Download Global Stability for the Disease Free Equilibrium of a Delayed Model

Int. Journal of Math. Analysis, Vol. 6, 2012, no. 38, 1877 - 1881
Global Stability for the Disease Free Equilibrium
of a Delayed Model for Malaria Transmission
Chunqing Wu1 and Zhongyi Jiang
School of Mathematics and Physics, Changzhou University
Changzhou, Jiangsu, 213164, China
Abstract
The global asymptotic stability for the disease free equilibrium of
a mathematical model for malaria transmission with two delays is obtained by constructing appropriate Lyapunov functions.
Mathematics Subject Classification: 34K20, 92B05
Keywords: Global asymptotic stability, Lyapunov function, Time delay,
Malaria transmission
1
Introduction
Malaria is a vector-borne infectious disease which is transmitted between human population by mosquitoes and is now endemic in over 90 countries including both tropical and subtropical regions, in which 40% of the world’s
population live [1]. Malaria is caused by a parasite called Plasmodium, which
is transmitted via the bites of infected mosquitoes. In the human body, the
parasites multiply in the liver, and then infect red blood cells. Symptoms of
malaria usually appear between 10 and 15 days after the mosquito bite [2].
R. Ross was the pioneer to establish mathematical models to understand
the dynamics of malaria transmission quantitatively [3]. His basic model was
extended by Macdonald [4], Anderson and May [5], Smith and McKenzie [6],
Auger [7], Ruan et al.[1], etc.
In [1], a delay differential equation system was established, which included
two delays: one is to describe the considerable incubation time of the parasites
in human body, and the other is to describe the incubation time in mosquito.
Moreover, the basic reproduction number for the model was defined and the
local stability both for the disease free and the endemic equilibrium of the
1
arnold [email protected]
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C. Q. Wu and Z. Y. Jiang
model was discussed via the eigenvalues of the characteristic equation associated with the linearized system for the model. However, the global stability
for the equilibria is not concerned in [1].
Global stability for the equilibria of a biomathematical model is often discussed in the literature [8]. Usually for mathematical models about infectious
diseases, the disease free equilibrium is globally asymptotically stable when the
reproduction number is less than 1, and the endemic equilibrium exists, which
is globally asymptotically stable when the reproduction number is greater than
1 [9].
The aim of this paper is to prove the global asymptotic stability for the
disease free equilibrium of the model established in [1] by constructing appropriate Lyapunov functions.
2
The model and Preliminaries
The model considered for the transmission of malaria between human and
mosquito in this paper is the following
dx
dt
dy
dt
= −γx(t) + abm[1 − x(t − τ1 )]y(t − τ1 ) exp(−γτ1 ),
= −μy(t) + acx(t − τ2 )[1 − y(t − τ2 )] exp(−μτ2 ).
(1)
In model (1), γ is the average per capita rate of recovery in humans, a is
the bites of a single mosquito on human per day, b is the rate of infected bites
that produces new infectious human individual, c is the rate of infected bites
that produces new infectious mosquito, τ1 , τ2 is the incubation period of the
parasite in human host and the vector mosquito, respectively, μ is the death
rate of mosquito. It is assumed that the total populations of both human
and mosquito in the considered region are constants, x(t) and y(t) are the
proportion of infected humans and mosquitoes at time t, respectively, m is
the number of mosquitoes per human. One can refer to [1] for more details of
model (1).
The initial values of model (1) is given by
x(t) = φ(t), y(t) = ψ(t), t ∈ [−τ, 0], τ = max{τ1 , τ2 },
(2)
where 0 ≤ φ(t) < 1, 0 ≤ ψ(t) < 1, t ∈ [−τ, 0) and 0 < φ(0) < 1, 0 < ψ(0) <
1, and both of φ(t), ψ(t) are continuous in the interval [−τ, 0].
The basic reproduction number of model (1) is defined by
R0 =
a2 bcm exp(γτ1 ) exp(μτ2 )
.
γμ
(3)
The following theorem, which is obtained by direct computation, is about
the equilibria of model (1).
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Global stability for equilibrium
Theorem 2.1 In the planar domain G = {(x, y)|x ≥ 0, y ≥ 0}, one has,
(i) if R0 ≤ 1, then model (1) has a unique disease free equilibrium E0 =
(0, 0).
(ii) if R0 > 1, then model (1) has two equilibria, the disease free equilibrium
E0 and the unique positive equilibrium (endemic equilibrium) E ∗ = (x∗ , y ∗)
where
a2 bcm exp(−γτ1 ) exp(−μτ2 )−γμ
R0 −1
x∗ = ac
= R0 +ac exp(−μτ
,
exp(−μτ2 )(abm exp(−γτ1 )+γ)
2 )/μ
(4)
2
a bcm exp(−γτ1 ) exp(−μτ2 )−γμ
R0 −1
∗
y = abm exp(−μτ2 )(ac exp(−μτ2 )+μ) = R0 +abm exp(−γτ1 )/γ .
3
Main Results
In this section, we first denote
Ω = {(a, b, c, m, γ, μ, τ1 , τ2 )|such that the solutions of (1)
with initial values (2) are positive}.
