ARE/ESP 175: Problem Set 5 The Economics of Infectious Diseases

ARE/ESP 175: Problem Set 5

The Economics of Infectious Diseases

Instructor: Matt Reimer

Problem Set prepared by Franc¸ois M. Castonguay

Problem 1: Covid-19 Vaccination Campaign [6 Points]

You were just hired by the CDC to lead the Covid-19 vaccination campaign. The private payoff from getting vaccinated is π = −c, where c is cost parameter that incorporates the risk of complications associated with vaccination (vaccines per se are free of charge). The private payoff associated with not getting vaccinated is π = −bR0(V c − V ), where:

– b is a cost parameter that incorporates the health risks associated with catching Covid-19;

– R0 is the basic reproduction number (i.e. how transmissible the disease is);

– Vc is the critical level of immunity, or vaccination, the population needs to reach to eventually eradi-cate Covid-19 (i.e. the herd immunity threshold);

– V is the fraction of the population that is currently immune/vaccinated.

Together, “R0(V c − V )” represents the probability of catching Covid-19. For simplicity, assume that the immunity one develops to Covid-19 is permanent, and that catching the disease provides you with the same immunity that the vaccine does, so that V includes both vaccinated and infected individuals.

1.A. [0.5 point] Let S denote the fraction of the population that is susceptible. Find R, the effective repro-ductionnumber and find Vc , the herd immunity threshold. How does V c change as new variants that have higher a R0 become the new dominant strain? Show your work and explain in words.

1.B. Private incentives of vaccination.

        i) [0.5 point] Find V ∗ , the equilibrium level of vaccination that private individuals would choose on their own. Show your work and explain in words.

        ii) [0.25 point] How does V ∗ change if people falsely believe that complication costs c are much higher than what they actually are? Show your work and explain in words why a good estimate of the c is impor-tant.

        iii) [0.25 point] For this subquestion only, suppose that a newly approved vaccine makes it such that c = 0. How does V ∗ change and how does it compare to V c ? Show your work and explain in words what this means and what it implies if c > 0.

1.C. Social incentives of vaccination.

        i) [1 point] Suppose that the total population is of size N. Find V ∗∗, the optimal level of vaccination in the population. How does it compare to the private vaccination level V ∗ and what does this imply for the optimality of V ∗?

        ii) [0.5 point] For this subquestion only, suppose that R0 = 3, c = 5, and b = 4. How does V ∗∗ compare to V c ? Is V ∗∗ optimal? Explain in words whether or not you can conclude using the value of V ∗∗ that Covid-19 should optimally be eradicated.

        iii) [0.5 point] For this subquestion only, suppose that R0 = 3, c = 5, and b = 8. How does V ∗∗ compare to V c ? Explain in words whether V ∗∗ optimal or not, and explain whether or not you can conclude using the value of V ∗∗ that Covid-19 should optimally be eradicated.

1.D. Suppose that R0 = 3, c = 5, and b = 4. Suppose that the CDC wants to eradicate Covid-19 and that they expect eradication will occur T years from now.

        i) [0.5 point] Use the Solver of choice (e.g. Excel, Google Sheets) and plot (1) the marginal private ben-efits of vaccination, (2) the marginal social benefits of vaccination, and (3) the marginal costs of vaccination as a function of the vaccination level V (i.e. benefits and costs are on the y-axis and vaccination level is on x-axis). You can also plot this by hand, but make sure to include value for the x-axis and y-axis, and to draw your graph to scale. Plot all curves on the same graph and make sure to identify V ∗, V ∗∗ and V c on your graph. Hint: Use the fact that the marginal social benefits of vaccination are equal to the private benefits plus the marginal benefits that the unvaccinated individual incur.

        ii) [1 point] What are the per-period costs of achieving eradication that the society incurs each year in the eradication phase (i.e. from t = 0 to t = T)? What are the per-period benefits that the society incurs each year during the eradication phase? What are the per-period benefits that the society incurs in the post-eradication phase (i.e. in t = T + 1 and after)?

        iii) [1 point] Experts confirm to you that vaccinating at a rate of V c for T = 50 years will eradicate Covid-19 in Period T. Assume the society’s discount rate is r = 0.1 or 10%. Should the CDC (a benevolent social planner) eradicate Covid-19? If so, how many years will it take for eradication to yield net benefits? If not, why should the CDC decide not to eradicate Covid-19?

