A Condition for Stability in an SIR Age Structured Disease Model with Decreasing Survival Rate.

A Condition for Stability in an SIR Age Structured Disease Model with

Decreasing Survival Rate

A.K. Supriatna1, Edy Soewono2

1Department of Mathematics, Universitas Padjadjaran, km 21 Bandung-Sumedang 45363, Indonesia fax: 062-22-7794696, email: asupriat@yahoo.com.au; aksupriatna@bdg.centrin.net.id

2 Financial and Industrial Mathematics Group ITB, Bandung 40132, Indonesia

Abstract

In this paper we present an SIR model for disease transmission with an assumption that individuals in the under-laying demographic population experience a monotonically decreasing survival rate. We show that the results in an analogous SI model for disease transmission are the special case of the SIR model in this paper. We found that there is a threshold for the disease transmission determining the existence and the absence of the endemic equilibrium. We investigate the stability of this equilibrium via a Gronwall-like inequality theorem. Unlike in the SI model, the threshold for the existence is not equivalent to the threshold for the stability of the equilibrium. We provide an additional condition which consistently generalizes the results in the SI model..

Keywords

: Disease Modeling, SIR Model, Threshold Number, Stability of an Equilibrium Point

I. Introduction

Age structure is among the important factors affecting the dynamics of a population in relation to the spread of contagious diseases. To study the effect of age structure in the dynamics of contagious diseases, at least there are two approaches, first by developing a population model with continuous age [1,2] and second by developing a population with age groups [3]. A model of SI disease transmission is studied in [1] and a model SIS disease transmission is studied in [2] by assuming continuous age.

An SI model only fits to diseases that cause an infective individual remains infective for life. To increase realism, in this paper we present a model for an SIR disease transmission by assuming continuous age. Here we assume individuals in the under-laying population experience a monotonically decreasing survival rate as their age goes by. We also assume that there is a density-dependent but age-independent birth rate. We show that there is an endemic threshold, below which the disease will stop, and above which the disease will stay endemic. The results in the SI disease model in [1] generalize into the SIR model.

II. The Mathematical Model

The model discussed here is the generalization of the model in [1] to include an R compartment as an attempt to increase the realism

of the model. Throughout the paper we use the following notations:

H

N =Total number of individuals in the population

H

S =The number of susceptible individuals in the population

H

I =The number of infective individuals in the population

H

R =The number of recover or immune individual in the population

H

B =The recruitment rate or the birth rate H

β =The transmission probability of the disease We assume that the population NH is divided into three compartments, SH, IH, and

H

R , such that NH =SH + +IH RH. To include age structure, suppose that there exists , a function of age describing the fraction of human population who survives to the age of a or more, such that,

( ) H

Q a

(0) 1 H

Q = and is a non-negative and monotonically decreasing for 0 . If it is assumed that life expectancy is finite, then

( ) H

Q a

a ≤ ≤ ∞

0QH( )a da L

∞

= < ∞

∫

and .

0 aQH( )a da

∞

< ∞

∫

1

Further, let also assume that ,

, , and denotes,

respectively, the numbers of

(0)( )

H

N t

(0)( )

H

S t IH(0)( )t RH(0)( )t

H

N (0), SH (0), H

(0) ₀

( ) ( ) t ( )

H H H H

N t =N t +

∫

B Q a da. 2 Since the per capita rate of infection in the population at time t isβ_HI_H( )t , then the number of susceptible at time t is given by

( ) (0) ₀

( ) ( ) ( )

t H H t a

t I s ds

H H H H

S t =S t +

∫

B Q a e−∫− β da_. 3

If the rate of recovery is γ then the number of infective at time is given by

t

( ) (0) ₀

( ) (0) ₀

( ) ( ) ( ) 1

( ) ( ) 1 .

t t

H H H

t a t a

t H H

t a H

t I s ds ds

H H H H

t I s ds _a

H H H

I t I t B Q a e e da

I t B Q a e e da

β γ

β γ

− −

−

− −

− ₋

⎡ _∫ ⎤ _∫

= + ⎢ − ⎥

⎣ ⎦

⎡ _∫ ⎤

= + _⎢ − _⎥

⎣ ⎦

∫

∫ 4

Furthermore, considering that then we have

( ) ( ) ( ) ( )

