Download Local and Global Stability of Host-Vector Disease Models

Local and Global Stability of Host-Vector Disease
Models
March 4, 2008
Cruz Vargas-De-León†1 , Jorge Armando Castro Hernández‡
†
Unidad Académica de Matemáticas, Universidad Autónoma de Guerrero, México
and Departamento de Matemáticas, Facultad de Ciencias, UNAM, México
‡
Departamento de Matemáticas, Escuela Superior de Fı́sica y Matemáticas, IPN,
México
Abstract
In this work we deal with global stability properties of two host-vector
disease models using the Poicaré-Bendixson Theorem and Second Method
of Lyapunov. We construct a Lyapunov function for each Vector-Host
model. We proved that the local and global stability are completely determined by the threshold parameter, R0 . If R0 ≤ 1, the disease-free
equilibrium point is globally asymptotically stable. If R0 > 1, a unique
endemic equilibrium point exists and is globally asymptotically stable in
the interior of the feasible region.
Key Words: Lyapunov Function, Dulac Function, Global Stability,
LaSalle’s invariant principle
1
Introduction
It is hard to overstate the medical importance and burden of vector-transmitted
infectious diseases (malaria, Chagas disease, leishmaniasis, dengue fever, lymphatic filariasis, West Nile virus and the encephalitis viruses).
The key elements involved in human vector-borne infectious diseases are the
infectious microorganism (virus, bacterium or parasite), the vector (mosquitoes,
tick or fly), and the reservoir from which the vector obtains the infection. Control strategies for these diseases should be informed by an understanding of
the complex dynamics of vector-host interactions and the ways in which the
environments of both the vector and host intersect to produce human disease.
The study of these dynamics using mathematical models has had a staggering
development in recent years, and has proven to be a valuable tool to understand
1 Corresponding
author. Email: [email protected]
1
epidemiological patterns and processes, provided that models are as close as
possible to real life situations and based on biological knowledge.
In the case of dengue fever and West Nile virus, the mathematical models
we have found in the literature propose compartmental dynamics in [3,4,5,6,7,8]
and [2,10,11], respectively.
We proved the global stability of host-vector disease models by means the
Poincaré-Bendixson theorem and second method of Lyapunov.
However it is difficult to construct a Dulac function and Lyapunov functions
to establish the global stability of the equilibrium E ∗ .
C. Vargas-De-León proved in [12], the global stability of endemic equilibrium
of host-vector diseaseP
models using the Lyapunov function:
b 1 , x2 , ...xn ) = n ci (xi − x∗ − x∗ log x∗i ), and LaSalle’s invariant prinL(x
i
i
i=1
xi
ciple.
2
Host-Vector Disease Model
In this section, we study a Host-Vector Model with constant population sizes in
the host and vector population and no immunity in host population.
We suppose that we dispose of a host population (respectively of vector
population) of size Nh (resp. Nv ) formed of Susceptibles Sh and Infectious Ih
(resp. Sv and Iv ). We consider a compartmental model that is to say that
every population is divided into classes, and that one individual of a population passes from one class to another with a suitable rate. We use the SIS
(Susceptible-Infectious-Susceptible) model for the host population and a SI
(Susceptible-Infectious) model for the vector population. Some vector-borne
infectious diseases confer no immunity then recovered with the possibility of
becoming susceptible. This type of disease can be modelled by this model, i.e.
malaria.
The dynamics of this disease in the host and vector populations are given
by the following system of non-linear differential equations:
dSh
dt
dIh
dt
dSv
dt
dIv
dt
= µNh − κSh Iv − µSh + φIh
= κSh Iv − (µ + φ)Ih
= ηNv − βSv Ih − ηSv
(1)
= βSv Ih − ηIv .
With the conditions: Nh = Sh + Ih and Nv = Sv + Iv . The parameters are
positive constants. Under the assumptions that transmission term is mass action. Mass action assumes that biting rates are limited by the densities of
both vectors and hosts. The vector infectious-to-host susceptible transmission
rate κSh Iv , where as the host infectious-to-vector susceptible transmission rate
βSv Ih . The host total population Nh is constant, µ is the per capita natural
2
mortality rate of host. The recovered individuals are removed from the susceptible class (Sh ) at the rate, φ. The vector total population Nv is constant, η is
the per capita natural mortality rate of vector.
We thus study system (1) in the following feasible region:
Γ = (Sh , Ih , Sv , Iv ) ∈ R4+ : Sh , Ih , Sv , Iv ≥ 0, Sh + Ih = Nh , Sv + Iv = Nv ,
where Γ is positively invariant with respect to (1). Direct calculation shows
that system (1) has two possible equilibrium points in R4+ : the disease-free
equilibrium point E 0 (Sh0 , Ih0 , Sv0 , Iv0 ), where Sh0 = Nh , Ih0 = 0, Sv0 = Nv , Iv0 = 0,
and a unique endemic equilibrium point E ∗ (Sh∗ , Ih∗ , Sv∗ , Iv∗ ) with
Sh∗ =
Sv∗ =
NH ((µ+φ)R0 +κNv )
,
(κNv +µ+φ)R0
Nv (κNv +µ+φ)
((µ+φ)R0 +κNv )
Ih∗ =
and Iv∗ =
η(µ+φ)(R0 −1)
β(κNv +µ+φ) ,
Nv (µ+φ)(R0 −1)
((µ+φ)R0 +κNv ) .
√
h Nv
c
R0 is called
Let R0 = κβN
η(µ+φ) is threshold parameter. The quantity R0 =
the basic reproductive number of the disease, since it represents the average
number of secondary cases that one case can produce if introduced into a susceptible population.
Then, we have proved the following proposition:
Proposition 1. System (1) always has the disease free equilibrium E 0 (Nh , 0, Nv , 0).
If R0 > 1 there is also a unique endemic equilibrium E ∗ (Sh∗ , Ih∗ , Sv∗ , Iv∗ ) in the
interior of Γ.
The human and vector populations remain constant in Γ, therefore, without
loss of generality, since Sh = Nh −Ih y Sv = Nv −Iv , system (1) in the invariant
space Γ can be written as the equivalent two-dimensional non-linear system of
ODEs:
dIh
dt
dIv
dt
= κ(Nh − Ih )Iv − (µ + φ)Ih
= β(Nv − Iv )Ih − ηIv
We thus study system (2) in the following feasible region:
Ω = (Ih , Iv ) ∈ R2+ : 0 < Ih ≤ Nh , 0 < Iv ≤ Nv
and the equilibrium points are defined: the trivial equilibrium point
P 0 (0, 0), and an interior equilibrium point P ∗ (Ih∗ , Iv∗ ) with
Ih∗ =
η(µ+φ)(R0 −1)
β(κNv +µ+φ) ,
and Iv∗ =
3
Nv (µ+φ)(R0 −1)
((µ+φ)R0 +κNv ) .
(2)
2.1
Asymptotic stability analysis
In this subsection, we study the local stability properties of (2).
Theorem 1. i) If R0 < 1, then the disease-free equilibrium point P 0 is locally
asymptotically stable. ii) If R0 > 1 , P 0 is unstable and P ∗ is locally asymptotically stable.
Proof
We study the local stability of the trivial equilibrium point P 0 . The Jacobian
matrix J = (Ih , Iv ) of system (2) is given by


