A modified optimal control for the mathematical model of dengue virus with vaccination

Puntipa Pongsumpun; Jiraporn Lamwong; I-Ming Tang; Puntani Pongsumpun; Puntipa Pongsumpun; Jiraporn Lamwong; I-Ming Tang; Puntani Pongsumpun

doi:10.3934/math.20231405

AIMS Mathematics

2023, Volume 8, Issue 11: 27460-27487. doi: 10.3934/math.20231405

Previous Article Next Article

Research article

A modified optimal control for the mathematical model of dengue virus with vaccination

1.
Department of Mathematics, School of Science, King Mongkut's Institute of Technology Ladkrabang, Bangkok 10520, Thailand
2.
Department of Applied Basic Subjects, Thatphanom College, Nakhon Phanom University, Nakhon Phanom 48000, Thailand
3.
Department of Physics, Faculty of Science, Mahidol University, Bangkok 10400, Thailand

Received: 24 July 2023 Revised: 18 August 2023 Accepted: 08 September 2023 Published: 27 September 2023
MSC : 00A71

The dengue viruses (of which there are four strains) are the causes of three illnesses of increasing severity; dengue fever (DF), dengue hemorrhagic fever (DHF) and dengue shock syndrome (DSS). Recently, dengue fever has reached epidemic proportion in several countries. Strategies or preventative methods have to be developed to combat these epidemics. This can be done by development of vaccines or by preventing the transmission of the virus. The latter approach could involve the use of mosquito nets or insecticide spraying. To determine which strategy would work, we test the strategy using mathematical modeling to simulate the effects of the strategy on the dynamics of the transmission. We have chosen the Susceptible-Exposed-Infected-Recovered (SEIR) model and the SusceptibleExposed-Infected (SEI) model to describe the human and mosquito populations, repectively. We use the Pontryagin's maximum principle to find the optimal control conditions. A sensitivity analysis revealed that the transmission rate $ ({\gamma }_{h}, {\gamma }_{v}) $, the birth rate of human population ($ {\mu }_{h} $), the constant recruitment rate of the vector population ($ A $) and the total human population ($ {N}_{h} $) are the most influential factors affecting the disease transmission. Numerical simulations show that the optimal controlled infective responses, when implemented, cause the convergence to zero to be faster than that in uncontrolled cases.
- dengue,
- mathematical model,
- optimal control,
- simulations,
- Lyapunov function,
- stabilities,
- sensitivity
Citation: Puntipa Pongsumpun, Jiraporn Lamwong, I-Ming Tang, Puntani Pongsumpun. A modified optimal control for the mathematical model of dengue virus with vaccination[J]. AIMS Mathematics, 2023, 8(11): 27460-27487. doi: 10.3934/math.20231405

Related Papers:

Abstract

The dengue viruses (of which there are four strains) are the causes of three illnesses of increasing severity; dengue fever (DF), dengue hemorrhagic fever (DHF) and dengue shock syndrome (DSS). Recently, dengue fever has reached epidemic proportion in several countries. Strategies or preventative methods have to be developed to combat these epidemics. This can be done by development of vaccines or by preventing the transmission of the virus. The latter approach could involve the use of mosquito nets or insecticide spraying. To determine which strategy would work, we test the strategy using mathematical modeling to simulate the effects of the strategy on the dynamics of the transmission. We have chosen the Susceptible-Exposed-Infected-Recovered (SEIR) model and the SusceptibleExposed-Infected (SEI) model to describe the human and mosquito populations, repectively. We use the Pontryagin's maximum principle to find the optimal control conditions. A sensitivity analysis revealed that the transmission rate $ ({\gamma }_{h}, {\gamma }_{v}) $, the birth rate of human population ($ {\mu }_{h} $), the constant recruitment rate of the vector population ($ A $) and the total human population ($ {N}_{h} $) are the most influential factors affecting the disease transmission. Numerical simulations show that the optimal controlled infective responses, when implemented, cause the convergence to zero to be faster than that in uncontrolled cases.

