iosr-Jm   Volume-3 ~ Issue-3

Abstract :The present paper considers a more practical problem of scheduling n jobs in a two machine specially structured open shop to minimize the rental cost. Further the processing time of jobs is associated with their respective probabilities including transportation time. In most of literature the processing times are always considered to be random, but there are significant situations in which processing times are not merely random but bear a well defined structural relationship to one another. The objective of this paper is to minimize the rental cost of machines under a specified rental policy. The algorithm is demonstrated through the numerical illustration.
Keywords - Open Shop Scheduling, Rental Policy, Processing Time, Utilization Time, Make span, Idle Time. Mathematical Subject Classification: 90B30, 90B35

[1] Anup (2002), "On two machine flow shop problem in which processing time assumes probabilities and there exists equivalent for
an ordered job block", JISSOR, Vol. XXIII No. 1-4, pp. 41-44.
[2] Bagga P C (1969), "Sequencing in a rental situation", Journal of Canadian Operation Research Society, Vol.7, pp.152-153.
[3] Baker, K. R. (1974), "Introduction of sequencing and scheduling," John Wiley and Sons, New York.
[4] Bellman, R. (1956), "Mathematical aspects of scheduling theory", J. Soc. Indust. Appl. Math. 4(3),168-205.
[5] Belwal & Mittal (2008), "n jobs machine flow shop scheduling problem with break down of machines,transportation time and
equivalent job block", Bulletin of Pure & Applied Sciences-Mathematics, Jan – June,2008, source Vol. 27, Source Issue 1.
[6] Chander S, K Rajendra & Deepak C (1992), "An Efficient Heuristic Approach to the scheduling of jobs in a flow shop",
European Journal of Operation Research, Vol. 61, pp.318-325.
[7] Chandramouli, A. B. (2005), "Heuristic approach for n-jobs, 3-machines flow-shop scheduling problem involving transportation
time, breakdown time and weights of jobs", Mathematical and ComputationalApplications 10(2), pp 301-305.
[8] Chandrasekharan R (1992), "Two-Stage Flowshop Scheduling Problem with Bicriteria " O.R. Soc. ,Vol. 43, No. 9, pp.871-84.
[9] D. Rebaine, V.A. Strusevich(1998), Two-machine open shop scheduling with special transportation times, CASSM R&D Paper 15,
University of Greenwich, London, UK..
[10] Gupta Deepak (2005), "Minimizing rental cost in two stage flow shop , the processing time associated with probabilities including
job block", Reflections de ERA, Vol 1, No.2, pp.107-120.

Abstract :This research deals with mathematical modelling of malaria transmission in North Senatorial Zone of Taraba State, Nigeria. The SIR proposed by Kermack and McKendrick and data obtained from Essential Programme on Immunisation (EPI) unit, F.M.C., Jalingo, Taraba state were used to analyse the rate of infection of malaria in the zone. From our analysis, we found out that the reproduction ratio (R0 )  0 . Based on the reproduction ratio 0 R , which is greater than 0, implies that the force of malaria infection in Taraba North Senatorial Zone is high. The researchers also make recommendations for the reduction of malaria in the zone. ,br> Keywords: Malaria, stability, equilibrium states, epidemics.

[1] Anderson, R. M., May, R. M. (1991). Infectious Diseases of Humans: Dynamics and Control.Oxford University Press, Oxford.
[2] Hyun, M. Y. (2000). Malaria transmission model for different levels of acquired immunity and temperature dependent parameters
vector. Rev. Saude Publica., 34(3): 223-231.
[3] Isao, K., Akira, S., Motoyoshi, M. (2004). Combining Zooprophylaxis and insecticide spraying A malaria-control strategy limiting
the development of insecticides resistance in vector mosquitoes. Proc. R. Soc. Lond., 271: 301-309.
[4] Jia, L. (2008). A malaria model with partial immunity in humans. Math. Bios. Eng., 5(4): 789- 801.
[5] Kermack, W. O. and Mckendric, A.G. ( 1927). A contribution to the mathematical theory of epidemics: preceedings of the Royal
society of London. Series A, Containing papers of a mathematical and physical character, 115:700-721.
[6] Makinde, O. D., Okosun, K. O. (2011). Impact of chemo-theraphy on optimal control of malaria disease with infected immigrants.
BioSystems. 104:32-41
[7] McDonald, G. (1957) The epidemiology control of malaria, Oxford university press, London.
[8] Okosun, K. O. (2010). Mathematical epidemiology of Malaria Disease Transmission and its Optimal Control Analyses, Ph.D.
thesis, University of the Western Cape, South Africa Puntani, P. I-ming, T. (2010). Impact of cross-border migration on disease
epidemics: case of the P. falciparum and P. vivax malaria epidemic along the Thai-Myanmar border. J. Bio. Sys., 18(1): 55-73
[9] Rafikov, M., Bevilacqua, L., Wyse, A. A. P. (2009). Optimal control strategy of malaria vector using genetically modified
mosquitoes. J. Theore. Bio., 258: 418-425.
[10] RollbackMalaria.What is Malaria?http://woo.rollback-malaria.org/cmc- upload/0/000/015/372/RBMinfosheet-1.pdf.(2010-05-
10).

