Sahand Communications in Mathematical Analysis 2322-5807 University of Maragheh 19749 Hepatitis B virus (HBV) Basic reproduction number ($R_0$) Compound matrices Disease-Free equilibrium state Global stability The analysis of a disease-free equilibrium of Hepatitis B model Reza Akbari Ali Vahidian Kamyad Ali akbar Heydari Aghileh Heydari Department of Mathematical Sciences, Payame Noor University, P.O.Box 19395-3697, Tehran, Iran. Department of Mathematics Sciences, University of Ferdowsi, Mashhad, Iran. Research Center for Infection Control and Hand Hygiene, Mashhad University Of Medical Sciences, Mashhad, Iran. Department of Mathematical Sciences, Payame Noor University, P.O.Box 19395-3697, Tehran, Iran. 06 2016 03 2 1 11 Full Text

In this paper we study the dynamics of Hepatitis B virus (HBV) infection under administration of a vaccine and treatment, where the disease is transmitted directly from the parents to the offspring  and also through contact with infective individuals. Stability of the disease-free steady state is investigated. The basic reproductive rate, $R_0$, is derived. The results show that the dynamics of the model is completely determined by the basic reproductive number $R_0$. If $R_0<1$, the disease-free equilibrium is globally stable and the disease always dies out and if $R_0>1$, the disease-free equilibrium is unstable and the disease is uniformly persistent.

 R.M. Anderson and R.M. May, Infectious Disease of Humans: Dynamics and Control, Oxford University Press, Oxford, 1991.  S. Bhattacharyyaa and S. Ghosh, Optimal control of vertically transmitted disease, Computational and Mathematical Methods in Medicine. 11(4) (2010) 369-387.  S. Bowong, J.J. Tewa, and J.C. Kamgang, Stability analysis of the transmission dynamics of tuberculosis models, World Journal of Modelling and Simulation. 7(2) (2011) 83-100.  P.V.D. Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Math Biosci. 180 (2002) 29-48.  Y. Enatsu, Y. Nakata, and Y. Muroya, Global stability of SIRS epidemic models with a class of nonlinear incidence rates and distributed delays, Published in Acta Mathematica Scientia, 32 (2012) 851-865.  H. Hethcote, The mathematics of infectious diseases, SIAM Review, 42 (2000) 599-653.  J.C. Kamgang and G. Sallet, Computation of threshold conditions for epidemio-logical models and global stability of the disease-free equilibrium (DFE), Mathematical Biosciences, 213 (2008) 1-12.  A.V. Kamyad, R. Akbari, A.A. Heydari, and A. Heydari, Mathematical Modeling of Transmission Dynamics and Optimal Control of Vaccination and Treatment for Hepatitis B Virus, Computational and Mathematical Methods in Medicine. Volume 2014, Article ID 475451, 15 pages.  T.K. Kar and S. Jana, A theoretical study on mathematical modelling of an infectious disease with application of optimal control, Bio Systems, 111(1) (2013) 3750.  T.K. Kar and A. Batabyal, Stability analysis and optimal control of an SIR epidemic model with vaccination, Biosystems. 104 (2011) 127-135.  X. Liu and L. Yang, Stability analysis of an SEIQV epidemic model with saturated incidence rate, Nonlinear Analysis: Real World Applications, 13 (2012) 2671-2679.  J. Mann and M. Roberts, Modelling the epidemiology of hepatitis B in New Zealand, Journal of Theoretical Biology. 269 (2011) 266-272.  G.F. Medley and N.A. Lindop, Hepatitis-B virus endemicity: heterogeneity, catastrophic dynamics and control, Nature Medicine, 7(5) (2001) 619-624.  A.A. Momoh, M.O. Ibrahim, and B.A. Madu, Stability Analysis of an Infectious Disease Free Equilibrium of Hepatitis B Model, Research Journal of Applied Sciences, Engineering and Technology, 3(9) (2011) 905-909.  J.S. Muldowney, Compound matrices and ordinary differential equations, Rocky Mountain J. Math. 20 (1990) 857-872.  J. Pang, J.A. Cui, and X. Zhou, Dynamical behavior of a hepatitis B virus transmission model with vaccination, Journal of Theoretical Biology. 265 (2010) 572-578.  S. Sun, Global Dynamics of a SEIR Model with a Varying Total Population Size and Vaccination, Int. Journal of Math. Analysis, 6(40) (2012) 1985-1995.  C. Sun, Y. Lin, and S. Tang, Global stability for an special SEIR epidemic model with nonlinear incidence rates, Chaos, Solitons and Fractals, 33 (2007) 290-297.  S. Thornley, C. Bullen, and M. Roberts, Hepatitis B in a high prevalence New Zealand population: A mathematical model applied to infection control policy, Journal of Theoretical Biology. 254 (2008) 599-603.  P. Van Den Driessche and J. Watmough, Reproduction numbers and subthreshold endemic equilibrium for compartmental models of disease transmission, Mathe- matical Biosciences, 180 (2002) 29-48.  K. Wanga, A. Fan, and A. Torres, Global properties of an improved hepatitis B virus model, Nonlinear Analysis: Real World Applications, 11 (2010) 3131-3138.  WHO, Hepatitis B Fact Sheet No. 204, The World Health Organisation, Geneva, Switzerland, 2013, http://www.who.int/mediacentre/factsheets/fs204/en/.  S. Zhang and Y. Zhou, The analysis and application of an HBV model, Applied Mathematical Modelling, 36 (2012) 1302-1312.  S.J. Zhao, Z.Y. Xu, and Y. Lu, A mathematical model of hepatitis B virus transmission and its application for vaccination strategy in China, Int.J.Epidemiol. 29 (2000) 744-752.  L. Zou, W. Zhang, and S. Ruan, Modeling the transmission dynamics and control of hepatitis B virus in China, Journal of Theoretical Biology, 262 (2010) 330-338.
Sahand Communications in Mathematical Analysis 2322-5807 University of Maragheh 19750 Growth analysis of entire functions of two complex variables Sanjib Kumar Datta Tanmay Biswas Department of Mathematics, University of Kalyani, P.O.-Kalyani, Dist-Nadia, PIN-741235, West Bengal, India. Rajbari, Rabindrapalli, R. N. Tagore Road, P.O.-Krishnagar, Dist-Nadia, PIN-741101, West Bengal, India. 06 2016 03 2 13 24 Full Text

