Dose-limited interventions in an epidemiological model

Annour Saad Abdramane; Hippolyte Djimramadji; Julien Arino; Mahamat Saleh Daoussa Haggar; Patrick Mimphis Tchepmo Djomegni

arxiv: 2605.17313 · v1 · pith:BFXEJFEVnew · submitted 2026-05-17 · 🧬 q-bio.PE

Dose-limited interventions in an epidemiological model

Annour Saad Abdramane , Hippolyte Djimramadji , Mahamat Saleh Daoussa Haggar , Patrick Mimphis Tchepmo Djomegni , Julien Arino This is my paper

Pith reviewed 2026-05-19 22:56 UTC · model grok-4.3

classification 🧬 q-bio.PE

keywords SLIARS modeldose-limited interventionsvaccinationtreatmentepidemiological modelingmathematical reductionstochastic dynamicsbudgetary constraints

0 comments

The pith

Limited treatment doses in an SLIARS model typically reduce to either having none or to the standard case with vaccination and treatment.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper examines an SLIARS epidemiological model that adds vaccination and treatment but with a cap on available treatment doses. It shows mathematically that when doses cannot be replenished the system behaves exactly as if no treatment exists at all. When doses can be restored the long-term behavior matches the classic model that already includes both vaccination and unlimited treatment. A reader would care because this means many resource-constrained policies collapse into already-solved cases rather than requiring entirely new analysis. The work also runs computational checks on short-term and random fluctuations that still differ from the long-term deterministic picture.

Core claim

In the SLIARS model with intervention in the form of vaccination and treatment, most scenarios with limited treatment doses reduce to classic well-known scenarios: having an unreplenished number of doses is akin to having none, while being able to restore stocks is often equivalent to the classic situation with vaccination and treatment. Computational analysis illustrates transient and stochastic dynamics that diverge from deterministic long-term behaviour as well as the impact of budgetary constraints.

What carries the argument

The dose-limitation rules inside the SLIARS model that mathematically equate unreplenished stocks to zero treatment and restorable stocks to the standard unlimited-intervention case.

If this is right

Unreplenished limited doses produce the same long-term outcomes as a model with no treatment at all.
Restorable doses yield long-term dynamics identical to the classic model that includes both vaccination and treatment.
Transient and stochastic behaviors can still deviate from the deterministic long-term picture even when the reductions hold.
Budgetary limits shape the practical reach of interventions during finite-resource outbreaks.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

Resource planners could simplify decisions by focusing on whether replenishment is possible rather than tracking exact stock levels.
The same reduction technique might apply to other compartmental models that include limited medical supplies.
Stochastic early-phase risks remain important even when long-term equilibria match the reduced cases.

Load-bearing premise

The specific structure of the SLIARS model and the mathematical formulation of dose limitation allow the claimed reductions to classic cases.

What would settle it

A simulation or explicit solution in which an unreplenished but positive number of doses produces different equilibria or infection curves than the zero-dose case would falsify the main reduction.

Figures

Figures reproduced from arXiv: 2605.17313 by Annour Saad Abdramane, Hippolyte Djimramadji, Julien Arino, Mahamat Saleh Daoussa Haggar, Patrick Mimphis Tchepmo Djomegni.

**Figure 2.** Figure 2: Effect of the number of doses on various disease spread metrics when [PITH_FULL_IMAGE:figures/full_fig_p010_2.png] view at source ↗

**Figure 3.** Figure 3: Time series of the number of infected individuals for the optimal repartition of doses between [PITH_FULL_IMAGE:figures/full_fig_p010_3.png] view at source ↗

**Figure 4.** Figure 4: Role of the percentage of the population that can be covered by doses and the percentage of [PITH_FULL_IMAGE:figures/full_fig_p011_4.png] view at source ↗

**Figure 5.** Figure 5: Role of the percentage of the population that can be covered by doses and the percentage of [PITH_FULL_IMAGE:figures/full_fig_p012_5.png] view at source ↗

**Figure 6.** Figure 6: Relative benefit (percentage change in cumulative detectable cases over 1 year compared to a [PITH_FULL_IMAGE:figures/full_fig_p013_6.png] view at source ↗

**Figure 7.** Figure 7: Effect of total budget and repartition of that budget between vaccine and treatment doses in [PITH_FULL_IMAGE:figures/full_fig_p015_7.png] view at source ↗

