An elaboration of theory about preventing outbreaks in homogeneous populations to include heterogeneity or preferential mixing

Abstract

The goal of many vaccination programs is to attain the population immunity above which pathogens introduced by infectious people (e.g., travelers from endemic areas) will not cause outbreaks. Using a simple meta-population model, we demonstrate that, if sub-populations either differ in characteristics affecting their basic reproduction numbers or if their members mix preferentially, weighted average sub-population immunities cannot be compared with the proportionally-mixing homogeneous population-immunity threshold, as public health practitioners are wont to do. Then we review the effect of heterogeneity in average per capita contact rates on the basic meta-population reproduction number. To the extent that population density affects contacts, for example, rates might differ in urban and rural sub-populations. Other differences among sub-populations in characteristics affecting their basic reproduction numbers would contribute similarly. In agreement with more recent results, we show that heterogeneous preferential mixing among sub-populations increases the basic meta-population reproduction number more than homogeneous preferential mixing does. Next we refine earlier results on the effects of heterogeneity in sub-population immunities and preferential mixing on the effective meta-population reproduction number. Finally, we propose the vector of partial derivatives of this reproduction number with respect to the sub-population immunities as a fundamentally new tool for targeting vaccination efforts.

Keywords: Heterogeneity; Population-immunity threshold; Vaccine coverage.

Figure A1

Contour surfaces for 3 sub-groups when mixing is preferential (A) or proportional (B). Plot C superimposes the surfaces in A and B. Parameter values are a₁ = 8, a₂ = 12, a₃ = 10, N₁ = N₂ = N₃ = 500, and ε₁ = ε₂ = ε₃ = ε with ε = 0.4 in A and ε = 0 in B.

Figure A2

Contour surfaces for 3 sub-populations. The surfaces are for ε⃑ = 0 (bottom), ε⃑ = 0.4 (middle) and ε⃑ = 0.8 (top). Other parameter values are the same as in Figure A1.

Figure 1

Immunity to measles in the United States among a) children aged 19 to 35 months and b) adolescents aged 13 to 17 years for all 50 states and the District of Columbia from the 2012 National Immunization Surveys (http://www.cdc.gov/vaccines/imz-managers/coverage/nis/child/2012-released.html). Immunity was estimated as proportions of children with at least one and adolescents with two or more doses of MMR vaccine times efficacies of 92% and 95%, respectively. The curves are fitted beta distributions having shape parameters α = 208.39 and β = 40.65 for children and α = 83.98 and β = 12.52 for adolescents. Insofar as some adolescents have had one dose of MMR, figure 1b under-estimates their immunity to measles.

Figure 2

The meta-population ℜ₀ as a function of fractions of the contacts that members of two sub-populations reserve for others within their own sub-populations (ε₁, ε₂) when their activities (average contact rates) are more or less heterogeneous. ℜ₀ decreases from the top surface (a₁ = 4, a₂ = 16), through the middle (a₁ = 8, a₂ = 12), to the bottom (a₁ = a₂ = 10). See table 1 for other parameter values. As heterogeneity in ε(e.g., ratio of the variance and mean) increases away from the line ε₁ = ε₂, heterogeneous preferential mixing also increases ℜ₀.

Figure 3

The function ℜ_v for scenario B of table 1 with a) proportional, and b) preferential, mixing. The dark blue planes represent ℜ_v = 1 and the lighter blue plane and curved (rainbow) surface represent ℜ_v for other tabulated parameters at all possible (p₁, p₂) pairs when a) ε₁ = ε₂ = 0 and b) ε₁ = ε₂ = 0.5, respectively. ℜ_v ≤ 1 for all combinations of p_i (i=1, 2) at or below the dark blue plane. While there is no single population-immunity threshold when n > 1, ℜ_v retains its utility as a threshold for outbreak prevention and, ultimately, disease elimination.