(5)
Followed, the main results on the global asymptotic stability for E0 in Ω
are given by
Theorem 3.1 If R0 < 1, then the disease free equilibrium E0 is globally
asymptotically stable in Ω.
Proof. For simplicity, we denote α = exp(−γτ1 ) and β = exp(−μτ2 ).
Note that
R0 =
abmα acβ
a2 bcm exp(−γτ1 ) exp(−μτ2 )
=
·
,
γμ
γ
μ
there are three cases corresponding to R0 < 1:
abmα
Case 1
< 1, acβ
< 1;
γ
μ
Case 2
abmα
γ
> 1,
acβ
μ
< 1;
Case 3
abmα
γ
< 1,
acβ
μ
> 1.
Next we prove that the disease free equilibrium E0 is globally asymptotically stable under these three cases by constructing appropriate Lyapunov
functions.
Case 1. The following function
V1 (t) =
t
t
1
1
abmα
acβ
x+
ydt + y +
xdt,
γ
γ
μ
t−τ1
t−τ2 μ
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C. Q. Wu and Z. Y. Jiang
is positive definite in Ω. Further, calculating the differentiate of V1 (t) along
the solution of model (1), we have
dV1 (t)
dt
= −x + abmα
(1 − x(t − τ1 ))y(t − τ1 ) + abmα
y − abmα
y(t − τ1 )
γ
γ
γ
acβ
acβ
acβ
−y + μ (1 − y(t − τ2 ))x(t − τ2 ) + μ x − μ x(t − τ2 )
= −x − abmα
x(t − τ1 )y(t − τ1 ) + abmα
y
γ
γ
acβ
acβ
−y − μ y(t − τ2 )x(t − τ2 ) + μ x
acβ
−
1
y
+
−
1
x
< abmα
γ
μ
< 0.
Therefore, the disease free equilibrium E0 is globally asymptotically stable
in Ω.
Case 2. Constructing the following function
acβ
1
γ
1
V2 (t) =
·
x+ ·
y+
abmα μ
μ abmα
t
t−τ1
γ
ydt +
abmα
t
t−τ2
γ
acβ
·
xdt,
μ abmα
and calculating the differentiate of V2 (t) along the solution of model (1), we
have
dV2 (t)
dt
γ
= − abmα
· acβ
x + acβ
y(n − τ1 ) − acβ
x(t − τ1 )y(t − τ1 ) +
μ
μ
μ
γ
acβ
γ
acβ
γ
x(n − τ2 )
− abmα · μ x(t − τ2 ) − abmα y + μ · abmα
acβ
γ
γ
γ
y(t − τ1 )
− μ · abmα x(t − τ2 )y(t − τ2 ) + abmα y − abmα
acβ
acβ
γ
= − μ x(t − τ1 )y(t − τ1 ) − μ · abmα x(t − τ2 )y(t − τ2 )
γ
+ acβ
− abmα
y(t − τ1 ).
μ
γ
abmα
·
acβ
x
μ
γ
Note that R0 = acβ
· abmα
< 1, hence, acβ
< abmα
which yields dV2 (t)/dt < 0.
μ
γ
μ
Again the disease free equilibrium E0 is globally asymptotically stable in Ω.
Case 3. Constructing the following function
V3 (t) =
1 abmα
1 μ
·
x+
·
y+
γ acβ
acβ
γ
t
t−τ1
abmα μ
·
ydt +
γ
acβ
t
t−τ2
μ
xdt,
acβ
and calculating the differentiate of V3 (t) along the solution of model (1), we
have
dV3 (t)
dt
μ
μ
μ
= − acβ
x + acβ
· abmα
y(t − τ1 ) − acβ
· abmα
x(t − τ1 )y(t − τ1 )
γ
γ
μ
μ
μ
abmα
abmα
y + abmα
x(t − τ2 )
+ acβ · γ y − acβ · γ y(t − τ1 ) − acβ · abmα
γ
γ
μ
μ
abmα
− γ x(t − τ2 )y(t − τ2 ) + acβ x − acβ x(t − τ2 )
μ
· abmα
x(t − τ1 )y(t − τ1 ) − abmα
x(t − τ2 )y(t − τ2 )
= − acβ
γ
γ
μ
+ abmα
− acβ x(t − τ2 ).
γ
μ
Note that R0 = acβ
· abmα
< 1, therefore, abmα
< acβ
which yields dV3 (t)/dt < 0.
μ
γ
γ
The disease free equilibrium E0 is also globally asymptotically stable in Ω.
Thus completes the proof of Theorem 3.1.
Global stability for equilibrium
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ACKNOWLEDGEMENTS. This work is supported by the National
Natural Science Foundation of China (No. 11071011), the Qing Lan Project of
Jiangsu Province (QLG1102002) and the Foundation of Changzhou University
(czupy1002).
References
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[3] R. Ross, The prevention of malaria, 2nd edition, Lodon, Murray, 1911.
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[5] R.M. Anerson and R.M. May, Infectious diseases of human: dynamics
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[6] D.L. Smith and M.E. McKenzie, Statics and dynamics of malaria infection
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Received: March, 2012