Solve the problem using exponential growth and decay of antiderivative.

Question

Direction: Read and analyze the problem carefully. Solve the problem using exponential growth and decay of antiderivative. Show your solution and answer the following question below.

1. Suppose that the treat of covid-19 grows exponentially. In the Philippines, it has 10,000 COVID patients. Two thousand COVID patients are confine in hospitals. After a year it increased to 3200 COVID patient. Assume that the COVID patients follow a limited growth model. Sketch the graph.

a. How many COVID patients could we have after three years?

b. Describe the graph, was there an increase or decrease of COVID patient after 3 years?

c. Use what you know about the exponential growth and decay of antiderivatives with regard to COVID

patients, and explain the probable cause why this happens.

Answer 1

Hi Lily M.,

There is some information missing in your question, no doubt supplied elsewhere in your course. If you were to assume that (all data refer to COVID only) patients in hospitals exactly keeps pace with patients total, then you could set up an exponential growth equation for patients in hospitals, and apply the same exponential coefficient (for the ^(+)k₁t) then also to the total patient population. However, this does not address the "decay of the antiderivative" portion of the problem.

Let's see what "derivative" and "antiderivative" refer to here. I think, derivative logically refers to the rate of new infections, and antiderivative refers to the current population of infected patients. (This is not, however, the only way these terms might be applied here; derivative might refer to the current rate of cure, and antiderivative to the total population of cured, post-infected patients.)

You might well approximate that the individual infections and recoveries are almost balanced, and you are dealing with a "moving average" incidence of infection. In that case, as single exponential growth value does it, until a significant portion of the entire population of the Philippines has been infected once. The infection rate then must taper, as naive subjects become scarcer, even though infected populations remain elevated as they "ride down" the infection "antiderivative".

But in order to successfully model, you still need some data on average time for (infection -> cured). Why is that? If the time to be cured is very long, that implies that the rate of new infections is low, but that the numbers of infected (and contagious) patients will continue to rise for a very long time, because the population is not cycling through COVID rapidly. But if the time to be cured is very short, that implies that the Phiippine population will cycle through COVID rapidly, and then the numbers of infected patients will decline presently as herd immunity grows.

Unfortunately, data so far indicate (to me) that populations are cycling through COVID, all right, but that the rise of new variants will just keep the rates up -- after all, the variants are like new diseases, they can infect people who have recovered from Delta, and so forth.

It's a great reason to make sure you have up-to-date boosters! So you can soon say Kamusta to your friends again!

--Cheers, --Mr. d.