−(κIv + µ + φ) κ(Nh − Ih )
,
J =
β(Nv − Iv )
−(βIh + η)
Then, the local stability of P 0 is governed by the eigenvalues of the matrix


−(µ + φ) κNh
,
JP 0 = 
βNv
−η
and the characteristic polynomial of JP 0 is:
τ 2 + (µ + φ + η)τ + η(µ + φ)(1 − R0 ) = 0.
(3)
The two eigenvalues of (3) have negative real parts if and only if the coefficients of (3) are positive, and this occurs if and only if R0 < 1. Therefore P 0
is locally asymptotically stable for R0 < 1. If for R0 > 1 the equilibrium P 0
becomes an unstable hyperbolic point. This proves Theorem 1. (i).
The local stability of P ∗ is given by the Jacobian evaluated in this point:


−(κIv∗ + µ + φ) κ(Nh − Ih∗ )
,
JP ∗ = 
β(Nv − Iv∗ )
−(βIh∗ + η)
that can be rewritten as


JP ∗ = 
−(κIv∗ + µ + φ)
η
I∗
(µ + φ) Ih∗

v

,
−(βIh∗ + η)
Iv∗
∗
Ih
when we take into account ∗the identities:
I
I∗
κ(Nh − Ih∗ ) = (µ + φ) Ih∗ and β(Nv − Iv∗ ) = η Iv∗ which are obtained from
v
h
equations of system (2) with the right-hand side equal to zero. Hence, the
characteristic polynomial of the linealized system is given by
Φ(τ ) = τ 2 + (κIv∗ + βIh∗ + µ + φ + η)τ + βIh∗ (κIv∗ + µ + φ) + ηκIv∗ .
4
(4)
The coefficients of (4) are positive, then the two eigenvalues of JP ∗ have
negative real part. Therefore P ∗ is locally asymptotically stable for R0 > 1.
This proves Theorem 1. (ii).
2.2
Global stability of interior equilibrium of (2)
In this subsection, we study the global stability of interior equilibrium of system
(2) using the Poicaré-Bendixson Theorem.
Theorem 2. If R0 > 1 , then P ∗ is globally asymptotically stable in the interior
of Ω.
Proof
Denote the right-hand side of (2) by (f, g) and choose a Dulac function as
D(Ih , Iv ) =
1
.
Ih Iv
(5)
Then we have
βNv
κNh
∂(Df ) ∂(Dg)
+
=− 2 − 2 ,
∂Ih
∂Iv
Ih
Iv
when Ih , Iv > 0. Thus, system (2) does not have a limit cycle in the interior of
Ω. Then, it is easy to see that P ∗ is globally stable in the interior of Ω.
2.3
Global stability of systems (1)
In this subsection, we study the global stability properties of system (1) using
the Second Method Lyapunov.
Theorem 3. i) If R0 ≤ 1, then the disease-free equilibrium point E 0 is globally
asymptotically stable in Γ. ii)If R0 > 1 , then E ∗ is globally asymptotically
stable in the interior of Γ.
Proof of Theorem 3 (i): Define the Lyapunov function
U : {(Sh , Ih , Sv , Iv ) ∈ Γ : Sh , Sv > 0} → R by
Sh
Sh − Nh − Nh log
+ Ih
Nh
(µ + φ)
Sv
+
Sv − Nv − Nv log
+ Iv .
βNv
Nv
U (Sh , Ih , Sv , Iv ) =
(6)
Then U is C 1 on the interior of Γ, E 0 is the global minimum of U on Γ, and
U (Sh0 , Ih0 , Sv0 , Iv0 ) = 0. The time derivative of U computed along solutions of (1)
is
5
dU
Nh dSh
dIh
(µ + φ)
Nv dSv
dIv
= 1−
+
+
1−
+
,
dt
Sh
dt
dt
βNv
Sv
dt
dt
dU
Nh
= 1−
[µNh − κSh Iv − µSh + φIh ] + [κSh Iv − (µ + φ)Ih ]
dt
Sh
(µ + φ)
Nv
+
1−
[ηNv − βSv Ih − ηSv ] + [βSv Ih − ηIv ].
βNv
Sv
From Ih = Nh − Sh , Sh0 = Nh and Sv0 = Nv , we have
dU
dt
2
h)
−
= −(µ + φ) (Sh −N
Sh
(µ+φ) (Sv −Nv )2
βNv
Sv
−
(µ+φ)
βNv (1
− R0 )Iv ≤ 0.
dU
Therefore, if R0 ≤ 1, dU
dt ≤ 0 for all Sh ,Ih ,Sv , Iv > 0. Note that, dt = 0 if
0
0
0
and only if Sh = Sh , Sv = Sv and Iv = 0, or ifR0 = 1, Sh = Sh and Sv = Sv0 .
Therefore the largest compact invariant set in (Sh , Ih , Sv , Iv ) ∈ Γ : dU
dt = 0 is
the singleton {E 0 }, where E 0 is the disease-free equilibrium point. LaSalle’s
invariant principle [9] then implies that E 0 is globally asymptotically stable in
Γ. This proves Theorem 3 (i).
Proof of Theorem 3 (ii): Define the Lyapunov function
Define L : {(Sh , Ih , Sv , Iv ) ∈ Γ : Sh , Ih , Sv , Iv > 0} → R by
Sh
Ih
L(Sh , Ih , Sv , Iv ) = c1 Sh − Sh∗ − Sh∗ log ∗ + c2 Ih − Ih∗ − Ih∗ log ∗
Sh
Ih
Sv
Iv
∗
∗
∗
∗
+ c3 Sv − Sv − Sv log ∗ + c4 Iv − Iv − Iv log ∗ , (7)
Sv
Iv
where c1 = c2 = βSv∗ Ih∗ and c3 = c4 = κSh∗ Iv∗ . Then L is C 1 on the interior of
Γ, P ∗ is the global minimum of L on Γ, and L(Sh∗ , Ih∗ , Sv∗ , Iv∗ ) = 0. The time
derivative of L computed along solutions of system (1) is
dL
S ∗ dSh
I ∗ dIh
= βSv∗ Ih∗ 1 − h
+ βSv∗ Ih∗ 1 − h
dt
Sh
dt
Ih
dt
∗
∗
S
dSv
I
dIv
+ κSh∗ Iv∗ 1 − v
+ κSh∗ Iv∗ 1 − v
Sv
dt
Iv
dt
∗
dL
S