References

[1]	Combating Dengue Outbreak and Addressing Overlapping Challenges with COVID-19, World Health Organization (WHO), 2023. Available from: https://www.who.int/thailand/news/detail/30-06-2023-combating-dengue-outbreak-and-addressing-overlapping-challenges-with-covid-19.
[2]	S. Zaheer, M. J. Tahir, I. Ullah, A. Ahmed, S. M. Saleem, S. Shoib, et al., Dengue outbreak in the times of COVID-19 pandemic: Common myths associated with the dengue, Ann. Med. Surg., 81 (2022), 104535. https://doi.org/10.1016/j.amsu.2022.104535 doi: 10.1016/j.amsu.2022.104535
[3]	A. Tangsathapornpong, U. Thisyakorn, Dengue amid COVID-19 pandemic, PLOS Glob Public Health, 3 (2023), e0001558. https://doi.org/10.1371/journal.pgph.0001558 doi: 10.1371/journal.pgph.0001558
[4]	Dengue and severe dengue, World Health Organization (WHO), 2023. Available from: https://www.who.int/news-room/fact-sheets/detail/dengue-and-severe-dengue.
[5]	Ten threats to global health in 2019, World Health Organization (WHO), 2023. Available from: https://www.who.int/news-room/spotlight/ten-threats-to-global-health-in-2019.
[6]	World NTD Day: Dengue Fever tops Thailand's agenda, Thai Public Broadcasting Service, 2023. Available from: https://www.thaipbsworld.com/world-ntd-day-dengue-fever-tops-thailands-agenda.
[7]	M. G. Guzman, E. Harris, Dengue, Lancet, 385 (2015), 453–465. https://doi.org/10.1016/S0140-6736(14)60572-9 doi: 10.1016/S0140-6736(14)60572-9
[8]	A. K. Supriatna, H. Husniah, E. Soewono, B. Ghosh, Y. Purwanto, E. Nurlaelah, Age-Dependent Survival Rates in SIR-SI Dengue Transmission Model and Its Application Considering Human Vaccination and Wolbachia Infection in Mosquitoes, Mathematics, 10 (2022), 3950. https://doi.org/10.3390/math10213950 doi: 10.3390/math10213950
[9]	World Health Organization, Dengue hemorrhagic fever: diagnosis, treatment, prevention and control, 2 Eds., Geneva: WHO, 1997. https://apps.who.int/iris/handle/10665/41988.
[10]	A. K. Supriatna, N. Nuraini, E. Soewono, Mathematical Models of Dengue Transmission and Control: A Survey, in Dengue Virus: Detection, Diagnosis and Control, 1 Eds., New York: Nova Publishers, 2010,187–208. https://www.academia.edu/7204876/Mathematical_Models_of_ Dengue_Transmission_and_Control_A_Survey
[11]	D. J. Gubler, Dengue and dengue haemorrhagic fever, Clin. Microbiol., 11 (1998), 480–496. https://doi.org/10.1128/CMR.11.3.480 doi: 10.1128/CMR.11.3.480
[12]	H. Nishiura, Mathematical and Statistical Analyses of the Spread of Dengue, Dengue Bull., 30 (2006), 51–67. https://apps.who.int/iris/handle/10665/170261
[13]	R. Isea, H. d.l. Puerta, Analysis of an SEIR-SEI four-strain epidemic dengue model with primary and secondary infections, CLIC., 7 (2014), 3–7. https://doi.org/10.48550/arXiv.1406.4155 doi: 10.48550/arXiv.1406.4155
[14]	G. R. Phaijoo, D. B. Gurung, Mathematical Model of Dengue Fever with and without awareness in Host Population, IJAERA., 1 (2015), 239–245. https://www.ijaera.org/manuscript/20150106003.pdf
[15]	P. Pongsumpun, The Dynamical Model of Dengue Vertical Transmission. KMITL Sci. Tech. J., 7 (2017), 48–61. https://li01.tci-thaijo.org/index.php/cast/article/view/128709
