On asymptotic stability of a class of time delay systems

PURE MATHEMATICS RESEARCH ARTICLE On asymptotic stability of a class of time delay systems Serbun Ufuk Değer 1, * and Yaşar Bolat 1, Received 31 January 1 Accepted 3 April 1 First Published 14 May 1 *Corresponding author Serbun Ufuk Değer, Department of Mathematics, Institute of Sciences, Kastamonu University, Kastamonu, Turkey; Department of Mathematics, Faculty of Art & Science, Kastamonu University, Kastamonu, Turkey E-mail sudeger@kastamonu.edu.tr Reviewing editor Hari M. Srivastava, University of Victoria, Canada Additional information is available at the end of the article Abstract In this paper, we give some new necessary and sufficient conditions for the asymptotic stability of a class of time delay systems of the form x ðþ t ð1 axt ðþ Axt ð ð kþxt ð l ¼ ; t ; where a is a real number, A is a real constant matrix, and k, l are positive numbers such that k > l. Subects Advanced Mathematics; Applied Mathematics; Dynamical Systems Keywords asymptotic stability; delay differential equations; characteristic equation 1. Introduction In this paper, we study the asymptotic stability of the solutions of time delay systems of the form x ðþ t ð1 axt ðþaxt ð ð kþxt ð l ¼ ; t ; (11) where a is a real number, A is a real constant matrix, and k, l are positive numbers such that k > l. Time delay systems are a type of differential equations in which the derivative of the unknown function at a certain time is given in terms of the values of the function at previous times. Also, they are called delay differential equations, retarded differential equations or differential-difference equations. On the other hand, since asymptotic stability is an interdisciplinary material, the asymptotic stability of these systems has a wide range of applications as biology, ABOUT THE AUTHOR My key research activities include Delay Differential Equations, Delay Difference Equations, Neutral Differential Equations, Neutral Difference Equations and stability of these equations. More generally, we can say Differential Equations, Difference Equations and stability of these equations. The research reported in this paper relates to the stability of the systems that can be modeled by delay differential equations such as biology, physics and medicine. Thus, it is useful to use qualitative approaches to investigate the asymptotic stability of these systems. Our research represents a generalized method for describing the asymptotic stability of systems within of science branches such as biology and physics. PUBLIC INTEREST STATEMENT In this paper, asymptotic stability of a class of time delay systems was investigated. This system is a generalized version of modeling compound optical resonators by using a matrix instead of a scaler. In addition, since asymptotic stability is an interdisciplinary material, the asymptotic stability of these systems has a wide range of applications as biology, physics and medicine. We know that for the time delay equations, an equation is asymptotically stable if and only if all roots of the associated characteristic equation have negative real parts. Stability analysis, however, does not require the exact calculation of the characteristic roots. This analysis can be performed by D-subdivision method, which gives a necessary and sufficient condition for stability based on the coefficients or delay parameters of the characteristic equation. Thus, this method reveals a qualitative approach in order to prove the asymptotic stability of the systems. Consequently, we have created new necessary and sufficient conditions for the systems, which are asymptotically stable. 1 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4. license. Page 1 of 11

