AMApplied Mathematics2152-7385Scientific Research Publishing10.4236/am.2016.710096AM-67580ArticlesPhysics&Mathematics Stability Analysis of SIQS Epidemic Model with Saturated Incidence Rate O.Adebimpe1L.M. Erinle-Ibrahim2A.F. Adebisi3Department of Mathematics, Tai Solarin University of Education, Ijebu-Ode, NigeriaDepartment of Mathematical and Physical Science, Osun State University, Osogbo, NigeriaDepartment of Physical Sciences, Landmark University, Omu-Aran, Nigeria07062016071010821086May 10 2015July 29 2015 June 22 2016© Copyright 2014 by authors and Scientific Research Publishing Inc. 2014This work is licensed under the Creative Commons Attribution International License (CC BY). http://creativecommons.org/licenses/by/4.0/

A SIQS epidemic model with saturated incidence rate is studied. Two equilibrium points exist for the system, disease-free and endemic equilibrium. The stability of the disease-free equilibrium and endemic equilibrium exists when the basic reproduction number R0, is less or greater than unity respectively. The global stability of the disease-free and endemic equilibrium is proved using Lyapunov functions and Poincare-Bendixson theorem plus Dulac’s criterion respectively.

SIQS Epidemic Model Saturated Incidence Rate Basic Reproduction Number Lyapunov Function Poincare-Bendixson Dulac Criterion
1. Introduction

The isolation and treatment of symptomatic individuals coupled with the quarantining of individuals that have a high risk of having been infected, constitute two commonly used epidemic control measures. Mass quarantine can inflict significant social, psychological and economic costs without resulting in the detection of many infected individuals. Day et al.  , Hethcote et al.  considered SIQS and SIQR epidemic models with three forms of incidence, which include the bilinear, standard and quarantined-adjusted incidences.

*Corresponding author.

Feng and Thieme  considered SEIQR models with arbitrarily distributed periods of infection, including quarantine and a general incidence assumed that all infected individuals go through the quarantine stage and investigated the model dynamics. Settapat and Wirawah  discussed the SIQ epidemic model with constant immigration. Yang et al.  also studied an SIQ epidemic model with isolation and nonlinear incidence rate. El-Marouf and Alihaby  studied the equilibrium points and their local stability for SIQ and SIQR epidemic models with three forms of incidence rates. They also studied the global stability of the equilibrium by constructing the new forms of Lyapunov functions.

Gbadamosi and Adebimpe investigated an SIQ epidemic model with nonlinear incidence rate. They introduced the concept that describes the present and past states of the disease.

We extended the work of Gbadamosi and Adebimpe to include the rates at which individuals recover and return to susceptible compartment from compartments I and Q respectively and we apply Lyapunov functions and Poincare-Bendixson theorem plus Dulac’s criterion to prove the global stability of disease-free and endemic equilibria respectively.

2. The Model

The model that governs a system of differential equation is presented as follows:

d S d t = ( 1 − p ) A − β S I 1 + m I − d S + γ I + ε Q d I d t = β S I 1 + m I + p A − ( γ + δ + d + α ) I d Q d t = δ I − ( ε + d + α ) Q (1)

Subject to initial conditions

S ( 0 ) = S 0 ≥ 0 ,           I ( 0 ) = I 0 ≥ 0 ,         Q ( 0 ) = Q 0 ≥ 0 (2)

The parameters with their descriptions are presented in Table 1.

The addition of the system (1), gives

d N d t = A − d N − α I − α Q         where     N = S + I + Q

From above equation, we get

0 ≤ lim t → ∞ sup N ( t ) ≤ N 0 with     lim t → ∞ sup N ( t ) = N 0       if     and   only     if   lim t → ∞ sup I ( t ) = 0

From the first equation of the system (1), it follows

0 ≤ lim t → ∞ sup S ( t ) ≤ S 0

And the second equation gives

Descriptions of parameters
ParameterDescription
( 1 − p ) ARecruitment of the susceptible corresponding to Births and Immigration
pAconstant recruitment of the infected compartment
DPer capita natural mortality rate
δThe rate constant for individuals leaving the compartment I for the quarantine compartment Q
αDisease-related death rate constant in compartments I and Q
γRate at which individuals recover and return to Susceptible (S) from compartment I
εRate at which individuals recover and return to Susceptible (S) from compartment Q
MThe saturation constant

0 ≤ lim t → ∞ sup I ( t ) ≤ I 0 .