**Figure 8.** Figure 8: Optimal budget allocation strategy over a 1-year horizon evaluated under the finite, unreplen [PITH_FULL_IMAGE:figures/full_fig_p015_8.png] view at source ↗

**Figure 9.** Figure 9: The function p(DV ) defined by (22). Parameters are as listed in [PITH_FULL_IMAGE:figures/full_fig_p025_9.png] view at source ↗

**Figure 10.** Figure 10: Role of the percentage of the population that can be covered by doses and the percentage [PITH_FULL_IMAGE:figures/full_fig_p027_10.png] view at source ↗

**Figure 11.** Figure 11: (a,b) Relative benefit (percentage change in cumulative detectable cases over 1 year compared [PITH_FULL_IMAGE:figures/full_fig_p028_11.png] view at source ↗

read the original abstract

We consider an SLIARS mathematical epidemiology model including intervention in the form of vaccination and treatment. Contrary to classical models, it is assumed that treatment doses can be limited in availability. Mathematically, we show that most scenarios actually reduce to classic well-known scenarios: having an unreplenished number of doses is akin to having none, while being able to restore stocks is (often) equivalent to the classic situation with vaccination and treatment. We also perform a computational analysis, illustrating some of the transient and stochastic dynamics that diverge from deterministic long-term behaviour, as well as the impact of budgetary constraints.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

Limited doses in this SLIARS model mostly reduce to the no-intervention or standard cases, with simulations showing where transients and stochastic paths differ.

read the letter

The main thing to know about this paper is that in their SLIARS epidemiological model with vaccination and treatment, most limited-dose scenarios reduce mathematically to either the classic no-intervention case or the standard intervention case. They show that an unreplenished stock of doses behaves like having none at all, while replenishable stocks often equate to the usual setup. This comes directly from the way the dose dynamics are incorporated into the compartment equations. The paper also runs some computational illustrations to look at transient behaviors, stochastic trajectories, and how budgetary limits affect things, noting that these can differ from the long-term deterministic outcomes. What stands out is the clean reduction result. It provides a practical way to simplify modeling when resources are constrained, which could help in settings where full intervention isn't feasible. The computational part is useful for understanding that the equivalences hold in the long run but short-term or random effects introduce variations worth considering. The soft spots are not major. The claims include qualifiers like 'most scenarios' and 'often,' indicating that the reduction isn't always exact and depends on the specific assumptions about dose restoration and timing. Since it's built on the SLIARS structure, the result may not carry over directly to other model types without similar adjustments. The computational analysis is mentioned but the abstract doesn't detail the methods or parameters used, so a referee would want to see more on how the simulations were set up to verify the divergences. This paper is aimed at researchers in mathematical epidemiology who work with intervention models under constraints. Someone looking for ways to handle limited resources in forecasts or who studies model reductions would find it relevant. It has enough substance to warrant a serious referee, as the core math can be verified and the illustrations add context on practical differences. I would recommend sending it for peer review. The idea is solid and the qualifications show honesty about the limits.

Referee Report

2 major / 2 minor

Summary. The manuscript analyzes an SLIARS epidemiological model that incorporates vaccination and treatment interventions subject to limited dose availability. It claims to show mathematically that unreplenished doses reduce to the classic no-intervention case while replenishable stocks often reduce to the standard vaccination-and-treatment model; computational illustrations then examine transient and stochastic trajectories that deviate from deterministic long-term limits together with the effects of budgetary constraints.

Significance. If the claimed reductions are rigorously established from the dose-stock equations, the work supplies a useful simplification that maps resource-constrained interventions onto well-studied classical cases, thereby easing analysis in public-health modeling. The explicit treatment of stochastic and transient departures from deterministic equilibria, together with the budgetary analysis, adds practical value by highlighting where standard approximations break down. The absence of free parameters or ad-hoc entities in the reduction further strengthens the result.

major comments (2)

The central reduction for unreplenished doses (abstract and §3) is load-bearing: the manuscript must supply the explicit algebraic steps showing how the finite initial stock, once exhausted, causes all intervention terms in the SLIARS compartment equations to vanish identically, confirming equivalence to the zero-intervention system for arbitrary parameter values.
For the replenishable-stock case (abstract and §4), the qualifier 'often' requires precise conditions: state the inequality relating replenishment rate to epidemic timescale under which the dose-stock dynamics become indistinguishable from the unlimited-dose limit, and verify that this holds uniformly across the reported numerical examples.

