Sains Malaysiana 49(5)(2020): 1191-1200

http://dx.doi.org/10.17576/jsm-2020-4905-24

 

Mathematical Model of Dengue Virus with Predator-Prey Interactions

 (Model Matematik Virus Denggi dengan Interaksi Pemangsa-Mangsa)

SARINAH BANU MOHAMED SIDDIK1*, FARAH AINI ABDULLAH2 & AHMAD IZANI MD. ISMAIL2

 

1Institute of Engineering Mathematics, Universiti Malaysia Perlis, 02600 Arau, Perlis, Malaysia

 

2School of Mathematical Sciences, Universiti Sains Malaysia, 11800 USM Pulau Pinang, Malaysia

 

Diserahkan: 8 Ogos 2019/Diterima: 15 Januari 2020

 

ABSTRACT

In this paper, a mathematical model of dengue incorporating two sub-models that: describes the linked dynamics between predator-prey of mosquitoes at the larval stage, and describes the dengue spread between humans and adult mosquitoes, is formulated to simulate the dynamics of dengue spread. The effect of predator-prey dynamics in controlling the dengue disease at the larval stage of mosquito populations is investigated. Stability analysis of the equilibrium points are carried out. Numerical simulations results indicate that the use of predator-prey dynamics of mosquitoes at the larval stage as biological control agents for controlling the larval stage of dengue mosquito assists in combating dengue virus contagion.

Keywords: Dengue virus; endemic equilibrium; numerical simulation; predator-prey

 

ABSTRAK

Dalam kertas ini, satu model matematik denggi yang menggabungkan dua sub-model iaitu: menerangkan dinamik antara nyamuk pemangsa-mangsa pada peringkat jejentik dan menerangkan penyebaran denggi antara manusia dan nyamuk dewasa, diformulasikan untuk mensimulasi dinamik penyebaran denggi.  Kesan dinamik pemangsa-mangsa untuk mengawal penyakit denggi pada peringkat jejentik populasi nyamuk diselidik. Analisis kestabilan titik kesimbangan dijalankan. Simulasi berangka menunjukkan bahawa penggunaan dinamik pemangsa-mangsa nyamuk pada peringkat jejentik sebagai agen kawalan biologi untuk mengawal tahap jejentik nyamuk denggi membantu dalam memerangi penularan virus denggi.

Kata kunci: Keseimbangan endemik; pemangsa-mangsa; simulasi berangka; virus denggi

 

RUJUKAN

Ali, T.M., Kamil, A.A. & Karim, M.F.A. 2015. Deterministic mathematical model of dengue disease spread. Far East Journal of Mathematical Sciences 96(4): 419-436.

Al-Sulami, H., El-Shahed, M., Nieto, J.J. & Shammakh, W. 2014. On fractional order dengue epidemic model. Hindawi Publishing Corporation 2014: 456537.

Andraud, M., Hens, N., Marais, C. & Beutels, P. 2012. Dynamic epidemiological models for dengue transmission: A systematic review of structural. PLoS Comput. Biol. 7(11): 332-346.

Bailey, N.T.J. 1975. The Mathematical Theory of Infectious Diseases and Its Applications. London: Griffin.

Benelli, G., Jeffries, C.L. & Walker, T. 2016. Biological control of mosquito vectors: Past, present and future. Insects 7(4): e52.

Cooke, K.L. & Van Den Driessche, P. 1996. Analysis of an SEIRS epidemic model with two delays. Journal of Mathematical Biology 35(2): 240-260.

Derouich, M. & Boutayeb, A. 2006. Dengue fever: Mathematical modelling and computer simulation. Applied Mathematics and Computation 177(2): 528-544.

Diekmann, O., Heesterbeek, J. & Metz, J.A. 1990. On the definition and the computation of the basic reproduction ratio  in models for infectious diseases in heterogeneous populations. Journal of Mathematical Biology 28(4): 365-382.

Erikson, R.A., Presley, S.M., Allen, L.J.S., Long, K.R. & Cox, S.B. 2011a. A stage-structured, Aedes albopictus population model. Ecological Modelling 221(9): 1273-1282.

