A Stability Test for Non Linear Systems of Ordinary Differential Equations Based on the Gershgorin Circles
|
|
- Brent Haynes
- 5 years ago
- Views:
Transcription
1 Contemporary Engineering Sciences, Vol. 11, 2018, no. 91, HIKARI Ltd, A Stability Test for Non Linear Systems of Ordinary Differential Equations Based on the Gershgorin Circles Danilo Alonso Ortega Bejarano, Eduardo Ibargüen-Mondragón Departamento de Matemáticas y Est., Facultad de Ciencias Exactas y Nat. Grupo de Investigación en Biología Matemática y Matemática Aplicada (GIBIMMA) Universidad de Nariño, Pasto, Colombia Enith Amanda Gómez-Hernández Maestría en Biomatemáticas, Facultad de Ciencias Exactas y Nat. Universidad del Quindío, Armenia, Colombia Copyright c 2018 Danilo Alonso Ortega Bejarano et al. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. Abstract The Gershgorin Circles Theorem (GCT) is a very useful tool to characterize the regions of the complex plane in which the eigenvalues of a matrix are found. Within the analysis of local stability to equilibrium solution x of a system of ordinary differential equations is vital to determine the sign of the real part of the eigenvalues of the Jacobian matrix evaluated in x. For this reason, a local stability test is formulated for equilibrium solutions, based on the indirect method of Lyapunov and GCT. Keywords: Circles of Gershgorin, Indirect method of Lypunov, local stability, dynamical systems. 1 Introduction The analytical solutions of nonlinear systems of ordinary differential equations can not always be explicitly determined. Moreover, it could be said
2 4542 Danilo Alonso Ortega Bejarano et al. that in a few cases the explicit solution can be determined. In 1892, A. M. Lyapunov established a criterion that characterizes the local behavior of the trajectories of the following dynamical system dx = f(x), (1) where f : D R n is a C 1 map and D R n. The criterion is known as the first method of Lyapunov or indirect method of Lyapunov (IML) and use the technique of linearization to determine the behavior of solutions near equilibrium points. This method allows us to analyze the stability of equilibrium solutions of the system (1) by studying the stability of the trivial solution for the linearized system dy = Df( x)y + G(y), (2) where G(y) is O ( y 2 ). The solutions of nonlinear systems near equilibrium points resemble those of their linear parts only in the case where the linearized system is hyperbolic; that is, when neither of the eigenvalues of the system has zero real part [3]. In this sense, we say that an equilibrium point x of a nonlinear system (1) is hyperbolic if all of the eigenvalues of Df( x) have nonzero real parts. If all the eigenvalues of Df( x) have a negative real part, then x is called a sink, if all the eigenvalues of Df( x) have a positive real part, then x is called a source and if Df( x) has at least an eigenvalue with a positive real part and an eigenvalue with a negative real part, x is called a saddle [2]. On the other hand, the following result establishes a connection between the hyperbolic equilibrium x of (1) and the equilibrium y = 0 of the linearization (2). Proposition 1.1 Suppose the n-dimensional system dx/ = f(x) has an hyperbolic equilibrium point at x. Then the nonlinear flow is conjugate to the flow of the linearized system in a neighborhood of x. See [2] for the proof of the Proposition 1.1, above proposition implies the asymptotic local stability of x when it is a sink and instability when it is a source or a saddle. The following result is a corollary of Proposition 1.1 Proposition 1.2 Suppose the n-dimensional system dx/ = f(x) has an hyperbolic equilibrium point at x. If all eigenvalue of Df( x) have negative real part then x is locally asymptotically stable. If any eigenvalue of Df( x) have positive real part then x is unstable. To use this criterium, the sign of the eigenvalues of the Jacobian matrix Df( x) must be determined, generally the estimation of these signs is a very difficult task to carry out for matrices of dimension greater than or equal to 3. For this reason, in this paper we focus on determining conditions that allow us to establish the sign of the eigenvalues of Df( x) through the Gershgorin circles.
3 A test of stability based on Gershgorin circles Gershgorin circles Theorem Proposition 2.1 Let A = (a ij ) be a square complex matrix. Then every eigenvalue of A lies in one of the Gershgorin circles D i = {z C : z a ii R i } D j = {z C : z a jj R j }, (3) where R i = n j=1,j i a ij and R j = n i=1,i j a ij. The union of the n Gersgorin disks is called the Gersgorin set, n D = D i. (4) i=1 We observe that D is closed and bounded in C, and all eigenvalues of A are elements of D, [18]. 3 Test of stability Proposition 3.1 Let x an equilibrium point of (1), J 11 J 12 J 1n J 21 J 22 J 2n Df( x) =.....,. J n1 J n2 J nn the Jacobian matrix of (1) evaluated in x and n R i = J ij (6) j=1,j i for i = 1,..., n. If J ii < 0 and R i < J ii for i = 1,..., n then x is locally asymptotically stable. Proof: The Gersgorin circles for Df( x) are given by (5) C i = {z C : z (J ii, 0) R i }. (7) From Proposition 1 we conclude that every eigenvalue of Df( x) lies in C k for some 1 k n. Therefore, all eigenvalues of Df( x) are in the Gersgorin set C = n i=1 C i. Since J ii < 0 for i = 1,..., n then the centers (J ii, 0) are located on the negative real half-axis of the complex plane, and the distance between (J ii, 0) and (0, 0) is J ii for i = 1,..., n. By hypothesis R i < J ii which implies C i {z C : z (J ii, 0) < J ii } for i = 1,..., n. In consequence, every Gersgorin cirle C i is contained in the union of the second and third quadrants of the complex plane, then the Gersgorin set C is also contained in above region. Therefore, all eigenvalues of Df( x) have negative real part. Now, from Proposition 1.2 we conclude that x is locally asymptotically stable
