Global Asymptotic Stability of SIRI Diseases Models Using ...m-hikari.com/.../p/kabliAMS97-100-2015.pdf · 4 Stability of the system In thispart of thepaperwewill studythelocal andglobal

Applied Mathematical Sciences, Vol. 9, 2015, no. 98, 4881 - 4896HIKARI Ltd, www.m-hikari.com

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 MACSMathematics and Computing Department

Ain Chock Science Faculty, Km 8 Route El JadidaBOP 5366-Maarif, Casablanca, Morocco

Copyright c© 2015 Khadija Niri, Karima Kabli and Soumia El moujaddid. This articleis distributed under the Creative Commons Attribution License, which permits unrestricteduse, distribution, and reproduction in any medium, provided the original work is properlycited.

Abstract

In this paper, we consider the global asymptotic stability of the freeand 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 Σ of the spaceR3. Moreover, we prove that the compactness condition (T ) alwaysholds, and consequently the free and the endemic equilibrium are glob-ally asymptotically stable on Σ. At the end of this work, numericalsimulations are presented to illustrate analytical results.

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

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

1 Introduction

The relapse phenomenon in some infectious diseases is characterized by theacquisition of quiescent state of the individuals that have previously been in-fected, and with subsequent relapse to the infectious state. This recurrenceof infections including diseases such as bovine tuberculosis and human herpes

4882 Khadija Niri, Karima Kabli and Soumia El moujaddid

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 modelincorporates, bi-linear incidence rate and constant total population. This sys-tem was extended to include nonlinear incidence functions by Moreira andWang [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 denDriessche and coauthors, [29], formulated a SEIRI model for a disease with ageneral 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 developsin [7] the monotone dynamical systems theory. The theory was extended byMatano [17] to the infinite dimensional case. After that, a synthesis of theHirsch and Matano’s approach was presented in Smith and Thieme’s articles,in this important work, Hirsch and Matano’s arguments were simplified andstreamlined [22],[23],[24],[25].The paper is organized as follow: In the next section, some preliminary resultsconcerning monotone dynamical systems theory for ordinary differential equa-tions. The analysis of model SIRI is presented in section 3, we give some resultsabout positivity of solutions, basic reproduction number, existence of equilib-rium and monotonicity of the system. The focus of section 4 is to study localand global stability of free and endemic disease equilibrium and we supportour results by numerical simulations. Finally, the conclusion is summarized insection 5.

2 Preliminary

Consider the autonomous system of ordinary differential equations

x = f(x) (1)

Where f is continuously differentiable on an open D ⊂ Rn.We write φt(x0) for the solution of (1) that starts at the point x0 at t = 0,φt(x0) = x (t, x0).φt(x) will be referred to as the flow corresponding to (1), we sometimes referto f as the vector field generating the flow φt(x).If φt(x) is defined for all t > 0 we write

γ+(x) = {φt(x), t ≥ 0}

Global asymptotic stability of SIRI diseases models 4883

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

φs(x)

We use the following notation for the vector order in Rn:

• x ≤ y if xi ≤ yi for all i.

• x < y if x ≤ y and xi < yi for some i.

• x << y if xi < yi for all i.

Definition 2.1. A semiflow on the set D is a continuous map Φ : D ×R+ −→ D which satisfies:

1. Φ0 = idD, where idD is the identity map in D.

2. Φt ◦ Φs = Φt+s 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 φt(e) = e for all t ≥ 0.The set of equilibria is denoted by E = {e ∈ D : φt(e) = e ∀t ≥ 0}.

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

{x ∈ D : φt(x)

t→∞−→ e; e ∈ E}

the set of allconvergent points.

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

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

Definition 2.4. 1. An n× n matrix A = (aij) is irreducible if for everynonempty proper subset I of the set N = 1, 2, ..., n, there is an i ∈ I andj ∈ J ≡ N\I such that aij 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 xn in D satisfying xn < xn+1 < x(x < xn+1 < xn) for n ≥ 1 and xn → x as n→∞ .


Lemma 2.6. Let D be an open subset of Rn and x ∈ D. Then x can beapproximated from below or above in D.

Theorem 2.7. Let U a closure of an open subset D of Rn and x, y ∈ Uwith x < y then there exist sequences xn, yn ∈ D satisfying xn < yn, xn −→ xand yn −→ 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 justverify 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∂fi∂xj≥ 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 φt(x) ≤φt(y) whenever x ≤ y and t ≥ 0.

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

Theorem 2.11. [8]

• if (1) is cooperative then φt is monotone.

• if (1) is cooperative and irreducible then φt is strongly monotone.

We introduce following mild compactness condition for our next result.

