Keywords and phrases: deterministic modelling, stochastic modelling, multiple strains, Pontryagin’s maximum principle, ensemble averages.
Received: April 25, 2024; Accepted: July 20, 2024; Published: August 22, 2024
How to cite this article: Andrew Kayanja, Benard Abola, Cliff R. Kikawa, Benedict Oyo and Amos Ssematimba, Optimal control harmony: navigating deterministic and stochastic realms with a two-strain model using Pontryagin’s maximum principle, Universal Journal of Mathematics and Mathematical Sciences 20(2) (2024), 93-142. http://dx.doi.org/10.17654/2277141724007
This Open Access Article is Licensed under Creative Commons Attribution 4.0 International License
References:
[1]D. V. Parums, Editorial: Revised World Health Organization (WHO) terminology for variants of concern and variants of interest of SARS-CoV-2, Medical Science Monitor 27 (2021), e933622. doi:10.12659/msm.933622.
[2]D. Duong, What’s important to know about the new covid-19 variants?, Canadian Medical Association Journal 193(4) (2021), E141-E142.
doi:10.1503/cmaj.1095915.
[3]A. Blut, et al., Influenza virus, Transfusion Medicine and Hemotherapy 36(1) (2009), 32-39. doi:10.1159/000197314.
[4]D. J. Alexander, An overview of the epidemiology of avian influenza, Vaccine 25(30) (2007), 5637-5644. doi:10.1016/j.vaccine.2006.10.051.
[5]R. Rodriguez-Roche and E. A. Gould, Understanding the dengue viruses and progress towards their control, BioMed Research International 2013 (2013), 1-20. doi:10.1155/2013/690835.
[6]M. Ciotti, S. Angeletti, M. Minieri, M. Giovannetti, D. Benvenuto, S. Pascarella, C. Sagnelli, M. Bianchi, S. Bernardini and M. Ciccozzi, Covid-19 outbreak: An overview, Chemotherapy 64(5-6) (2019), 215-223. doi:10.1159/000507423.
[7]R. Sah, A. Siddiq, B. K. Padhi, A. Mohanty, A. A. Rabaan, D. Chandran, C. Chakraborty and K. Dhama, Dengue virus and its recent outbreaks: current scenario and counteracting strategies, International Journal of Surgery 109(9) (2023), 2841-2845. doi:10.1097/js9.0000000000000045.
[8]Z. Gao, Y. Xu, C. Sun, X. Wang, Y. Guo, S. Qiu and K. Ma, A systematic review of asymptomatic infections with COVID-19, Journal of Microbiology, Immunology and Infection 54(1) (2021), 12-16. doi:10.1016/j.jmii.2020.05.001.
[9]C. J. Feare and M. Yasué, Asymptomatic infection with highly pathogenic avian influenza H5N1 in wild birds: how sound is the evidence?, Virology Journal 3(1) (2006), 96. doi:10.1186/1743-422x-3-96.
[10]S. Ly, C. Fortas, V. Duong, T. Benmarhnia, A. Sakuntabhai, R. Paul, R. Huy, S. Sorn, K. Nguon, S. Chan, S. Kimsan, S. Ong, K. S. Kim, S. Buoy, L. Voeung, P. Dussart, P. Buchy and A. Tarantola, Asymptomatic dengue virus infections, Cambodia, 2012-2013, Emerging Infectious Diseases 25(7) (2019), 1354-1362. doi:10.3201/eid2507.181794.
[11]S. Moore, E. M. Hill, M. J. Tildesley, L. Dyson and M. J. Keeling, Vaccination and non-pharmaceutical interventions for COVID-19: a mathematical modelling study, The Lancet Infectious Diseases 21(6) (2021), 793-802.
doi:10.1016/s1473-3099(21)00143-2.
[12]J. Y. T. Mugisha, J. Ssebuliba, J. N. Nakakawa, C. R. Kikawa and A. Ssematimba, Mathematical modeling of COVID-19 transmission dynamics in Uganda: Implications of complacency and early easing of lockdown, PLOS ONE 16(2) (2021), e0247456. doi:10.1371/journal.pone.0247456.
