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

Human populations are heterogeneous, but all differences need not be modeled to answer any specific question. Immunity to vaccine-preventable diseases, for example, is heterogeneous within the United States (_{0}. As this quantity is derived from a mathematical model, ascertaining its adequacy amounts to determining if the model from which it was derived is sufficiently detailed.

Mechanistic models are hypotheses about processes underlying natural phenomena. Simplicity is a virtue because it facilitates their evaluation. But the only way to ensure that one's model is not too simple is to compare results with those from models that include additional details that might affect them. In transmission modeling, one generally distinguishes sub-populations whose members have characteristics with which their risks of being infected or infecting others vary (e.g., age, gender, location). Levins (

Recently, Ball et al. (

Rates of person-to-person contact may vary with population density (e.g., be greater in urban than rural areas). They may also vary with personal characteristics (e.g., be greater among schoolchildren than younger and older people). The effect of such heterogeneity on ℜ_{0}, defined as the average number of secondary infections caused by a newly infectious person on introduction to a wholly susceptible population, was studied by Dietz (_{0} varies with the contact rate's variance and mean.

Nold (_{0} attains its maximum when individuals having high average

May and Anderson (

Recently, Fine et al. (^{1} which was derived from a model of a proportionally-mixing homogeneous population. In the next section, we define a meta-population model and mixing function with which to evaluate the utility of this threshold when mixing is preferential or sub-populations are heterogeneous with respect to characteristics affecting their basic reproduction numbers.

We employ the simplest meta-population model capable of informing vaccination policy to illustrate the effects of heterogeneity in sub-population contact rates and immunities, together with preferential mixing, on the effective reproduction number. A glossary accompanies this section.

Our model comprises _{i}_{i}_{i}_{i}_{i}

The force of infection among susceptible members of sub-population _{i}_{ij}^{th} sub-population's contacts that are with members of the ^{th} sub-population, _{j}/N_{j}

We follow Jacquez et al. (_{ij}_{i}δ_{ij}_{i}_{j}_{i}^{th} sub-population reserve for others in sub-population _{ij}_{i}_{j}

We limit ^{2} where

For ease of reference, we define several terms based on the properties of (_{1}, _{2}, …, _{n}

Mixing is _{i}

Mixing is _{i}_{i}

The ^{th} sub-population is _{i}

The _{i}

The _{i}_{j}

First we derive the reproduction numbers for our model meta-population, whose sub-populations may differ in subscripted variables (numbers susceptible, infectious and recovered; proportions vaccinated at birth; contact rates and fractions within and among groups). Consequently, our model is more general than that of Goldstein et al. (_{i}_{i}_{i}

In our model, the basic and effective reproduction numbers for sub-population _{0}_{i}_{vi}

Meta-population reproduction numbers are properties of next-generation matrices (

We derive the next-generation matrix

where _{v}_{1}_{11}, _{v}_{1}_{12}, _{v}_{2}_{21}, _{v}_{2}_{22}. If 0 < _{v}_{vi}_{ij}

As shown originally by Dietz (

where ℜ̄_{0} is the mean ℜ_{0}, and ^{2} are the mean and variance of activity among sub-populations. We observe that similar expressions could be derived for any characteristic affecting ℜ_{0}_{i}_{i}_{i}

We find the spectral radius (eigenvalue of greatest magnitude) of the matrix _{0} even when mixing is proportional, and also because some of its many variations are incorrect. We also provide an example where two meta-populations have the same mean activity, but differ in their variances. In _{0} from 3.50 to 3.72 for proportional mixing (_{1} = _{2} = 0) and preferential mixing (_{1} = _{2} = 0.5) further increases it to 3.88. Evidently preferential mixing magnifies the effect of heterogeneity in activity, one sub-population characteristic affecting ℜ_{0}_{i}

