A mathematical model for equitable in-country COVID-19 vaccine allocation

Figures & data

Figure 1. An example vaccine distribution network for CVAP.

Figure of in-country vaccine distribution flow.

Table 1. KPIs for evaluating equity and effectiveness of CVAP solutions.

	KPI No.	KPI description	KPI notation
Equity	KPI 1a	Standard deviation of total coverage (among municipalities)	StDev [${\hat{Z}}_m$]
	KPI 2a	Range of total coverage (among municipalities)	Range [${\hat{Z}}_m$]
	KPI 3a	Minimum total coverage (among municipalities)	Min [${\hat{Z}}_m$]
	KPI 4a	Gini coefficient for the total coverage (among municipalities)	Gini [${\hat{Z}}_m$]
	KPI 5a	Average of the absolute value of difference between total fair coverage and total achieved coverage (among municipalities)	Avg [$\left|{\hat{\mathrm{\Delta }}}_m\right|$]
	KPI 6a	Maximum difference between total fair coverage and total achieved coverage (among municipalities)	Min [${\hat{\mathrm{\Delta }}}_m-$]
	KPI 1b	Standard deviation of coverage for each priority group (among municipalities)	StDevP [$Z_{mp}$]
	KPI 2b	Range of coverage for each priority group (among municipalities)	RangeP [$Z_{mp}$]
	KPI 3b	Minimum coverage for each priority group (among municipalities)	MinP [$Z_{mp}$]
	KPI 4b	Gini coefficient for the coverage for each priority group (among municipalities)	GiniP [$Z_{mp}$]
	KPI 5b	Average of the absolute value of difference between fair coverage and achieved coverage for each priority group (among municipalities)	AvgP [$\left|{\mathrm{\Delta }}_{mp}\right|$]
	KPI 6b	Maximum difference between fair coverage and achieved coverage for each priority group (among municipalities)	MinP [${\mathrm{\Delta }}_{mp}-$]
Effectiveness	KPI 7	Total ${\alpha }_{mp}$-weighted average coverage	(${\alpha }_{mp}$) Wavg [$Z_{mp}$]
	KPI 8.d	Total ${\lambda }_m^d$-weighted average coverage	(${\lambda }_m^d$) Wavg [${\hat{Z}}_m$]
	KPI 9	Total ${\tau }_p$-weighted average coverage	(${\tau }_p$) Wavg [${\hat{Z}}_p$]

Figure 2. Attained and fair coverage levels for Plans 1, 2 and 3.

A bar chart showing attained coverages and fair coverages of 8 municipalities under plans 1, 2 and 3.

Table 2. Equity and effectiveness KPIs for the three allocation plans.

KPI No.	KPI	Plan 1	Plan 2	Plan 3
KPI 1a	StDev [${\hat{Z}}_m$]	0.0%	21.2%	18.5%
KPI 2a	Range [${\hat{Z}}_m$]	0.0%	60.0%	50.0%
KPI 3a	Min [${\hat{Z}}_m$]	60.0%	30.0%	40.0%
KPI 4a	Gini [${\hat{Z}}_m$]	0.00	0.19	0.16
KPI 5a	Avg [$\left|{\hat{\mathrm{\Delta }}}_m\right|$]	0.0%	0.0%	4.1%
KPI 6a	Min [${\hat{\mathrm{\Delta }}}_m-$]	0.0%	0.0%	$-$12.5%
KPI 7	(${\alpha }_{mp}$) Wavg [$Z_{mp}$]	60.0%	67.5%	67.2%
KPI 8.1	(${\lambda }_m^1$) WAvg [${\hat{Z}}_m$]	60.0%	67.5%	67.2%

Table 3. KPIs of solution with sufficient and restricted capacity for municipality M5.

KPI No.	KPI	Sufficient capacity	Restricted capacity
KPI 1a	StDev [${\hat{Z}}_m$]	21.2%	19.8%
KPI 2a	Range [${\hat{Z}}_m$]	60.0%	60.0%
KPI 3a	Min [${\hat{Z}}_m$]	30.0%	30.0%
KPI 4a	Gini [${\hat{Z}}_m$]	0.19	0.18
KPI 5a	Avg [$\left|{\hat{\mathrm{\Delta }}}_m\right|$]	0.0%	3.8%
KPI 6a	Min [${\hat{\mathrm{\Delta }}}_m-$]	0.0%	−15.0%
KPI 7	(${\alpha }_{mp}$) Wavg [$Z_{mp}$]	67.5%	66.6%
KPI 8.1	(${\lambda }_m^1$) Wavg [$Z_{mp}$]	67.5%	66.6%

Figure 3. Coverage values with and without ECF model.

