A Multi-Stage Stochastic Programming Approach to Epidemic Resource Allocation with Equity Considerations
AA Multi-Stage Stochastic Programming Approach toEpidemic Resource Allocation with EquityConsiderations † Xuecheng Yin and ˙I. Esra B¨uy¨uktahtakın ‡ Department of Mechanical and Industrial Engineering, New JerseyInstitute of TechnologyFebruary 24, 2021
Abstract
Existing compartmental models in epidemiology are limited in terms of optimiz-ing the resource allocation to control an epidemic outbreak under disease growth un-certainty. In this study, we address this core limitation by presenting a multi-stagestochastic programming compartmental model, which integrates the uncertain diseaseprogression and resource allocation to control an infectious disease outbreak. The pro-posed multi-stage stochastic program involves various disease growth scenarios andoptimizes the distribution of treatment centers and resources while minimizing thetotal expected number of new infections and funerals. We define two new equity met-rics, namely infection and capacity equity, and explicitly consider equity for allocatingtreatment funds and facilities over multiple time stages. We also study the multi-stage value of the stochastic solution (VSS), which demonstrates the superiority of the † Cite as: Yin, X. and B¨uy¨uktahtakın ˙I. E., 2021. A multi-stage stochastic programming approach toepidemic resource allocation with equity considerations. Health Care Management Science, pp. 1–57. ‡ Corresponding author email: [email protected] a r X i v : . [ s t a t . A P ] F e b roposed stochastic programming model over its deterministic counterpart. We applythe proposed formulation to control the Ebola Virus Disease (EVD) in Guinea, SierraLeone, and Liberia of West Africa to determine the optimal and fair resource-allocationstrategies. Our model balances the proportion of infections over all regions, even with-out including the infection equity or prevalence equity constraints. Model results alsoshow that allocating treatment resources proportional to population is sub-optimal,and enforcing such a resource allocation policy might adversely impact the total num-ber of infections and deaths, and thus resulting in a high cost that we have to pay forthe fairness. Our multi-stage stochastic epidemic-logistics model is practical and canbe adapted to control other infectious diseases in meta-populations and dynamicallyevolving situations. Keywords . Epidemic diseases, fair resource allocation, compartmental models, un-certainty in disease growth, multi-stage stochastic mixed-integer programming model,optimization, epidemic logistics and supply chains, equity constraints, Ebola VirusDisease (EVD), epidemiological data, West Africa, COVID-19.
An epidemic is the rapid spread of an infectious disease that impacts a large number ofpeople. Epidemic diseases can disperse widely in a short time period, usually, two weeks orless, such as influenza, meningitis, and cholera, impacting populations either in a specificarea, or become a pandemic affecting the lives of millions at a global scale, such as theongoing the coronavirus pandemic. All over the world, outbreaks continue to take lives, ruinthe economy, and weaken the health-care system. Unfortunately, the toll is higher in theless-developed countries because millions of people in poor regions of the world do not havethe opportunity to receive sufficient treatment in case of an outbreak.Ebola virus disease (EVD) is a prime example of a devastating epidemic. The EVD,also known as Ebola hemorrhagic fever, is a severe, often fatal illness affecting humans andother primates (WHO, 2019a). The 2014-2016 outbreak in West Africa was the biggestEbola outbreak in history, causing more than 28,600 cases and 11,325 deaths by the end ofJune 2016 (CDC, 2019). The virus started in Guinea, then moved across countries to SierraLeone and Liberia. The tenth outbreak of the Ebola virus disease has been ongoing in the2emocratic Republic of the Congo (DRC) since August 2018. The outbreak has startedfrom the northeast region of the country, centered in the North Kivu and Ituri provinces.More than 3000 cases have been verified by March 2020 (MSF, 2020), and it is the country’slargest-ever Ebola outbreak.There are no specific cure, vaccine, or treatment for Ebola-infected individuals. Althoughmultiple investigational Ebola vaccines have been developed and tested in numerous clinicaltrials around the world, none of them have yet been licensed to prevent the Ebola virusdisease (NIH, 2019). Short term intervention strategies, including quarantine, isolation, con-tact tracing, and safe burial, have been helpful to Ebola control. Moreover, Ebola treatmentcenters (ETCs), which mainly isolate and treat infected individuals, play a significant rolein controlling the Ebola virus disease.The optimization problem of allocating resources to control an epidemic, such as Ebola,is an immense challenge, especially in the regions where available treatment facilities andfunds are scarce. The decision-maker has to make difficult decisions to allocate limitedresources to the right locations and in the right amount for slowing down the outbreak andminimize its impacts. Due to the insufficiency of intervention resources, some regions maynot receive their fair share of treatment resources, compared to other regions that are alsoimpacted by the disease. Furthermore, the EVD can spread from one individual to anotherthrough multiple mechanisms, such as through person-to person-contact or by touching thedead body infected by the EVD. The rates of disease transmission can change under variousconditions and thus could be highly unpredictable.Operations Research (OR) and mathematical modeling methods have been widely usedto determine optimal resource allocation strategies to control an epidemic disease. Thoseapproaches include simulations (Ajelli et al., 2016; Kurahashi and Terano, 2015; Siettoset al., 2015; Wells et al., 2015), differential equations (Craft et al., 2005; Kaplan et al., 2003),network models (Berman and Gavious, 2007; Longini Jr et al., 2007; Porco et al., 2004; Rileyand Ferguson, 2006), resource allocation analysis (Nguyen et al., 2017; Shaw and Schwartz,2010; Tebbens and Thompson, 2009; Zaric et al., 2000), and stochastic compartmental models(Funk et al., 2017; Kibis et al., 2020; Lekone and Finkenst¨adt, 2006; Tanner et al., 2008).The majority of previous work focuses on analyzing the impact of different interven-tion strategies on disease transmission. Most of those studies consider disease growth andresource allocation problems separately in different models or enumerate each resource allo-cation policy in a simulation model one by one. Moreover, few studies incorporate fairness inresource allocation optimization models. The former epidemic-logistics model presented in3¨uy¨uktahtakın et al. (2018) has incorporated the logistics of treatment into a disease spreadmodel, which foresees the disease growth over a spatial scale, and at the same time allocateslimited resources to control the spread of the disease. B¨uy¨uktahtakın et al. (2018) consid-ered the varying treatment capacity based on a limited budget. The mathematical modelof B¨uy¨uktahtakın et al. (2018) was deterministic and assumed expected values for diseasetransmission rates. However, in reality, the disease transmission rate could be quite uncer-tain, changing over time and space under various scenarios. Thus, a stochastic OR modelis necessary to represent the uncertainty in transmission in a more realistic way. Moreover,the majority of the OR models do not consider equity and fairness in resource allocation,resulting in solutions that may provide few or no resources to some regions impacted by thedisease, especially when resources are quite limited.The objective of this paper is to develop a multi-stage stochastic programming exten-sion of the deterministic epidemic-logistics model of B¨uy¨uktahtakın et al. (2018) with equityconsiderations and present realistic insights into controlling the EVD under disease trans-mission uncertainty. Considering different budget levels and various tightness of the equityconstraints, we analyze the optimal resource allocation strategies in a meta-population overthree countries in West Africa. In our paper, the stochastic program incorporates variousscenarios of disease transmission rates through person-to-person contact, thus capturing theuncertainty and variability in the infection transmission rate better compared to its de-terministic counterpart. The objective function of our multi-stage stochastic programmingepidemic-logistics model is to minimize the expected number of new infections and deceasedindividuals overall scenarios, all time periods, and all regions considered. We study theValue of the Stochastic Solution (VSS), a well-known stochastic programming measure thatcompares the efficiency of the deterministic and the stochastic models. Furthermore, we in-troduce two new equity metrics for fair resource allocation in epidemics control and analyzethe impact of various budget distribution strategies on the number of infected people anddeceased individuals under each of these equity metrics.
The majority of mathematical models in the epidemiological literature have used simulationmethods to forecast the progression of the epidemic and to study the efficacy of severalinterventions (Dasaklis et al., 2017; Meltzer et al., 2014; Onal et al., 2019; Pandey et al.,2014; Rivers et al., 2014; Siettos et al., 2015). Several studies have considered stochastic4ompartmental models to analyze different strategies for controlling epidemic diseases, suchas vaccination strategies, behavioral changes that impact the interaction between differentgroups, and regional intervention strategies (Funk et al., 2017; Lekone and Finkenst¨adt,2006). Some other studies used simulation and network models to explore Ebola vaccinationstrategies (Ball et al., 1997; Nguyen et al., 2017; Shaw and Schwartz, 2010).Previous operations research models that study the epidemic diseases and resource al-location mainly focused on the logistics and operation management to control the diseasein optimal ways (B¨uy¨uktahtakın et al., 2018; Ekici et al., 2013; Liu et al., 2019; Zaric andBrandeau, 2001). Regarding the capacity of hospitals and logistics issues, B¨uy¨uktahtakınet al. (2018) developed a new epidemic-logistics mixed-integer programming model of theepidemic control problem. Their model considered the dynamic spread of an epidemic overmultiple regions and the allocation of Ebola treatment centers and resources to control thedisease simultaneously. Different than the classical epidemiological models, the transmissionrate between the infected and treated compartment was not constant but instead dependedon the treatment capacity and the number of infected people receiving treatment. Later, Liuet al. (2019) adapted the epidemics-logistics model of B¨uy¨uktahtakın et al. (2018) to studythe control of the 2009 H1N1 outbreak in China and presented similar results for the H1N1epidemic.In the sensitivity analysis of B¨uy¨uktahtakın et al. (2018), the disease transmission ratewithin the community was found to be the most critical parameter impacting infected andfunerals. While the disease transmission rates are highly uncertain, relatively fewer studiesin the OR community take into account the uncertain parameters for resource allocation inan effort to control the disease. Those OR models that integrate resource allocation withepidemics control use either stochastic dynamic programming (SDP), partially observableMarkov decision process (POMDP) framework (Suen et al., 2018) or two-stage stochasticprogramming (Co¸sgun and B¨uy¨uktahtakın, 2018; Long et al., 2018; Ren et al., 2013; Tanneret al., 2008; Yarmand et al., 2014).Most resource allocation models on epidemic control computed the optimal solution with-out considering fairness in resource allocation. Fair resource allocation has been studied inthe literature, but mainly with different applications. For example, Orgut et al. (2016) con-sidered a food allocation model with equitable and effective distribution of donated foodunder capacity constraints. Davis et al. (2015) developed a multi-period linear optimizationmodel for improving geographical equity in kidney allocation while also respecting trans-plant system constraints and priorities. Moreover, Lane et al. (2017) gave a systematic5eview of equity in health-care resource allocation decision-making. Marsh and Schilling(1994) presented a literature review of various mathematical methods for equity measuresin facility-location decision models. To our knowledge, fairness has not been studied beforewithin the context of resource allocation for epidemic control over large spatial scales.
