Effectivity of the vaccination strategy for a fractional-order discrete-time SIC epidemic model
Abstract
Indirect disease transmission is modeled via a fractional-order discretetime Susceptible-Infected-Contaminant (SIC) model vaccination as a control strategy. Two control actions are considered, giving rise to two different models: the vaccine efficacy model and the vaccination impact model. In the first model, the effectiveness of the vaccine is analyzed by introducing a new parameter, while in the second model, the impact of the vaccine is studied incorporating a new variable into the model. Both models are studied giving population thresholds to ensure the eradication of the disease. In addition, a sensitivity analysis of the Basic Reproduction Number has been carried out with respect to the effectiveness of the vaccine, the fractional order, the vaccinated population rate and the exposure rate. This analysis has been undertaken to study its effect on the dynamics of the models. Finally, the obtained results are illustrated and discussed with a simulation example related to the evolution of the disease in a pig farm.
Keyword : epidemic process, discrete fractional-order, indirect transmission, vaccination, sensitivity analysis
How to Cite
Coll, C., Ginestar, D., Herrero, A., & Sánchez, E. (2024). Effectivity of the vaccination strategy for a fractional-order discrete-time SIC epidemic model. Mathematical Modelling and Analysis, 29(3), 525–545. https://doi.org/10.3846/mma.2024.19354
This work is licensed under a Creative Commons Attribution 4.0 International License.
References
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L.J.S. Allen and P. van den Driessche. The basic reproduction number in some discrete-time epidemic models. J. Difference Equ. Appl., 14(10–11):1127–1147, 2008. https://doi.org/10.1080/10236190802332308
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A.D.C. Berriman, D. Clancy, H.E. Clough and R.M. Christleyz. Semi-stochastic models for salmonella infection within finishing pig units in the UK. Math. Biosci., 245(2):148–156, 2013. https://doi.org/10.1016/j.mbs.2013.06.004
A.D.C. Berrmann. Mathematical modelling of the dynamics and control of Salmonella on UK pig farms. PH.D. thesis, University of Liverpool, UK, 2012.
F. Brauer. A new epidemic model with indirect transmission. J. Biol. Dyn., 11:285–293, 2017. https://doi.org/10.1080/17513758.2016.1207813
Y.M. Chu, M.F. Khan, S. Ullah, S.A.A. Shah, M. Farooq and M. Mamat. Mathematical assessment of a fractional-order vector-host disease model with the Caputo-Fabrizio derivative. Math. Methods Appl. Sci., 46(1):232–247, 2023. https://doi.org/10.1002/mma.8507
Y.M. Chu, S. Rashid, A.O. Akdemir, A. Khalid, D. Baleanu, B.R. Al-Sinan and O.A.I. Elzibar. Predictive dynamical modeling and stability of the equilibria in a discrete fractional difference COVID-19 epidemic model. Results in Physics, 49(106467), 2023. https://doi.org/10.1016/j.rinp.2023.106467
Y.M. Chu, R. Zarin, A. Khan and S. Murtaza. A vigorous study of fractional order mathematical model for SARS-CoV-2 epidemic with Mittag-Leffler kernel. Alexandria Engineering J., 71:565–579, 2023. https://doi.org/10.1016/j.aej.2023.03.037
C. Coll, D. Ginestar, A. Herrero and E. Sánchez. The discrete fractional order difference applied to an epidemic model with indirect transmission. Appl. Math. Model., 103:636–648, 2022. https://doi.org/10.1016/j.apm.2021.11.002
C. Coll and E. Sánchez. Epidemic spreading by indirect transmission in a compartmental farm. Appl. Math. Comput., 386(125473), 2020. https://doi.org/10.1016/j.amc.2020.125473
