All materials on our website are shared by users. If you have any questions about copyright issues, please report us to resolve them. We are always happy to assist you.

Description

"In this paper, we study the global stability conditions of two epidemiological model with relapse, and bilinear and standard incidence rates, respectively, that includes recruitment rate of susceptible individuals into the community and that

Tags

Transcript

Abstraction & Application
9
(2013) 50
−
61 UADY
On the global stability of infectious diseases models with relapse
Cruz Vargas-De-Le´on
Unidad Acad´emica de Matem´aticas,Universidad Aut´onoma de Guerrero,Guerrero, M´exico.leoncruz82@yahoo.com.mx
Abstract
In this paper, we study the global stability conditions of two epidemiological model with relapse,and bilinear and standard incidence rates, respectively, that includes recruitment rate of susceptibleindividuals into the community and that the disease produces non-negligible death in the infectiousclass. The models are appropriate for tuberculosis, including tuberculosis in human and bovine, andfor human herpes virus. It is also included the dynamics of tobacco and alcohol use. We presentsthe construction of Lyapunov functions for the systems mentioned above using suitable combinationsof known functions, common quadratic and Volterra–type, and of a novel function we call compositeVolterra–type function.
Keywords and phrases
: Epidemiological model, Relapse, Global stability, Lyapunov’s second method, LaSalle’s invarianceprinciple.2010
Mathematics Subject Classification
: 92D30, 34D23.
1 Introduction
The relapse phenomenon in some infectious diseases is characterized by the acquisition of quiescent stateof the individuals that have previously been infected, and with subsequent relapse (or reactivation) to theinfectious state. This recurrence of infections including diseases such as bovine tuberculosis and humanherpes virus. These types of diseases can be modelled by
SIRI
epidemiological systems.Tudor [15] formulated one of the ﬁrst epidemic models with relapse, this model incorporates, bilinearincidence rate and constant total population. This system was extended to include nonlinear incidencefunctions by Moreira and Wang [12]. Blower [3] developed a compartmental model for genital herpes,assuming standard incidence for the disease transmission and constant recruitment rate.Van den Driessche and Zou [5], developed a
SIRI
model in a constant population with standard incidenceand a general relapse distribution. Van den Driessche and coauthors, [6], formulated a
SEIRI
model for adisease with a general exposed distribution in a constant population.In [4, 14] modeling the dynamics of tobacco and alcohol use, respectively, as a contagious disease withrelapse. In the smoking and drinking models formulation proposed in [4, 14], assumed that the populationis constant, standard incidence and, linear and nonlinear relapse rates.In this paper, we present the construction of Lyapunov functions for two epidemiological models withrelapse, constant recruitment and disease-induced death.50
Cruz Vargas-De-Le´on
51
Λ
↓
S
βSI
−→
I
κI
−→
R
γR
−→
I
↓
µS
↓
(
α
+
µ
)
I
↓
µR
Figure 1: The transfer diagram for the
SIRI
modelIn recent years, Lyapunov
s
second method has been a popular technique to study global stability of epidemiological models. A Volterra-type Lyapunov function has been used in [2, 8] to prove global stability of the steady states of classic
SIS
,
SIR
and
SIRS
epidemiological models with bilinear incidence rate. Vargas-De-Le´on in [16, 17] made some ingenious linear combination of functions, to obtain a suitable Lyapunovfunctions to prove global stability of steady states of
SIS
,
SIR
and
SIRS
epidemiological models withbilinear and standard incidence rates, respectively.The most popular types of Lyapunov functions are the common quadratic and Volterra-type functions.The common quadratic functions are of the form