[13]L. A. Grohskopf, L. H. Blanton, J. M. Ferdinands, J. R. Chung, K. R. Broder, H. K. Talbot, R. L. Morgan and A. M. Fry, Prevention and control of seasonal influenza with vaccines: Recommendations of the advisory committee on immunization practices - United States, 2022-23 influenza season, MMWR Recommendations and Reports 71(1) (2022), 1-28. doi:10.15585/mmwr.rr7101a1.
[14]C. Warren-Gash, E. Fragaszy and A. C. Hayward, Hand hygiene to reduce community transmission of influenza and acute respiratory tract infection: a systematic review, Influenza and Other Respiratory Viruses 7(5) (2012), 738-749. doi:10.1111/irv.12015.
[15]B. J. Hoye, V. J. Munster, H. Nishiura, M. Klaassen and R. A. Fouchier, Surveillance of wild birds for avian influenza virus, Emerging Infectious Diseases 16(12) (2010), 1827-1834. doi:10.3201/eid1612.100589.
[16]A. Ssematimba, T. Hagenaars, J. de Wit, F. Ruiterkamp, T. Fabri, J. Stegeman and M. de Jong, Avian influenza transmission risks: Analysis of biosecurity measures and contact structure in Dutch poultry farming, Preventive Veterinary Medicine 109(1-2) (2013), 106-115. doi:10.1016/j.prevetmed.2012.09.001.
[17]E. Kumaran, D. Doum, V. Keo, L. Sokha, B. Sam, V. Chan, N. Alexander, J. Bradley, M. Liverani, D. B. Prasetyo, A. Rachmat, S. Lopes, J. Hii, L. Rithea, M. Shafique and J. Hustedt, Dengue knowledge, attitudes and practices and their impact on community based vector control in rural Cambodia, PLOS Neglected Tropical Diseases 12(2) (2018), e0006268. doi:10.1371/journal.pntd.0006268.
[18]T. Alamo, P. Millán, D. G. Reina, V. M. Preciado and G. Giordano, Challenges and future directions in pandemic control, IEEE Control Systems Letters 6 (2022), 722-727. doi:10.1109/lcsys.2021.3085700.
[19]M. E. Kretzschmar, B. Ashby, E. Fearon, C. E. Overton, J. Panovska-Griffiths, L. Pellis, M. Quaife, G. Rozhnova, F. Scarabel, H. B. Stage, B. Swallow, R. N. Thompson, M. J. Tildesley and D. Villela, Challenges for modelling interventions for future pandemics, Epidemics 38 (2022), 100546.
doi:10.1016/j.epidem.2022.100546.
[20]M. L. Barreto, M. G. Teixeira, F. I. Bastos, R. A. Ximenes, R. B. Barata and L. C. Rodrigues, Successes and failures in the control of infectious diseases in brazil: social and environmental context, policies, interventions, and research needs, The Lancet 377(9780) (2011), 1877-1889.
doi:10.1016/s0140-6736(11)60202-x.
[21]R. J. Coker, B. M. Hunter, J. W. Rudge, M. Liverani and P. Hanvoravongchai, Emerging infectious diseases in southeast Asia: regional challenges to control, The Lancet 377(9765) (2011), 599-609. doi:10.1016/s0140-6736(10)62004-1.
[22]F. Brauer, C. Castillo-Chavez and Z. Feng, Mathematical models in epidemiology, Vol. 32, Springer, 2019.
[23]W. O. Kermack and A. G. McKendrick, A contribution to the mathematical theory of epidemics, Proceedings of the Royal Society of London. Series A, Containing Papers of a Mathematical and Physical Character 115(772) (1927), 700-721.
doi:10.1098/rspa.1927.0118.
[24]A. El Koufi and N. El Koufi, A stochastic epidemic model with general incidence rate control approach, Commun. Math. Biol. Neurosci. 2022 (2022), Article-ID 34.
