Marchuk's models of infection diseases: new developments
Irina Volinsky, Alexander Domoshnitsky, Marina Bershadsky, Roman Shklyar
MMARCHUK’S MODELS OF INFECTION DISEASES:NEW DEVELOPMENTS
IRINA VOLINSKY, ALEXANDER DOMOSHNITSKY, MARINABERSHADSKY AND ROMAN SHKLYAR
Abstract.
We consider mathematical models of infection diseasesbuilt by G.I. Marchuk in his well known book on immunology.These models are in the form of systems of ordinary delay differentialequations. We add a distributed control in one of the equationsdescribing the dynamics of the antibody concentration rate. Distributedcontrol looks here naturally since the change of this concentrationrather depends on the corresponding average value of the differenceof the current and normal antibody concentrations on the timeinterval than on their difference at the point t only. Choosingthis control in a corresponding form, we propose some ideas of thestabilization in the cases, where other methods do not work. Themain idea is to reduce the stability analysis of a given integro-differential system of the order n , to one of the auxiliary systemof the order n + m , where m is a natural number, which is easilyfor this analysis in a corresponding sense. Results for this auxiliarysystems allow us to make conclusions for the given integro-differentialsystem of the order n . We concentrate our attempts in the analysisof the distributed control in an integral form. An idea of reducingintegro-differential systems to systems of ordinary differential equationsis developed. We present results about the exponential stability ofstationary points of integro-differential systems, using the methodbased on the presentation of solution with the help of Cauchymatrix. Various properties of integro-differential systems are studiedby this way. Methods of general theory of functional differentialequations developed by N.V.Azbelev and his followers are used.One of them is the Azbelev W -transform. We propose ideas allowingto achieve faster convergence to stationary point using a distributedcontrol. We obtain estimates of solutions, using estimates of theCauchy matrices. Introduction
Mathematical models in the form of systems of nonlinear ordinarydifferential equations, are used in many fields of science and technologyto describe various phenomena. In medicine the purpose of mathematicalmodeling is the analysis and prediction of the development of diseasesand their possible treatment. A comprehensive work on mathematicalmodels in the field of immunology was summarized by Marchuk in hisbook [7]. The models constructed there reflect the most significantpatterns of the immune system acting during these diseases. These a r X i v : . [ q - b i o . CB ] J u l IRINA VOLINSKY, ALEXANDER DOMOSHNITSKY, MARINA BERSHADSKY AND ROMAN SHKLYAR model was studied in many works. Note, for example, the recentpapers [20], [21] and the bibliography therein. The adding control wasproposed, for example, in [6], [9], [10], [11], [12], [13], [14], [16], [24]. Inthe works [5],[8] the basic mathematical model that takes into accountthe discrete control of the immune response is proposed. See also therecent papers [6], [11], [13], where distributed control was considered.It can be noted that the use of information about behavior of a diseaseand the immune system for a long time (defined by distributed control,for example, in the form of an integral term) looks very natural inchoosing strategy of a possible treatment. Optimal control in the basicmodel of the infection disease was considered in the work [5], where thecontrol function characterizing realization of an immunotherapy whichincludes in administration of immunoglobulin or donor antibodies isproposed. In the work [22], the model of influence of an immunotherapyon dynamics of an immune response which represents generalization ofbasic model was considered. On the basis of the proposed model, theproblem of determination of coefficients on the basis of laboratory dateswas considered and a suitable management was proposed in [5], [23].Such task was called control in uncertain conditions [9]. A controlalgorithm in the uncertain conditions was proposed in the work ([5],see pages 71-73).In the recent papers [6], [11], [13] we present new approach for thestudy of the model of infection diseases. In this paper we summarizetheir results and formulate mathematical problems which look verynatural from the medical point of view.Our contribution in the modeling is a distributed feedback controlwhich is added to the equation describing the concentration of antibodies.This step transforms these systems to functional differential ones. Asa result, we have to study the properties of solutions of these systemssuch as asymptotic behavior in the neighborhood of stationary pointsand stability of the stationary points. Importance of stationary pointsshould be stressed. These points describe the conditions of the healthybody or the chronic disease. The aim of the treatment is to lead theprocess to one of the stationary points. Further we try to obtainestimates of solutions of linear and nonlinear systems of functionaldifferential equations. One of the ways to these estimates is constructionof the Cauchy matrix. First steps in this direction were proposed inthe recent paper [13].2.