4 4544 Danilo Alonso Ortega Bejarano et al. 4 Application of main result The test can be used to verify the equilibrium solution of different models such as those developed in [4, 5, 6, 7, 8, 9, 10, 11, 12, 13, 14, 15, 16, 17]. However, in this section we will analyze the stability of the SEIR model, a multi-patch model and a model on immunology of tuberculosis. 4.1 The SEIR model The SEIR model describes the interaction dynamics of individuals between four compartments: susceptible (S), exposed (E), infected (I), and Recovered (R). The model is given by ds = Λ βsi µs, de = βsi (ɛ + µ)e (8) di = ɛe (γ + µ)i, dr = γi µr, where Λ, β, ɛ, γ and µ are positive constat parameters. solutions of (9) are given by E 0 = ( ) Λ, 0, 0, 0 µ, E 1 = The equilibrium ( Λ Λ, µr 0 ɛ + µ R 0(R 0 1), µ β (R 0 1), γ ) β (R 0 1),(9) Λβɛ where R 0 =. The elements of the diagonal of Df(E µ(ɛ+µ)(γ+µ) 0) are J 11 = J 44 = µ, J 22 = (ɛ + µ) and J 33 = (γ + µ). Note that J ii < 0 for i = 1,..., 4 which satisfies the first hypothesis of the Proposition 3.1. The second hypothesis, R i < J ii for i = 1,..., 4, is verified if and only if the following inequalities βλ µ < µ, βλ µ < ɛ + µ, ɛ < γ + µ and µ < γ, (10) are satisfied. Therefore, if (10) is taken, then E 0 is locally asymptotically stable. Multiplying the second and third inequality of (10) we obtain Λβɛ/µ < (ɛ + µ)(γ + µ) or equivalently R 0 < 1. Similarly, we verify that all elements of the diagonal of Df(E 1 ) are negative real number, and the second hypothesis is verified if and only if the following inequalities βλ µr 0 < µr 0, µ(r 0 1) + βλ µr 0 < ɛ + µ, ɛ < γ + µ and γ < µ, (11) Λ. There- µ are satisfied. From (11) we obtain the following condition R 0 > β µ fore, if (11) is taken then E 1 is locally asymptotically stable. γ µ
5 A test of stability based on Gershgorin circles Two-patch model for the dynamics of the anopheles mosquito Two non-identical patches are assumed, each of then with the same dynamics of the model described in [1]. S i and I i for i = 1, 2 are populations of non carrier and carrier mosquitoes in patch 1 and patch 2, respectively. In the model, it is supposed that there is migration from the zone one to zone two, for which it is defined ψ21 S 0 and ψ21 I 0, as the rate of migration from patch 1 to patch 2 of susceptible and infected mosquitoes, respectively. ( ds 1 = γ 1(S 1 + I 1 ) 1 S ) 1 + I 1 β 1 S 1 µ 1 S 1 ψ k 21S S 1 ( 1 ds 2 = γ 2(S 2 + I 2 ) 1 S ) 2 + I 2 β 2 S 2 µ 2 S 2 + ψ k 21S S 1 2 (12) di 1 = β 1S 1 µ 1 I 1 ψ21i I 1 di 2 = β 2S 2 µ 2 I 2 + ψ21i I 1. The trivial equilibrium solution of (12) is given by E 0 = (0, 0, 0, 0). The Jacobian matrix at point E 0 is γ 1 β 1 µ 1 ψ21 S 0 γ 1 0 J(E 0 ) = ψ21 S γ 2 µ 2 β 2 0 γ 2 β 1 0 µ 1 ψ21 I 0. (13) 0 β 2 ψ21 I µ 2 The first hypothesis J ii < 0 for i = 1,..., 4 of the Proposition 3.1 is verified if and only if γ 1 < β 1 + µ 1 + ψ S 21 and γ 2 < µ 2 + β 2. (14) The radius of the circle of Gerhsgorin R i is calculated by adding all absolute values of the components of the i-th row of the matrix, except the i- th component of the row. An interesting property is that the radius of the Gershgorin circle can also be calculated by summing the components of the j th column, except for the j th component of the column; that is to say, R j = n i=1,i j J ij. In consequence the second hypothesis is equivalent to R j < J jj for j = 1,..., 4 which is verified if and only if ψ S 21+β 1 < γ 1 β 1 µ 1 ψ S 21, β 2 < γ 2 µ 2 β 2, γ 1 +ψ I 21 < µ 1 +ψ I 21, γ 2 < µ 2. (15) From (14) and (15) we obtain that γ 1 < µ 1 and γ 2 < µ 2. (16) Therefore, if (16) is taken then E 1 is locally asymptotically stable.
6 4546 Danilo Alonso Ortega Bejarano et al. 4.3 Cellular Immunology of Tuberculosis Following [7], we denote by M U (t), MI (t), B(t), and T (t) the populations densities at time t of non infected macrophages, infected macrophages, bacilli Mtb, T cells, respectively. The model is given by d M U d M I d B d T = Λ U µ U MU β B M U. (17) = β B M U ᾱ T MI T µi MI ( = rµ I MI + ν 1 B ) B γ U MU B µb B K ( = k I 1 T ) M I µ T T, T max where Λ U, β, ᾱ T, r, ν, γ U, k I, µ U, µ I, µ B, µ T, K and T max are positive constant parameters. For the infection free equilibrium P 0 = (1, 0, 0, 0), the Jacobian is given by where J (P 0 ) = β = βk, µ U 0 β 0 0 µ I β 0 0 r ν (γ U + µ B ) 0 0 k I 0 µ T Λ U γ U = γ U, r = r µ U K µ Λ U I, k I = k I Λ U. µ U µ U, (18) The first hypothesis J ii < 0 for i = 1,..., 4 of the Proposition 3.1 is satisfied if and only if ν < γ U + µ B, and the second hypothesis R i < J ii for i = 1,..., 4 is verified if and only if β < µ U, β < µ I, r < ν (γ U + µ B ) and k I < µ T. In consequence, P 0 is locally asymptotically stable when all of above inequalities are satisfied. 5 Discussion Local stability conditions are determined for equilibrium solutions of dynamical systems by means of the test. The criterion is very practical given that it allows us to establish stability conditions without the need to calculate the eigenvalues of the Jacobian matrix associated with the dynamic system. In certain areas of applied mathematics such as Biomathematics, the qualitative analysis of the solutions of dynamical systems defined by ordinary differential equations is fundamental to understand problems in biology. In this sense, the test can be very useful for researchers in performing the qualitative analysis of dynamic systems.