_{0}_{i}_{T}_{1} + _{2}, and all other parameters identical for the two sub-populations except _{i}

for _{1} = _{2} = 0, ℜ_{0}(0, 0) is minimized when _{1} = _{2} = _{T}_{2} − _{1}| (i.e., ℜ_{0} is maximized when heterogeneity in activity is greatest);

a similar result as in (i) holds when either _{1} = 1 or _{2} = 1. That is, ℜ_{0}(_{1}, 1) = ℜ_{0}(1, _{2}) as shown in _{1} = _{2} = _{T}_{2} − _{1}|. Therefore, ℜ_{0}(_{1}, 1) = ℜ_{0}(1, _{2}) is maximized when heterogeneity in activity is greatest.

Preference, the fraction of contacts reserved for one's own sub-population, ranges over the unit interval, but we can derive explicit limiting conditions (i.e.,

Case 1 (isolated sub-populations): When _{1} = _{2} = 1, we have _{11} = 1, _{12} = 0, _{21} = 0, _{22} = 1. Thus, in the expression above for the largest eigenvalue of the next-generation matrix

That is, R_{v}_{v}_{1}, R_{v}_{2}}. When _{1} = _{2} = 0, we have ℜ_{vi}_{0}_{i}_{0}= max{R_{01}, R_{02}}.

Case 2 (proportionally mixing sub-populations): When _{1} = _{2} = 0, we have _{11} = _{1}, _{12} = _{2}, _{21} = _{1}, _{22} = _{2}. Thus, _{v}_{1}
_{1}, _{v}_{1}_{2}, _{v}_{2}_{1}, _{v}_{2}_{2}, and

When _{1} = _{2} = 0, we have ℜ_{vi}_{0}_{i}_{0} = R_{01}_{1} + R_{02}_{2}. Chow et al. (_{v}_{i}_{i}

Goldstein et al. (_{v}_{v}_{1}, _{2}) as a function of _{1} and _{2}, we can solve for the critical value of _{1}_{c}_{v}_{1}_{c}_{2}) = 1 for any given value of _{2}. Recall that

where _{1}, _{2}) = R_{v}_{1}_{11}, _{1},_{2}) = R_{v}_{1}_{12}, _{1},_{2}) = R_{v}_{2}_{21}, _{1},_{2}) = _{v}_{2}_{22} with

Note that ℜ_{v}_{1}, _{2}) is an increasing function of _{1} (_{v}_{2}) > 1, then ℜ_{v}_{1}, _{2}) > 1 for all _{1}. If ℜ_{v}_{2}) < 1, however, we can solve the equation ℜ_{v}_{1}, _{2}) = 1 for _{1} and arrive at

This expression provides a formula for _{1}_{c}_{1}_{c}_{1}_{c}_{v}_{v}_{0} (with ℜ_{vi}_{0}_{i}_{0}(0, _{2}) < 1.

Colizza and Vespignani (_{i}_{i}_{2} = 0.5) among sub-populations differing in ℜ_{0}_{i}_{0} compared to proportional mixing (_{i}_{2} = 0), heterogeneous preferential mixing (_{i}_{2} = 0.75 or vice versa) increases ℜ_{0} compared to homogeneous preferential mixing (from 3.88 to 3.92 or 3.98). When _{1} = 0, _{2} = 1 or vice versa), both are isolated (_{0} = max{ℜ_{01}, ℜ_{02}} = 4.37 as noted in the previous section. Apolloni et al. (

_{i}_{v}_{1}, _{2}) = 0 and (_{1}, _{2}) = 1. Values of ℜ_{v}_{i}_{v}_{i}

Because mixing affects ℜ_{v}_{1} = _{2} = _{1}-_{2} plane divides the region 0 ≤ _{1} ≤ 1, 0 ≤ _{2} ≤ 1 into sub-regions such that ℜ_{v}_{1}, _{2}) below (above) the curve (

When _{v}_{v}_{0}_{i}_{1} and _{2}) in our meta-population model. Vaccinating the sub-population whose members have higher _{v}

