On the existence and stability of periodic solutions in the absence of immunity in an impulsive model based on Gompertzi
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THE EXISTENCE AND STABILITY PERIODIC SOLUTIONS IN THE ABSENCE IMMUNITY IN AN IMPULSIVE MODEL BASED GOMPERTZIAN DYNAMICS V. P. Martsenyuka† and I. S. Gvozdetskaa‡
UDC 519.876.2:611.018.4
Abstract. The existence of the periodic solution of a system with the absence of immunity is proved in the paper. The conditions for its global asymptotic stability are obtained. The periodic solution of the system with the absence of immunity is proved to be a global attractor. Keywords: immunity, stability, impulsive differential equations, systems with delay. DEVELOPING AN IMPULSIVE MODEL The simplified model of antitumor immunity proposed in [1] employs the Gompertzian dynamics for the growth of a population of proliferating cells and the Marchuk model of immune system [2]. A correct setting up of models of dynamic systems in biology and medicine involves the boundedness analysis of solutions. Moreover, problems of the qualitative analysis of dynamic systems need constructive estimates of the solutions of models. The conditions that guarantee the local existence and continuity of the solution of such systems are obtained in [3]. The purpose of the present paper is to prove the existence of a periodic solution in the absence of immunity where plasmatic cells and antibodies are absent and to obtain the conditions of its global asymptotic stability. We will consider the following model: q dL( t ) = a L L( t ) ln L - g L F ( t )L( t ), dt L( t ) dC ( t ) = x( m) aL( t - t )F ( t - t ) - mC (C ( t ) - C 0 ) , dt dF ( t ) = b f C ( t ) - ( m f + hg L L( t ))F ( t ), dt dm( t ) = sL( t ) - m m m( t ) , t ¹ nT , n Î N , dt
DL( t ) = - pL( t ), 0 < p < 1,ü ï DC ( t ) = DF ( t ) = 0, ý ï Dm( t ) = - pm( t ) þ
(1)
t = nT , n Î N .
The initial conditions of system (1) are specified: ( f1 ( s ), f 2 ( s ), f 3 ( s ), f 4 ( s )) Î C + = C ([ -t ,0], R +4 ) , f i ( 0) > 0, i = 1, 3.
(2)
System (1) is considered in the biologically significant domain D = {( L, C , F , m )| L, C , F , m ³ 0}. The values of the variables and coefficients of the model are presented in [1]. a
I. Ya. Horbachevsky Ternopil State Medical University, Ternopil, Ukraine, †[email protected]; [email protected]. Translated from Kibernetika i Sistemnyi Analiz, No. 4, July–August, 2012, pp. 126–131. Original article submitted November 28, 2011.
‡
586
1060-0396/12/4804-0586
©
2012 Springer Science+Business Media, Inc.
PERIODIC SOLUTION IN THE ABSENCE OF IMMUNITY Let us consider the existence of a periodic solution in the absence of immunity when plasmatic cells and antibodies are absent in the system, i.e., C ( t ) = F ( t ) = 0, t ³ 0. Under such conditions, growth of tumoral cells is observed on a time interval nT £ t £ ( n + 1)T . Some main properties of subsystem (1) will be analyzed: q dL( t ) = a L L( t ) ln L , t ¹ nT , n Î N , dt L( t ) dm( t ) (3) = sL( t ) - m m m( t ) , dt DL( t ) = - pL( t ),ü ý t = nT , n Î N . Dm( t ) = - pm( t )þ In system (3) we assume that ln L( t ) = v ( t ) and obtain the impulsive inhomogeneous linear equat
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