Strategic interactions in multi-institutional epidemics of antibiotic resistance

Abstract

The increasing frequency of antibiotic resistance in hospital-acquired infections is a major public health concern that has both biological and economic causes. Here we develop conceptual mathematical models that couple the economic incentives and population biology of hospital infection control (HIC). We show that the optimal investment by a hospital for HIC changes with the proportion of patients already colonized with antibiotic-resistant bacteria (ARB) at the time of admission. As that proportion increases, the optimal behavior of a hospital is to increase spending to control ARB with low transmissibility and decrease spending on those with high transmissibility. In some cases, the global optimum investment in HIC can shift discontinuously from one that contains transmission to a do-nothing policy once the proportion already colonized at the time of admission becomes too great. We also show that investments in HIC are determined by a strategic game when several hospitals share patients. Hospitals acting selfishly and rationally will free-ride on the investments of other hospitals, and the level of free-riding should increase with the number of other hospitals in the area. Thus, in areas with many hospitals, the rational strategy for each hospital is to spend less than in areas with few hospitals. Thus, we predict that transmission rates and the prevalence of ARB should be higher in urban hospitals, for instance, compared with rural hospitals. We conclude that regional coordination and planning for HIC is an essential element of public health planning for hospital-acquired infections.

Fig. 1.

As the proportion already colonized at the time of admission increases, the costs and benefits of HIC change. When the proportion of patients already colonized at the time of admission is low, it is cost-effective to invest enough in HIC to eliminate the pathogen. Once that proportion exceeds ≈12%, two minima exist. A local minimum is to continue to invest at levels that would eliminate the pathogen if no patients were colonized at the time of admission, but the globally cost-effective strategy is to abandon HIC and do nothing. Hospitals spending near the no-colonization optimum would see total costs increase if they spent slightly less on infection control, but they would see total costs decrease if they abandon it entirely. Once the proportion colonized on admission reaches 18%, the local minimum disappears, and hospitals would see total costs decrease as they spent less on HIC. The point at which abandoning HIC becomes optimal depends on the particular function that describes the relationship between costs and transmission; here, we use the function S(c) = 4e^-0.03c and D = $100 per patient per day.

Fig. 2.

Optimal expenditures on HIC (solid line) vary depending on the transmissibility of the ARB S(0) and the proportion already colonized at the time of admission κ; at the optimum, transmission rates decrease [shown as S(c^*), the dotted line]. For κ = 0, the optimum investment is to eliminate the ARB if S(0) < ≈2.2. Otherwise, the optimum reduces transmission but does not eliminate ARB. For κ = 4%, it is not possible to eliminate ARB, so the optimum response curve is not as sharp. For S(0) < ≈1.5, the optimum spent on HIC increases with the proportion already colonized at the time of admission but not for ARB with higher intrinsic transmission rates. The shifts in optimum investment depend on the specific function used; here, S(c) = S(0)(1 + 0.2√c)^-1, a function that does not predict sudden shifts in spending.

Fig. 3.

A fraction of the total secondary cases prevented are expected to occur during the initial visit (below the dotted line), but some occur on subsequent visits. As the number of other hospitals that share a population increases (n), the benefits of reducing transmission are increasingly shared among hospitals (dark gray) as the number of other hospitals in a region increases. Thus, infection control is an economic game. The fraction of benefits that accrue to other hospitals increases when the interval between hospitalization is short and persistence times are long. We have illustrated the relationship for p = 0.3, equal expenditures (i.e., c = c̃) and values of n that range from a rural hospital to a hospital in a city of several million people.

Fig. 4.

The strategic responses and corresponding dynamics change with the number of hospitals that interact. (a) The response curves (solid) for increasing n. The strategic optimum, the points at which a focal hospital's optimal investment matches the investment of other hospitals, decreases with n (the dashed line shows equal investments). Note also that the coordinated optimum for many hospitals is the same as the optimum investment for a single hospital, n = 1. (b) These decisions affect the rate at which resistance increases. For n = 10, an epidemic occurs if hospitals allocate at the strategic optimum, whereas the coordinated optimum prevents emergence for more than a decade. Hence, the epidemic will be delayed and less severe in isolated areas. The transmission function here is the same one used for Fig. 2, with 1/λ = 2,000 days and 1/r ≈ 1,500 days.