A public choice framework for controlling transmissible and evolving diseases

Benjamin M. Althousea, Theodore C. Bergstromb, and Carl T. Bergstroma,1

aDepartment of Biology, University of Washington, Seattle, WA 98195-1800; and bDepartment of Economics, University of California, Santa Barbara, CA 93106

Edited by Peter T. Ellison, Harvard University, Cambridge, MA, and approved November 17, 2009 (received for review July 23, 2009) Control measures used to limit the spread of infectious disease tended to be illustrative rather than predictive, and thus we have often generate externalities. for transmissible dis- put a high premium on simplicity; in each case, we use the most eases can reduce the of disease even among the unvacci- basic models possible with simple approximate parameters. Faced nated, whereas chemotherapy can lead to the with an actual policy decision for a specific disease, one would do evolution of and thereby limit its own well to sacrifice the simplicity here for the realism conferred by effectiveness over time. We integrate the economic theory of more complex disease models. Extension is straightforward to any public choice with mathematical models of infectious disease to type of disease model, deterministic or stochastic, analytical provide a quantitative framework for making allocation decisions or simulation-driven, well-mixed, age-structured, explicitly spatial, in the presence of these externalities. To illustrate, we present a or network-based. As far as the economic modeling is concerned, series of examples: vaccination for tetanus, vaccination for the disease model is simply a “black box” that generates the measles, treatment of otitis media, and antiviral treat- marginal public benefit curves by specifying the hazard of ment of influenza. and its derivatives as a function of how broadly and to whom in- terventions are allocated. Even when these quantities cannot be antibiotic resistance | externalities | vaccination | emerging infectious obtained analytically, numerical methods can be used for all of the disease | pandemic influenza calculations described herein. Economic Method ontrol measures used to limit the spread of infectious disease Coften generate both direct effects on the targeted individuals The economic theory of public choice provides a mathematical and indirect effects on other parties. As we shall see, these side framework for making decisions in the presence of externalities consequences, or externalities, may be either beneficial or detri- (18, 19). In this paper, we are concerned with the allocation of mental, depending on the biology of the disease and the nature public health interventions: , , and anti- of the interventions. virals. Due to externalities that arise with these goods, the pri- An effective framework for making health policy decisions must vately obtained levels are not always socially efficient. Some sort account for both direct effects and externalities. This requirement of government intervention such as subsidies or taxes may be situates the control of infectious disease as a problem in public needed to achieve efficiency. choice economics. Several economic studies have considered the We begin by describing the basic economic formulation, using role of externalities in public health. Brito et al. (1) first posed the vaccination as an example. As we will show, this framework can problem, addressing the positive externalities associated with vac- also be adapted to deal with antimicrobial therapy. Consider a N q cination programs. Subsequent analyses (2, 3) use this approach to large population in which a fraction of the population is h q illustrate the difficulty of entirely eradicating a disease by vacci- vaccinated. The annual hazard of infection is (1, ) for those h q nation. Gersovitz and Hammer further expand the economic who are vaccinated and (0, ) for those who are not. Let in- i U h x h models and extend the analysis to public goods associated with dividual have a utility function i( i, i), where i is the hazard x i control (4–6). Cook et al. (7) propose a graphical approach rate and i is the amount of goods that consumes in time pe- t i w p similar to that presented here and work out a detailed case study for riod .If has initial wealth i and pays to be vaccinated, then x w – p i’ fi cholera vaccination. However, none of these analyses address the i = i . Individual s private bene t from vaccination is a v q fi i negative externalities associated with the evolutionary process, function i( ), which is de ned as the most money that would be v q fi namely the evolution and spread of antimicrobial resistance. Within willing to pay to be vaccinated. Thus i( ) is implicitly de ned the field of infectious disease epidemiology, there is a growing re- by the equation alization that game-theoretic problems arise around vaccine uptake U ½hð ; qÞ; w − v ðqÞ ¼ U ½hð ; qÞ; w : [1] (8–14) and antibiotic resistance evolution (15, 16). However, epi- i 1 i i i 0 i demiologists have yet to adopt a general framework for handling the economic externalities associated with disease control measures. Although the method generalizes easily to heterogeneous Here we fuse these two lines of research to sketch a general populations, as we will see in the otitis media example, for framework for considering the consequences of control inter- now consider the special case where consumers have identical ventions on evolving infectious diseases. To illustrate the fun- utility functions of the form damental issues that arise, we will step through a sequence of four infectious diseases. We begin by describing tetanus, for which vaccination is a pure private good and no externalities This paper results from the Arthur M. Sackler Colloquium of the National Academy of Sciences, “Evolution in Health and Medicine” held April 2–3, 2009, at the National arise. Next we use measles to illustrate a case in which the Academy of Sciences in Washington, DC. The complete program and audio files of externalities, namely reduced infectious pressure due to herd most presentations are available on the NAS web site at www.nasonline.org/Sackler_ (17), are entirely positive. We then consider otitis Evolution_Health_Medicine. media, a case in which the externalities, namely selection for Author contributions: B.M.A., T.C.B., and C.T.B. designed research, performed research, antibiotic resistance, are due to evolutionary processes and are analyzed data, and wrote the paper. purely negative. We conclude with a model of pandemic influ- The authors declare no conflict of interest. enza, in which antiviral treatment simultaneously generates both This article is a PNAS Direct Submission. positive and negative externalities. Each of our examples is in- 1To whom correspondence should be addressed. E-mail: [email protected].

