75021054928J Math BiolJ Math BiolJournal of mathematical biology0303-68121432-141629445854609226410.1007/s00285-018-1216-zHHSPA982020ArticleConstrained minimization problems for the reproduction number in meta-population modelsPoghotanyanGayane1FengZhilan1GlasserJohn W.2HillAndrew N.3 Department of Mathematics, Purdue University, West Lafayette, IN, USA Centers for Disease Control and Prevention, National Center for Immunization and Respiratory Diseases, Atlanta, GA, USA Centers for Disease Control and Prevention, National Center for HIV/AIDS, Viral Hepatitis, STD, and TB Prevention, Atlanta, GA, USA Zhilan Feng, fengz@purdue.edu

Gayane Poghotanyan, gpoghota@purdue.edu; John W. Glasser, jglasser@cdc.gov; Andrew N. Hill, ahill2@cdc.gov

2172018142201812201801122019776-717951831

The basic reproduction number (R0) can be considerably higher in an SIR model with heterogeneous mixing compared to that from a corresponding model with homogeneous mixing. For example, in the case of measles, mumps and rubella in San Diego, CA, Glasser et al. (Lancet Infect Dis 16(5):599–605, 2016. https://doi.org/10.1016/S1473-3099(16)00004-9), reported an increase of 70% in R0 when heterogeneity was accounted for. Meta-population models with simple heterogeneous mixing functions, e.g., proportionate mixing, have been employed to identify optimal vaccination strategies using an approach based on the gradient of the effective reproduction number (Rv), which consists of partial derivatives of Rv with respect to the proportions immune pi in sub-groups i (Feng et al. in J Theor Biol 386:177–187, 2015. https://doi.org/10.1016/j.jtbi.2015.09.006; Math Biosci 287:93–104, 2017. https://doi.org/10.1016/j.mbs.2016.09.013). These papers consider cases in which an optimal vaccination strategy exists. However, in general, the optimal solution identified using the gradient may not be feasible for some parameter values (i.e., vaccination coverages outside the unit interval). In this paper, we derive the analytic conditions under which the optimal solution is feasible. Explicit expressions for the optimal solutions in the case of n = 2 sub-populations are obtained, and the bounds for optimal solutions are derived for n > 2 sub-populations. This is done for general mixing functions and examples of proportionate and preferential mixing are presented. Of special significance is the result that for general mixing schemes, both R0 and Rv are bounded below and above by their corresponding expressions when mixing is proportionate and isolated, respectively.

Meta-population modelConvexity of reproduction numberOptimization problemVaccination strategyEpidemiology37N2549J1534H0592D30Introduction

Mechanistic models of pathogen transmission are key public health tools for identifying optimal interventions that can mitigate outbreaks or perhaps even eliminate infectious diseases. However, the utility and credibility of such models hinge on incorporating realistic mixing between sub-populations (i.e., means by which infectious members of one sub-population infect susceptible members of others), which typically is not uniformly random due to preference among age groups, genders, or spatial locations. In fact, models that do not sufficiently account for differences among relevant sub-populations can generate biased or misleading results in situations where evaluations of intervention strategy require incorporation of such heterogeneity and realistic mixing. For example, in the case of measles, mumps and rubella in San Diego, CA, Glasser et al. (2016), reported an increase of 70% in R0 when heterogeneity was accounted for.

Recently, progress has been made (Glasser et al. 2012; Feng et al. 2017) in extending realistic mixing functions based on earlier work (Nold 1980; Jacquez et al. 1988). The effective reproduction numbers Rv derived from these meta-population models with non-homogeneous mixing functions are used to identify optimal vaccination strategies by using methods based on the gradients of Rv (partial derivatives with respect to control parameters) (Feng et al. 2015, 2017). These are constrained optimization problems with the objectives of either minimizing Rv given limited number of vaccine doses, or minimizing vaccine doses needed to reduce Rv to a given level. However, the examples considered in these studies focus only on cases where an optimal solution exists and is feasible in the sense that the vaccine coverages lie between 0 and 1. Conditions have not yet been identified to determine parameter regions within which optimal mathematical solutions are indeed feasible. This is the objective of the current paper. Similar optimization problems using Rv in the context of age-dependent vaccination strategies have been considered in Castillo-Chavez and Feng (1998) and Hadeler and Müller (1996a, b).

When heterogeneous mixing is considered in epidemiological models, preferred mixing is among the commonly used mixing structures (Nold 1980; Jacquez et al. 1988; Glasser et al. 2012), of which proportionate mixing is a special case. In this paper, results and proofs are presented for both preferred and more general mixing. Conditions are determined under which optimal solutions for vaccination strategies exist, some of which are given in explicit expressions depending on model parameters. The proofs for the existence and uniqueness of the optimal vaccination strategy when there are n = 2 sub-populations involve some fundamental properties that we establish for the reproduction number Rv as a function of vaccination coverage p = (p1, p2); namely, homogeneity and convexity. For the case of n > 2 sub-populations, the proofs of results are based on a convexity result of Friedland (1980/81) for the spectral radius over a class of positive matrices.

For n = 2, explicit analytical expressions are derived for the optimal allocation of vaccine P∗=(p1∗,p2∗) as well as the minimized reproduction number, Rv. A formula for the critical vaccine doses η* to achieve Rv ≤ 1 is also derived for n = 2. For the case of n > 2, lower and upper bounds for the minimum of Rv as well as the minimum vaccine doses are derived.

The organization of this paper is as follows. The main problem is described in Sect. 2, which consists of two constrained optimization problems of Lagrange type. Sect. 3 presents the main results of the optimization problem for n = 2 sub-populations. Results for n > 2 sub-populations are provided in Sect. 4. In Sect. 5, we discuss the results. Some detailed proofs are included in the “Appendix”.

Description of the problem

The models considered by Glasser et al. (2016) and Feng et al. (2015, 2017) are of the SIR or SEIR type, i.e., the population is apportioned into disjoint states including susceptible (S), exposed (E), infectious (I), and removed or immune (R), and the models consist of systems of ordinary differential equations (ODEs). These models include one or more types of population heterogeneity (e.g., age, spatial location, activity level, vaccination coverage, preferential mixing, population density, etc.), so they are meta-population models with each sub-population model being an SIR or SEIR type linked by a mixing function. We use the simplest of these models as an example, but similar results apply to other models. The model considered in this paper is described by the ODE system dSidt=(1−pi)θNi−(λi+θ)SidIidt=λiSi−(γ+θ)IidRidt=piθNi+γIi−θRiNi=Si+Ii+Riλi=βai∑j=1ncijIj∕Nj,i=1,2,…,n, where pi are proportions immunized at entry into sub-population i, γ is the per capita recovery rate, θ is the per capita rate for entering and leaving sub-population i so that the population size Ni remains constant. The function λi is the force of infection, i.e., per capita hazard rate of infection of susceptible individuals in sub-population i, in which β is the probability of infection upon contacting an infectious person, ai is average contact rate (activity) in sub-population i, cij is the proportion of ith sub-population’s contacts that are with members of jth sub-population, and Ij/Nj is the probability that a randomly encountered member of sub-population j is infectious.

One of the most influential factors affecting the reproduction number is the mixing function cij. Denote the mixing matrix by C = (cij). Typically, the matrix C has to satisfy the following conditions of Busenberg and Castillo-Chavez (1991): cij≥0,i,j=1,…,n, ∑j=1ncij=1,i=1,…,n, aiNicij=aiNjcji,i,j=1,…,n. A commonly used non-homogeneous mixing function that satisfies conditions (2.2)–(2.4) is the preferred mixing function of Jacquez et al. (1988) given by: cij=ϵiδij+(1−ϵi)(1−ϵj)ajNjΣk=1n(1−ϵk)akNk,i,j=1,…,n, where δij is the Kronecker delta function (taking value 1 when i = j, 0 otherwise) and ϵi ∈ [0, 1] is the fraction of contacts of group i that is reserved for itself (preferential mixing), whereas the complement (1 − ϵi) is distributed among all sub-populations in proportion to the unreserved contacts, including i (proportionate mixing). Special cases arise when: ϵi = 1 for all i whence C is the identity matrix (exclusively preferential mixing); ϵi = 0 for all i whence cij=ajNj∕ΣkakNk (exclusively proportionate mixing). We will refer to this mixing structure throughout the manuscript as Jacquez mixing. More complex examples of mixing matrices C = (cij), such as two-level preferential mixing can be found in Feng et al. (2017).

