Mathematical models of vaccine inventory design for a breakout of epidemic disease
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Mathematical models of vaccine inventory design for a breakout of epidemic disease Yifan Liu 1, ∗ 1 Department of Systems Engineering and Operations Research, George Mason University, Fairfax, VA 22030, USA. We aim at building up the mathematical models, which could be used to derive the necessary stockpile levels for the vaccine, in order to meet the urgent need, due to a natural breakout of an epidemic disease, or due to a bio-terror attack. 1 Introduction In this paper, we consider a scenario of the vaccine shortage due to a demand resulting from an urgent incident, while the production is normal. We aim at building up the mathematical models to derive the optimal stockpile levels for vaccine, in order to meet the future urgent need, due to a natural breakout of an epidemic disease, or due to a bio-terror attack. When an epidemic disease breaks out in a certain region, the most feasible way to alleviate the regional vaccine shortage is to immediately ship the vaccine from nationwide, while increasing the production to a necessary level. Although finally the demand will be met, there will be a substantial time delay during which a large number of people may get infected. We need to design the vaccine stockpiles at different highly-populated areas in a country, to get it better prepared for this potential shortage. 2 An Over-simplified Inventory Model for Emergency To illustrate the main idea, we first consider an over-simplified model, and later incorporate more details. Suppose in a single region with population p, there is only one manufacturer continuously producing a certain kind of vaccine at a deterministic rate r p , equal to the demand rate r d doses/hour. The inventory maintains a constant level at I doses, and the inventory holding cost is h dollars per dose per hour. For each dose, it takes d hours in the whole supply chain, which includes the time in production d p , and in transportation d t . If an emergent disease breaks out, all people need to be vaccinated within T hours, that is, the demand suddenly increases to R = p/T >> r d . As a response, the manufacturer increases its production rate to R. Due to the d hours delay in the supply chain, the supply rate will remain at r d for the first d hours, when the inventory level decreases with a rate of R − r d . Therefore, to ensure that no shortage occurs during these d hours, we need I ≥ (R − r d )d. We define this level as the “safety level”, denoted by I s . There is one important factor to be taken into account, which is the shelf-life of the vaccine. No vaccine dose can be stored in the inventory for an infinitely long period. This means, if we set the inventory level too high, then part of the vaccine will be wasted before they can really be used. Suppose the shelf-life of the vaccine is l hours after it is produced, then it can only be stored in the inventory for l − d p hours. If the doses of vaccine follow the first-in-first-out policy in the inventory, to maintain I s as the inventory level at usual time without any waste, while consumption rate is at r d , we need I s ≤ r d (l − d p ) to ensure that each dose will be consumed within the shelf-life. We define this level as the “no-waste” level, denoted by I w . Therefore, we need I w ≥ I s to ensure that no shortage may happen while no waste will occur. If this is not satisfied, then to ensure no shortage, we have to allow a waste rate r w , and the production rate will be r p = r d + r w , and the inventory level is H s = r p (l − d p ). Our goal is to minimize the shortage, given a fixed total cost, including both the waste and the inventory holding cost. If we measure the cost in unit of dose·hour, then after some arithmetics, we reach the following mathematical model, which is a non-linear programming with linear constraint. min S = 1 2 (I s − I ) 2 ( 1 R − r p + 1 R )+(T − d)(I s − I ) subject to the cost constraint hI + c(r p − r d ) ≤ C, the inventory constraint I ≤ r p (l − d t ), and the production rate constraint r d ≤ r p ≤ R. ∗ E-mail: [email protected], Phone: +001 703 993 4620, Fax: +001 703 993 1521 PAMM · Proc. Appl. Math. Mech. 7, 2150013–2150014 (2007) / DOI 10.1002/pamm.200700367 © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim © 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
Transcript of Mathematical models of vaccine inventory design for a breakout of epidemic disease
Mathematical models of vaccine inventory design for a breakout ofepidemic disease
Yifan Liu1,∗
1 Department of Systems Engineering and Operations Research, George Mason University, Fairfax, VA 22030, USA.
We aim at building up the mathematical models, which could be used to derive the necessary stockpile levels for the vaccine,in order to meet the urgent need, due to a natural breakout of an epidemic disease, or due to a bio-terror attack.