www.pnas.org/cgi/doi/10.1073/pnas.0906078107 PNAS Early Edition | 1of6 Downloaded by guest on October 1, 2021 Tetanus Uðh; xÞ¼x − kh: [2] With our economic framework in place, we turn to a series of examples. We begin with one of the few cases in which disease When utility is of the form in Eq. 2, it follows from Eq. 1 that prevention is a purely private good. Tetanus is the severe pro- fi vðqÞ¼k½hð0; qÞ − hð1; qÞ: [3] longed contraction of skeletal muscle bers caused by the neu- rotoxin tetanospasmin produced by the bacterium Clostridium tetani. Whereas tetanus kills hundreds of thousands of individuals If individuals must pay a price c for vaccination, they will prefer fi v q worldwide every year, it is virtually nonexistent in industrialized to be vaccinated if and only if the private bene ts ( ) of vacci- nations, with a reported incidence of 50–100 cases per year in the nation exceed the private cost c. q United States over the past 30 years. This low incidence is due Assume that the total cost of vaccinating a fraction of the directly to vaccination use (20, 21). C qN fi q population is ( ). In this case, the ef cient level of is that Tetanus is not contagious, making it the only vaccine-preventable which maximizes the sum of individual utilities subject to the disease that is infectious but not transmissible human-to-human constraint that total consumption of goods equals the sum of (20). As a result, there is no from vaccination. In individual wealths minus total resource costs of vaccinations. terms of our model, the hazard rate for an unvaccinated in- This sum is equal to the sum of the utilities of all of the vacci- dividual to become infected is simply a constant, independent of nated individuals plus the sum of the utilities of the unvacci- q, and thus ∂h(0, q)/∂q = 0. Furthermore, the vaccine targets the nated, which is tetanospasmin toxin rather than the bacterium that produces it, thereby minimizing the selective consequences of vaccination on ∑ Uiðh; wiÞ¼∑ wi − CðqNÞ − Nqkhð1; qÞ the bacterium itself and minimizing the risk of resistance evo- i i lution, even relative to other vaccines (22). Hence, the hazard − Nð1 − qÞkhð0; qÞ rate for a vaccinated individual to become infected is also in- dependent of q and so ∂h(1, q)/∂q = 0 as well. Thus, for tetanus, X ¼ ∑ wi − CðqNÞþNqv q − Nkhð0; qÞ: q i ( ) = 0, and there are no externalities associated with vacci- nation. Tetanus toxoid vaccine is a purely private good; an in- fi q dividual will choose to vaccinate if his or her benefitislargerthan It follows that the ef cient level of vaccination e is that which — maximizes the cost, irrespective of what others are doing and this decision has no side effects on other individuals’ risks of disease. NqυðqÞ − CðqNÞ − Nkhð0; qÞ: [4] Measles Next we consider a disease for which control generates positive If we differentiate expression 4 with respect to q and divide by N fi externalities from reduced . Measles is caused by a , we have the following rst-order necessary condition for paramyxovirus and kills an estimated 242,000 people globally per efficiency: year, despite an effective and readily available vaccine (23). Vacci- nation is the primary form of control, and generates a positive ex- ∂hð1; qÞ ∂hð0; qÞ vðqÞ − C′ðqNÞ − k q þð − qÞ ¼ : [5] ternality because high vaccination levels induce “herd immunity” ∂q 1 ∂q 0 (17) that reduces the risk of infection to nonvaccinated individuals. To investigate the dependence of hazard rates on vaccination, Let us define we consider the stationary equilibrium of a basic susceptible- infectious-recovered (SIR) model of vaccination (see ref. 24). ∂hð1; qÞ ∂hð0; qÞ Expected lifespan is T years and population size is constant, with XðqÞ¼ − k q þð1 − qÞ : [6] ∂q ∂q birth rate μ =1/T equal to the death rate. Individuals who are vaccinated will be vaccinated at birth. We assume that the vaccine h q q Thus, X(q) measures the marginal value to the population of the is perfectly protective, so that (1, ) = 0 for all .Atequilibrium, indirect effects of vaccination. This is the externality generated thosewhoarenotvaccinatedwillfaceaconstanthazardrateof by vaccinating a single individual. Note that it is composed of two contracting measles, but after recovering will not be susceptible. γ β terms: the ∂h(1, q)/∂q term reflects the effect of treating one Let be the recovery rate and the infection transmission rate. S t additional individual on the other treated individuals, and the Let ( ) be the fraction of the population that is susceptible to I t ∂h(0, q)/∂q term reflects the effect of treating one additional infection, ( ) be the fraction of the population that is infectious, R t individual on the untreated individuals. In the remainder of this and ( ) be the fraction that is recovered from the disease and no S_ I_ R_ paper, we look at how these terms operate for four illustrative longer infectious. Where , ,and indicate derivatives with re- diseases. For tetanus, both terms are zero and no externalities spect to time, the governing differential equations are are generated. For measles vaccination, which generates a pos- _ itive externality for the unvaccinated (but not the vaccinated), S ¼ μð1 − qÞ − β SI− μ S the ∂h(1, q)/∂q is zero and the ∂h(0, q)/∂q is negative. For anti- I_ ¼ β SI− ðγ þ μÞI [7] biotic treatment of otitis media, which generates a negative ex- _ ternality for the other treated individuals (but not the untreated), R ¼ γ I þ μ q − μ R: the ∂h(1, q)/∂q is positive and the ∂h(0, q)/∂q is zero. For antiviral fl treatment of pandemic u, which generates both positive and We define basic reproductive number R0 = β/(γ + μ). To find the negative externalities, both terms are nonzero. equilibrium level of infection I*(q) when a fraction q is vacci- _ Suppose that individuals are required to pay the marginal cost nated, we take S ¼ I_ ¼ R_ ¼ 0. We find that of their vaccination. Because the private benefit of vaccination is v q q v μðR ð − qÞ − Þ R ð − qÞ≥ ( ), the equilibrium level of vaccination will be p such that I∗ðqÞ¼ β 0 1 1 if 0 1 1 [8] q C′ Nq fi ( p)= ( p). To induce an ef cient outcome, the government 0ifR0ð1 − qÞ < 1: can provide a so-called Pigouvian subsidy, subsidizing each vac- cination by an amount X(qe) equal to the marginal value of the We see that the equilibrium fraction of infected individuals indirect effects at the efficient point. decreases linearly as the vaccinated proportion of the population