When Model (2.1) is used, the basic and effective sub-population reproduction numbers, denoted respectively by R0i and Rvi, for sub-population i (i = 1, 2, … , n) are given by R0i=ρai,Rvi=R0i(1−pi),i=1,2,…,n, where ρ=βγ+θ, see e.g. Brauer and Castillo-Chavez (2012, Chapter 10). Following Diekmann et al. (1990) and van den Driessche and Watmough (2002), the next generation matrix (NGM) corresponding to this meta-population model is Kv=(Rv1c11Rv1c12⋯Rv1c1nRv2c21Rv2c22⋯Rv2c2n⋮⋮⋮⋮Rvncn1Rvncn2⋯Rvncnn). Then the effective reproduction number for the meta-population is given as Rv=r(Kv), which is the spectral radius [and the dominant eigenvalue, by the Perron–Frobenius Theorem (Seneta 1973)] of the nonnegative matrix Kv. Let p = (p1, p2, … , pn). Naturally, Rv=Rv(p) is a function of p. The total number of vaccine doses, denoted by η, is η=∑i=1npiNi. For demonstration purposes, we will assume that vaccine efficacy is 100%. In this paper, we focus on identifying the most efficient allocation of vaccine p = (p1, p2, … , pn) ∈ [0, 1]n for reducing Rv with limited vaccine doses η or using fewest doses to achieve Rv < 1 (to prevent outbreaks). More specifically, we consider the following two constrained optimization problems:

Minimize Rv=Rv(p), subject to ℓ(p)≔∑i=1npiNi=η, for p ∈ [0, 1]n.

Minimize η=∑i=1npiNi subject to Rv(p)≤1, for p ∈ [0, 1]n.

In this study, we consider the optimization problems only for the case of R0=Rv(0)≥1, as there will be no outbreak if R0<1.

Because of the continuity of Rv(p) and the compactness of [0, 1]n, Problem (I) has a solution for any fixed η ∈ [0, N] , where N = N1 + N2 + … + Nn is the total population. If P* = P*(η) and Rv{min}(η) denote the optimal vaccination allocation and the corresponding minimum reproduction number, respectively, then we have P∗(η)=(p1∗(η),p2∗(η),…,pn∗(n))∈Ωp(n)(η)∩[0,1]n,Rv{min}(η)=minΩp(n)(η)∩[0,1]nRv=Rv∣P∗(η), where Ωp(n)(η)≔{(p1,p2,…,pn):ℓ(p)=η}. An optimal solution P*(η) to Problem (I) that lies in the interior (0, 1)n of the unit hypercube must also satisfy the following equations: ∇Rv∣P∗(η)=λ∇ℓ=λ(N1,…,Nn),ℓ∣P∗(η)=∑i=1npi∗(η)Ni=η, where the constant λ is the Lagrange multiplier.

Similarly, noting that the constraint set {Rv(p)≤1}∩[0,1]n is compact and nonempty (as Rv(1, 1, … , 1) = 0 ≤ 1), Problem (II) always has a solution. The minimum value of η, which we denote by η*, signifies the smallest number of vaccine doses that can prevent outbreaks under an optimal vaccination policy. It is useful practically to have an explicit expression or estimate of the bounds for η*.

To find η*, notice that Rv(p) is a monotonically decreasing function of pi, and thus, a decreasing function of η=∑i=1npiNi. Therefore, recalling also the assumption Rv(0)=R0≥1, the inequality constraint Rv(p)≤1 can be replaced by an equality constraint Rv(p)=1, and thus, η∗=min{Rv(p)=1}∩[0,1]nℓ(p). It follows that η* is the minimum of η ∈ [0, N] such that Rv{min}(η)=1 and can be found by solving the equation: Rv{min}(η∗)=Rv∣P∗(η∗)=1.

Notation

Below we introduce some mathematical concepts that will be used throughout the paper. For ease of reference, we also list the quantities that appear, along with their definitions, in Table 1.

A set E in a vector space is called convex if for any x0, x1 ∈ E the convex combination (1 − t)x0 + tx1 ∈ E for any t ∈ [0, 1].

We say that a real-valued function f on a convex set E is convex if for any x0, x1 ∈ E f((1−t)x0+tx1)≤(1−t)f(x0)+tf(x1),t∈[0,1].

We say f is strictly convex if the above inequality is strict for all t ∈ (0, 1).

Recall that if f is twice continuously differentiable, then f is convex on an open convex set if and only if the Hessian Hess f = (∂2 f/∂xi ∂xj) is a nonnegative semi-definite matrix for any x ∈ E. Moreover, f will be strictly convex if Hess f is positive definite. (The converse of this statement is false.)

All matrices considered in this paper will be over the field of real numbers.

If C = (cij) is a matrix, then we write C > 0 (C ≥ 0) if cij > 0 (cij ≥ 0) for any pair i, j . We also say that such C is positive (nonnegative). This should not be confused with the notion of positive definite (nonnegative semi-definite) matrices, which are related to the positivity (nonnegativity) of the quadratic form x ↦ xTCx associated with C.

We say that a square matrix C = (cij) is essentially nonnegative1 if its off-diagonal elements are nonnegative; i.e., cij ≥ 0 if i ≠ j.

We say that a nonnegative squarematrix C is irreducible if for any pair (i, j), i, j = 1, … , n, there exists a natural number m = m(i, j) such that the entry in the ith row and jth column of Cm is positive.

Results for <italic>n</italic> = 2 sub-populations

In the case of n = 2, the NGM (2.7) is a 2 × 2 matrix. From this matrix we obtain the following explicit expression for the reproduction number Rv as a function of p = (p1, p2): Rv(p)=12[Rv1c11+Rv2c22+(Rv1c11−Rv2c22)2+4Rv1c12Rv2c21]. One condition needed for proving the existence of an optimal solution to Problem (I) is that the mixing matrix satisfies C>0,∣C∣=c11c22−c12c21>0. It is easy to verify that condition (3.1) holds for Jacquez mixing as given in (2.5): ∣C∣=∣c11c12c21c22∣=ϵ1ϵ2+ϵ1(1−ϵ2)2a2N2+ϵ2(1−ϵ1)2a1N1(1−ϵ1)a1N1+(1−ϵ2)a2N2>0, provided that ϵi ∈ (0, 1), ai > 0, and Ni > 0, i = 1, 2.

Statements of the main results

Before we state the main results on the existence and uniqueness of the optimal solutions to Problems (I) and (II), we give the following critical properties of Rv(p1, p2):

Theorem 3.1 (Key properties of Rv(p1, p2)) Consider Rv=Rv(p1,p2) as a function of p1 and p2, and assume that condition (3.1) holds.

Rv(1,1)=0 and Rv grows linearly on the rays emanating from (1, 1) into the square [0, 1]2 (See Fig. 1 for illustration).

Rv(p1,p2) is convex on [0, 1]2 and strictly convex on the constraint set Ωp(2)(η)∩[0,1]2.

The proof of Theorem 3.1 is provided in “Appendix A.1”.

The following theorem describes the solution to Problem (I). For ease of presentation, we introduce the following notation: η0≔N−κ1N1+κ2N2max{κ1,κ2},κ1≔c22N1N2R02−N2c12c21R01R02,κ2≔c11N1N2R01−N1c12c21R01R02,Γ:(p1,p2)=(1,1)−s(κ1,κ2),s>0[orΓ:1−p21−p1=κ2κ1]. The set Γ describes the ray emanating from (1, 1) in the direction of −(κ1, κ2), to which we will refer as the critical ray (see Fig. 1).

Theorem 3.2 (Optimal solution to Problem (I) when n = 2) Consider Rv=Rv(p1,p2) as a function of p1 and p2, and let η0 and κi be given in (3.2). Assume that condition (3.1) holds.

For any given η ∈ [0, N], the optimal point P*(η) exists and is unique.

The point P*(η) lies in the interior of the unit square if and only if κ1>0,κ2>0 and η ∈ (η0, N).

For η ∈ (η0, N), all points P*(η) lie on the critical ray Γ, defined in (3.2) (see Fig. 1).

For each η ∈ (η0, N), the explicit formulae for P*(η) and Rv{min}(η) are P∗(η)=(1,1)−N−ηκ1N1+κ2N2(κ1,κ2) Rv{min}(η)=∣C∣R01R02N1N2N−ηκ1N1+κ2N2. A proof for Theorem 3.2 is given in “Appendix A.1”.

Remarks A few remarks are in order.

If either of the conditions in (3.3) is violated, or equivalently κ1 ≤ 0 or κ2 ≤ 0, then the ray Γ does not intersect the interior of the square (0, 1)2.

The minimum point P*(η) is the intersection of the critical ray Γ and the constraint set Ωp(2)(η) for each η ∈ (η0, N).

If η ∈ (0, η0), the intersection of Γ and Ωp(2)(η) lies outside the square [0, 1]2. When η = η0, the intersection lies on the boundary of the square.

If η ∈ (0, η0), the minimum point P*(η) is one of the boundary points (η/N1, 0) or (0, η/N2) and hence Rv{min}(η)=min{Rv(η∕N1,0),Rv(0,η∕N2)}.

Overall, Rv{min}(η) is a strictly decreasing convex function of η and linear on (η0, N) (see Fig. 2).

An explicit expression for the optimal solution η* to Problem (II) can be obtained by using Eq. (2.10). We consider two cases depending on the value of Rv{min}(η0)=∣C∣R01R02N1N2max{κ1,κ2}.

Theorem 3.3 (Critical number of vaccine doses) If condition (3.3) is satisfied, then the minimum value of η* in Problem (II) is given below.