7 A test of stability based on Gershgorin circles 4547 Acknowledgements. E. Ibargüen-Mondragón acknowledge support from project No /10/2017 (VIPRI-UDENAR). References [1] E.A. Gómez-Hernández and E. Ibargüen-Mondragón, Modeling the Dynamics of the Mosquito Anopheles calderoni Transmitters of Malaria, Contemporary Engineering Sciences, 11 (2018), [2] M.W. Hisrch, S. Smale and R.L. Devaney, Differential Dquations, Dynamical Systems, and An Introduction to Chaos, 2ed, Vol. 60, Elsevier, Newyork, [3] E. Ibargüen-Mondragón, M. Cerón and J.P. Romero-Leiton, A simple test for asymptotic stability in some dynamical systems,revista de Ceincias- Univalle, 18 (2014), [4] E. Ibargüen-Mondragón, J.P. Romero-Leiton, L. Esteva and E.M. Burbano-Rosero, Mathematical modeling of bacterial resistance to antibiotics by mutations and plasmids, Journal of Biological Systems, 24 (2016), [5] E. Ibargüen-Mondragón and L. Esteva, On CTL response against Mycobacterium tuberculosis, Applied Mathematical Sciences, 8 (2016), [6] E. Ibargüen-Mondragón, L. Esteva and L. Chávez-Galán, A mathematical model for cellular immunology of tuberculosis, Mathematical Biosciences and Engineering, 8 (2011), [7] E. Ibargüen-Mondragón, L. Esteva, E.M. Burbano-Rosero, Mathematical model for the growth of Mycobacterium tuberculosis in the granuloma, Mathematical Biosciences and Engineering, 15 (2018), [8] E. Ibargüen-Mondragón and L. Esteva, On the interactions of sensitive and resistant Mycobacterium tuberculosis to antibiotics, Mathematical Biosciences, 246 (2013), [9] E. Ibargüen-Mondragón and L. Esteva, Un modelo matemático sobre la dinámica del Mycobacterium tuberculosis en el granuloma, Revista Colombiana de Matemáticas, 46 (2012),
8 4548 Danilo Alonso Ortega Bejarano et al. [10] L Esteva, E Ibargüen-Mondragón, Modeling basic aspects of bacterial resistance of Mycobacterium tuberculosis to antibiotics, Ricerche di Matematica, 67 (2018), [11] E. Ibargüen-Mondragón and L. Esteva, Simple mathematical models on macrophages and CTL responses against Mycobacterium tuberculosis, Sigma, 12 (2016), [12] J.M. Montoya Aguilar, J.P. Romero-Leiton, E. Ibargüen-Mondragón, Qualitative analysis of a mathematical model applied to malaria disease transmission in Tumaco (Colombia), Applied Mathematical Sciences, 12 (2018), [13] J.P. Romero-Leiton, J.M. Montoya Aguilar, E. Ibargüen-Mondragón, An optimal control problem applied to malaria disease in Colombia, Applied Mathematical Sciences, 12 (2018), [14] J.P. Romero-Leiton, E. Ibargüen-Mondragón and L. Esteva, Un modelo matemático sobre bacterias sensibles y resistentes a antibióticos, Matemáticas: Enseñanza Universitaria, 19 (2011), [15] J.P. Romero-Leiton and E. Ibargüen Mondragón, Sobre la resistencia bacteriana hacia antibióticos de acción bactericida y bacteriostática, Revista Integración, 32 (2014), [16] J.P. Romero-Leiton, J.M. Montoya Aguilar, M. Villaroel and E. Ibargüen- Mondragón, Influencia de la fuerza de infección y la transmición vertical de la malaria: Modelado Matemático, Revista Facultad de Ciencias Básicas- Unimilitar, 13 (2017), [17] J.P. Romero-Leiton, E. Ibargüen-Mondragón, A. Pulgarín, D. Cordero, I. P Castaño, Análisis de un modelo planta-herbívoro aplicado a la interacción gramínea-bovino, Revista de Matemática: Teoría y Aplicaciones, 23 (2016), [18] R.S. Varga, Ger sgorin and His Circles, Springer, Berling, Received: October 3, 2018; Published: November 1, 2018
On CTL Response against Mycobacterium tuberculosis
Applied Mathematical Sciences, Vol. 8, 2014, no. 48, 2383-2389 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.43150 On CTL Response against Mycobacterium tuberculosis Eduardo Ibargüen-Mondragón
More informationAn Optimal Control Problem Applied to Malaria Disease in Colombia
Applied Mathematical Sciences, Vol. 1, 018, no. 6, 79-9 HIKARI Ltd, www.m-hikari.com https://doi.org/10.1988/ams.018.819 An Optimal Control Problem Applied to Malaria Disease in Colombia Jhoana P. Romero-Leiton
More informationA Mathematical Model for Transmission of Dengue
Applied Mathematical Sciences, Vol. 10, 2016, no. 7, 345-355 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2016.510662 A Mathematical Model for Transmission of Dengue Luis Eduardo López Departamento
More informationAedes aegypti Population Model with Integrated Control
Applied Mathematical Sciences, Vol. 12, 218, no. 22, 175-183 HIKARI Ltd, www.m-hiari.com https://doi.org/1.12988/ams.218.71295 Aedes aegypti Population Model with Integrated Control Julián A. Hernández
More informationMathematical Model of Tuberculosis Spread within Two Groups of Infected Population
Applied Mathematical Sciences, Vol. 10, 2016, no. 43, 2131-2140 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2016.63130 Mathematical Model of Tuberculosis Spread within Two Groups of Infected
More informationContemporary Engineering Sciences, Vol. 11, 2018, no. 48, HIKARI Ltd,
Contemporary Engineering Sciences, Vol. 11, 2018, no. 48, 2349-2356 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ces.2018.85243 Radially Symmetric Solutions of a Non-Linear Problem with Neumann
More informationEffective Potential Approach to the Dynamics of the Physical Symmetrical Pendulum
Contemporary Engineering Sciences, Vol. 11, 018, no. 104, 5117-515 HIKARI Ltd, www.m-hikari.com https://doi.org/10.1988/ces.018.811593 Effective Potential Approach to the Dynamics of the Physical Symmetrical
