Optimal Control for Resource Allocation in a Multi-Patch Epidemic Model with Human Dispersal Behavior
DOI:
https://doi.org/10.5614/cbms.2025.8.1.1Keywords:
Multi-patch model, human dispersal, optimal control, effective reproduction number, vaccination, social distancingAbstract
A multi-patch epidemic compartmental model with human dispersal behavior studies the spread of the disease and it sets the model to real-world situations. The mobility matrix (M) applies human dispersal behavior to the model. The optimal control theory assists in controlling the disease burden while minimizing the cost of infected individuals and implementing control measures. We formulate a multi-patch SIR model with human dispersal behavior to control and reduce communicable disease outbreaks such as COVID-19 by optimizing resource allocation in Sri Lanka. Results are represented by using the reproduction number (R0), effective reproduction number (Rt), and final epidemic size (ci). Compared to the basic reproduction number (R0), the effective reproduction number (Rt) represents the significant result in the epidemiological model incorporated with control measures. The average number of secondary cases concerning the current susceptible population is represented by Rt and the final epidemic size represents the patched-specified cost value for infected individuals. According to the results, the disease burden can be controlled by vaccination relative to social distancing.
References
Lee, S., Baek, O. and Melara, L., Resource allocation in introduction-patch epidemic model with state-dependent dispersal behaviors using optimal control, Processes, 8(9), p. 1087, 2020.
Brauer, F., Castillo-Chavez, C., Feng, Z., Brauer, F., Castillo-Chavez, C. and Feng, Z., Epidemiological models incorporating mobility, behavior, and time scales, Mathematical Models in Epidemiology, pp. 477-504, 2019.
Funk, S., Salath?e, M. and Jansen, V.A., Modelling the influence of human behaviour on the spread of infectious diseases: a review, Journal of the Royal Society Interface, 7(50), pp. 1247?1256, 2010.
Martcheva, M., An introduction to mathematical epidemiology, Springer, 61, pp. 9-31, 2015.
Asano, E., Gross, L.J., Lenhart, S. and Real, L.A., Optimal control of vaccine distribution in a rabies metapopulation model, Mathematical Biosciences and Engineering, 5(2), pp. 219-238, 2008.
Okyere, E., Olaniyi, E. and Bonyah, E., Analysis of zika virus dynamics with sexual transmission route using multiple optimal controls, Scientific African, 9, p. e00532, 2020.
Erandi, K., Mahasinghe, A.C., Perera S. and Jayasinghe, S., Effectiveness of the strategies implemented in Sri Lanka for controlling the Covid-19 outbreak, Journal of Applied Mathematics, 2020(1), pp. 1?10, 2020.
Ahmad, M.D., Usman, M., Khan, A. and Imran, M., Optimal control analysis of ebola disease with control strategies of quarantine and vaccination, Infectious Diseases of Poverty, 5, pp. 1?12, 2016.
Makinde O.D. and Okosun, K.O., Impact of chemo-therapy on optimal control of malaria disease with infected immigrants, BioSystems, 104(1), pp. 32-41, 2011.
Singh, A. and Chattopadhyay, A., COVID-19 recovery rate and its association with development, Indian Journal of Medical Sciences, 73(1), p. 8, 2021.
Swarnamali, H., Francis, T.V., Sooriyaarachchi, P. and Jayawardena, R., COVID-19 vaccine hesitancy in Sri Lanka: A national level survey, International Journal of Health Sciences, 17(1), p. 3, 2023.
Lenhart, S. and Workman, J.T., Optimal control applied to biological models, CRC Press, 2007.
ElHassan, A., AbuHour, Y. and Ahmad, A., An optimal control model for covid-19 spread with impacts of vaccination and facemask,Heliyon, 9(9), 2023.
Mahasinghe, A., Erandi, K. and Perera, S., An optimal lockdown relaxation strategy for minimizing the economic effects of covid-19 outbreak, International Journal of Mathematics and Mathematical Sciences, 2021(1), pp. 1-10, 2021.
Guner, H.R., Hasanoglu, I. and Aktas., F., COVID-19: Prevention and control measures in community, Turkish Journal of Medical Sciences, 50(9), pp. 571-577, 2020.
WHO, Advice for the public on Covid-19. https://www.who.int/emergencies/diseases/novel-coronavirus-2019/advice-for-public, Accessed on December 7, 2024.
WHO, Covid-19 weekly epidemiological update. https://www.epid.gov.lk/storage/post/pdfs/en 64f33bb8771c9sitrep-gl-en-02-09 10 23.pdf,2023, Accessed on December 6, 2023.
Amaratunga, D., Fernando, N., Haigh, R. and Jayasinghe, N., The Covid-19 outbreak in Sri Lanka: A synoptic analysis focusing on trends, impacts, risks and science policy interaction processes, Progress in Disaster Science, 8, p. 100133, 2020.
Mahasinghe, A.C., Erandi, K. and Perera, S., An optimal lockdown relaxation strategy for minimizing the economic effects of covid-19 outbreak in sri lanka, This Version Posted June, 9, p. 2020, 2020.
