When Does Overuse of Antibiotics Become a Tragedy of the Commons?

Background Over-prescribing of antibiotics is considered to result in increased morbidity and mortality from drug-resistant organisms. A resulting common wisdom is that it would be better for society if physicians would restrain their prescription of antibiotics. In this view, self-interest and societal interest are at odds, making antibiotic use a classic “tragedy of the commons”. Methods and Findings We developed two mathematical models of transmission of antibiotic resistance, featuring de novo development of resistance and transmission of resistant organisms. We analyzed the decision to prescribe antibiotics as a mathematical game, by analyzing individual incentives and community outcomes. Conclusions A conflict of interest may indeed result, though not in all cases. Increased use of antibiotics by individuals benefits society under certain circumstances, despite the amplification of drug-resistant strains or organisms. In situations where increased use of antibiotics leads to less favorable outcomes for society, antibiotics may be harmful for the individual as well. For other scenarios, where a conflict between self-interest and society exists, restricting antibody use would benefit society. Thus, a case-by-case assessment of appropriate use of antibiotics may be warranted.


Model 1
For Table 1 in the main text, we computed the quantities A, B, C, and D using Equation (17). In this section, we first show that B > A when R SS > 1 and R SS > R RR , and then show that the result B > A holds under the assumption R RR > R SS as well, provided R RR < R 0,SS . This includes the case where the drug-sensitive and drug-resistant organisms behave the same (except for transmission). Specifically, denoting the payoff for an individual as P (θ i , θ) = X (i) * , where θ i is the treatment rate for the individual and θ the treatment rate for the community, we have A = P (0, 0) = ρ S /β S , B = P (0, θ max ), C = P (θ i,max , 0), and where λ 0 S (θ) and λ 0 R (θ) are defined as in Equations (19) and (20), respectively. Then we have In fact, it follows R SS = β S /(ρ S + θ) > 1 and Moreover, an individual who chooses not to treat will benefit more from higher community treatment (i.e., ∂B/∂θ > 0) provided that ρ R − δρ S > 0, R SS > 1 and R SS > R RR . After some straightforward but lengthy algebraic calculations, we find that ∂B/∂θ > 0 is equivalent to It suffices to show that We now assume that R RR > R SS and and R 0,SS = β S /ρ S When θ i = 0 and θ = 0, we have λ 0 When ρ R − δρ S < 0, we have a sufficiently long duration for resistant infection that it is always best for an individual not to be treated. We assume coexistence conditions R SS > 1 and R SS > R RR , i.e. the community treatment rate θ being considered is not at a high enough rate to eradicate sensitive infection.
In this case, individual incentives favor undertreatment.

Model 2
For the community dynamics of Model 2, the equilibrium fraction of individuals in each state for the coexistence equilibrium can be calculated explicitly. The fraction in the susceptible class is given by When R SS < 1, this equation yields a fraction greater than 1, which is impossible; the coexistence equilibrium cannot exist when R SS < 1. Moreover, because this increases as θ increases, the higher the rate of treatment of the mild stage, the lower the prevalence of infection. (However, it is possible to increase the fraction in the severe state, as will be shown). For the fraction with mild infection with the drug-sensitive organism, where and c 2 = γ S (δθ +ρ R ) +ρ R (ρ S +θ).
The fraction with mild infection with the drug-resistant organism may be expressed in terms of Y * S /N (Equation (1)): The fraction with severe infection due to the drug-sensitive organism is as given directly from Equation (??). Note that this ratio is independent of the treatment rate of mild infection.
Finally, we express the fraction with severe infection due to the drug-resistant organism in terms of where The equilibrium force of infection for the drug-sensitive organism is then given by substituting Equations (1) and (3) into the following: Similarly, the equilibrium force of infection for the drug-resistant organism is given by substituting Equations (2) and (4) into the following: The overall fraction of time spent in the severe state for the community level analysis for Model 2 is then The value of θ which minimizes V (θ) is the utilitarian optimum mild treatment level, i.e. the treatment rate which minimizes the fraction of severe disease for the population.