There is one exception. The curves intersect the straight line in _{01}, 1 − 1/ℜ_{02}), at which ℜ_{v}_{1} = ℜ_{v}_{2} = 1, whereupon the next-generation matrix

whose dominant eigenvalue is 1∀

Using a model meta-population composed of a city and several villages, May and Anderson (_{v}_{0} (and, as ℜ_{v}_{0}, on ℜ_{v}_{1} = _{2} ≠ 0) than proportional (_{1} = _{2} = 0) if sub-population sizes are equal (e.g., both 0.5

In _{v}_{1}-_{2} plane that otherwise would be straight lines. Consequently, uniform immunization (_{1} = _{2}) of meta-populations composed of sub-populations that are identical in characteristics affecting ℜ_{0}_{i}_{v}_{0}_{i}

Fine et al. (_{01} = 5 and ℜ_{02} = 1 and ℜ_{0} = 5, so their sub-populations must be isolated (i.e., _{1} = _{2} = 1, whereupon _{12} = _{21} = 0). They state, “Because the high-risk group is responsible for any increase in incidence, outbreaks could in theory be prevented by vaccinating 80% of the high-risk group alone, thus, < 80% of the entire population.” This is true, but only because their meta-population is composed of isolated sub-populations. Keeling and Rohani (_{0}, if vaccination is applied at random,” which is true only if ℜ_{0} is correctly specified. And ℜ_{0} in structured and unstructured populations differ. Assuming, in the example of Fine et al., that _{1} = 15 and _{2} = 3. Averaging these contact rates, which is tantamount to assuming a proportionally-mixing homogeneous population, _{0} = 3 and the naïve population-immunity threshold, _{c}_{1} = 15 and _{2} = 3) increases ℜ_{0} to 4.3, while preferential mixing (_{1} = _{2} = 1) further increases it to 5.

Evidently ℜ_{v}_{i}_{i}_{j}_{j}_{j}_{i}_{v}_{v}_{i}

For any meta-population immunity (_{1}_{c}_{2}_{c}_{1}, _{2})-plane) along which the rate of change in ℜ_{v}_{1}_{c}_{2}_{c}_{v}_{v}

A familiar analogy may clarify this concept: Consider a topographic map with elevations represented by contour lines and axes latitude and longitude. In the equation above, Δℜ_{v}_{v}_{1}, Δ_{2}). In our analogy, this vector equals (Δ_{lat}, Δ_{lon}). Our |∇ℜ_{v}_{1}, _{2}), is that in which this rate of increase is greatest (i.e., the steepest route).

Chow et al. (_{v}_{1} and _{2}. For example,

and similarly, ∂_{v}_{2} < 0. As increasing any element of a non-negative matrix decreases its dominant eigenvalue according to Perron-Frobenius theory, these partials are negative for arbitrary _{1}_{c}_{2}_{c}_{1} and _{2} for which the rate of change in ℜ_{v}

The gradient provides a means of identifying which sub-population to vaccinate to affect ℜ_{v}_{v}_{1} and Δ_{2} of 5%. _{v}_{1} = 1 and _{2} = 0, the arrow is almost vertical (i.e., the gradient direction has only a small _{1} component), indicating that changes in _{2} would affect ℜ_{v}

In the example of Fine et al. (_{1} = 15 and _{2} = 12 so that ℜ_{01} = 5 and ℜ_{02} = 4 (the values of _{1} = _{2} = 1, then for all (_{1}, _{2}) with _{2} > (5/4) _{1} − 1/4, we have that ℜ_{v}_{v}_{1} = 5(1 − _{1}), which implies that the gradient direction is horizontal; thus, only population 1 need be vaccinated to attain ℜ_{v}_{1} = _{2} = 0.3, _{1} = 1100, and _{2} = 900. In this case, illustrated in _{1}, _{2}). We observe also that the trajectories in the _{1}-_{2} plane, when the gradient direction is followed, are not straight lines, as shown in