2of6 | www.pnas.org/cgi/doi/10.1073/pnas.0906078107 Althouse et al. Downloaded by guest on October 1, 2021 increases, so long as R0(1 – q) > 1. If R0(1 – q) ≤ 1, the disease Measles Vaccination will be eradicated. The annual hazard rate for an unvaccinated, susceptible individual is given by 15

∗ hð0; qÞ¼βI ðqÞ¼μ½R0ð1 − qÞ − 1: [9] 10 We assume that vaccination at the beginning of life results in

lifetime immunity. Suppose that individuals who are vaccinated 5 C are asked to pay the marginal cost of their own vaccination in Marginal Cost or Benefit A equal installments over their lifetimes. Let c be the marginal cost B of vaccination; then the annual payment would be c/T = μ c. The 0 0.5 0.6 0.7 0.8 0.9 1.0 annual utility gain from vaccination is v(q)=kh(0, q). At an Proportion Vaccinated interior equilibrium, individuals would be indifferent about being Fig. 1. Measles vaccination with a cost subsidy. The annual marginal cost vaccinated or not. Thus the privately supported equilibrium fi q or bene t is plotted. Solid lines: marginal public cost (red) and marginal proportion of vaccinated individuals would be p, such that public benefit (blue) of vaccination. Once 90% of the population is vac- cinated, the disease is eradicated and no further benefit accrues to vac- vðqpÞ¼khð0; qpÞ¼k μ½R0ð1 − qpÞ − 1¼μ c: [10] cination. Dashed lines: net private cost (red) is the cost minus the subsidy; net private benefit (blue) of vaccination. Point A: private optimum without This implies that at equilibrium the fraction of the population subsidy. Point B: private level of vaccination with subsidy. Point C: efficient that is unvaccinated is level of vaccination. An annual subsidy brings the level of vaccination at the private optimum into line with the efficient level. The shaded region c þ k illustrates the net welfare gain due to the subsidy. The parameters in this − q ¼ : [11] example, with time measured in years and costs in dollars, are as follows. 1 p kR 0 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 fi The efficient level of vaccination qe is that which maximizes the of vaccination ( nancial cost plus perceived risk of being vaccinated) is c =200. sum of individual utilities subject to the constraint that total Given a death rate of μ = 0.02, the annual cost is c μ = 4 and the annual subsidy μ consumption of goods equals the sum of total wealths minus is (c + k)/2 = 3. total resource costs of vaccination. The total annual cost of C q μ N vaccinations is ( ) and the total annual cost of infection is Typically caused by ineffective clearing of Streptococcus pneumo- niae or Haemophilus influenzae bacteria from the duct, otitis me- ð1 − qÞNkhð0; qÞ¼ð1 − qÞNkμ½R ð1 − qpÞ − 1: [12] 0 dia is the leading cause of antibiotic prescription in children and fi adolescents in the United States (26). Due to the position of in- An ef cient outcome, therefore, is one that minimizes fection, human-to-human transmission of the infection-causing bacteria is unlikely. With asymptomatic carriage in the naso- Cðq μ NÞþð1 − qÞNkμ½R ð1 − qÞ − 1: [13] 0 pharynx extremely common and transmission much more likely from this site than from the eustachian tube, treatment of clinical c fi If the marginal cost of an additional vaccination is , the rst- infection has little or no effect in reducing transmission. fi q order condition for an ef cient rate of vaccination e is The economics of otitis media treatment differ from measles μNc¼ μNk½ R ð − q Þ − : [14] vaccination, as there exists a negative externality of antibiotic 2 0 1 e 1 resistance caused by overprescription of antibiotics (27–29). Although most cases of otitis media are caused by bacterial in- This implies that fection, some are viral in etiology and often the two are difficult c þ k to distinguish. This commonly leads to unnecessary antibiotic − q ¼ : [15] treatment. Moreover, most cases resolve spontaneously without 1 e kR 2 0 treatment. The American Academy of Pediatrics now recom- mends that no antibiotic treatment be given for most cases in We see from Eqs. 11 and 15 that in an efficient outcome, the children over the age of 2 in the absence of severe illness (30). fraction of the population that is left unvaccinated is just half of We adjust our interpretation of the model parameters to fit the equilibrium unvaccinated fraction if individuals pay the the specifics of otitis media. We now interpret N as the total marginal cost of their vaccinations. number of clinical cases and q as the fraction of cases that re- In this model, the annual externality from a vaccination is ceive antibiotic therapy, and we replace the hazard rate h with a X(q)=k μ R0(1 – q). Thus, an annual subsidy of k μ R0(1 – qe)= function π that gives the chance of developing complications that μ(c + k)/2 for those who choose vaccination would be sufficient require further intervention beyond any initial antibiotic therapy. to induce an efficient outcome. Fig. 1 illustrates. There is considerable variation across individuals in their need As seen in previous analyses, complete eradication can be for antibiotic treatment. To account for the individual variation difficult even with government subsidy, due to the so-called in need for antibiotic treatment, we extend our economic model elasticity in private demand of vaccine: Prevalence of to consider the case in which individual utility functions differ. a disease declines through increased vaccination use; the will- Suppose that there is a continuum of consumers such that the ingness to pay for vaccination decreases as well (2, 25). However, utility function of consumer t is in the framework defined above, increasing vaccine prevalence through government intervention still provides a positive ex- Uðπ; xÞ¼x − kðtÞπ; ternality in reducing the prevalence of measles (4). where t ∈ [0, 1] and k is a nondecreasing function of t. Otitis Media In an efficient allocation, all of those who are treated must Otitis media is an inflammation of the duct known as the eusta- have at least as high a value of k as all of those who are not. If chian tube, which connects the middle ear to the nasopharynx. a fraction q of the cases are treated and the persons treated