W
(
x
1
,x
2
,...x
n
) =
ni
=1
c
i
2
(
x
i
−
x
∗
i
)
2
,
and the Volterra-type functions
W
(
x
1
,x
2
,...x
n
) =
ni
=1
c
i
x
i
−
x
∗
i
−
x
∗
i
ln
x
i
x
∗
i
.
We introduce a novel family of functions [17, 18]
W
(
x
1
,x
2
,...,x
n
) =
c
ni
=1
(
x
i
−
x
∗
i
)
−
ni
=1
(
x
∗
i
)ln
ni
=1
(
x
i
)
ni
=1
(
x
∗
i
)
,
we call this, composite-Volterra function.Generally speaking, within infectious disease models with bilinear incidence exist methods for construct-ing Lyapunov functions (see [7, 8, 9, 10, 11]), there are no systematic methods for constructing Lyapunovfunctions for epidemic models with standard incidence rate. In [17, 18] author reported the construction of Lyapunov functions in the classic epidemiological models with standard incidence.We will make ingenious linear combination of common quadratic, Volterra–type and composite–Volterra–type functions, to obtain a suitable Lyapunov functions to study the global stability conditions of two
SIRI
models with bilinear and standard incidence rates, respectively.In Sections 2 and 3, we shall construct Lyapunov functions for
SIRI
models. Section 4 includes thediscussion and concluding remarks.
2 The epidemiological model with relapse and bilinear incidence
In most compartmental epidemiological models, the total population is often divided into several disjointclasses. The model divides the total population into the following sub-groups that are susceptible indi-viduals
S
, infectious individuals
I
, and individuals previously infectious but temporarily reverted to thenon-infectious state,
R
. 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 Fig. 1. The transfer diagram leads to thefollowing system of diﬀerential equations:
52
infectious diseases models
dS dt
= Λ
−
βSI
−
µS,dI dt
=
βSI
−
(
α
+
κ
+
µ
)
I
+
γR,
(2.1)
dRdt
=
κI
−
(
µ
+
γ
)
R.
The parameters are positive constants. The constant Λ is the recruitment rate of susceptibles corre-sponding to births and immigration,
µ
is the natural death rate of population. The parameter
β
is thedisease transmission coeﬃcient. In most classic disease transmission models, the disease-induced mortalityis neglected, we add the parameter
α
, that is the disease related death rate. The parameter
κ
describesthe rate that the infectious individuals becomes no-infectious individuals and
γ
denotes the rate that theno-infectious individuals are reverted to the infectious state.The diﬀerential equation of the total population of (2.1) is:
ddt
(
S
+
I
+
R
) = Λ
−
µ
(
S
+
I
+
R
)
−
αI.
(2.2)Thus the total population size may vary in time. In the absence of disease, the population size convergesto the steady state Λ
/µ
. We thus study (2.1) in the following feasible region:Ω =
(
S,I,R
)
∈
R
3+
:
S
≥
0
, I
≥
0
, R
≥
0
, S
+
I
+
R
≤
Λ
/µ
which can be shown to be positively invariant with respect to (2.1). Direct calculation shows that system (2.1)has two possible steady states in the non-negative octant
R
3+
: the disease–free steady state
E
0
= (
S
0
,
0
,
0)where
S
0
= Λ
/µ
, and a unique endemic steady state
E
∗
= (
S
∗
,I
∗
,R
∗
) with
S
∗
= Λ
µR
0
, I
∗
=
µβ
(
R
0
−
1)
, R
∗
=
κµβ
(
γ
+
µ
) (
R
0
−
1)
.
(2.3)Where
R
0
,
the basic reproduction number, is
R
0
= (
µ
+
γ
)
βS
0
κµ
+ (
µ
+
γ
)(
α
+
µ
) = (
µ
+
γ
)
β
Λ
µ
(
κµ
+ (
µ
+
γ
)(
α
+
µ
))
.
2.1 Global stability of disease–free steady state
The global stability of the disease–free steady state
E
0
is proved by using common quadratic and linearLyapunov functions and LaSalle’s invariance principle.
Theorem 1.
If
R
0
≤
1
, then the disease–free steady state
E
0
of (2.1) is globally asymptotically stable in
Ω
.Proof.
V
:
(
S,I,R
)
∈
R
3+
:
S >
0
→
R
by
V
(
S,I,R
) = (
µ
+
γ
)2
S
0
(
S
−
S
0
)
2
+ (
µ
+
γ
)
I
+
γR.
(2.4)
Cruz Vargas-De-Le´on
53It is clear that at
E
0
the function
V
(
S,I,R
) reaches its global minimum in
R
3+
, and hence
V
(
S,I,R
) isa Lyapunov function. The derivative of (2.4) with respect to
t
along solution curves of (2.1) is given by
