A STABILITY THEOREM FOR EQUILIBRIA OF DELAY DIFFERENTIAL EQUATIONS IN A CRITICAL CASE WITH APPLICATION TO A MODEL OF CELL EVOLUTION

In this paper the stability of the zero equilibrium of a system with time delay is studied. The critical case of a multiple zero root of the characteristic equation of the linearized system is treated by applying a Malkin type theorem and using a complete Lyapunov-Krasovskii functional. An application to a model for malaria under treatment considering the action of the immune system is presented. Mathematics Subject Classification. 93A30, 93B18, 93C10, 93D05, 93D30. Received August 31, 2020. Accepted April 12, 2021.

includes the process of erythropoiesis, the evolution of the parasites, the action of the immune system and the effect of the treatment.
A different model where the same critical case is encountered is the one introduced in [3].

A general theorem on a critical case
For general results on stability for delay differential equation we refer to [5,8,9,11].
. From general properties related to Lyapunov-Krasovskii functionals, it follows that there exists N 2 > 0 so that ||ξ(t)|| 2 ≤ N 2 in [0, t 1 ) and N 2 can be made arbitrary small if η is small.

Cell evolution in Malaria under treatment
One DDE model of malaria can be found in the paper [6]. It contains a simplified equation for erythrocytes' evolution with respect to the model in [7]. We consider a more physiological model for erythropoiesis, following [1,2,7,19]. We only concentrate on the evolution of merozoites during malaria, since their number considerably overcomes that of the gametocytes and their impact is responsible for the damaging effects of the disease. The influence of the immune system in the evolution of malaria is well recognized [15,17], so it is also considered in the model in the second stage of the illness, the blood stage. The model for the action of the immune system is based on [4,16,17]. The following equations describe the evolution of the disease induced by Plasmodium falciparum under treatment with Artemisinin.

The mathematical model
Recent studies [12] show that Plasmodium falciparum acts on both young and mature erythrocytes. With p the invasion rate, the low of masses results in the presence of the term −pv 2 v 6 that is accountable for the infection process.
The model that takes into consideration the response to the treatment is: v Here r > 1, β = β 1 − β d , β 1 being the burst size in absence of treatment and β d the effect of treatment with Artemisinin. Also,Ã e = A e (2η 2 + η 1 ), with A e the amplification factor. S accounts for the mortality of infected RBCs, and is influenced by treatment (see [6]) It is clear that E 1 = (0, 0,v 3 ,v 4 , 0, 0,v 7 , 0,v 9 , 0, 0, 0) is an equilibrium point, that can be interpreted as closed to the death of the patient.
Let A = [a i,j ] be the matrix in the linear approximation around E 1 corresponding to undelayed terms, B = [b i,j ] the matrix corresponding to terms with the delay τ 1 , C = [c i,j ] the matrix that corresponds to the terms with the delay τ 2 , D = [d i,j ] the matrix that corresponds to the terms with the delay τ 3 , E = [e i,j ] the matrix that corresponds to the terms with the delay τ 4 , F = [f i,j ] the matrix that corresponds to the terms with the delay τ 5 , G = [g i,j ] the matrix that corresponds to the terms with the delay τ 6 and H = [h i,j ] the matrix that corresponds to the terms with the delay τ 7 . Then, as can be easily checked, the following nonzero elements appear in the matrices defined above: for A = ∂f ∂v a 11 = − γ0 Accordingly, the characteristic equation will be: ·(λ − a 77 )(λ − a 88 )(λ − a 99 )(λ − a 10,10 )(λ − a 11,11 ) = 0, and one can see that a critical case for stability by the first approximation theory appears.

Analysis of the critical case of cell evolution in Malaria under treatment
We perform a translation to zero by p i = v i −v i , for i = 3, 4, 7, 9. The new system becomes . p i =f i (p, p τ1 , p τ2 , · · · , p τ7 ), i = 1, 12, The matrices of partial derivatives into zero are as before, The characteristic equation for the zero solution of the new system is exactly the one forẼ. We have: Since we do not have the linear part equal to zero, the Theorem 2.2 is not directly applicable. We will bring the system to the canonical form to which Theorem 2.2 can be applied.
For β 12 = 1, it follows that β 6 = −â 12,6 a66 . Then ξ 2 = β 6 u 6 + u 12 , so the equation of ξ 2 has no linear part, that isξ 4 containing only terms of order greater or equal to two. Take Replace the twelfth equation byξ 2 so this equation has a zero linear term. Substitute p 12 in the equations of the system in u and define: Remark that the linear part of g 6 does not contain ξ 2 . and the other equations do not contain ξ 2 at all. Redenote the other right-hand components of the system in u by the corresponding g. A new system is obtained for ζ = (u1, u2, u3, ξ1, u5, . . . , u11, ξ2)ζ = g(ζ, ζ τ1 , . . . , ζ τ7 ) (3.2) ∂g ∂ζ c 21 =Ã e k(u 3 +v 3 ).
From these calculations we conclude that Theorem 2.2 can be applied to study the stability of the zero solution of system (11) and its conclusions transferred to the study of stability of the equilibrium point E 1 of system (10).

Conclusions
A qualitative study of the solutions of models described by delay differential equations is an important step towards the validation of the model. One important property is the stability of equilibrium points. A difficult problem in this direction is the stability study when zero eigenvalues exist in the spectrum of the Jacobian matrix calculated in the equilibrium. This makes the theorem on stability by linear approximation inapplicable. This critical case, discussed in this pape, is not uncommon in DDEs models and gives rise to a number of problems.
Although the theorem we present can be applied to a specific class of systems of DDEs, there is a considerable amount of models which can fit the required type, either in life sciences or in engineering. The original biological model for which we use the above theorem has a standalone importance. We considered a widely spread disease (with available treatment), which continues to impact human life. The need to have available tools to study such models is undeniable.
It is also worth noticing that the mathematical model we introduced might be adapted to capture other diseases involving blood cells evolution (see [3]).