Former stochastic programming approaches on epidemic control involved a time domainwith only two periods. Furthermore, there is a need for analyzing the equity and efficiencytradeoff in a mathematical programming formulation when allotting resources for controllinginfectious diseases. Our approach contributes to the epidemiology and OR literature in thefollowing ways.
Modeling Contributions.
Firstly, to the best of our knowledge, our study presents the firstmulti-stage stochastic programming (SP) model for infectious disease control, consideringthe uncertainty in the disease transmission parameter. Multi-stage SP is superior over two-stage SP models because disease transmission dynamically changes over multiple time stages.Our stochastic programming approach is also preferable to probabilistic sensitivity analysis,which considers a single scenario at a time and also to robust optimization (RO), whichcould provide highly conservative results by focusing on the worst-set of outcomes in ahostile environment (Defourny et al., 2012). Due to the temporal and spatial dimensionsconsidered in our resource allocation model, multi-stage SP is also computationally moreamenable to dynamic programming, which cannot tackle such a high-dimensional problem.Second, we present the multi-stage VSS, which shows that the proposed stochastic pro-gramming model is superior to its deterministic counterpart.Third, we introduce and formulate two new equity metrics and incorporate equity mea-sures as a constraint in the mathematical formulation to balance efficiency and equity for fairresource allocation in epidemics control. To our knowledge, this study is the first one thatmodels equity in a multi-stage stochastic programming formulation. Our multi-stage modelprovides an advantage of adjusting the level of equity over time with respect to evolvingdisease dynamics, as opposed to using a standard equity measure, which is not updated overtime. Furthermore, unlike former work, we address equity in both establishing treatmentcenters and allocating treatment resources over metapopulations and multiple periods using6athematical optimization.The infection equity constraint is also easier to implement than using standard equitymetrics, such as the absolute difference between regional prevalence (cases per populationin a region) and the overall prevalence (cases per population over all regions), because theabsolute gap value using the prevalence metric could be tiny and difficult to adjust comparedto the absolute gap value defined by the infection equity constraint. Furthermore, compu-tational results imply that our model balances the proportion of infections in each region,even without including the infection equity or prevalence equity constraint.Fourth, while we tailor our epidemics-logistics stochastic programming modeling frame-work for the EVD, it can be adapted to study different diseases to determine the optimaland fair resource-allocation strategies among various regions and multiple planning periodsto curb the spread of an epidemic.
Applied Contributions and Policy Insights:
Our mathematical model could be used asa decision support tool to aid policymakers in understanding disease dynamics and makingthe most effective decisions to fight epidemics under uncertainty. In particular, our modelcould be used by the stakeholders in epidemic control (e.g., governments, UN entities, non-profit organizations) to determine the optimal location and timing of ETCs opened andtreatment resources allocated to minimize the total expected infections and deaths overmetapopulations in multiple locations and over multiple time periods.Our model provides significant insights into the control of the Ebola Virus Disease in WestAfrica that would not be possible with existing models and methods in infectious diseasecontrol. Our multi-stage stochastic program foresees various disease growth scenarios tooptimize resource allocation, as opposed to solving the problem for an average scenario andmyopically for one stage at a time with fixed periodic budgets, which could provide sub-optimal solutions and thus less effective resource allocation. Specifically, our study providesthe following several policy insights and recommendations to decision-makers:( i ) Our analysis emphasizes that quick response, such as allocating treatment centers andresources in the early stages of the epidemic, is critical for minimizing the total numberof infected individuals and deaths related to the disease.( ii ) The value of the stochastic solution demonstrates that the optimal timing and locationof resource allocation vary with respect to the disease transmission scenario, and thuspossible disease growth scenarios should be considered when planning for an epidemicinstead of considering a single scenario of the expected value.7 iii ) Our results show that the infection level (“the number of infected people in a region” /“the total number of infected people” - “population in a region” / “total population”)is a key factor for resource allocation.( iv ) Our analysis suggests that the region with the highest infection level has the priority toreceive the majority of the resources at the beginning of the time horizon to minimizethe number of infections and funerals.( v ) Model results also show that allocating treatment resources proportional to populationis sub-optimal.( vi ) While equitable resource allocation is important in decision making, too much focus onthe equity of resource allocation might adversely impact the total number of infectionsand deaths and thus resulting in a high cost that we have to pay for fairness. Therefore,decision makers are advised to be cautious about enforcing fairness when allocatingresources to multiple regions. This section gives the formulation of a multi-stage stochastic programming model, includingthe compartmental model, description of the scenario tree, formulation, equity constraints,and their explanation. Model notations that will be used throughout the rest of this paperare presented in Appendix A. 8 .1 Compartmental Disease Model Description
Figure 1: One-Step Disease Compartmental ModelFigure 1 shows the transmission dynamics of the Ebola Virus Disease (EVD) in a region r of a country located in West Africa. The disease spreads among the susceptible population(S), by either person-to-person contact at a periodic rate of χ ω ,r under scenario ω or throughtouching Ebola-related dead bodies that are not yet buried during traditional funerals at aperiodic rate of χ ,r . Thus, susceptible individuals (S) are infected and become infected (I)with a rate of χ ω ,r as a function of I and with a rate of χ ,r as a function of funerals (F), whorepresent dead but unburied people. Without treatment, some of the infected individualsin the compartment (I) will die and move to the funeral (F) compartment with the rate of λ ,r , while some of the infected individuals will recover with a rate of λ ,r , moving into therecovered compartment (R). However, the number of individuals that will be hospitalized fortreatment (T) is based on the treatment capacity variable C ωj,r , which gives the total availablenumber of beds in the ETCs in region r under scenario ω in period j . Thus, there is noconstant transition rate from I to T . Meanwhile, individuals who did not receive treatmentwill remain in the community and continue to spread the disease. In treated compartment(T), some of the individuals will recover with a periodic rate of λ ,r , and a fraction of themwill die with a periodic rate of λ ,r . The deceased individuals in the funeral compartmentare safely buried at a rate of λ ,r , moving into the buried compartment (B). Thus, weassume that every death (F) leads to a safe burial (B). In order to describe the migration ofsusceptible and infected individuals within a given country, we define ( α l → r , υ r → l ) as the ratesof migration of susceptible individuals into and out of region r , respectively, and ( ψ l → r , ρ r → l )as the rates of migration of infected individuals into and out of region r , respectively. The9ulti-stage stochastic programming epidemic-logistics model is defined in detail in the nextsection.The latent period for the EVD is highly variable, changing from 2 to 21 days (WHO,2020d). In our model, we assume that each time stage represents two weeks, in whichan infected but asymptomatic individual can become symptomatic and infect others. Forthis reason, and to avoid computational complexity, we do not include an explicit latentcompartment in the model; instead, we fit those individuals within the infected compartment.Similarly, the Ebola modeling literature focusing on logistics usually omit the latent periodto avoid further computational complexity (see, e.g., B¨uy¨uktahtakın et al. (2018), Long et al.(2018)). It is beyond the scope of this work to introduce a new methodology for multi-period scenariotree generation; we refer the reader to Heitsch and R¨omisch (2009), Le¨ovey and R¨omisch(2015), and Pflug and Pichler (2015) for different approaches to generate scenario trees.To generate the scenario tree for our case, we follow a similar approach presented in thestudy of Alonso-Ayuso et al. (2018). Here, we focus on the most uncertain parameter: thecommunity transmission rate based on former research stating that transmission rates impactthe infections and deaths the most among all different input parameters based on sensitivityanalysis (B¨uy¨uktahtakın et al., 2018).We model the future uncertainties regarding the progression of the disease by a discreteset of scenarios, denoted ω ∈ Ω. Each scenario has a probability, p ω , where (cid:80) ω ∈ Ω p ω = 1.We assume that the uncertain community transmission rate follows a normal distribution.The data regarding the distribution of the community transmission rate parameter is notavailable. Thus, we use the lower and upper bounds on the transmission rate in the commu-nity based on the data gathered from literature (Table 1) to generate the normal distributionfunction for the transmission rate parameter at time zero. The upper and lower bounds, thusthe distribution functions for the uncertain parameter, are specified for each country and aredifferent at each node of the scenario tree. Accordingly, the mean µ nr is defined for each re-gion r ∈ R and node n ∈ N . The lower bound and upper bound is considered as the value of0.001- and 0.999-quantiles of the normal distribution, respectively. The standard deviation10 r is defined according to a normal distribution using the initial lower and upper boundsprovided for each region r ∈ R . Also, we use Q h to represent the value of the h -quantile ofthe normal distribution.Table 1: The range (lower and upper bounds), mean, and standard deviation of communitytransmission rate in each country. Data is gathered from Althaus (2014) and Towers et al.(2014). Region Rate Range Mean Standard DeviationGuinea [0.24, 0.84] 0.54 0.10Sierra Leone [0.24, 0.88] 0.66 0.07Liberia [0.24, 0.64] 0.44 0.0711igure 2: Scenario tree generation example for Guinea, where each circle, denoted by n , n := { , . . . , } , represents a node of the scenario tree.As shown in an example scenario tree in Figure 2, a particular scenario could give thecommunity transmission rates ( χ ω ,r ) into the future for the next two stages in all consideredregions. In our model, we consider three realizations for each node of the scenario tree,namely as low, medium, and high. The low and high realizations have a probability of 0 . .