Description of model
In this paper we deal with the system of functional differential equations x (cid:48) ( t ) + ( Ax )( t ) = (Φ T x )( t ) , t ∈ [0 , ∞ ) , x = col { x , , x n } (2 . ARCHUK’S MODELS OF INFECTION DISEASES: NEW DEVELOPMENTS 3 where the operators T and A are linear continuous. T, A : C n [0 , ∞ ) → L n ∞ [0 , ∞ ) ( C n [0 , ∞ ) , L n ∞ [0 , ∞ ) , are the spaces of continuous, and essentiallybounded vector functions x : [0 , ∞ ) → R n respectively), F : L n ∞ [0 , ∞ ) → L n ∞ [0 , ∞ ) can be a linear or nonlinear bounded operator. We couldanalyze various boundary value problems for equation (2.1). One ofthem is the initial value problem. One of the main questions is thestability of this system [4]. We consider the stationary points forcorresponding operators in the spaces of continuous functions C n [0 , ∞ ) oressentially bounded functions L n ∞ [0 , ∞ ) . We use our theoretical resultsin application to Marchuks model of infection diseases. This modelreflects the most significant patterns of the immune system functioningduring infectious diseases and focuses on the interactions between antigensand antibodies at different levels. We try to investigate the stabilityof stationary points of the immune system and its response to thetreatment. We propose the control in the distributed form and obtainstabilization in the neighborhood of the stationary point in the modelof infection diseases. From the applications’ point of view, the goal ofthe control in the system can be interpreted as a possibility to providea corresponding immune response. It is noted in [9] that the immuneresponse mechanisms provides a key to understanding disease processesand methods of effective medical treatment [7]. We try to combineour theoretical results with possible applications. Let us start with adescription of one of these applications. Consider, for example, theMarchuk model of infection diseases: dVdt = βV ( t ) − γF ( t ) V ( t ) dCdt = ζ ( m ) αF ( t ) V ( t ) − µ c ( C ( t ) − C ∗ ) dFdt = ρC ( t ) − ηγF ( t ) V ( t ) − µ f F ( t ) dmdt = σV ( t ) − µ m m ( t ) (2 . V ( t ) the antigen concentration rate, C ( t ) - the plasma cellconcentration rate, F ( t ) - the antibody concentration rate, m ( t ) therelative features of the body. It is clear that system (2.2) can bepresented in the form of general system (2.1). Let us describe thecoefficients: β - coefficient describing the antigen activity, γ - theantigen neutralizing factor, µ f - coefficient inversely proportional to thedecay time of the antibodies, µ m - coefficient inversely proportional tothe organ recovery time, µ c -coefficient of reduction of plasma cells dueto aging (inversely proportional to the lifetime), σ - constant relatedwith a particular disease, ρ - rate of production of antibodies by oneplasma cell. Denote C ∗ and F ∗ - the plasma rate concentration andantibody concentration of the healthy body, respectively. It is assumedthat during a certain period of time τ , the plasma is restored as a resultof the interaction between the antigen and the antibody cells. Theproduct ζ ( m ) αF ( t ) V ( t ) includes the following coefficients: α is the IRINA VOLINSKY, ALEXANDER DOMOSHNITSKY, MARINA BERSHADSKY AND ROMAN SHKLYAR stimulation factor of the immune system. The function ζ ( m ) = 1 , ≤ m < m ∗ − m − m ∗ , m ∗ ≤ m ≤ , is a continuous function, characterizing the health of the organ, whichdepends on the relative characteristics m of the body, where m ∗ is themaximum proportion of cells destroyed by antigens in the case thatthe normal functioning of the immune system is still possible. Thisfunction is non-negative and does not increase. The function m ( t ) canbe described as 1 − − M ( t )1 − M ∗ ( t ) , where M ( t ) is the characteristic of a healthyorgan (mass or area) and M ∗ ( t ) is the corresponding characteristic of ahealthy part of the affected organ. Let us discuss now every equation inthe model (2.2) in more detail form. The first equation dVdt = βV ( t ) − γF ( t ) V ( t ) presents the block of the virus dynamics. It describes thechanges in the antigen concentration rate and includes the amountof