Global asymptotic stability of SIRI diseases models using monotone dynamical systems theory

http://dx.doi.org/10.12988/ams.2015.56428

Global Asymptotic Stability of SIRI Diseases Models Using Monotone Dynamical Systems Theory

Khadija Niri, Karima Kabli and Soumia El moujaddid Laboratory MACS

Mathematics and Computing Department Ain Chock Science Faculty, Km 8 Route El Jadida

BOP 5366-Maarif, Casablanca, Morocco

Copyright c2015 Khadija Niri, Karima Kabli and Soumia El moujaddid. This article is distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

In this paper, we consider the global asymptotic stability of the free and the endemic equilibrium of an SIRI epidemiological model with de- mographic structure. We show that the system is cooperative and ir- reducible on the closure of an open order convex subset

of the space

. Moreover, we prove that the compactness condition

always holds, and consequently the free and the endemic equilibrium are glob- ally asymptotically stable on

simulations are presented to illustrate analytical results.

Mathematics Subject Classification: 37N30, 92D30, 70K20

Keywords: Monotone dynamical systems, Compartmental model, Basic reproduction number, Global asymptotic stability, Relapse, Infectious disease

1 Introduction

The relapse phenomenon in some infectious diseases is characterized by the

acquisition of quiescent state of the individuals that have previously been in-

fected, and with subsequent relapse to the infectious state. This recurrence

of infections including diseases such as bovine tuberculosis and human herpes

virus. These types of diseases can be modeled by SIRI epidemiological sys- tems.

Tudor [27] formulated one of the first epidemic models with relapse, this model incorporates, bi-linear incidence rate and constant total population. This sys- tem was extended to include nonlinear incidence functions by Moreira and Wang [19].

Blower [1] developed a compartmental model for genital herpes, assuming stan- dard incidence for the disease transmission and constant recruitment rate.

Van den Driessche and Zou [28], developed a SIRI model in a constant pop- ulation with standard incidence and a general relapse distribution. Van den Driessche and coauthors, [29], formulated a SEIRI model for a disease with a general exposed distribution in a constant population.

Monotone methods were introduced by Miiller(1926), Kamke(1932), Krasnoe- selskli and others for ordinary differential equations. Then, Hirsch develops in [7] the monotone dynamical systems theory. The theory was extended by Matano [17] to the infinite dimensional case. After that, a synthesis of the Hirsch and Matano’s approach was presented in Smith and Thieme’s articles, in this important work, Hirsch and Matano’s arguments were simplified and streamlined [22],[23],[24],[25].

The paper is organized as follow: In the next section, some preliminary results concerning monotone dynamical systems theory for ordinary differential equa- tions. The analysis of model SIRI is presented in section 3, we give some results about positivity of solutions, basic reproduction number, existence of equilib- rium and monotonicity of the system. The focus of section 4 is to study local and global stability of free and endemic disease equilibrium and we support our results by numerical simulations. Finally, the conclusion is summarized in section 5.

2 Preliminary

Consider the autonomous system of ordinary differential equations

˙

x = f(x) (1)

Where f is continuously differentiable on an open D ⊂ R

.

We write φ

(x

) for the solution of (1) that starts at the point x

at t = 0, φ

(x

) = x (t, x

).

φ

(x) will be referred to as the flow corresponding to (1), we sometimes refer to f as the vector field generating the flow φ

(x).

If φ

(x) is defined for all t > 0 we write

γ

(x) = {φ

(x), t ≥ 0}

for the positive orbit through the point x.

The omega limit set, ω(x) of x ∈ D is defined by ω(x) = \

t≥0

[

s≥t

φ

(x)

We use the following notation for the vector order in R

:

• x ≤ y if x

≤ y

for all i.

• x < y if x ≤ y and x

< y

for some i.

• x << y if x

< y

for all i.

Definition 2.1. A semiflow on the set D is a continuous map Φ : D × R

−→ D which satisfies:

1. Φ

= id

, where id

is the identity map in D.