physics, and medicine. For instance, Naresh, Tripathi, Tchuenche, and Sharma (9) formulated a nonlinear mathematical model to study the framework of disease Epidemiology. As another example Ikeda and Watanabe (14) were investigated the stochastic differential equations and diffusion processes in their study about physic. Kruthika, Mahindrakar, and Pasumarthy (17) were studied stability analysis of nonlinear time delayed systems with application to biological models. They analyzed the local stability of a gene-regulatory network and immunotherapy for cancer modeled as nonlinear time delay systems. Many authors have also focused on the asymptotic stability of time delay systems. Some important studies about related subect can be examined from the below authors Bellman and Cooke (1963), Cooke and Grossman (19), Cooke and van den Driessche (196), Stepan (199), Freedman and Kuang (1991), Kuang (1993), Ruan and Wei (3), Elaydi (5), Matsunaga (), Smith (1), Gray, Greenhalgh, Hu, Mao, and Pan (11), Khokhlovaa, Kipnis, and Malygina (11), Xu (1), Hrabalova (13), Nakaima (14), Liu, Jiang, Shi, Hayat, and Alsaedi (16), and Li, Ma, Xiao, and Yang (17). Also, an equation which is the special case of our system which is investigated by Kuang (1993) and he demonstrated that the zero solution of the delay differential equation with two delays of the form x ðþ t qxt ð ð kþ xt ð l ¼ ; t ; (1) where k, l and q is positive constant and is asymptotically stable if and only if qðkþ lcos k l π π (13) k þ l. Preliminaries It is known that for the time delay equations, an equation is asymptotically stable if and only if all roots of the associated characteristic equation have negative real parts. Stability analysis, however, does not require the exact calculation of the characteristic roots; only the sign of the real part of the critical root must be determined. This analysis can be performed by D-subdivision method (see, e.g., Insperger & Stépán, 11; Stepan, 199), which gives a necessary and sufficient condition for stability based on the coefficients of the characteristic equation. The aim of this paper is to obtain new results for the asymptotic stability of zero solution of system (1.1), while the characteristic equation of system (1.1) has roots on the imaginary axis when A is a constant matrix. If we obtain xt ð¼pyðforaregularmatrixp t in (1.1), then we have the following system y ðþ t ð1 ayt ðþp 1 APðyðt kþyt ð l ¼ ; t Thus, matrix A can be given one of the following two matrices in Jordan form ½7Š ða I ¼ q 1 r ; q q 1 ; q and r are real constants; ðii A ¼ q cos θ sin θ sin θ cos θ ; q; θ are real constants and θ π Here we consider the case ðii; the other case should be considered similarly. The characteristic equation of system (1.1) is given as Fðλ ¼ det λi þ ð1 ai þ A e λk þ e λl ¼ ; (1) where I is the identity matrix. Using ð1; we obtain Fðλ ¼ λ þ ð1 aþq cos θ e λk þ e λl þ q sin θ e λk þ e λl Page of 11

¼ λ þ ð1 aþq cose θ λk þ e λl iq sine θ λk þ e λl ¼ λ þ ð1 aþqe i θ e λk þ e λl λ þ ð1 aþqe i θ e λk þ e λl If we let fð¼ λ λ þ ð1 aþq e λkþi θ þ e λlþi θ ; () then we have Fðλ ¼fðf λ ð; λ where λ is the complex conugate of any complex λ Note that fð¼ λ implies fð¼ λ 3. Some auxiliary lemmas In this section, we will investigate the distribution of the zeros of the characteristic equation of system (1.1). Thus, we state and prove some basic results on the roots of the characteristic equation of system (1.1). Lemma 1 (Stepan, 199) The zero solution of (1.1) is asymptotically stable if and only if all the roots of equation fðλ; k; l ¼ λ þ ð1 aþq e λkþi θ þ e λlþi θ Lie in the left half of the complex plane. ¼ ; (31) Since f is an analytic function of λ; k and l for the fixed numbers a; q and θ; one can regard the root λ ¼ λðk; l of (3.1) as a continuous function of k and l The next lemma plays very important role for the main theorem. Lemma (Cooke & Grossman, 19) As k and l vary, the sum of the multiplicities of the roots of (3.1) in the open right half-plane can change only if a root appears on or crosses the imaginary axis. Consequently, we claim that (3.1) has only imaginary roots iω. We will determine how the value of k and l change as Equation (3.1) has roots on the imaginary axis. Now, we can write the characteristic Equation (3.1) as follows λ þ ð1 a þq e λkþiθ þ e λlþiθ ¼ (3) At the same time, we take λ ¼ iω such that ω R Firstly, since fð ¼ ð1 aþqe iθ ; we see that ω For ω, we obtain fðiω ¼iω þ ð1 aþq e iωkþiθ þ e iωlþiθ ¼ (33) Using the real part and the imaginary part of (3.3) ω ¼ qðsinðωk θþsinðωl θ; (34) a 1 ¼ qðcosðωk θþcosðωl θ; (35) which is equivalent to ω ¼ q sin ωðk þ l θ cos ωðk ; (36) Page 3 of 11