[16]	P. Pongsumpun, I. M. Tang, N. Wongvanich, Optimal control of the dengue dynamical transmiss ion with vertical transmission, Adv. Differ. Equ., 176 (2019). https://doi.org/10.1186/s13662-019-2120-6 doi: 10.1186/s13662-019-2120-6
[17]	R. Sungchasit, P. Pongsumpun, Mathematical Model of Dengue Virus with Primary and Secondary Infection, Curr. Appl. Sci. Technol. 19 (2019), 154–176. https://li01.tci-thaijo.org/index.php/cast/article/view/188624
[18]	M. A. Khan, Fatmawati, Dengue infection modeling and its optimal control analysis in East Java, Indonesia, Heliyon, 7 (2021), e06023. https://doi.org/10.1016/j.heliyon.2021.e06023 doi: 10.1016/j.heliyon.2021.e06023
[19]	P. Affandi, K. M. Ahsar, E. Suhartono, J. Dalle, Systematic Review: Math-ematics Model Epidemiology of Dengue Fever, Univers. J. Public Health, 10 (2022), 419–429. https://doi.org/10.13189/ujph.2022.100415 doi: 10.13189/ujph.2022.100415
[20]	A. Schaum, R. B. Jaquez, C. Torres-Sosa, G. Sánchez-González, Modeling the spreading of dengue using a mixed population model, IFAC-PapersOnLine, 55 (2022), 582–587. https://doi.org/10.1016/j.ifacol.2022.09.158 doi: 10.1016/j.ifacol.2022.09.158
[21]	T. Li, Y. Guo, Modeling and optimal control of mutated COVID-19 (Delta strain) with imperfect vaccination, Chaos Soliton. Fract., 156 (2022), 111825. https://doi.org/10.1016/j.chaos.2022.111825 doi: 10.1016/j.chaos.2022.111825
[22]	Y. Guo, T. Li, Modeling and dynamic analysis of Novel Coronavirus Pneumonia (COVID-19) in China, J Appl. Math. Comput., 68 (2022), 2641–2666. https://doi.org/10.1007/s12190-021-01611-z doi: 10.1007/s12190-021-01611-z
[23]	Y. Guo, T. Li, Dynamics and optimal control of an online game addiction model with considering family education, AIMS. Math., 7 (2022), 3745–3770. https://doi.org/10.3934/math.2022208 doi: 10.3934/math.2022208
[24]	Y. Guo, T. Li, Fractional-order modeling and optimal control of a new online game addiction model based on real data, Commun. Nonlinear Sci., 121 (2023), 107221. https://doi.org/10.1016/j.cnsns.2023.107221 doi: 10.1016/j.cnsns.2023.107221
[25]	Y. Guo, T. Li, Modeling the competitive transmission of the Omicron strain and Delta strain of COVID-19, J. Math. Anal. Appl., 526 (2023), 127283. https://doi.org/10.1016/j.jmaa.2023.127283 doi: 10.1016/j.jmaa.2023.127283
[26]	D. Fever, Bureau of Epidemiology Department of Disease Control, 2022. Available from: http://www.boe.moph.go.th/boedb/surdata/disease.php
[27]	D. Rodriguez, C. Major, L.Sánchez-González, E. Jones, M. Delorey, C. Alonso, et al., Dengue vaccine acceptability before and after the availability of COVID-19 vaccines in Puerto Rico, Vaccine, 41 (2023), 3627–3635. https://doi.org/10.1016/j.vaccine.2023.04.081 doi: 10.1016/j.vaccine.2023.04.081
[28]	S. R. Hadinegoro, J. L. Arredondo-Garcia, M. R. Capeding, C. Deseda, T. Chotpitayasunondh, R. Dietze, et. al., Efficacy and Long-Term Safety of a Dengue Vaccine in Regions of Endemic Disease, N. Engl. J. Med., 373 (2015), 1195–1206. https://doi.org/10.1056/NEJMoa1506223 doi: 10.1056/NEJMoa1506223