2. Φ

◦ Φ

= Φ

for t, s ≥ 0.

Definition 2.2. 1. A point e is called an equilibrium of the system (1) if it satisfies f(e) = 0 or the equivalent way if φ

(e) = e for all t ≥ 0.

The set of equilibria is denoted by E = {e ∈ D : φ

(e) = e ∀t ≥ 0}.

2. A point x is called a convergent point if ω(x) consists of a single point of E. We denote by C = n

x ∈ D : φ

(x)

−→ e; e ∈ E o

the set of all convergent points.

Definition 2.3. 1. We say that D is p-convex if tx + (1 − t)y ∈ D for all t ∈ [0, 1] whenever x, y ∈ D and x ≤ y.

2. We say that a subset A of D is positively invariant if φ

A ⊂ A.

Definition 2.4. 1. An n × n matrix A = (a

) is irreducible if for every nonempty proper subset I of the set N = 1, 2, ..., n, there is an i ∈ I and j ∈ J ≡ N \I such that a

6= 0.

2. The vector field f is irreducible in D provided the Jacobian matrix Df(x) is an irreducible matrix for every x ∈ D.

Definition 2.5. If x ∈ D, we say that x can be approximated from below

(above) in D if there exists a sequence x

in D satisfying x

< x

< x

(x < x

< x

) for n ≥ 1 and x

→ x as n → ∞ .

Lemma 2.6. Let D be an open subset of R

and x ∈ D. Then x can be approximated from below or above in D.

Theorem 2.7. Let U a closure of an open subset D of R

and x, y ∈ U with x < y then there exist sequences x

, y

∈ D satisfying x

< y

, x

−→ x and y

−→ y as n −→ ∞.

We will see how a cooperative and irreducible system can be strongly mono- tone. As it’s not easy to verify the property of monotonicity, so we will just verify if the system is cooperative and irreducible.

Definition 2.8. Let D be a p-convex set, the system (1) is said to be coop- erative if ∂f

∂x

≥ 0; i 6= j, i = 1...n and j = 1...n, x ∈ D.

Definition 2.9. The semi-flow φ is said to be monotone proved φ

(x) ≤ φ

(y) whenever x ≤ y and t ≥ 0.

Definition 2.10. The semi-flow φ is said to be strongly monotone if φ is monotone and whenever x < y and t > 0 then φ

(x) << φ

(y).

Theorem 2.11. [8]

• if (1) is cooperative then φ

is monotone.

• if (1) is cooperative and irreducible then φ

is strongly monotone.

We introduce following mild compactness condition for our next result.

(T ) : for each x ∈ D : φ

(x) is defined for all t ≥ 0 and φ

(x) ∈ D. Moreover for each bounded subset A of D there exists a closed bounded subset B = B(A) of D such that for each x ∈ A, φ

(x) ∈ B for each large t.

The following result should be viewed as prototypical of the generic conver- gence result that may be proved using general results in [21].

Theorem 2.12. Let (1) be a cooperative and irreducible system on a domain D and assume that (T ) holds. Assume also that each point x ∈ D can be approximated either from above or from below by a sequence {x

} in D.

Then the set C of convergent points contains an open and dense subset of D.

Remark 2.13. [21, 26]

In the theorem above, we assume that either the domain D is an open subset

of R

or it is the closure of an open subset of R

. In the latter case, it is

assumed that the vector field extends to a continuously differentiable one on a

neighborhood of D.

3 Study of the model

b

-

S

? -

d

λSI

I γ

?

d

?

α

R

?

d

Figure 1: The transfer diagram from SIRI model

In most compartmental epidemiological models, the total population is of- ten divided into several disjoint classes. The model divides the total population into the following sub-groups that are susceptible individuals S, infectious in- dividuals I, and individuals previously infectious but temporarily reverted to the non-infectious state R. With the total population N (t) = S(t)+I(t)+R(t).