H H H H

R t =N t −S t −I t

(0) ₀

( ) (0) ₀

( ) (0) ₀

( ) (0) ₀

( ) ( ) ( )

( ) ( )

( ) ( ) 1

( ) ( ) 1 1 .

t H H t a

t H H

t a H

t H H

t a H

t

H H H H

t I s ds

H H H

t I s ds _a

H H H

t I s ds a

H H H

R t N t B Q a da

S t B Q a e da

I t B Q a e e da

R t B Q a e e da

β

β _γ

β _γ

−

−

−

−

− ₋

− ₋

= +

∫

− −

⎡ _∫ ⎤

− − ⎢− ⎥

⎣ ⎦

⎡ _∫ ⎤_{⎡ ⎤}

= + _⎢− _{⎥ ⎣} − _⎦

⎣ ⎦

∫ ∫ ∫

∫ 5

It is clear that

(0)

lim

( ) 0 H

N t

t→ ∞ = , (0)

lim

( ) 0 H

S t

t→ ∞ = ,

(0)

lim

( ) 0 H

I t

t→ ∞ = , and (0)

lim

( ) 0 H

R t

t→ ∞ = .

6 Hence, equations (3), (4), (5), and (6) constitute an SIR age structured disease model.

III. The Existence of a Threshold

Number

In this section we will show that there is a threshold number for the model discussed above. Let us consider the following limit system of equations which has the same behavior with the system (3) to (6) whenevert→ ∞:

0

( ) ( )

H H H

N t =

∫

∞B Q a da, 7

( ) 0

( ) ( )

t H H t a I s ds

H H H

S t =

∫

∞B Q a e−∫−β da_, 8

( ) 0

( ) ( ) 1

t H H

t a H

I s ds _a

H H H

I t = ∞B Q a ⎡_⎢ −e−∫−β ⎤_⎥e−γ da

⎣ ⎦

∫

9

( ) 0

( ) ( ) 1 1

t H H

t a I s ds aH

H H H

R t = ∞B Q a⎡_⎢ −e−∫−β ⎤⎡ −e−γ ⎤

⎥ ⎣ ⎦

⎣ ⎦

∫ da 10

Equations (7) show that the value of is constant, hence the equations for the age-structured SIR model reduce to three equations, (8) to (10).

( ) H

N t

The Equilibrium of the system is given by with

* * *

(SH,IH,RH)

*

H

I satisfying

* *

0 ( ) 1

H HI a aH

H H H

I =

∫

∞B Q a _⎣⎡ −e−β _⎦⎤e−γ da. ₁₁

It is easy to see that is the disease-free equilibrium. To find a non-trivial equilibrium (an endemic equilibrium), we could observe the following.

* * * *

(SH,IH,RH)=(NH, 0, 0)

*

* 0

1

( ) 1.

H HI a a H

H H H

H H

e e

B Q a da

I

β γ

β

β

− −

∞ ⎡_⎢⎣ − ⎤_⎥⎦

=

∫

12

The LHS of (12) is a function of I*H, say

*

*

* 0

1

( ) ( )

H HI a a H

H H H H

H H

e e

f I B Q a

I

β γ

β

β

− −

∞ ⎡_⎢⎣ − ⎤_⎥⎦

=

∫

da

13 This function is monotonically decreases with

* *

*

* ₀ *

0 lim

( ) 0

1 lim

( ) 0

( )

H H H

H

H H

I a a

H H H

H H H

a

H H H

f I I

e e

B Q a da

I I

B aQ a e da

β γ

γ

β

β β

+

− −

∞ +

∞ ₋

→

⎡ ₋ ⎤

⎢ ⎥

⎣ ⎦

= → =

∫

∫

14

* *

*

* ₀ *

lim

( )

lim ₁

( )

0

H H H

H H

I a a

H H H

H H H

f I I

e

B Q a e da

I I

β _γ

β

β

−

−

∞ ₋

−

→ ∞

− =

→ ∞

=

∫

15

Therefore, a unique non-trivial value of occurs if and only if

* H

I

0

0 ( ) 1

H

a

H H H

R =β B

∫

∞aQ a e− γ da> . 16

Since the LHS of (16) determines the occurrence of the non-trivial value of I*H, then it will be

refereed as a threshold number R0 of the model. Hence, an endemic equilibrium

occurs if and only

if .