1 Introduction
In this paper, we consider a scenario of the vaccine shortage due to a demand resulting from an urgent incident, while theproduction is normal. We aim at building up the mathematical models to derive the optimal stockpile levels for vaccine, inorder to meet the future urgent need, due to a natural breakout of an epidemic disease, or due to a bio-terror attack.
When an epidemic disease breaks out in a certain region, the most feasible way to alleviate the regional vaccine shortageis to immediately ship the vaccine from nationwide, while increasing the production to a necessary level. Although finallythe demand will be met, there will be a substantial time delay during which a large number of people may get infected. Weneed to design the vaccine stockpiles at different highly-populated areas in a country, to get it better prepared for this potentialshortage.
2 An Over-simplified Inventory Model for Emergency
To illustrate the main idea, we first consider an over-simplified model, and later incorporate more details.Suppose in a single region with population p, there is only one manufacturer continuously producing a certain kind of
vaccine at a deterministic rate rp, equal to the demand rate rd doses/hour. The inventory maintains a constant level at I doses,and the inventory holding cost is h dollars per dose per hour. For each dose, it takes d hours in the whole supply chain, whichincludes the time in production dp, and in transportation dt.
If an emergent disease breaks out, all people need to be vaccinated within T hours, that is, the demand suddenly increasesto R = p/T >> rd. As a response, the manufacturer increases its production rate to R. Due to the d hours delay in thesupply chain, the supply rate will remain at rd for the first d hours, when the inventory level decreases with a rate of R − rd.Therefore, to ensure that no shortage occurs during these d hours, we need I ≥ (R − rd)d. We define this level as the “safetylevel”, denoted by Is.
There is one important factor to be taken into account, which is the shelf-life of the vaccine. No vaccine dose can be storedin the inventory for an infinitely long period. This means, if we set the inventory level too high, then part of the vaccine will bewasted before they can really be used. Suppose the shelf-life of the vaccine is l hours after it is produced, then it can only bestored in the inventory for l − dp hours. If the doses of vaccine follow the first-in-first-out policy in the inventory, to maintainIs as the inventory level at usual time without any waste, while consumption rate is at rd, we need Is ≤ rd(l − dp) to ensurethat each dose will be consumed within the shelf-life. We define this level as the “no-waste” level, denoted by Iw.
Therefore, we need Iw ≥ Is to ensure that no shortage may happen while no waste will occur. If this is not satisfied, thento ensure no shortage, we have to allow a waste rate rw, and the production rate will be rp = rd + rw, and the inventory levelis Hs = rp(l − dp).
Our goal is to minimize the shortage, given a fixed total cost, including both the waste and the inventory holding cost. Ifwe measure the cost in unit of dose·hour, then after some arithmetics, we reach the following mathematical model, which is anon-linear programming with linear constraint.
min S =12(Is − I)2(
1R − rp
+1R
) + (T − d)(Is − I)
subject to the cost constraint hI + c(rp − rd) ≤ C, the inventory constraint I ≤ rp(l− dt), and the production rate constraintrd ≤ rp ≤ R.
∗ E-mail: [email protected], Phone: +001 703 993 4620, Fax: +001 703 993 1521
PAMM · Proc. Appl. Math. Mech. 7, 2150013–2150014 (2007) / DOI 10.1002/pamm.200700367
© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
3 A More Realistic Model
Having some basic ideas from the over-simplified model, we incorporate some more details to build a more complicated butrealistic model.