= βSv∗ Ih∗ 1 − h [µNh − κSh Iv − µSh + φIh ]
dt
Sh
I∗
+ βSv∗ Ih∗ 1 − h [κSh Iv − (µ + φ)Ih ]
Ih
S∗
+ κSh∗ Iv∗ 1 − v [ηNv − βSv Ih − ηSv ]
Sv
I∗
+ κSh∗ Iv∗ 1 − v [βSv Ih − ηIv ]
Iv
6
Using Ih = Nh − Sh , Ih∗ = Nh∗ − Sh∗ , µNh = κSh∗ Iv∗ + (µ + φ)Sh∗ , (µ + φ) = κ
ηNv = βSv∗ Ih∗ + ηSv∗ and η = β
∗
Sv∗ Ih
Iv∗ ,
∗ ∗
Sh
Iv
∗ ,
Ih
to rewrite this, we get
dL
(Sh − Sh∗ )2
(Sv − Sv∗ )2
= −(µ + φ)βSv∗ Ih∗
− ηκSh∗ Iv∗
dt
Sh
Sv
∗
∗
∗
∗
S
I
I
S
S
I
I
S
v
h
h
v
− βκSh∗ Ih∗ Sv∗ Iv∗ h + v + ∗ h ∗ + ∗ v ∗ − 4 ≤ 0.
Sh
Sv
Sh Ih Iv
Sv Iv Ih
Thus, using the arithmetic–geometric means inequality, one can see that dL
dt
is less or equal to zero with equality only if Sh = Sh∗ , Ih = Ih∗ , Sv = Sv∗ and Iv =
Iv∗ . Therefore the largest compact invariant set in (Sh , Ih , Sv , Iv ) ∈ Γ : dL
dt = 0
is the singleton {E ∗ }, where E ∗ is the endemic equilibrium point. LaSalle’s
invariant principle [9] then implies that E ∗ is globally asymptotically stable in
the interior of Γ. This proves Theorem 3 (ii).
3
Host-Vector Disease Model with Transmission
Horizontal in Host Population
In this section, we formulate a model with constant population sizes with transmission horizontal in the host population. The Chagas diseases involve transmission by vectors and blood transfusion. The following system is a simple
model for Chagas disease:
dSh
dt
dIh
dt
dSv
dt
dIv
dt
= µNh − κSh Iv − λSh Ih − µSh
= κSh Iv + λSh Ih − µIh
= ηNv − βSv Ih − ηSv
(8)
= βSv Ih − ηIv
The variables and parameters are the same as in the previous model. We thus
study system (8) in Γ. The host infectious-to-host susceptible transmission rate
λSh Ih .
For system (8) the region Γ is positively invariant. Our next result concerns the existence of equilibrium points. We observe that E 0 (Nh , 0, Nv , 0) is
the disease-free equilibrium point. From the equations of system (8) with the
right-hand side equal to zero, can be seen that the endemic equilibrium point
E ∗ (Sh∗ , Ih∗ , Sv∗ , Iv∗ ) with 0 < Ih∗ < Nh and 0 < Iv∗ < Nv must satisfy the following
relations
Sh∗ =
∗
µ(βIh
+η)
∗,
κβNv +λη+λβIh
Sv∗ =
ηNv
∗ +η
βIh
and Iv∗ =
∗
βηNv Ih
∗ +η)η .
(βIh
and Ih∗ is solution of the following quadratic polynomial
7
P (Ih ) = λβIh2 + (µβ + λη + βκNv − λβNh )Ih − µη(R0 − 1),
(9)
v +λη)
where R0 = Nh (κβN
.
ηµ
We are looking for solutions of equation (9) between zero and Nh . For this,
notice that P (0) < 0 if and only if R0 > 1, and therefore a necessary and
sufficient condition to have a unique zero 0 < Ih∗ < Nh is that P (Nh ) > 0.
Then, we have proved the following proposition:
Proposition 2. System (8) always has the disease free equilibrium E 0 (Nh , 0, Nv , 0).
If R0 > 1 there is also a unique endemic equilibrium E ∗ (Sh∗ , Ih∗ , Sv∗ , Iv∗ ) in the
interior of Γ.
Theorems 1, 2 and 3, hold for system (8) as well. Theorems 1 is proved with
an analysis made similar to the system (2) using the same Dulac function (5) to
establish the global stability of a two-dimensional model as the equivalent fourdimensional system (8). Theorems 2 and 3 are proved using the second method
of Lyapunov based in the same Lyapunov functions (6) and (7) to establish the
global stability of the two equilibrium points of four-dimensional model (8).
4
Discussions and Conclusions
This paper presents a mathematical study on the global stability of standard
host-vector models using a Dulac functions and Lyapunov functions. It is established in Theorem 3 that R0 is a sharp threshold parameter and completely
determines the global stability of (1) and (8) in the feasible region. If R0 ≤ 1,
the disease-free equilibrium point is globally asymptotically stable in the feasible region and the disease will die out. If R0 > 1, a unique endemic equilibrium
point is globally asymptotically stable in the interior of the feasible region and
the disease persists.
The results of this work indicate that the Lyapunov functions of the form
b 1 , x2 , ...xn ) = Pn ci (xi − x∗ − x∗ log x∗i ) can be especially useful for comL(x
i
i
i=1
xi
partmental host-vector models with any number of compartments.
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