Figure 4

Contour plots of the threshold ℜ_v = 1 in the p₁-p₂ plane for different per capita contact rates, a, and proportions within-group, ε. For proportional mixing (ε⃑ = 0), the threshold pairs (p₁, p₂) for outbreak prevention or control form a (dark blue) line, p₂ = −bp₁ + r, where r > 0 is a constant and b=1 when a) sub-populations are identical in characteristics affecting ℜ₀ (here a₁ = a₂ = 10) and b) b≠1 when they differ (here a₁ = 5, a₂ = 15). At the other extreme, isolated sub-populations (ε⃑ = 1), the region in which ℜ_v < 1 is a rectangle. In between, dashed curves represent selected 0 < ε⃑ < 1 (red, ε₁ = ε₂ = 0.5; green, ε₁ = ε₂ = 0.75). These thresholds divide the plane into sub-regions such that ℜv >1 (ℜ_v < 1) below (above) the line or curve. Preferential mixing increases the difficulty of achieving ℜ_v =1. When ε= 1, pairs (p₁, p₂) must be within a relatively small rectangular area in the upper right quadrant. When ε = 0, pairs (p₁, p₂) need only be in the larger area above the solid line.

Figure 5

Comparison of homogeneous (p₁ = p₂) and heterogeneous immunity (p₁ ≠ p₂) when mixing is preferential (0 < ε₁, ε₂ ≤ 1). The parameter values are β = 0.05, γ= 1/7, ε₁ = ε₂ = 0.6, and N₁ = N₂ = 500. In figure a), a₁ = a₂ = 10, in which case ℜ₀ = 3.5. The solid curves are contours of the function ℜ_v(p₁, p₂) with the thicker (red) curve corresponding to ℜ_v = 1. The lighter (dotted) p₂ = p₁ line indicates homogeneous coverage; its intersections with the contour curves represent corresponding ℜ_v values. The homogeneous coverage required to achieve ℜ_v = 1 is p₁ = p₂ = 1−1/ℜ₀ = 0.71. The thicker dot-dashed line passing through point (p₁, p₂) = (0.71, 0.71) identifies all (p₁, p₂) pairs requiring the same number of vaccine doses in sub-populations of the same size (i.e., they satisfy p₁ + p₂ = 2×0.71). Its intersections with the contour curves also represent corresponding ℜ_v values. Note that ℜ_v > 1 for all (p₁, p₂) pairs other than (0.71, 0.71). In figure b), where a₁ = 8, a₂ = 12, in which case ℜ₀ = 3.8. The homogeneous coverage required to achieve ℜ_v = 1 is p₁ = p₂ = 1−1/ℜ₀ = 0.74. We observe some pairs with p₁ < 0.74 for which ℜ_v < 1, for one of which ℜ_v = 0.86.

Figure 6

The gradient, the n partial derivatives of ℜ_v with respect to the p-variables, and the magnitude and direction of this vector-valued function. a) The ASRΔℜ_v approximates the change in ℜ_v at points (p₁, p₂) corresponding to increases in Δp₁ and Δp₂, here both equal to 0.05. The more negative the value of Δℜ_v, the larger the reduction. b) Lengths, |∇ℜ_v|, or magnitudes of the gradient (i.e., rates of change in ℜ_v) at points (p₁, p₂). c) Directions of the negative gradient ∇ℜ at evenly spaced points (p₁, p₂) where arrows indicate the changes in p₁ and p₂ that would yield the greatest reductions in ℜ_v. Equivalently, at any point (p₁, p₂), increasing p₁ and p₂ in the direction of ∇ℜ_v would most efficiently (i.e., require the fewest doses of vaccine) achieve any particular Δℜ_v. See the text for derivation. Other parameter values are ε₁ = 0.3, ε₂ = 0.1, a₁ = 5, a₂ = 10, N₁ = 750, and N₂ = 250.

Figure 7

The a) negative gradient ∇ℜ at evenly spaced points (p₁, p₂) and b) optimal path from arbitrary starting points for a modification of the example of Fine et al. solely to increase transparency. The arrows in figure 7a, a vector plot, indicate the changes in p₁ and p₂ that would yield the greatest reductions in ℜ_v, represented by their sizes, while those in figure 7b, a stream plot, indicate only direction. Because these two sub-populations are not isolated, the gradient is defined over the entire parameter space. And, because their contact rates are not as disparate as those of Fine et al., the gradient directions do not seem to be either horizontal or vertical.

Figure 8

A numerical solution to the Lagrange problem. We observe that the line p₂ = (c − p₁N₁)/N₂ (dotted) is tangent only to the contour curve ℜ_v(p₁, p₂) = 2.2. They intersect at the point (p₁, p₂) = (0.66, 0.29), the optimal solution (marked with a red dot). The parameter c = 0.9 × N₁ with others the same as in figure 7. For some parameter values, the optimal solution might be outside the unit square in the p₁-p₂ plane.