The gradient also describes the most efficient means of attaining any programmatic goal: To illustrate this, we fix Δℜ_{v}_{v}_{v}_{v}_{1}_{c}_{2}_{c}_{v}_{1}_{c}_{2}_{c}

If we increase the fractions immune by Δ_{1} and Δ_{2} along a unit direction, _{1}, Δ_{2}) = _{1}, Δ_{2}). It follows that Δℜ_{v}_{1}_{c}_{2}_{c}_{v}_{1}_{c}_{2}_{c}_{1}, _{2}) and the unit length vector _{1}_{c}_{2}_{c}_{v}_{i}_{1}_{c}_{2}_{c}_{1}_{c}_{2}_{c}_{1}_{c}_{2}_{c}_{1}, Δ_{2}) ≈ _{1}_{c}_{2}_{c}_{1}_{c}_{2}_{c}_{1}_{1} + Δ_{2}_{2} ≈ _{1}_{c}N_{1} + _{2}_{c}N_{2})/| (_{1}_{c}_{2}_{c}_{1}_{1} + Δ_{2}_{2} is smallest when the vector (Δ_{1},Δ_{2}) is parallel to the gradient vector ∇ℜ_{v}_{1}_{c}_{2}_{c}_{v}

The gradient can also be used to devise optimal allocation strategies for limited vaccines. We can minimize ℜ_{v}_{1}, _{2}) for fixed total vaccine doses _{1}_{1} + _{2}_{2} = _{1} and _{2} being fixed constants. In this case, we solve the equation ∇ℜ_{v}_{1}, _{2}) = 0, subject to _{1}_{1} + _{2}_{2} = _{2} = (_{1}_{1})/_{2}. As this line is orthogonal to ∇ℜ_{v}_{1}, _{2}), its intersection with the contour curve ℜ_{v}_{1}, _{2}) to which it is tangent (_{1} = 0.66 and _{2} = 0.29 reduces ℜ_{0} = 4.6 (when _{1} = _{2} = 0) to ℜ_{v}

Using the simplest meta-population model that is capable of informing vaccination policy, we reproduce earlier results concerning the effect of heterogeneity in inter-personal contact rates – attributable, for example, to disparate sub-population densities – on the basic meta-population reproduction number. We observe that this reasoning extends to any variable affecting sub-population reproduction numbers. Preferential mixing, especially if heterogeneous, also affects the basic meta-population reproduction number. We refine earlier results on effects of heterogeneity in sub-population immunities and preferential mixing on the effective meta-population reproduction number. Heterogeneity in immunity can result from disparate socioeconomic circumstances, religious or philosophical beliefs, or information about the risks and benefits of vaccination, to name but a few possible causes.

Were populations proportionally-mixing or homogeneous, the population-immunity threshold would inform vaccination programs to mitigate the risk of outbreaks upon the introduction of infectious people. For diseases that had been eliminated domestically, these would be travelers infected abroad. This threshold is defined as the immunity at which an average infectious person infects only one susceptible person.

If individuals differed only in the sizes of their sub-populations, one could use the meta-population reproduction number to determine a single population-immunity threshold, as Goldstein et al. (_{0}_{i}_{i}

Meta-population reproduction numbers (glossary, ℜ_{0}, ℜ_{v}

Our first result of programmatic import is that, in assessing the impact of heterogeneity, one cannot ignore mixing. Mixing not only modifies the effects of heterogeneity, but identifies relevant sub-populations (i.e., mixing is proportional within and preferential between them). While others have recognized its importance (

Our second such result is that differences among sub-populations in characteristics affecting their basic reproduction numbers – heterogeneity – may substantially increase meta-population reproduction numbers, especially when combined with preferential mixing. Sub-population members can differ, but not in these characteristics. We began by questioning whether sub-populations themselves were needed. In reaching this conclusion, we extend the contributions of many authors, some of whom studied characteristics affecting basic reproduction numbers (

