Abstract
This paper deals with a deterministic mathematical model of dengue based on a system of fractional-order differential equations (FODEs). In this study, we consider dengue control strategies that are relevant to the current situation in Malaysia. They are the use of adulticides, larvicides, destruction of the breeding sites, and individual protection. The global stability of the disease-free equilibrium and the endemic equilibrium is constructed using the Lyapunov function theory. The relations between the order of the operator and control parameters are briefly analysed. Numerical simulations are performed to verify theoretical results and examine the significance of each intervention strategy in controlling the spread of dengue in the community. The model shows that vector control tools are the most efficient method to combat the spread of the dengue virus, and when combined with individual protection, make it more effective. In fact, the massive use of personal protection alone can significantly reduce the number of dengue cases. Inversely, mechanical control alone cannot suppress the excessive number of infections in the population, although it can reduce the Aedes mosquito population. The result of the real-data fitting revealed that the FODE model slightly outperformed the integer-order model. Thus, we suggest that the FODE approach is worth to be considered in modelling an infectious disease like dengue.
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References
Agarwal P, Singh R, Rehman A (2020) Numerical solution of hybrid mathematical model of dengue transmission with relapse and memory via Adam-Bashforth-Moulton predictor-corrector scheme. Chaos Solitons and Fractals 143
Aguiar M, Anam V, Blyuss K, Estadilla C, Guerrero B, Knopoff D, Kooi B, Srivastav A, Steindorf V, Stollenwerk N (2022) Mathematical models for dengue fever epidemiology: A 10-year systematic review. Phys Life Rev 40:65–92
Aldila D, Gotz T, Soewono E (2013) An optimal control problem arising from a dengue disease transmission model. Math Biosci 242(1):9–16
Alexander L, Ben-Shachar R, Katzelnick L, Kuan, G., Balmaseda A, Harris E, Boots M (2021) Boosting can explain patterns of fluctuations of ratios of inapparent to symptomatic dengue virus infections. In: Proceedings of the National Academy of Sciences, vol 118(14)
Al-Sulami H, El-Shahed M, Nieto J, Shammakh W (2014) On fractional order dengue epidemic model. Math Probl Eng
Antonio M, Yoneyama T (2001) Optimal control and sub-optimal control in dengue epidemics. Optim Control Appl Methods 22(2):63–73
Atangana A, Bildik N (2013) Approximate solution of tuberculosis disease population dynamics model. Abst Appl Anal
Bartley L, Donnely C, Garnett G (2002) The seasonal patterns of dengue in endemic areas: mathematical models of mechanism. Trans R Soc Trop Med Hyg 96:387–397
Blayneh K, Gumel A, Lenhart S, Clayton T (2010) Backward bifurcation and optimal control in transmission dynamics of West Nile virus. Bull Math Biol 72(4):1006–1028
Bosch P, Gomez-Aguilar J, Rodriguez J, Sigarreta J (2020) Analysis of dengue fever outbreak by generalized fractional derivative. Fractals 28(8)
Burattini M, Chen M, Chow A, Coutinho F, Goh K, Lopez L, Ma S, Massad E (2008) Modelling the control strategies against dengue in Singapore. Epidemiol Infect 136:309–319
Bustamam A, Aldila D, Yuwanda A (2018) Understanding dengue control for short-and long-term intervention with a mathematical model approach. J Appl Math, iD 9674138
Caputo M (1967) Linear model of dissipation whose Q is almost frequency independent-II. Geophys J R Astron Soc 13:529–539
Carvalho S, da Silva S, Charret I (2019) Mathematical modeling of dengue epidemic: control methods and vaccination strategies. Theory Biosci 138(2):223–239
Carvalho A, Pinto C, Baleanu D (2018) HIV/HCV coinfection model: a fractional-order perspective for the effect of the HIV viral load. Adv Differ Equ (2)