More informationSolving Homogeneous Systems with Sub-matrices
Pure Mathematical Sciences, Vol 7, 218, no 1, 11-18 HIKARI Ltd, wwwm-hikaricom https://doiorg/112988/pms218843 Solving Homogeneous Systems with Sub-matrices Massoud Malek Mathematics, California State
More informationOn the Coercive Functions and Minimizers
Advanced Studies in Theoretical Physics Vol. 11, 17, no. 1, 79-715 HIKARI Ltd, www.m-hikari.com https://doi.org/1.1988/astp.17.71154 On the Coercive Functions and Minimizers Carlos Alberto Abello Muñoz
More informationNon Isolated Periodic Orbits of a Fixed Period for Quadratic Dynamical Systems
Applied Mathematical Sciences, Vol. 12, 2018, no. 22, 1053-1058 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.87100 Non Isolated Periodic Orbits of a Fixed Period for Quadratic Dynamical
More informationPID Controller Design for DC Motor
Contemporary Engineering Sciences, Vol. 11, 2018, no. 99, 4913-4920 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ces.2018.810539 PID Controller Design for DC Motor Juan Pablo Trujillo Lemus Department
More informationExplicit Expressions for Free Components of. Sums of the Same Powers
Applied Mathematical Sciences, Vol., 27, no. 53, 2639-2645 HIKARI Ltd, www.m-hikari.com https://doi.org/.2988/ams.27.79276 Explicit Expressions for Free Components of Sums of the Same Powers Alexander
More informationThursday. Threshold and Sensitivity Analysis
Thursday Threshold and Sensitivity Analysis SIR Model without Demography ds dt di dt dr dt = βsi (2.1) = βsi γi (2.2) = γi (2.3) With initial conditions S(0) > 0, I(0) > 0, and R(0) = 0. This model can
More informationPoincaré`s Map in a Van der Pol Equation
International Journal of Mathematical Analysis Vol. 8, 014, no. 59, 939-943 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.1988/ijma.014.411338 Poincaré`s Map in a Van der Pol Equation Eduardo-Luis
More informationApplications in Biology
11 Applications in Biology In this chapter we make use of the techniques developed in the previous few chapters to examine some nonlinear systems that have been used as mathematical models for a variety
More informationOn a Certain Representation in the Pairs of Normed Spaces
Applied Mathematical Sciences, Vol. 12, 2018, no. 3, 115-119 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.712362 On a Certain Representation in the Pairs of ormed Spaces Ahiro Hoshida
More informationThe Solution of the Truncated Harmonic Oscillator Using Lie Groups
Advanced Studies in Theoretical Physics Vol. 11, 2017, no. 7, 327-335 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/astp.2017.7521 The Solution of the Truncated Harmonic Oscillator Using Lie Groups
More informationA New Mathematical Approach for. Rabies Endemy
Applied Mathematical Sciences, Vol. 8, 2014, no. 2, 59-67 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.39525 A New Mathematical Approach for Rabies Endemy Elif Demirci Ankara University
More informationHopf Bifurcation Analysis of a Dynamical Heart Model with Time Delay
Applied Mathematical Sciences, Vol 11, 2017, no 22, 1089-1095 HIKARI Ltd, wwwm-hikaricom https://doiorg/1012988/ams20177271 Hopf Bifurcation Analysis of a Dynamical Heart Model with Time Delay Luca Guerrini
More informationStability Analysis of a SIS Epidemic Model with Standard Incidence
tability Analysis of a I Epidemic Model with tandard Incidence Cruz Vargas-De-León Received 19 April 2011; Accepted 19 Octuber 2011 leoncruz82@yahoo.com.mx Abstract In this paper, we study the global properties
More informationGlobal Stability Analysis on a Predator-Prey Model with Omnivores
Applied Mathematical Sciences, Vol. 9, 215, no. 36, 1771-1782 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ams.215.512 Global Stability Analysis on a Predator-Prey Model with Omnivores Puji Andayani
More informationChaos Control for the Lorenz System
Advanced Studies in Theoretical Physics Vol. 12, 2018, no. 4, 181-188 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/astp.2018.8413 Chaos Control for the Lorenz System Pedro Pablo Cárdenas Alzate
More informationSolitary Wave Solution of the Plasma Equations
Applied Mathematical Sciences, Vol. 11, 017, no. 39, 1933-1941 HIKARI Ltd, www.m-hikari.com https://doi.org/10.1988/ams.017.7609 Solitary Wave Solution of the Plasma Equations F. Fonseca Universidad Nacional
More informationAustralian Journal of Basic and Applied Sciences
AENSI Journals Australian Journal of Basic and Applied Sciences ISSN:1991-8178 Journal home page: www.ajbasweb.com A SIR Transmission Model of Political Figure Fever 1 Benny Yong and 2 Nor Azah Samat 1
More informationIntroduction to SEIR Models
Department of Epidemiology and Public Health Health Systems Research and Dynamical Modelling Unit Introduction to SEIR Models Nakul Chitnis Workshop on Mathematical Models of Climate Variability, Environmental
More informationAlternate Locations of Equilibrium Points and Poles in Complex Rational Differential Equations
International Mathematical Forum, Vol. 9, 2014, no. 35, 1725-1739 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2014.410170 Alternate Locations of Equilibrium Points and Poles in Complex
More informationResearch Article Global Dynamics of a Competitive System of Rational Difference Equations in the Plane