In this SIRI model, the flow is from the S class to the I class, and then either directly to the R class and then back to I class as shown in Figure 1. The transfer diagram leads to the following system of differential equations:



 

 



 

 



S ˙ = bN − λ

− dS I ˙ = λ

− (d + γ) I + αR R ˙ = γI − (d + α) R

(2)

Where the parameters b > 0 and d > 0 are the birth rate and death rate constants, respectively, λ > 0 is the average number of effective contacts of an infectious individual per unit time, and the parameter γ > 0 describes the rate that the infectious individuals becomes no-infectious individuals and α > 0 de- notes the rate that the no-infectious individuals are reverted to the infectious state.

We consider a closed community in which the birth rate and death rate con- stants are equal, thus b = d.

In such a case the total population remains a constant.

Denote s =

and i =

and r =

the fractions of the number of individuals

in compartments S, I and R in the total population N , respectively. It is easy

to verify that s, i and r satisfy the system



 

 



 

 



˙

s = d − λsi − ds

˙ i = λsi − (d + γ ) i + αr

˙

r = γi − (d + α) r

(3)

where solutions are restricted to the hyperplane s + i + r = 1.

To verify if all solutions are defined we have to prove that all solutions are nonnegative.

We introduce following condition:

(P): ∀x ∈ D and x ≥ 0. If x

= 0, then f

(x) ≥ 0.

Theorem 3.1. Consider the system (1). If f is Lipschitz on an open subset D of R

and the property (P) is verified, then the solution x (t) ≥ 0.

Theorem 3.2. The octant R

= {(x, y, z) ∈ R

: x ≥ 0, y ≥ 0, z ≥ 0} is positively invariant with respect (3).

Proof. We can easily verify, from the system (3), that the function f is con- tinuous and Lipschitz, and we have:

• If s = 0, s ˙ = d > 0.

• If i = 0, i ˙ = αr ≥ 0.

• If r = 0, r ˙ = γi ≥ 0.

Then the octant R

is positively invariant by theorem 3.1.

In the remainder of this work we can attack (3) by studying the system:







˙

r = γi − (d + α) r

i ˙ = λi (1 − i − r) − (d + γ) i + αr

(4)

and determining s from s = 1 − i − r.

We study (4) in the closed set Σ =

(r; i) ∈ R

; 0 ≤ r + i ≤ 1

Where R

denotes the non-negative cone of R

.

In the next section, the reduced proportionate system (4) will be analyzed.

We establish the positivity of solution, basic reproduction and existence of

equilibrium.

Theorem 3.3. The set Σ =

(r; i) ∈ R

; 0 ≤ r + i ≤ 1 is positively in- variant with respect to (4).

Proof. Analysis similar to that in the proof of theorem 3.2 with s + i + r = 1 shows that Σ is positively invariant with respect to (4).

3.1 The basic reproduction number

In order to study the stability of equilibrium points we need to calculate the basic reproduction rate R

. the basic reproduction rate is the number of people that an infective person can infect.

The basic reproduction number of system (4) is given by:

R

= λ(d + α) d(d + α + γ )

3.2 Existence of equilibrium points

In this section, we show that there exists a disease-free equilibrium and one infection equilibrium which represents the endemic equilibrium.

It is not hard to see that if R

≤ 1, the disease-free equilibrium E

= (0; 0) is the unique equilibrium point corresponding to the extinction of the free virus.

The following result presents the existence and uniqueness of endemic equilib- rium when R

> 1.

Theorem 3.4. Let R

=

1. If R

≤ 1, then the system (4) has a unique disease-free equilibrium of the form E

= (0; 0).

2. If R

> 1,the disease-free equilibrium is still present and the system (4) has a unique endemic equilibrium of the form E

= (r

, i

) where r

=

and i

=

.

Proof. At any equilibrium, the following equation hold:







γi

− (d + α) r

= 0

λi

(1 − i

− r

) − (d + γ) i

+ αr

= 0

(5)

By the first equation we have r

=

i

And we replaced r

in the second equation we have i

= 0 or i

=

λ(d+α+γ)

=

If i

= 0 then r

= 0

If i

=

then r

=

i

which proves the theorem and proof is complete.