the antigen in the blood. The antigen concentration decreases asa result of the interaction with the antibodies. The immune processcharacterizes the antibodies, whose concentration changes with time(destruction rate), is described by the equation: dFdt = ρC ( t ) − ηγF ( t ) V ( t ) − µ f F ( t ). The amount of the antibody cells also decreases as a result ofthe natural destruction. However, the plasma restores the antibodiesand therefore the plasma state plays an important role in the immuneprocess. Thus, the change in concentration rate of the plasma cell isincluded in several differential equations describing this system. Takinginto account the healthy body level of plasma cells and their naturalaging, the term µ c ( C ( t ) − C ∗ ) is included in the second equation ofsystem (2.2). The second and third equations present the humoralimmune response dynamics. Concerning the last equation of system(2.2): dmdt = σV ( t ) − µ m m ( t ). The following can be noted 1) thevalue of m increases with the antigen’s concentration rate V ( t ); 2)the maximum value of m is unity, in the case of 100% organ damageor zero for a fully healthy organ. The coefficient µ m describes therate of generation of the target organ. This model was considered inthe recent work of Skvortsova [10]. Adding the control in the modelintroduced in Marchuk’s book [7] is proposed, for example, in theworks by Rusakov and Chirkov [8],[9] where the importance of thisdevelopment is explained.3. Stabilization through a support of the immune system
Our first goal is in stabilization of the process in the neighborhood ofa suitable stationary solution. We make a corresponding linearizationand then use the concepts of the stability theory proposed by N.V.Azbelevand his followers in the well-known books [2],[3],[4] for linear functionaldifferential systems. The main idea is to choose close in a correspondingsense auxiliary linear system, to solve it and to construct its Cauchy
ARCHUK’S MODELS OF INFECTION DISEASES: NEW DEVELOPMENTS 5 matrix (see, for example [11], [13], [15]). Then the scheme of theAzbelev W -transform is used. We propose new ideas in choosing closesystems. For system of the order n , a corresponding close system canbe of the order n + m . Our main idea here is to reduce the analysisof a given system of the order n to one of the auxiliary system of theorder n + m , which is easily in a corresponding sense. Results for theauxiliary systems allows us to make conclusions for the given system ofthe order n . We essentially concentrate our attempts in the analysis ofthe distributed control in an integral form. The integral terms reflectan orientation on average values in the construction of the control.Another reason of appearance of the integral terms is in the use of the”history of the process” to choose a strategy of a possible treatment.In our model, we demonstrate among other ideas that observation onthe process of diseases can be very important in a treatment. It shouldbe also noted that the proposed control can be realized practically. Tosum up all these consequences, we can conclude that the control in theintegral form is reasonable from the medical point of view. Stabilityproperties of integro-differential systems are studied.Modifying model (2 . u ( t ) = − b (cid:90) t ( F ( s ) − F ∗ − (cid:15) ) e − k ( t − s ) ds. (3 . . dVdt = βV ( t ) − γF ( t ) V ( t ) dCdt = ζ ( m ) αF ( t ) V ( t ) − µ c ( C ( t ) − C ∗ ) dFdt = ρC ( t ) − ηγF ( t ) V ( t ) − µ f F ( t ) + u ( t ) dmdt = σV ( t ) − µ m m ( t ) (3 . u ( t ) is defined by (3 . F ∗ be the value of the antibodyconcentration rate for a healthy body. While the case of F ∗ > βγ is considered by G.I. Marchuk in the book [7]. We try to considerthe ”bad” case where F ∗ < βγ . It is clear that system (2.2) couldnot be stable in this case in the neighborhood of the stationary point(0 , C ∗ , F ∗ , dVdt = βV ( t ) − γF ( t ) V ( t ) dCdt = ζ ( m ) αF ( t ) V ( t ) − µ c ( C ( t ) − C ∗ ) dFdt = ρC ( t ) − ηγF ( t ) V ( t ) − µ f F ( t ) + u ( t ) dmdt = σV ( t ) − µ m m ( t ) dudt = − b ( F ( t ) − F ∗ − ε ) − ku ( t ) . (3 . . . Lemma 3.1.
