key: cord-0899716-4qxz84ih
authors: Rao, Isabelle J.; Brandeau, Margaret L.
title: Optimal allocation of limited vaccine to minimize the effective reproduction number()
date: 2021-06-30
journal: Math Biosci
DOI: 10.1016/j.mbs.2021.108654
sha: 6a59fe19f03bf09b9ed6450c0d3b46d59020f56d
doc_id: 899716
cord_uid: 4qxz84ih

We examine the problem of allocating a limited supply of vaccine for controlling an infectious disease with the goal of minimizing the effective reproduction number [Formula: see text]. We consider an SIR model with two interacting populations and develop an analytical expression that the optimal vaccine allocation must satisfy. With limited vaccine supplies, we find that an all-or-nothing approach is optimal. For certain special cases, we determine the conditions under which the optimal [Formula: see text] is below 1. We present an example of vaccine allocation for COVID-19 and show that it is optimal to vaccinate younger individuals before older individuals to minimize [Formula: see text] if less than 59% of the population can be vaccinated. The analytical conditions we develop provide a simple means of determining the optimal allocation of vaccine between two population groups to minimize [Formula: see text].

A natural objective in minimizing the outbreak of an infectious disease is to minimize the effective reproduction number, R e ; this is the average number of secondary cases per infectious case in a population with both susceptible and infected individuals. If a vaccine is available for the disease, vaccination is one means of controlling epidemic spread. However, vaccine supplies may be limited, particularly for newly identified diseases such as COVID-19 [1] . Here we consider the problem of allocating a limited supply of vaccine for controlling an infectious disease with the goal of minimizing R e .

A number of studies have considered the vaccine allocation problem. Some researchers have proposed a mixed-integer or linear programming formulation to minimize the number or cost of vaccines under the constraint that the reproduction number is below 1 [2, 3, 4] . Other researchers use optimal control to determine the allocation of vaccine which minimizes vaccination cost plus the cost of infection [5, 6] . Some studies have considered vaccination for seasonal influenza, typically using age-structured compartmental models and numerical simulation of alternative policies [7, 8] or numerical optimization to determine the optimal allocation between different groups [9, 10] . Recent studies have focused on optimal vaccination policies for COVID-19 using age-structured compartmental models. One study finds that the optimal vaccine allocation should prioritize age-based fatality rates rather than occupation-based infection rates in order to minimize the cost of infections plus economic losses [11] . Other studies find that vaccinating older groups averts more deaths, whereas vaccinating younger groups averts more infections [12, 13, 14] . Here we consider the optimal allocation of vaccine between two population groups with the goal of minimizing the effective reproduction number.

We develop an SIR model of a population with two interacting groups in which an infectious disease is spreading ( Figure 1 ). Individuals in each group i can be susceptible (S i ), infected (I i ), recovered (R i ), or dead (D i ). Individuals in group i can acquire infection from contact with individuals in their own population group (at rate β ii > 0) or the other population group j (at rate β ij > 0). Infected individuals in group i either recover (at rate γ i > 0) or die (at rate µ i > 0). We consider a relatively short time horizon and thus do not include births, non-infection-related deaths, or other forms of entry into and exit from the population. We assume that a preventive vaccine with effectiveness η > 0 is available and that vaccination of susceptible individuals moves them to a recovered health state. We assume that vaccination takes place at time 0. Vaccination does not affect the transmission rates between infected and susceptible individuals (β ij ) nor the recovery rates of infected individuals (γ i ). We denote by P the population size, v = (v 1 , v 2 ) ∈ R 2 the proportion of individuals vaccinated, S i (0), I i (0), R i (0), D i (0) the proportion of the entire population in each compartment at time 0 without vaccination, and S i (v; t), I i (v; t), R i (v; t), D i (v; t) the proportion of individuals in each compartment at time t in the presence of vaccination v. Since v i is the proportion of the entire population that is vaccinated and belongs to group i, we have the constraints v i ≤ S i (0) for i = 1, 2.