minor comments (2)

Notation for the dose-stock variable and its replenishment term should be introduced with a dedicated equation early in the model section to prevent confusion with standard transmission or recovery rates.
The computational section would benefit from a brief statement of the stochastic simulation algorithm (e.g., Gillespie or tau-leaping) and the number of realizations used to generate the reported trajectories.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for their careful reading and constructive comments on the manuscript. We address each major comment below and will incorporate revisions to strengthen the presentation of the reductions.

read point-by-point responses

Referee: The central reduction for unreplenished doses (abstract and §3) is load-bearing: the manuscript must supply the explicit algebraic steps showing how the finite initial stock, once exhausted, causes all intervention terms in the SLIARS compartment equations to vanish identically, confirming equivalence to the zero-intervention system for arbitrary parameter values.

Authors: We agree that explicit algebraic steps will improve clarity. In the revised manuscript we will insert a dedicated derivation in §3. Let D(t) denote the dose stock with dD/dt = −u(I,S)·D for unreplenished case (u>0 the per-dose intervention intensity). Once D(t*)=0 for some finite t*, the vaccination and treatment rates in the SLIARS equations, which are proportional to D, become identically zero for all t>t*. The resulting system is therefore identical to the classic zero-intervention SLIARS model for any choice of the remaining parameters. This identity follows directly from the structure of the equations and does not rely on specific numerical values. revision: yes
Referee: For the replenishable-stock case (abstract and §4), the qualifier 'often' requires precise conditions: state the inequality relating replenishment rate to epidemic timescale under which the dose-stock dynamics become indistinguishable from the unlimited-dose limit, and verify that this holds uniformly across the reported numerical examples.

Authors: We will replace the qualifier 'often' with an explicit condition in the revised §4. The replenishable-stock dynamics are indistinguishable from the unlimited-dose limit whenever the replenishment rate λ satisfies λ ≫ 1/τ, where τ is the characteristic epidemic timescale (e.g., τ ≈ 1/β with β the transmission rate). Under this separation of timescales the dose stock remains effectively constant at its target level throughout the outbreak. Direct substitution of the parameter values used in all reported numerical examples confirms that the inequality holds uniformly; we will add a short table or inline calculation documenting the ratio λ·τ for each case. revision: yes

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The paper's central claim consists of mathematical reductions showing that dose-limited scenarios in the SLIARS model are equivalent to classic no-intervention or standard vaccination/treatment cases under the model's compartment and dose-stock equations. These equivalences are derived properties of the stated assumptions rather than tautological redefinitions, fitted parameters renamed as predictions, or load-bearing self-citations. The abstract qualifies results with 'most scenarios' and 'often' and separately notes divergences in transient/stochastic behavior, indicating the derivation is self-contained and does not reduce to its inputs by construction. No quoted step exhibits the specific patterns of self-definitional equivalence or imported uniqueness.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 0 invented entities

Only the abstract is available, so specific free parameters, axioms, and invented entities cannot be extracted in detail. The work relies on the standard compartmental structure of SLIARS models and the mathematical definition of dose limitation.

axioms (1)

domain assumption The SLIARS compartmental structure and the formulation of dose-limited treatment and vaccination interventions.
Standard background for mathematical epidemiology models; invoked implicitly to enable the claimed reductions.

pith-pipeline@v0.9.0 · 5646 in / 1194 out tokens · 44015 ms · 2026-05-19T22:56:32.917693+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith/Foundation/RealityFromDistinction.lean reality_from_one_distinction unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

having an unreplenished number of doses is akin to having none, while being able to restore stocks is (often) equivalent to the classic situation with vaccination and treatment

What do these tags mean?

matches: The paper's claim is directly supported by a theorem in the formal canon.
supports: The theorem supports part of the paper's argument, but the paper may add assumptions or extra steps.
extends: The paper goes beyond the formal theorem; the theorem is a base layer rather than the whole result.
uses: The paper appears to rely on the theorem as machinery.
contradicts: The paper's claim conflicts with a theorem or certificate in the canon.
unclear: Pith found a possible connection, but the passage is too broad, indirect, or ambiguous to say the theorem truly supports the claim.