The components of the solution-vector y ( t ) = col ( v ( t ) , s ( t ) , f ( t ) , m ( t )) of system (3 . and four first components of the solution-vector x ( t ) = IRINA VOLINSKY, ALEXANDER DOMOSHNITSKY, MARINA BERSHADSKY AND ROMAN SHKLYAR col ( v ( t ) , s ( t ) , f ( t ) , m ( t ) , ˜ u ( t )) of system (3 . satisfying the initialcondition u (0) = 0 coincide. Theorem 3.1.
Let the inequality εγ > β − γF ∗ , k > , b > befulfilled, then the stationary solution (0 , C ∗ , F ∗ + ε, , of system (3 . is exponentially stable .To prove Theorem 3.1, we reduce the analysis of system (3.2) to oneof system (3.3) by Lemma 3.1, linearize in the neighborhood of thestationary pointand then the negativity of roots to the characteristic polynomial ofsystem (3.3) is demonstrated (see, for example, [6]).Thus, we can stabilize the process at the point (0 , C ∗ , F ∗ + ε, , F ∗ + ε , where ε > β − γF ∗ γ .4. Distributed Control and the Lyapunov CharacteristicExponents
To use the control in order to make convergence to set stationarystate faster is the second goal. Note that the stationary points presentthe condition of the healthy body or at least chronical process of diseasewhich we try to reach. This problem is directly related to the durationof a possible treatment. In many cases, this may have an importantinfluence on the choice of treatment method and on the decision on theacceptability of such treatment in principle.The goal of this part to obtain faster tending to set stationary state.Consider the system dvdt = βv ( t ) − γF ∗ f ( t ) v ( t ) dsdt = αV m F ∗ C ∗ ζ ( m ) f ( t ) v ( t ) − µ c ( s ( t ) − dfdt = ρC ∗ F ∗ s ( t ) − ηγV m f ( t ) v ( t ) − µ f f ( t ) − b ˜ u ( t ) dmdt = σV m v ( t ) − µ m m ( t ) d ˜ udt = f ( t ) − − k ˜ u ( t ) (4 . u = (cid:90) t ( f ( s ) − e − k ( t − s ) ds . Denoting in (4 . α = β, α = γF ∗ , α = αV m F ∗ C ∗ , α = µ f = ρC ∗ F ∗ , α = µ c , α = σV m , α = µ m , α = ηγV m , (4 . .
1) in the neighborhood of stationary point v = m = ˜ u = 0, s = f = 1, we can write system (4 .
1) in the form
ARCHUK’S MODELS OF INFECTION DISEASES: NEW DEVELOPMENTS 7 dx dt = ( α − α ) x dx dt = α x − α x dx dt = − α x + α x − α x − bx dx dt = α x − α x dx dt = x − kx , (4 . .