[29]	Y. J. Hertanto, B. D. Novita, Efficacy of Live Attenuated Dengue Vaccines: CYD-TDV, TDV (TAK-003), and TV003/TV005, Folia. Med. Indonesiana, 57 (2021), 365–371. https://doi.org/10.20473/fmi.v57i4.21741 doi: 10.20473/fmi.v57i4.21741
[30]	T. Vianney, E. Susannah, R. Mahadev, C. Paul, M. Zenaida, L. Edde, et al., A randomized phase 3 trial of the immunogenicity and safety of coadministration of a live-attenuated tetravalent dengue vaccine (TAK-003) and an inactivated hepatitis a (HAV) virus vaccine in a dengue non-endemic country, Vaccine, 41 (2023), 1398–1407. https://doi.org/10.1016/j.vaccine.2023.01.007 doi: 10.1016/j.vaccine.2023.01.007
[31]	J. M. Torres-Flores, A. Reyes-Sandoval, M. I. Salazar, Dengue Vaccines: An Update, BioDrugs, 36 (2022), 325–336. https://doi.org/10.1007/s40259-022-00531-z doi: 10.1007/s40259-022-00531-z
[32]	J. Lamwong, N. Wongvanich, I. M. Tang, T. Changpuek, P. Pongsumpun, Global stability of the transmission of hand-foot-mouth disease according to the age structure of the population, Curr. Appl. Sci. Technol., 21 (2021), 351–369. https://li01.tci-thaijo.org/index.php/cast/article/view/248058
[33]	S. M. Guo, X. Z. Li, M. Ghosh, Analysis of dengue disease model with nonlinear incidence, Discret. Dyn. Nat. Soc., 2013, 320581. https://doi.org/10.1155/2013/320581 doi: 10.1155/2013/320581
[34]	Y. Yaacob, Analysis of a dengue disease transmission model without immunity, MATEMATIKA Malays. J. Ind. Appl. Math., 23 (2007), 75–81. https://doi.org/10.11113/matematika.v23.n.524 doi: 10.11113/matematika.v23.n.524
[35]	H. M. Yang, The basic reproduction number obtained from Jacobian and next generation matrices—A case study of dengue transmission modelling, Biosyst., 126 (2014), 52–75. https://doi.org/10.1016/j.biosystems.2014.10.002 doi: 10.1016/j.biosystems.2014.10.002
[36]	M. Z. Ndii, N. Anggriani, J. J. Messakh, B. S. Djahi, Estimating the reproduction number and designing the integrated strategies against dengue, Results Phys., 27 (2021), 104473. https://doi.org/10.1016/j.rinp.2021.104473 doi: 10.1016/j.rinp.2021.104473
[37]	J. J. Xiang, J. Wang, L. M. Cai, Global stability of the dengue disease transmission models, Discrete Cont. Dyn–B, 20 (2015), 2217–2232. https://doi.org/10.3934/dcdsb.2015.20.2217 doi: 10.3934/dcdsb.2015.20.2217
[38]	A. Abidemia, J. Ackora-Prah, H. O. Fatoyinbo, J. K. K. Asamoah, Lyapunov stability analysis and optimization measures for a dengue disease transmission model, Physica A, 602 (2022), 127646. https://doi.org/10.1016/j.physa.2022.127646 doi: 10.1016/j.physa.2022.127646
[39]	P. Chanprasopchai, I. M. Tang, P. Pongsumpun, Effect of rainfall for the dynamical transmission model of the dengue disease in Thailand, Comput. Math. Methods Med., 2017, 2541862. https://doi.org/10.1155/2017/2541862/ doi: 10.1155/2017/2541862/
[40]	P. Chanprasopchai, I. M. Tang, P. Pongsumpun, SIR Model for Dengue Disease with Effect of Dengue Vaccination, Comput. Math. Methods Med., 2018, 9861572. https://doi.org/10.1155/2018/9861572 doi: 10.1155/2018/9861572
[41]	D. Fever, Ministry of Public Health, 2021. Available from: http://www.boe.moph.go.th/boedb/surdata/disease.php?dcontent=old&ds=66.