* * * *

(SH,IH,RH)≠(NH, 0, 0)

0 1

R >

IV. The Stability of the Equilibria

LEMMA 4.1. (BRAUER, 2001). Let f t( )be a bounded non-negative function which satisfies an estimate of the form

0 ₀

( ) ( ) t ( ) ( ) f t ≤ f t +

∫

f t−a R a da ,

where f t₀( )is a non-negative function with and is a non-negative

function with Then

.

0

lim_t→∞ f t( )=0 R a( )

0 R a da( ) 1.

∞

<

∫

lim_t→∞ f t( )=0

PROOF. See [1]. It is also showed in [1] that the lemma is still true if the inequality in the lemma is replaced by

0 ₀

( ) ( ) tsupt a s t ( ) ( )

f t ≤ f t +

∫

_{− ≤ ≤} f s R a da. 17 The following lemma is the extension of Brauer’s lemma.

LEMMA 4.2. Let be bounded

non-negative functions satisfying ( ), 1, 2 j

f t j=

1( ) 10( ) ₀sup 1( ) 1( )

t

t a s t

f t ≤ f t +

∫

_{− ≤ ≤} f s R a da ,

2( ) 20( ) ₀sup 1( ) 2( )

t

t a s t

f t ≤ f t +

∫

_{− ≤ ≤} f s R a da ,

where fj₀( )t is non-negative with and is non-negative with

Then li .

0

lim_t→∞ f_j ( )t =0 R a_j( )

0 R a daj( ) 1.

∞

<

∫

mt→∞ f tj( )=0, j=1, 2 PROOF.

1 10 20

1 2 1 2

0

( ) sup{ ( ), ( )}

tsup_{t a s t}sup{ ( ), ( )}sup{ ( ), ( )}

f t f t f t

f s f s R a R a da

− ≤ ≤

≤

+

∫

2 10 20

1 2 1 2

0

( ) sup{ ( ), ( )}

tsup_{t a s t}sup{ ( ), ( )}sup{ ( ), ( )}

f t f t f t

f s f s R a R a da

− ≤ ≤

≤

+

∫

and hence,

1 2 10 20

1 2 1 2

0

sup{ ( ), ( )} sup{ ( ), ( )}

tsupt a s tsup{ ( ), ( )}sup{ ( ), ( )}

f t f t f t f t

f s f s R a R a da

− ≤ ≤

≤

+

∫

From Lemma 4.1 we conclude that

, and this is suffice to show that li

1 2

lim_t→∞sup{ ( ),f t f t( )}=0

mt→∞ f tj( )=0, j=1, 2.

4.1. The stability of the disease-free equilibrium. We investigate the stability of the disease-free equilibrium for the case ofR0 <1.

Consider the following inequalities.

( )

1 (

sup ( ) t

H H t a I s ds t

H H t a

H t a s t H

e I s

a

β

β β −

−

−

− ≤ ≤

∫

− ≤ ∫

≤

)ds

I s 18

Hence we have,

(0)

( ) 0

(0)

0

( ) ( )

( )(1 )

( )

( )( sup ( ))

t H H

t a H

H

H H

t I s ds _a

H H H

t _a

H H H t a s t H

I t I t

B Q a e e da

I t

B Q a a I s e da

β _γ

γ

β −

− ₋

− − ≤ ≤

=

∫

+ −

≤ +

∫

∫

19

And

(0)

( ) 0

(0)

0

( ) ( )

( ) 1 1

( )

( )( sup ( )) 1

t H H

t a H

H

H H

t I s ds _a

H H

H

t _a

H H H t a s t H

R t R t

B Q a e e da

R t

B Q a a I s e da

β _γ

γ β

−

− ₋

− − ≤ ≤

=

⎡ _∫ ⎤_{⎡ ⎤}

+ _⎢ − _{⎥ ⎣} − _⎦

⎣ ⎦

≤

⎡ ⎤

+ _⎣ − _⎦

∫

∫

20

Moreover, since (0)

lim

( ) 0 H

I t

t→ ∞ = and

0

0 ( ) 1

H a

H H H

B Q a aβ e γ da R

∞ ₋

= <

∫

then using

Lemma 4.1 we conclude that limt→∞IH( )t =0.