a 1 ¼ q cos ωðk þ l θ cos ωðk ; (37) is obtained. Lemma 3. Suppose that, q > and θ π π Let λ ¼ iω be a root ð31 where ω n o nπþθ kþl ; nπþθ kþl for n ¼ ; 1; ;..., then the following conditions hold ðif i q cos ωðkl ða 1 ; then there exists no real number ω. kl ; kl π ðif ii q cos ωðkl ða 1 > ; then the real numbers ω; q and the delays ðk þ l are as follows r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω ¼γ ¼ q cos ωðkl ða 1 a1 ; q ¼ cosð ωðkþl θcosð ωðkl and k þ l ¼ ð k n þ l n for n ¼ ; 1; ;... where ðk n þ l n þ and ðk n þ l n ; ðk n þ l n ðk n þ l n þ ¼ a 1 nπ þ A þ θ γ q cos ωðkl ; ¼ a 1 nπ þ A θ γ q cos ωðkl ; ; and iγ or iγ is a root of ð31 for the sum of delays ðk n þ l n þ or ðk n þ l n Proof. By squaring both sides of (3.6) and (3.7), and adding them together, we obtain ω þ ða 1 ¼ q cos ωðk (3) If q cos ωðkl ða 1, then statement (3.) implies ω ; contradicts with ω > Since ω, condition ðis i verified; that is, (3.1) has no root on the imaginary axis for all k > l > On the other hand, if q cos ωðkl ða 1 > ; statement ð3 implies that ω ¼ q cos ωðk ða 1 (39) If we let γ ¼ q cos ωðk ða 1 ; then we can write ω ¼γ. By (3.7), we have q ¼ cos ωðkþl a 1 θ cos ωðkl Now, we will show that iγ is a root of (3.1). In the case of ω ¼ γ; since cos ωðkl n o > for ω ; π kl nπþθ kþl and using ð36 sin ωðk þ l θ > ; (31) Page 4 of 11

is obtained. Thus, we use (3.6) and (3.7), we have 1 ωðk þ l a 1 θ ¼ nπ þ A n ¼ ; 1; ;...; q cos ωðkl which yields ðk n þ l n þ Also, from (3.6) and (3.7), we obtain 11 a 1 sin@ AA γ ¼ q cos ωðkl q cos ωðkl because of 1 a 1 > arcsin A ¼ q cos ωðkl > π arcsin γ q cosð ωðkl if a 1 γ if a 1 ð ð q cos ω kl Hence, for the case k þ l ¼ ðk n þ l n þ ; iγ is a root of (3.1). Indeed, fiω ð ¼ iω þ ð1 aþq e iðωkθ þ e iðωlθ ¼ i q cos ωðk ða 1 þ ð1 aþq cos ωðk ¼ i q cos ωðk ða 1 þ ð1 aþ þ q cos ωðk e a1 i nπþarccos q cosð ωðkl ¼ i q cos ωðk ða 1 þ ð1 aþq cos ωðk ; ð ; e i ωðkþl θ 11 119 a 1 cos@ AA a 1 i sin@ AA q cos ωðkl q cos ωðkl ; ; ¼ i q cos ωðk ða 1 i q cos ωðk ða 1 ; ¼ Thus, this implies that iγ is a root of (3.1). Similarly, in the case ω ; it can be shown that iγ is a root of (3.1) for the sum of delays ðk n þ l n The proof is completed. When q > ; we have the following analogous result. Lemma 4. Suppose that, q and θ π Let λ ¼ iω be a root of (3.1) where ω π n o for n ¼ ; 1; ;..., then the following conditions hold nπþθ kþl ; nπþθ kþl kl ; π kl Page 5 of 11

ðif i q cos ωðkl ða 1 ; then there exists no real number ω. ðif ii q cos ωðkl ða 1 > ; then the real numbers ω; q and the delays ðk þ l are as follows r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi ω ¼γ ¼ q cos ωðkl ða 1 a1 ; q ¼ cosð ωðkþl cosð ωðkl and k þ l ¼ ð α n þ β n for n ¼ ; 1; ;..., where ðα n þ β n þ and ðα n þ β n ; ðα n þ β n þ ¼ a 1 nπ A þ θ γ q cos ωðkl ; ðα n þ β n θ ¼ a 1 ðn þ π A θ γ q cos ωðkl ; ; and iγ or iγ is a root of (3.1) for the sum of delays ðα n þ β n þ or ðα n þ β n Proof. The proof is similar to Lemma 3. Remark 1. For q >, from the definitions of ðk n þ l n ; n o > ðk þ l ifða 1 q cos θ cos ωðkl min ðk n þ l n n ¼ ; 1;... ¼ > ðk þ l þ ifða 1q cos θ cos ωðkl ; is obtained. Similarly, for q,we obtain from the definitions of ðα n þ β n as follows n o ð min ðα n þ β n n ¼ ; 1;... ¼f α þ β þ ifða 1q cos θ cos ωðkl ðα þ β ifða 1q cos θ cos ωðkl Lemma 5. Suppose that θ π ; ω fg and ω þ q ωðk þ l sinðωðl k > π kl ; π kl Also, the following conditions sinðωkθ lsinωðkl > qif > q ω sinðωlθ ksinωðkl > q if > q ω and sinðωkθ lsinωðkl q if q ω sinðωlθ ksinωðkl qif q ω ; (311) are provided. Then all the roots of Equation (3.1) on the imaginary axis move in the right half-plane as k and l increase. Proof. Let λ ¼ iω be a root of (3.1) where ω π kl ; π kl fg is a real number. It will be enough to show Re @λ @k > and Re @λ @l > Firstly, we take the derivative of λ with respect to k on Equation (3.1), we have @λ @k qλeλkþiθ qkλe λkþiθ @λ @k qleλlþiθ @λ @k ¼ ; Page 6 of 11