Hindawi Publishing Corporation Advances in Difference Equations Volume 009 Article ID 1380 30 pages doi:101155/009/1380 Research Article Global Dynamics of a Competitive System of Rational Difference Equations
More informationLie Symmetries Analysis for SIR Model of Epidemiology
Applied Mathematical Sciences, Vol. 7, 2013, no. 92, 4595-4604 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2013.36348 Lie Symmetries Analysis for SIR Model of Epidemiology A. Ouhadan 1,
More informationTheoretical Analysis of an Optimal Control Model
Applied Mathematical Sciences, Vol. 9, 215, no. 138, 6849-6856 HKAR Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ams.215.57483 Theoretical Analysis of an Optimal Control Model Anibal Muñoz L., M. John
More informationMath 5490 November 5, 2014
Math 549 November 5, 214 Topics in Applied Mathematics: Introduction to the Mathematics of Climate Mondays and Wednesdays 2:3 3:45 http://www.math.umn.edu/~mcgehee/teaching/math549-214-2fall/ Streaming
More informationDetermination of the Theoretical Stress. Concentration Factor in Structural Flat Plates with. Two Holes Subjected to Bending Loads
Contemporary Engineering Sciences, Vol. 11, 2018, no. 98, 4869-4877 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ces.2018.810536 Determination of the Theoretical Stress Concentration Factor in
More informationProblem set 7 Math 207A, Fall 2011 Solutions
Problem set 7 Math 207A, Fall 2011 s 1. Classify the equilibrium (x, y) = (0, 0) of the system x t = x, y t = y + x 2. Is the equilibrium hyperbolic? Find an equation for the trajectories in (x, y)- phase
More information3 Stability and Lyapunov Functions
CDS140a Nonlinear Systems: Local Theory 02/01/2011 3 Stability and Lyapunov Functions 3.1 Lyapunov Stability Denition: An equilibrium point x 0 of (1) is stable if for all ɛ > 0, there exists a δ > 0 such
More informationExact Solutions for a Fifth-Order Two-Mode KdV Equation with Variable Coefficients
Contemporary Engineering Sciences, Vol. 11, 2018, no. 16, 779-784 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ces.2018.8262 Exact Solutions for a Fifth-Order Two-Mode KdV Equation with Variable
More informationA Solution of the Spherical Poisson-Boltzmann Equation
International Journal of Mathematical Analysis Vol. 1, 018, no. 1, 1-7 HIKARI Ltd, www.m-hikari.com https://doi.org/10.1988/ijma.018.71155 A Solution of the Spherical Poisson-Boltzmann quation. onseca
More informationMathematical Analysis of Epidemiological Models: Introduction
Mathematical Analysis of Epidemiological Models: Introduction Jan Medlock Clemson University Department of Mathematical Sciences 8 February 2010 1. Introduction. The effectiveness of improved sanitation,
More informationProx-Diagonal Method: Caracterization of the Limit
International Journal of Mathematical Analysis Vol. 12, 2018, no. 9, 403-412 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijma.2018.8639 Prox-Diagonal Method: Caracterization of the Limit M. Amin
More informationThe Greatest Common Divisor of k Positive Integers
International Mathematical Forum, Vol. 3, 208, no. 5, 25-223 HIKARI Ltd, www.m-hiari.com https://doi.org/0.2988/imf.208.822 The Greatest Common Divisor of Positive Integers Rafael Jaimczu División Matemática,
More informationBehavior Stability in two SIR-Style. Models for HIV
Int. Journal of Math. Analysis, Vol. 4, 2010, no. 9, 427-434 Behavior Stability in two SIR-Style Models for HIV S. Seddighi Chaharborj 2,1, M. R. Abu Bakar 2, I. Fudziah 2 I. Noor Akma 2, A. H. Malik 2,
More informationMethod of Generation of Chaos Map in the Centre Manifold
Advanced Studies in Theoretical Physics Vol. 9, 2015, no. 16, 795-800 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/astp.2015.51097 Method of Generation of Chaos Map in the Centre Manifold Evgeny
More informationModeling and Global Stability Analysis of Ebola Models
Modeling and Global Stability Analysis of Ebola Models T. Stoller Department of Mathematics, Statistics, and Physics Wichita State University REU July 27, 2016 T. Stoller REU 1 / 95 Outline Background
More informationThe Dynamic Properties of a Deterministic SIR Epidemic Model in Discrete-Time
Applied Mathematics, 05, 6, 665-675 Published Online September 05 in SciRes http://wwwscirporg/journal/am http://dxdoiorg/046/am056048 The Dynamic Properties of a Deterministic SIR Epidemic Model in Discrete-Time
More informationCalibration of Analog Scales Based on. a Metrological Comparison Between. the SIM Guide and OIML R 76-1
Contemporary Engineering Sciences, Vol. 11, 2018, no. 82, 4049-4057 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ces.2018.88447 Calibration of Analog Scales Based on a Metrological Comparison
More informationKKM-Type Theorems for Best Proximal Points in Normed Linear Space
International Journal of Mathematical Analysis Vol. 12, 2018, no. 12, 603-609 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijma.2018.81069 KKM-Type Theorems for Best Proximal Points in Normed
More informationDiophantine Equations. Elementary Methods
International Mathematical Forum, Vol. 12, 2017, no. 9, 429-438 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2017.7223 Diophantine Equations. Elementary Methods Rafael Jakimczuk División Matemática,