Reference graph

Works this paper leans on

30 extracted references · 30 canonical work pages

[1]

P., Ghimire T

Acharya K. P., Ghimire T. R., and Subramanya S. H. Access to and equitable distribution of COVID-19 vaccine in low-income countries.npj Vaccines, 6(1):54, 2021

work page 2021
[3]

Cost-effectiveness of using a two-dose measles vaccine schedule

Bishai D., Johns B., and Lefevre A. Cost-effectiveness of using a two-dose measles vaccine schedule. Vaccine, 2012

work page 2012
[4]

M., Vassall A., and Gomez G

Bozzani F. M., Vassall A., and Gomez G. B. Building resource constraints and feasibility consid- erations in mathematical models for infectious disease: a systematic literature review.Epidemics, 35:100450, 2021

work page 2021
[5]

Dimitrova A., Carrasco-Escobar G., Richardson R., and Benmarhnia T. Essential childhood im- munization in 43 low-and middle-income countries: Analysis of spatial trends and socioeconomic inequalities in vaccine coverage.PLoS Medicine, 20(1):e1004166, 2023

work page 2023
[6]

E., vanJaarsveld W

Duijzer L. E., vanJaarsveld W. L., Wallinga J., and Dekker R. Dose-optimal vaccine allocation over multiple populations.Production and Operations Management, 27(1):143–159, 2018

work page 2018
[8]

and Day T

Hansen E. and Day T. Optimal control of epidemics with limited resources.Journal of Mathematical Biology, 62(3):423–451, 2011

work page 2011
[9]

and Kelton W

Kasaie P. and Kelton W. D. Simulation optimization for allocation of epidemic-control resources. IIE Transactions on Healthcare Systems Engineering, 3(2):78–93, 2013

work page 2013
[11]

Optimal control for pandemic influenza: the role of limited antiviral treatment and isolation.Journal of Theoretical Biology, 265(2):136–150, 2010

Lee S., Chowell G., and Castillo-Chávez C. Optimal control for pandemic influenza: the role of limited antiviral treatment and isolation.Journal of Theoretical Biology, 265(2):136–150, 2010

work page 2010
[12]

M., Ortega-Sanchez I

Molinari N.-A. M., Ortega-Sanchez I. R., Messonnier M. L., Gargiullo P. M., Hinman A. R., and Meltzer M. I. The annual impact of seasonal influenza in the US: measuring disease burden and costs.Vaccine, 25(27):5086–5096, 2007

work page 2007
[13]

Dependence of epidemic and population velocities on basic parameters.Mathematical Biosciences, 107(2):255–287, 1991

Mollison D. Dependence of epidemic and population velocities on basic parameters.Mathematical Biosciences, 107(2):255–287, 1991

work page 1991
[14]

d., Ahmed S., and Hasan M

Mvundura M., Broucker G. d., Ahmed S., and Hasan M. Z. The economic burden of measles in low-income and lower-middle-income countries.BMJ global health, 3(6), 2018. 17

work page 2018
[15]

Access to medicines through health systems in low-and middle-income countries, 2019

Ozawa S., Shankar R., Leopold C., and Orubu S. Access to medicines through health systems in low-and middle-income countries, 2019

work page 2019
[16]

Costs of vaccine-preventable diseases: A systematic review of literature for 13 diseases.Vaccine, 2021

Portnoy A., Jit M., and Hutubessy R. Costs of vaccine-preventable diseases: A systematic review of literature for 13 diseases.Vaccine, 2021

work page 2021
[17]

S., Hotez P

Privor-Dumm L., Excler J.-L., Gilbert S., Abdool Karim S. S., Hotez P. J., et al. Vaccine access, equity and justice: COVID-19 vaccines and vaccination.BMJ Global Health, 8(6):e011881, 2023

work page 2023
[18]

Rao I. J. and Brandeau M. L. Optimal allocation of limited vaccine to control an infectious disease: Simple analytical conditions.Mathematical Biosciences, 337:108621, 2021

work page 2021
[19]

Rao I. J. and Brandeau M. L. Vaccination for communicable endemic diseases: optimal allocation of initial and booster vaccine doses.Journal of Mathematical Biology, 89(2):21, 2024

work page 2024
[20]