Althouse et al. PNAS Early Edition | 3of6 Downloaded by guest on October 1, 2021 are chosen efficiently, those who are treated must have k(t) ≥ k equilibrium frequency of resistant bacteria [i.e., Ir/(Iw + Ir)at (1 – q) and those who are not treated must have k(t) ≤ k(1 – q). equilibrium] is p(q)=α q σ/(κ – α q γt) and the derivative with 2 The total cost of treatment is C(Nq), and the integral of con- respect to q is p′(q)=ακσ/(κ – α q γt) . sumers’ utilities is We can now compute ∂π(1, q)/∂q and ∂π(0, q)/∂q. The latter is Z Z straightforward: Assuming that there is no difference in the 1 1 − q virulence of resistant and sensitive strains, the frequency of W − CðNqÞ − N πð ; qÞ kðtÞdt − N πð ; qÞ kðtÞdt; [16] 1 0 resistance does not directly impact untreated individuals and 1 − q 0 thus ∂π(0, q)/∂q = 0. The former term depends on the fre- R þ 1 − quency of treatment failure, which in turn is proportional to the with W ¼ ∑ wi. Let us define K ðqÞ¼ kðtÞdt and K ðqÞ¼ ρ R i 1 − q frequency of resistant bacteria. Let be the chance that 1 − q kðtÞdt 16 treatment failure leads to complications. Then ∂π(1, q)/∂q = 0 . Then expression can be written as p′(q)ρ and X(q)isnegative.Theefficient level of treatment qe W − CðNqÞ − Nπð1; qÞKþðqÞ − Nπð0; qÞK − ðqÞ: [17] lies below the private equilibrium, and some government in- tervention is required to discourage usage if we are to reach the 17 q efficient outcome. In this case, a tax of X(qe) will shift the Taking the derivative of expression with respect to , and fi dividing by N, the first-order condition for efficiency is private equilibrium to the ef cient point. This optimal taxation level is illustrated in Fig. 2. ∂πð ; qÞ ∂πð ; qÞ C′ðqÞ¼kðqÞ½πð ; qÞ − πð ; qÞ − KþðqÞ 1 − K − ðqÞ 0 Pandemic Influenza 0 1 ∂q ∂q Novel influenza A (H1N1) was first reported in Mexico City in late April 2009, and within 6 weeks (June 11) was declared a full If the fraction q of cases are treated, the private benefits to an k t pandemic by the World Health Organization. The most widely individual to whom the cost of infection is ( ) will be used antiviral agents, neuraminidase inhibitors oseltamivir and zanamivir, have demonstrated beneficial effects on H1N1 infec- vðt; qÞ¼kðtÞ½πð0; qÞ − πð1; qÞ: tion and are the WHO-recommended first-line and chemo- prophylactic treatment for these viruses (33, 34). However, c If marginal cost is a constant , and individuals purchase their resistance to antiviral agents is a concern for public health plan- own treatment at a price equal to the marginal cost, an equili- fi kðqÞðπð ; qÞ − πð ; qÞÞ ¼ c: ners (35). Resistance can evolve readily with minimal tness costs brium will be an outcome in which 0 1 (36), and the use of neuraminidase inhibitor antivirals has the Thus, the marginal effect of a change in q on the population, – potential to decrease the long-term effectiveness of these drugs. X(q), is much as before but with K+(q) and K (q) replacing the k q k – q In this section, we develop a simple illustrative model of and (1 ) terms: pandemic influenza. Our analysis is based on ref. 37, with the simplification that we model only antiviral treatment and not ∂πð1; qÞ ∂πð0; qÞ XðqÞ¼ − KþðqÞ − K − ðqÞ : [18] antiviral prophylaxis. Our state variables are as follows: X is the ∂q ∂q fraction of susceptible individuals in the population; Ysu is the fraction of individuals infected with sensitive virus but untreated Antibiotic use typically selects for resistance; as usage increases, with the antiviral; Yst is the fraction of individuals infected with we expect the prevalence of antibiotic-resistant strains to sensitive virus and treated with the antiviral; Yr is the fraction of increase and, with it, the probability of treatment failure (31). individuals infected with resistant virus; and Z is the fraction of In our model, this means that the probability of complications removed (recovered or dead) individuals in the population. Let γ π q q despite treatment (1, ) will be an increasing function of .To be the recovery rate and let σ be the rate at which sensitive determine the form of this function, we again can turn to math- ematical models in disease epidemiology. Here we use a simple susceptible-infected-susceptible (SIS) model of antibiotic use in the community to treat a human- Antibiotics 150 commensal bacterium, following ref. 32. Only a small fraction α of individuals carrying the bacterium experience symptoms and 100 q are candidates for treatment; the treatment fraction now refers B 50 to the fraction of these symptomatic cases treated. For a pop- A ulation of size N, we let S be the proportion of individuals not 0 C carrying the bacterial species of interest, and let Iw and Ir be the proportion of people carrying wild-type bacteria or resistant -50 γ γ bacteria, respectively. Let w and r be the rates of spontaneous Marginal-100 Cost or Benefit clearance for wild-type and resistant bacterial carriage, and let γt -150 be the clearance rate due to treatment for wild-type carriage. Let 0.0 0.2 0.4 0.6 0.8 β be the and let σ be the rate at which treated Fraction Treated with Antibiotic wild-type individuals develop de novo resistance. The governing Fig. 2. Antibiotic treatment for otitis media. Solid lines: marginal public differential equations are cost (red) and marginal public benefit (blue) of antibiotic therapy. Dashed fi _ lines: net private cost with tax (red) and net private bene t (blue) of anti- S ¼ − β SðIw þ IrÞþðγw þ α q γtÞIw þ γr Ir biotic therapy. Point A: private optimum without tax. Point B: private level fi I_ ¼ β SI − ðγ þ α q γ þ α q σÞI [19] of antibiotic therapy with tax. Point C: ef cient level of antibiotic use. The w w w t w 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 Ir ¼ β SIr − γr Ir þ α q σIw: tax. Parameters with time in days and costs in dollars are as follows: γw = 0.05 and fitness cost of resistance is a 10% increase in clearance rate so that γr = At steady state, sensitive and resistant strains will coexist 0.055. We have γt = 0.5, σ = 0.05, α = 0.01, β =1,ρ = 0.1. The cost of antibiotics provided that β > γw + α q γt + α q σ and κ > α q(γt + σ), where is c = 10. The function k(t) specifying the consumer values of treatment is an –5(1–t) κ = γr – γw is the differential clearance rate. In this case, the exponential curve: k(t) = 5000 e .

