In class, we discussed the use of difference equations for modeling continuous systems. This practice assignment will task you with building a spreadsheet-based simulation (using difference equations) to model the spread of the Hong Kong Flu through New York City in the winter of 1968-1969.
The equations below can be used to describe the cycle of a disease. Equation 1 shows that the susceptible population decreases as more people within the susceptible group become infected (more people become infected based on the infection rate and how many people are already infected). Equation 2 shows that the infected group changes based on the number of people who are susceptible to be infected and how many people are recovering, and Equation 3 shows that the number of people in the recovered group changes based on the recovery rate and the number of already infected people. The state variables in this system are St, It, and Rt. The definition (and units) for each of the variables in the three equations are given below the equations themselves.
Eq 1: St+∆t = St-λStIt∆t
Eq 2: It+∆t = It+[λStIt-(δIt)] ∆t
Eq 3: Rt+∆t = Rt+δIt∆t
Where
St = # in susceptible group at time t (people)
It = # in infected group at time t (people)
Rt = # in recovered group at time t (people)
δ = recovery rate (time / person)
λ = infection rate (time / person)
∆t = the length of time between examinations of the state variables (time)
Use Equations 1, 2, and 3 to build a model of the disease progression over time in Excel. As a hint, you will likely want to use different columns in Excel for: the number of days that have passed, the Susceptible Population (St), Infected Population (It), Recovered Population (Rt). In such a set-up, each row will represent t (the number of days that have passed), and each row will contain a value for “days into disease cycle”, St, It, and Rt. You’ll also need to have placeholders for some input model parameters: infection rate (λ), recovery rate (δ), and ∆t, which you’ll use to fill in the equations that calculate the values for St, It, and Rt. Start your model at “0” days into the disease cycle (this row will represent the initial conditions).
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