A bar chart with coverage levels on the y axis and municipality - priority group pair on x axis that shows the fair coverage levels and coverage levels attained with and without ECF model side by side for each municipality and priority group.

Table 4. KPIs of CVAP solutions with and without ECF model.

KPI No.	KPI	Without ECF		With ECF
KPI 1a	StDev [${\hat{Z}}_m$]	10.21%		3.24%
KPI 2a	Range [${\hat{Z}}_m$]	32.25%		9.25%
KPI 3a	Min [${\hat{Z}}_m$]	17.75%		24.75%
KPI 4a	Gini [${\hat{Z}}_m$]	0.19		0.06
KPI 5a	Avg [$\left|{\hat{\mathrm{\Delta }}}_m\right|$]	11.66%		24.47%
KPI 6a	Min [${\hat{\mathrm{\Delta }}}_m-$]	−10.00%		−4.00%
		P1	P2	P1	P2
KPI 1b	StDevP [$Z_{mp}$]	20.41%	0.00%	6.48%	0.00%
KPI 2b	RangeP [$Z_{mp}$]	64.50%	0.00%	18.50%	0.00%
KPI 3b	MinP [$Z_{mp}$]	25.50%	1.00%	49.50%	0.00%
KPI 4b	GiniP [$Z_{mp}$]	0.23	0.00	0.06	0.00
KPI 5b	AvgP [$\left|{\mathrm{\Delta }}_{mp}\right|$]	1.46%	0.18%	3.06%	0.38%
KPI 6b	MinP [${\mathrm{\Delta }}_{mp}-$]	0.00%	$-$10.50%	0.00%	$-$15.50%
KPI 7	(${\alpha }_{mp}$) Wavg [$Z_{mp}$]	29.86%		31.20%
KPI 8.1	(${\lambda }_m^1$) Wavg [${\hat{Z}}_m$]	31.82%		35.96%
KPI 9	(${\tau }_p$) Wavg [${\hat{Z}}_p$]	36.67%		40.00%

Figure 4. Coverage values with and without capacity restriction.

Five bar charts with coverage levels on the y axis and municipality - priority group pair on x axis that shows the fair coverage levels and coverage levels attained in with and without capacity restriction settings for CVAP and the alternative benchmark models.

Figure 5. Coverage levels of municipalities under different plans for the case study instance.

Four maps of Turkey and associated histogram graphs that shows the coverage density of municipalities, where lighter color indicates lower coverage levels and darker color indicates higher coverage levels.

Figure 6. Coverage values for each municipality and priority group for different plans for the case study.

Four different bar charts with coverage levels on the y axis and municipality numbers on the x axis, showing the fair coverage levels of municipalities and the cumulative coverage of each priority group within a municipality with different colors in a single bar.

Table 5. KPIs for different plans for the case study.