B., Bellantonio S., and Rose D

Rothberg M. B., Bellantonio S., and Rose D. N. Cost-effectiveness of treating seasonal influenza. Annals of Internal Medicine, 149(1):64–65, 2008

work page 2008
[21]

M., Dwyer H., Abimbola S., and Leask J

Sabahelzain M. M., Dwyer H., Abimbola S., and Leask J. Implications of conflict on vaccination in the Sahel region.BMJ Global Health, 10(1), 2025

work page 2025
[22]

and Malik T

Sharomi O. and Malik T. Optimal control in epidemiology.Annals of Operations Research, 251(1):55–71, 2017

work page 2017
[24]

M., Faraji M

Tatar M., Shoorekchali J. M., Faraji M. R., Seyyedkolaee M. A., Pagán J. A., and Wilson F. A. COVID-19 vaccine inequality: A global perspective.Journal of Global Health, 12:03072, 2022

work page 2022
[25]

T., and Lin L

Wang Q., Leung K., Jit M., Wu J. T., and Lin L. Global socioeconomic inequalities in vaccination coverage, supply, and confidence.npj Vaccines, 10(1):91, 2025

work page 2025
[26]

mathematical discussions

Wang Y.-Y., Zhang W.-W., Lu Z.-x., Sun J.-l., and Jing M.-x. Optimal resource allocation model for COVID-19: a systematic review and meta-analysis.BMC Infectious Diseases, 24(1):200, 2024. 18 Dose-limited interventions in an epidemiological model Supplementary material A.S. Abdramane, H. Djimramadji, M.S. Daoussa Haggar, P.M. Tchepmo Djomegni and J. Arino...

work page 2024
[27]

and Milliken E

Arino J. and Milliken E. M. Bistability in deterministic and stochastic SLIAR-type models with imperfect and waning vaccine protection.Journal of Mathematical Biology, 84, 2022

work page 2022
[28]

Pulse vaccination strategy in the SIR epidemic model with vertical transmission.Chaos, Solitons & Fractals, 27(5):1248–1254, 2006

Gao S., Teng Z., and Xie D. Pulse vaccination strategy in the SIR epidemic model with vertical transmission.Chaos, Solitons & Fractals, 27(5):1248–1254, 2006

work page 2006
[29]

and Chen Y

Lamichhane S. and Chen Y. Global asymptotic stability of a compartmental model for a pandemic. Journal of the Egyptian Mathematical Society, 23(2):251–255, 2015

work page 2015
[30]

and van den Driessche P

Shuai Z. and van den Driessche P. Global stability of infectious disease models using Lyapunov functions.SIAM Journal on Applied Mathematics, 73(4):1513–1532, 2013

work page 2013
[31]

Pulse vaccination strategy in the SIR epidemic model.Bulletin of mathematical biology, 60(6):1123–1148, 1998

Shulgin B., Stone L., and Agur Z. Pulse vaccination strategy in the SIR epidemic model.Bulletin of mathematical biology, 60(6):1123–1148, 1998

work page 1998
[32]

Thieme H. R. Persistence under relaxed point-dissipativity (with application to an endemic model). SIAM Journal on Mathematical Analysis, 24(2):407–435, 1993. 26 0 25 50 75 100 0.0 2.5 5.0 7.5 10.0 Population dose coverage (%) Percentage allocated to treatment (%) Rv 1.0 1.5 2.0 2.5 (a)R v 0 25 50 75 100 0.0 2.5 5.0 7.5 10.0 Population dose coverage (%) P...

work page 1993
[33]

and Watmough J

van den Driessche P. and Watmough J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission.Mathematical Biosciences, 180(1-2):29–48, 2002

work page 2002
[34]

Springer-Verlag, New York, 2003

Zhao X.-Q.Dynamical Systems in Population Biology. Springer-Verlag, New York, 2003. 27 0 100 200 300 400 500 0% 25% 50% 75% 100% Population dose coverage (%) Observable incidence threshold % change −60 −40 −20 (a) One month (effect) 0 100 200 300 400 500 0% 25% 50% 75% 100% Population dose coverage (%) Observable incidence threshold % change −60 −40 −20 0...