Watson, O.J., Barnsley, G., Toor, J., Hogan, A.B., Winskill, P., and Ghani, A.C., Global impact of the first year of Covid-19 vaccination: a mathematical modelling study, The Lancet Infectious Diseases, 22(9), pp. 1293?1302, 2022.
Epidemiology Unit, Vaccination summary. https://www.epid.gov.lk/vaccination-summary, Accessed on September 1, 2023.
King, W.G., The prevention of malaria, British Medical Journal, 1(2631), p. 1348, 1911.
Hsieh, Y.H., Van den Driessche, P. and Wang, L., Impact of travel between patches for spatial spread of disease, Bulletin of Mathematical Biology, 69, pp. 1355-1375, 2007.
Rushton, S. and Mautner, A.J., The deterministic model of a simple epidemic for more than one community, Biometrika, 42(1/2), pp. 126-132, 1955.
Bichara, D., Kang, Y., Castillo-Chavez, C., Horan, R. and Perrings, C., SIS and SIR epidemic models under virtual dispersal, Bulletin of Mathematical Biology, 77, pp. 2004-2034, 2015
Nuha, A.R., Achmad, N., Rahman, G.A., Abdullah, S., Chasanah, S.I.U. and Valentika, N., Analysis of optimum control on the implementation of vaccination and quarantine on the spread of covid-19, BAREKENG: Jurnal Ilmu Matematika dan Terapan, 16(4), pp. 1139-1146, 2022.
Firmansyah, F. and Rangkuti, Y.M., Sensitivity Analysis and Optimal Control of Covid 19 Model, Jambura Journal of Biomathematics (JJBM), 4(1), pp. 95-102, 2023.
Sattenspiel, L. and Dietz, K., A structured epidemic model incorporating geographic mobility among regions, Mathematical Biosciences, 128(1-2), pp. 71-91, 1995.
Lee, S. and Castillo-Chavez, C., The role of residence times in two-patch dengue transmission dynamics and optimal strategies, Journal of Theoretical Biology, 374, pp. 152-164, 2015.
Bock, W. and Jayathunga, Y., Optimal control of a multi-patch dengue model under the influence of Wolbachia bacterium, Mathematical Biosciences, 315, p. 108219, 2019.
Erandi, K.K.W.H., Mahasinghe, A.C., Ganegoda, N.C., Perera, S.S.N. and Jayasinghe, S., Identifying long?term optimal vaccination strategies for mitigating a pandemic: a computational modeling approach using covid-19 data in srilanka, Communications in Combinatorics, Cryptography and Computer Science, 2023(1), pp. 20-28, 2022.
Department of Census and Statistics - Sri Lanka, Census map - Sri Lanka, 2023. http://map.statistics.gov.lk:8080/LankaStatMap/apps/gs/censusmap/index.html, Accessed on December 6, 2023.
Heffernan, J.M., Smith, R.J. and Wahl, L.M., Perspectives on the basic reproductive ratio, Journal of the Royal Society Interface, 2(4), pp. 281-293, 2005.
Wallinga, J. and Teunis, P., Different epidemic curves for severe acute respiratory syndrome reveal similar impacts of control measures, American Journal of Epidemiology, 160(6), pp. 509-516, 2004.
Caicedo-Ochoa, Y., Rebellon-Sanchez, D.E., Penaloza-Rallon, M., Cortes-Motta, H.F. and Mendez-Fandino, Y.R., Effective Reproductive Number estimation for initial stage of COVID-19 pandemic in Latin American Countries, International Journal of Infectious Diseases, 95, pp. 316-318, 2020.
Barratt, H., Kirwan, M., and Shantikumar, S., Epidemic theory (effective and basic reproduction numbers, epidemic thresholds) and techniques for analysis of infectious disease data (construction and use of epidemic curves, generation numbers, exceptional reporting and identification of significant clusters), Health Knowledge, 2018.
Epidemiology Unit, Covid-19 disease surveillance and vaccination. https://www.epid.gov.lk/covid-19-data, Accessed on December 6, 2023.
Ministry of Finance Sri Lanka, https://www.treasury.gov.lk/budget#estimates, Accessed on December 6, 2023.
SciPy Developers, scipy.optimize.least squares-scipy v1.8.0 reference guide, https://docs.scipy.org/doc/scipy/reference/generated/scipy.optimize.least squares.html, Accessed on March 14, 2024.
National Transport Commission, https://www.transport.gov.lk/web/index.php?option=com content&view=article&id=27&Itemid=147&lang=en#private-bus-operational-data-inter-intra-provincial, Accessed on December 6, 2023.
Sri Lanka Transport Board, https://www.transport.gov.lk/web/index.php?option=com content&view=article&id=25&Itemid=145&lang=en#no-of-passengers-transportedl, Accessed on December 6, 2023.
Downloads
Published
How to Cite
Issue
Section
License
Copyright (c) 2025 Communication in Biomathematical Sciences
This work is licensed under a Creative Commons Attribution-NoDerivatives 4.0 International License.