(Interior minimum) If Rv{min}(η0) ≥ 1, then η* ≥ η0 and η∗=N−κ1N1+κ2N2∣C∣R01R02N1N2.

(Boundary minimum) If Rv{min}(η0) ≤ 1, then 0 ≤ η* ≤ η0 and η∗=min{N1−(1−c22R02)N1c11R01−∣C∣R01R02,N2−(1−c11R01)N2c22R02−∣C∣R01R02}. The proof is given in “Appendix A”.

Results for <italic>n</italic> ≥ 2 sub-populations

In this section, we extend the results for n = 2 sub-populations presented in Sect. 3 to the case of n > 2. For general mixing matrices C = (cij), due to the complexity of the optimization problems (I) and (II) when n > 2, we are unable to obtain explicit expressions for the optimal solutions. Nevertheless, we can derive lower and upper bounds for the minimum reproduction number Rv{min}(η) and the minimum vaccine doses η*. We first present results for general mixing (cij) satisfying (2.2)–(2.4). We then illustrate that some of the key necessary properties of the mixing matrix can be verified for the Jacquez mixing given in (2.5).

Preliminaries

Rewrite the NGM given in (2.7) as Kv(p)=(Rv1Rv2⋱Rvn)(c11c12⋯c1nc21c22⋯c2n⋮⋮⋮⋮cn1cn2⋯cnn)=diag(R01(1−p1),…,R0n(1−pn))C The effective reproduction number Rv for the meta-population is the spectral radius (and the dominant eigenvalue, by Perron–Frobenius Theorem) of the nonnegative matrix Kv(p), i.e., Rv(p) = r(Kv(p)).

Although the focus of this study is on optimal solutions to Problems (I) and (II), the results presented in the following theorem about the bounds of Rv(p) and R0=Rv(0) are significant in more general applications.

Theorem 4.1 (Bounds for Rv(p)) Let C be a nonnegative, invertible, irreducible matrix such that −C−1 is essentially nonnegative and the conditions (2.2)–(2.4) are satisfied. Then

The lower and upper bounds of Rv(p) are: ∑i=1nωiRvi≤Rv≤max{Rv1,…,Rvn},whereωi=aiNiΣk=1nakNk.

The lower and upper bounds of Rv(p) correspond to the cases of proportionate mixing and isolated mixing, respectively.

 The proof of Theorem 4.1 is given after the proof of Theorem 4.9.

Remarks Theorem 4.1 is stated using the effective reproduction number Rv(p) for 0 ≤ p ≤ 1. The results holds in particular for the basic reproduction number R0=Rv(0) that the the lower and upper bounds are ∑i=1nωiR0i and {R01,…,R0n}.

For the ease of presentation, we introduce the ‘reflected’ variables qi=1−pi,i=1,…,n,q=(q1,…,qn)∈(0,1)n. Note that qi represents the unvaccinated portion of sub-population i = 1, … , n. We also introduce the ‘reflected’ function R¯v(q)=Rv(1−q1,…,1−qn)=r(diag(R01q1,…,R0nqn)C). Note that the formula (4.2) can be used to extend R¯v to [0,∞)n. The constraint hyperplanes Ωp(n)(η) in Problem (I) will transform to Ωq(n)(η¯)≔{(q1,q2,…,qn):ℓ(q)=η¯}, where η¯≔N−η and the optimal point P*(η) will become Q∗(η¯)≔(1,…,1)−P∗(η)=(1,…,1)−P∗(N−η¯). For the minimum value of R¯v(q) on Ωq(n)(η¯)∩[0,1]n we will have R¯v{min}(η¯)=R¯v∣Q∗(η¯)=Rv∣P∗(η)=Rv{min}(η).

The following result states key properties of R¯v(q), generalizing Theorem 3.1 in the case n = 2.

Theorem 4.2 (Convexity and homogeneity of R¯v(q)) The function R¯v(q) is homogeneous of degree 1 on q ∈ [0, 1]n, i.e., R¯v(sq)=sR¯v(q),for allq∈[0,1]nands>0such thatsq∈[0,1]n.

Moreover, R¯v is convex if the matrix C = (cij) is invertible and −C−1 is essentially nonnegative. If additionally C is irreducible, then R¯v(q) is strictly convex on the constraint set Ωq(n)(η¯)∩[0,1]n (see Fig. 3).

Theorem 4.2 immediately implies the following property.

Theorem 4.3 (Critical ray) Let C be a nonnegative square invertible irreducible matrix such that −C−1 is essentially nonnegative. Let η¯ ∈ (0, N) be such that R¯v has an interior relative minimum point Q∗(η¯) on Ωq(n)(η¯)∩[0,1]n. Then Q∗(η¯) is the unique point that satisfies the Lagrange multiplier condition R¯v∣Q∗(η¯)=λ(N1,N2,…,Nn).

If s > 0 is such that sη¯ ∈ (0, N), then the unique relative minimum point on Ωq(n)(sη¯)∩[0,1]n is given by Q∗(sη¯)=sQ∗(η¯), provided that this point still lies in the interior of the unit hypercube.2 Thus, all interior relative minimum points lie on the critical ray Γ¯ emanating from the origin.

We start with the following facts about the spectral radius r (A) of a nonnegative matrix A. The first one is rather simple, if not obvious.

Lemma 4.4 If A is a square matrix and s > 0 then r(sA)=sr(A).

The second one is also well-known, see e.g. Hill and Longini (2003); Nussbaum (1986).

Lemma 4.5 If A, B are nonnegative irreducible matrices such that A ≤ B, then r(A)≤r(B).

The next one is more subtle and is based on a theorem of Friedland (1980/81, Theorem 4.3); see also generalizations of this result in Nussbaum (1986, Sect. 1).

Lemma 4.6 Let C be a nonnegative invertible squarematrix such that −C−1 is essentially nonnegative. Then the mapping rC:D↦r(DC) is convex on the set of positive diagonal matrices D = diag(d1, … , dn), di > 0, i.e., rC((1−t)D1+tD2)≤(1−t)rC(D1)+trC(D2) for any positive matrices D1 and D2 and t ∈ [0, 1]. Moreover, if additionally C is irreducible,3 then the inequality above is strict for t ∈ (0, 1), unless D2 = sD1 for some s > 0.

Proof of Theorem 4.2 If we denote K¯v(q)=Kv((1,…,1)−q)=diag(R01q1,…,R0nqn)C, then for homogeneity we just note that K¯v(sq)=sK¯v(q) and therefore by Lemma 4.4, we have R¯v(sq)=r(K¯v(sq))=r(sK¯v(q))=sr(K¯v(q))=sR¯v(q). The convexity of R¯v follows from the fact that R¯v(q)=rC(diag(R01q1,…,R0nqn)), i.e., R¯v is a composition of a linear mapping q↦diag(R01q1,…,R0nqn) and a convex function rC, and is therefore convex, as we impose the condition that −C−1 exists and is essentially nonnegative.

Finally, the strict convexity of R¯v on Ωq(n)(η¯)∩[0,1]n follows from the strict convexity property of rC in Lemma 4.6, because no two points on Ωq(n)(η¯) lie on the same ray emanating from the origin.

Upper and lower bounds of optimal solutions

In this section, we establish bounds on quantities relevant to Problems (I) and (II) for general mixing matrices C, satisfying (2.2)–(2.4) with an additional property that −C−1 is essentially nonnegative. As we saw in Theorem 4.2, the latter condition is needed to guarantee the convexity of R¯v. On the other hand, conditions (2.2)–(2.4) provide important information on positive eigenvectors of the mixing matrix C, which is instrumental in deriving our bounds.

More specifically, we prove upper and lower bounds for Q∗(η¯), R¯v{min}(η¯), and η¯∗ (or equivalently, P*(η), Rv{min}(η), and η*). An interesting feature is that these bounds are independent of the functional form of mixing cij.

<italic>Equal</italic> per capita <italic>contact rates</italic>

We start with a special case when all per capita contact rates ai are the same. We show that the minimum of R¯v(q) on Ωq(n)(η¯)∩[0,∞)n will occur on the diagonal q1 = q2 = … = qn, under the conditions on C that guarantee the convexity of R¯v(q). In particular, this will hold for simple mixing matrices given by (2.5) for any choice of ϵi ∈ (0, 1) and Ni > 0.

Theorem 4.7 (Equal per capita contact rates) Let C be a nonnegative, invertible, irreducible matrix such that −C−1 is essentially nonnegative and for which the conditions (2.2)–(2.4) are satisfied. Assume additionally that ai = a > 0, i = 1, … , n. Then the minimum of R¯v on the intersection of Ωq(n)(η¯)∩[0,1]n for η¯∈[0,N] is achieved at the point q1=q2=⋯=qn=η¯N. Thus, the critical ray Γ¯ is given by Γ¯:q1=q2=⋯=qn.

We will need the following characterization of Friedland (1980/81, Theorem 3.4) for the spectral radius.