Next, let us see the

expression which,

if , can be written in the

form

0 ( ) 1

H a

H H H

B Q a aβ e γ da

∞ ₋

⎡ − ⎤

⎣ ⎦

∫

0S ₀ H H( ) H R =

∫

∞B Q a aβ da

0 0S

R −R . Hence, if R0S < +1 R0 then

0 ( ) 1 1

H a

H H H

B Q a aβ e γ da

∞ ₋

⎡ − ⎤ <

⎣ ⎦

∫

(Appendix 1).

Furthermore, since then using Lemma 4.2 we conclude that

(0)

lim

( ) 0 H

R t

t→ ∞ =

limt→∞RH( )t =0. Consequently,

This shows that the disease-free equilibrium is globally stable.

*

limt→∞SH( )t =limt→∞(NH( )t −RH( )t −IH( ))t =NH

0)

* * * *

(S IH H,IV)=(NH0,

4.2. The stability of the endemic

equilibrium. The endemic equilibrium

appears only if . Let us see the perturbations of

* * *

(SH,IH,RH) R0>1

*

H

I and *

H

R , respectively, by and . Define

( )

u t

( )

* * (0) [ ( )] 0 ( ) ( )

( ) 1 t

H H

t a H

H H

t I u s ds _a

H H

I u t I t

B Q a e− −β + e−γ da

+ = ⎛ _∫ ⎞ + _⎜ − _⎟ ⎝ ⎠

∫

* * * * (0) ( ) 0 (0) 0 ( ) 0 ( ) ( )

( ) 1

( )(1 ) ( )

( ) 1

t t

H H H

t a t a H

H H H

t H t a

H H H

H H

t I ds u s ds _a

H H

I a a

H H H

t _{I a} u s ds _a

H H

u t I I t

B Q a e e e da

B Q a e e da I t

B Q a e e e da

β β _γ β γ β β γ − − − − − ₋ ∞ _{− −} − − − = − + ⎛ _{∫ ∫} ⎞ + _⎜ − _⎟ ⎝ ⎠ = − − + ⎛ _∫ ⎞ + _⎜ − _⎟ ⎝ ⎠

∫

∫

∫

* * * (0) 0 ( ) 0

( ) ( )(1 ) ( )

( )(1 )

( ) 1

H H H

H H

t H t a

H H H

I a a

H H H

t t

I a H H

t _{I a} u s ds _a

H H

u t B Q a e e da I t

B Q a e da

B Q a e e e da

β γ β β β − γ ∞ _{− −} − − − − = − − + − − ⎛ _∫ ⎞ + _⎜ − _⎟ ⎝ ⎠

∫

∫

∫

* * * * (0) ( ) 0 (0)

( ) ( )(1 ) ( )

( ) 1

( )(1 ) ( )

( ) sup ( )

H H H

t H t a

H H H

H H H

H H H

I a a

H H H

t

t _{I a} u s ds _a

H H

I a a

H H H

t

I a a

H H H t a s t

u t B Q a e e da I t

B Q a e e e da

B Q a e e da I t

B Q a e ae u s

β γ β β γ β γ β _β γ − ∞ _{− −} − − − ∞ _{− −} − − − ≤ ≤ = − − + ⎛ _∫ ⎞ + _⎜ − _⎟ ⎝ ⎠ ≤ − − + +

∫

∫

∫

0 t da

∫

Hence, we have

*

*

(0)

0

( ) ( )(1 ) ( )

sup ( ) ( )

H H H

H H H

I a a

H H H

t t

I a a

t a s t H H H

u t B Q a e e da I t

u s B Q a e ae da

β γ β _β γ ∞ _{− −} − − − ≤ ≤ ≤ − − + +

∫

∫

By defining f t₁( )= u t( ),

* 1( ) ( )

H HI a a

H H H

R a =B Q a e−β β ae−γH_{, and}

*

10( ) ( )(1 ) (0)( )

H HI a aH

H H H

t

f t = −

∫

∞B Q a −e−β e−γ da+I t , then we have

1( ) 10( ) ₀sup 1( ) 1( )

t

t a s t

f t ≤ f t +

∫

_{− ≤ ≤} f s R a da. We see that and it can be shown that is non-negative with

10

limt→∞ f ( )t =0

1( )

R a ₁

0 R a da( ) 1

∞

<

∫

(see Appendix 2). Then by Lemma 4.1 we have , means that

1

limt→∞ f t( )=0

*

limt→∞IH( )t =IH.