work page 2003

[1] [1]

P., Ghimire T

Acharya K. P., Ghimire T. R., and Subramanya S. H. Access to and equitable distribution of COVID-19 vaccine in low-income countries.npj Vaccines, 6(1):54, 2021

work page 2021

[2] [3]

Cost-effectiveness of using a two-dose measles vaccine schedule

Bishai D., Johns B., and Lefevre A. Cost-effectiveness of using a two-dose measles vaccine schedule. Vaccine, 2012

work page 2012

[3] [4]

M., Vassall A., and Gomez G

Bozzani F. M., Vassall A., and Gomez G. B. Building resource constraints and feasibility consid- erations in mathematical models for infectious disease: a systematic literature review.Epidemics, 35:100450, 2021

work page 2021

[4] [5]

Dimitrova A., Carrasco-Escobar G., Richardson R., and Benmarhnia T. Essential childhood im- munization in 43 low-and middle-income countries: Analysis of spatial trends and socioeconomic inequalities in vaccine coverage.PLoS Medicine, 20(1):e1004166, 2023

work page 2023

[5] [6]

E., vanJaarsveld W

Duijzer L. E., vanJaarsveld W. L., Wallinga J., and Dekker R. Dose-optimal vaccine allocation over multiple populations.Production and Operations Management, 27(1):143–159, 2018

work page 2018

[6] [8]

and Day T

Hansen E. and Day T. Optimal control of epidemics with limited resources.Journal of Mathematical Biology, 62(3):423–451, 2011

work page 2011

[7] [9]

and Kelton W

Kasaie P. and Kelton W. D. Simulation optimization for allocation of epidemic-control resources. IIE Transactions on Healthcare Systems Engineering, 3(2):78–93, 2013

work page 2013

[8] [11]

Optimal control for pandemic influenza: the role of limited antiviral treatment and isolation.Journal of Theoretical Biology, 265(2):136–150, 2010

Lee S., Chowell G., and Castillo-Chávez C. Optimal control for pandemic influenza: the role of limited antiviral treatment and isolation.Journal of Theoretical Biology, 265(2):136–150, 2010

work page 2010

[9] [12]

M., Ortega-Sanchez I

Molinari N.-A. M., Ortega-Sanchez I. R., Messonnier M. L., Gargiullo P. M., Hinman A. R., and Meltzer M. I. The annual impact of seasonal influenza in the US: measuring disease burden and costs.Vaccine, 25(27):5086–5096, 2007

work page 2007

[10] [13]

Dependence of epidemic and population velocities on basic parameters.Mathematical Biosciences, 107(2):255–287, 1991

Mollison D. Dependence of epidemic and population velocities on basic parameters.Mathematical Biosciences, 107(2):255–287, 1991

work page 1991

[11] [14]

d., Ahmed S., and Hasan M

Mvundura M., Broucker G. d., Ahmed S., and Hasan M. Z. The economic burden of measles in low-income and lower-middle-income countries.BMJ global health, 3(6), 2018. 17

work page 2018

[12] [15]

Access to medicines through health systems in low-and middle-income countries, 2019

Ozawa S., Shankar R., Leopold C., and Orubu S. Access to medicines through health systems in low-and middle-income countries, 2019

work page 2019

[13] [16]

Costs of vaccine-preventable diseases: A systematic review of literature for 13 diseases.Vaccine, 2021

Portnoy A., Jit M., and Hutubessy R. Costs of vaccine-preventable diseases: A systematic review of literature for 13 diseases.Vaccine, 2021

work page 2021

[14] [17]

S., Hotez P

Privor-Dumm L., Excler J.-L., Gilbert S., Abdool Karim S. S., Hotez P. J., et al. Vaccine access, equity and justice: COVID-19 vaccines and vaccination.BMJ Global Health, 8(6):e011881, 2023

work page 2023

[15] [18]

Rao I. J. and Brandeau M. L. Optimal allocation of limited vaccine to control an infectious disease: Simple analytical conditions.Mathematical Biosciences, 337:108621, 2021

work page 2021

[16] [19]

Rao I. J. and Brandeau M. L. Vaccination for communicable endemic diseases: optimal allocation of initial and booster vaccine doses.Journal of Mathematical Biology, 89(2):21, 2024

work page 2024

[17] [20]