4. Each path from the root node to the leaf12ode of the scenario tree represents a scenario ω . In the example shown in Figure 2, we havetwo stages, and thus 3 = 9 scenarios. In addition, two scenarios are inseparable at stage j if they share the same scenario path up to that stage. This implication is modeled usingnon-anticipativity constraints, as described in Appendix B. For example, for scenarios ω to ω , the decision at node 0 should be the same as we do not know the values of the uncertainparameters at stage 0. Similarly, for scenarios ω to ω , the decision at node 1 should be thesame because these scenarios cannot be differentiated at stage 1 due to uncertainty.The probability of a scenario ω , p ω , is calculated as the multiplication of probabilities onthe scenario path. For example, the probability of scenario ω , which corresponds to a lowrealization in the first and second stages, is 0 .
09, while the probability of scenario ω , whichcorresponds to a medium realization in the first stage and a high realization in the secondstage, is 0 . n ∈ N in the scenario tree, the low realization value of the random variable ξ nr is given by the value of the 0.15-quantile ( µ nr,low = E ( ξ nr | Q . ≤ ξ nr ≤ Q . ) = Q . ),the medium realization is given by the value of the 0.50-quantile ( µ nr,medium = E ( ξ nr | Q . ≤ ξ nr ≤ Q . ) = Q . ), and the high realization is equal to the value of the 0.85-quantile of thenormal distribution ( µ nr,high = E ( ξ nr | Q . ≤ ξ nr ≤ Q . ) = Q . ). In our example, at node 0the normal distribution of the uncertain community transmission rate parameter in Guineahas µ r = 0 .
54 and σ r = 0 .
10. The low, medium, and high realizations of the uncertainparameter at nodes 0 and 1 are given in Table 2 below.Table 2: The 0.15-, 0.50-, 0.85-quantiles of the normal distribution of the random variable ξ nr at nodes 0 and 1 of the scenario tree in Figure 2.Low Medium Highnode 0: Q . =0.44 Q . =0.54 Q . =0.64node 1: Q . =0.26 Q . =0.44 Q . =0.62The normal distribution of community transmission rate associated with nodes 1, 2, and3 at stage 1 have a mean of Q . = 0 . Q . = 0 .
54, and Q . = 0 .
64, respectively. Whilescenarios ω , ω , and ω at stage 1 has a single realization value of 0 .
54 for the randomparameter at node 1, the realizations of scenarios ω , ω , and ω at stage 2 correspond tonodes 4, 5, and 6, with a mean of Q . = 0 . Q . = 0 .
44, and Q . = 0 .
62, respectively.13 .3 Model Features and Assumptions
In this study, we have considered six regions, each consisting of multiple districts, in the threecountries most affected by the 2014-16 EVD. West Africa is poor and the budget for theEbola treatment comes from an international consortium of partners, including governments,international financial Institutions, regional organizations, and private foundations (UnitedNations, 2020). Those funding is either directly provided to the affected governments or theUnited Nations (UN) entities. In this paper, we took the perspective of the UN entities, suchas the World Health Organization (WHO), where the total funding is collected centrally andallocated among those three countries to optimize the use of treatment resources and thedonated funding.The actual capacity of ETCs varies from 20 to 200 operational beds (WHO, 2020c);however, we used 50 and 100-bed ETCs in our model to reduce the computational complexity.It is essential to differentiate the small and large ETCs in the model because each ETC typehas a different setup cost, which impacts the optimal allocation of resources. We assumethat each Ebola patient will receive the same treatment in either a large or small capacityETC. The treatment capacity parameter is cumulative and only reflects total ETC beds.Furthermore, the cost of burying dead bodies safely is shown to be minor compared to theETC and treatment cost (B¨uy¨uktahtakın et al., 2018; WHO, 2020b). In addition, changingthe burial rate into a variable that is optimized in the model would have complicated themodel considerably, and so we only focus on adjusting the variable values of treatmentresources. Thus, we assume that the burial rate is constant, and burials and treatment areoperated separately using different budgets.
Following the convention of B¨uy¨uktahtakın et al. (2018), the multi-stage stochastic program-ming epidemic-logistics model (1) can be formulated as follows:min (cid:88) j ∈ J − (cid:88) r ∈ R (cid:88) ω ∈ Ω p ω (( I ωj +1 ,r − I ωj,r ) + F ωj +1 ,r ) (1a)s.t. S ω ,r = π r , I ω ,r = (cid:36) r , T ω ,r = θ r , R ω ,r = ϑ r ,F ω ,r = υ r , B ω ,r = τ r , C ω ,r = ζ r , r ∈ R, ∀ ω ∈ Ω (1b) S ω ( j +1) ,r = S ωj,r + ˆ S ωj,r − (cid:101) S ωj,r − χ ω ,r I ωj,r − χ ,r F ωj,r ,j ∈ J \ { J } , r ∈ R, ∀ ω ∈ Ω , (1c)14 ω ( j +1) ,r = I ωj,r + ˆ I ωj,r − (cid:101) I ωj,r + χ ω ,r I ωj,r + χ ,r F ωj,r − ( λ ,r + λ ,r ) I ωj,r − I ωj,r ,j ∈ J \ { J } , r ∈ R, ∀ ω ∈ Ω , (1d) T ω ( j +1) ,r = T ωj,r + I ωj,r − ( λ ,r + λ ,r ) T ωj,r ,j ∈ J \ { J } , r ∈ R, ∀ ω ∈ Ω , (1e) R ω ( j +1) ,r = R ωj,r + λ ,r T ωj,r + λ ,r I ωj,r ,j ∈ J \ { J } , r ∈ R, ∀ ω ∈ Ω , (1f) F ω ( j +1) ,r = F ωj,r + λ ,r I ωj,r + λ ,r T ωj,r − λ ,r F ωj,r ,j ∈ J \ { J } , r ∈ R, ∀ ω ∈ Ω , (1g) B ω ( j +1) ,r = B ωj,r + λ ,r F ωj,r , j ∈ J \ { J } , r ∈ R, ∀ ω ∈ Ω , (1h)ˆ S ωj,r = (cid:88) l ∈ M r α l → r S ωj,l , j ∈ J, r ∈ R, ∀ ω ∈ Ω , (1i)ˆ I ωj,r = (cid:88) l ∈ M r φ l → r I ωj,l , j ∈ J, r ∈ R, ∀ ω ∈ Ω , (1j) (cid:101) S ωj,r = (cid:88) l ∈ M r ν r → l S ωj,r , j ∈ J, r ∈ R, ∀ ω ∈ Ω , (1k) (cid:101) I ωj,r = (cid:88) l ∈ M r ρ r → l I ωj,r , j ∈ J, r ∈ R, ∀ ω ∈ Ω , (1l) (cid:88) r ∈ R ( (cid:88) j ∈ J \{ ,j } (cid:88) a ∈ A g aj,r y ωaj,r + (cid:88) j ∈ J b j,r T ωj,r ) ≤ ∆ ∀ ω ∈ Ω , (1m) C ωj,r = j (cid:88) m =1 (cid:88) a ∈ A k a y ωamj,r + C ,r , j ∈ J \ J , r ∈ R, ∀ ω ∈ Ω , (1n) I ωj,r = min { I ωj,r , C ωj,r − T ωj,r } , j ∈ J \ J , r ∈ R, ∀ ω ∈ Ω , (1o) S ωj,r I ωj,r T ωj,r R ωj,r F ωj,r B ωj,r I ωj,r ≥ , j ∈ J, r ∈ R, ∀ ω ∈ Ω , (1p) y ωaj,r ∈ { , , , . . . } ; y ωaj,r ≤ I ωj,r , a ∈ A, j ∈ J \ { J } , r ∈ R, ∀ ω ∈ Ω , (1q) y ωat ( n ) ,r − y an,r = 0 , I ωt ( n ) ,r − I n,r = 0 , C ωt ( n ) ,r − C n,r = 0 ,a ∈ A, ∀ ω ∈ β ( n ) , ∀ n ∈ N, (1r)The objective function (1a) minimizes the total expected number of newly infected in-dividuals plus funerals over all scenarios, in all regions throughout the planning horizon.Constraints (1b) represent the number of individuals in susceptible, infected, treated, recov-ered, funeral, and buried compartments and the total ETC capacity, respectively, in eachregion r at the beginning of the planning horizon. Equations (1c)–(1h) represent the dy-namics of the population in each disease compartment, as shown in Figure 1. Specifically,15onstraint (1c) implies that the number of susceptible individuals in region r at the end ofperiod j + 1 under scenario ω is equal to the number of susceptible individuals from theprevious year plus the number of susceptible individuals who immigrate into region r minusthe number of susceptible individuals who emigrate from region r and minus the number ofnewly infected individuals at the end of period j under scenario ω . Constraint (1d) givesthe number of infected individuals at the end of period j + 1 in region r under scenario ω ,which is equal to the number of infected individuals from the previous year plus immigratedinfected individuals minus emigrated infected individuals, plus newly infected individualsand minus individuals who recovered, died, or were accepted for treatment at the end ofperiod j under scenario ω . Constraint (1e) describes the total number of treated individualsin region r at the end of time period j + 1 under scenario ω , which is equal to the number oftreated individuals at the end of period j plus infected individuals who accepted treatmentbased on the availability of beds minus treated individuals who died or recovered. Constraint(1f) ensures that the cumulative number of recovered individuals in region r at the end ofthe period j + 1 under scenario ω is equal to the number of recovered individuals from theprevious year plus newly recovered individuals. Constraint (1g) defines the total numberof unburied funerals in region r at the end of time period j + 1 under scenario ω , which isequal to the infected and treated individuals who moved to the funeral compartment minusthe buried dead bodies. Constraint (1h) gives the cumulative number of buried dead bodiesat the end of the period j under scenario ω . Constraints (1i)–(1l) present the number ofimmigrated and emigrated individuals in susceptible and infected compartments. Specifi-cally, constraints (1i) and (1j) give the number of susceptible and infected individuals whoimmigrated into region r from region l ∈ M r under scenario ω . Constraints (1k) and (1l) rep-resent the number of susceptible and infected individuals, who emigrated from region r intoneighboring region l ∈ M r under scenario ω . Constraints (1m)–(1o) represent the restric-tions regarding logistics and operation management. Specifically, constraint (1m) denotesthe budget limitation on the sum of the fixed costs of opening ETCs and the variable cost oftreating infected individuals over all regions r in all periods j under scenario ω . Constraint(1n) shows the total capacity in region r at the end of period j under scenario ω . Constraint(1o) ensures that the number of hospitalized individuals is limited by the number of availablebeds in ETCs in region r . In particular, the number of hospitalized individuals ( I ) is equalto the minimum of the number of infected individuals and the capacity available at estab-lished ETCs after considering currently hospitalized individuals in ETCs. Constraints (1p)present non-negativity restrictions on the number of susceptible, infected, treated, funeral,16uried, and recovered individuals, respectively, under scenario ω . Constraints (1q) denotethe integer requirements on the number of type- n ETCs to be opened in region r at the endof period j under scenario ω . In addition, if the number of infected individuals is less than1 in a region r , the value of the integer variable corresponding to opening an n -bed ETC isforced to be zero, and thus no ETC will be opened in that region. Constraints (1r) representnonanticipativity restrictions, which state that if two scenarios share the same path up tostage j , the corresponding decisions should be the same, as described in Appendix B. Equitable resource allocation has long been studied in health-care resource allocation decisionmaking (Lane et al., 2017). Some examples include equity in facility location (Ares