V
(
S,I,R
) = (
µ
+
γ
)
S
0
(
S
−
S
0
)
dS dt
+ (
µ
+
γ
)
dI dt
+
γ dRdt
= (
µ
+
γ
)
S
0
(
S
−
S
0
)(Λ
−
µS
−
βSI
)+ (
µ
+
γ
)(
βSI
−
(
α
+
κ
+
µ
)
I
+
γR
) +
γ
(
κI
−
(
µ
+
γ
)
R
)
,
= (
µ
+
γ
)
S
0
(
S
−
S
0
)(
µS
0
−
µS
−
βSI
)+ (
µ
+
γ
)
βSI
−
(
κµ
+ (
α
+
µ
)(
µ
+
γ
))
I.
Using the expression
βSI
(
S
−
S
0
)
S
0
=
βI
(
S
−
S
0
)
2
S
0
+
βI
(
S
−
S
0
)
.
We obtain,
V
(
S,I,R
) =
−
(
γ
+
µ
)(
µ
+
βI
)(
S
−
S
0
)
2
S
0
+ (
κµ
+ (
µ
+
γ
)(
α
+
µ
))
I
(
µ
+
γ
)
βS
0
κµ
+ (
µ
+
γ
)(
α
+
µ
)
−
1
,
=
−
(
γ
+
µ
)(
µ
+
βI
)(
S
−
S
0
)
2
S
0
−
(
κµ
+ (
µ
+
γ
)(
α
+
µ
))
I
(1
−
R
0
)
.
Therefore,
R
0
≤
1 ensures that
V
(
S,I,R
)
≤
0 for all
S
,
I
,
R >
0, and that
V
(
S,I,R
) = 0 holds when
S
=
S
0
and
I
= 0, or if
R
0
= 1 and
S
=
S
0
. It is easy to verify that the disease–free steady state
E
0
is thelargest invariant set in
{
(
S,I,R
)
∈
Ω :
V
(
S,I,R
) = 0
}
, and hence by the LaSalle’s invariance principle [13],the steady state
E
0
is globally asymptotically stable.
2.2 Global stability of endemic steady state
The globally asymptotic stability of the endemic steady state is proved by constructing a global Lyapunovfunction. We obtain the Lyapunov function of a suitable combination of common quadratic and Volterratype functions.
Theorem 2.
If
R
0
>
1
, then the unique endemic steady state
E
∗
of (2.1) is globally asymptotically stable in the interior of
Ω
.Proof.
Deﬁne
L
:
{
(
S,I,R
)
∈
Ω :
S,I,R >
0
}→
R
by
L
(
S,I,R
) = (
S
−
S
∗
)
2
2
S
∗
+
I
−
I
∗
−
I
∗
ln
I I
∗
+
γR
∗
κI
∗
R
−
R
∗
−
R
∗
ln
RR
∗
.
This function is deﬁned, continuous and positive deﬁnite for all
S
,
I
,
R >
0. It can be veriﬁed that thefunction
L
(
S,I,R
) takes the value
L
(
S,I,R
) = 0 at the steady state
E
∗
, and thus, the global minimumof
L
(
S,I,R
) occurs at the endemic steady state
E
∗
. Since (
S
∗
,I
∗
,R
∗
) is an endemic steady state point of (2.1), we have
54
infectious diseases models
Λ =
βS
∗
I
∗
+
µS
∗
,
(
α
+
κ
+
µ
) =
βS
∗
+
γ R
∗
I
∗
,
(2.5)(
µ
+
γ
) =
κ I
∗
R
∗
.
Computing the derivative of
L
(
S,I,R
) along the solutions of system (2.1), we obtain
L
(
S,I,R
) = (
S
−
S
∗
)
S
∗
dS dt
+ (
I
−
I
∗
)
I dI dt
+
γR
∗
κI
∗
1
−
R
∗
R
dRdt
= (
S
−
S
∗
)
S
∗
(Λ
−
βSI
−
µS
)+ (
I
−
I
∗
)
βS
−
(
α
+
κ
+
µ
) +
γ RI
+
γR
∗
κI
∗
1
−
R
∗
R
(
κI
−
(
µ
+
γ
)
R
)
.
Using (2.5), we obtain
L
(
S,I,R
) =
−
(
S
−
S
∗
)
S
∗
(
µ
(
S
−
S
∗
) +
β
(
SI
−
S
∗
I
∗
))+ (
I
−
I
∗
)
β
(
S
−
S
∗
) +
γ
RI
−
R
∗
I
∗
+
γR
∗
κI
∗
1
−
R
∗
R
κI
−
κI
∗
RR
∗
.
Notice that
SI
−
S
∗
I
∗
=
S
∗
(
I
−
I
∗
) +
I
(
S
−
S
∗
)
.
Thus,
L
(
S,I,R
) =
−
(
S
−
S
∗
)
S
∗
(
µ
(
S
−
S
∗
) +
β
(
S
∗
(
I
−
I
∗
) +
I
(
S
−
S
∗
)))+
β
(
I
−
I
∗
)(
S
−
S
∗
) +
γR
∗
RR
∗
−
I I
∗
−
I
∗
RIR
∗
+ 1
+
γR
∗
I I
∗
−
RR
∗
−
IR
∗
I
∗
R
+ 1
,
=
−
(
µ
+
βI
)(
S
−
S
∗
)
2
S
∗
+
γR
∗
2
−
I
∗
RIR
∗
−
IR
∗
I
∗
R
,
=
−
(
µ
+
βI
)(
S
−
S
∗
)
2
S
∗
−
γR
∗
I
∗
RIR
∗
−
IR
∗
I
∗
R
2
.
Therefore,
L
(
S,I,R
)
≤
0 for all
S,I >
0, where the equality
L
(
S,I,R
) = 0 holds only when
S
=
S
∗
and
IR
∗
=
I
∗
R
. It is easy to see that the endemic steady state
E
∗
is the only largest invariant set in
{
(
S,I,R
)
∈
Ω :
L
(
S,I,R
) = 0
}
. Therefore, by LaSalle’s invariance principle [13], the endemic steady state
E
∗
is globally asymptotically stable in the interior of Ω.

Related Search

Eradication Of Infectious DiseasesTreaty On The Non Proliferation Of Nuclear WeMolecular Immunology of Infectious Diseases..Ecology of infectious diseasesImmunology of infectious diseases, immunizatiEpidemiology of infectious diseasesThesis on the Positive Impact of Forensic AccMolecular Epidemiology of Infectious DiseasesMathematical Modeling of Infectious DiseasesConvention On The Rights Of The Child

Similar documents

We Need Your Support

Thank you for visiting our website and your interest in our free products and services. We are nonprofit website to share and download documents. To the running of this website, we need your help to support us.

Thanks to everyone for your continued support.

No, Thanks