Next, define and

substitute these quantities into equation (5) to obtain the following calculations:

*

( ) ( )

H H

R t =R +v t

* * (0) [ ( )] 0 ( ) ( )

( )(1 ) 1

t H H

t a H

H H

t I u s ds _a

H H

R v t R t

B Q a e− −β + e−γ da

+ = ∫ _{⎡ ⎤} +

∫

− _⎣ − _⎦ * * * * (0) ( ) 0 (0) 0 ( ) 0 ( ) ( )

( ) 1 1

( )(1 ) 1 ( )

( ) 1 1

t t

H H H

t a t a H

H H H

t H t a

H H H

H H

t I ds u s ds _a

H H

I a a

H H H

t _{I a} u s ds _a

H H

v t R R t

B Q a e e e da

B Q a e e da R t

B Q a e e e da

β β _γ β γ β β γ − − − − − ₋ ∞ _{− −} − − − = − + ⎛ _{∫ ∫} ⎞_{⎡ ⎤} + _⎜ − _⎟⎣ − _⎦ ⎝ ⎠ ⎡ ⎤ = − − _⎣ − _⎦ + ⎛ _∫ ⎞_{⎡ ⎤} + _⎜ − _⎟⎣ − _⎦ ⎝ ⎠

∫

∫

∫

* * * (0) 0 ( ) 0

( ) ( )(1 ) 1 ( )

( )(1 ) 1

( ) 1 1

H H H

H H H

t H t a

H H H

I a a

H H H

t

t _{I a a}

H H

t _{I a} u s ds _a

H H

v t B Q a e e da R t

B Q a e e da

B Q a e e e da

β γ β γ β β − γ ∞ _{− −} − − − − − ⎡ ⎤ = − − _⎣ − _⎦ + ⎡ ⎤ − − _⎣ − _⎦ ⎛ _∫ ⎞_{⎡ ⎤} + _⎜ − _⎟⎣ − _⎦ ⎝ ⎠

∫

∫

∫

* * * * (0) ( ) 0 (0)

( ) ( )(1 ) 1 ( )

( ) 1 1

( )(1 ) 1 ( )

( ) 1

H H H

t H t a

H H H

H H H

H H

I a a

H H H

t

t _{I a} u s ds _a

H H

I a a

H H H

t

I a

H H H

v t B Q a e e da R t

B Q a e e e da

B Q a e e da R t

B Q a e a

β γ β β γ β γ β _β − ∞ _{− −} − − − ∞ _{− −} − ⎡ ⎤ = − − _⎣ − _⎦ + ⎛ _∫ ⎞_{⎡ ⎤} + _⎜ − _⎟⎣ − _⎦ ⎝ ⎠ ⎡ ⎤ ≤ − − _⎣ − _⎦ + +

∫

∫

∫

0 sup ( )

H

t _a

t a s t

e−γ _{− ≤ ≤}u

⎡ − ⎤

⎣ ⎦

∫

s da

Hence, we have

*

*

(0)

0

( ) ( )(1 ) 1 ( )

sup ( ) ( ) 1

H H H

H H H

I a a

H H H

t t

I a a

t a s t H H H

v t B Q a e e da I t

u s B Q a e a e da

β γ β _β γ ∞ _{− −} − − − ≤ ≤ ⎡ ⎤ ≤ − − _⎣ − _⎦ + ⎡ ⎤ + _⎣ − _⎦

∫

∫

By defining f t1( )= u t( ), f t2( )= v t( )

, and

*

2( ) ( ) H H 1 H

I a a

H H H

R a =B Q a e−β β a⎡_⎣ −e−γ ⎤_⎦

*

20( ) ( )(1 ) 1 (0)( )

H HI a aH

H H H

t

f t = −

∫

∞B Q a −e−β ⎡_⎣ −e−γ ⎤_⎦da+I t

, then we have

2 20 1 2

0

( ) ( ) tsupt a s t ( ) ( ) f t ≤ f t +

∫

_{− ≤ ≤} f s R a da. We can show that in Appendix 3 that

20

lim_t→∞ f ( )t =0and R a₂( )is non-negative with

2

0 R a da( ) 1

∞

<

∫

. Then by Lemma 4.2 we have

2

limt→∞ f t( )=0, means that .