et al.,2016; Marsh and Schilling, 1994), organ allocation for kidney transplantation (Bertsimaset al., 2013; Su and Zenios, 2006), vaccine coverage (Enayati and ¨Ozaltın, 2020), and health-care fleet management (McCoy and Lee, 2014).In the health-care sector, an equity metric compares two or more populations based onthe service or utility the health system provides to the different populations. The compar-ison of various populations could be based on the health status, distribution of resources,expenditures, utilization, and access (Culyer and Wagstaff, 1993; Goddard and Smith, 2001).While it is essential to clearly define equity to be used for fair resource allocation, there isno universal consensus on the definition and measurement of equity in public health decisionmaking (Stone, 2002). Lane et al. (2017) find a large disparity in the description of equityin health care resource allocation based on their review of the related literature.Among numerous definitions of equity, Young (1995) defines three equity concepts onresource allocation: parity (claimants should be treated equally), proportionality (goodsshould be divided in proportion to differences among claimants), and priority (the personwith the greatest claim to the good should get it). Savas (1978) describes equity as fairness,impartiality, or equality of service. Culyer (2001) discusses utilitarian principles dictatingthat resources should be allocated in such a way as to maximize the overall health andwellbeing of a society, and egalitarian principles dictating that all people are equal andthat inequalities between groups should be removed. McCoy and Lee (2014) use utilitarian,proportionally fair, and egalitarian principals to incorporate equity into optimal resourceallocations.Marsh and Schilling (1994) present a list of 20 equity measures within the context of17acility location. Among the most commonly-used equity measures are the sum of absolutedeviations (SAD), the mean absolute deviation (MAD), the minimum effect (ME), andthe Gini coefficient (GC). Love-Koh et al. (2020) categorize methods used to define equitymeasures into five: 1) gap measures, regression-based measures, Lorenz and concentrationcurves, measures incorporating inequality aversion, and health-related social welfare. Theequity measures defined by absolute and relative gaps are commonly used by internationalagencies, such as the WHO, to distribute resources, such as vaccines and medical treatment,between population groups in low- and middle-income countries (Casey et al., 2017).Equitable resource allocation has also been studied considering the tradeoff between theefficiency and equity in resource allocation for infectious diseases, such as HIV and influenza(e.g., Mbah and Gilligan (2011), Zaric and Brandeau (2007), Kaplan and Merson (2002),Enayati and ¨Ozaltın (2020)). For example, Earnshaw et al. (2007) develop a linear program-ming planning tool to help policymakers understand the effectiveness of different allocationsof HIV prevention funds under fairness constraints. Enayati and ¨Ozaltın (2020) proposean equity constraint in a mathematical program to help public health authorities considerfairness when making vaccine distribution decisions. In a food allocation problem, Orgutet al. (2016) present a deterministic linear programming model to optimize the allocation ofdonated food, considering objectives of both equity and effectiveness.Similar to these works, we will follow an approach that would balance the efficiencyand equity in epidemics resource allocation. Specifically, we will focus on equity over meta-populations and multiple spatial dimensions. We define our equity measures within thecontext of proportionality and priority, as described in Young (1995). Our formulationsof equity are gap-based, combining absolute and relative gaps. Our approach is seekinga balance between utilitarian and egalitarian objectives studied in Culyer and Wagstaff(1993) and McCoy and Lee (2014) by determining a resource allocation strategy that willminimize total infections and deaths but at the same time incorporates equality dimensionsas a constraint. Unlike former work, we address equity in the resource allocation for bothtreatment centers and treatment resources using mathematical optimization.Our definition of equity is similar to the descriptions of Mbah and Gilligan (2011), whodefines social equity as the equal opportunity for infected individuals to access treatment,Marsh and Schilling (1994), who define equity within the context of facility location, andOrgut et al. (2016) who study equity in the fair allocation of food. Specifically, we defineequity as the case where each region and country receives its fair share of the ETCs andmedical treatment resources during an epidemic outbreak.18he majority of studies on fair resource allocation define the equity as a one-periodmetric, which does not change over time. In our multi-stage stochastic programming model,the equity standard is adjusted over time with respect to the changing disease dynamicsthroughout the planning horizon, increasing the efficiency of the resource allocation. To thebest of our knowledge, our study is the first to model the fair resource allocation using amulti-stage stochastic programming model, which captures disease dynamics over multipletime periods.
In the first formulation, we will address the objective of equity by limiting the absolutedeviation between a region’s relative number of infections and its relative population withrespect to all regions, while effectiveness corresponds to minimizing the expected number ofinfections and deaths. In this equity measure, namely infection equity constraint, we considerpriority concerning the proportions of infections and enforce resource allocation to limit theproportion of infections with respect to the population for each region. The infection equityconstraint is given as follows: | (cid:80) j ∈ J (cid:80) ω ∈ W p ω I ωj,r (cid:80) j ∈ J (cid:80) r ∈ R (cid:80) ω ∈ W p ω I ωj,r − u r (cid:80) r ∈ R u r | ≤ k (2)The infection equity constraint (2) gives a bound on the total number of infections in eachregion relative to the total infections in all regions. Specifically, constraint (2) implies thatthe absolute value of the number of infected individuals in region r divided by the totalnumber of infected individuals over all regions minus the ratio of the population of region r , u r , over the total population over all regions should be less than or equal to a specific value k .Because the EVD case fatality rate is high [50% on average (WHO, 2020d)] and theEVD is highly contagious, having the lowest infections system-wide will lead to the lowestmortality for the EVD. Thus, we consider the number of infections instead of deaths as themain parameter for resource allocation in our equity metric. The number of infections inconstraint (2) could also be adjusted to the number of fatalities.19 .5.2 Capacity Equity Constraint In the second formulation, we will formulate equity by limiting the absolute deviation be-tween the proportion of treatment capacity established in a region and proportion of thepopulation in a region relative to all regions while again, effectiveness corresponds to min-imizing expected deaths and infections. The capacity equity constraint enforces allocatingresources considering the proportionality based on the relative population and is formulatedas follows: | (cid:80) j ∈ J (cid:80) ω ∈ W p ω C ωj,r (cid:80) j ∈ J (cid:80) r ∈ R (cid:80) ω ∈ W p ω C ωj,r − u r (cid:80) r ∈ R u r | ≤ k, (3)Similarly, we define the capacity equity constraint (3) to bound the absolute value of thedifference between the proportion of the capacity at region r over the total capacity withthe proportion of the population at region r over the total population with a predefinedparameter k . We also study a widely-used equity metric, known as prevalence (Kedziora et al., 2019; Lasryet al., 2008). Here, we define the prevalence equity constraint to limit the absolute differencebetween the regional prevalence (cases per population in a region) and the country prevalence(cases per population over all regions) by the parameter k , and formulate it as follows: | (cid:80) j ∈ J (cid:80) ω ∈ W p ω I ωj,r u r − (cid:80) j ∈ J (cid:80) r ∈ R (cid:80) ω ∈ W p ω I ωj,r (cid:80) r ∈ R u r | ≤ k (4)The prevalence equity constraint (4) bounds the proportion of infections in each regionrelative to the proportion of infections in all regions. In the mathematical formulation (1), we have two types of non-linearity. The first non-linearequation corresponds to the capacity-availability constraint (1o), and the second correspondsto the equity constraints (2) and (3) (see Appendix C for linearization of (1o), (2), and (3)).The non-linear multi-stage stochastic programming epidemic–logistics model (1) is converted20nto an equivalent MIP formulation by replacing the non-linear capacity availability con-straint (1o) with constraints (8), (9a)-(9d) and (10a)-(10d), the non-linear infection equityconstraint (2) with constraints (12a) and (12b), and the non-linear capacity equity constraint(3) with constraints (13a) and (13b), as given in Appendix C.We apply the MIP model to a case study involving the control of the 2014–2015 Ebola out-break in the three most-affected West African countries, Guinea, Sierra Leone, and Liberia.The details of the 2014–2015 Ebola outbreak data used as an input into the mathematicalmodel, including population and migration data, resource cost data, and epidemiologicaldata are presented in Appendix D.The MIP model is solved using CPLEX 12.7 on a desktop computer running with Inteli7 CPU and 64.0 GB of memory. A time limitation of 7,200 CPU seconds was imposed forsolving the test instances without equity constraints, while the time limit is increased to72,000 CPU seconds for the instances with equity constraints due to their computationaldifficulty. The multi-stage stochastic model is solved over eight stages for the base case witheach stage representing a 2-week period, thus for a total of the 16-week planning horizon.Since we consider three outcomes on each branch of the scenario tree, we solve for 3 = 6561scenarios in the mathematical model. In this section, we present computational results for the multi-stage stochastic MIP modelpresented in Section 2 for the considered case study instance in West Africa. Our goal inthis section is to provide insights into the optimal and fair resource allocation for controllingthe Ebola disease outbreak under the uncertainty of disease transmission.