In a more realistic model, we should assume there are n different regions nationwide. Suppose each region j has populationpj . During normal situation when there is no break-out of the disease, we can assume the supply and demand are in equilib-rium, which means the overall production rate and the consumption rate are equal, so that there is neither shortage nor waste.We denote this demand rate by rd
j . When there is an epidemic disease breakout, the demand rate for region j is assumed to beRj .
Different from the over-simplified model, we assume m suppliers, each having maximum capacity rmaxi . We further assume
that in daily life, one supplier may cover multi-regions, but each region has only one supplier. We use the notation j ∈ S(i) tomean that region j is covered by supplier i. As in the over-simplified model, there is time delay in supply for region j, whichis denoted by dj , during which dt
j is in transportation and dpj is in production. Also, we assume the cost for producing and
shipping one dose for region j is cj .Something more complicated in this realistic model is – when an epidemic disease breaks out, it is possible to ship the
vaccine from the inventory of other regions to the region suffering from the disease, to help alleviate the shortage. We assumethe time delay during the transportation from region j to region k is djk, and the cost for shipping each dose is cjk.
3.1 Decision Variables and Constraints
To formulate and solve the problem with operations research methodology, we need first to choose our decision variables.In our problem, we have the following decision variables: Inventory level Ij , j = 1, ..., n; Production rate rp
j , i = 1, ...,m;Whether to ship from region j to k when the disease breakout: Binary variable bjk, 0 or 1.
Besides the trivial sign constraints for all variables and the binary constraints, We have the following essential constraints:1. Capacity Constraints –
∑j∈S(i) rp
j ≤ rmaxi
2. Shelf-life Constraints – rpj (l − dt
j) ≥ Ij
3. Cost Constraints – At usual time when there is no disease, the inventory holding cost∑
j hIj and dose waste cost∑j cj(r
pj − rd
j ) should be within a level.
3.2 Building-up the Objective Function
In the over-simplified model, we measure the objective function in the unit of dose·hour, which implies the loss is proportionalto the number of people who are not immunized timely, and the time delay in this immunization. However, this linearityassumption is not realistic, and we need to figure out a more detailed relation between the shortage and the cost.
Step 1. We need to compute the immunization rate vj for region j, which is related to the value of all decision variables, andall known parameters. This relation should be complicatedly, but not difficultly, set up as a function, denoted by vj(Ij , r
pj , bkj).
If vj(Ij , rpj , bkj) < Rj , shortage happens.
Step 2. We need to find how the number of infectious people will increase with time, in case of both vaccine shortage andno shortage. Our main idea is to use Anderson’s epidemic model [1] with a few variations. Anderson’s model categorizesall people in an area into 4 types, susceptible, infecteds who are not yet infectious (incubation), infectious individuals, andthose who are recovered and immune, denoted with variables X, H, Y and Z respectively. The model assumes they follow anODE system. A few later papers [2, 3] modified it with some parameters (σ, β, γ) to include inhomogeneity. In our paper, wefurther break X into two categories Xv and Xn, representing those vaccinated and not vaccinated. Then the ODE system canbe written as
⎧⎪⎪⎪⎪⎨⎪⎪⎪⎪⎩
dXn
dt = −(βnXn)εY δ − VdXv
dt = −(βvXv)εY δ + VdHdt = ((βnXn)ε + (βvXv)ε)Y δ − σHdYdt = σH − γYdZdt = γY
Solving the ODE system, we can get a relation between Yj(t) and vj(t).Step 3. Assume the Loss be proportional to the number of infectious people Y (t) times the period when they are infectious,
i.e.,∫ T
0Y (t)dt. Then
∫ T
0Y (t)dt is our objective function. Therefore, we complete the whole model.
References[1] R. M. Anderson, Population Dynamics of Infectious Diseases (Chapman & Hall, New York, 1982).[2] C. Lefevre, Operations Research, 29, (1981).[3] N. C. Severo, Math. Biosci., 4, 393-402 (1969).
© 2007 WILEY-VCH Verlag GmbH & Co. KGaA, Weinheim
ICIAM07 Contributed Papers 2150014
