A Stability Test for Non Linear Systems of Ordinary Differential Equations Based on the Gershgorin Circles

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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.

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