In this paper, we introduce the idea of generalized relative order (respectively generalized relative lower order) of entire functions of two complex variables. Hence, we study some growth properties of entire functions of two complex variables on the basis of the definition of generalized relative order and generalized relative lower order of entire functions of two complex variables.

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Sahand Communications in Mathematical Analysis 2322-5807 University of Maragheh 19784 Menger probabilistic normed space is a category topological vector space Ildar Sadeqi Farnaz Yaqub Azari Department of Mathematics, Faculty of Science, Sahand University of Technology, Tabriz, Iran. University of Payame noor, Tabriz, Iran. 06 2016 03 2 25 32 Full Text

In this paper, we formalize the Menger probabilistic normed space as a category in which its objects are the Menger probabilistic normed spaces and its morphisms are fuzzy continuous operators. Then, we show that the category of probabilistic normed spaces is isomorphicly a subcategory of the category of topological vector spaces. So, we can easily apply the results of topological vector spaces in probabilistic normed spaces.

 T. Bag and S.K. Samanta, Finite dimensional fuzzy normed linear spaces, Fuzzy Math. 11 (2003) 687-705.  G. Constantin and I. Istratfescu, Elements of probabilistic analysis, Kluwer Academic Publishers, 1989.  P. Freyd, Abelian categories, An Introduction to the theory of functors, Happer & Row, New York, Evanston & London and John Weatherhill, INC, Tokyo, 1964.  A. George and P. Veeramani, On some results in fuzzy metric spaces, Fuzzy Sets and Systems, 64 (1994) 395-399.  U. Hohle, A note on the hyporgraph functor, Fuzzy Sets and Systems, 131 (2002) 353-356.  O. Kaleva and S. Seikala, On fuzzy metric spaces, Fuzzy Sets and Systems, 12 (1984) 143-154.  I. Kramosil and J. Michalek, Fuzzy metric and statistical metric spaces, Kybernetika, 11 (1975) 326-334.  K. Menger, Statistical metrics. Proc. Nat. Acad. Sci, USA, 28 (1942) 53-57.  S.E. Rodabaugh, Fuzzy addition in the L-fuzzy real line, Fuzzy Sets and Systems, 8 (1982) 39-51.  S.E. Rodabaugh, Complete fuzzy topological hyper elds and fuzzy multiplication in the fuzzy real lines, Fuzzy Sets and Systems, 15 (1985) 285-310.  S.E. Rodabaugh, A theory of fuzzy uniformities with applications to the fuzzy real lines, J. Math. Anal. Appl., 129 (1988) 37-70.  W. Rudin, Functional Analysis, Tata McGraw-Hill Publishing Company, 1990.  I. Sadeqi and F. Solaty kia, Fuzzy normed linear space and it's topological structure, Chaos, fractal, solution & Fractals, 40 (2007) 2576-2589.  I. Sadeqi and F. Solaty kia, The category of fuzzy normed linear space, The journal of fuzzy mathematics, 3 (2010) 733-742.  I. Sadeqi, F. Solaty kia and F. Yaqub azari, Menger probabilistic normed linear spaces and its topological structure, Fuzzy Inteligent and Systems, (On published data).  E.S. Santos, Topology versus fuzzy topology, preprint, Youngstown State University, 1977.  B. Schweizer and A. Sklar, Probablisitic metric spaces, North-Holand, Amesterdam, 1983.
Sahand Communications in Mathematical Analysis 2322-5807 University of Maragheh 19751 Marichev-Saigo-Maeda fractional calculus operators Generalized Mittag-Leffler function Generalized Wright hypergeometric function On certain fractional calculus operators involving generalized Mittag-Leffler function Dinesh Kumar Department of Mathematics \& Statistics, Jai Narain Vyas University, Jodhpur - 342005, India. 06 2016 03 2 33 45 Full Text

