you are exposed to and (2) the chance that being exposed actually leads to you being sick which is determined by the aggressiveness of the virus. The first condition would be met when P(1) = 1 or P(3) = 1. The second condition can only happen if the first condition occurs. The second condition would be determined by taking the number of sick neighbors (1 in the above example) and insert the value into the equation below: ChanceSick = 0.25*(Number of Sick Neighbors) This would then be the probability that you get sick. Thus if one person around you is sick, you have a 25% chance of getting sick. If two people around you are sick, then you have a 50% chance of being sick. The more aggressive the virus the higher these probabilities would be. Part A. One-Dimensional Model of Virus Spreading Your task in Part A is to expand this model to include more than 3 elements in the vector, P. Below is the scenario for Day 1 and how to determine the state of the population going into Day 2.
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