Synthesizing epidemiological and economic optima for control of immunizing infections

Abstract

Epidemic theory predicts that the vaccination threshold required to interrupt local transmission of an immunizing infection like measles depends only on the basic reproductive number and hence transmission rates. When the search for optimal strategies is expanded to incorporate economic constraints, the optimum for disease control in a single population is determined by relative costs of infection and control, rather than transmission rates. Adding a spatial dimension, which precludes local elimination unless it can be achieved globally, can reduce or increase optimal vaccination levels depending on the balance of costs and benefits. For weakly coupled populations, local optimal strategies agree with the global cost-effective strategy; however, asymmetries in costs can lead to divergent control optima in more strongly coupled systems--in particular, strong regional differences in costs of vaccination can preclude local elimination even when elimination is locally optimal. Under certain conditions, it is locally optimal to share vaccination resources with other populations.

Fig. 1.

(A) One-patch SIR model without immigration: vaccination costs (black line) and disease costs for two different values of ( = 1.5 in blue lines, = 5 in green lines) for two different values of per-capita cost of infection (solid lines, moderate cost; dashed lines, high cost). Vertical colored lines indicate the critical level of coverage, p_c = 1 − 1/ , needed to eliminate the disease in the absence of economic constraints. Parameters: 1/ν = 14, 1/μ = 80 y. (B) Total costs for the case when there is no immigration (combined vaccination and disease costs). Solid (moderate cost) and dashed (high cost) blue and green lines represent the total coverage for, respectively, = 1.5 and = 5, until the elimination coverage indicated by colored vertical lines. Dotted line represents the coverage cost after the elimination. Black vertical lines indicate the level of coverage that minimizes total costs (solid, moderate cost; dashed, high cost) given by Eq. 3. (C) Optimal coverage as a function of immigration likelihood in the one-patch SIR model with immigration. Solid line, moderate infection cost; dashed line, high infection cost. Vertical colored lines indicate the level of immigration for which equilibrium prevalence and total costs are shown in D and E: blue, η = 0; green, η = 0.00065; red, η = 0.01.

Fig. 2.

Two-patch SIR model. Colored lines show contours of the total costs (both patches) as a function of coverage levels in each patch. The coverage levels for which the global costs are minimized are indicated with a blue circle. Black lines indicate the cost minimizing coverage values at one patch for a given coverage in the other patch, and the black circle is the Nash equilibrium. (A) Weak coupling, η = μ/10, asymmetric vaccination costs (a₁ > a₂), = = 5, c_I1 = c_I2 = 5. (B) Same as A but strong coupling, η = 10μ. (C) Asymmetric disease costs, c_I1 > c_I2, = = 5, a₁ = a₂ = 0.1, η = μ. (D) Asymmetry in values, > , c_I1 = c_I2, = 5, a₁ = a₂ = 0.1, η = μ.

Fig. 3.

Optimal allocation of globally and locally available resources. (A) Long-term optimal allocation of global (or external) resources. Shaded area represents a range of equally cost-effective strategies that lead to elimination; solid and dashed vertical red lines indicate the budget required to reach the elimination threshold in patch 1 and patch 2, respectively (p_c1 and p_c2); green vertical line indicates global elimination threshold, and black vertical line indicates the budget required to vaccinate everyone. (B) Short-term optimal allocation of patch-1 resources. Red vertical line indicates the level of budget required to reach the elimination threshold in patch 1, and black vertical line indicates budget required to vaccinate everyone in patch 1. (C) Long-term optimal allocation of patch-1 budget. Shaded area represents multiple optimal strategies for patch 1, where increasing allocation makes patch 2 better off while patch 1 remains at optimum (Pareto improvement). Red curve is the Pareto optimal strategy (no further Pareto improvements can be made). Red vertical line indicates the level of budget required to reach elimination threshold in patch 1, and black vertical line indicates budget required to vaccinate everyone in patch 1. (D) Magnification of the threshold area in C. Parameters: = = 3; a₁ = 1, a₂ = 0.5; c_I1 = c_I2 5; η = 10μ.