In this subsection, we validate our model against the real outbreak data (WHO, 2016) interms of the cumulative number of infections from August 30, 2014, to December 19, 2014.The values of parameters used in the model are obtained from the literature (Camacho et al.,2014; WHO, 2019b; WHO E. R. Team, 2014).We fix the number of ETCs at each stage according to the number and timing of theETCs established in reality (B¨uy¨uktahtakın et al., 2018). For instance, according to theoutbreak data, one 50-bed ETC was established on September 15, 2014, in northern Liberia,21nd so the value of the related variable is fixed to one in stage one in the model. Once theETCs are fixed in the model based on their opening time and the capacity throughout theplanning horizon, the model is solved and validated by comparing the predicted number ofinfections with the real outbreak data given in the WHO database (WHO, 2016).According to the visual comparison of the predicted results and real outbreak data inFigure 3, our model provides a good fit for the cumulative number of infected individuals inGuinea, Sierra Leone, and Liberia during the considered time period. In addition, we applythe paired t-test to analyze the difference between the pairs of weekly predicted cases andthe actual data. As shown in Table 3, all p-values are greater than 0.05, indicating that ourmodel provides statistically similar results to the real outbreak data from August 30, 2014,to December 19, 2014.Figure 3: Comparison of predicted cases with real outbreak data for cumulative infectionsin Guinea, Liberia, and Sierra LeoneTable 3: Statistical analysis to compare bi-weekly predicted cases and real outbreak data.Country Mean Two-tailed paired t-testOutbreak Predicted t-stat t-critical p-valueGuinea 221.0 266.8 0.41 1.89 0.65Infections Sierra Leone 866.3 910.1 0.65 0.73Liberia 471.1 534.5 0.45 0.6722 .2 The Value of Stochastic Solution (VSS)
To demonstrate the value of using a stochastic program over a deterministic (expected value)model, we use a standard measure in stochastic programming, known as the value of stochas-tic solution (VSS) (Birge, 1982). The VSS gives the expected gain from solving a stochasticmodel over its deterministic counterpart, in which random parameters are replaced by theirexpected values.
WS is the wait-and-see problem objective value, which is the expected value of using theoptimal solution for each scenario. EEV is the expected result of using the solution of thedeterministic model (EV), which replaces all uncertain parameters by their expected values,and RP is the optimal value of our stochastic programming model, i.e., the minimizationrecourse problem. Then the following inequalities are satisfied for the minimization problems(Madansky, 1960):
W S ≤ RP ≤ EEV,
The VSS can then be formulated as follows:
V SS = EEV − RP A large value of the VSS implies that incorporating uncertainty is important to representthe problem realistically, and the solution of the deterministic problem is not “so good.” Onthe other hand, if the VSS value is small, replacing uncertain parameters with their expectedvalues might be a good choice.
For the multi-stage problem, the value of the stochastic solution is introduced as a chain ofvalues
V SS t for t = 1 , . . . , T , where T is the final period of the planning horizon (Escuderoet al., 2007). In order to calculate the V SS t , the solution up to stage t − EEV t value, and RP valueis subtracted from EEV t . Consider a stochastic model, which only contains decision variables x and recourse variables y , and let ( ˆ x t , ˆ y t ) be the optimal solution of the corresponding EVmodel. The EEV t can then be formulated as: EEV t : RP model23.t. x ω = ˆ x ∀ ω ∈ Ω , · · · x ωt − = ˆ x t − ∀ ω ∈ Ω . The
V SS t for each t = 1 , . . . , T is then given as: V SS t = EEV t − RP As an example, we calculate the
V SS t for an 8-stage problem for t = 1 , . . . ,
4. Since
EEV = RP , the value of the V SS is zero. We solve the model under a $
24M budget andpresent the results in Table 4 below.Table 4:
V SS t values up to 4 stages for the 8-stage problem with EEV t values V SS ( RP ) V SS V SS V SS V SS t value is increasing asthe stage t increases, thus a multi-stage stochastic model is needed to obtain a better resultcompared to the deterministic problem. We notice that under the $
24M budget level, themodel allocates almost all the ETCs in the first stage. Thus, the
V SS t value will not changesignificantly when t ≥
3. For varying budget cases or disease dynamics, we expect that themodel will allocate ETCs in the stages following the first stage, and thus the
V SS t valuesmay become larger than the values in this instance. The results for solving the 8-stage modelhighlight the importance of using a multi-stage stochastic model for the epidemic-logisticsproblem over its deterministic counterpart. The columns of Table 5 present results for each
Budget level ( $ $ $ Country and
Region , Stage-1 Budget allocated,
Total Budget allocated,
Stage-1 ETC (50/100) representing the number of 50- and 100-bed ETCs allocated in the firststage of the planning horizon, and
Total ETC (50/100) indicating the total number of50- and 100-bed ETCs allocated throughout the planning horizon. Here, expected values ofthe optimal budget and the number of ETCs allocated at the first stage and throughout theplanning horizon over 6561 scenarios are presented for each budget level. Correspondingly,24he expected value of the total number of infections and funerals for different budget levels ispresented in Figure 6. The CPU time used to solve the model is 7230 s for the $
12M budget,7232 s for the $
24M budget, and 7228 s for the $
48M budget. The optimality gap for all thecases are 0.1%.The fifth column of Table 5 and Figure 4 show the allocation of the total budget amongthree different countries. Due to the high initial number of infected individuals, Sierra Leonegets the most budget allocation under all different budget levels. Although the transmissionrate of Guinea is higher than Liberia, the second highest budget goes to Liberia under the $
48M budget case because the initial state of the infection in this country is high, and thus,the allocated budget will provide a more significant impact on Liberia compared to Guineawhen the budget is ample. According to the results of ETC allocation at all budget levels,most of the beds are allocated in the first period (stage-1) of the planning horizon undertight budget cases, as shown in Table 5. Figure 5 shows the total capacity allocation underdifferent budget levels.Figure 6 shows the total number of infections and funerals in those three countries underdifferent budget levels. According to the result under the $
0M budget level, the case inwhich no intervention action is taken, the number of infections and funerals in Sierra Leonewould be extremely large if we do not take any intervention action. As shown in Figure 6,the total number of infections and funerals in all three countries, especially in Liberia andSierra Leone, drops significantly from $
12M to $
48M budget level.25able 5: Budget and bed allocated under different budget levels
Budget Country Region Stage-1 Total Stage-1 Total ( $ M ) Budget Budget ETC ETC( $ M) ( $ M) (50/100) (50/100)
12 Guinea UG 0.06 0.13 1/1 1/1MG 0.01 0.02 1/0 1/0LG 0.03 0.06 1/0 1/0Sierra Leone S 4.33 11.67 1/4 1/4Liberia NL 0.04 0.09 1/1 1/1SL 0.01 0.03 1/1 1/1
Total 4.47 11.99 6/7 6/7
24 Guinea UG 0.72 1.83 1/1 1/1MG 0.52 1.21 1/0 1/1LG 0.62 1.53 1/1 1/1Sierra Leone S 5.35 15.50 1/5 1/5Liberia NL 0.83 2.31 1/1 1/1SL 0.57 1.60 1/1 1/1
Total 8.62 23.98 6/9 6/10
48 Guinea UG 1.11 2.52 1/1 1/1MG 0.91 1.91 1/1 1/1LG 1.01 2.32 1/1 1/1Sierra Leone S 6.85 18.89 4/5 5/5Liberia NL 3.94 10.42 3/3 3/3SL 2.40 6.03 2/2 2/2
Total 16.22 42.09 12/13 13/13
In this section, we present results regarding the budget, and ETC allocation as well asthe corresponding total number of infections and funerals for five specific scenarios under abudget level of $ Scenario defined above underthe $
24M budget level.The first-stage budget allocation is presented in the fourth column of Table 6, while thetotal budget is presented in both the fifth column of Table 6 and Figure 7. In terms of bedallocation, all the regions have the same number of bed allocation for stage-1 and for thetotal stages under all scenarios. This result implies that it is optimal to open treatmentcenters early in all the locations, in particular, in the initial stages.Figure 8 represents the total capacity allocation under different scenarios. According tothe results, the total capacity allocated under the worse scenario group is higher than thecapacity allocated for the better group. This result implies that under the worse scenariogroup, more budget is allocated to build new Ebola treatment centers. Also, as shown inFigure 9, the total number of new infections and funerals under the “High-Low” case is muchhigher than the corresponding number under the “Low-High” case. Thus, a scenario wherethe disease starts with a low transmission rate and then progresses fast is better than ascenario in which the disease progression is fast and then slows down. This may be becausediseases that initially progress less aggressively give us more time to get prepared, establishthe ETCs and treatment resources, and thus reduce the number of infections immediately.29able 6: Budget and bed allocated under different scenarios
Scenario Country Region Stage-1 Total Total ( $ M ) Budget Budget Bed( $ M) ( $ M) (50/100)
All Low Guinea UG 0.60 1.15 2/0MG 0.60 1.02 2/0LG 0.60 1.13 2/0Sierra Leone S 5.39 12.44 0/5Liberia NL 2.15 5.46 0/2SL 1.08 2.79 0/2
Total 10.41 24.00 6/9
All Medium Guinea UG 0.60 1.64 2/0MG 0.60 1.43 2/0LG 0.60 1.64 2/0Sierra Leone S 5.39 16.13 0/5Liberia NL 0 0 0/0SL 1.08 3.16 0/2
Total 8.26 24.00 6/7
All High Guinea UG 1.08 2.75 0/2MG 0.60 1.51 2/0LG 1.08 2.23 0/2Sierra Leone S 7.06 17.51 2/7Liberia NL 0 0 0/0SL 0 0 0/0
Total 9.82 24.00 4/11
Low-High Guinea UG 0.60 1.44 2/0MG 0.60 1.18 2/0LG 0.60 1.34 2/0Sierra Leone S 4.31 12.02 0/5Liberia NL 1.68 4.91 2/2SL 1.08 3.11 0/2
Total 8.86 24.00 8/9
High-Low Guinea UG 1.08 2.44 0/2MG 0.60 1.29 2/0LG 1.08 2.23 0/2Sierra Leone S 6.46 15.47 0/7Liberia NL 1.08 2.56 0/2SL 0 0 0/0
Total 10.29 24.00 2/13