So, from the above, if N > N 0 , then d N d t < 0 .

We can now write

Ω = { ( S , I , Q ) ∈ R + 3 : S + I + Q ≤ N 0 ,   S ≤ S 0 ,   I ≤ I 0 }

Equilibria

The system (1) has always the disease-free equilibrium at E 0 = ( S 0 , I 0 , Q 0 ) = ( ( 1 − p ) A d , 0 , 0 )

Endemic Equilibrium: E * = ( S * , I * , Q * )

3. Local Stability

In this section, we discussed the local stability of the disease-free equilibrium and endemic equilibrium for the system (1).

We state and prove the following results:

Theorem 1: At E 0 , the disease-free equilibrium of the system (1) is locally asymptotically stable when R 0 < 1 .

Proof: The Jacobian matrix at the point E 0 through linearization is given by

J 0 = ( − d − ( β S 0 − γ ) ε 0 β S 0 − ( γ + δ + d + α ) 0 0 δ − ( ε + d + α ) )

By finding the eigenvalues, we have the following λ s :

λ 1 = − d ,             λ 2 = β S 0 − ( γ + δ + d + α ) ,         λ 3     = − ( ε + d + α )

For λ 2 to be negative β S 0 < ( γ + δ + d + α )

That is, β S 0 γ + δ + d + α < 1

Let R 0 = β ( 1 − p ) A d ( γ + δ + d + α )

If R 0 = β ( 1 − p ) A d ( γ + δ + d + α ) < 1 ,     λ 2 < 0

Since λ 1 < 0 , λ 3 < 0 and λ 2 < 0 if R 0 < 1 , the disease-free equilibrium is locally asymptotically stable.

Theorem 3.1: The system (1) is locally asymptotically stable at E * if R 0 > 1 , otherwise unstable.

Proof: At the endemic equilibrium E * , the Jacobian matrix of the system (1) is given by:

J * = ( − ( β I * + d ) − β S 0 + γ ε β I * β S * − ( γ + δ + d + α ) 0 0 δ − ( ε + d + α ) )

The characteristic equation of the Jacobian matrix J * is given by

λ 3   + a 1 λ 2 + a 2 λ + a 3

where a 1 = 3 α + 2 δ + γ + 3 d + β I * + ε − 2 β S *

a 2 = 2 β I * α + β I * d + β I * δ + 3 d 2 + 4 α d + α 2 + γ α + δ α + 2 δ d + 2 γ d     + 3 d ε + γ δ + β I * ε + β I * d − β S * d − 2 β S * ε

a 3 = β I * d ε + β I * α ε + γ d ε + δ d ε + d 2 ε + α d ε + β I * δ d + β I * α d + γ d 2 + d 3 + 2 α d 2     + β I * α δ + β I * d α + β I * α 2 + γ α d + δ α d + α 2 d − β S * α d

as a 1 > 0 , a 2 > 0 ,   a 3 > 0     and     a 1 a 2 − a 3 > 0

If a 1 a 2 > a 3 , by Routh Hurwitz criterion, all the eigenvalues of the system (1) has negative real part. Therefore, the endemic equilibrium of the system (1) at E * is locally asymptotically stable.

4. Global Stability

In this section, we study the global stability of the disease-free equilibrium and endemic equilibrium by Lyapunov function and Poincare-Bendixson theorem respectively.

