Problem Set 1 for CaSB 150
Due April 17, 2025
(Late problem sets will lose 10 points per day)
1. A. When an exponentially growing number of cases for a disease—imagine a disease spreading like COVID-19—is increasing every 5 days by a factor of e=Exp(1)=2.71828..., express the total number of cases, N(t), as the solution for continuous exponential growth. Assume the infection started with one person: N(0)=1. What is the value for the per-capita (a.k.a. intrinsic) growth rate, r, of the disease? Let the units of time, t, be in days for your answer.
B. Write down an expression for how many new cases you should expect to appear on day t using your expression from part A. What is the fraction of new cases on day t relative to the total number of cases on day t?
C. Assuming for now that everyone who is infected gets tested immediately and that the result is processed and reported 3 days later, write down a formula for how many positive test results, R(t), will be reported on day t.
2. A. If 0.5% of cases result in death and if death always happens 3 weeks after
infection, write down a formula for the number of deaths, D(t), on day t.
B. Evaluate the number value of your expressions from problems 1 and 2.A
when the infection has been spreading for 5 days and for 100 days.
Comment on the meaning of the numbers and any unusual values.
3. Consider the case that growth rate is exponential but modified by being multiplied by another function with a negative exponential
where τ is a time constant, r is a constant, N(t) is population size, and t is time.
a. At early times when N(t)<< N(τ), what does this equation simplify to? Solve the simplified equation for N(t) and describe the growth for this case. (Hint: To simplify, recall that Taylor series can often be helpful.)
b. At long times when N(t)>>N(τ), what does this equation simplify to? Solve the simplified equation for N(t) and describe the growth for this case.
c. Sketch a graph for dN(t)/dt versus time based on your answers to parts a. and b.?
d. For part a. is there a different mathematical space in which you can find a function that is linear versus time to obtain a perfect approximation to the equation? If so, what is that space? Or is the original linear space already the best space possible? (This is equivalent to looking at ln(N(t)) versus t space for exponential growth as we did in class, which works because the lnN(t) is linear with time, meaning dln(N(t))/dt is constant.)
4. A. Analyze the data for new confirmed cases per day of COVID-19 in the
table below to see if it is exponential? If not, formulate and solve a differential equation that would better describe these data? Repeat the same analysis but for the data for testing capacity—maximum number of tests processed per day—in the same table. Comparing the growth rate of testing capacity with the growth rate of new confirmed cases, what can you conclude about the driving cause of the growth rate for the data for confirmed cases?
B. Consider the cases where the number of new cases of infected individuals each day has flattened and is now constant at about 2,000,000 per day in the nation. If 20% eventually show symptoms and 50% of those with symptoms eventually get tested, how many new cases per day would eventually be reported?
C. In contrast to B., if we can only conduct 150,000 tests per day in the
nation, we only test people with respiratory symptoms, and about 30% of people with respiratory symptoms have COVID-19 (as opposed to allergies, flu, etc.), how many new cases per day would be reported?
Given this testing capacity and approach, could you distinguish between new cases due to exponential growth of disease spread versus a flattened curve (such as a constant number of new cases each day as in B.)?