Our third result of programmatic import is the gradient, a fundamentally new tool for exploring the combined effects of heterogeneity in characteristics affecting sub-population reproduction numbers and mixing among the members of different sub-populations on the average number of secondary infections per infectious person. The gradient evaluated at the point (_{1},…, _{n}) is a vector giving the direction in which to move away from (_{1},…, _{n}_{v}

We demonstrate that, at any point in the

We are grateful to Lance Rodewald for suggesting that the impact of heterogeneity in vaccine coverage on the population-immunity threshold might be amenable to modeling, to Dan Higgins for improving our illustrations, and to Aaron Curns and several anonymous reviewers for constructively critiquing earlier drafts of the manuscript.

Funding: ZF's research is supported in part by NSF grant DMS-1022758.

We reserve the term “herd immunity” for the indirect effect of vaccination, a reduction in the force of infection experienced by unvaccinated members of a population by virtue of the vaccination of others.

In the United States, elementary schools are located in the neighborhoods where most of their students reside. Thus, neighborhoods are the smallest sub-populations for which immunity to specific vaccine-preventable diseases can be calculated from proportions vaccinated, routinely surveyed at school-entry and exit, and vaccine efficacy.

Contributions: ZF and AH performed the analyses and critiqued earlier drafts of the manuscript, JG conceived the study and wrote the manuscript, PS estimated vaccine coverage from National Immunization Surveys and critiqued earlier drafts of the manuscript. All authors have approved submission of this draft.

Conflicts: The authors declare that they have no conflicts of interest.

Disclaimer: The findings and conclusions in this report are those of the authors and do not necessarily represent the official position of the Centers for Disease Control and Prevention or other institutions with which they are affiliated.

Given that

Letting

At the disease-free equilibrium, _{i}_{i}

Substituting,

The Jacobian,

We can rewrite J as F-V, where F are infection and V other terms

The next-generation matrix ^{–1}

The reproduction number is the dominant eigenvalue of

We restrict calculations in the main text to

The characteristic equation is

where

And ℜ_{v}

Consider ℜ_{v}_{1}, _{2} and _{3} with all other parameter values fixed. _{v}_{v}_{v}

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 _{1} = 8, _{2} = 12, _{3} = 10, _{1} = _{2} = _{3} = 500, and _{1} = _{2} = _{3} =

Contour surfaces for 3 sub-populations. The surfaces are for

To prove (i), notice that when _{1} = _{2} = 0, we have

Thus,

Let _{2} = _{T}_{1} and let

Then ℜ_{0} = _{1})_{1}) is a decreasing (increasing) function on 0 < _{1} < _{T}_{T}_{1} < _{T}_{T}_{1}) has a minimum at _{1} = _{T}_{1} = 0 and _{1} = _{T}_{0}(0,0) is minimized when _{1} = _{2}, and maximized when _{1} = 0 and _{1} = _{T}

To prove (ii), notice that for _{1} = 1 or _{2} = 1 we have _{11} = _{22} = 1 and _{12} = _{21} = 0. Therefore, ℜ_{0}(_{1},1) = ℜ_{0}(1, _{2}) = max{ℜ_{01},ℜ_{02}}. It is easy to see that max{ℜ_{01},ℜ_{02}} is the smallest when _{1} = _{2} = _{T}_{1} = 0 (in which case ℜ_{0} = ℜ_{02} = _{T}_{1} = _{T}_{0} = ℜ_{01} = _{T}

Together with the fact that ℜ_{0}(_{1}, _{2}) increases with _{1} and _{2} for all 0 < _{1}, _{2} < 1, we would expect the behavior shown in _{0}(_{1}, _{2}) surface moves up as heterogeneity in