4of6 | www.pnas.org/cgi/doi/10.1073/pnas.0906078107 Althouse et al. Downloaded by guest on October 1, 2021 evolve de novo resistance. Let βsu, βst, and βr be the Antivirals transmission parameters for untreated sensitive, treated sensi- 600 tive, and resistant virus, respectively. As in the model of ref. 37, treatment reduces the infectiousness but not the duration of 400 infection. Let q be the fraction of drug-sensitive infections that receive treatment. The governing differential equations are then 200 _ A C X ¼ − ðβsuYsu þ βstYstÞX − βrYrX B _ 0 Y su ¼ βsuYsu þ βstYst 1 − q X − γYsu Marginal Cost or Benefit _ Y st ¼ðβ Ysu þ β YstÞqð1 − σÞX − γYst [20] su st 0.0 0.1 0.2 0.3 0.4 0.5 _ Fraction Treated with Antiviral Y r ¼ β Ysu þ β Yst qσX þ β YrX − γYr su st r _ Fig. 4. Antiviral treatment for pandemic flu, flu kit scenario. Solid lines: Z ¼ γ Ysu þ Yst þ Yr : marginal public cost (red) and marginal public benefit (blue) of antiviral therapy. Dashed lines: net private cost with subsidy (red) and net private In the previous examples, we looked at how the treatment benefit (blue) of antiviral therapy. Point A: private optimum without sub- fraction influenced the steady-state prevalence of the disease. sidy. Point B: private level of antiviral therapy with subsidy. Point C: efficient Such an approach is not appropriate for pandemic influenza, level of antiviral use. The subsidy brings the level of treatment at the private which sweeps through a population and then is eradicated, rather optimum into line with the efficient level. The shaded region illustrates the than reaching a steady level. Thus, we will focus in this net welfare gain due to the subsidy. Disease parameters are as in Fig. 3, with time in days and costs in dollars. Cost of antivirals is c = 100, and disease example on the time course of infection, tracking the number of valuation parameters are k = 850, k = 1000. individuals infected over time. t u Fig. 3 shows how the fractions of individuals infected with resistant and sensitive strains, over the course of an , antiviral therapy will be available during the epidemic to those depend on the fraction of cases treated with antivirals. As seen in who did not choose to purchase them initially. In the pharmacy previous analyses, intermediate levels of antiviral therapy mini- distribution scenario, no advance purchase is made. Instead, mize the total number of cases (37) by reducing the degree to when an individual is infected, he or she has the option to pur- which the epidemic overshoots its eradication threshold (38). chase a course of antivirals for immediate use from the phar- In our economic analysis, individuals have incentive to pur- macy, but this decision is made without knowing whether the c chase antivirals at cost because treatment ameliorates symp- infection is sensitive or resistant. fl k k toms of in uenza, such that t and u represent the dollar costs of The πu and πt values can be computed directly from the fl a case of treated and untreated in uenza, respectively. Under fraction of the population that has been infected by each strain U π π x these assumptions, utility functions are of the form ( t, u, )= at the end of the epidemic, conditioned on the treatment choices x – k π – k π x π t t u u, where is the amount spent on other goods, t is of the individual. Let πr(q) be the fraction of the population that the probability that the individual contracts a sensitive case and has been infected by resistant strains at the end of the epidemic π treats it, and r is the probability that an individual contracts a and let πs(q) be the fraction of the population infected by sen- case and does not treat it, either because he or she had chosen sitive strains. In the flu kit scenario, the purchase is made in not to purchase the antiviral or because the case is resistant. advance of infection. Ignoring resale and exchange of antivirals, We consider two different ways in which antivirals might be the fraction of sensitive infections treated, q, will be equal to the flu kits dispensed. In the scenario (39), the consumer has an fraction of the population who chose to purchase the flu kit. In advance option to purchase one course of antiviral therapy to be this case, the private benefitisv(q)=(ku – kt)πs(q). In the used in the event that he or she is infected in a pandemic. For pharmacy distribution scenario, the purchase is deferred until simplicity, in this scenario, we assume that no further courses of the individual is known to be infected. The fraction of sensitive infections treated will be equal to the fraction of individuals who choose to purchase treatment once infected. In this case, the Prevalence of Resistant and Sensitive Infections private benefit is conditioned on infection by some strain of flu: 0.7