KPI No.	KPI	Plan 1			Plan 2			Plan 3			Plan 4
KPI 1a	StDev [${\hat{Z}}_m$]	4.5%			5.5%			21.3%			17.5%
KPI 2a	Range [${\hat{Z}}_m$]	35.6%			41.4%			90.0%			83.2%
KPI 3a	Min [${\hat{Z}}_m$]	41.2%			41.2%			10.0%			8.6%
KPI 4a	Gini [${\hat{Z}}_m$]	0.01			0.03			0.17			0.13
KPI 5a	Avg [$\left|{\hat{\mathrm{\Delta }}}_m\right|$]	0.9%			0.9%			0.1%			0.0%
KPI 6a	Min [${\hat{\mathrm{\Delta }}}_m-$]	$-$33.7%			$-$32.0%			$-$0.2%			$-$0.1%
		P1	P2	P3	P1	P2	P3	P1	P2	P3	P1	P2	P3
KPI 1b	StDevP [$Z_{mp}$]	0.0%	1.1%	9.0%	0.0%	5.2%	7.0%	21.3%	22.7%	19.8%	18.2%	19.9%	14.2%
KPI 2b	RangeP [$Z_{mp}$]	0.1%	10.0%	69.0%	0.1%	46.7%	44.3%	90.0%	90.0%	90.0%	91.7%	93.6%	65.0%
KPI 3b	MinP [$Z_{mp}$]	74.8%	64.9%	10.0%	99.9%	42.7%	10.0%	10.0%	10.0%	10.0%	8.3%	6.4%	10.0%
KPI 4b	GiniP [$Z_{mp}$]	0.00	0.00	0.03	0.00	0.01	0.03	0.13	0.17	0.20	0.09	0.13	0.16
KPI 5b	AvgP [$\left|{\mathrm{\Delta }}_{mp}\right|$]	0.0%	0.0%	0.0%	0.0%	0.0%	0.0%	0.0%	0.0%	0.0%	0.0%	0.0%	0.0%
KPI 6b	MinP [${\mathrm{\Delta }}_{mp}-$]	0.0%	$-$2.8%	$-$30.9%	0.0%	$-$13.0%	$-$19.0%	0.0%	$-$0.2%	$-$0.2%	0.0%	$-$0.1%	$-$0.1%
KPI 7	(${\alpha }_{mp}$) Wavg [$Z_{mp}$]	74.3%			87.6%			81.8%			83.1%
KPI 8.1	(${\lambda }_m^1$) Wavg [${\hat{Z}}_m$]	74.9%			77.0%			83.2%			80.1%
KPI 8.2	(${\lambda }_m^2$) Wavg [${\hat{Z}}_m$]	73.6%			75.5%			66.0%			68.2%
KPI 9	(${\tau }_p$) Wavg [${\hat{Z}}_p$]	74.9%			93.5%			82.1%			87.2%

Figure A1. Worldwide vaccination coverages by December 9, 2021 (Ritchie et al. 2020).

A bar chart with number of countries on the y axis and coverage level intervals on the x axis, showing the number of countries that are fully and partly vaccinated for each coverage interval.

Table B1. Small instance parameters.

Municipality	Demand (Susceptible)	Infected	Infected/ Population	${\alpha }_{m1}$ (Plan 1)	${\alpha }_{m1}$ (Plan 2&3)	$f_{m1}$ (Plan 1)	$f_{m1}$ (Plan 2&3)
M1	100	12	10.71%	0.125	0.187	60	90
M2	100	6	5.66%	0.125	0.125	60	60
M3	100	3	5.66%	0.125	0.125	60	60
M4	100	3	2.91%	0.125	0.063	60	30
M5	400	48	10.71%	0.125	0.187	240	360
M6	400	24	5.66%	0.125	0.125	240	240
M7	400	24	5.66%	0.125	0.125	240	240
M8	400	12	2.91%	0.125	0.063	240	120

Table B2. Parameters for small instance with priority groups.

Municipality	Priority Group	Demand	${\tau }_p$	${\lambda }_m^1$	${\alpha }_{mp}$	$f_{mp}$	Initial $g_{mp}$	Adjusted $g_{mp}$
1	1	50	1	1.00	0.09	22	90.0%	68.0%
	2	50	0.5	1.00	0.06	15	10.0%	0.0%
2	1	50	1	0.67	0.07	18	90.0%	62.0%
	2	50	0.5	0.67	0.05	12	10.0%	0.0%
3	1	50	1	0.67	0.07	18	90.0%	62.0%
	2	50	0.5	0.67	0.05	12	10.0%	0.0%
4	1	50	1	0.33	0.05	13	90.0%	50.0%
	2	50	0.5	0.33	0.04	8	10.0%	0.0%
5	1	200	1	1.00	0.09	88	90.0%	67.0%
	2	200	0.5	1.00	0.06	62	10.0%	0.0%
6	1	200	1	0.67	0.07	72	90.0%	61.5%
	2	200	0.5	0.67	0.05	51	10.0%	0.0%
7	1	200	1	0.67	0.07	72	90.0%	61.5%
	2	200	0.5	0.67	0.05	51	10.0%	0.0%
8	1	200	1	0.33	0.05	51	90.0%	49.5%
	2	200	0.5	0.33	0.04	35	10.0%	0.0%

Figure A2. The framework for calculation of the composite prioritization weight ${\alpha }_{mp}$.

