We have already estimated the average period of infectiousness at three days, so that would suggest kВ =В 1/3. We don't know values for the parameters b and k yet, but we can estimate them, and then adjust them as necessary to fit the excess death data. The trace level of infection is so small that this won't make any difference.) Our complete model is (Note: The sum of our starting populations is not exactly N, nor is the sum of our fractions exactly 1.

In terms of the scaled variables, these initial conditions are s(0)В =В 1 2В Thus, our initial values for the population variables are S(0)В =В 7,900,000 We will assume that there was a trace level of infection in the population, say, 10 people. For this particular virus - Hong Kong flu in New York City in the late 1960's - hardly anyone was immune at the beginning of the epidemic, so almost everyone was susceptible. Why is it reasonable that the rate of flow from the infected population to the recovered population should depend only on i(t) ?ДЇinally, we complete our model by giving each differential equation an initial condition.What assumption about the model does this reflect? Now explain carefully how each component of the equationДЇollows from what you have done thus far. Explain how the corresponding differential equation for r(t),ДЇollows from one of the assumptions preceding Step 4. Now explain how this equation leads to the following differential equation for s(t). Explain carefully how each component of the differential equationДЇollows from the text preceding this step. Let's see what these assumptions tell us about derivatives of our dependent variables. A 'recovered' person can still feel miserable, and might even die later from pneumonia.) (Strictly speaking, what we mean by 'infected' is really 'infectious,' that is, capable of spreading the disease to a susceptible person. For example, if the average duration of infection is three days, then, on average, one-third of the currently infected population recovers each day. We also assume that a fixed fraction k of the infected group will recover during any given day.Thus, on average, each infected individual generates bВ s(t) new infected individuals per day. If we assume a homogeneous mixing of the population, the fraction of these contacts that are with susceptibles is s(t). Not all these contacts are with susceptible individuals. In particular, suppose that each infected individual has a fixed number b of contacts per day that are sufficient to spread the disease. We assume that the time-rate of change of S(t), the number of susceptibles, 1 depends on the number already susceptible, the number of individuals already infected, and the amount of contact between susceptibles and infecteds. The only way an individual leaves the susceptible group is by becoming infected. No one is added to the susceptible group, since we are ignoring births and immigration. Next we make some assumptions about the rates of change of our dependent variables: Sketch on a piece of paper what you think the graph of each of these functions looks like.

Under the assumptions we have made, how do you think s(t) should vary with time? How should r(t) vary with time? How should i(t) vary with time?.

The two sets of dependent variables are proportional to each other, so either set will give us the same information about the progress of the epidemic. It may seem more natural to work with population counts, but some of our calculations will be simpler if we use the fractions instead. The recovered fraction of the population. The infected fraction of the population, and The susceptible fraction of the population, So, if N is the total population (7,900,000 in our example), we have The second set of dependent variables represents the fraction of the total population in each of the three categories. Is the number of infected individuals, and Is the number of susceptible individuals, The first set of dependent variables counts people in each of the groups, each as a function of time: We consider two related sets of dependent variables. The independent variable is time t, measured in days. As the first step in the modeling process, we identify the independent and dependent variables.

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