# zbMATH — the first resource for mathematics

Application of three controls optimally in a vector-borne disease – a mathematical study. (English) Zbl 1306.92059
Summary: We have proposed and analyzed a vector-borne disease model with three types of controls for the eradication of the disease. Four different classes for the human population namely susceptible, infected, recovered and vaccinated and two different classes for the vector populations namely susceptible and infected are considered. In the first part of our analysis the disease dynamics are described for fixed controls and some inferences have been drawn regarding the spread of the disease. Next the optimal control problem is formulated and solved considering control parameters as time dependent. Different possible combination of controls are used and their effectiveness are compared by numerical simulation.

##### MSC:
 92D30 Epidemiology 49N90 Applications of optimal control and differential games
Full Text:
##### References:
 [1] Chiyaka, C.; Garira, W.; Dube, S., Effects of treatment and drug resistance on the transmission dynamics of malaria in endemic areas, Theor Popul Biol, 75, 14-29, (2009) · Zbl 1210.92005 [2] Kermack, W. O.; McKendrick, A. G., Contributions to the mathematical theory of epidemics, Proc R Soc Lond A, 141, 94-122, (1933) · Zbl 0007.31502 [3] Okosun, K. O.; Makinde, O. D., Modelling the impact of drug resistance in malaria transmission and its optimal control analysis, Int J Phys Sci, 6, 28, 6479-6487, (2011) [4] Nakul, C.; Hyman, J. M.; Cushing, J. M., Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model, Bull Math Biol, 70, 1272-1296, (2008) · Zbl 1142.92025 [5] Makinde, O. D.; Okosun, K. O., Impact of chemo-therapy on optimal control of malaria disease with infected immigrants, Biosystems, 104, 1, 32-41, (2011) [6] Lashari, A. A.; Zaman, G., Optimal control of a vector-borne disease with horizontal transmission in host population, Comput Math App, 61, 745-754, (2011) · Zbl 1217.34064 [7] Okosun, K. O.; Ouifki, R.; Marcus, N., Optimal control analysis of a malaria disease transmission model that includes treatment and vaccination with waning immunity, Biosystems, 106, 136-145, (2011) [8] Birkoff, G.; Rota, G. C., Ordinary differential equations, (1982), Ginn Boston [9] Lashari, A. A.; Zaman, G., Optimal control of a vector-borne disease with horizontal transmission, Nonlinear Anal Real World Appl, 13, 203-212, (2012) · Zbl 1238.93066 [10] Zaman, G.; Kang, Y. H.; Jung, I. H., Stability analysis and optimal vaccination of an SIR epidemic model, Biosystems, 93, 240-249, (2008) [11] Joshi, H. R., Optimal control of an HIV immunology model, Optim Cont Appl Methods, 23, 199-213, (2002) · Zbl 1072.92509 [12] Kar, T. K.; Batabyal, A., Stability analysis and optimal control of an SIR epidemic model with vaccination, Biosystems, 104, 127-135, (2011) [13] Lukes, D. L., Differential equations: classical to controlled, (Mathematics in science and engineering, vol. 162, (1982), Academic Press New York) · Zbl 0509.34003 [14] Pontryagin, L. S.; Boltyanskii, V. G.; Gamkrelidze, R. V.; Mishchenko, E. F., The mathematical theory of optimal processes, (1962), Wiley New York · Zbl 0102.32001 [15] Amaku, M.; Coutinho, F. A.B.; Massad, E., Why dengue and yellow fever coexist in some areas of the world and not in others?, Biosystems, 106, 111-120, (2011) [16] Lenhart, S.; Workman, J. T., Optimal control applied to biological models, Mathematical and computational biology series, (2007), Chapman & Hall/CRC [17] Sachs, J., A new infected global effort to control malaria, Science, 298, 122-124, (2002) [18] Moulay, D.; Aziz-Alaoui, M. A.; Kwon, H. D., Optimal control of chikungunya disease: larvae reduction, treatment and prevention, Math Biosci Eng, 9, 2, 369-392, (2012) · Zbl 1260.92068 [19] Li, G.; Wang, W.; Jin, Z., Global stability of an SEIR epidemic model with constant immigration, Chaos Solitons Fractals, 30, 1012-1019, (2006) · Zbl 1142.34352 [20] Yi, N.; Zhang, Q.; Mao, K.; Yang, D.; Li, Q., Analysis and control of an SEIR epidemic system with nonlinear transmission rate, Math Comput Modell, 50, 1498-1513, (2009) · Zbl 1185.93101 [21] Meng, X.; Chen, L., Global dynamical behaviors for an SIR epidemic model with time delay and pulse vaccination, Taiwanese J Math, 12, 5, 1107-1122, (2008) · Zbl 1166.34046 [22] Kar, T. K.; Mondal, P. K., Global dynamics and bifurcation in delayed SIR epidemic model, Nonlinear Anal Real world Appl, 12, 2058-2068, (2011) · Zbl 1235.34216 [23] Kar, T. K.; Jana, S., A theoretical study on mathematical modeling of an infectious disease with application of optimal control, Biosystems, 111, 1, 37-50, (2013) [24] Kar, T. K.; Ghorai, A.; Jana, S., Dynamics of pest and its predator model with disease in the pest and optimal use of pesticide, J Theor Biol, 310, 187-198, (2012) · Zbl 1337.92180 [25] Kar, T. K.; Ghosh, B., Sustainability and optimal control of an exploited prey predator system through provision of alternative food to predator, Biosystems, 109, 220-232, (2012)
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. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.