[42]	A. Chamnan, P. Pongsumpun, I. M. Tang, N. Wongvanich, Optimal Control of Dengue Transmission with Vaccination, Mathematics, 9 (2021), 1833. https://doi.org/10.3390/math9151833 doi: 10.3390/math9151833
[43]	M. L'Azou, A. Moureau, E. Sarti, J. Nealon, B. Zambrano, T. A. Wartel, et al., Symptomatic dengue in children in 10 Asian and Latin American countries, N. Engl. J. Med., 374 (2016), 1155–1166. https://doi.org/10.1056/NEJMoa1503877 doi: 10.1056/NEJMoa1503877
[44]	A. Chamnan, P. Pongsumpun, I. M. Tang, N. Wongvanich, Effect of a Vaccination against the Dengue Fever Epidemic in an Age Structure Population: From the Perspective of the Local and Global Stability Analysis, Mathematics, 10 (2022), 904. https://doi.org/10.3390/math10060904 doi: 10.3390/math10060904
[45]	S. Lenhart, J. T. Workman, Optimal Control Applied to Biological Models, 1 Eds., London: Chapman & Hall/CRC, 2007. https://doi.org/10.1201/9781420011418
[46]	L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze, E. F. Mishchenko, The Mathematical Theory of Optimal Processes, 1 Eds., New York: Wiley, 1962. https://doi.org/10.1002/zamm.19630431023
[47]	D. Olajumoke, S. O. Falowo, T. O. Abiodun, Optimal control assessment of Rift Valley fever model with vaccination and environmental sanitation in the presence of treatment delay, Model. Earth Syst. Environ., 9 (2023), 457–471. https://doi.org/10.1007/s40808-022-01508-1 doi: 10.1007/s40808-022-01508-1
[48]	J. P. Romero-Leiton, J. E. Castellanos, E. Ibargüen-Mondragón, An optimal control problem and cost-effectiveness analysis of malaria disease with vertical transmission applied to San Andrés de Tumaco (Colombia), Comp. Appl. Math., 38 (2019), 1–24. https://doi.org/10.1007/s40314-019-0909-2 doi: 10.1007/s40314-019-0909-2
[49]	A. Abidemi, N.A.B. Aziz, Optimal control strategies for dengue fever spread in Johor, Malaysia, Comput. Methods Programs Biomed., 196 (2020), 105585. https://doi.org/10.1016/j.cmpb.2020.105585 doi: 10.1016/j.cmpb.2020.105585
[50]	O. A. Adepoju, S. Olaniyi, Stability and optimal control of a disease model with vertical transmission and saturated incidence, Sci. Afri., 12 (2021), e00800. https://doi.org/10.1016/j.sciaf.2021.e00800 doi: 10.1016/j.sciaf.2021.e00800
[51]	N. Chitnis, J. M. Hyman, J. M. Cushing, Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bullet. Math. Biol., 70 (2008), 1272–1296. https://doi.org/10.1007/s11538-008-9299-0 doi: 10.1007/s11538-008-9299-0
[52]	S. Rashid, F. Jarad, S. A. A. El-Marouf, S. K. Elagan, Global dynamics of deterministic-stochastic dengue infection model including multi specific receptors via crossover effects, AIMS Math., 8 (2023), 6466–6503. https://doi.org/10.3934/math.2023327 doi: 10.3934/math.2023327
[53]	J. Lamwong, P. Pongsumpun, I. M. Tang, N. Wongvanich, Vaccination role in combatting the Omicron Variant outbreak in Thailand: An optimal control approach, Mathematics, 10 (2022), 3899. https://doi.org/10.3390/math10203899 doi: 10.3390/math10203899

Reader Comments

Your name:*

Email:*
© 2023 the Author(s), licensee AIMS Press. This is an open access article distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/by/4.0)