2) and write it in the form dx dt = ( α − α ) x dx dt = α x − α x dx dt = − α x + α x − α x dx dt = α x − α x (4 . λ i , i = { , } the roots of the characteristic polynomial ofsystems (4 . λ = max ≤ i ≤ λ i , ˜ λ ∗ = max ≤ j ≤ Re ( λ ∗ j ) of (4 . Theorem 4.1. If β < γF ∗ , b > and k > , then integro-differentialsystem (4 . is exponentially stable and if in addition the inequality k > α is fulfilled then ˜ λ ≥ ˜ λ ∗ . To prove Theorem 4.1, after reducing analysis of system (3.2) to one ofsystem (3.3) by Lemma 3.1 and linearizing in the neighborhoods of thestationary points of system (2.2) and (3.3) respectively, we comparethe roots of characteristic polynomials of system (4.3) and (4.4) (see,for example [11]).On Figures 1-4 the solution of model of the pneumonia with thenatural flow of data without the control of disease are presented bycurves of red color, disease in the case of considered distributed control-by curves of green color.Figure 1 demonstrates the dynamics in antigen concentration duringthe course of the disease. The insert detailing the process in thefirst two days was performed on a different scale and demonstratesthe fact that the management transfers the disease from the acuteform to the subclinical one (the antigen concentration only decreasesafter injetion). Figure 2 demonstrates the dynamics in plasma cellconcentration during the disease process. It can be seen from the figurethat control leads to a faster increase in the concentration of plasmacells, which in this case ensures a transition to the subclinical formof the disease. In addition, it is necessary to note a fourfold increasein the maximum concentration of plasma cells in the case of control,compared with the option without control. Figure 3 demonstratesthe dynamics in antibody concentration during the disease process.The graph shows that the concentration of antibodies in the solutionwith control practically does not change, because in this case theyare replaced by donor antibodies, which is what the control actually
IRINA VOLINSKY, ALEXANDER DOMOSHNITSKY, MARINA BERSHADSKY AND ROMAN SHKLYAR
Figure 1.
Dynamics of the immune response: antigen
Figure 2.
Dynamics of the immune response: plasma
Figure 3.
Dynamics of the immune response: antibodiesconsists of. The dynamics in the proportion of target organ cellsdestroyed by antigen during the disease process is presented on Figure4. The values for the variant with control are given with an increase of10 times. Thus, control allows to reduce the maximum proportion ofaffected cells of the target organ by more than 2 . × times.5. Cauchy Matrix
To estimate the size of the neighborhood of the stationary solutionswhich usually describe the states of the healthy body is the third goalof our research. In practical problems it is necessary since we have to
ARCHUK’S MODELS OF INFECTION DISEASES: NEW DEVELOPMENTS 9
Figure 4.
Dynamics of the immune response: rate ofthe destroyed cellshold process in a corresponding zone. Process going beyond a certainadmissible neighborhood of a stationary solution may be dangerous forpatients.In constructing every model, the influences of various additionalfactors that have seemed to be nonessential were neglected. The influenceeffect of choosing nonlinear terms by their linearization in neighborhoodof stationary solution is also neglected. Even in the frame of linearizedmodel, only approximate values of coefficients instead of exact ones areused. Changes of these coefficients with respect to time are not usuallytaken into account. It looks important to estimate an influence of allthese factors.In order to make this we have to obtain estimates of the elements ofthe Cauchy matrix of corresponding linearized (in a neighborhood of astationary point) system. Consider the system x (cid:48) ( t ) = P ( t ) x ( t ) + G ( t ) , where P ( t ) is a ( n × n )-matrix, G ( t ) is n -vector. Its general solution x ( t ) = col { x ( t ) , ...x n ( t ) } can be represented in the form (see, forexample, [2]) x ( t ) = (cid:90) t C ( t, s ) G ( s ) ds + C ( t, x (0) , where n × n -matrix C ( t, s ) is called the Cauchy matrix. Its j -th column( j = 1 , ..., n ) for every fixed s as a function of t , is a solution of thecorresponding homogeneous system x (cid:48) ( t ) = P ( t ) x ( t ) , satisfying the initial conditions x i ( s ) = δ ij , where δ ij = (cid:26) , i = j, , i (cid:54) = j, i = 1 , ..., n, This Cauchy matrix C ( t, s ) satisfies the following symmetric properties C ( t, s ) = X ( s ) X − ( s ) , where X ( t ) is a fundamental matrix, C ( t,
0) = C ( t, s ) C ( s, P ( t ) = P, we have X ( t − s ) = C ( t, s ) is a fundamental matrix for every s ≥ . Thesedefinition and properties allow us to construct and estimate C ( t, s ).The construction of the Cauchy matrix of system (4 .