Without loss of generality, we assume S 1 (0) > S 2 (0). We further assume that a limited number of vaccines, N , are available to be distributed at time 0, where N/P < S 1 (0) and v 1 + v 2 ≤ N P . Since vaccination only impacts the initial conditions, we have

case in a fully susceptible population. The model has two infected host compartments, x = I 1 I 2 . Let F i be the rate at which new infected individuals enter compartment i, and let V i be the transfer of individuals into and out of compartment i. We define two matrices F and V , where

∂x j , and x 0 is the disease-free equilibrium. Using this notation, we have dx dt = (F − V )x. For our model, F and V are as follows:

R 0 is given by the largest eigenvalue of the next generation operator F V −1 , where the entry (i, j) represents the expected number of of secondary cases in compartment i caused by an individual in compartment j.

The largest eigenvalue of F V −1 is:

We next derive the effective reproduction number R e . With vaccination v = (v 1 , v 2 ), the starting susceptible population in group i becomes S i − ηv i , and the effective reproduction number is:

If

then we have:

Because (3) provides a closed-form expression for R e , we can find the optimal solution numerically.

We can also solve the problem analytically for a certain range of N . For notational simplicity, we let N = N P , S 1 = S 1 (0), and S 2 = S 2 (0). We want to allocate all available vaccines, so we have

we can write R e (v) as a univariate function:

is a piecewise concave function, and therefore the minimum is at an extreme point:

In both cases

Therefore, the function is piecewise concave: the minimum is global and will be at an extreme point: v * 2 ∈ {0, φ(N ), min(N , S 2 )}. We can further refine the solution when φ(N ) < 0 or φ(N ) > 1. In that case, v * 2 ∈ {0, min(N , S 2 )}, since 0 ≤ v * 2 ≤ 1. When φ(N ) < 0 ≤ v 2 , we can establish two additional results. Proof. When φ(N ) < 0, since v 2 ≥ 0, we have φ(N ) < v 2 for all feasible v 2 , and from Proposition 1, the optimal solution v * 2 can only be 0 or min(N , S 2 ). From (6), we have

We first consider the case where N ≤ S 2 . We calculate

and it is optimal to vaccinate group 1.

We now consider the case S 2 < N ≤ S 1 . We have

Let R * e be the optimal effective reproduction number. We establish conditions under which R * e ≤ 1. ηN ) . After algebraic manipulation, we find that

Solving this quadratic inequality, we have ∆ 1 = β 12 β 21 S 2 (1 − η) + 4 β 11 γ 1 +µ 1 > 0. The two roots are

Let ∆ 2 = β 12 β 21 S 2 + 4 β 11 γ 1 +µ 1 > 0. The two roots of this quadratic equation are

J o u r n a l P r e -p r o o f Similar to Rao and Brandeau [12] , we consider the case of COVID-19 spreading in two interacting populations, and use data reflective of the initial COVID-19 outbreak in New York. Group 1 (84% of the population) comprises individuals younger than age 65, and group 2 (16% of the population) comprises individuals age 65 and older. We assume that S 1 = 80.9%, S 2 = 16.0%, γ 1 = 0.079, γ 2 = 0.064, µ 1 = 0.00012, µ 2 = 0.00460, β 11 = 0.403, β 12 = 0.071, β 21 = 0.154, β 22 = 0.613, η = 0.9 [12] . With these parameter values, the reproduction number with no intervention is 4.31, which is consistent with other studies that aim to estimate R 0 in an initial outbreak while taking into account transmission from unconfirmed cases [17, 18, 19, 20] .

The optimal allocation can be determined numerically from (6) . Plugging in the parameter values, we find that for N/P ≤ 0.59 we have φ(N/P ) ≤ 0, and thus

Evaluating the conditions from Proposition 2, for N/P ≤ S 2 , we find

and for S 2 ≤ N/P ≤ 0.59, we find

Thus, from the analytical conditions we find that all vaccine should be allocated to individuals in group 1 for 0.0003 ≤ N/P ≤ 0.59. From Proposition 3, we find that R * e ≤ 1 if and only if N/P ≥ 0.85. Therefore, for N/P ≤ 0.59, R e cannot be less than 1. We can show numerically that for any amount of vaccine up to N/P ≤ 0.59 (including 0 ≤ N/P ≤ 0.0003), it is optimal to vaccinate group 1 only (Figure 2 ). For N/P > 0.59, allocating a portion of the vaccines to individuals in group 2 is optimal: for example, (v * 1 , v * 2 ) = (0.65, 0.03) for N/P = 0.68 and (v * 1 , v * 2 ) = (0.69, 0.09) for N/P = 0.78. In these cases, we find that v * 2 = φ(N/P ). We compare the values of the effective reproduction number under optimal allocation, R * e , and equal allocation, R eq e = R e ( N P S 1 (0) Table 1) . We find that R * e is up to 23% lower than R eq e . Figure 3 : R e as a function of constant daily vaccination at different levels (λ, colored lines) and different allocations between groups 1 and 2 (αλ, (1 − α)λ).