In this subsection, we present results by adding each of the three equity constraints (2), (3),and (4), as introduced in Section 2.5, separately into the linearized multi-stage stochasticprogramming epidemic-logistics model (1). Equity constraints impose a bound on the totalnumber of infections in each region and thus enforcing that each region considered in WestAfrica receives a more equitable share of resources, including ETCs and treatment funds,while minimizing the total number of infections and deaths.According to the results, imposing the infection equity constraint (2) or the prevalenceequity constraint (4) does not significantly change the optimal budget allocation and thetotal number of new infections and funerals (see Appendix E for detailed results). Without32ntroducing the infection equity constraint into the mathematical model (1), the absolutevalue of the difference between the infection ratio and the population ratio in Guinea, SierraLeone, and Liberia is 0.42, 0.04, and 0.38, respectively, based on the optimal solution valuesimilar to the k values considered here. This result implies that our model balances the totalnumber of infections in each region with its population and population, even without theinfection equity constraint.Similar to the infection equity case, we introduce the capacity equity constraint (3) intothe multi-stage stochastic programming epidemic-logistics model (1) for an 8-stage instancewith the $
24M budget level under different values of k . Table 7 represents the run timespecifics regarding the mathematical model (1) with the capacity equity constraint (3), whileFigures 10 and 11 present the budget allocation and the total number of infections andfunerals over the three considered countries for varying k values. When k is larger than 0.4,we observe no significant change in the results. However, a small k value can impact theresults significantly. For example, when k = 0 .
05, all three regions have a similar budgetallocation. If k increases from 0.05 to 0.2, the total number of infections and funerals inGuinea is slightly increased, but it is decreased when k is further increased. Thus, allocatingthe majority of resources to Guinea may not be necessary, and some of those resources wouldbe wasted. As we relax the equity capacity constraint by increasing the k value from 0.05to 0.4 and above, we observe a significant drop in the number of infected individuals andfunerals in Sierra Leone. The total number of infected people and funerals over all threecountries is the largest (12,769) when the capacity equity constraint is strictly enforced, andit is the smallest (10,995) when the capacity equity constraint is relaxed. This result impliesthat enforcing a tight equity constraint might adversely impact the total number of infectionsand deaths, and thus resulting in a high cost that we have to pay for fairness.33able 7: Model run specifics with the capacity equity constraint (3) k value Solution Time (CPU sec) Optimality Gap (%)0.05 72,103 70.1 72,121 80.2 72,053 60.4 72,031 2A large k value 7,232 0(no-equity-constraint case)Figure 10: Optimal budget allocation under different k values for an 8-stage problem with $
24M budget 34igure 11: Total number of new infections and funerals under different k values for an 8-stageproblem with $
24M budget
In this paper, we extended the epidemic-logistics model of B¨uy¨uktahtakın et al. (2018)to study an epidemic control problem in a large-scale population where the transmissionrate of the disease is uncertain. To our knowledge, this is the first multi-stage stochasticepidemic-logistic model that takes into account both the uncertain disease growth and equi-table resource allocation simultaneously. We consider various disease progression scenariosresulted from the realization of the community transmission rates. Our objective is to min-imize the total expected number of infected individuals and funerals over all scenarios, allperiods, and all regions considered. We study the value of the stochastic solution and intro-duce the equity constraints to analyze the fair resource allocation among different countriesand multiple regions of a country. Our multi-stage VSS analysis suggests that the stochasticmodel considerably improves the solution of the deterministic model, and the considerationof uncertainty in a multi-stage disease-transmission model is necessary.We define the infection level as the difference between the ratio of the number of infectedpeople in a region to the total number of infected people over all regions and the ratio of thepopulation in a region to the total population over all regions. Under tight budget levels,most of the budget would be allocated to the region that has the highest initial infectionlevel, while other regions would receive ETCs and treatment resources according to theirinfection level as the available budget increases. This indicates that the initial infection level35s a key factor in resource allocation. Additionally, more 100-bed ETCs would be allocatedto the country that has a high infection level since more capacity will be needed to treatinfected people while saving from the fixed cost of opening new ETCs.According to the results, our model allocated most of ETCs in the first stage to providea quick response to the epidemic and reduce a large number of unnecessary infections andfunerals. Our results showed that the number of untreated infections dropped quickly whenearly actions were taken with a sufficiently large budget, and the disease was controlled muchfaster than the report date of the World Health Organization (WHO). The uncertainty indisease transmission is a critical factor that makes it challenging to manage an outbreakin a real-life situation. To be more specific, the transmission rate might suddenly becomehigh after a latent period, and the existing resources may not be sufficient to handle suchunexpected situations. Consequently, a large number of unisolated and untreated individualscould stay in the community and continue to spread the disease, as in the case of the currentoutbreak of Coronavirus (COVID-19) disease (WHO, 2020a). Thus, the preparedness andearly action to handle the uncertain disease transmission are crucial, and we would rather“the beds waiting for people” than “people waiting for the beds.” Our findings are consistentwith several other articles that also report the importance of early action for epidemic control(Jacobsen et al., 2016; Lekone and Finkenst¨adt, 2006; Siedner et al., 2015). The lessonslearned from the EVD control in West Africa by WHO and Centers for Disease Control andPrevention (CDC) also indicate that an early action will have a significant improvement inslowing down an epidemic and eventually stopping it (CDC, 2019; WHO, 2019b).Different than the former literature, the solutions of our multi-stage stochastic program-ming model show that the optimal timing of the resource allocation might vary if we have arelatively ample budget. For instance, in both $
24M and $
48M budget levels, some resourceswere allocated throughout the planning horizon in some locations, such as Guinea and SierraLeone. This is because we have more budget to take action when the transmission of thedisease gets worse. This result shows that the timing of the resource allocation should be de-cided dynamically and based on the predicted disease growth scenario and budget, and thusimplying the superiority of a multi-stage stochastic programming model over a two-stage orstatic model again.We analyzed five specific disease growth scenarios and studied resource allocation strate-gies under each scenario. Under the scenarios in which the disease moves faster, more numberof ETCs are allocated compared to the scenarios in which the disease moves slower to treatmore people. In addition, if the disease moves faster, the majority of the capacity is allo-36ated to the region that has the highest initial infection level. If the disease consistentlymoves at a slow rate, the treatment capacity is allocated more equally among regions tohelp fight against the disease. In the “Low-High” case, in which the disease moves in aslow rate first and then starts to be more aggressive in the following time stages, the modelallocates budget immediately to the regions with a high infection level and knocks down thenumber of infected individuals to low values, which will lessen the impacts of a high diseasetransmission rate later in the planning horizon. Because an initially slow-moving diseasegives us more time to get prepared to control the disease spread, the “Low High” case canbe considered as a better scenario compared to the “High-Low” case.We introduced the infection and capacity equity constraints separately into our modelto analyze the impact of enforcing fairness in resource allocation. Solutions obtained withthe infection equity constraint imply that the original optimal solution balances the resourceallocation among multiple regions in a similar fashion to the infection equity constraint.Thus, our model takes into account the ratio of infection to the total infection level as wellas the ratio of the population to the total population level over all three countries whilemaking the resource allocation decision.When a tight capacity equity constraint is enforced, the budget is allocated equally tothe three regions. However, in this case, some of the budget may be wasted, and no obviouseffects are brought out by providing additional capacity to a region based solely on itspopulation. This result shows that allocating treatment resources proportional to populationis sub-optimal, which is also consistent with the findings of Ren et al. (2013). When thecapacity equity constraint is relaxed, the number of infections and funerals in Guinea andLiberia is slightly changed, but this number decreased significantly for Sierra Leone, and thetotal three countries. For both tight and ample budget cases, the total number of infectionsand funerals is much higher when the capacity equity constraint is strictly forced, resultingin a heavy price we would have to pay for perfect equity in resource allocation. This resultimplies that the decision maker should be cautious about enforcing fairness when allocatingresources to multiple regions.There are several important future research directions that arise out of this study. Forexample, the impact of vaccinations currently used to prevent the spread of the diseasecould be analyzed in a future study. The influence of vaccination is group-specific, and thussusceptible individuals can be divided into different groups according to their age, sex, race,and health status. Due to the lack of available data, the transmission rate from susceptibleindividuals to infected individuals would be more difficult to predict under vaccination.37urthermore, different kinds of vaccines used, the amount of vaccine allocated to each region,and the time when vaccination becomes accessible might impact the disease transmissionrate significantly. Our model could be extended by adding a compartmental class named as“vaccinated” to study the various dimensions of vaccination.Moreover, our multi-stage stochastic program only includes the expectation criterion inthe objective function when it compares random variables to find the best decisions. Thus,our study provides a risk-neutral approach. In a future extension of this work, risk measures,such as Conditional Value at Risk (
CV aR ), could be incorporated into the objective functionto reflect the perspectives of a risk-averse decision maker.
