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equation must also include positive term to account for the newly infected.

The values of β and γ do depend on biological as pects of the particular disease pathogen, but they also depend on properties of the population prior to the outbreak and the response of public health offi cials. If the underlying health of a community is poor, γ could be lower than for a generally healthy community facing an outbreak of the same disease. Conversely, effective treatments and a robust health care system could increase γ. The transmis sion rate β is affected by whether or not the patho gen can be aerosolized and how long it survives on surfaces, but it could be reduced through preventa tive measures such as mask wearing that would re duce the likelihood that a single encounter would result in a new infection or effective social distanc ing, which would reduce the number of contacts. As derived, the model (1) - (3) is governed by three parameters, β, γ, and N . However, we can reduce this model to only one parameter without losing any aspects of the behavior of the system though a process called nondimensionalization . We now ex plain this process and then analyze the behavior of the solution directly from the equations without actually solving the system. The first step is to switch our unknown functions from the total people in each compartment to the proportion of the total population in each compart ment. That is, we rescale the unknowns in the sys tem by the total population as: Nondimensionalization and R 0 Below, we will see how this eliminates N as ex plicit parameter without effectively changing the model. Also, since of these new variables represent fractions of the population and everyone in the population is in one of these groups, we have s + i + r = 1. The key to the second step lies in carefully choos ing the problem ’ s timescale. There is no inherent link between the disease outbreak and how fast the earth turns on its axis. Therefore, instead of meas-

Infection is more complicated to model. Recovery of an individual is unaffected by the total number of infectious or recovered people. The growth of new infections, however, is driven by interactions between the infectious and the susceptible. The more infectious people there are, the more people who are able to someone else, and so it is clear that the rate of change of I is proportional to I . Howev er, no matter how many people are infectious, new infections may only occur if there remain suscepti ble people to catch the disease, and so the more susceptible people there are, the more who will be come infected. Thus, the rate of new infections must be also directly proportional to S . Together, these mean the new infection term transferring sus ceptible people to infectious must be proportional to the product IS . From another point of view, the rate of change of S must be proportional to S , but with a proportionali ty constant that actually depends on I . This “ constant ” consists of two factors. The transmis sion rate or contact rate β measures the number of contacts with another person per unit of time that were close enough to result in a new infection if the contact had been with an infectious person. The other factor in the proportionality is therefore not truly the number of infectious people I , but instead it is the fraction of the total population that is infec tious, or I / N . This is important because it represents the likelihood that the susceptible person ’ s contact was infectious. Thus, β ( IS / N ) represents people leaving the susceptible population and would be negative in the dS/dt equation, but it would be posi tive in the dI/dt equation as these are the people entering the infectious compartment. Therefore, we have the other two equations to complete our sys tem:

Virginia Mathematics Teacher vol. 47, no. 1

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