More informationRosenzweig-MacArthur Model. Considering the Function that Protects a Fixed. Amount of Prey for Population Dynamics
Contemporary Engineering Sciences, Vol. 11, 18, no. 4, 1195-15 HIKAI Ltd, www.m-hikari.com https://doi.org/1.1988/ces.18.8395 osenzweig-macarthur Model Considering the Function that Protects a Fixed Amount
More informations-generalized Fibonacci Numbers: Some Identities, a Generating Function and Pythagorean Triples
International Journal of Mathematical Analysis Vol. 8, 2014, no. 36, 1757-1766 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ijma.2014.47203 s-generalized Fibonacci Numbers: Some Identities,
More informationk-pell, k-pell-lucas and Modified k-pell Numbers: Some Identities and Norms of Hankel Matrices
International Journal of Mathematical Analysis Vol. 9, 05, no., 3-37 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/0.988/ijma.05.4370 k-pell, k-pell-lucas and Modified k-pell Numbers: Some Identities
More informationPeriodic and Soliton Solutions for a Generalized Two-Mode KdV-Burger s Type Equation
Contemporary Engineering Sciences Vol. 11 2018 no. 16 785-791 HIKARI Ltd www.m-hikari.com https://doi.org/10.12988/ces.2018.8267 Periodic and Soliton Solutions for a Generalized Two-Mode KdV-Burger s Type
More informationA Mathematical Model of Malaria. and the Effectiveness of Drugs
Applied Matematical Sciences, Vol. 7,, no. 6, 79-95 HIKARI Ltd, www.m-ikari.com A Matematical Model of Malaria and te Effectiveness of Drugs Moammed Baba Abdullai Scool of Matematical Sciences Universiti
More informationRestrained Independent 2-Domination in the Join and Corona of Graphs
Applied Mathematical Sciences, Vol. 11, 2017, no. 64, 3171-3176 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2017.711343 Restrained Independent 2-Domination in the Join and Corona of Graphs
More informationOn Symmetric Bi-Multipliers of Lattice Implication Algebras
International Mathematical Forum, Vol. 13, 2018, no. 7, 343-350 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/imf.2018.8423 On Symmetric Bi-Multipliers of Lattice Implication Algebras Kyung Ho
More informationThe death of an epidemic
LECTURE 2 Equilibrium Stability Analysis & Next Generation Method The death of an epidemic In SIR equations, let s divide equation for dx/dt by dz/ dt:!! dx/dz = - (β X Y/N)/(γY)!!! = - R 0 X/N Integrate
More informationDynamical System of a Multi-Capital Growth Model
Applied Mathematical Sciences, Vol. 9, 2015, no. 83, 4103-4108 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2015.53274 Dynamical System of a Multi-Capital Growth Model Eva Brestovanská Department
More informationLinearization of Differential Equation Models
Linearization of Differential Equation Models 1 Motivation We cannot solve most nonlinear models, so we often instead try to get an overall feel for the way the model behaves: we sometimes talk about looking
More informationA Note on Multiplicity Weight of Nodes of Two Point Taylor Expansion
Applied Mathematical Sciences, Vol, 207, no 6, 307-3032 HIKARI Ltd, wwwm-hikaricom https://doiorg/02988/ams2077302 A Note on Multiplicity Weight of Nodes of Two Point Taylor Expansion Koichiro Shimada
More informationGlobal Analysis of a HCV Model with CTL, Antibody Responses and Therapy
Applied Mathematical Sciences Vol 9 205 no 8 3997-4008 HIKARI Ltd wwwm-hikaricom http://dxdoiorg/02988/ams20554334 Global Analysis of a HCV Model with CTL Antibody Responses and Therapy Adil Meskaf Department
More informationA Delayed HIV Infection Model with Specific Nonlinear Incidence Rate and Cure of Infected Cells in Eclipse Stage
Applied Mathematical Sciences, Vol. 1, 216, no. 43, 2121-213 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ams.216.63128 A Delayed HIV Infection Model with Specific Nonlinear Incidence Rate and
More informationModels of Infectious Disease Formal Demography Stanford Spring Workshop in Formal Demography May 2008
Models of Infectious Disease Formal Demography Stanford Spring Workshop in Formal Demography May 2008 James Holland Jones Department of Anthropology Stanford University May 3, 2008 1 Outline 1. Compartmental
More informationLocal and Global Stability of Host-Vector Disease Models
Local and Global Stability of Host-Vector Disease Models Marc 4, 2008 Cruz Vargas-De-León 1, Jorge Armando Castro Hernández Unidad Académica de Matemáticas, Universidad Autónoma de Guerrero, México and
More informationEquivalent Multivariate Stochastic Processes
International Journal of Mathematical Analysis Vol 11, 017, no 1, 39-54 HIKARI Ltd, wwwm-hikaricom https://doiorg/101988/ijma01769111 Equivalent Multivariate Stochastic Processes Arnaldo De La Barrera
More informationA Class of Multi-Scales Nonlinear Difference Equations
Applied Mathematical Sciences, Vol. 12, 2018, no. 19, 911-919 HIKARI Ltd, www.m-hiari.com https://doi.org/10.12988/ams.2018.8799 A Class of Multi-Scales Nonlinear Difference Equations Tahia Zerizer Mathematics
More informationDiameter of the Zero Divisor Graph of Semiring of Matrices over Boolean Semiring
International Mathematical Forum, Vol. 9, 2014, no. 29, 1369-1375 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2014.47131 Diameter of the Zero Divisor Graph of Semiring of Matrices over
More informationSolution of the Hirota Equation Using Lattice-Boltzmann and the Exponential Function Methods