A scheme that represents the different components included in the composite prioritization weight emerging through arrows, and the equation that is used to calculate the composite prioritization weight on top of the arrow.

Table C1. Summary of relevant vaccine allocation studies.

		Allocations among:		Problem setting
Article	Context	Regions	Priority groups	Vaccine types	Vial Sizes	Dose regimen	Regional capacity differences	Equity concern	Equity metric	Methodology	Objective
Teytelman and Larson (2013)	Influenza	✓	x	One type	x	Single dose	x	✓	• regional lower bounds for vaccine allocations	Disease spread model	Maximize the infections averted by a single vaccine
Yarmand et al. (2014)	Influenza	✓	x	One type	x	Single dose	x	✓	• regional lower bounds for vaccine allocations	Stochastic optimisation model + Disease spread model	Minimize the total vaccination cost
Huang et al. (2017)	Influenza	✓	✓	Multiple types	✓	Single dose	x	✓	• (minimize) location, priority group wise weighted unsatisfied demand • ensure similar allocation proportions across regions	Mathematical optimisation models (Primary-secondary)	Primary model (proportional fairness) Secondary model (policy simplicity + geographic equity)
Enayati and Özaltın (2020)	Influenza	✓	✓	One type	x	Single dose	x	✓	• Gini coefficient (coverage equity among subpopulations)	Disease spread model + Mathematical optimisation models	Minimize the amount of vaccines distributed to contain the outbreak
Tavana et al. (2021)	COVID-19	✓	✓	Multiple types	x	Single/multiple dose	✓	✓	• maximise the minimum coverage for each priority group and region • set a lower coverage thresheold level for each priority group in each region	Mathematical optimisation model	Maximize the minimum delivery-to-demand ratio
Munguía-López and Ponce-Ortega (2021)	COVID-19	✓	✓	One type	x	Single dose	x	✓	• regional lower bounds for vaccine allocations • maximise the lower bound (Rawlsian justice scheme)	Mathematical optimisation models	Different objectives analysed (social welfare, Rawlsian justice scheme, Nash scheme)
Jadidi et al. (2021)	COVID-19	x	✓	One type	x	Single dose	x	x		Mathematical optimisation model +Disease spread model	Minimize the total number of infections
Jarumaneeroj et al. (2022)	COVID-19	✓	✓	Multiple types	x	Single dose	✓	✓	• regional lower bounds for vaccine allocations	Mathematical optimisation model + Disease spread model	Minimize the total weighted strain on the healthcare system
Anahideh, Kang, and Nezami (2022)	COVID-19	✓	✓	One type	x	Single dose	x	✓	• minimise the allocation differences that are made based on group attributes and regardless of group attributes	Multi-objective optimisation model	Provide social fairness + geographic equity
Goodarzian et al. (2022)	COVID-19	✓	✓	Multiple types	x	Single dose	✓	✓	• minimise the maximum unmet demand	Multi-objective optimisation model	Minimize the total vaccination cost, maximum unmet demand, delivery time of vaccines and environmental impacts
Fadaki et al. (2022)	COVID-19	✓	✓	One type	✓	Single dose	✓	✓	• minimise the weighted risk of unvaccinated population	Multi-objective optimisation model	Minimize the weighted risk of unvaccinated population considering the direct shipment of vaccines from distribution centres to vaccination sites, transshipment among sites, the return of remaining vaccines to the distribution centres
Walker et al. (2022)	COVID-19	x	✓	One type	x	Multiple dose	x	x		Disease spread model	Observe the number of deaths, infections, hospitalisations averted according to different allocation schemes
Yang et al. (2022)	COVID-19	✓	x	One type	x	Multiple dose	x	x		Disease spread model + Particle swarm optimisation algorithm	Minimize the total number of confirmed cases
Our Study	COVID-19	✓	✓	Multiple types	✓	Multiple dose	✓	✓	• propose a fair coverage level which is based relative importance of regions and priority groups • set a lower coverage threshold level for each priority group in each region	Integer Programming	Minimize the weighted sum of deviations from fair coverage levels (primary objective)

Table D1. Example $f_{mp}$ calculation.