Acknowledgments
The authors thank two anonymous referees, the associate editor, and the editor, whoseremarks helped to improve the content and clarity of our exposition.
Appendices
A Notation
Model notations that are used throughout the rest of this paper are presented in Tables 8–12below. 38able 8: Sets and indices.
Notation Description J Set of time periods, J = { , ..., J } . A Set of ETC types, A = { , ..., A } . R Set of regions, R = { , ..., R } . M r Set of all surrounding regions of region r .Ω Set of scenarios, Ω = { , ..., Ω } . j Index for time period where j ∈ J . r Index for region where r ∈ R . a Index defining type of ETC, where a ∈ A . ω Index for scenario where ω ∈ Ω. Table 9: Transition parameters describing the rate of movement between disease compart-ments.
Notation Description λ ,r Disease fatality rate without treatment in region r . λ ,r Disease fatality rate while receiving treatment in region r . λ ,r Disease survival rate without treatment in region r . λ ,r Disease survival rate with treatment in region r . λ ,r Safe burial rate of Ebola-related dead bodies in region r . χ ω ,r Transmission rate per person due to community interaction in region r under scenario ω . χ ,r Transition rate per person during traditional funeral ceremony in region r . Notation Description b j,r Unit cost of treatment for an infected individual in region r at end of period j . b j,r Unit cost of safe burial for a dead body in region r at end of period j . g aj,r Fixed cost of establishing type a ETC in region r at end of period j . k a Capacity (number of beds) of type a ETC. u r The population in region r .∆ Total available budget for treatment. π r Initial number of susceptible individuals in region r . (cid:36) r Initial number of infected individuals in region r . θ r Initial number of treated individuals in region r . ϑ r Initial number of recovered individuals in region r . υ r Initial number of unburied dead bodies (funerals) in region r . τ r Initial number of buried dead bodies (safe burials) in region r . ς r Initial treatment capacity in terms of number of ETC beds in region r . ε l → r Migration rate of susceptible individuals from surrounding regions l ∈ M r to region r . φ l → r Migration rate of infected individuals from surrounding regions l ∈ M r to region r . ν r → l Migration rate of susceptible individuals from region r to surrounding regions l ∈ M r . ρ r → l Migration rate of infected individuals from region r to surrounding regions l ∈ M r . Notation Description S ωj,r Number of susceptible individuals in region r at end of period j under scenario ω . I ωj,r Number of infected individuals in region r at end of period j under scenario ω . T ωj,r Number of individuals receiving treatment in region r at end of period j under scenario ω . R ωj,r Number of recovered individuals in region r at end of period j under scenario ω . F ωj,r Number of deceased individuals due to the epidemic in region r at end of period j under scenario ω . B ωj,r Number of buried individuals in region r at end of period j under scenario ω .ˆ S ωj,r Number of susceptible individuals migrating into region r at end of period j under scenario ω . (cid:101) S ωj,r Number of susceptible individuals emigrating from region r at end of period j under scenario ω .ˆ I ωj,r Number of infected individuals migrating into region r at end of period j under scenario ω . (cid:101) I ωj,r Number of infected individuals emigrating from region r at end of period j under scenario ω . Table 12: Decision variables.
Notation Description C ωj,r Total capacity (number of beds) of established ETCs in region r at end of period j under scenario ω . I ωj,r Number of infected individuals hospitalized (and quarantined) in region r at end of period j under scenario ω . y ωaj,r Number of type a ETCs established in region r at end of period j under scenario ω . Non-Anticipativity
Two scenarios should have the same decision variables at a stage j if they share the samescenario path up to that stage. Corresponding decisions up to stage j of two inseparable sce-narios should be the same. These implications are named as non-anticipativity constraints,and can be formulated as follows. Consider the node marked n in the scenario tree, anddenote the corresponding stage as t ( n ). Let the set of scenarios that pass through node n be β ( n ). We must ensure that decision variables at stage t ( n ) that are associated with node n (for example: x ωt ( n ) ) have identical values for ω ∈ β ( n ). One way to do this is to add thenon-anticipativity constraint as in the following form: x ωt ( n ) − x n = 0 ∀ ω ∈ β ( n ) . As an example, consider the first three stages of the multi-stage problem shown in Fig-ure 2. The set of nodes of this scenario tree is given by N = { , , , , ..., , } , where t (0) = 0 , t (1) = 1 , t (2) = 1 , t (3) = 2 , t (4) = 2 , t (5) = 2 , t (6) = 2 , t (7) = t (8) = t (9) = t (10) = t (11) = t (12) = t (13) = 3. The set of scenarios that share node n = 2 is given by β (2) = { , , , } . Let x ωt (2) represent decision variables for ω ∈ β (2). The non-anticipativityconstraint for those variables can be written as: x ωt (2) − x = 0 ∀ ω ∈ β (2) . C Linearization
We first linearize the logical constraint that describes the number of hospitalized individualsin equation (1o). Following the method of Kıbı¸s and B¨uy¨uktahtakın (2017), for each j ∈ J \ J , r ∈ R , and ω ∈ Ω, constraint (1o) can be written as: I ωj,r = ( C ωj,r − T ωj,r ) z ωj,r + I ωj,r (1 − z ωj,r ) , (6)where z ωj,r is a binary variable, which takes the value 1 if the number of infected individualsto be hospitalized is restricted by the number of available beds in ETCs, and the value 0 ifthe number of beds in ETCs is sufficiently large to hospitalize all infected individuals. Inorder to ensure that I ωj,r takes the minimum value of ( C ωj,r − T ωj,r ) and I ωj,r , we should havethe following inequalities satisfied for each j ∈ J \ J , r ∈ R , and ω ∈ Ω: I ωj,r ≤ C ωj,r − T ωj,r (7a)42 ωj,r ≤ I ωj,r . (7b)However, constraint (6) is still non-linear due to quadratic terms. Therefore, two auxiliaryvariables U ωj,r and W ωj,r are introduced to be substituted with ( C ωj,r − T ωj,r ) z ωj,r and I ωj,r (1 − z ωj,r ),respectively. In this case, for each j ∈ J \ J , r ∈ R , and ω ∈ Ω, constraint (6) can be writtenas: I ωj,r = U ωj,r + W ωj,r (8)We then introduce a lower bound ( H LB ) and upper bound ( H UB ) for C ωj,r − T ωj,r , such that H LB ≤ C ωj,r − T ωj ≤ H UB and add the following constraints to the model for each j ∈ J \ J , r ∈ R , and ω ∈ Ω: U ωj,r ≤ H UB z ωj,r (9a) U ωj,r ≥ H LB z ωj,r (9b) U ωj,r ≤ ( C ωj,r − T ωj,r ) − H LB (1 − z ωj,r ) (9c) U ωj,r ≥ ( C ωj,r − T ωj,r ) − H UB (1 − z ωj,r ) . (9d)Similarly, we introduce a lower bound ( I LB ) and an upper bound ( I UB ) for I ωj,r , such that I LB ≤ I ωj,r ≤ I UB , and add the following four constraints for each j ∈ J \ J , r ∈ R , and ω ∈ Ω to the model: W ωj,r ≤ I UB (1 − z ωj,r ) (10a) W ωj,r ≥ I LB (1 − z ωj,r ) (10b) W ωj,r ≤ I ωj,r − I LB z ωj,r (10c) W ωj,r ≥ I ωj,r − I UB z ωj,r . (10d)Thus the constraint (1o) can be equivalently linearized by replacing it with constraints(8), (9a)-(9d) and (10a)-(10d).We then linearize the equity constraint given by equation (2). By multiplying the twodenominators on the left side of (2) by each other and multiplying the right side of (2) by (cid:80) r ∈ R u r (cid:80) j ∈ J (cid:80) r ∈ R (cid:80) ω ∈ W P ω I ωj,r , we obtain the following inequality: | (cid:88) r ∈ R u r (cid:88) j ∈ J (cid:88) ω ∈ W P ω I ωj,r − u r (cid:88) j ∈ J (cid:88) r ∈ R (cid:88) ω ∈ W P ω I ωj,r | ≤ k (cid:88) r ∈ R u r (cid:88) j ∈ J (cid:88) r ∈ R (cid:88) ω ∈ W P ω I ωj,r (11)The absolute value in inequality (11) could be linearized using the following two constraints: (cid:88) r ∈ R u r (cid:88) j ∈ J (cid:88) ω ∈ W P ω I ωj,r − u r (cid:88) j ∈ J (cid:88) r ∈ R (cid:88) ω ∈ W P ω I ωj,r − k (cid:88) r ∈ R u r (cid:88) j ∈ J (cid:88) r ∈ R (cid:88) ω ∈ W P ω I ωj,r ≤ r ∈ R u r (cid:88) j ∈ J (cid:88) ω ∈ W P ω I ωj,r − u r (cid:88) j ∈ J (cid:88) r ∈ R (cid:88) ω ∈ W P ω I ωj,r + k (cid:88) r ∈ R u r (cid:88) j ∈ J (cid:88) r ∈ R (cid:88) ω ∈ W P ω I ωj,r ≥ (cid:88) r ∈ R u r (cid:88) j ∈ J (cid:88) ω ∈ W P ω C ωj,r − u r (cid:88) j ∈ J (cid:88) r ∈ R (cid:88) ω ∈ W P ω C ωj,r − k (cid:88) r ∈ R u r (cid:88) j ∈ J (cid:88) r ∈ R (cid:88) ω ∈ W P ω C ωj,r ≤ (cid:88) r ∈ R u r (cid:88) j ∈ J (cid:88) ω ∈ W P ω C ωj,r − u r (cid:88) j ∈ J (cid:88) r ∈ R (cid:88) ω ∈ W P ω C ωj,r + k (cid:88) r ∈ R u r (cid:88) j ∈ J (cid:88) r ∈ R (cid:88) ω ∈ W P ω C ωj,r ≥ D Ebola Case Study Data
This section presents the data used to formulate the model, including population and mi-gration data, resource cost data, and epidemiological data. All data provided in this sectionwas collected using literature resources and given bi-weekly. Data pertaining to the 2014-2015 Ebola outbreak and the deterministic epidemics-logistics model have been validated byB¨uy¨uktahtakın et al. (2018).