Advanced Studies in Theoretical Physics Vol. 11, 2017, no. 7, 307-315 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/astp.2017.7418 Solution of the Hirota Equation Using Lattice-Boltzmann and the
More informationPólya-Szegö s Principle for Nonlocal Functionals
International Journal of Mathematical Analysis Vol. 12, 218, no. 5, 245-25 HIKARI Ltd, www.m-hikari.com https://doi.org/1.12988/ijma.218.8327 Pólya-Szegö s Principle for Nonlocal Functionals Tiziano Granucci
More informationApproximation to the Dissipative Klein-Gordon Equation
International Journal of Mathematical Analysis Vol. 9, 215, no. 22, 159-163 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/1.12988/ijma.215.5236 Approximation to the Dissipative Klein-Gordon Equation Edilber
More informationExperimental Analysis of Random Errors for. Calibration of Scales Applying. Non-Parametric Statistics
Contemporary Engineering Sciences, Vol. 11, 2018, no. 86, 4293-4300 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ces.2018.88481 Experimental Analysis of Random Errors for Calibration of Scales
More informationA note on linear differential equations with periodic coefficients.
A note on linear differential equations with periodic coefficients. Maite Grau (1) and Daniel Peralta-Salas (2) (1) Departament de Matemàtica. Universitat de Lleida. Avda. Jaume II, 69. 251 Lleida, Spain.
More informationOn Permutation Polynomials over Local Finite Commutative Rings
International Journal of Algebra, Vol. 12, 2018, no. 7, 285-295 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ija.2018.8935 On Permutation Polynomials over Local Finite Commutative Rings Javier
More informationA Characterization of the Cactus Graphs with Equal Domination and Connected Domination Numbers
International Journal of Contemporary Mathematical Sciences Vol. 12, 2017, no. 7, 275-281 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijcms.2017.7932 A Characterization of the Cactus Graphs with
More informationIntroduction: What one must do to analyze any model Prove the positivity and boundedness of the solutions Determine the disease free equilibrium
Introduction: What one must do to analyze any model Prove the positivity and boundedness of the solutions Determine the disease free equilibrium point and the model reproduction number Prove the stability
More information2.10 Saddles, Nodes, Foci and Centers
2.10 Saddles, Nodes, Foci and Centers In Section 1.5, a linear system (1 where x R 2 was said to have a saddle, node, focus or center at the origin if its phase portrait was linearly equivalent to one
More informationOn the Deformed Theory of Special Relativity
Advanced Studies in Theoretical Physics Vol. 11, 2017, no. 6, 275-282 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/astp.2017.61140 On the Deformed Theory of Special Relativity Won Sang Chung 1
More informationGlobal Analysis of a Mathematical Model of HCV Transmission among Injecting Drug Users and the Impact of Vaccination
Applied Mathematical Sciences, Vol. 8, 2014, no. 128, 6379-6388 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ams.2014.48625 Global Analysis of a Mathematical Model of HCV Transmission among
More informationOn the Power of Standard Polynomial to M a,b (E)
International Journal of Algebra, Vol. 10, 2016, no. 4, 171-177 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/ija.2016.6214 On the Power of Standard Polynomial to M a,b (E) Fernanda G. de Paula
More informationMath 312 Lecture Notes Linearization
Math 3 Lecture Notes Linearization Warren Weckesser Department of Mathematics Colgate University 3 March 005 These notes discuss linearization, in which a linear system is used to approximate the behavior
More informationConstruction of Lyapunov functions by validated computation
Construction of Lyapunov functions by validated computation Nobito Yamamoto 1, Kaname Matsue 2, and Tomohiro Hiwaki 1 1 The University of Electro-Communications, Tokyo, Japan yamamoto@im.uec.ac.jp 2 The
More informationComparison of the Concentration Factor of. Stresses on Flat Sheets with Two Holes. with Low and High Speed Voltage Test
Contemporary Engineering Sciences, Vol. 11, 2018, no. 55, 2707-2714 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ces.2018.86288 Comparison of the Concentration Factor of Stresses on Flat Sheets
More informationLAW OF LARGE NUMBERS FOR THE SIRS EPIDEMIC
LAW OF LARGE NUMBERS FOR THE SIRS EPIDEMIC R. G. DOLGOARSHINNYKH Abstract. We establish law of large numbers for SIRS stochastic epidemic processes: as the population size increases the paths of SIRS epidemic
More informationOn Annihilator Small Intersection Graph
International Journal of Contemporary Mathematical Sciences Vol. 12, 2017, no. 7, 283-289 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijcms.2017.7931 On Annihilator Small Intersection Graph Mehdi
More informationCaristi-type Fixed Point Theorem of Set-Valued Maps in Metric Spaces
International Journal of Mathematical Analysis Vol. 11, 2017, no. 6, 267-275 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijma.2017.717 Caristi-type Fixed Point Theorem of Set-Valued Maps in Metric
More informationA Survey of Long Term Transmission Expansion Planning Using Cycles
