×

Numerical treatments of the transmission dynamics of West Nile Virus and it’s optimal control. (English) Zbl 1409.49005

Summary: In this paper, numerical studies for transmission dynamics of West Nile Virus mathematical model are presented. The nonstandard finite difference method is introduced to solve the posed model. Positivity, boundedness, and convergence of the nonstandard finite difference scheme are studied. Also, numerical stability analysis of fixed points is studied. An optimal control problem is formulated and studied theoretically using the Pontryagin’s maximum principle. The obtained results by using nonstandard finite difference method are compared with standard finite difference method. It can be concluded that the nonstandard finite difference method is more efficient and preserves the stability and positivity of the solutions in large regions.

MSC:

49J15 Existence theories for optimal control problems involving ordinary differential equations
65L12 Finite difference and finite volume methods for ordinary differential equations
34D05 Asymptotic properties of solutions to ordinary differential equations
PDFBibTeX XMLCite
Full Text: Link

References:

[1] B. Adams, H. Banks, M. Davidian, H. Kwon, H. Tran, S. Wynne, and E. Rosenberg, HIV Dynamics: Modeling, Data Analysis and Optimal Treatment Protocols, Journal of Computational Applied Mathematics, 184, 10-49, 2005. · Zbl 1075.92030
[2] J. Anderson, T.G. Andreadis, C. Vossbrink, S. Tirrell, E. Wakem, R. French, A. Germendia, and H. Van Kruiningen, Isolation of West Nile Virus from Mosquitoes, Crows, and A Cooper’s Hawk in Connecticut, Science, 286, 2331-2333, 1999.
[3] R. Anguelov and J. M. S. Lubuma, Nonstandard Finite Difference Method by Nonlocal Approximation, Mathematics and computer in Simulation, 61, 36, 465-475, 2003. · Zbl 1015.65034
[4] A. J. Arenas, G. Gonz´alez-Parra., and B. M. Caraballo, A Nonstandard Finite Difference Scheme for a Nonlinear Black-Scholes Equation, Mathematical and Computer Modelling, 57, 1663-1670, 2013. · Zbl 1305.91238
[5] K.A. Bernard, J.G. Maffei, S.A. Jones, E.B. Kauffman, G.D. Ebel, A.P. Dupuis, Jr, K.A. Ngo, D.C. Nicholas, D.M. Young, P.-Y. Shi, V.L. Kulasekera, M. Eidson, D.J. White, W.B. Stone, EJMAA-2019/7(2)NUMERICAL TREATMENTS OF WNV AND IT’S OPTIMAL CONTROL 33 (a) Susceptibles (b) Infected (c) Exposed, Hospitalized and Recovered Humans Figure 3. Numerical simulations of the system (1) when R0> 1 with time step size ∆t = 1 by using SFD method. Team N.S.W.N.V.S., and L.D. Kramer, West Nile Virus Infection in Birds and Mosquitoes, New York State, 2000. Emerging Infectious Diseases, 7, 679-685, 2001.
[6] K. Blayneh, Y. Cao, and H. Kwon, Optimal Control of Vector-Borne Disease: Treatment and Prevention, Discrete and Continuous Dynamical Systems Series, 11, 3, 587-611, 2009. · Zbl 1162.92034
[7] C. Bowman, A. B. Gumel, P. Van den Driessche, J. Wu, and H. Zhu, A Mathematical Model for Assessing Control Strategies against West Nile Virus, Bulletin of Mathematical Biology, 67, 1107-1133, 2005. · Zbl 1334.92392
[8] G. L. Campbell, A. A. Marfin, R. S. Lanciotti, and D. J. Gubler, West Nile Virus: Reviews. Lancet Infectious Diseases, 2, 519-529, 2002.
[9] Center for Disease Control and Prevention (CDC), West Nile Virus: Virus History and Distribution, 2002a. http://www.cdc.gov/ncidod/dvbid/westnile/background.html.
[10] Centers for Disease Control and Prevention (CDC), Intrauterine West Nile Virus InfectionNew York, 2002, MMWR Morb. Mortal. Wkly Rep., 51, 1135-1136, 2002.
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.