The analytical conditions we develop provide a simple means of determining the optimal allocation of vaccine between two population groups to minimize R e . Our analysis shows that the optimal vaccination strategy depends on the number of vaccines available: an all-or-nothing approach is only optimal when vaccine supplies are limited. Therefore, before determining the vaccine allocation, policy makers must first estimate the proportion of the population that can be vaccinated, taking into account not only vaccine supply but also other limiting factors such as operational constraints and vaccine hesitancy. For instance, recent polls suggest that approximately 30% of the U.S. population is hesitant about COVID-19 vaccination, suggesting that N/P ≤ .70 for COVID-19 vaccination [21, 22] .

We assumed that N/P ≤ S 1 (0) in order to simplify calculations. It is straightforward to extend the analysis to the case N/P ≤ S 1 (0) + S 2 (0), a less limited vaccine supply. To do so, one must separately consider the case N/P ≥ S 1 (0), and add the constraint v 2 ≥ N −S 1 (0) to the univariate problem (6); otherwise R e (N −v 2 , v 2 ) would not be well defined. The minimum will again be at an extreme point: v * 2 ∈ {N −S 1 (0), φ(N ), S 2 (0)}. Our analysis is based on a relatively simple SIR model. We illustrate the model with an example of COVID-19. Although COVID-19 may be more accurately modeled with an SEIR model, several studies have used an SIR model for COVID-19 and have obtained a good fit to the data [12, 23, 24, 25] . Further work is needed to extend our analytical approach to more complex compartmental models that can capture more details of disease transmission and progression.

Finally, we consider a static policy with a single allocation of vaccine at time 0. In numerical simulations, we consider a constant daily vaccination rate and find that the same solution as for one-time allocation is still optimal. In practice, because vaccination efforts will occur over time, the vaccination policy can evolve. A heuristic dynamic solution would be to recalculate R e at certain points in time and then adjust the vaccine allocation using the optimization criteria we provide. Further work is needed to extend our analysis to a multi-period setting.

uitable Allocation of Vaccine for the Novel Coronavirus, Equitable Allocation of Vaccine for the Novel Coronavirus

Optimal vaccination strategies for a community of households

Finding optimal vaccination strategies under parameter uncertainty using stochastic programming

Optimal influenza vaccine distribution with equity

Vaccination models and optimal control strategies to dengue

Optimal control of vaccination in a vector-borne reaction-diffusion model applied to Zika virus

Optimal allocation of pandemic influenza vaccine depends on age, risk and timing

Optimal pandemic influenza vaccine allocation strategies for the Canadian population

Optimizing vaccine allocation at different points in time during an epidemic

Optimizing influenza vaccine distribution

The optimal allocation of COVID-19 vaccines

Optimal allocation of limited vaccine to control an infectious disease: Simple analytical conditions

Vaccine optimization for COVID-19: Who to vaccinate first?

Allocation of COVID-19 vaccines under limited supply

Perspectives on the basic reproductive ratio

On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations

High contagiousness and rapid spread of severe acute respiratory syndrome coronavirus 2

State-by-state estimates of R0 at the start of COVID-19 outbreaks in the USA

Estimation of the transmission risk of the 2019-nCoV and its implication for public health interventions

Quantifying asymptomatic infection and transmission of COVID-19 in New York City using observed cases, serology, and testing capacity

Vaccine hesitancy for COVID-19: State, county, and local estimates

National population by characteristics

A SIR model assumption for the spread of COVID-19 in different communities

Measuring and preventing COVID-19 using the SIR model and machine learning in smart health care

An adaptive social distancing SIR model for COVID-19 disease spreading and forecasting

Highlights • We examine how to allocate limited vaccine to minimize the effective reproduction number

• We consider an SIR model with two interacting populations

• We develop an analytical expression that the optimal vaccine allocation must satisfy

With limited vaccine supplies, we find that an all-or-nothing approach is optimal

• For some special cases