0.6 Total

0.5 Antivirals Sensitive 200 B 0.4 100 0.3 C A 0 0.2 Fraction of Population Resistant -100 0.1 -200 0.0 0.0 0.2 0.4 0.6 0.8 1.0 -300 Fraction Treated Marginal Cost or-400 Benefit Fig. 3. Fraction of resistant (red) and sensitive (blue) infections for pan- fl -500 demic u, as a function of the fraction of infections treated. Total fraction 0.26 0.28 0.30 0.32 0.34 infected is indicated by the dashed yellow curve. Parameters follow Lipsitch Fraction Treated with Antiviral et al. (37) 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. Fig. 5. Antiviral treatment for pandemic flu, pharmacy distribution sce- Treatment reduces transmission of sensitive virus by 67%. The infectious nario. Parameters are as in Fig. 4. Here the private market overuses antivirals period is 3.3 days, and each treated case has a 1/500 chance of developing de and a tax is required to bring the level of treatment at the private optimum

novo resistance. These values correspond to γ = 1/3.3, σ = 0.002, βsu = 1/1.8, into line with the efficient level. The shaded region illustrates the net wel- βr =0.9βsu, βst = 0.33βsu. fare gain due to the tax.

Althouse et al. PNAS Early Edition | 5of6 Downloaded by guest on October 1, 2021 v(q)=(ku – kt)πs(q)/[πs(q)+πr(q)]. In each case, we can solve Summary numerically for the private equilibrium fraction q of individuals When coupled with mathematical models of disease, the eco- who choose to purchase antiviral therapy. As in our previous ex- nomic theory of public choice provides a robust framework for amples, the social optimum is the point that maximizes the sum of assessing the economic impact of public health interventions. As the utilities. This can also be computed numerically. Fig. 4 and we have illustrated, this framework allows decision makers to Fig. 5 illustrate. account for the positive and negative externalities associated This model illustrates an interesting tradeoff. When antiviral with control measures such as vaccination or antimicrobial che- usage is low, increasing use generates a positive externality in the motherapy. Depending on the biology of the disease and the form of reduced total cases. When antiviral use gets higher, this nature of the intervention, the private market may under- or positive externality is reduced and, moreover, a negative exter- overallocate. Taxes or subsidies can guide the private market to a more efficient level of intervention. Although the models pre- nality arises in the form of reduced antiviral effectiveness. sented here have been deliberately simple, the framework illus- Depending on the distribution technology, the private market fi trated can be fruitfully applied to more sophisticated quantitative may over- or underuse antivirals relative to the ef cient level. models of infectious disease. Doing so should help public health For the particular parameter values here, too few people planners more efficiently control evolving transmissible diseases. stockpile antivirals if required to purchase them in advance of the pandemic, whereas too many use antivirals if allowed to ACKNOWLEDGMENTS. The authors thank Marc Lipsitch for helpful discus- purchase them at the time of infection. Corresponding subsidies sions. This work was supported by the National Institute of General Medical Sciences Models of Infectious Disease Agent Study Program cooperative or taxes can bring the private equilibrium into line with the agreement 5U01GM07649 and the MIDAS Center for Communicable Disease efficient point, as illustrated in Fig. 4 and Fig. 5. Dynamics 1U54GM088588 at Harvard University.