Lemma 4.8 If C is a nonnegative, invertible matrix, such that −C−1 is essentially nonnegative, then its spectral radius r (C) is given by infξ∈Pnsupx>0∑i=1nξixi(Cx)i=1r(C), where Pn={ξ=(ξ1,…,ξn):ξi≥0,∑i=1nξi=1}. Moreover, if C is irreducible, and u = (u1, … , un)T > 0, v = (v1, … vn)T > 0 are right and left eigenvectors of C, i.e., Cu=r(C)u,CTv=r(C)v,∑i=1nuivi=1, then for ξ = (u1v1, – , unvn) one has supx>0∑i=1nξixi(Cx)i=1r(C). Proof of Theorem 4.7 When ai = a, i = 1, … , n, we have the following essential properties of the matrix C ∑j=1ncij=1,∑i=1nNicij=Nj.

This means that u = (1, … , 1)T and v = (N1/N, … , Nn/N)T are normalized right and left eigenvectors of C: Cu=u,CTv=v,Σuivi=1.

By Perron–Frobenius Theorem, we also have r (C) = 1. (Note that this holds for any nonnegative matrix satisfying (2.3)). Then by Lemma 4.8 we have 1=1r(C)=supx>0∑i=1nNiNxi(Cx)i.

Suppose now that q = (q1, … , qn) ∈ (0, 1)n is such that ∑i=1nqiNi=η¯∈(0,N). Then β = (β1, … , βn) ∈ Pn, where βi=qiNi∕η¯,i=1,…,n. Note, that we also have (K¯v(q)x)i=R0qi(Cx)i, where R0=ρa is the common value of R0i , i = 1, … , n, which also is the meta-population basic reproduction number. Then, by Lemma 4.8, 1r(K¯v)(q)≤supx>0∑i=1βixi(K¯v(q)x)i=supx>0∑i=1nqiNiη¯xiR0qi(Cx)i=NR0η¯supx>0∑i=1nNiNxi(Cx)i=NR0η¯. Note that the use of Lemma 4.8 above is justified because, as with C, K¯v(q) is positive, invertible, irreducible, and −K¯v(q)−1=−C−1diag((R0q1)−1,…,(R0qn)−1) is essentially nonnegative. Hence, R¯v(q)≥R0η¯N,on{ℓ(q)=η¯}∩(0,1)n. On the other hand, K¯v(η¯N,…,η¯N)=R0η¯NC and therefore R¯v(η¯N,…,η¯N)=R0η¯N. By continuity of R¯v, this completes the proof.

<italic>Arbitrary</italic> per capita <italic>contact rates</italic>

In this section, we first establish the upper and lower bounds for R¯v(q) (see Theorem 4.9) and equivalently Rv(p). The lower bound can be proved using the arguments similar to the proof of Theorem 4.7, and the upper bound follows from the monotonicity of the spectral radius as a function of nonnegative matrices. We then proceed to obtain bounds for the relative minima R¯v{min}(η¯) and equivalently Rv{min}(η), as well as a bound for the critical value η¯=η¯∗ that makes R¯v{min}(η¯) ≤ 1.

Let amin denote the minimum of the activities of the sub-populations, i.e., amin=min{a1,…,an}. Note that amin > 0. The results beloware for general mixing matrices (not just Jacquez mixing given in (2.5)).

Theorem 4.9 (Bounds for R¯v(q)) Let C be a nonnegative, invertible, irreducible matrix such that −C−1 is essentially nonnegative and the conditions (2.2)–(2.4) are satisfied. Then the bounds of R¯v(q) are: ρΣi=1nai2NiqiΣi=1naiNi≤R¯v(q)≤ρmax{a1q1,…,anqn},forq∈[0,1]n. Moreover, equalities hold if q=s(1a1,…,1an) for s ∈ [0, amin].

Proof We start with the lower bound in (4.3). The key observation is that ∑j=1ncij=1,∑i=1naiNicij=ajNj, which gives positive eigenvectors for C and CT. That is, if u = (1, … , 1)T and v = (a1N1/A, … , anNn/A)T with A=∑i=1naiNi, then Cu=u,CTv=v,u⋅v=∑i=1nuivi=1. Then, by Lemma 4.8, 1=1r(C)=supx>0∑i=1naiNiAxi(Cx)i=supx>0Σi=1naiNixi(Cx)iΣk=1nakNk. Then, for any q ∈ (0, 1)n, let βi=ai2NiqiΣk=1nak2Nkqk and note that βi > 0 and ∑i=1nβi=1; i.e., β = (β1, … , βn) ∈ Pn. Hence, by Lemma 4.8, 1R¯v(q)=1r(K¯v(q))≤supx>0∑i=1nβixi(K¯v(q)x)i=supx>0∑i=1nai2Niqixiρaiqi(Cx)i∑k=1nak2Nkqk=supx>0∑i=1naiNixi(Cx)iρ∑k=1nak2Nkqk=∑k=1nakNkρ∑k=1nak2Nkqk, or equivalently R¯v(q)≥ρ∑i=1nai2Niqi∑i=1naiNi,for anyq∈(0,1)n, and by continuity also for all q ∈ [0, 1]n. This proves the lower bound.

The upper bound of R¯v(q) can be obtained by noticing that K¯v(q)=diag(Rv1,…,Rvn)C≤max{Rv1,…,Rvn}C and applying Lemma 4.5: R¯v(q)=r(K¯v(q))≤max{Rv1,…,Rvn}r(C)=max{Rv1,…,Rvn}=ρmax{a1q1,…,anqn}, where we have used that r (C) = 1.

To establish the equalities in (4.3), it is easy to see that the upper and lower bounds are the same when q = s(1/a1, … , 1/an) for constant s ∈ (0, amin). The restriction on s guarantees that the point q is in (0, 1)n. Then, by continuity, equality holds also for the endpoint values of s.

The proof is completed.

We now use the results above to prove Theorem 4.1.

Proof of Theorem 4.1 To prove Theorem 4.1(a), i.e., the bounds for Rv(p) in (4.1), recall that qi = 1 – pi, Rv(p)=R¯v(q), and Rvi=ρaiqi for i = 1, 2, … , n. Substitution of these relationships into (4.3) yields (4.1).

For (b), note that when themixing C = (cij) is proportionate, cij = ωj. In this case, the NGM has rank 1 and its dominant eigenvalue is the sum of the diagonal elements. That is, Rv(p)=∑i=1nωiRvi. Note also that in this case of isolated mixing, i.e., ϵi = 1 for all i in (2.5), C = I and the NGM is diag(Rv1,Rv2,…,Rvn), for which Rv=max{Rv1,Rv1,…,Rv1}.

It is clear that for the proportionate mixing and isolated mixing, the corresponding reproduction numbers coincide with the lower and upper bounds of Rv given in (4.1). Thus, while Theorem 4.1, as stated, is not formally applicable to the proportionate and isolated mixing functions, the bounds of Rv in (4.1) correspond to these two extreme cases.

This completes the proof.

It is easy to verify that, when a1 = a2 = ⋯ = an, Theorem 4.9 contains the conclusion of Theorem 4.7.

Theorems 4.1 and 4.9 can be used to derive the lower and upper bounds for the minimum reproduction number Rv{min}(η). Introduce the following notation: πi≔Ni∕N,1≤i≤nPopulation fraction of sub-populationi;S≔∑i=1n(1−pi)πiPopulation fraction unvaccinated;R^0≔∑i=1nR0iπiPopulation weighted reproduction number;R0⋄≔(∑i=1n1R0iπi)−1Harmonic mean ofR0iweighted by sub-population fractionsπi;R~0≔miniR0i2∕R^0Analogous to a scaled reproduction number

The following results provide the lower and upper bounds for the minimum Rv{min}(η) in Problem (I):

Theorem 4.10 Assume that the conditions of Theorem 4.9 hold. Let η < N (or η¯ > 0), and let S, R^0, R0⋄, and R~0 be defined in (4.5).

The bounds of R¯v{min}(η¯) for q∈Ωq(n)(η¯)∩[0,1]n are: ρamin2η¯Σi=1naiNi≤R¯v{min}(η¯)≤ρη¯Σi=1nNi∕aiforη¯≤mini{R0i}R0⋄N.

The bounds of Rv{min}(η) for p∈Ωp(n)(η)∩[0,1]n are: R~0S≤Rv{min}(η)≤R0⋄S.

Proof (a) To prove the lower bound, note first that R¯v{min}(η¯) ≤ R¯v(q) for all q ∈ q∈Ωq(n)(η¯)∩[0,1]n. Note also that from the inequality in Theorem 4.9 we have R¯v(q)≥ρΣi=1nai2NiqiΣi=1naiNi≥ρamin2Σi=1nNiqiΣi=1naiNi=ρamin2η¯Σi=1naiNi. This proves the lower bound in (4.6).

For the upper bound in (a), choose s > 0 so that Q = s(1/a1, … , 1/an) ∈ Ωq(n)(η¯); i.e., s=η¯∕(∑i=1nNi∕ai). Note that Q ∈ [0, 1]n if 0 ≤ s ≤ amin or 0 ≤ η¯ ≤ amin ∑i=1nNi∕ai. Then, by the case of equality in (a) of Theorem 4.9, R¯v{min}(η¯)≤R¯v(Q)=ρs=ρη¯Σi=1nNi∕ai. This completes the proof of (a).