Population group	$d_{mp}$	${\alpha }_{mp}$	$f_{mp}$ by (14(14) $\begin{array}{rl}{f}_{mp}& =\{\begin{array}{ll}{\displaystyle \frac{\sum _{k=1}^K{s}_k{d}_{mp}{\alpha }_{mp}}{\begin{array}{c}\sum _{m=1}^M\sum _{p=1}^P\\ d_{mp}{\alpha }_{mp}\end{array}}},& \mathrm{i}\mathrm{f}{\displaystyle \frac{\sum _{k=1}^K{s}_k{d}_{mp}{\alpha }_{mp}}{\begin{array}{c}\sum _{m=1}^M\sum _{p=1}^P\\ d_{mp}{\alpha }_{mp}\end{array}}}\le d_{mp}\\ d_{mp},& \mathrm{o}\mathrm{t}\mathrm{h}\mathrm{e}\mathrm{r}\mathrm{w}\mathrm{i}\mathrm{s}\mathrm{e}\end{array}\\ & \mathrm{\forall }m,p\end{array}$(14) )	Fair coverage	Final $f_{mp}$ by Algorithm 1	Fair coverage
M1P1	100	0.4	100	100.0%	100	100.0%
M2P1	50	0.4	50	100.0%	50	100.0%
M1P2	100	0.1	26	26.0%	34	34.0%
M2P2	50	0.1	13	26.0%	16	32.0%

Table

Algorithm 1:	Iterative Calculation of the Fair Amounts
Step 0.	Initialization of parameters:
	Set the fair amount, composite prioritization weights, and remaining demand for each priority group in each municipality as $f_{mp}$:=0; ${\overline{\alpha }}_{mp}$:=${\alpha }_{mp}$; ${\overline{d}}_{mp}$:=$d_{mp}$; respectively
	Set the vaccine amount to distribute as $V:={\displaystyle \sum _{k=1}^K{s}_k}$; total weighted eligible demand as W:=0
	Set the additional allocation amount in iteration t for each priority group and municipality as $f_{mp}^t$:=0
Step 1.	If $V\ge 0$, go to Step 1.1. If not, end the algorithm.
Step 1.1.	Set $f_{mp}^t$:=0 for each priority group in each municipality.
Step 1.2.	Specify the eligible priority groups in municipalities that can receive vaccine at this iteration, where $f_{mp}<d_{mp}$.
Step 1.3.	Normalize the composite prioritization weights of the eligible priority groups in municipalities as ${\overline{\alpha }}_{mp}:={\overline{\alpha }}_{mp}/{\displaystyle \sum _{m=1}^M\sum _{p=1}^P{\overline{\alpha }}_{mp}}$.
Step 1.4.	Calculate $W:={\displaystyle \sum _{m=1}^M\sum _{p=1}^P{\overline{d}}_{mp}{\overline{\alpha }}_{mp}}$.
Step 1.5.	Calculate $f_{mp}^t:=\lfloor ({\overline{d}}_{mp}\ast {\overline{\alpha }}_{mp}\ast V)/W\rfloor$ for the eligible priority groups in municipalities.
Step 1.6.	Set $f_{mp}^t:=min\{d_{mp},f_{mp}^t\}$ for the eligible priority groups in municipalities.
Step 1.7.	Update $f_{mp}:=f_{mp}+f_{mp}^t$ for the eligible priority groups in municipalities.
Step 1.8.	Update ${\overline{d}}_{mp}:={\overline{d}}_{mp}-f_{mp}^t$ for the eligible priority groups in municipalities.
Step 2.	Update $V:=V-{\displaystyle \sum _{m=1}^M\sum _{p=1}^P{f}_{mp}^t}$.
Step 3.	Set t: = t + 1.
Step 4.	If $f_{mp}^t=0$ for all priority groups in all municipalities due to rounding, and $V\ge 0$, go to Step 4.1. If not, go to Step 1.
Step 4.1.	Starting from the highest prioritised group in the first municipality, choose the next priority group and municipality.
Step 4.2.	Update $f_{mp}:=f_{mp}+1$ for the specified priority group and municipality (to ensure the allocation amounts for the same priority group among municipalities are similar).
Step 5.	If $V\ge 0$, go to Step 4.1. If not, end the algorithm.