The object of this paper is to establish certain generalized fractional integration and differentiation involving generalized Mittag-Leffler function defined by Salim and Faraj . The considered generalized fractional calculus operators contain the Appell's function $F_3$ [2, p.224] as kernel and are introduced by Saigo and Maeda . The Marichev-Saigo-Maeda fractional calculus operators are the generalization of the Saigo fractional calculus operators. The established results provide extensions of the results given by Gupta and Parihar , Saxena and Saigo , Samko et al. . On account of the general nature of the generalized Mittag-Leffler function and generalized Wright function, a number of known results can be easily found as special cases of our main results.

 J. Choi and D. Kumar, Certain uni ed fractional integrals and derivatives for a product of Aleph function and a general class of multivariable polynomials, Journal of Inequalities and Applications, 2014 (2014), 15 pages.  A. Erdelyi, W. Magnus, F. Oberhettinger, and F.G. Tricomi, Tables of Integral Transforms, McGraw-Hill, New York, 1, 1954.  A. Gupta and C.L. Parihar, Fractional di erintegral operators of the generalized Mittag-Leer function, Bol. Soc. Paran. Math., 33(1) (2015), 137-144.  H.J. Haubold, A.M. Mathai, and R.K. Saxena, Mittag-Leer functions and their applications, J. Appl. Math. (Article ID 298628) (2011), 1-51.  A.A. Kilbas and M. Saigo, Fractional integrals and derivatives of Mittag-Leer type function, Doklady Akad. Nauk Belarusi, 39(4) (1995), 22-26.  A.A. Kilbas, M. Saigo and R.K. Saxena, Generalized Mittag-Leer function and generalized fractional calculus operators, Integral Transform Special Function, 15 (2004), 31-49.  A.A. Kilbas, M. Saigo and J.J. Trujillo, On the generalized Wright function, Fract. Calc. Appl. Anal., 5(4) (2002), 437460.  Y.C. Kim, K.S. Lee and H.M. Srivastava, Some applications of fractional integral operators and Ruscheweyh derivatives, J. Math. And. Appl., 197(2) (1996), 505-517.  V. Kiryakova, All the special functions are fractional di erintegrals of elementary functions, Journal of Physics A: Mathematical and General, 30(14) (1997), 5085-5103.  D. Kumar and J. Daiya, Fractional calculus pertaining to generalized H-functions, Global Journal of Science Frontier Research: F Mathematics and Decision Sciences, 14(3) (2014), 25-36.  D. Kumar and S. Kumar, Fractional Calculus of the Generalized Mittag-Leer Type Function, International Scholarly Research Notices 2014 (2014), Article ID 907432, 6 pages.  D. Kumar and S.D. Purohit, Fractional di erintegral operators of the generalized Mittag-Leer type function, Malaya J. Mat., 2(4) (2014), 419-425.  D. Kumar and R.K. Saxena, Generalized fractional calculus of the M-Series involving F3 hypergeometric function, Sohag J. Math., 2(1) (2015), 17-22.  O.I. Marichev, Volterra equation of Mellin convolution type with a Horn function in the kernel, Izvestiya Akademii Nauk BSSR. Seriya Fiziko-Matematicheskikh Nauk, 1 (1974), 128-129, (Russian).  A.M. Mathai and H.J. Haubold, Special Functions for Applied Scientists, Springer, New York, 2008.  K.S. Miller and B. Ross, An Introduction to the Fractional Calculus and Fractional Di erential Equations, John Wiley & Sons, New York, NY, USA, 1993.  G.M. Mittag-Leer, Sur la nouvelle fonction E (x), C.R. Acad. Sci. Paris 137 (1903), 554-558.  