[25]M. El Fatini, I. Sekkak, R. Taki and T. El Guendouz, A control treatment for a stochastic epidemic model with relapse and Crowly-Martin incidence, The Journal of Analysis 29(3) (2020), 713-729. doi:10.1007/s41478-020-00276-4.
[26]Y. Li and Z. Wei, Dynamics and optimal control of a stochastic coronavirus (COVID-19) epidemic model with diffusion, Nonlinear Dynamics 109(1) (2021), 91-120. doi:10.1007/s11071-021-06998-9.
[27]K. Fackeldey, M. Oster, L. Sallandt and R. Schneider, Approximative policy iteration for exit time feedback control problems driven by stochastic differential equations using tensor train format, Multiscale Modeling and Simulation 20(1) (2022), 379-403. doi:10.1137/20m1372500.
[28]L. Bourdin and E. Trélat, Pontryagin maximum principle for finite dimensional nonlinear optimal control problems on time scales, SIAM Journal on Control and Optimization 51(5) (2013), 3781-3813.
[29]J. Yong and X. Y. Zhou, Stochastic controls: Hamiltonian systems and HJB equations, Springer, New York, 1999. doi:10.1007/978-1-4612-1466-3.
[30]M. J. Mardanov and Y. A. Sharifov, Pontryagin’s maximum principle for the optimal control problems with multipoint boundary conditions, Abstract and Applied Analysis 2015 (2015), 1-6. doi:10.1155/2015/428042.
[31]M. Serhani and N. Raïssi, Nonconvex duality and semicontinuous proximal solutions of HJB equation in optimal control, RAIRO - Operations Research 43(2) (2009), 201-214. doi:10.1051/ro/2009012.
[32]S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, Chapman and Hall/CRC, 2007. doi:10.1201/9781420011418.
[33]D. Arpit, H. Wang, Y. Zhou and C. Xiong, Ensemble of averages: Improving model selection and boosting performance in domain generalization, Advances in Neural Information Processing Systems 35 (2022), 8265-8277.
[34]J. M. Murphy, The impact of ensemble forecasts on predictability, Quarterly Journal of the Royal Meteorological Society 114(480) (1988), 463-493.
doi:10.1002/qj.49711448010.
[35]A. K. Srivastav, P. K. Tiwari, P. K. Srivastava, M. Ghosh and Y. Kang, A mathematical model for the impacts of face mask, hospitalization and quarantine on the dynamics of COVID-19 in India: deterministic vs. stochastic, Mathematical Biosciences and Engineering 18(1) (2021), 182-213. doi:10.3934/mbe.2021010.
[36]R. Datko, Unconstrained control problems with quadratic cost, SIAM Journal on Control 11(1) (1973), 32-52. doi:10.1137/0311003.
[37]L. Lemecha Obsu and S. Feyissa Balcha, Optimal control strategies for the transmission risk of COVID-19, Journal of Biological Dynamics 14(1) (2020), 590-607. doi:10.1080/17513758.2020.1788182.
[38]S. Olaniyi, K. O. Okosun, S. O. Adesanya and R. S. Lebelo, Modelling malaria dynamics with partial immunity and protected travellers: optimal control and cost-effectiveness analysis, Journal of Biological Dynamics 14(1) (2020), 90-115. doi:10.1080/17513758.2020.1722265.
[39]M. Martcheva, An Introduction to Mathematical Epidemiology, Vol. 61 of Texts in Applied Mathematics, Springer, 2015.
[40]S. Ji, S. Peng, Y. Peng and X. Zhang, Solving stochastic optimal control problem via stochastic maximum principle with deep learning method, Journal of Scientific Computing 93(1) (2022), 30.
[41]A. W. Edridge, J. Kaczorowska, A. C. Hoste, M. Bakker, M. Klein, K. Loens, M. F. Jebbink, A. Matser, C. M. Kinsella, P. Rueda, et al., Seasonal coronavirus protective immunity is short-lasting, Nature Medicine 26(11) (2020), 1691-1693. doi:10.1038/s41591-020-1083-1.