@λ @k ¼ 1 qke ð λkþiθ þ le λlþiθ qλe λkþiθ Substituting λ ¼ iω into the above equation, we obtain @λ @k iω þ 1 a ¼ 1 qke ð i ð ωkθ þ le i ð ωlθ iω þ 1 a ¼ 1 qkcos ð ðωk θþl cosðωl θþiqðk sinðωk θþl sinðωl θ If we multiply with the complex conugate of the denominator in the above equation, then we can write Re @λ @k where ¼ qω sin ð ωk θ q ωl sinðωðk ; (31) M M ¼ ð1 qkcos ð ðωk θþl cosðωl θ þ q ðk sinðωk θþl sinðωl θ Since (3.11), we can write Re @λ @k to l on Equation (3.1), similar to (3.1) Re @λ @l > On the other hand, we take the derivative of λ with respect qω sin ωl θ ¼ ð q ωk sinðωðk ; (313) M is obtained. From (3.11), we obtain Re @λ @l together, we have > Moreover, by adding both (3.1) and (3.13) Re @λ @k þre @λ @l ¼ ω þ q ωðk þ lsinðωðl k M > (314) Hence, the proof is completed. Now we can state and prove main theorems. 4. Main results We will show that the stability analysis with a qualitative approach, as we have already mentioned in section 1. Theorem 1. Suppose that θ π and the conditions of Lemma 5 are satisfied. Let the matrix A of system (1.1) be in the form ðii. Then system (1.1) is asymptotically stable if and only if either ða 1q cos θ cos ωðkl (41) q cos ωðkl ða 1 ; or ða 1q cos θ cos ωðk q cos ωðk > ða 1 > sgnðq a 1 k þ l r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A θ q cos ωðkl ða 1 q cos ωðkl ; > (4) Page 7 of 11

Proof. In the case of k ¼ and l ¼ ; the root of (3.1) is only λð; ¼ a 1 b cos θ ib sin θ Thus, the root of the Equation (3.1) has a negative real part. By the continuity of the roots with respect to k and l; we can say that all the roots of (3.1) lie in the left half plane for k > and l > sufficiently small. For the sufficiency, here our claim is If either condition (4.1) or (4.) holds, then (3.1) does not have a root on the imaginary axis. By condition (4.1) and Lemma 3, our claim is true for k > and l > Now, suppose that condition (4.) holds Since sgnðq a 1 r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A θ q cos ωðkl ða 1 q cos ωðkl ; ¼ ðk þ l if q > ðα þ β þ if q ; and Remark 1, we obtain ðk þ l ðk n þ l n, ðk þ l ðα n þ β n for n ¼ ; 1; ;... Thus, we obtain the contraposition with Lemma 1, our other claim is also true. By the above argument and Lemma, we can say that if either condition (4.1) or (4.) holds, then all the roots of (3.1) lie in the left half plane. For the necessity, we will show the following contraposition either ða 1q cos θ cos ωðk ; (43) or > q cos ωðkl ða 1 > sgnðq k þ l qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a1 arccos > ðq cosð ωðkl ð ð ða1 q cos ω kl θ (44) Thus, if (4.3) and (4.4) hold, then there exists roots λ of (3.1) such that Re λ > for ¼ 1; Assume that (4.3) holds and let λ 1 ðk; l be the branch of the root of satisfying λ 1 ð; ¼ a 1 b cos θ ib sin θ Then, Lemma 5 or the continuity of λ 1 ðk; l implies that Reðλ 1 > for k > and l > sufficiently small. From here, we can say that λ 1 ðk; lcannot move in the left half-plane crossing on the imaginary axis as k and l increase. Hence, we have Reðλ 1 > for all k > and l > Assume that (4.4) holds and let λ ðk; l be the branch of the root of rffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi satisfying λ ðk ; l ¼ isgnðq q cos ωðkl ða 1 Then, Lemma 5 or the continuity of λ ðk; l implies that Reðλ > for k k > and l l > sufficiently small. From here, we can say that λ ðk; l cannot move in the left half-plane crossing on the imaginary axis as k and l increase. Hence, we have Reðλ > for all k > k and l > l The proof is completed. Remark. We consider the delay differential system (1.1) where matrix A is given as in case ð, I i.e., x ðþ t ð1 axt ðþ q 1 r ðxt ð kþxt ð l ¼ ; t (45) q Then, characteristic equation of (4.5) is as follows λ þ ð1 aþq 1 e λk þ e λl λ þ ð1 aþq e λk þ e λl ¼ (46) It is obvious that for a ¼ 1 and i ¼ 1;, the equation λ þ q i e λk þ e λl ¼ is the characteristic equation of (4.6) with q ¼ q i, and so one can immediately obtain the following corollary from the previous result given by Kuang (1993). Page of 11