G.M. Mittag-Leer, Sur la representation analytique d'une branche uniforme d'une function monogene, Acta Math. 29 (1905), 101-181.  J. Paneva-Konovska, Inequalities and asymptotic formulae for the three para-metric Mittag-Leer functions, Math. Balkanica, 26 (2012), 203-210.  J. Paneva-Konovska, Convergence of series in three parametric Mittag-Leer functions, Math. Slovaca 62, 2012.  T.R. Prabhakar, A singular integral equation with a generalized Mittag-Leer function in the Kernel, Yokohama Math. J. 19 (1971), 7-15.  M. Saigo, A remark on integral operators involving the Gauss hypergeometric functions, Math. Rep., College General Ed. Kyushu Univ. 11 (1978), 135-143.  M. Saigo and N. Maeda, More generalization of fractional calculus Transform Methods and Special Functions, Varna, Bulgaria, (1996), 386-400.  T.O. Salim, Some properties relating to the generalized Mittag-Leer function, Adv. Appl. Math. Anal., 4 (2009), 21-30.  T.O. Salim and A.W. Faraj, A generalization of Mittag-Leer function and Integral operator associated with fractional calculus, Journal of Fractional Calculus and Application, 3(5) (2012), 1-13.  S.G. Samko, A.A. Kilbas and O.I. Marichev, Fractional Integrals and Derivatives, Theory and Applications, Gordon and Breach, Yverdon et alibi, 1993.  R.K. Saxena and D. Kumar, Generalized fractional calculus of the Aleph-function involving a general class of polynomials, Acta Mathematica Scientia, 35(5) (2015), 1095-1110.  R.K. Saxena, J. Ram and D. Kumar, Generalized fractional di erentiation of the Aleph-Function associated with the Appell function F3 , Acta Ciencia Indica, 38M(4) (2012), 781-792.  R.K. Saxena, J. Ram and D. Kumar, On the Two-Dimensional Saigo-Maeda fractional calculus associated with Two-Dimensional Aleph Transform, Le Matematiche, 68 (2013), 267-281.  R.K. Saxena and M. Saigo, Certain properties of the fractional calculus operators associated with generalized Mittag-Leer function, Fract. Calc. Appl. Anal., 8(2) (2005), 141-154.  H.M. Srivastava and Z. Tomovski, Fractional calculus with an integral operator containing a generalized Mittag-Leer function in the kernel, Appl. Math. Comput., 211 (2009), 198-210.  A. Wiman, Uber de fundamental satz in der theorie der funktionen E (x), Acta Math. 29 (1905), 191-201.  E.M. Wright, The asymptotic expansion of generalized hypergeometric function, J. London Math. Soc., 10 (1935), 286-293.
Sahand Communications in Mathematical Analysis 2322-5807 University of Maragheh 19832 Delay integral equations Collocation method Multistep collocation method Convergence ‎Multistep collocation method for nonlinear delay integral equations Parviz Darania Department of Mathematics, Faculty of Science, Urmia University, P.O.Box 5756151818, Urmia-Iran. 06 2016 03 2 47 65 Full Text

‎The main purpose of this paper is to study the numerical solution of nonlinear Volterra integral equations with constant delays, based on the multistep collocation method. These methods for approximating the solution in each subinterval are obtained by fixed number of previous steps and fixed number of collocation points in current and next subintervals. Also, we analyze the convergence of the multistep collocation method when used to approximate smooth solutions of delay integral equations. Finally, numerical results are given showing a marked improvement in comparison with exact solution.