3) can be found,for example, in [13].6.
Stabilization with the use of uncertain coefficient inthe control
Consider the following system of equations with uncertain coefficientin the control dVdt = βV ( t ) − γF ( t ) V ( t ) dCdt = ζ ( m ( t )) αF ( t ) V ( t ) − µ c ( C ( t ) − C ∗ ) dFdt = ρC − ηγF ( t ) V ( t ) − µ f F ( t ) − ( b + (cid:52) b ( t )) u ( t ) dmdt = σV ( t ) − µ m m ( t ) dudt = F ( t ) − F ∗ − ku ( t ) (6 . u ( t ) = t (cid:82) ( F ( s ) − F ∗ ) e − k ( t − s ) ds This system can be rewritten in the form x (cid:48) = ( a − a ) x + g ( x ( t ) , x ( t )) x (cid:48) = a x − a x + g ( x ( t ) , x ( t )) x (cid:48) = − a x + a x − a x − ( b + (cid:52) b ( t )) x + g ( x ( t ) , x ( t )) x (cid:48) = a x − a x x (cid:48) = x − kx , (6 . g i ( x ( t ) , x ( t )) ( t ) , ≤ i ≤ X (cid:48) = AX + ∆ B ( t ) X + F ( t ) , (6 . X ( t ) = x ( t ) x ( t ) x ( t ) x ( t ) x ( t ) , ∆ B ( t ) = − (cid:52) b ( t )0 0 0 0 00 0 0 0 0 . On the basis of the estimates of the elements of the Cauchy matrixwe obtain the following assertions on the stability of system (6.2).Denoting Q j = ess sup t ≥ t (cid:82) (cid:80) i =1 (cid:12)(cid:12)(cid:12) (∆ B ( t ) C ( t, s )) ij (cid:12)(cid:12)(cid:12) ds and (cid:52) b ∗ = ess sup t ≥ |(cid:52) b ( t ) | ,we obtain the estimates: ARCHUK’S MODELS OF INFECTION DISEASES: NEW DEVELOPMENTS 11 Q ≤ (cid:52) b ∗ (cid:12)(cid:12)(cid:12) α ( α − α ) − α ( α − α ) α α ( α − α ) (cid:12)(cid:12)(cid:12) | λ | + (cid:12)(cid:12)(cid:12) α ( α − α ) − α ( α − α ) α α ( α − α ) (cid:12)(cid:12)(cid:12) | λ | + (cid:12)(cid:12)(cid:12) α a α α (cid:12)(cid:12)(cid:12) ,Q ≤ (cid:52) b ∗ (cid:104)(cid:12)(cid:12)(cid:12) α − α α ( α − α ) (cid:12)(cid:12)(cid:12) | λ | + (cid:12)(cid:12)(cid:12) α − α α ( α − α ) (cid:12)(cid:12)(cid:12) | λ | + | a α | (cid:105) ,Q ≤ (cid:52) b ∗ (cid:104) | α − α | | λ | + | α − α | | λ | (cid:105) ,Q = 0 ,Q ≤ (cid:52) b ∗ (cid:104)(cid:12)(cid:12)(cid:12) α α − α (cid:12)(cid:12)(cid:12) | λ | + (cid:12)(cid:12)(cid:12) α α − α (cid:12)(cid:12)(cid:12) | λ | (cid:105) . (6 . Theorem 6.1. [13]