Castillo-Chavez C, Song B (2004) Dynamical models of tuberculosis and their applications. Math Biosci Eng 1:361–404
Delavari H, Baleanu D, Sadati J (2012) Stability analysis of Caputo fractional-order nonlinear systems revisited. Nonlinear Dynam 67(4):2433–2439
Derouich M, Boutayeb A, Twizell E (2003) A model of dengue fever. BioMedical Engineering OnLine 2
Diethelm K (2013) A fractional calculus based model for the simulation of an outbreak of dengue fever. Nonlinear Dyn 71(4):613–619
Diethelm K, Freed A (1999) The FracPECE subroutine for the numerical solution of differential equations of fractional order
DOSM (2018) Press statement: Life expectancy at birth (2016-2018), https://www.dosm.gov.my
Du M, Wang Z, Hu H (2013) Measuring memory with the order of fractional derivative. Sci Rep 3:1–3
Dumont Y, Chiroleu F (2010) Vector control for the Chikungunya disease. Math Biosci 7:313–345
Dumont Y, Chiroleu F, Domerg C (2008) On a temporal model for the Chikungunya disease: modeling, theory and numerics. Math Biosci 213:80–91
Esteva L, Vargas C (1998) Analysis of dengue transmission model. Math Biosci 15(2):131–151
Fatmawati M. Khan (2021) Analysis of dengue fever outbreak by generalized fractional derivative. Alex Eng J 60(1):321–336
Fischer A, Chudej K, Pesch H (2019) Optimal vaccination and control strategies against dengue. Math Meth Appl Sci 1–12
Garappa R (2010) On linear stability of predictor-corrector algorithms for fractional differential equations. Int J Comput Math 87(10):2281–2290
Garba S, Gumel A, Abu Bakar M (2008) Backward bifurcations in dengue transmission dynamics. Math Biosci 215:11–25
Garrappa R (2015) Trapezoidal methods for fractional differential equations: theoretical and computational aspects. Math Comput Simul 11:96–112
Hamdan N, Kilicman A (2018) A fractional order SIR epidemic model for dengue transmission. Chaos Solitons Fractals 114:55–62
Hamdan N, Kilicman A (2021) The development of a deterministic dengue epidemic model with the influence of temperature: a case study in Malaysia. Appl Math Model 90:547–567
Hamdan N, Kilicman A Analysis of the fractional order dengue transmission model: a case study in Malaysia. Adv Differ Equ
Huang D, Tang Y, Arshad S, Baleanu D, Al-qurashi M (2016) Dynamical analysis of fractional order model of immunogenic tumors. Adv Mech Eng 8(7)
Islam M, Peace A, Medina D, Oraby T (2020) Integer versus fractional order SEIR deterministic and stochastic models of measles. Environ Res Public Health 17
Jajarmi A, Arshad S, Baleanu D (2019) A new fractional modelling and control strategy for the outbreak of dengue fever. Physica. https://doi.org/10.1016/j.physa.2019.122524
Jan R, Khan M, Kumam P, Thounthong P (2019) Modeling the transmission of dengue infection through fractional derivatives. Chaos Solitons Fractals 127:189–216
Jan R, Khan M, Gomez-Aguilar J (2020) Asymptomatic carriers in transmission dynamics of dengue with control interventions. Optim Control Appl Methods 41(2):430–447
Khatua A, Kar T Dynamical behaviour and control strategy of a dengue epidemic model. Eur Phys J Plus 135(643)
Lin W (2007) Global existence theory and chaos control of fractional differential equations. J Math Anal Appl 332:709–726
Lizarralde-Bejarano D, Arboleda-Sanchez S, Puerta-Yepes M (2017) Understanding epidemics from mathematical models: details of the (2010) dengue epidemic in Bello (Antioquia, Colombia). Appl Math Model 43:566–578
Martcheva M (2015) An Introduction to Mathematical Epidemiology, Text in Applied Mathematics, vol 61. Springer, New York
Matignon D (1996) Stability results for fractional differential equations with applications to control processing. Proc Comput Eng Syst Appl 2:963–968
McCall P, Kelly D (2002) Learning and memory in disease vectors. Trends Parasitol 18(10):429–433
MOH (2018) Health facts 2018. http://www.moh.gov.my
Nikin-Beers R, Blackwood J, Childs L, Ciupe S (2018) Unraveling within-host signatures of dengue infection at the population level. J Theor Biol 446:79–86