To prove (b), note that Rv{min}(η)=R¯v{min}(η¯). Note also that the left-hand side (LHS) and right-hand side (RHS) of inequality (4.6) can be re-expressed in terms of sub-population reproduction numbers R0i=ρai. For the LHS of (4.6), LHS=min1≤i≤nR0i2R^0×∑i=1n(1−pi)πi=R~0S. It follows that Rv{min}(η)≥R~0S. For the RHS of (4.6), note that Rv{min}(η)≤1Σi=1nπi∕R0i×∑i=1n(1−pi)πi=R0⋄S. From (4.8) to (4.9) we obtain (4.7). This completes the proof of (b).

We can now deduce the results for the upper bound of the critical number of vaccine doses η* in Problem (II):

Theorem 4.11 (Critical number of vaccine doses) Let C satisfy the same conditions as in Theorem 4.9, and let R0⋄ be defined in (4.5).

the upper bound for the minimum vaccine dose is given by: η∗≤N−min{1∕ρ,a1,…,an}∑i=1nNi∕ai.

If R0i>1 for all i , then the inequality (4.10) can be re-written as η∗N≤1−1R0⋄.

Proof (a) We prove the inequality (4.10) by considering two cases.

Case 1: Assume first that 1/ρ ≤ amin. In that case, the upper bound in Theorem 4.10 is applicable for η¯=(1∕ρ)∑i=1nNi∕ai, which gives R¯v{min}(η¯)≤1. Thus, η¯∗≥1ρ∑i=1nNi∕ai.

Case 2: Assume now that 1/ρ ≤ amin. Then the upper bound in Theorem 4.10 for η¯=amin∑i=1nNi∕ai gives R¯v{min}(η¯)≤ρamin≤1. Thus, η¯∗≥amin∑i=1nNi∕ai. Combining cases 1 and 2, we obtain η¯∗≥min{1∕ρ,a1,…,an}∑i=1nNi∕ai, and recalling that η¯∗ = N − η*, we complete the proof of (a).

For (b), note that the RHS of (4.10) can be re-written as N−1ρ∑i=1nNiai=N−∑i=1nNiR0i=N(1−∑i=1n1R0iπi). Thus, the inequality (4.10) can be re-written as η∗N≤1−∑i=1n1R0iπi=1−1R0⋄. This completes the proof.

Remarks

Note that R0⋄ and R~0 (see (4.5)) are weighted basic reproduction numbers, and the factor S is the fraction of the overall population that remains susceptible. In light of this, we see that the lower and upper bounds for Rv{min}(η) in (4.7) take the familiar form of an effective reproduction number.

The lower and upper bounds in (4.7) are equal if the activities ai for subpopulations i are all the same. Note that R0i = ρai = βai /(θ + γ). Thus, R0i are the same when ai are the same for all i. Then, from (4.5) we see that R~0=R0⋄, which implies that the inequalities in (4.7) become equalities.

For the upper bound of η*, if ai = a are all the same, we have R0⋄=R0, in which case the upper bound in (4.11) becomes 1 − 1/R0. This is similar to the usual formula for the critical vaccination fraction pc = 1 − 1/R0, for which the number of vaccinated is ηc = pcN = N(1 − 1/R0).

4.3 Example: the case of Jacquez mixing

As an example ofmixing functions that satisfy the conditions described in Lemma 4.6, we consider the cij for the meta-population model in Feng et al. (2015), which is the Jacquez mixing as given in (2.5).

Proposition 4.12 Let the matrix C = (ci j) be given by (2.5) with ϵi ∈ (0, 1), i = 1, … , n. Then C is invertible and B = C−1 = (bij) is given by bij=δijϵi−1+(1−ϵi−1)(1−ϵj−1)ajNjΣk=1n(1−ϵk−1)akNk,i,j=1,…,n. In particular, −C−1 is essentially nonnegative.

Remark 4.13 It is also clear that under conditions of Proposition 4.12, C is positive and thus irreducible.

Proof It will be sufficient to show that CB = I (the identity matrix), or ∑j=1ncijbjk=δik,for alli,k=1,…,n. To simplify computations, let μc=∑l=1n(1−ϵl)alNl,μb=∑l=1n(1−ϵl−1)alNl. Then ∑j=1ncijbjk=∑j=1n[δijϵi+(1−ϵi)(1−ϵj)ajNjμc]×[δjkϵj−1+(1−ϵj−1)(1−ϵk−1)akNkμb]=∑j=1nδijδjkϵiϵj−1+1μc∑j=1nδjk(1−ϵi)(1−ϵj)ϵj−1ajNj+1μb∑j=1nδijϵi(1−ϵj−1)(1−ϵk−1)akNk+1μcμb∑j=1n(1−ϵi)[(1−ϵi)(1−ϵj−1)](1−ϵk−1)ajNjakNk=δik−(1−ϵi)(1−ϵk−1)akNkμc−(1−ϵi)(1−ϵk−1)akNkμb+Σj=1n[(1−ϵj)+(1−ϵj−1)]ajNjμcμb(1−ϵi)(1−ϵk−1)akNk=δik+(−1μc−1μb+μc+μbμcμb)(1−ϵi)(1−ϵk−1)akNk=δik, where we have used above that (1−ϵj)(1−ϵj−1)=(1−ϵj)+(1−ϵj−1). This completes the proof.

Discussion

The main goal of this study was to solve Problems (I) and (II), which identify the most efficient allocation of limited vaccines using a meta-population model for vaccine-preventable infectious diseases. Although we demonstrated the results using Model (2.1), the approach can be applied to other meta-population models for vaccine-preventable diseases. Model (2.1) incorporates various heterogeneities such as in activity, contacts between sub-populations (mixing), vaccination coverage, and size of each sub-population. We considered general mixing functions that satisfy conditions (2.2)–(2.4), including the special case of Jacquez mixing in (2.5) and special cases of this namely, proportionate mixing (ϵi = 0 for all i) and preferential mixing ( ϵi = 1 for all i). However, it would be more challenging to consider a similar approach to meta-population models that are less-tractable analytically than Model (2.1), particularly when the NGM has a more complicated structure. For example, when a model includes additional factors such as aging from one age-group to the next, multi-level mixing (e.g., age and spatial), and heterogeneity in infectivity and susceptibility, the effective reproduction number Rv as a function of vaccination coverage will be more difficult to study from an analytic point of view.

The optimization problem is based on reducing the effective reproduction number Rv (if R0 > 1) by determining the optimal combination of vaccine coverages p = (p1, p2, … , pn). Because the parameters pi must be between 0 and 1, the optimal solution P*(η) needs to be in the unit hypercube. Even in the case of n = 2 sub-populations, the solution of Problems (I) and (II) is not trivial. For n > 2, the most challenging task is to show the convexity of Rv(p); Theorem 4.2. This proves a conjecture of Hill and Longini (2003), although those authors did not consider the structure of the mixing matrix C specified by conditions (2.2)–(2.4). Our proofs are facilitated by using the ‘reflected’ quantities qi = 1 − pi and R¯v(q1, q2, … , qn) = Rv(p1, p2, … , pn). For ease of presentation, we first illustrated results for the simpler case of n = 2 sub-populations, and then extended them to n > 2 sub-populations.

In the case of n = 2, explicit formulae are obtained for the optimal solutions when mixing is proportionate or preferential. For Problem (I), the optimal solution P*(η) for a given number of vaccine doses η and the minimized reproduction number Rv{min}(η) are described as functions of model parameters (Theorem 3.2). For Problem (II), an analytical formula for the minimum vaccine doses η* that will reduce Rv to below 1 is provided (Theorem 3.3).

Another interesting finding is that, for any number of vaccine doses η in the constraint, the optimal vaccine coverage P*(η) lies along the ‘critical’ ray Γ . In addition, when n = 2, for the optimal solution P*(η) to be in the unit square [0, 1]2, available vaccine doses must satisfy η0 < η < N, where the lower bound η0 is determined by model parameters. For η ≤ η0, the optimal strategy will be to vaccinate only one sub-population.

Extension of these results to the case n > 2 is complicated by the fact that no explicit formulae are available. Nevertheless, using results for the spectral radius of nonnegative matrices, in Sects. 4.1 and 4.2, we obtain bounds of Rv for an arbitrary mixing C that satisfies the conditions in Theorem 4.1. An interesting finding is that for a large class of mixing matrix C (not just Jacquez), the proportionate mixing gives the smallest Rv while the isolated mixing (no mixing between sub-populations) gives the largest Rvi given Rvi (see Theorem 4.1). It is clear that these conclusions hold particularly for R0=Rv(0). The facts that population heterogeneities tend to increase R0 and that models assuming proportionate mixing generate lower values of R0 have been suggested by other researchers (Adler 1992; Andersson and Britton 1998; Diekmann et al. 2012).

