A Multi-Stage Stochastic Programming Approach to Epidemic Resource Allocation with Equity Considerations

Abstract

Existing compartmental models in epidemiology are limited in terms of optimizing the resource allocation to control an epidemic outbreak under disease growth uncertainty. In this study, we address this core limitation by presenting a multi-stage stochastic programming compartmental model, which integrates the uncertain disease progression and resource allocation to control an infectious disease outbreak. The proposed multi-stage stochastic program involves various disease growth scenarios and optimizes the distribution of treatment centers and resources while minimizing the total expected number of new infections and funerals. We define two new equity metrics, namely infection and capacity equity, and explicitly consider equity for allocating treatment funds and facilities over multiple time stages. We also study the multistage value of the stochastic solution (VSS), which demonstrates the superiority of the proposed stochastic programming model over its deterministic counterpart. We apply the proposed formulation to control the Ebola Virus Disease (EVD) in Guinea, Sierra Leone, and Liberia of West Africa to determine the optimal and fair resource-allocation strategies. Our model balances the proportion of infections over all regions, even without including the infection equity or prevalence equity constraints. Model results also show that allocating treatment resources proportional to population is sub-optimal, and enforcing such a resource allocation policy might adversely impact the total number of infections and deaths, and thus resulting in a high cost that we have to pay for the fairness. Our multi-stage stochastic epidemic-logistics model is practical and can be adapted to control other infectious diseases in meta-populations and dynamically evolving situations.

Risk-Averse Multi-Stage Stochastic Programming to Optimizing Vaccine Allocation and Treatment Logistics for Effective Epidemic Response

“…Specifically, the inequality (2m) represents the budget constraint on the sum of the fixed costs of opening ETCs and the variable cost of treating infected individuals, and the cost of allocating vaccines over all regions r in all periods j under scenario ω. Constraint (2n) denotes the total capacity in region r at the end of period j under scenario ω. Constraint (2o)-(2q) are linear constraints that ensure the number of available beds in ETCs limit the number of hospitalized individuals in region r. Particularly, linear equations (2o)-(2q) are 18 equivalent to the non-linear constraints implying that the number of hospitalized individuals (I) is equal to the minimum number of infected individuals and the capacity available at established ETCs after considering currently hospitalized individuals in ETCs (see Yin and Büyüktahtakın (2021) for the details of the linearization). Constraint (2r) represents that the number of vaccines supplied to region r at period j under scenario ω is limited by the number of close contacts in region r at period j under scenario ω. Constraint (2s) ensures that the total number of vaccines allocated over all regions can not exceed the available supply at each time period.…”

“…Their optimization model simultaneously considered the logistics problem and the disease growth and represented the transitions from the infections to the treatment compartment as a variable since this transition depends on the treatment center’s capacity. Yin and Büyüktahtakın (2021) extended the deterministic epidemics-logistics model of Büyüktahtakın et al (2018) to a multi-stage stochastic mixedinteger programming model. They introduced the value of the stochastic solution (VSS) to study the advantages of the stochastic model compared to the deterministic model.…”

“…Constraint (2o)–(2q) are linear constraints that ensure the number of available beds in ETCs limit the number of hospitalized individuals in region r . Particularly, linear equations (2o)–(2q) are equivalent to the non-linear constraints implying that the number of hospitalized individuals ( Ī ) is equal to the minimum number of infected individuals and the capacity available at established ETCs after considering currently hospitalized individuals in ETCs (see Yin and Büyüktahtakın (2021) for the details of the linearization). Constraint (2r) represents that the number of vaccines supplied to region r at period j under scenario ω is limited by the number of close contacts in region r at period j under scenario ω .…”

“…Those mainly use stochastic and approximate dynamic programming (Coşgun and Büyüktahtakın, 2018; Long et al, 2018) and two-stage stochastic programming (Ren et al, 2013; Tanner et al, 2008; Yarmand et al, 2014). Because the growth of an infectious disease dynamically changes over time, Yin and Büyüktahtakın (2021) present a multi-stage stochastic programming model to capture the dynamics of an evolving disease for effective epidemic control under the uncertainty of disease transmission.…”

“…Thus a large number of infections and losses could happen in shorter time periods than expected, as in the case of COVID-19 (Lazzerini and Putoto, 2020; Li et al, 2020). The former epidemics control multi-stage stochastic programming model of Yin and Büyüktahtakın (2021) only considered an expectation criterion in the objective function. To alleviate the adverse impacts of experiencing a disastrous disease transmission scenario, we consider a risk measure in the objective function in addition to the expectation criterion.…”