Erikson, R.A., Presley, S.M., Allen, L.J.S., Long, K.R. & Cox, S.B. 2011b. A dengue model with a dynamic Aedes albopictus vector population. Ecological Modelling 221(24): 2899-2908.

Esteva, L. & Vargas, C. 1998. Analysis of a dengue disease transmission model. Mathematical Biosciences 150(2): 131-151.

Ghosh, M., Lashari, A.A. & Li, X.Z. 2013. Biological control of malaria: A mathematical model. Applied Mathematics and Computation 219(15): 7923-7939.

Heffernan, J.M., Smith, R.J. & Wahl, L.M. 2005. Perspectives on the basic reproductive ratio. Journal of the Royal Society Interface 2(4): 281-293.

Huang, Y.J.S., Stephens, H. & Vanlandingham, D.L. 2017. Biological control strategies for mosquito vectors of arboviruses. Insect 8(1): 21-28.

Hove-Musekwa, S.D. 2008. Determining effective spraying periods to control malaria via indoor residual spraying in Sub-Saharan Africa. Journal of Applied Mathematics and Decision Sciences 2008: 745463.

Lou, Y. & Zhao, X.Q. 2011. Modelling malaria control by introduction of larvivorous fish. Bulletin of Mathematical Biology 73(10): 2384-2407.

Menach, A.L., McKenzie, F.E., Flahault, A. & Smith, D.L. 2005. The unexpected importance of mosquitoes oviposition behavior for malaria: Non-productive larval habitats can be sources for malaria transmission. Malaria Journal 4(1): e23.

Moore, S.M., Borer, E.T. & Hosseini, P.R. 2010. Predators indirectly control vector borne disease: Linking predator-prey and host-pathogen models. Journal of the Royal Society Interface 7(42): 161-176.

Nuraini, N., Tasman, H., Soewono, E. & Sidarto, K.A. 2009. A with-in host dengue infection model with immune response. Mathematical and Computer Modelling 49(5-6): 1148-1155.

Nyamah, M.A., Sulaiman, S. & Omar, B. 2011. Field observation on the efficacy of Toxorhynchites splendens (wiedemann) as a biocontrol agent against Aedes albopictus (skuse) larvae in a cemetery. Trop. Biomed.28(2): 312-319.

Ong, S.Q. 2016. Dengue vector control in Malaysia: A review for current and alternative strategies. Sains Malaysiana 45(5): 777-785.

Pandey, A., Mubayi, A. & Medlock, J. 2013. Comparing vector-host and SIR models for dengue transmission. Mathematical Biosciences 246(2): 252-259.

Rodrigues, H.S., Monteiro, M.T.T., Torres, D.F.M. & Zinober, A. 2012. Dengue disease, basic reproduction number and control. International Journal of Computer Mathematics 89(3): 334-346.

Steffan, W.A. & Evenhuis, N.L. 1981. Biology of Toxorhynchites. Annual Review of Entomology 26: 159-181.

Wen, T.H., Tsai, C.T. & Chin, W.C.B. 2016. Evaluating the role of disease importation in the spatiotemporal transmission of indigenous dengue outbreak. Applied Geography 76: 137-146.

World Health Organization (WHO). 2016. Dengue Report 2016. http://www.who.int/dengue/publications/world_dengue_report_2016/report/en.

Yang, H.M. & Ferreira, C.P. 2008. Assessing the effects of vector control on dengue transmission. Applied Mathematics and Computation 198: 401-413.

Zaini, Z.I.I., Othman, H., Karim, N., Rashid, N.A.A., Abas, M.B.H., Sahani, M., Hod, R. & Nordin, S.A. 2019. Knowledge and practices regarding Aedes control amongst residents of dengue hotspot areas in Selangor: A cross-sectional study. Sains Malaysiana 48(4): 841-849.

Zuharah, W.F., Fadzly, N., Yusof, N.A. & Dieng, H. 2015. Risky behaviors: Effects of Toxorhynchites splendens (Diptera:culicidae) predator behavior of three mosquito species. Journal of Insect Sciences 15(1): 128-134.

 

*Pengarang untuk surat-menyurat; email: sarinah@unimap.edu.my

 

 

sebelumnya