Corollary 1. Suppose that a ¼ 1 for system (4.6) Let the matrix A of system (4.6) is written as the form ð I Then system (4.6) is asymptotically stable if and only if for i ¼ 1; q i ðk þ lcos k l π π (47) k þ l Theorem. Suppose that conditions of Lemma 5 are satisfied. Let the matrix A of system (1.1) be in the form ð I Then system (1.1) is asymptotically stable if and only if for i ¼ 1; either ða 1q i cos ωðkl (4) q i cos ωðkl ða 1 ; or > k þ l > ða 1q i cos ωðkl q i cos ωðkl ða 1 > sgnðq qffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi a1 arccos ðq i cosð ωðkl ð ð ða1 q i cos ω kl (49) Proof. The proof is similar to Theorem 1. 5. An extension to a system of higher dimension Finally, a higher dimensional linear delay differential system with two delays is considered x ðþ t ð1 axt ðþaxt ð ð kþxt ð l ¼ t ; (51) where a is a real number, A is a d d real constant matrix, and k, l are positive numbers such that k > l. Theorem 3. Let q e iθ ð ¼ 1; ;...; d be the eigenvalues of matrix A. Then system (5.1) is asymptotically stable iff ða 1q cos θ cos ωðkl (5) q cos ωðkl ða 1 ; or ða 1q cos θ cos ωðk q cos ωðk > ða 1 > sgnðq a 1 k þ l r ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi A θ q cos ωðkl ða 1 q cos ωðkl ; > where q, θ are real numbers and θ π. (53) Proof. Since q e iθ ð ¼ 1; ;...; d be the eigenvalues of matrix A the characteristic equation of system (5.1) is given by fð¼ λ Yd ¼1 λ þ ð1 aþq e λkþi θ þ e λlþiθ ¼ Thus, Theorem 3 can be seen as a result of Theorems 1 and 3. Page 9 of 11