We also establish bounds on the relative minima Rv{min}(η) and the critical vaccine dose η* (see Theorems 4.10 and 4.11). Interpretations of those bounds are provided in terms of biological quantities such as weighted reproduction numbers. In particular, we see that the lower and upper bounds are products of the weighted reproduction number R~0 and the harmonic mean R0⋄, respectively, with the fraction unvaccinated S (see (4.7)). Thus, both bounds are in the familiar form of effective reproduction numbers. Moreover, the bounds are equal when all sub-populations have the same activity (ai). A similar interpretation holds for the upper bound of η*, in which case the usual basic reproduction number R0 is replaced by the harmonic mean R0⋄ of sub-population reproduction numbers the R0i weighted by sub-population fractions πi (see also (4.11)).

Acknowledgements

The findings and conclusions in this report are those of the author(s) and do not necessarily represent the official position of the Centers for Disease Control and Prevention or other institutions with which they are affiliated. We thank the anonymous reviewers for comments and suggestions, which helped improve the presentation of the manuscript.

The findings and conclusions in this paper are those of the authors and do not necessarily represent the views of the Centers for Disease Control and Prevention.

Note that terminology differs among authors. Here, we use the terminology of Nussbaum (1986); the negatives of such matrices are called M-matrices by Friedland (1980/81) and are said to have a Z sign pattern by van den Driessche and Watmough (2002).

This will automatically hold for 0 < s < 1.

See the definition in Sect. 2.1.

This will automatically hold for 0 < s < 1.

A Appendix

In this appendix, we provide detailed proofs for Theorems 3.1–3.3 in Sect. 3 and illustrate an example of these results in the case of Jacquez mixing.

Proofs of Theorems <xref ref-type="disp-formula" rid="FD17">3.1</xref>–<xref ref-type="disp-formula" rid="FD20">3.3</xref>

To prove these theorems, we first prove several propositions. Instead of working with the function Rv(p1, p2) along the rays emanating from the point (p1, p2) (1, 1), it is much easier to consider the ‘reflected’ variables: qi=1−pi,i=1,2,q=(q1,q2), and the corresponding rays emanating from the point (q1, q2) = (0, 0) into the unit square [0, 1]2. The ‘reflected’ function R¯v is then given by R¯v(q1,q2)=Rv(1−q1,1−q2). The quantities corresponding to those mentioned directly after Problem (I) in terms of q and the reflected function are η¯≔q1N1+q2N2=N−η,ℓ(q)≔q1N1+q2N2=η¯,Q∗(η¯)≔(1,1)−P∗(η)=(1,1)−P∗(N−η¯). In addition, at the optimal points P* or Q*, we have R¯v{min}(η¯)=R¯v∣Q∗(η¯)=Rv∣P∗(η)=Rv{min}(η), and we know that R¯v satisfies the equation ∇R¯v∣Q∗(η¯)=λ(N1,N2), or, equivalently, ∇R¯v∣Q∗(η¯)⋅(N2,−N1)=0, provided Q∗(η¯) is in the interior of the unit square.

Reflected function <inline-formula><mml:math display="inline" id="M287" overflow="scroll"><mml:mrow><mml:msub><mml:mover accent="true"><mml:mi mathvariant="script">R</mml:mi><mml:mo>¯</mml:mo></mml:mover><mml:mi>v</mml:mi></mml:msub></mml:mrow></mml:math></inline-formula> and its properties

The reflected function R¯v is given explicitly by R¯v(q)=12[R01c11q1+R02c22q2+(R01c11q1−R02c22q2)2+4R01R02c12c21q1q2] for q ∈ [0, 1]2. Note that the formula (A.1) can be used to naturally extend R¯v to the first quadrant q ≥ 0 (i.e., q1, q2 ≥ 0). It can be checked that the function R¯v is homogeneous of degree 1, i.e., R¯v(sq)=sR¯v(q),s>0,q≥0. Geometrically, this means that R¯v grows linearly on the rays emanating from the origin, as illustrated in Fig. 4.

We next compute the first derivatives of R¯v: ∂R¯v∂qi=12[R0icii+R0i2cii2qi+(2c12c21−c11c22)R01R02qj(R01c11q1−R02c22q2)2+4R01R02c12c21q1q2] for i, j = 1, 2 and i ≠ j. Note that the function ∇R¯v=(∂R¯v∕∂q1,∂R¯v∕∂q2) is homogeneous of degree 0: ∇R¯v(sq)=∇R¯v(q),s>0,q≥0, i.e., ∇R¯v is constant on rays emanating from the origin.

Further, we compute the second derivatives of R¯v. By direct computation, we have the following formula for the Hessian HessR¯v=k(q22−q1q2−q1q2q12), where k=c12c21∣C∣R012R022[(R01c11q1−R02c22q2)2+4R01R02c12c21q1q2]3∕2. Note that k > 0 by (3.1) and therefore Hess R¯v is a nonnegative semi-definite matrix, by Sylvester’s criterion. Consequently, R¯v is a convex function of (q1, q2). We explicitly note here that R¯v is not a strictly convex function, as it grows linearly on the rays emanating from the origin. However, as we show below, it is strictly convex in certain directions. To be more precise, let u = (u1, u2) be a unit vector, and consider the second derivative of R¯v in the direction u. We have ∂2R¯v∂u2=k(u12q22+u22q12−2u1u2q1q2)=k(u1q2−u2q1)2≥0, whereupon ∂2R¯v∂u2>0unlessu∥(q1,q2). (Here, u ∥ v indicates that vectors u and v are parallel.) In particular, R¯v is strictly convex in the direction (N2, − N1).

To proceed, recall that Ωq(2)η¯≔{(q1,q2):ℓ(q)=η¯}, and consider again the constraint set Ωq(2)(η¯)∩[0,1]2,forη¯∈[0,N]. This is a line segment parallel to the vector (N2, − N1) with endpoints on the boundary of [0, 1]2. We will denote the left and right endpoints (with respect to the direction (N2, − N1)) by Q1(η¯) and Q2(η¯), respectively. It is easy to see that Q1(η¯)∈({0}×[0,1])∪([0,1]×{1}),Q2(η¯)∈([0,1]×{0})∪({1}×[0,1]).

Proposition A.1 The function R¯v(q) is strictly convex on the intersections Ωq(2)(η¯)∩[0,1]2, for η¯ ∈ (0, N) if condition (3.1) holds. Consequently its minimum will occur at an interior point Q∗(η¯) if and only if ∇R¯v∣Q1(η¯)⋅(N2,−N1)<0, ∇R¯v∣Q2(η¯)⋅(N2,−N1)>0, where Q1(η¯) and Q2(η¯) are the left and right endpoints of Ωq(2)(η¯)∩[0,1]2 (see Fig. 5). Moreover, Q∗(η¯) is the unique point on Ωq(2)(η¯)∩[0,1]2 such that ∇R¯v∣Q∗(η¯)⋅(N2,−N1)=0,

From the homogeneity properties (A.2) and (A.4), we also have the following proposition.

Proposition A.2 Under the assumptions of Proposition A.1, let η¯ ∈ (0, N) be such that Q∗(η¯) is an interior point. Then, for s > 0 such that sη¯ ∈ (0, N), the minimum point of R¯v on Ωq(2)(sη¯)∩[0,1]2 is given by Q∗(sη¯)=sQ∗(η¯), provided that this point still lies in the interior of the unit square.4 In other words, all interior minimum points lie on a ray emanating from the origin. We will denote this ray by Γ¯ and call it the (reflected) critical ray. Moreover, by Proposition A.1 Γ¯∩[0,1]2={q:∇R¯v(q)⋅(N2,−N1)=0}∩[0,1]2. Proof For Q∗(η¯) we have (see Fig. 5) ℓ(Q∗(η¯))=η¯and∇R¯v∣Q∗(η¯)⋅(N2,−N1)=0. But then ℓ(sQ∗(η¯))=sη¯ and by (A.4) ∇R¯v∣sQ∗(η¯)⋅(N2,−N1)=∇R¯v∣Q∗(η)⋅(N2,−N1)=0, implying that Q∗(η¯) is the critical point on the constraint with constant s η¯ and thus that Q∗(sη¯)=sQ∗(η¯). The rest then follows from Proposition A.1.

Endpoint conditions

We now write the endpoint conditions (A.5)–(A.6) using the explicit formulae for the derivatives of R¯v. Because of Proposition A.2, to verify (A.5)–(A.6), it will be sufficient to verify them for the constraint value s η¯ with a small s > 0. Thus, without loss of generality, we may assume that η¯ itself is small. In that case, the intersection points Q1(η¯) and Q2(η¯) of Ωq(2)(η¯) with ∂([0, 1]2) will lie on the left and bottom sides of the square, i.e., Q1(η¯)∈{0}×(0,1),Q2(η¯)∈(0,1)×{0}. From (A.3) we have ∇R¯v∣q1=0=(c12c21c22R01,R02c22),∇R¯v∣q2=0=(R01c11,c12c21c11R02) and, therefore, the conditions (A.5)–(A.6) will take the form c12c21c22R01N2−R02c22N1<0,R01c11N2−c12c21c11R02N1>0 which is equivalent to (3.3).