Contemporary Engineering Sciences, Vol. 11, 218, no. 12, 547-555 HIKARI Ltd, www.m-hikari.com https://doi.org/1.12988/ces.218.81562 A Survey of Long Term Transmission Epansion Planning Using Cycles Pedro
More informationStability of the feasible set in balanced transportation problems
Vol XVIII, N o 2, Diciembre (21) Matemáticas: 75 85 Matemáticas: Enseñanza Universitaria c Escuela Regional de Matemáticas Universidad del Valle - Colombia Stability of the feasible set in balanced transportation
More informationQualitative Theory of Differential Equations and Dynamics of Quadratic Rational Functions
Nonl. Analysis and Differential Equations, Vol. 2, 2014, no. 1, 45-59 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/nade.2014.3819 Qualitative Theory of Differential Equations and Dynamics of
More informationA Solution of the Cylindrical only Counter-ions Poisson-Boltzmann Equation
Contemporary Engineering Sciences, Vol. 11, 2018, no. 54, 2691-2697 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ces.2018.86280 A Solution of the Cylindrical only Counter-ions Poisson-Boltzmann
More informationMATH 215/255 Solutions to Additional Practice Problems April dy dt
. For the nonlinear system MATH 5/55 Solutions to Additional Practice Problems April 08 dx dt = x( x y, dy dt = y(.5 y x, x 0, y 0, (a Show that if x(0 > 0 and y(0 = 0, then the solution (x(t, y(t of the
More informationA Family of Nonnegative Matrices with Prescribed Spectrum and Elementary Divisors 1
International Mathematical Forum, Vol, 06, no 3, 599-63 HIKARI Ltd, wwwm-hikaricom http://dxdoiorg/0988/imf0668 A Family of Nonnegative Matrices with Prescribed Spectrum and Elementary Divisors Ricardo
More informationDepartment of Mathematics IIT Guwahati
Stability of Linear Systems in R 2 Department of Mathematics IIT Guwahati A system of first order differential equations is called autonomous if the system can be written in the form dx 1 dt = g 1(x 1,
More informationHyperbolic Functions and. the Heat Balance Integral Method
Nonl. Analysis and Differential Equations, Vol. 1, 2013, no. 1, 23-27 HIKARI Ltd, www.m-hikari.com Hyperbolic Functions and the Heat Balance Integral Method G. Nhawu and G. Tapedzesa Department of Mathematics,
More informationSome Properties of D-sets of a Group 1
International Mathematical Forum, Vol. 9, 2014, no. 21, 1035-1040 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2014.45104 Some Properties of D-sets of a Group 1 Joris N. Buloron, Cristopher
More informationA Note on the Variational Formulation of PDEs and Solution by Finite Elements
Advanced Studies in Theoretical Physics Vol. 12, 2018, no. 4, 173-179 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/astp.2018.8412 A Note on the Variational Formulation of PDEs and Solution by
More informationContra θ-c-continuous Functions
International Journal of Contemporary Mathematical Sciences Vol. 12, 2017, no. 1, 43-50 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ijcms.2017.714 Contra θ-c-continuous Functions C. W. Baker
More informationSolution of the Maxwell s Equations of the Electromagnetic Field in Two Dimensions by Means of the Finite Difference Method in the Time Domain (FDTD)
Contemporary Engineering Sciences, Vol. 11, 2018, no. 27, 1331-1338 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ces.2018.83114 Solution of the Maxwell s Equations of the Electromagnetic Field
More informationDevaney's Chaos of One Parameter Family. of Semi-triangular Maps
International Mathematical Forum, Vol. 8, 2013, no. 29, 1439-1444 HIKARI Ltd, www.m-hikari.com http://dx.doi.org/10.12988/imf.2013.36114 Devaney's Chaos of One Parameter Family of Semi-triangular Maps
More informationA Trivial Dynamics in 2-D Square Root Discrete Mapping
Applied Mathematical Sciences, Vol. 12, 2018, no. 8, 363-368 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.8121 A Trivial Dynamics in 2-D Square Root Discrete Mapping M. Mammeri Department
More information2D-Volterra-Lotka Modeling For 2 Species
Majalat Al-Ulum Al-Insaniya wat - Tatbiqiya 2D-Volterra-Lotka Modeling For 2 Species Alhashmi Darah 1 University of Almergeb Department of Mathematics Faculty of Science Zliten Libya. Abstract The purpose
More informationDouble Total Domination in Circulant Graphs 1
Applied Mathematical Sciences, Vol. 12, 2018, no. 32, 1623-1633 HIKARI Ltd, www.m-hikari.com https://doi.org/10.12988/ams.2018.811172 Double Total Domination in Circulant Graphs 1 Qin Zhang and Chengye
More informationNew Nonlinear Conditions for Approximate Sequences and New Best Proximity Point Theorems
Applied Mathematical Sciences, Vol., 207, no. 49, 2447-2457 HIKARI Ltd, www.m-hikari.com https://doi.org/0.2988/ams.207.7928 New Nonlinear Conditions for Approximate Sequences and New Best Proximity Point
More informationModels of Infectious Disease Formal Demography Stanford Summer Short Course James Holland Jones, Instructor. August 15, 2005
Models of Infectious Disease Formal Demography Stanford Summer Short Course James Holland Jones, Instructor August 15, 2005 1 Outline 1. Compartmental Thinking 2. Simple Epidemic (a) Epidemic Curve 1:
More informationThe Standard Polynomial in Verbally Prime Algebras
International Journal of Algebra, Vol 11, 017, no 4, 149-158 HIKARI Ltd, wwwm-hikaricom https://doiorg/101988/ija01775 The Standard Polynomial in Verbally Prime Algebras Geraldo de Assis Junior Departamento
More information