 D. Conte and B. Paternoster, Multistep collocation methods for Volterra integral equations, Appl. Numer. Math., 59 (2009) 1721-1736.  V. Horvat, On collocation methods for Volterra integral equations with delay arguments, Mathematical Communications, 4 (1999) 93-109.  H. Brunner, Collocation Methods for Volterra Integral and Related Functional Equations, Cambridge Monographs on Applied and Computational Mathematics, Cambridge University Press, Cambridge, 2004.  H. Brunner, High-order collocation methods for singular Volterra functional equations of neutral type, Applied Numerical Mathematics, 57 (2007) 533-548.  H. Brunner, Iterated collocation methods for Volterra integral equations with delay arguments, Math. Comp., 62 (1994) 581-599.  H. Brunner, Implicitly linear collocation methods for nonlinear Volterra integral equations, Appl. Numer. Math., 9 (1992) 235-247.  I. Ali, H. Brunner, and T. Tang, Spectral methods for pantograph-type di erential and integral equations with multiple delays, Front. Math. China, 4 (2009) 49-61.  H. Brunner, The numerical solution of weakly singular Volterra functional integro-di erential equations with variable delays, Comm. Pure Appl. Anal. 5 (2006) 261-276.  Y. Liu, Stability analysis of θ-methods for neutral functional-di erential equations, Numer. Math., 70 (1995) 473-485.  T. Tang, X. Xu, and J. Cheng, On the spectral methods for Volterra type integral equations and the convergence analysis, J. Comput. Math., 26 (2008) 825-837.  V. Volterra, Lecons sur les equations integrals et les equations integro-di erentielles, Gauthier-Villars, Paris, 1913.  J. Cerha, On some linear Volterra delay equations, Casopis Pest Mat., 101 (1976) 111-123.
Sahand Communications in Mathematical Analysis 2322-5807 University of Maragheh 19748 On the topological centers of module actions Kazem Haghnejad Azar Department of Mathematics, University of Mohaghegh Ardabili, Ardabil, Iran. 06 2016 03 2 67 74 Full Text

In this paper, we  study the Arens regularity properties of module actions. We investigate some properties of topological centers of module actions ${Z}^\ell_{B^{**}}(A^{**})$ and  ${Z}^\ell_{A^{**}}(B^{**})$ with some conclusions in group algebras.

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Sahand Communications in Mathematical Analysis 2322-5807 University of Maragheh 17973 Inverse Sturm-Liouville problem Jump conditions Non self-adjoint operator Parameter dependent condition Inverse Sturm-Liouville problems with a Spectral Parameter in the Boundary and transmission conditions Mohammad Shahriari Department of Mathematics, Faculty of Science, University of Maragheh, Maragheh, Iran. 06 2016 03 2 75 89 Full Text

In this manuscript, we study the inverse problem for non self-adjoint Sturm--Liouville operator $-D^2+q$ with eigenparameter dependent boundary and discontinuity conditions inside a finite closed interval. By defining  a new Hilbert space and  using its spectral data of a kind, it is shown that the potential function can be uniquely determined by part of a set of values of eigenfunctions at some interior point and  parts of two  sets of eigenvalues.

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Sahand Communications in Mathematical Analysis 2322-5807 University of Maragheh 18507 Doubly stochastic matrix Linear preserver Multiplicative map On multiplicative (strong) linear preservers of majorizations Mohammad Ali Hadian Nadoshan Ali Armandnejad Department of Mathematics, Vali-e-Asr University of Rafsanjan, Zip Code: 7718897111, Rafsanjan, Iran. Department of Mathematics, Vali-e-Asr University of Rafsanjan, Zip Code: 7718897111, Rafsanjan, Iran. 06 2016 03 2 91 106 Full Text

‎In this paper, we study some kinds of majorizations on $\textbf{M}_{n}$ and their linear or strong linear preservers. Also, we find the structure of linear or strong linear preservers which are multiplicative, i.e.  linear or strong linear preservers like $\Phi$ with the property $\Phi (AB)=\Phi (A)\Phi (B)$ for every $A,B\in \textbf{M}_{n}$.

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Sahand Communications in Mathematical Analysis 2322-5807 University of Maragheh 19780 On $n$-derivations Mohammad Hossein Sattari Department of Mathematics, Faculty of Science, Azarbaijan Shahid Madani University, P.O.Box 53751-71379, Tabriz, Iran. 06 2016 03 2 107 115 Full Text

In this article, the notion of $n-$derivation is introduced for all integers $n\geq 2$. Although all derivations are $n-$derivations,  in general these notions are not equivalent. Some properties of ordinary derivations are  investigated for $n-$derivations. Also, we show that under certain mild condition  $n-$derivations are derivations.

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