_{0}in models for infectious diseases in heterogeneous populations

_{i}

_{i}

_{i}

_{i}

Numbers susceptible; numbers infected/infectious; numbers recovered/immune; total number in group (or sub-population)

Specific (or

_{i}

Force (or hazard rate) of infection per susceptible member of group (or sub-population)

Number of groups (or sub-populations), indexed by

_{i}

Activity (average

_{ij}

Mixing (or proportion of the contacts of members of group

_{j}

_{j}

Probability that a proportionally contacted member of group

_{j}

Proportional (or activity-weighted proportional) mixing, a function defined in the text

_{i}

Preference (average fraction of contacts reserved for others in one's own group)

_{ij}

Kronecker delta, equal to 1 when

For all groups

_{0}, ℜ

_{0}

_{i}

_{v}

_{vi}

Overall and group-specific intrinsic (or basic) reproduction numbers, functions defined in the text; Overall and group-specific effective (or control by vaccination, hence the subscript

_{i}

_{c}

Proportion of group _{v}

Symbols defined in the text solely to simplify notation

_{0}

Average of ℜ_{0}_{i}

^{2}

Mean and variance of _{i}

_{0}/∂

_{i}

_{v}

_{i}

Partial derivatives of the basic reproduction number with respect to preference and the control reproduction number with respect to immunity

_{v}

Gradient, a vector function defined in the text

_{v}

Magnitude of the gradient

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 (

The meta-population ℜ_{0} as a function of fractions of the contacts that members of two sub-populations reserve for others within their own sub-populations (_{1}, _{2}) when their activities (average contact rates) are more or less heterogeneous. ℜ_{0} decreases from the top surface (_{1} = 4, _{2} = 16), through the middle (_{1} = 8, _{2} = 12), to the bottom (_{1} = _{2} = 10). See _{1} = _{2}, heterogeneous preferential mixing also increases ℜ_{0}.

The function ℜ_{v}_{v}_{v}_{1}, _{2}) pairs when a) _{1} = _{2} = 0 and b) _{1} = _{2} = 0.5, respectively. ℜ_{v}_{i}_{v}

Contour plots of the threshold ℜ_{v}_{1}-_{2} plane for different _{1}, _{2}) for outbreak prevention or control form a (dark blue) line, _{2} = −_{1} + _{0} (here _{1} = _{2} = 10) and b) _{1} = 5, _{2} = 15). At the other extreme, isolated sub-populations (_{v}_{1} = _{2} = 0.5; green, _{1} = _{2} = 0.75). These thresholds divide the plane into sub-regions such that ℜ_{v}_{v}_{1}, _{2}) must be within a relatively small rectangular area in the upper right quadrant. When _{1}, _{2}) need only be in the larger area above the solid line.

Comparison of homogeneous (_{1} = _{2}) and heterogeneous immunity (_{1} ≠ _{2}) when mixing is preferential (0 < _{1}, _{2} ≤ 1). The parameter values are _{1} = _{2} = 0.6, and _{1} = _{2} = 500. In figure a), _{1} = _{2} = 10, in which case ℜ_{0} = 3.5. The solid curves are contours of the function ℜ_{v}_{1}, _{2}) with the thicker (red) curve corresponding to ℜ_{v}_{2} = _{1} line indicates homogeneous coverage; its intersections with the contour curves represent corresponding ℜ_{v}_{v}_{1} = _{2} = 1−1/ℜ_{0} = 0.71. The thicker dot-dashed line passing through point (_{1}, _{2}) = (0.71, 0.71) identifies all (_{1}, _{2}) pairs requiring the same number of vaccine doses in sub-populations of the same size (i.e., they satisfy _{1} + _{2} = 2×0.71). Its intersections with the contour curves also represent corresponding ℜ_{v}_{v}_{1}, _{2}) pairs other than (0.71, 0.71). In figure b), where _{1} = 8, _{2} = 12, in which case ℜ_{0} = 3.8. The homogeneous coverage required to achieve ℜ_{v}_{1} = _{2} = 1−1/ℜ_{0} = 0.74. We observe some pairs with _{1} < 0.74 for which ℜ_{v}_{v}