Problem 2: The Neverending HIV/AIDS Epidemic [4 Points]

You were hired as a consultant by WHO to tackle the HIV/AIDS epidemic. Thankfully in many parts of the world, the epidemic has reached a steady state where the growth in susceptible individuals is equal to the growth in infected individuals. Let I denote the fraction of individuals that are infected (i.e. 0 ≤ I ≤ 1) and (1−I) denote the fraction of individuals that are susceptible. Let β denote the effective contact rate between the susceptible and infected individuals, and let γ denote the rate of recovery (i.e., the rate at which infected individuals transition back into susceptible individuals).1 As such, the growth in I (that is, the change over time in the fraction of individuals that are infected, I˙ ; pronounced ”I-dot”) is given by:

I˙ = β(1 − I)I − γI

2.A [0.5 point] Find an expression for the steady-state value of infected individuals, ISS. Under which conditions does ISS = 0? Given your answer, how would WHO want its public health policies to affect β and γ? Show your work and explain in words.

2.B Suppose WHO is considering a nonpharmaceutical intervention in the form of a massive information campaign promoting the use of condoms, which would help reducing the effective contact rate β.

        i) [0.25 point] What will happen to the long-term steady-state level of infected individuals? Show your work in math and explain in words.

        ii) [1 point] Now suppose that, before the information campaign, β = 2 and γ = 1.25, and that experts predict the campaign would reduce contact rate β by 5% indefinitely. Assume the campaign costs $C today and would save lives that a have a constant value of a disability-adjusted life year (V-DALY) of $10,000. Assume the population contains 1,000 individuals and WHO’s discount rate is r = 0.1 or 10%. What is the value of ISS today? What would be the value of ISS if WHO decides to proceed with the information campaign? How high does C need to be for WHO to choose not to proceed with the campaign? Attach a print screen of your Excel sheet. Hint: Use a 300-year horizon to approximate the infection costs that are incurred indefinitely.

2.C Now suppose that after extensive research a vaccine for HIV/AIDS has finally been developed, ef-fectively creating a new class of individuals, the vaccinated individuals V . The vaccine is only effective against susceptible individuals (it provides them with indefinite immunity) and it has no effect on people that are already infected. Suppose that a proportion u of susceptible individuals are vaccinated every pe-riod. Because of this new class of individuals, I still represents the fraction of individuals that are infected, but (1 − I) now represents the fraction of individuals that are either susceptible S or vaccinated V . Hence, S + I + V = 1. Suppose that β = 2, γ = 1.25, that WHO’s discount rate is r = 0.1 or 10%, that the total population of interest is 1,000 people, and that the vaccine cost function is C(u) = 1000u2.

        i) [0.25 point] Write a new expression for I˙ that incorporates the fact that some people will now be vaccinated, write an expression for the growth of the fraction of individuals that are susceptible ˙S, and write an expression for the growth of the fraction of individuals that are vaccinated ˙V. Note: don’t input the parameter values yet—we’ll do that later directly in the Solver.

        ii) [1 point] Suppose WHO’s objective is to vaccinate systematically a fraction u = 0.1 of susceptible individuals each year. In what period T is the disease eradicated? What needs to be the value of a disability-adjusted life year (V-DALY) to make eradication a good investment? Attach a print screen of your Excel sheet. Hint: Input some arbitrary value of V-DALY and ask your Solver to find the one that makes it exactly worth it to eradicate HIV/AIDS.

        iii) [1 point] As the only economist working for WHO’s HIV/AIDS division, you are wondering if vaccinating systematically every year at a rate of u = 0.1 is the optimal thing to do. Using a solver, find the optimal vaccination level for each year, assuming that 0 ≤ u(t) ≤ 0.1. Does your answer differ from WHO’s proposed strategy in 2.C.(ii)? Explain why or why not your answer differs from WHO’s proposed strategy. Assume that vaccination cannot continue beyond T = 30 years and that V-DALY = $10,000. Attach a print screen of your Excel sheet showing the optimal vaccination level over 30 years.