*

limt→∞RH( )t =RH Finally, since limt→∞NH( )t is a constant,

, and

*

limt→∞RH( )t =RH H

*

limt→∞IH( )t =I then is globally stable.

* * *

V. Concluding Remarks

In this paper we have discussed an age-structured SIR disease model with a decreasing survival rate. We found a threshold number for the existence and uniqueness of an endemic

equilibrium, that is, . As is the case of

the SI disease model discuss in [1], an endemic equilibrium appears if and disappears if

. In the SI disease model, the threshold for the existence of the equilibrium is also the threshold for the stability of the equilibrium. However, in our case in which there is a recover compartment, there is an additional condition for the equilibrium to be stable. Here we found that there is a stable endemic equilibrium if

and , and there

is a stable disease-free equilibrium if

0

0 ( )

H

a

H H H

R =β B

∫

∞aQ a e−γ da

0 1

R >

0 1

R <

0 1

R >

0S H H ₀ H( ) 1 R =β B

∫

∞Q a ada< +R₀

0 1

R < and . We notice that this condition is consistent with that in [1] if the recovery rate

0S H H ₀ H( ) 1 0

R =β B

∫

∞Q a ada< +R

0

γ = , since in this case is equivalent to . Hence, we conclude that the SI model in [1] is naturally nested in the SIR model discussed in this paper.

0

R

0S

R

VI. References

[1]

F. Brauer. A model for an SI disease in an age-structured population. Discrete and Continuous Dynamical Systems – Series B. 2

(2002), 257-264.

[2] S. Busenberg, M. Ianelli, and H.R. Thieme. Global behavior of an age-structured epidemic model, SIAM J. Math. Anal. 22 (1991), 1065-1080.

[3] H.W. Hethcote. An age-structured model for pertussis transmission. Math. Biosc. 145

(1997), 89-136.

Appendix 1

If then

.

0S 1

R < +R₀

<

da da

1

0 ( ) 0 ( ) 1 1

H a

H H H

R a da B Q a β a e γ da

∞ ∞ ₋

⎡ ⎤

= _⎣ − _⎦

∫

∫

Proof:

It is straightforward from the definition that

and .

0

0 ( )

H a

H H H

R =

∫

∞B Q a aβ e−γ

0S ₀ H H( ) H R =

∫

∞B Q a aβ

Appendix 2

We claim that limt→∞ fj0( )t =0and

is non-negative with . ( )

j R a

0 R a daj( ) 1

∞

<

∫

Proof:

It is clear that, if X Y, ∈{ , }H V with X ≠Ythen

* 0

(0)

lim ( )

lim ( )(1 X Y ) ( ) 0.

t j

I a

t X X X

t f t

B Q a e β da I t

→∞

∞ ₋

→∞

=

−

∫

− + =

To witness that , let us proceed for as follows.

0 R a daj( ) 1

∞

<

∫

1( )

R a

Define

* 0

1

( ) ( )

H H V

H

ax I a

a

H H H

H

e a

g x B e Q a e da

x

β β

γ

β ∞ − − −_β−

=

∫

which is a decreasing function of

x

. We see that *

*

* 0

1

( ) ( )

H V H

I a a

V H H H

H V

e a

g I B e Q a d

I

β γ

β

β

−

∞ − −

− =

∫

a

1

. Since IV* is an equilibrium value, then we have

*

( _V)

g −I = .

Furthermore,

* 0

1 0

(0) ( )

( ) .

H V

H I a

a

H H H

g B e aQ a e

R a da

β γ

β ∞ − −

∞

=

=

∫

∫

da

Considering that

g

is decreasing function then

* 1

0 ( ) (0) ( V) 1

R a da g g I

∞

= < − =

∫

.

Appendix 3

If R₀S < +1 R0 then

is less than one.

* 2

0 ( ) 0 ( ) 1

H HI a aH

H H H

R a da B Q a e β β a e γ da

∞ ∞ _{− −}

⎡ ⎤

= _⎣ − _⎦

∫

∫

Proof:

Using the result in Appendix 1 we have

* 0

0

( ) 1

( ) 1 1.

H H H

H

I a a

H H H

a

H H H

B Q a e a e da

B Q a a e da

β γ

γ

β β

∞ _{− −}

∞ ₋

⎡ − ⎤

⎣ ⎦

⎡ ⎤

≤ _⎣ − _⎦ <