3.3 Monotony of the system

To verify if a system is cooperative, we calculate the Jacobian matrix and we check if the sign of the off diagonal is positive.

The Jacobian matrix of the system (4) is given by:

J =

−(d + α) γ

−λi + α −(d + γ) + λ − 2λi − λr

We obtain the following result.

Theorem 3.5. The system (4) is strongly monotone in U = [0; 1] × 0;

. Proof. We have the Jacobian matrix J is irreducible on U , therefore the system is irreducible.

The off diagonal of the matrix J is positive for each x ∈ U , moreover the system is cooperative by Definition 2.8.

The system (4) is cooperative and irreducible on U , then by Theorem 2.11 the system (4) is strongly monotone on U .

Theorem 3.6. Assume that α < λ < α + d + γ or R

< 1, then the set U = [0; 1] ×

0;

is positively invariant with respect (4).

Proof. On the border i =

Then we have ˙ i = −

(d + α + γ − λ) = −

λα dR₀

+

0

(1 − R

)

< 0 Since R

< 1 or λ < α + d + γ

Thus U = [0; 1] × 0;

is positively invariant.

Lemma 3.7. Let U be the closure of the open subset D and A a bounded subset of U .

If U is positively invariant, then there exists a closed bounded subset B(A) of U, such that ∀x ∈ A : φ

(x) ∈ B (A).

Proof. Let A be a bounded subset of U . We have φ

(A) ⊂ U ,then φ

(A) is a closed bounded subset of U (U is bounded set ).

Thus, with B (A) = φ

(A) the result follows.

To study the global stability of the equilibrium point on a set by monotone method, this set must be positively invariant. However according to Lemma 3.7 there are two cases to ensure that a set be positively invariant. In the next sections, we will treat this cases.

4 Stability of the system

In this part of the paper we will study the local and global stability of equi- librium points using the theorem of Smith 2.12 for global stability in cases R

< 1 and R

> 1.

4.1 Local stability of disease-free equilibrium

The characterization of the local stability of the disease-free equilibrium is given by the following statement.

Theorem 4.1. .

1. If R

< 1, then E

is locally asymptotically stable.

2. If R

> 1, then E

is unstable.

Proof. The Jacobian matrix of the system (4) at E

is given by:

J

=

−(d + α) γ α −(d + γ) + λ

tr J

= − (2d + γ + α − λ) = −

d +

0

+

0

(1 − R

) . If R

< 1, then tr J

< 0, and det J

= d (d + α + γ) (1 − R

) > 0.

Thus E

is locally asymptotically stable.

4.2 Local stability of endemic equilibrium

Now, we focus on local stability of endemic equilibrium E

. It is easy to verify that the point E

does not exists if R

< 1 and E

= E

when R

= 1, if R

> 1, then we have the following theorem.

Remark 4.2. If

> 1, we have Σ ⊂ U , then we can study the system in Σ.

Theorem 4.3. Assume

> 1. If R

> 1, then E

is locally asymptotically

stable in Σ.

Proof. The Jacobian matrix of the system (4) at E

is given by:

J

=

−(d + α) γ

α − d (R

− 1) −(d + γ) + λ − 2d (R

− 1) −

Then tr (J

) = − (2d + γ + α − λ) − 2d (R

− 1) −

< 0, since α > λ and R

> 1.

We have det (J

) = d (d + α + γ) (R

− 1) > 0, When R

> 1.

Thus E

is locally asymptotically stable.

Theorem 4.4. Assume that α < λ < α + d + γ. If R

> 1, then E

is locally asymptotically stable in U .

Proof. If R

> 1, we remark that tr (J

) < 0 since λ < α + d + γ, and det (J

) = d (d + α + γ) (R

− 1) > 0.

Thus E

is locally asymptotically stable.