Ritchie, H., E. Mathieu, L. Rodés-Guirao, C. Appel, C. Giattino, E. Ortiz-Ospina, J. Hasell, et al. 2020. “Coronavirus (COVID-19) Vaccinations.” Accessed December 28, 2021. https://ourworldindata.org/covid-vaccinations.

 Google Scholar

Teytelman, A., and R. C. Larson. 2013. “Multiregional Dynamic Vaccine Allocation During An Influenza Epidemic.” Service Science 5 (3): 197–215.

 Web of Science ®Google Scholar

Yarmand, H., J. S. Ivy, B. Denton, and A. L. Lloyd. 2014. “Optimal Two-Phase Vaccine Allocation to Geographically Different Regions Under Uncertainty.” European Journal of Operational Research 233 (1): 208–219.

 Web of Science ®Google Scholar

Huang, H.-C., B. Singh, D. P. Morton, G. P. Johnson, B. Clements, and L. A. Meyers. 2017. “Equalizing Access to Pandemic Influenza Vaccines Through Optimal Allocation to Public Health Distribution Points.” PLoS ONE 12 (8): Article ID e0182720.

 Web of Science ®Google Scholar

Enayati, S., and O. Y. Özaltın. 2020. “Optimal Influenza Vaccine Distribution with Equity.” European Journal of Operational Research 283 (2): 714–725.

 Web of Science ®Google Scholar

Munguía-López, A. d. C., and J. M. Ponce-Ortega. 2021. “Fair Allocation of Potential COVID-19 Vaccines Using an Optimization-Based Strategy.” Process Integration and Optimization for Sustainability 5 (1): 3–12.

 Google Scholar

Jadidi, M., S. Jamshidiha, I. Masroori, P. Moslemi, A. Mohammadi, and V. Pourahmadi. 2021. “A Two-Step Vaccination Technique to Limit COVID-19 Spread Using Mobile Data.” Sustainable Cities and Society 70: Article ID 102886.

 PubMed Web of Science ®Google Scholar

Jarumaneeroj, P., P. O. Dusadeerungsikul, T. Chotivanich, T. Nopsopon, and K. Pongpirul. 2022. “An Epidemiology-Based Model for the Operational Allocation of COVID-19 Vaccines: A Case Study of Thailand.” Computers & Industrial Engineering 167: Article ID 108031.

 PubMed Web of Science ®Google Scholar

Anahideh, H., L. Kang, and N. Nezami. 2022. “Fair and Diverse Allocation of Scarce Resources.” Socio-Economic Planning Sciences 80: Article ID 101193.

 PubMed Web of Science ®Google Scholar

Goodarzian, F., A. Navaei, B. Ehsani, P. Ghasemi, and J. Muñuzuri. 2022. “Designing An Integrated Responsive-Green-Cold Vaccine Supply Chain Network Using Internet-of-Things: Artificial Intelligence-Based Solutions.” Annals of Operations Research doi:10.1007/s10479-022-04713-4

 Web of Science ®Google Scholar

Fadaki, M., A. Abareshi, S. M. Far, and P. T.-W. Lee. 2022. “Multi-Period Vaccine Allocation Model in a Pandemic: A Case Study of COVID-19 in Australia.” Transportation Research Part E: Logistics and Transportation Review 161: Article ID 102689.

 PubMed Web of Science ®Google Scholar

Walker, J., P. Paul, K. Dooling, S. Oliver, P. Prasad, M. Steele, P. A. Gastañaduy, et al. 2022. “Modeling Strategies for the Allocation of SARS-CoV-2 Vaccines in the United States.” Vaccine 40 (14): 2134–2139.

 PubMed Web of Science ®Google Scholar

Yang, T., W. Deng, Y. Liu, and J. Deng. 2022. “Comparison of Health-Oriented Cross-Regional Allocation Strategies for the COVID-19 Vaccine: A Mathematical Modelling Study.” Annals of Medicine 54 (1): 941–952.

 PubMed Web of Science ®Google Scholar

Data availability statement

The authors confirm that the data supporting the findings of this study are available within the article [and/or] its supplementary materials.