Odibat Z, Shawagfeh N (2007) Generalized Taylor’s formula. Appl Math Comput 186(1):286–293
Pinho S, Ferreira C, Esteva L, Barreto F, Morato E, Silva V, Teixeira M (2010) Modelling the dynamics of dengue real epidemics. Philos Trans R Soc 368:5679–5693
Pliego-Pliego E, Vasilieva O, Velazquez-Castro J, Collar A (2021) Control strategies for a population dynamics model of Aedes aegypti with seasonal variability and their effects on dengue incidence. Appl Math Model 81:296–319
Pooseh S, Rodrigues H, Torres S, Delfilm F (2011) Fractional derivatives in dengue epidemics. In: AIP Conference of Proceedings 1389(739)
Rodrigues H, Monteiro M, Torres D (2014) Vaccination models and optimal control strategies to dengue. Math Biosci 247:1–12
Sardar T, Rana S, Chattopadhyay J (2015) A mathematical model of dengue transmission with memory. Commun Nonlinear Sci Numer Simul 22(1–3):511–525
Schoombie A, Bolton S, Cloot L, Slabbert J (2015) A proposed fractional-order Gompertz model and its application in tumor growth data. Math Med Biol 32(2):187–207
Shah K, Jarad F, Abdeljawad T (2020) On a linear fractional order model of dengue fever disease under Caputo-Fabrizio derivative. Alex Eng J 59:2305–2313
Sweilam N, Al-Mekhlafi S, Shatta S (2021) Optimal bang-bang control for variable-order dengue virus; numerical studies. J Adv Res 32:37–44
Takken W, Verhulst N (2013) Host preferences of blood-feeding mosquitoes. Annu Rev Entomol 58:433–453
Ting-Ting Z, Lin-Fei N (2018) Modelling the transmission dynamics of two-strain dengue in the presence awareness and vector control. J Theor Biol 443:82–91
Vargas-De-Leon C (2015) Voltera-type Lyapunov functions for fractional-order epidemic systems. Commun Nonlinear Sci Numer Simul 24(1–3):75–85
Vargas-De-Leon C (2015) Voltera-type Lyapunov functions for fractional-order epidemic systems. Commun Nonlinear Sci Numer Simul 24(1–3):75–85
WHO (2018) Vaccines and immunization: dengue. https://www.who.int/news-room/questions-and-answers/item/dengue-vaccines
WHO (2020) Dengue and severe dengue. http://www.who.int/news-room/fact-sheets/detail/dengue-and-severe-dengue
Woon Y, Hor C, Lee K, Mohd Anuar S, Mudin R, Sheikh Ahmad M, Komari S, Amin F, Jamal R, Chen W, Goh P, Yeap L, Lim Z, Lim T (2018) Estimating dengue incidence and hospitalization in Malaysia, 2001–2013. BMC Public Health 18(946)
Yang H, Ferreira C (2008) Assessing the effects of vector control on dengue transmission. Appl Math Comput 198(1):401–413
Yang H, da Graca Macoris M, Galvani K, Andrighetti M, Wanderley D (2009) Assessing the effects of temperature on the population of Aedes aegypti, the vector of dengue. Epidemiol Infect 137:1188–1202
Yang H, da Graca Macoris M, Galvani K, Andrighetti M (2011) Follow up estimation Aedes aegypti entomological parameters and mathematical modellings. Biosystems 103(3):360–371
Zafar Z, Mushtaq M, Rehan K (2018) A non-integer order dengue internal transmission model. Adv Differ Equ. https://doi.org/10.11186/s13662-018-1472-7
Acknowledgements
The authors are very grateful for partial financial support by the Universiti Putra Malaysia providing Putra Grant GP-IPS/2018/9625000. The authors also thank the Ministry of Education Malaysia and the Universiti Teknologi Mara (UiTM), and special appreciation to the School of Mathematical Sciences, College of Computing, Informatics and Media, Universiti Teknologi MARA. The authors would like to extend sincere gratitude for the constructive comments and suggestions in improving the manuscript by the reviewers.
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Hamdan, N.’., Kilicman, A. Mathematical Modelling of Dengue Transmission with Intervention Strategies Using Fractional Derivatives. Bull Math Biol 84, 138 (2022). https://doi.org/10.1007/s11538-022-01096-2
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DOI: https://doi.org/10.1007/s11538-022-01096-2