(T ) : for each x ∈ D : φt(x) is defined for all t ≥ 0 and φt(x) ∈ D. Moreoverfor each bounded subset A ofD there exists a closed bounded subset B = B(A)of D such that for each x ∈ A, φt(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 domainD and assume that (T ) holds. Assume also that each point x ∈ D can beapproximated either from above or from below by a sequence {xn} 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 subsetof Rn or it is the closure of an open subset of Rn. In the latter case, it isassumed that the vector field extends to a continuously differentiable one on aneighborhood 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 populationinto the following sub-groups that are susceptible individuals S, infectious in-dividuals I, and individuals previously infectious but temporarily reverted tothe 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 eitherdirectly to the R class and then back to I class as shown in Figure 1. Thetransfer diagram leads to the following system of differential equations:

S = bN − λSI

N− dS

I = λSIN− (d+ γ) I + αR

R = γI − (d+ α)R

(2)

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

Nand i = I

Nand r = R

Nthe 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 systems = 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 arenonnegative.We introduce following condition:

(P): ∀x ∈ D and x ≥ 0. If xi = 0, then fi (x) ≥ 0.

Theorem 3.1. Consider the system (1). If f is Lipschitz on an open subsetD of Rn and the property (P) is verified, then the solution x (t) ≥ 0.

Theorem 3.2. The octant R3+ = {(x, y, z) ∈ R3 : 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 R3+ 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) ∈ R2+; 0 ≤ r + i ≤ 1

}Where R2

+ denotes the non-negative cone of R2.

In the next section, the reduced proportionate system (4) will be analyzed.We establish the positivity of solution, basic reproduction and existence ofequilibrium.


Theorem 3.3. The set Σ ={

(r; i) ∈ R2+; 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 = 1shows 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 thebasic reproduction rateR0. the basic reproduction rate is the number of peoplethat an infective person can infect.The basic reproduction number of system (4) is given by:

R0 =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 oneinfection equilibrium which represents the endemic equilibrium.It is not hard to see that if R0 ≤ 1, the disease-free equilibrium Ef = (0; 0) isthe unique equilibrium point corresponding to the extinction of the free virus.The following result presents the existence and uniqueness of endemic equilib-rium when R0 > 1.

Theorem 3.4. Let R0 = λ(d+α)d(d+α+γ)

1. If R0 ≤ 1, then the system (4) has a unique disease-free equilibrium ofthe form Ef = (0; 0).

2. If R0 > 1,the disease-free equilibrium is still present and the system(4) has a unique endemic equilibrium of the form E∗ = (r∗, i∗) wherer∗ = γi∗

d+αand i∗ = d(R0−1)

λ.

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∗ = γd+α

i∗

And we replaced r∗ in the second equation we have

i∗ = 0 or i∗ = λ(d+α)−d(d+α+γ)λ(d+α+γ)

= d(R0−1)λ


If i∗ = 0 then r∗ = 0

If i∗ = d(R0−1)λ

then r∗ = γd+α

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 wecheck 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; α

λ

].

Proof. We have the Jacobian matrix J is irreducible on U , therefore the systemis irreducible.The off diagonal of the matrix J is positive for each x ∈ U , moreover thesystem is cooperative by Definition 2.8.The system (4) is cooperative and irreducible on U , then by Theorem 2.11 thesystem (4) is strongly monotone on U .

Theorem 3.6. Assume that α < λ < α + d + γ or R0 < 1, then the setU = [0; 1]×

[0; α

λ

]is positively invariant with respect (4).

Proof. On the border i = αλ

Then we have i = −αλ

(d+ α + γ − λ) = −(λαdR0

+ λR0

(1−R0))< 0

Since R0 < 1 or λ < α + d+ γThus U = [0; 1]×

[0; α

λ

]is positively invariant.

Lemma 3.7. Let U be the closure of the open subset D and A a boundedsubset of U .If U is positively invariant, then there exists a closed bounded subset B(A) ofU , such that ∀x ∈ A : φt (x) ∈ B (A).

Proof. Let A be a bounded subset of U . We have φt (A) ⊂ U ,then φt (A) is aclosed bounded subset of U (U is bounded set ).Thus, with B (A) = φt (A) the result follows.


To study the global stability of the equilibrium point on a set by monotonemethod, this set must be positively invariant. However according to Lemma3.7 there are two cases to ensure that a set be positively invariant. In the nextsections, 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 casesR0 < 1 and R0 > 1.

4.1 Local stability of disease-free equilibrium

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

Theorem 4.1. .