The gradient, the _{v}_{v}_{v}_{1}, _{2}) corresponding to increases in Δ_{1} and Δ_{2}, here both equal to 0.05. The more negative the value of Δℜ_{v}_{v}_{v}_{1}, _{2}). c) Directions of the negative gradient ∇ℜ at evenly spaced points (_{1}, _{2}) where arrows indicate the changes in _{1} and _{2} that would yield the greatest reductions in ℜ_{v}_{1}, _{2}), increasing _{1} and _{2} in the direction of ∇ℜ_{v}_{v}_{1} = 0.3, _{2} = 0.1, _{1} = 5, _{2} = 10, _{1} = 750, and _{2} = 250.

The a) negative gradient ∇ℜ at evenly spaced points (_{1}, _{2}) 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 _{1} and _{2} that would yield the greatest reductions in ℜ_{v}

A numerical solution to the Lagrange problem. We observe that the line _{2} = (_{1}_{1})/_{2} (dotted) is tangent only to the contour curve ℜ_{v}_{1}, _{2}) = 2.2. They intersect at the point (_{1}, _{2}) = (0.66, 0.29), the optimal solution (marked with a red dot). The parameter _{1} with others the same as in _{1}-_{2} plane.

Preferential Mixing Magnifies the Impact of Heterogeneity in Person-to-Person Contact Rates (activity) on ℜ_{0}. The number of sub-populations and their sizes also affect these results, but here _{1} = 500 and _{2} = 500, so that the mean activity is the same in scenarios A and B. Common parameters: _{1} = _{2} = 500 unless otherwise specified in figure or table legends.

Scenario A | Scenario B | |||
---|---|---|---|---|

Parameter | _{1} = 10 | _{2} = 10 | _{1} = 7.5 | _{2} = 12.5 |

ℜ_{0i} | 3.5 | 3.5 | 2.62 | 4.37 |

ℜ_{0} (_{1} = _{2} = 0) | 3.5 | 3.72 | ||

ℜ_{0} (_{1} = _{2} = 0.5) | 3.5 | 3.88 | ||

ℜ_{0} (_{1} = 0.25, _{2} = 0.75) | 3.5 | 3.92 | ||

ℜ_{0} (_{1} = 0.75, _{2} = 0.25) | 3.5 | 3.98 |

Sub-Population Sizes Affect the Impact of Heterogeneity in Activity and Preferential Mixing on ℜ_{0}. When sub-population sizes are unequal, as in scenario C, the larger dominates, but when they are equal, as in scenario D, heterogeneity and preferential mixing have greater impact. Other parameters: _{1} = _{2} = 10 (the village and city if _{1} ≪ _{2}).

Scenario C | Scenario D | |||
---|---|---|---|---|

Parameter | _{1} = 0.1 | _{2} = 0.9 | _{1} = 0.5 | _{2} = 0.5 |

ℜ_{0i} | 1.75 | 3.5 | 1.75 | 3.5 |

ℜ_{0} (_{1} = _{2} = 0) | 3.41 | 2.92 | ||

ℜ_{0} (_{1} = _{2} = 0.5) | 3.44 | 3.09 | ||

ℜ_{0} (_{1} = 0.25, _{2} = 0.75) | 3.42 | 3.12 | ||

ℜ_{0} (_{1} = 0.75, _{2} = 0.25) | 3.46 | 3.2 |

Population-immunity thresholds are useful only in homogeneous, proportionally-mixing populations

Meta-population effective reproduction numbers, ℜ_{v}

Heterogeneity in variables affecting sub-population reproduction numbers is relevant

Together with preferential mixing among sub-populations, such heterogeneity increases ℜ_{v}

The vector of partial derivatives of ℜ_{v}_{v}