Abstract :In this work, The researchers present Passively Immune Infant( ) m V -Susceptible class( ) m S - Infection class( ) m I -Recovery class( ) m R model to study the dynamic of tuberculosis transmission and vaccination impact in North Senatorial Zone, Taraba State, Nigeria. The compartment of the model is presented in a system of ordinary differential equations. Quantitative analysis of the model was done to investigate the equilibrium and stability of the model. An analytical approach was used to determine their Disease Free equilibrium and the Epidemic equilibrium state. The stability of the epidemic equilibrium is tested using Bellman and Cooke's theorem. The model had two equilibrium position: The disease free equilibrium which was asymptotically stable for ( ) 0 e R   and the endemic equilibrium which was locally asymptotically stable for as it satisfies the Bellman and Cooke's condition for stability i.e. J  0 .
Keywords -Tuberculosis (TB), Vaccination, Infection, Equilibrium analysis, Stability analysis.

[1] Aparicio, J.P., Capurri, A.F., Castillo-Chavez, C. (2000), "Transmission and dynamics of Tuberculosis on generalized household',
J. Theoretical Biology.
[2] Bellman R. and Cooke K.C. (1963), "Differential difference equation". London. Academic press.
[3] Castillo-Chavez, C. and Feng, Z., (1998), Global stability of an age-structure model for TB and its applications to optimal
vaccination strategies. Math Biosci,151(2):135–154.
[4] Dye, C., Garnett, G. P., Sleeman, K. and Williams, B. G.(1998), Prospects for Worldwide tuberculosis control under the who dots
strategy. Directly observed short-course therapy.
[5] Fine PEM (1988). BCG vaccination against tuberculosis and leprosy.Br Med Bull; 44:691-703.
[6] Fine PEM, Ponnighaus J.M., Maine N.P. (1986), The relationship between delayed type hypersensitivity and protective immunity
induced by mycobacterial vaccines in man.Symposium on the Immunology of Leprosy, Oslo, Norway.
[7] Heimbeck J. Sur la vaccination préventive de la tuberculose par injection sous-cutanée de BCG chez les élèves infirmières de
l'hôpital Ulleval à Oslo (Norvège).
[8] Milstien JB, Gibson JJ. (1989)., Quality control of BCG vaccines by the World Health Organization: a review of factors that may
influence vaccine effectiveness and safety. Bull WHO; 68:93-108.
[9] Myint TT, Win H, Aye HI-I, (1987), Case-control study on evaluation of BCG vaccination of newborn in Rangoon, Burma. Ann
Trop Paediatr 1987;7:159-166.
[10] Ndaman I.(2010) A deterministic mathematical model of Tuberculosis disease dynamics, M. TECH thesis, F.U.T Minna.

Abstract :The application of differential equations towards stability analysis of ferrofluids is analysis both in porous medium and nonporous medium and a comparative analysis is made. Weakly non- linear analysis is carried out. A mathematics model of the differential equations employed is presented. The non-dimensional thermal Rayleigh number Ra and magnetic Rayleigh number Rm are analysed with allowable range of parameter.
Keywords -Ferrofluids, Mathematical Model, Non Porous Medium, Porous Medium,Weakly Non Linear Equations.

[1]. Baily R.L (1983): Lessor known applications of ferrofluids:- J.M.M.M.,vol.39,pp.178-182.
[2]. Berkovskii B.M., Medvedev V.F. and Krakov M.S(1993): Magnetic fluids-engineering applications-oxford: oxford science
publications.
[3]. Chandrasekher S.(1961): Hydrodynamic and stability-oxford:Clarendon.
[4]. Finlayson B.A. (1970): Convective instability of ferromagnetic fluids –Journal of Fluid mech., Vol.40,pp.753-767.
[5]. E.R.Benton(1966). On the flow due to a rotating disk- Journal of Fluid mech. 24(4),pp.781-800.
[6]. H.Schlichting (1960), Boundary Layer Theory, McGraw-Hill Book company, New York..
[7]. H.A.Attia (2009),Steady flow over a rotating disk in porous medium with heat transfer. Non-Linear analysis modelling and
control.14(1)pp.21-26.
[8]. J.L.Newringer, R.E.Rosensweig(1964),Magnetic fluids, Physics of fluids ,1927.
[9]. M.I.Shliomis(2004), Ferrofluids as thermal ratchets. Physical Review Letters,92(18),188901.
[10]. P.D.S. Verma, M.Singh (1981),Magnetic fluid flow through porous annulus. Int.J.Non-linear Mechanics,16(3/4),pp.371-378.