Funding This work was supported by the Serbun Ufuk DEĞER. Author details Serbun Ufuk Değer 1, E-mail sudeger@kastamonu.edu.tr Yaşar Bolat 1, E-mail ybolat@kastamonu.edu.tr 1 Department of Mathematics, Institute of Sciences, Kastamonu University, Kastamonu, Turkey. Department of Mathematics, Faculty of Art & Science, Kastamonu University, Kastamonu, Turkey. Citation information Cite this article as On asymptotic stability of a class of time delay systems, Serbun Ufuk Değer & Yaşar Bolat, Cogent Mathematics & Statistics (1), 5 147379. References Bellman, R., & Cooke, K. L. (1963). Differential-difference equations. New York, NY Academic Press. Cooke, K. L., & Grossman, Z. (19). Discrete delay, distributed delay and stability switches. Journal of Mathematical Analysis Applications, 6, 59 67. doi1.116/-47x()943- Cooke, K. L., & van den Driessche, P. (196). On zeroes of some transcendental equations. Funkciala Ekvacio, 9, 77 9. (MR6515 (7m349)) Elaydi, S. (5). An introduction to difference equations (3rd ed.). New York, NY Springer-Verlag. Freedman, H. I., & Kuang, Y. (1991). Stability switches in linear scalar neutral delay equation. Funkciala Ekvacio, 34, 17 9. Gray, A., Greenhalgh, D., Hu, L., Mao, X., & Pan, J. A. (11). Stochastic differential equation SIS epidemic model. SIAM Journal on Applied Mathematics, 71(3), 76 9. doi1.1137/1156x Hrabalova, J. (13). Stability properties of a discretized neutral delay differential equation. Tatra Mountains Mathematical Publications, 54, 3 9. Ikeda, N., & Watanabe, S. (14). Stochastic differential equations and diffusion processes (Vol. 4). North Holland Elsevier. Insperger, T., & Stépán, G. (11). Semi-discretization for time-delay systems Stability and engineering applications (Vol. 17). New York, Dordrecht, Heidelberg, London Springer Science & Business Media. Khokhlovaa, T., Kipnis, M., & Malygina, V. (11). The stability cone for a delay differential matrix equation. Applied Mathematics Letters, 4, 74 745. doi1.116/.aml.1.1. Kruthika, H. A., Mahindrakar, A. D., & Pasumarthy, R. (17). Stability analysis of nonlinear time delayed systems with application to biological models. International Journal of Applied Mathematics and Computer Science, 7(1), 91 13. doi1.1515/amcs-17-7 Kuang, Y. (1993). Delay differential equations with applications in population dynamics. MR11 (94f341) Boston Academic Press. Li, Y., Ma, W., Xiao, L., & Yang, W. (17). Global stability analysis of the equilibrium of an improved timedelayed dynamic model to describe the development of T Cells in the thymus. Filomat, 31(), 347 361. doi1.9/fil17347l Liu, Q., Jiang, D., Shi, N., Hayat, T., & Alsaedi, A. (16). Asymptotic behaviors of a stochastic delayed SIR epidemic model with nonlinear incidence. Communications in Nonlinear Science and Numerical Simulation, 4,9 99. doi1.116/.cnsns.16.4.3 Matsunaga, H. (). Delay dependent and delay independent stability criteria for a delay differential system. American Mathematical Society, 4, 435 431. (136 Fields Inst. Commun.) Nakaima, H. (14). On the stability of a linear retarded differential-difference equation. Funkciala Ekvacio, 57, 43 56. doi1.1619/fesi.57.43 Naresh, R., Tripathi, A., Tchuenche, J. M., & Sharma, D. (9). Stability analysis of a time delayed SIR epidemic model with nonlinear incidence rate. Computers & Mathematics with Applications, 5(), 34 359. doi1.116/.camwa.9.3.11 Ruan, S.,&Wei, J.(3). on the zeros of transcendental functions with applications to stability of delay differential equations with two delays. Dynamic of Continuous, Discrete and Impulsive Systems, 1, 63 74. Smith, H. (1). An introduction to delay differential equations with applications to the life science. New York, NY Springer. Stepan, G. (199). Retarded dynamical systems Stability and characteristic functions, pitman research notes in mathematics series (Vol. 1). New York, NY Academic Press. Stepan, G. (199). Delay-differential equation models for machine tool chatter. Dynamics and Chaos in Manufacturing Processes, 47115935, 165 19. Xu, R. (1). Global stability of a delayed epidemic model with latent period and vaccination strategy. Applied Mathematical Modelling, 36(11), 593 53. doi1.116/.apm.11.1.37 Page 1 of 11

1 The Author(s). This open access article is distributed under a Creative Commons Attribution (CC-BY) 4. license. You are free to Share copy and redistribute the material in any medium or format. Adapt remix, transform, and build upon the material for any purpose, even commercially. The licensor cannot revoke these freedoms as long as you follow the license terms. Under the following terms Attribution You must give appropriate credit, provide a link to the license, and indicate if changes were made. You may do so in any reasonable manner, but not in any way that suggests the licensor endorses you or your use. No additional restrictions You may not apply legal terms or technological measures that legally restrict others from doing anything the license permits. Cogent Mathematics & Statistics (ISSN 574-55) is published by Cogent OA, part of Taylor & Francis Group. Publishing with Cogent OA ensures Immediate, universal access to your article on publication High visibility and discoverability via the Cogent OA website as well as Taylor & Francis Online Download and citation statistics for your article Rapid online publication Input from, and dialog with, expert editors and editorial boards Retention of full copyright of your article Guaranteed legacy preservation of your article Discounts and waivers for authors in developing regions Submit your manuscript to a Cogent OA ournal at www.cogentoa.com Page 11 of 11