The critical ray <inline-formula><mml:math display="inline" id="M350" overflow="scroll"><mml:mover accent="true"><mml:mi>Γ</mml:mi><mml:mo>¯</mml:mo></mml:mover></mml:math></inline-formula>

We next characterize the critical ray Γ¯. Namely, we identify the intersection point of Γ¯ with ∂([0, 1]2), which we denote by Q0.

For this, consider the function ϕ(q)=∇R¯v(q)⋅(N2,−N1). Because ϕ = 0 on Γ¯ (see Proposition A.1), we must have ϕ(Q0)=0. Note that conditions (A.5) and (A.6) are equivalent to ϕ(0,1)<0,ϕ(1,0)>0. Thus, the location of Q0 will depend on the sign of ϕ(1,1)=∇R¯v(1,1)⋅(N2,−N1). That is, Q0=(1,q20) ∈ {1} × (0, 1] if ϕ(1, 1) ≤ 0, and Q0 = (q10, 1) ∈ (0, 1] × {1} if ϕ(1, 1) ≥ 0. We will use this fact to find Q0 and characterize Γ¯.

Proposition A.3 Assume that conditions (3.1) and (3.3) hold, where κ1, κ2 are as in (3.2). Then the intersection Q0=(q10,q20) of the critical ray Γ¯ with ∂([0, 1]2) has the property q20q10=κ2κ1. Consequently, the critical ray Γ¯ is given by Γ¯:q2q1=κ2κ1. Proof Recall that ϕ(q)=∇R¯v(q)⋅(N2,−N1) and consider the case ϕ(1, 1) ≤ 0 first. In that case, Q0=(1,q20), where q20 is found from the equation ϕ(1, q2) = 0. This equation can be reduced to a quadratic equation for q2, which has two roots, given by the formulae q2(−)=λρc11ρ−c12c21λc22λ−c12c21ρ,q2(+)=λρc11ρ+c12c21λc22λ+c12c21ρ, where λ=N1N2,ρ=R01R02. Then q20 equals either q2(−) or q2(+) (the reduction to a quadratic equation may have introduced a false root). Plugging the formulae above into ϕ, it can be verified that ϕ(1,q2(−))=0 if and only if c22λ2+c11ρ2≥2λρc12c21. This inequality is indeed satisfied, because of the condition (3.1): c22λ2+c11ρ2≥2λρc22c11≥2λρc12c21. On the other hand, the verification of the equation ϕ(1,q2(+))=0 results in the condition c22λ2 = c11ρ2, which implies that q2(+)=q2(−). Thus, in either case, we can conclude that q20=q2(−). Hence, q20q10=q20=λρc11ρ−c12c21λc22λ−c12c21ρ, which is the same quantity as κ2/κ1 as in the statement of the proposition. This finishes the proof in this case.

The case ϕ(1, 1) ≥ 0 is considered similarly and we obtain exactly the same value for the ratio q20∕q10.

We next identify the value of the constraint η¯=η¯0, which corresponds to the intersection point Q0=(q10,q20) of Γ¯ and ∂([0, 1]2).

Proposition A.4 Let Q0 be as in Proposition A.3. Then η¯0=ℓ(Q0)=κ1N1+κ2N2max{κ1,κ2}. Proof If κ1 ≥ κ2 then q10=1 and q20=κ2∕κ1 and thus η¯0=ℓ(Q0)=N1+κ2κ1N2=κ1N1+κ2N2κ1. On the other hand, if κ1 ≤ κ2 then q20=1 and q10=κ1∕κ2 and thus η¯0=ℓ(Q0)=κ1κ2N1+N2=κ1N1+κ2N2κ2. Combining the two cases, we obtain the stated formula.

Explicit formulae for minima

Proposition A.5 For 0 < η¯ < η¯0, we have the following explicit formulae for Q∗(η¯) and the minimum of R¯v{min}(η¯): Q∗(η¯)=η¯κ1N1+κ2N2(κ1,κ2),R¯v{min}(η¯)=∣C∣R01R02N1N2η¯κ1N1+κ2N2. Proof We know that Q∗(η¯)=(κ1,κ2)s, where s > 0 can be found from the constraint ℓ(κ1s,κ2s)=η¯, which gives s=η¯κ1N1+κ2N2. This proves the first formula. To establish the second, we first use the homogeneity of R¯v: R¯v{min}(η¯)=R¯v∣Q∗(η¯)=R¯v(κ1s,κ2s)=R¯v(κ1,κ2)s. Furthermore, by direct calculations, one can show that R¯v(κ1,κ2)=∣C∣R01R02N1N2, which completes the proof.

Proofs of the Theorems

The proofs for Theorems 3.1 and 3.2 can be completed by combining Propositions A.1–A.5, and by writing their statements in terms of the original variables pi and the constraint value η.

To prove Theorem 3.3, we consider several cases. When Rv{min}(η0)≥1, we simply use the formula for Rv{min}(η) in Theorem 3.2 for η0 < η < N to find η*. When Rv{min}(η0)≤1, the minimum points P*(η) will be on the boundary of the square for 0 < η < η0 and we solve the equations Rv(p1,0)=1,Rv(0,p2)=1, or more precisely in variables (q1, q2): Rv(q1,1)=1,Rv(1,q2)=1, from which we get q1=(1−c22R02)c11R01−∣C∣R01R02,q2=(1−c11R01)c22R02−∣C∣R01R02. The corresponding values of η* are η∗=min{N1(1−q1),N2(1−q2)}.

Example: the case of Jacquez preferred mixing

The conditions in (3.3), which guarantee that the critical ray Γ passes through the interior of the unit square, hold for general mixing functions that satisfy (2.2)–(2.4). These conditions may simplify when specific functions are considered. In this section, we consider the Jacquez preferred mixing given in (2.5).

Interpretation of conditions for the interior critical ray

To verify the conditions (3.3), we consider two cases, one for homogeneous activity (a1 = a2) and other for heterogeneous activity (a1 ≠ a2).

Case 1: a1 = a2. In this case, the inequality κ1 > 0 can be rewritten as (1−ϵ1)(1−ϵ2)N2<ϵ2(1−ϵ1)N1+[ϵ2(1−ϵ2)+(1−ϵ2)2]N2 or −ϵ1(1−ϵ2)N2<ϵ2(1−ϵ1)N1, which is always satisfied. Similarly, one can verify that κ2 > 0 as well and therefore conditions (3.3) will always hold if a1 = a2.

Case 2: a1 ≠ a2. In this case, the inequality κ1 > 0 can be rewritten as (1−ϵ1)(1−ϵ2)a1N2<ϵ2[(1−ϵ1)a1N1+(1−ϵ2)a2N2]+(1−ϵ2)2a2N2 and simplifying further to (1−ϵ2)[(1−ϵ1)a1−a2]N2<ϵ2(1−ϵ1)a1N1.

Note that this inequality will readily hold if (1 − ϵ1)a1 − a2 ≤ 0 (which will happen, e.g., if a1 ≤ a2), as the left-hand side will be nonpositive and the right-hand side positive. If, however, (1 − ϵ1)a1 − a2 > 0, then the above inequality will transform to N2N1<ϵ21−ϵ2(1−ϵ1)a1(1−ϵ1)a1−a2.

By repeating this analysis with interchanged indices, we summarize the results above in the following proposition.

Proposition A.6 Suppose the mixing matrix C is given by (2.5) with n = 2.

If a1 = a2, then condition (3.3) holds for any Ni, ϵi ∈ (0, 1), i = 1, 2.

If 1 − ϵ1 ≤ a2/a1 ≤ 1/(1 − ϵ2), then (3.3) holds for any N1, N2.

If a2/a1 < 1 − ϵ1, then (3.3) becomes N2N1<ϵ21−ϵ2(1−ϵ1)a1(1−ϵ1)a1−a2.

(iv) If 1/(1 − ϵ2) < a2/a1, then (3.3) becomes 1−ϵ1ϵ1(1−ϵ2)a2−a1(1−ϵ2)a2<N2N1.

Simplified expressions at the optimal point

Some of the explicit expressions for the optimal solution provided in previous subsections hold for more general mixing functions C = (cij). These expressions may be simplified when the preferred mixing given in (2.5) is used. These simplified expressions are described in the following propositions. Let αi≔aj[(1−ϵj)(aj−(1−ϵj)ai)Nj+aiϵj(1−ϵi)Ni], for i, j = 1, 2 and i ≠ j. Note that the αi differ from the κi in (3.2) by a constant positive factor. In particular, α2/α1 = κ2/κ1.

Proposition A.7 Consider the mixing function given in (2.5). Let condition (3.3) be satisfied as described in Proposition A.6. Then the relative minima Q∗(η¯) will be interior points if and only if 0<η¯<η¯0=α1N1+α2N2max{α1,α2}, and will lie on the critical ray Γ¯:q2q1=α2α1. Moreover, the following explicit formulae hold for 0 < η¯ < η¯0: Q∗(η¯)=(α1,α2)α1N1+α2N2η¯,Rv{min}(η¯)=ρa1a2(a2N2ϵ1(1−ϵ2)+a1N1(1−ϵ1)ϵ2)α1N1+α2N2η¯.