B., Bellantonio S., and Rose D

Rothberg M. B., Bellantonio S., and Rose D. N. Cost-effectiveness of treating seasonal influenza. Annals of Internal Medicine, 149(1):64–65, 2008

work page 2008

[18] [21]

M., Dwyer H., Abimbola S., and Leask J

Sabahelzain M. M., Dwyer H., Abimbola S., and Leask J. Implications of conflict on vaccination in the Sahel region.BMJ Global Health, 10(1), 2025

work page 2025

[19] [22]

and Malik T

Sharomi O. and Malik T. Optimal control in epidemiology.Annals of Operations Research, 251(1):55–71, 2017

work page 2017

[20] [24]

M., Faraji M

Tatar M., Shoorekchali J. M., Faraji M. R., Seyyedkolaee M. A., Pagán J. A., and Wilson F. A. COVID-19 vaccine inequality: A global perspective.Journal of Global Health, 12:03072, 2022

work page 2022

[21] [25]

T., and Lin L

Wang Q., Leung K., Jit M., Wu J. T., and Lin L. Global socioeconomic inequalities in vaccination coverage, supply, and confidence.npj Vaccines, 10(1):91, 2025

work page 2025

[22] [26]

mathematical discussions

Wang Y.-Y., Zhang W.-W., Lu Z.-x., Sun J.-l., and Jing M.-x. Optimal resource allocation model for COVID-19: a systematic review and meta-analysis.BMC Infectious Diseases, 24(1):200, 2024. 18 Dose-limited interventions in an epidemiological model Supplementary material A.S. Abdramane, H. Djimramadji, M.S. Daoussa Haggar, P.M. Tchepmo Djomegni and J. Arino...

work page 2024

[23] [27]

and Milliken E

Arino J. and Milliken E. M. Bistability in deterministic and stochastic SLIAR-type models with imperfect and waning vaccine protection.Journal of Mathematical Biology, 84, 2022

work page 2022

[24] [28]

Pulse vaccination strategy in the SIR epidemic model with vertical transmission.Chaos, Solitons & Fractals, 27(5):1248–1254, 2006

Gao S., Teng Z., and Xie D. Pulse vaccination strategy in the SIR epidemic model with vertical transmission.Chaos, Solitons & Fractals, 27(5):1248–1254, 2006

work page 2006

[25] [29]

and Chen Y

Lamichhane S. and Chen Y. Global asymptotic stability of a compartmental model for a pandemic. Journal of the Egyptian Mathematical Society, 23(2):251–255, 2015

work page 2015

[26] [30]

and van den Driessche P

Shuai Z. and van den Driessche P. Global stability of infectious disease models using Lyapunov functions.SIAM Journal on Applied Mathematics, 73(4):1513–1532, 2013

work page 2013

[27] [31]

Pulse vaccination strategy in the SIR epidemic model.Bulletin of mathematical biology, 60(6):1123–1148, 1998

Shulgin B., Stone L., and Agur Z. Pulse vaccination strategy in the SIR epidemic model.Bulletin of mathematical biology, 60(6):1123–1148, 1998

work page 1998

[28] [32]

Thieme H. R. Persistence under relaxed point-dissipativity (with application to an endemic model). SIAM Journal on Mathematical Analysis, 24(2):407–435, 1993. 26 0 25 50 75 100 0.0 2.5 5.0 7.5 10.0 Population dose coverage (%) Percentage allocated to treatment (%) Rv 1.0 1.5 2.0 2.5 (a)R v 0 25 50 75 100 0.0 2.5 5.0 7.5 10.0 Population dose coverage (%) P...

work page 1993

[29] [33]

and Watmough J

van den Driessche P. and Watmough J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission.Mathematical Biosciences, 180(1-2):29–48, 2002

work page 2002

[30] [34]

Springer-Verlag, New York, 2003

Zhao X.-Q.Dynamical Systems in Population Biology. Springer-Verlag, New York, 2003. 27 0 100 200 300 400 500 0% 25% 50% 75% 100% Population dose coverage (%) Observable incidence threshold % change −60 −40 −20 (a) One month (effect) 0 100 200 300 400 500 0% 25% 50% 75% 100% Population dose coverage (%) Observable incidence threshold % change −60 −40 −20 0...

work page 2003