Thus, on average, each infected individual generates b 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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