4.3 Global asymptotic stability of disease-free equilib- rium

Our next result is one of the main results of this paper, it gives sufficient condition for a solution to converge to the free equilibrium if R

< 1 and to the endemic equilibrium if R

> 1.

Theorem 4.5. 1. Assume

> 1. If R

< 1 , then the disease free equilibrium E

is globally asymptotically stable in Σ.

2. Assume

< 1. If R

< 1 , then the disease free equilibrium E

is globally asymptotically stable in U .

Proof. 1. Suppose

> 1, then the system (4) is cooperative and irreducible on the set U by theorem 3.5. We have Σ ⊂ U , then the system (4) is cooperative and irreducible on Σ.

Σ is positively invariant by theorem 3.3 and it is bounded then the con- dition (T ) is verified with B (A) = φ

(A) by Lemma 3.7.

From Theorem 2.7, Theorem 2.12 , Remark 2.13 and Definition 2.5, we have E consists of one points E

which is locally asymptotically stable since R

< 1. Then all solutions with initial value in Σ converge to E

. Therefore E

is globally asymptotically stable in Σ.

2. If α < λ and R

< 1 then U is positively invariant by theorem 3.6 and the condition (T ) is verified.

The set E consists of one point E

which is locally asymptotically sta-

ble since. Thus all solutions with initial values in U converge to E

.

Therefore E

is globally asymptotically stable in U .

4.4 Global asymptotic stability of endemic equilibrium

Theorem 4.6. 1. Suppose

> 1. If R

> 1, then the endemic equilib- rium E

is globally asymptotically stable in Σ.

2. Suppose that α < λ < α +d + γ. If R

> 1, then the endemic equilibrium E

is globally asymptotically stable in U .

Proof. 1. The system (4) is strongly monotone on the set U . Since

> 1, we have Σ ⊂ U , then the system (4) is strongly monotone on Σ.

Since R

> 1 the endemic equilibrium is locally asymptotically stable.

Thus all solutions with initial value in Σ converge to E

. Global con- vergence to E

and local stability of E

imply the global asymptotic stability of E

in Σ.

2. The system (4) is cooperative and irreducible on the set U by theorem 3.5, we have α < λ < α + d + γ then U is positively invariant by theorem 3.6 and the condition (T ) is verified.

E consists of two points E

and E

. Since R

> 1 the endemic equi- librium is locally asymptotically stable. Thus all solutions with initial value in U converge to E

.

4.5 Numerical simulation

To complete the analytical results of the previous sections, we show some nu- merical simulation for model (4).

•

> 1

We adopt the following values for the parameters:

Figure 2: α = 0.5, γ = 0.5, λ = 0.25 and d = 0.9

We have R

= 0.2 < 1 that means all solutions converge to the free equilibrium.

Figure 3: α = 0.5, γ = 0.01, λ = 0.25 and d = 0.018

We have R

= 13.6 > 1 that mean all solution converge to the endemic equilibrium I

= 0.9 and R

= 0.01.

•

< 1 We adopt the following values for the parameters:

Figure 4: α = 0.25, γ = 0.5, λ = 0.8 and d = 0.9

We have R

= 0.5 < 1 that mean all solution converge to the free equilibrium.

Figure 5: α = 0.25, γ = 0.51, λ = 0.5 and d = 0.018

Figure 2: Convergence to the disease free equilibrium

Figure 3: Convergence to the endemic disease equilibrium

We have R

= 9.568 > 1 that mean all solution converge to the endemic

equilibrium I

= 0.308 and R

= 0.587.

Figure 4: Convergence to the disease free equilibrium

Figure 5: Convergence to the endemic disease equilibrium

5 Conclusion

In this work, we established the global asymptotic stability of SIRI infectious diseases models by using monotone dynamical system theory and we comple- mented this work by numerical simulations that support the theory results.

The study of global stability by monotone dynamical systems theory can be

generalized to other cooperatives epidemiological models. Its advantage is to

avoid the construction of a function Lyapunov often find ingenious.

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Received: June 25, 2015; Published: July 16, 2015