Let k > , b > and a i , ≤ i ≤ , arereal positive and different, a < a , ( a − k ) > b and the inequality max ≤ j ≤ {| Q j |} < be true. Then system (6 . is exponential stable. Denoting P j = ess sup t ≥ t (cid:82) (cid:80) i =1 (cid:12)(cid:12)(cid:12) (∆ B ( t ) C ( t, s )) ij (cid:12)(cid:12)(cid:12) ds , we obtain theestimates P ≤ (cid:52) b ∗ (cid:12)(cid:12)(cid:12) β β − β β β β β (cid:12)(cid:12)(cid:12) a + k ) + (cid:12)(cid:12)(cid:12) β ( β − β ) − β ( β − β ) β β β β (cid:12)(cid:12)(cid:12) | a − k | + (cid:12)(cid:12)(cid:12) β β β (cid:12)(cid:12)(cid:12) | a | + | β | | a − a | ,P ≤ (cid:52) b ∗ (cid:20)(cid:12)(cid:12)(cid:12) β β β (cid:12)(cid:12)(cid:12) a + k ) + (cid:12)(cid:12)(cid:12) β − β β β β (cid:12)(cid:12)(cid:12) | a − k | + | β | | a | (cid:21) ,P ≤ (cid:52) b ∗ (cid:20) | β | a + k ) + | β β | | a − k | (cid:21) ,P = 0 ,P ≤ (cid:52) b ∗ | β | | a − k | . (6 . Theorem 6.2. [13]
Let k > , b > and a i , ≤ i ≤ , arereal positive and different, a < a , ( a − k ) = 4 b and the inequality max ≤ j ≤ {| P j |} < be true. Then system (6 . is exponential stable. Denoting R j = ess sup t ≥ t (cid:82) (cid:80) i =1 (cid:12)(cid:12)(cid:12) (∆ B ( t ) C ( t, s )) ij (cid:12)(cid:12)(cid:12) ds we obtain estimates R ≤ (cid:52) b ∗ (cid:12)(cid:12)(cid:12) γ − γ γ γ (cid:12)(cid:12)(cid:12) | a + k | + (cid:12)(cid:12)(cid:12) γ (2 γ − a + k )+ γ ( a − a +3 k ) γ γ γ (cid:12)(cid:12)(cid:12) | a + k | + (cid:12)(cid:12)(cid:12) γ γ γ (cid:12)(cid:12)(cid:12) | a | + (cid:12)(cid:12)(cid:12) γ (cid:12)(cid:12)(cid:12) | a − a | ,R ≤ (cid:52) b ∗ (cid:104) | γ | | a + k | + (cid:12)(cid:12)(cid:12) a − k +2 a γ γ (cid:12)(cid:12)(cid:12) | a + k | + | γ | | a | (cid:105) ,R ≤ (cid:52) b ∗ | γ | | a + k | ,R = 0 ,R ≤ (cid:52) b ∗ (cid:104) | a + k | + (cid:12)(cid:12)(cid:12) a − kγ (cid:12)(cid:12)(cid:12) | a + k | (cid:105) . (6 . Theorem 6.3. [13]
Let k > , b > and a i , ≤ i ≤ , arereal positive and different, a < a , ( a − k ) < b and the inequality max ≤ j ≤ {| R j |} < be true. Then system (6 . is exponential stable. Remark 6.1.
Note that the approach presented here can be used inthe model of testosterone regulation (see, for example, [12] , [17] , [18] , [19] ). Acknowledgements
This paper is part of the third and fourthauthors Ph.D. thesis which is being carried out in the Department ofMathematics at Ariel University.
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Ariel university, Ariel 40700, Israel, Department of Mathematics
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