Theorem 3: (Dulac’s Criterion)

Consider the following general nonlinear autonomous system of de

x ( t ) = f ( x ) , x ∈ E (*)

Let f = C 1 ( E ) where E is a simple connected region in R2. If the exists a function it H ∈ C 1 ( E ) such that ∇ ⋅ ( H f ) is not identically zero and does not change sign in E, the system (*) has no close orbit lying entirely in E. if A is an annular region contained in E on which ∇ ⋅ ( H f ) does not change sign, then there is at most one limit cycle of the system (*) in A.

Theorem 4: (The Poincare-Bendixson Theorem):Suppose that f ∈ C 1 ( E ) where E is an open subset of Rn and that the system (*) has a rejecting Γ contained in a compact subset f of E. assume that the system (*) has only one unique equilibrium point x0 in f, then one of the following possibilities holds.

(a) w ( Γ ) is the equilibrium point x0

(b) w ( Γ ) is a periodic orbit

(c) w ( Γ ) is a graphic

Theorem 5: The disease-free equilibrium of the model (1) is globally asymptotically stable if R 0 < 1

Proof: To prove this result, we construct the following Lyapunov function

L = u 1 ( S − S 0 ) + u 2 ( I − I 0 ) + u 3 Q (3)

where u 1 ,   u 2   and   u 3 are positive constants to be determined later. Differentiating equation (3) with respect to t, we obtain

L ′ = u 1 [ ( 1 − p ) A − β S I 1 + m I − d S + γ I + ε Q ] + u 2 [ β S I 1 + m I + p A − ( γ + δ + d + α ) I ]     + u 3 [ δ I − ( ε + d + α ) Q ]

After rearrangements, we get

L ′ = β S I 1 + m I ( u 2 − u 1 ) + p A ( u 2 − u 1 ) + γ I ( u 2 − u 1 ) + ε Q ( u 3 − u 1 ) + δ I ( u 3 − u 1 )     + u 1 A − u 1 d S − u 2 d I − u 3 d Q − u 2   α I − u 3 α Q

Let us choose the constants u 1 = u 2 = u 3 = 1 . Finally, we obtain

L ′ = A − d ( S + I + Q ) − α ( I + Q ) L ′ = − ( d N − A ) − α ( N − S ) < 0

Thus, the disease-free equilibrium of the system (1) is globally asymptotically stable if R 0 < 1

In the next theorem, we present the global stability of the endemic equilibrium of the system (1) at E *

Theorem 6: The endemic equilibrium E * of the system (1) is globally asymptotically stable if R 0 > 1 .

Proof: In order to prove the result, we use Dulac plus Poincare Bendixson theorem as follow

H ( S , I , Q ) = 1 S I Q     where     S > 0 ,     I > 0 ,   Q > 0.

Then,

∇ ⋅ ( H F ) = ∂ ∂ S ( H ⋅ F 1 ) + ∂ ∂ I ( H ⋅ F 2 ) + ∂ ∂ Q ( H ⋅ F 3 )                                     = ∂ ∂ S [ 1 S I Q ( ( 1 − p ) A − β S I 1 + m I − d S + γ I + ε Q ) ] + ∂ ∂ I [ β S I 1 + m I + p A − ( γ + δ + d + α )     I ]                                             + ∂ ∂ Q [ δ I − ( ε + d + α ) Q ] ∇ ⋅ ( H F ) = − ( 1 − p ) A S 2 I Q − γ S 2 Q − ε S 2 I − p A S I 2 Q − δ S Q 2 < 0

Hence, by Dulac’s criterion, there is no closed orbit in the first quadrant. Therefore, the endemic equilibrium is globally asymptotically stable.

5. Discussion of Results

The mathematical and stability analysis of SIQS epidemic model with saturated incidence rate and temporary immunity has been presented. We investigated the local stability of the disease-free equilibrium and endemic equilibrium using the basic reproduction number, R 0 . We observed that, when R 0 < 1 , the disease-free equilibrium is stable at E 0 locally and endemic equilibrium is unstable which means there is tendency for the disease to die out in the long run. We proved the global stability of the disease free equilibrium and endemic equilibrium of the model using Lyapunov function and Dulac’s criterion plus Poincare-Bendixson theorem respectively.

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