Measles vaccination with a cost subsidy. The annual marginal cost or benefit is plotted. Solid lines: marginal public cost (red) and marginal public benefit (blue) of vaccination. Once 90% of the population is vaccinated, the disease is eradicated and no further benefit accrues to vaccination. Dashed lines: net private cost (red) is the cost minus the subsidy; net private benefit (blue) of vaccination. Point A: private optimum without subsidy. Point B: private level of vaccination with subsidy. Point C: efficient level of vaccination. An annual subsidy brings the level of vaccination at the private optimum into line with the efficient level. The shaded region illustrates the net welfare gain due to the subsidy. The parameters in this example, with time measured in years and costs in dollars, are as follows. The recovery rate is γ = 100. So that R₀ ≈ 10, the transmission parameter is γ = 1000. The annual valuation of reduced risk is k = 100; the total cost of vaccination (financial cost plus perceived risk of being vaccinated) is c = 200. Given a death rate of μ = 0.02, the annual cost is c μ = 4 and the annual subsidy is μ(c + k)/2 = 3.

Antibiotic treatment for otitis media. Solid lines: marginal public cost (red) and marginal public benefit (blue) of antibiotic therapy. Dashed lines: net private cost with tax (red) and net private benefit (blue) of antibiotic therapy. Point A: private optimum without tax. Point B: private level of antibiotic therapy with tax. Point C: efficient level of antibiotic use. The tax brings the level of treatment at the private optimum into line with the efficient level. The shaded region illustrates the net welfare gain due to the tax. Parameters with time in days and costs in dollars are as follows: γ_w = 0.05 and fitness cost of resistance is a 10% increase in clearance rate so that γ_r = 0.055. We have γ_t = 0.5, σ = 0.05, α = 0.01, β = 1, ρ = 0.1. The cost of antibiotics is c = 10. The function k(t) specifying the consumer values of treatment is an exponential curve: k(t) = 5000 e^–5(1–t).

Fraction of resistant (red) and sensitive (blue) infections for pandemic flu, as a function of the fraction of infections treated. Total fraction infected is indicated by the dashed yellow curve. Parameters follow Lipsitch et al. () as follows: the basic reproductive ratio R₀ is 1.8 for sensitive virus. Resistant virus suffers a 10% fitness cost in the form of reduced transmission. Treatment reduces transmission of sensitive virus by 67%. The infectious period is 3.3 days, and each treated case has a 1/500 chance of developing de novo resistance. These values correspond to γ = 1/3.3, σ = 0.002, β_su = 1/1.8, β_r = 0.9β_su, β_st = 0.33β_su.

Antiviral treatment for pandemic flu, flu kit scenario. Solid lines: marginal public cost (red) and marginal public benefit (blue) of antiviral therapy. Dashed lines: net private cost with subsidy (red) and net private benefit (blue) of antiviral therapy. Point A: private optimum without subsidy. Point B: private level of antiviral therapy with subsidy. Point C: efficient level of antiviral use. The subsidy brings the level of treatment at the private optimum into line with the efficient level. The shaded region illustrates the net welfare gain due to the subsidy. Disease parameters are as in , with time in days and costs in dollars. Cost of antivirals is c = 100, and disease valuation parameters are k_t = 850, k_u = 1000.

Antiviral treatment for pandemic flu, pharmacy distribution scenario. Parameters are as in . Here the private market overuses antivirals and a tax is required to bring the level of treatment at the private optimum into line with the efficient level. The shaded region illustrates the net welfare gain due to the tax.