We next note that, for αi defined in (A.8), we have α2−α1=(a1−a2)[a1N1(1−ϵ1)+a2N2(1−ϵ2)], and therefore, max{α1,α2}={α1,a1≤a2α2,a1≥a2.} In particular, α1 = α2 if a1 = a2. This implies the following particular case of Proposition A.7, which is especially interesting as the minimum points and values do not depend on ϵi (i = 1, 2) although the function R¯v and its level sets do (see Fig. 6). We also remark that this is essentially a version of Theorem 4.7 for n = 2, but its proof is more elementary.

Proposition A.8 Let C be the Jacquez mixing given in (2.5), and let a1=a2≕a. Then for all possible values of constants ϵi ∈ (0, 1) and Ni > 0 (i = 1, 2), the critical ray coincides with the diagonal Γ¯:q1=q2, and the expressions in (A.9) simplify to Q∗(η¯)=η¯N,R¯v{min}(η¯)=ρaη¯N,0<η¯<N.

ReferencesAdlerFR (1992) The effects of averaging on the basic reproduction ratio. Math Biosci 111(1):89–981515741AnderssonH, BrittonT (1998) Heterogeneity in epidemic models and its effect on the spread of infection. J Appl Probab 35(3):651–661BrauerF, Castillo-ChavezC (2012) Mathematical models in population biology and epidemiology, texts in applied mathematics, 2nd edn. Springer, New York https://doi.org/10.1007/978-1-4614-1686-9BusenbergS, Castillo-ChavezC (1991) A general solution of the problem of mixing of subpopulations and its application to risk- and age-structured epidemic models for the spread of AIDS. IMA J Math Appl Med Biol 8(1):1–291875096Castillo-ChavezC, FengZ (1998) Global stability of an age-structure model for TB and its applications to optimal vaccination strategies. Math Biosci 151(2):135–1549711046DiekmannO, HeesterbeekJAP, MetzJAJ (1990) On the definition and the computation of the basic reproduction ratio R0 in models for infectious diseases in heterogeneous populations. J Math Biol 28(4):365–382. https://doi.org/10.1007/BF001783242117040DiekmannO, HeesterbeekH, BrittonT (2012) Mathematical tools for understanding infectious disease dynamics. Princeton University Press, PrincetonFengZ, HillAN, SmithPJ, GlasserJW (2015) An elaboration of theory about preventing outbreaks in homogeneous populations to include heterogeneity or preferential mixing. J Theor Biol 386:177–187. https://doi.org/10.1016/j.jtbi.2015.09.00626375548FengZ, HillAN, CurnsAT, GlasserJW (2017) Evaluating targeted interventions via meta-population models with multi-level mixing. Math Biosci 287:93–104. https://doi.org/10.1016/j.mbs.2016.09.01327671169FriedlandS (1980/81) Convex spectral functions. Linear Multilinear Algebra 9(4):299–316. https://doi.org/10.1080/03081088108817381GlasserJ, FengZ, MoylanA, Del ValleS, Castillo-ChavezC (2012) Mixing in age-structured population models of infectious diseases. Math Biosci 235(1):1–7. https://doi.org/10.1016/j.mbs.2011.10.00122037144GlasserJW, FengZ, OmerSB, SmithPJ, RodewaldLE (2016) The effect of heterogeneity in uptake of the measles, mumps, and rubella vaccine on the potential for outbreaks of measles: a modelling study. Lancet Infect Dis 16(5):599–605. https://doi.org/10.1016/S1473-3099(16)00004-926852723HadelerK, MüllerJ (1996a) Vaccination in age structured populations I: the reproduction number. Models Infect Human Dis Struct Relat Data 6:90HadelerK, MüllerJ (1996b) Vaccination in age structured populations II: Optimal strategies. Models for infectious human diseases: their structure and relation to data pp 102–114HillAN, LonginiIMJr (2003) The critical vaccination fraction for heterogeneous epidemic models. Math Biosci 181(1):85–106. https://doi.org/10.1016/S0025-5564(02)00129-312421553JacquezJA, SimonCP, KoopmanJ, SattenspielL, PerryT (1988) Modeling and analyzing HIV transmission: the effect of contact patterns. Math Biosci 92(2):119–199. https://doi.org/10.1016/0025-5564(88)90031-4NoldA (1980) Heterogeneity in disease-transmission modeling. Math Biosci 52(3–4):227–240. https://doi.org/10.1016/0025-5564(80)90069-3NussbaumRD (1986) Convexity and log convexity for the spectral radius. Linear Algebra Appl 73:59–122. https://doi.org/10.1016/0024-3795(86)90233-8SenetaE (1973) Non-negative matrices. An introduction to theory and applications. Halsted Press, New Yorkvan den DriesscheP, WatmoughJ (2002) Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Dedicated to the memory of John Jacquez. Math Biosci 180(1–2):29–48. https://doi.org/10.1016/S0025-5564(02)00108-612387915

Plot of contour curves of Rv(p1,p2) in the case of n = 2 sub-populations and depiction of the optimal point P*(η). Γ is the critical ray. Ωp(2)(η) is the constraint line. η0 is the greatest lower bound of 0 < η < N such that P*(η) ∈ (0, 1)2 only if η0 < η < N. This figure illustrates that P*(η) always lies on Γ for such η

Depiction of the minimized reproduction number Rv{min}(η) as a function of η. The illustration corresponds to the case when Rv{min}(η0) ≥ 1

a is a contour plot of R¯v(q)=Rv((1,1,1)−q) in [0, 1]3. b illustrates the restriction of level sets to Ωq(3)(η¯)∩[0,1]3

The graph of R¯v(q1,q2). R¯v grows linearly on the rays emanating from the origin

Level sets of a R¯v and b ϕ=∇R¯v⋅(N2,−N1). The point Q∗(η¯) in (b) is the relative minimum point of R¯v on Ωq(2)(η¯)∩[0,1]2. Γ¯ is the set of all such minima

The critical ray Γ¯ coincides with the main diagonal when a1 = a2 as in Proposition A.8, for any values of ϵi ∈ (0, 1), Ni > 0, i = 1, 2. This figure illustrates two cases with values a ϵ1 = 0.1, ϵ2 = 0.5, N1 = 400, N2 = 1000 and b ϵ1 = 0.8, ϵ2 = 0.2, N1 = 1200, N2 = 300

Parameters and symbols with their definitions

SymbolDescription
ai Per capita contact rate of members of sub-population i
γ Per capita rate of recovery
θ Per capita rate of entering and exiting a sub-population
β Probability of infection on contact
ρ = β/(γ + θ)
Ni Size of sub-population i
N = N1 + N2 + … + Nn. Total population
cij Proportion of contacts of individuals in group i that are with group j
C = (cij). Mixing matrix
r(A) Spectral radius of the matrix A
ϵ i Fraction of contacts of group i reserved for itself
n Number of sub-populations in the meta-population
pi Proportion of sub-population i that is vaccinated
p = (p1, p2,…, pn)
R0i = ρai. Basic reproduction number of sub-population i
Rvi = R0i(1−pi). Effective reproduction number of sub-population i
R0 Meta-population basic reproduction number
Kv(p) diag(Rv1,…,Rvn)C. Next generation matrix (NGM)
Rv(p) = r(Kv(p)). Meta-population effective reproduction number
η = ∑i=1npiNi. Total number of vaccine doses
Ωp(n)(η) = {(p1, p2, …, pn) : ℓ(p) = η}
P*(η)= (p1∗(η), p2∗(η), … , pn∗(η)). Optimal allocation of vaccine
Rv{min}(η) Minimum of Rv for a given number of vaccine doses η
η * Minimum doses for achieving Rv≤1
η o Infimum of all η ∈ (0, N) such that P*(η) ∈ (0, 1)n
qi = 1 − pi
q = (q1, q2, …, qn)
η¯ = ∑i=1nqiNi=N−η
Ωq(n)(η¯) = {(q1, q2, … ,qn): ℓ(q)=η¯}
Q∗(η¯) = (1,1, … ,1) − p*(η)
R¯v(q) = Rv(1 − q1, 1 − q2, … , 1 − qn)
R¯v{min}(η¯) = Rv{min}(η)
Γ, Γ¯Critical and ‘reflected’ critical rays (see Theorems 3.2 and 4.3)
π i = Ni/N. Fraction of population belong to sub-population i
S = ∑i=1n(1−pi)πi. Fraction of the population that is unvaccinated
R0⋄ = (∑i=1nπi∕R0i)−1. Harmonic mean of R0i weighted by πi
R^0 = ∑i=1nR0iπi≥R0⋄. Population weighted reproduction number
R~0 = miniR0i2∕R^0. Analogous to a scaled reproduction number

i, j = 1, 2, … , n.