1. Brito DL, Sheshinski E, Intriligator MD (1991) Externalities and compulsory vaccinations. 22. Read AF, Mackinnon MJ (2008) Pathogen evolution in a vaccinated world. JPublicEcon45:69–90. Evolution in Health and Disease, eds Stearns SC, Koella JC (Oxford University Press, 2. Geoffard P-Y, Philipson T (1997) Disease eradication: Private versus public vaccination. Oxford), 2nd Ed. Am Econ Rev 87:222–230. 23. World Health Organization (2007) Measles Fact Sheet (World Health Organization, 3. Francis PJ (1997) Dynamic epidemiology and the market for vaccinations. J Public Econ Geneva). – 63:383 406. 24. Keeling MJ, Rohani P (2008) Modeling Infectious Diseases in Humans and Animals 4. Gersovitz M, Hammer JS (2003) Infectious diseases, public policy, and the marriage of (Princeton University Press, Princeton, NJ). – economics and epidemiology. World Bank Res Obs 18:129 157. 25. Philipson T (1996) Private vaccination and public health: An empirical examination for 5. Gersovitz M, Hammer JS (2004) The economic control of infectious diseases. Econ J U.S. measles. J Hum Resour 31:611–630. 114:1–27. 26. McCaig LF, Besser RE, Hughes JM (2002) Trends in antimicrobial prescribing rates for 6. Gersovitz M, Hammer JS (2004) Tax/subsidy policies toward vector-borne infectious children and adolescents. JAMA 287:3096–3102. diseases. J Public Econ 89:647–674. 27. Goossens H, Ferech M, Vander Stichele R, Elseviers M; ESAC Project Group (2005) 7. Cook J, et al. (2009) Using private demand studies to calculate socially optimal vaccine Outpatient antibiotic use in Europe and association with resistance: A cross-national subsidies in developing countries. J Policy Anal Manage 28:6–28. – 8. Bauch CT, Galvani AP, Earn DJD (2003) Group interest versus self-interest in smallpox database study. Lancet 365:579 587. vaccination policy. Proc Natl Acad Sci USA 100:10564–10567. 28. Lipsitch M (2001) Measuring and interpreting associations between antibiotic use and – 9. Bauch CT, Earn DJD (2004) Vaccination and the theory of games. Proc Natl Acad Sci penicillin resistance in Streptococcus pneumoniae. Clin Infect Dis 32:1044 1054. USA 101:13391–13394. 29. Lipsitch M, Samore MH (2002) Antimicrobial use and antimicrobial resistance: A 10. Reluga TC, Bauch CT, Galvani AP (2006) Evolving public perceptions and stability in population perspective. Emerg Infect Dis 8:347–354. vaccine uptake. Math Biosci 204:185–198. 30. American Academy of Pediatrics Subcommittee on Management of Acute Otitis 11. Galvani AP, Reluga TC, Chapman GB (2007) Long-standing influenza vaccination Media (2004) Diagnosis and management of acute otitis media. Pediatrics 113: policy is in accord with individual self-interest but not with the utilitarian optimum. 1451–1465. Proc Natl Acad Sci USA 104:5692–5697. 31. Bergstrom CT, Feldgarden M (2008) The ecology and evolution of antibiotic-resistant 12. Vardavas R, Breban R, Blower S (2007) Can influenza be prevented by bacteria. Evolution in Health and Disease, eds Stearns SC, Koella JC (Oxford University voluntary vaccination? PLoS Comput Biol 3:e85. Press, Oxford), 2nd Ed. 13. van Boven M, Klinkenberg D, Pen I, Weissing FJ, Heesterbeek H (2008) Self-interest 32. Bonhoeffer S, Lipsitch M, Levin BR (1997) Evaluating treatment protocols to prevent versus group-interest in antiviral control. PLoS One 3:e1558. antibiotic resistance. Proc Natl Acad Sci USA 94:12106–12111. 14. Medlock J, Galvani AP (2009) Optimizing influenza vaccine distribution. Science 325: 33. Centers for Disease Control (2009) Interim Guidance on Antiviral Recommendations – 1705 1708. for Patients with Novel Influenza A (H1N1) Virus Infection and Their Close Contacts 15. Smith DL, Levin SA, Laxminarayan R (2005) Strategic interactions in multi-institutional (Centers Dis Control, Atlanta). – epidemics of antibiotic resistance. Proc Natl Acad Sci USA 102:3153 3158. 34. World Health Organization (2006) WHO Rapid Advice Guidelines on Pharmacological 16. Foster KR, Grundmann H (2006) Do we need to put society first? The potential for Management of Humans Infected with Avian Influenza A (H5N1) Virus (World Health tragedy in antimicrobial resistance. PLoS Med 3:e29. Organization, Geneva). 17. Fine PEM, Clarkson JA (1986) Individual versus public priorities in the determination 35. Centers for Disease Control (2009) Interim Recommendations for the Use of Influenza of optimal vaccination policies. Am J Epidemiol 124:1012–1020. Antiviral Medications in the Setting of Oseltamivir Resistance Among Circulating In- 18. Samuelson PA (1954) The pure theory of public expenditure. Rev Econ Stat 36: fl – fl 387–389. uenza A (H1N1) Viruses, 2008 09 In uenza Season (Centers Dis Control, Atlanta). fl 19. Samuelson PA (1955) Diagrammatic exposition of a theory of public expenditure. Rev 36. Weinstock DM, Zuccotti G (2009) The evolution of in uenza resistance and treatment. – Econ Stat 37:350–356. JAMA 301:1066 1069. 20. Centers for Disease Control (2008) Epidemiology and Prevention of Vaccine-Preventable 37. Lipsitch M, Cohen T, Murray M, Levin BR (2007) Antiviral resistance and the control of Diseases (The Pink Book) (Public Health Foundation, Washington, DC), 10th Ed. pandemic influenza. PLoS Med 4:e15. 21. Centers for Disease Control (1991) Diphtheria, tetanus, and pertussis: Recom- 38. Handel A, Longini IM (2007) Jr, Antia R (2007) What is the best control strategy for mendations for vaccine use and other preventive measures: Recommendations of the multiple infectious disease outbreaks? Proc Biol Sci 274:833–837. Practices Advisory Committee (ACIP). MMWR Morb Mortal Wkly Rep 40 39. Traynor K (2008) Antiinfluenza medication kits need work, FDA advisers conclude. (RR-10):1–28. Am J Health Syst Pharm 65:2314–2316.

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