Logistic & Gompertz Functions

  1. Consider the logistic function defined by the formula \[ \ell(x) = \frac{530}{1+8\mathrm{e}^{-0.02x}}\,. \]
    1. What is the domain and range of the function \(\ell\,?\)
    2. What is the value of \(\ell(200)\,?\) Write down an exact expression for the value, but also express the value as a decimal rounded to three decimal places.
    3. For what value(s) of \(x\) does \(\ell(x) = 100\,?\) Write down an exact expression for the value(s), but also express the value(s) as a decimal rounded to three decimal places.
  2. Consider the Gompertz function defined by the formula \[ g(x) = 530(0.1)^{\left(0.96^x\right)} \]
    1. What is the domain and range of the function \(g\,?\)
    2. What is the value of \(g(50)\,?\) Write down an exact expression for the value, but also express the value as a decimal rounded to three decimal places.
    3. For what value(s) of \(x\) does \(g(x) = 300\,?\) Write down an exact expression for the value(s), but also express the value(s) as a decimal rounded to three decimal places.
  3. In March 2014, the World Health Organization responded to a large outbreak of Ebola in Western Africa. Here is the data they reported for the total (cumulative) number of cases in Guinea, Liberia, and Sierra Leone, by month for the first year and a half of the epidemic, according to the CDC:
    Months since March 2014 0 1 2 3 4 5 6 7 …
    Total Ebola Cases 120 234 309 599 1322 3052 6553 13540 …
    … 8 9 10 11 12 13 14 15 16 17
    … 17099 20371 22257 23894 25378 26498 27345 27740 28040 28265
    The total number of cases could initially be accurately modelled by an exponential function. But as the spread of the disease began to slow, and the data began to take on a sigmoid shape, a different model was needed.
    1. Sketch a scatter plot of the data on the axes below. (plan the scaling of your axes before you begin!)
      Then using technology capable of regression, ideally Desmos, find an equation \(P(t)\) for the logistic function that best models this data, and carefully sketch it on the plot.
    2. One of the defining features of a logistic function is that is has a strict upper-limit, usually called the carrying capacity. What is the upper-limit that your specific logistic model for this data indicates, and what is does this number mean in the context of the Ebola epidemic?

    3. The inflection point of a logistic curve is the point that is vertically half-way between zero and the upper-limit, where the curve “turns” and begins to level-out. I.e. it’s the point where the logistic function stops increasing exponentially. In relation to this situation, the inflection point corresponds to the time when the number of new Ebola cases each month starts to decrease rather than increase.

      During which month after March 2014 is the inflection point of our logistic model?

    4. There’s an obvious inconsistency between the data and the upper-limit this logistic model suggests; what is it?
    5. According to the CDC, the final tally of Ebola cases in west Africa once the pandemic was declared over, was \(28616.\) With this fact in mind we can find a better model. Plot the following Gompertz function along with your logistic model and the data to visually verify it’s a reasonable model for the total number of Ebola cases. \[ G(t) = 28616\mathrm{e}^{-18(0.65)^t} \]
    6. (Extrapolate) According to this Gompertz model, how many total Ebola cases will have been reported by the 18th month since March 2014?
    7. According to this Gompertz model, what month did the case-total hit \(28600?\)
  4. Here is data from the US Census Bureau for the population of the United States (in millions) for select decades in the past century.
    Year 1920 1930 1940 1950 1960 1970 1980 1990 2000 2010 2020
    US Pop. (millions) 106 123 132 151 179 203 227 249 281 309 331
    If you were to plot this data, you’d notice that the data trends along a very slight sigmoid-shaped curve, as population data from high-income countries often does.
    1. Find a logistic function to model the US population over time. I recommend that to make your independent variable something like “decades since 1920”.
    2. What is the upper-limit on the US population that your logistic model suggests?
    3. What are the coordinates of the inflection point of your logistic curve that models the population? I.e. during what decade was the US population growing the fastest?