Mathematics for Life Sciences and Medicine - Takeuchi Iwasa and Sato

From (16), (21) and the second of equations (7), for su ciently large t > 0 and small > 0, we have

If we choose

¯ ¯ ¯ ¯ ¯

M = max(XU , XI , B, IR, AR) ,

then there exists M > 0 such that the first inequality in (19) holds true. Consider now the “liminf”, i. e. the second in (19).

(i)Consider AR, IR.

From (13) and (14) we have

(ii)Consider B.

From (23), for su ciently large t > 0 and small > 0 we have

(iii)Consider XU .

For large t > 0, small > 0 we have

where the constants on the right side of (28) are positive. Thus, if we choose

m = min(XU , XI , B, IR, AR) , m > 0

considering that lim inf x(t) ≤ lim sup x(t), we have found two positive constants m, M, m ≤ M such that (19) hold true.

188 Edoardo Beretta et al.

Remark 1. As we will see in Theorem 1, if ∆(τ1) = 0 the permanence condition (20) becomes the existence condition of the positive equilibrium EP . Furthermore, if ∆(τ1) = 0, then B = B .

Concerning the equilibria of (7), we can give the following result (we omit

which is interior to the positive cone in R5.

We observe that the positive equilibrium EP exists whenever the parameter

8.4 Characteristic equation and local stability

System (7) linearized around any of the equilibria gives

If we define by x(t) = col (XU (t), XI (t), B(t), IR (t), AR (t)), then by inspection of (7) we get that L R5×5 is the matrix

and K (θ) : [−h, 0] → R5×5is the matrix function

0 0 w˜1(θ) 0 0 0 0 w2(θ) 0 0

where I R5×5 is the identity matrix and λ are the characteristic roots. If

we define by

0

Fi (λ) := wi(θ) eλθ dθ , i = 1, 2 (36)

−τi

then we get the following explicit structure for the characteristic equation:

Thus the study of the characteristic equation (37) is reduced to the study of (39), the remaining characteristic roots being negative.

We remark that F2 (λ) does not appear in (39), then the characteristic

roots in (39) are independent of the second delay τ2 of the model, i. e. the

0

term w (θ) B (t + θ) dθ does not play any role in the local stability of

−τ2

the equilibria. This implies that the first delay, that of innate immunity, is determinant in disease outcome. This likely follows because the adaptive response, AR, does not feedback into (3). Recall this was one formulation of a delay, and in other works we consider others.

Regarding local stability of the boundary equilibrium, we can prove:

Theorem 2. The boundary equilibrium EB is:

1. asymptotically stable if

α20 − α3 (sIR /µIR ) < 0 ;

2. linearly neutrally stable if

α20 − α3 (sIR /µIR ) = 0 ;

with one real vanishing characteristic root, while others characteristic roots are negative;

3. unstable (with one positive real root) if

α20 − α3 (sIR /µIR ) > 0 .

Proof. It follows immediately from (39) (since at the boundary equilibrium EB , B = 0 and IR = (sIR /µIR ) which gives two characteristic roots: one negative λ = −µIR and the other equal to the threshold parameter R0 for the existence of interior equilibrium EP :

λ = α20 − α3 (sIR /µIR ) .

We now study the local stability of the positive equilibrium EP . Assume

R0 > 0.

At EP , B satisfies

α20 − α3IR − ασ20 B = 0

and is therefore dependent on the choice of the delay kernel w1(θ).

8.4.1 Uniform delay kernel

Since F1(λ) regards the delay in immune response it is reasonable, as stated in the introduction, to assume that the delay kernel w1 is uniform, i. e.

which is defined since λ = 0 is not a root of (41). In fact, if λ = 0 then F1(0) = ∆(τ1) and (41) becomes

where by B (τ1) we emphasize the dependence on delay τ1, as it is evident from the equilibrium components (30).

Now remark that if τ1 = 0, then F1(λ) = 0 and (41) becomes

which has two negative roots, i. e. EP is asymptotically stable at τ1 = 0. We have thus the general problem to find the delay values τ1, if they exist,

at which for increasing τ1 the stability of EP changes or, in other words, at which EP undergoes a stability switch.

Since λ = 0 cannot be a root of (41) for any τ1 ≥ 0 a stability switch for EP can only occur at delay values τ1 at which a pair of pure imaginary roots λ = ±iω(τ1), ω(τ1) > 0, crosses the imaginary axis.

Let ω = ω(τ ), τ I R+0 be a solution of (52). We assume that ω = ω(τ ) is a continuous and di erentiable function of τ I.

For each solution ω = ω(τ ), τ I, of (50) we find the angle θ = θ(τ ), τ I

which are continuous and di erentiable for τ I.

Finally, by any mathematical software such as Maple or Matlab, we draw

Theorem 3. All the roots λ = ±iω(τ ), ω(τ ) > 0 of (50) occur at the delay values τ if and only if τ is a zero of one of the functions in the sequence

Sn, n N0.

At each τ I a pair of roots of (50) λ = ±iω(τ ) is crossing the imagi-

(57)

A stability switch occurs at τ = τ I if the total multiplicity on the right side of the imaginary axis changes from 0 to 2 or from 2 to 0 when τ increases through τ .

Theorem 4. If τ is the lowest positive zero of the function S0(τ ) and the transversality condition

!

dReλ

sign = 1

dτ λ=±iω(τ )

holds, (50) has