D.1 Population and Migration Data
Table 13 presents the distribution of the population in Guinea, Liberia, and Sierra Leone,all located in West Africa. We consider six regions: three of them are located in Guinea(Upper Guinea (UG), Middle Guinea (MG), and Lower Guinea (LG)), two of them are inLiberia (Northern Liberia (NL) and Southern Liberia (SL)) and the last one, Sierra Leone,is a county itself (S). Table 14 shows the total number of initial infections in each country.44able 15 gives the migration rates from each of the five regions (UG,MG,LG,NL,SL) to theother four regions. There is no migration in Sierra Leone because it is considered as a regionby itself. Rapidly after the initial recognition of the Ebola outbreak, those three countriesclosed the national borders, so we only consider the migration within a country.Table 13: Regions, population size and rate in West Africa
Guinea Population Ratio Liberia Population Ratio Sierra Leone Population Ratio(millions) (millions) (millions)UG 4,3 0.41 NL 2,2 0.64 S 4,9 1.00MG 2,7 0.25 SL 1,2 0.36LG 3,7 0.34
Total 10,7
Table 14: The number of infected people at the beginning of the planning horizon (August30, 2014) in West Africa Guinea Sierra Leone Liberia218 604 685Table 15: Bi-weekly migration rate between regions of Guinea and Liberia, original dataacquired from Wesolowski et al. (2014).
From \ To UG MG LG NL SLUG 0.0032 0.0010MG 0.0052 0.0025LG 0.0012 0.0018NL 0.0007SL 0.0011
D.2 Resource Allocation Cost Data
The fixed cost of locating Ebola treatment centers (ETCs) and the per-person cost of Ebolatreatment for either 50 or 100-bed ETC are given below in Table 16. The treatment costincludes the fixed cost for establishing each type of ETCs, isolation unit center, and labo-ratory diagnosis. Additionally, each facility has a variable running cost mainly composed45f treating infected people and contact tracing of the infected individuals. There is also asafe burial cost for safely burying infected dead bodies. Fixed costs are one-time; however,all other costs are given for a 2-week period in Table 16. For example, the variable cost ofthe Ebola treatment center represents the cost of treating one infected individual over twoweeks. Table 16: Summary of Ebola treatment cost for 50 (100)-bed ETC.
Cost description Fixed Cost Variable cost * Safe burial cost * Ebola treatment center $ $ $ $ $ $ $ $ $ $ Total $ ( $ ) $ $ * Variable and safe burial costs are bi-weekly.
D.3 Epidemiological Data
Table 17 presents the data values for transmission parameters for each of the three consideredcountries in West Africa. The data contains the fatality rate with and without treatment,recovery rate with and without treatment, safe burial rate, and transmission rates. Becausethe transmission rate in the community is an uncertain parameter, we present its value undereach of the two realizations as low and high. Moreover, we considered the expected value ofthe transmission rate at a traditional funeral for each country.46able 17: Transmission parameters and bi-weekly rates for the Ebola outbreak.
ParameterDescription Data ReferenceGuinea Sierra LeoneLiberia λ Rate of fatality without treatment 0.428 0.124 0.176 WHO (2019b), WHO E. R. Team (2014) λ Rate of fatality with treatment 0.350 0.096 0.128 WHO (2019b), WHO E. R. Team (2014) λ Rate of recovery without treatment 0.240 0.242 0.232 WHO (2019b), WHO E. R. Team (2014) λ Rate of recovery with treatment 0.416 0.327 0.312 WHO (2019b), WHO E. R. Team (2014) λ Safe burial rate 0.730 0.710 0.740 WHO (2019b), WHO E. R. Team (2014) χ l ,r Transmission rate in community (Low) 0.660 0.632 0.560 Camacho et al. (2014) χ h ,r Transmission rate in community (High)0.990 0.940 0.840 Camacho et al. (2014) χ ,r Transmission rate at traditional funeral 1.460 1.420 1.480 Camacho et al. (2014)
E Analysis of Infection and Prevalence Equity Con-straints
The infection equity constraint (2) limits the difference between the proportion of infectionsin each region over the total number of infections and the proportion of the population ateach region over the total population with a specific k value. Introducing the infection equityconstraint to the mathematical model with 8 stages increased the average CPU solution timefrom 7200 seconds to 10 hours when k = 0 .
2, and the average optimality gap from 1% to29%. Table 18 gives the run time specifics regarding the mathematical model (1) with eightstages and the infection equity constraint (2). As seen in Table 18, for k values between 0.2and 0.4, the computational complexity significantly increases compared to the case wherethe infection equity constraint is relaxed, i.e., k is set to a large number.Figures 12 and 13 show the budget allocation and the total number of infections andfunerals over the three considered countries for different k values. According to the results,varying k values does not significantly change the optimal budget allocation and the totalnumber of infections and funerals. Without introducing the infection equity constraint intothe mathematical model (1), the absolute value of the difference between the infection ra-tio and the population ratio in Guinea, Sierra Leone, and Liberia is 0.42, 0.04, and 0.38,respectively, based on the optimal solution value similar to the k values considered here.47able 18: Model run specifics with the infection equity constraint (2) k value Solution Time (CPU sec) Optimality Gap (%)0.2 36,068 290.3 7,213 10.4 7,214 1A large k value 7232 0(no-equity-constraint case)Figure 12: Optimal budget allocation under different k values for an 8-stage problem with $
24M budget 48igure 13: Total number of new infections and funerals under different k values for an 8-stageproblem with $
24M budgetAs a comparison, we also test the prevalence equity constraint (4) and compare it tothe infection equity constraint (2). The prevalence equity constraint bounds the absolutedifference between the regional prevalence (cases per population in a region) and the countryprevalence (cases per population over all regions). Without the prevalence equity constraint(4) constraint, the absolute value of the difference between the infection ratio over a regionand the infection ratio over all regions in Guinea, Sierra Leone, and Liberia is 4 . × − ,8 . × − , and 1 . × − , respectively, based on the optimal solution value.We test the prevalence equity constraint under the $
24M budget level. Table 19 presentsthe run-time and optimality gap specifics for each k value in inequality (4) . Figures 14 and 15show the optimal budget allocation and the number of infections and funerals under different k values, respectively. Note that the k values used in the prevalence equity constraint (4)are much smaller than the k values used in the infection equity constraint (2). Similar tothe infection equity constraint, the optimal budget allocation does not show any significantdifference among each k value, but the number of infections and funerals slightly reduceswhen we relax the prevalence equity constraint. These results imply that our model balancesthe proportion of infections in each region, even without imposing the infection equity (2)or (4) prevalence equity constraints. 49able 19: Model run specifics with the prevalence equity constraint k value Solution Time (CPU sec) Optimality Gap (%)3 × − × − × − × − k value 7,232 0(no-equity-constraint case)Figure 14: Optimal budget allocation under different k values for an 8-stage problem with $
24M budget 50igure 15: Total number of new infections and funerals under different k values for an 8-stageproblem with $
24M budget
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