A. Dababneh, N. Djenina, A. Ouannas, G. Grassi, I.M. Batiha and I.H. Jebril. A new incommensurate fractional-order discrete COVID-19 model with vaccinated individuals compartment. Fractal and Fractional, 6(8):456, 2022. https://doi.org/10.3390/fractalfract6080456
N. Djenina, A. Ouannas, I.M. Batiha, G. Grassi, T.E. Oussaeif and S. Momani. A novel fractional-order discrete SIR model for predicting COVID-19 behavior. Mathematics, 10(13):2224, 2022. https://doi.org/10.3390/math10132224
A. Dzielinski and D. Sierociuk. Stability of discrete fractional order state-space systems. IFAC Proceedings Volumes, 39(11):505–648, 2006. https://doi.org/10.3182/20060719-3-PT-4902.00084
Z.Y. He, A. Abbes, H. Jahanshahi, N.D. Alotaibi and Y. Wang. Fractionalorder discrete-time SIR epidemic model with vaccination: Chaos and complexity. Mathematics, 10(2):165, 2022. https://doi.org/10.3390/math10020165
L.V.C. Hoan, M.A. Akinlar, M. Inc, J.F. Gómez-Aguilar, Y.M. Chu and B. Almohsen. A new fractional-order compartmental disease model. Alexandria Engineering J., 59(5):3187–3196, 2020. https://doi.org/10.1016/j.aej.2020.07.040
J. Huo and H. Zhao. Dynamical analysis of a fractional SIR model with birth and death on heterogeneous complex networks. Physica A, 448:41–56, 2016. https://doi.org/10.1016/j.physa.2015.12.078
A.N. Jensen, A. Dalsgaard, A. Stockmarr, E.M. Nielsen and D.L. Baggesen. Survival and transmission of salmonella enterica serovar typhimurium in an outdoor organic pig farming environment. Appl. Environ. Microbiol., 72(3):1833–1842, 2006. https://doi.org/10.1128/AEM.72.3.1833-1842.2006
A. Khan, H.M. Alshehri, T. Abdeljawad, Q.M. Al-Mdallal and H. Khan. Stability analysis if fractional nabla difference COVID-19 model. Results in Physics, 22(10388), 2021. https://doi.org/10.1016/j.rinp.2021.103888
A. Khan, H. Khan, J.F. Gómez-Aguilar and T. Abdeljawad. Existence and Hyers-Ulam stability for a nonlinear singular fractional differential equations with Mittag-Leffler kernel. Chaos, Solitons & Fractals, 127:422–427, 2019. https://doi.org/10.1016/j.chaos.2019.07.026
R. Pakhira, U. Ghosh and S. Sarkar. Study of memory effects in an inventory model using fractional calculus. Appl. Math. Sci., 12(17):797–824, 2018. https://doi.org/10.12988/ams.2018.8578
P. Pandey, Y.M. Chu, J.F. Gómez-Aguilar, H. Jahanshahi and A.A. Aly. A novel fractional mathematical model of COVID-19 epidemic considering quarantine and latent time. Results in Physics, 26(104286), 2021. https://doi.org/10.1016/j.rinp.2021.104286
Md. Samsuzzoha, M. Singh and D. Lucy. Uncertainity and sensitivity analysis of the basic reproduction number of a vaccinated epidemic model of influenza. App. Math. Model., 37(3):903–915, 2013. https://doi.org/10.1016/j.apm.2012.03.029
S.R. Saratha, G. Sai Sundara Krishnan and M. Bagyalakshmi. Analysis of a fractional epidemic model by fractional generalised homotopy analysis method using modified Riemann-Liouville derivative. Appl. Math. Model., 92:525–545, 2021. https://doi.org/10.1016/j.apm.2020.11.019
A. Scherer and A. McLean. Mathematical models of vaccination. British Medical Bulletin, 62:187–199, 2002. https://doi.org/10.1093/bmb/62.1.187
S. Siriprapaiwan, E.J. Moore and S. Koonprasert. Generalized reproductions numbers, sensitivity analysis and critical immunity levels of an SEQIJR disease model with immunization and vaying total population size. Math. Comput. Simulation, 146:70–89, 2018. https://doi.org/10.1016/j.matcom.2017.10.006
N. Ziyadi and A. Yakubu. Local and global sensitivity analysis in a discrete-time seis epidemic model. Adv. Dyn. Sys. and App., 11:15–33, 2016.