1. If R0 < 1, then Ef is locally asymptotically stable.

2. If R0 > 1, then Ef is unstable.

Proof. The Jacobian matrix of the system (4) at Ef is given by:

JEf =

(−(d+ α) γ

α −(d+ γ) + λ

)

tr(JEf

)= − (2d+ γ + α− λ) = −

(d+ λα

dR0+ λR0

(1−R0)).

If R0 < 1, then tr(JEf

)< 0, and det

(JEf

)= d (d+ α + γ) (1−R0) > 0.

Thus Ef 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 verifythat the point E∗ does not exists if R0 < 1 and E∗ = Ef when R0 = 1, ifR0 > 1, then we have the following theorem.

Remark 4.2. If αλ> 1, we have Σ ⊂ U , then we can study the system in

Σ.

Theorem 4.3. Assume αλ> 1. If R0 > 1, then E∗ is locally asymptotically

stable in Σ.


Proof. The Jacobian matrix of the system (4) at E∗ is given by:

JE∗ =

(−(d+ α) γ

α− d (R0 − 1) −(d+ γ) + λ− 2d (R0 − 1)− γd(R0−1)d+α

)Then tr (JE∗) = − (2d+ γ + α− λ)− 2d (R0 − 1)− γd(R0−1)

d+α< 0, since α > λ

and R0 > 1.We have det (JE∗) = d (d+ α + γ) (R0 − 1) > 0, When R0 > 1.Thus E∗ is locally asymptotically stable.

Theorem 4.4. Assume that α < λ < α + d + γ. If R0 > 1, then E∗ islocally asymptotically stable in U .

Proof. If R0 > 1, we remark that tr (JE∗) < 0 since λ < α + d + γ, anddet (JE∗) = d (d+ α + γ) (R0 − 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 sufficientcondition for a solution to converge to the free equilibrium if R0 < 1 and tothe endemic equilibrium if R0 > 1.

Theorem 4.5. 1. Assume αλ> 1. If R0 < 1 , then the disease free

equilibrium Ef is globally asymptotically stable in Σ.

2. Assume αλ< 1. If R0 < 1 , then the disease free equilibrium Ef is globally

asymptotically stable in U .

Proof. 1. Suppose αλ> 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) iscooperative 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) = φt (A) by Lemma 3.7.From Theorem 2.7, Theorem 2.12 , Remark 2.13 and Definition 2.5, wehave E consists of one points Ef which is locally asymptotically stablesince R0 < 1. Then all solutions with initial value in Σ converge to Ef .Therefore Ef is globally asymptotically stable in Σ.

2. If α < λ and R0 < 1 then U is positively invariant by theorem 3.6 andthe condition (T ) is verified.The set E consists of one point Ef which is locally asymptotically sta-ble since. Thus all solutions with initial values in U converge to Ef .Therefore Ef is globally asymptotically stable in U .


4.4 Global asymptotic stability of endemic equilibrium

Theorem 4.6. 1. Suppose αλ> 1. If R0 > 1, then the endemic equilib-

rium E∗ is globally asymptotically stable in Σ.

2. Suppose that α < λ < α+d+γ. If R0 > 1, then the endemic equilibriumE∗ is globally asymptotically stable in U .

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

we have Σ ⊂ U , then the system (4) is strongly monotone on Σ.Since R0 > 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 asymptoticstability of E∗ in Σ.

2. The system (4) is cooperative and irreducible on the set U by theorem3.5, we have α < λ < α+d+γ then U is positively invariant by theorem3.6 and the condition (T ) is verified.E consists of two points Ef and E∗. Since R0 > 1 the endemic equi-librium is locally asymptotically stable. Thus all solutions with initialvalue 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).

• αλ> 1

We adopt the following values for the parameters:Figure 2: α = 0.5, γ = 0.5, λ = 0.25 and d = 0.9We have R0 = 0.2 < 1 that means all solutions converge to the freeequilibrium.Figure 3: α = 0.5, γ = 0.01, λ = 0.25 and d = 0.018We have R0 = 13.6 > 1 that mean all solution converge to the endemicequilibrium I∗ = 0.9 and R∗ = 0.01.

• αλ< 1 We adopt the following values for the parameters:

Figure 4: α = 0.25, γ = 0.5, λ = 0.8 and d = 0.9We have R0 = 0.5 < 1 that mean all solution converge to the freeequilibrium.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 R0 = 9.568 > 1 that mean all solution converge to the endemicequilibrium 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 infectiousdiseases 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 begeneralized 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

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Global Asymptotic Stability of SIRI Diseases Models Using ...m-hikari.com/.../p/kabliAMS97-100-2015.pdf · 4 Stability of the system In thispart of thepaperwewill studythelocal andglobal