[42]E. G. Levin, Y. Lustig, C. Cohen, R. Fluss, V. Indenbaum, S. Amit, R. Doolman, K. Asraf, E. Mendelson, A. Ziv, et al., Waning immune humoral response to BNT162b2 Covid-19 vaccine over 6 months, The New England Journal of Medicine 385(24) (2021), e84. doi:10.1056/NEJMoa2114583.
[43]H. Xin, Y. Li, P. Wu, Z. Li, E. H. Lau, Y. Qin, L. Wang, B. J. Cowling, T. K. Tsang and Z. Li, Estimating the latent period of coronavirus disease 2019 (COVID-19), Clinical Infectious Diseases 74 (9) (2022), 1678-1681.
doi:10.1093/cid/ciab746.
[44]G. Gonzalez-Parra, D. Martínez-Rodríguez and R. J. Villanueva-Micó, Impact of a new SARS-CoV-2 variant on the population: A mathematical modeling approach, Mathematical and Computational Applications 26(2) (2021), 25.
doi:10.3390/mca26020025.
[45]S. Zhao, B. Tang, S. S. Musa, S. Ma, J. Zhang, M. Zeng, Q. Yun, W. Guo, Y. Zheng, Z. Yang, Z. Peng, M. K. Chong, M. Javanbakht, D. He and M. H. Wang, Estimating the generation interval and inferring the latent period of COVID-19 from the contact tracing data, Epidemics 36 (2021), 100482.
doi:10.1016/j.epidem.2021.100482.
[46]S. Ma, J. Zhang, M. Zeng, Q. Yun, W. Guo, Y. Zheng, S. Zhao, M. H. Wang and Z. Yang, Epidemiological parameters of coronavirus disease 2019: a pooled analysis of publicly reported individual data of 1155 cases from seven countries, Medrxiv (2020), 2020-03. doi:10.1101/2020.03.21.20040329.
[47]Y. Xiang, Y. Jia, L. Chen, L. Guo, B. Shu and E. Long, COVID-19 epidemic prediction and the impact of public health interventions: A review of COVID-19 epidemic models, Infectious Disease Modelling 6 (2021), 324-342.
doi:10.1016/j.idm.2021.01.001.
[48]Center for Disease Control, Ending isolation and precautions for people with covid-19: Interimguidance, https://www.cdc.gov/coronavirus/2019-ncov/hcp/duration-isolation.html. Last accessed 30 March, 2024.
[49]World Health Organization, Life expectancy, https://data.who.int/countries/124. Last accessed 20 March, 2024.
[50]World Population Review, Quebec city population 2024, https://worldpopulationreview.com/canadian-cities/quebec-city-population. Last accessed 20 March 2024.
[51]H. T. Huynh, I. Soumare, et al., Stochastic simulation and applications in finance with MATLAB programs, John Wiley & Sons, 2011.
[52]E. A. Coddington, N. Levinson and T. Teichmann, Theory of Ordinary Differential Equations, American Institute of Physics, 1956.
[53]W. Boyce and R. DiPrima, Elementary differential equations and boundary value problems, textbook and student solutions manual set, 796 (2009).
[54]A. Abidemi, S. Olaniyi and O. A. Adepoju, An explicit note on the existence theorem of optimal control problem, Journal of Physics: Conference Series 2199(1) (2022), 012021. doi:10.1088/1742-6596/2199/1/012021.
[55]A. Bahar and X. Mao, Stochastic delay Lotka-Volterra model, Journal of Mathematical Analysis and Applications 292(2) (2004), 364-380.
doi:10.1016/j.jmaa.2003.12.004.
[56]T. Burton, Liapunov functions and boundedness, Journal of Mathematical Analysis and Applications 58(1) (1977), 88-97.
doi:10.1016/0022-247x(77)90230-x.
|