Download Stability Analysis and Optimal Vaccination Strategies for an SIR

ISSN 1749-3889 (print), 1749-3897 (online)
International Journal of Nonlinear Science
Vol.16(2013) No.4,pp.323-333
Stability Analysis and Optimal Vaccination Strategies for an SIR Epidemic Model
with a Nonlinear Incidence Rate
Hassan Laarabi1 ∗ , Abdelhadi Abta2 , Mostafa Rachik1 ,
Jamal Bouyaghroumni1 , El Houssine Labriji1
1
Université Hassan II Mohammadia Faculté des Sciences Ben M’Sik Département de Mathématiques et Informatique
Casablanca, Morocco
2
Université Chouaib Doukkali Faculté des Sciences Département de Mathématiques et Informatique El Jadida, Morocco
(Received 5 March 2013, accepted 17 November 2013)
Abstract: An SIR epidemic model with saturated incidence rate and a vaccination program is formulated,
where the susceptibles are assumed to satisfy the logistic equation. The incidence term is of saturated form
with the infected individuals. First, we have discussed the existence and the stability of both the disease free
and endemic equilibrium. Second, the impact of vaccination in reducing R(u) is tackled. Then, to achieve
control of the disease, a control problem is formulated and it is shown that an optimal control exists for our
model. The optimality system is derived and solved. Finally, numerical simulations are performed to illustrate and verify the analytical results.
Keywords: SIR model; Optimal Control; Vaccination; Stability.
1
Introduction
Mathematical modeling is of considerable importance in the study of epidemiology because it may provide understanding
of the underlying mechanisms which influence the spread of diseases and may suggest control strategies. Mathematical
models take into account main factors that govern the development of a disease, such as transmission and recovery rates,
and predict how the disease will spread over a period of time. In recent years, many attempts have been made to develop
realistic mathematical models for investigating the transmission dynamics of infectious diseases, and the asymptotic
behaviors of these epidemic models are studied [39]. A key role is played by the incidence rate, namely a function
describing the mechanism of transmission of the disease. Generally, such a function depends on both the susceptible
and infected classes. In many epidemic models, the bilinear incidence is frequently used [18,34]. This incidence rate is
based on the law of mass action. More specifically, if S and I are the numbers of susceptible and infected individuals
in a population, and if β is the per capita contact rate, then it is assumed that the infection spreads with the rate β SI.
This contact law is more appropriate for communicable diseases such as influenza, etc., but not for sexually transmitted
diseases. Furthermore, the mass action incidence may not yield appropriate results for a variety of reasons. For example,
the underlying assumption of homogeneous mixing may not always hold. In this case, the necessary population structure
and the heterogeneous mixing may be incorporated into a model with a specific form of non-linear transmission. The
bilinear incidence rate can also be affected by the control policies adopted by public health authorities. For instance, during
the outbreak of SARS in 2003, protection measures such as closing schools, closing restaurants, postponing conferences,
isolating the infected, etc., were taken by the Chinese government [37] and greatly reduced the contact rate per unit
time. Therefore, a number of non-linear incidence rates was suggested by researchers. Thus, after studying the cholera
epidemic spread in Bari in 1973, Capasso and Serio[9] introduced the saturated incidence Sg(I) into epidemic models.
This is important because the number of effective contacts between infected individuals and susceptible individuals may
be saturated at high infection levels due to the crowding of infected individuals or due to the protection measures by the
susceptible individuals. To incorporate the
effect of the behavioral changes of the susceptible individuals, Liu et al.[33]
βSI p
,
proposed the general incidence rate 1+kI
q where p and q are positive constants and k is nonnegative. The special cases
∗ Corresponding
author.
E-mail address:[email protected]:
c
Copyright⃝World
Academic Press, World Academic Union
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324
International Journal of Nonlinear Science, Vol.16(2013), No.4, pp. 323-333
when p, q and k take different values have been used by many authors. Many mathematical models, including vaccination,
aim at deciding on
( vaccination strategy and determining changes in a qualitative behavior that could result from such a
control measure Farrington, )2003; Gumel and Moghadas, 2003; Kribs-Zaleta and Velasco-Hernandez, 2000; Shulgin et
al., 1998; Zaman et al., 2008 . Our interest is to investigate the consequences of providing vaccination to the susceptible
population on the SIR dynamics. Through this study, it is possible to develop strategies that manipulate the level of
vaccination. It is found that, by providing the population with appropriate vaccination, it is possible to either reduce or
eradicate the disease. In this paper we use optimal control strategies in the form of vaccination to control the number
of susceptible individuals and increase the number of recovered individuals. Consequences of providing a susceptible
population with vaccination on SIR epidemic model have been receiving much attention by researchers with their main
concern being control and eradication of diseases. With the help of the Maximum Principle of Pontryagin (Morton and
Nancy, 2000) and an iterative method we shall develop a new model. The goal of this work is not to consider a special
disease but to present a method of how to treat this class of optimization problems. This problem is formulated as an
optimal control problem with u (vaccination coverage) as a control variable. The necessary conditions for optimal control
problem are applied to derive properties of the needed optimal strategies. The rest of the paper is organized as follows.
Section 2 describes the SIR model with vaccination, its equilibrium points and their stability analysis. The necessary
conditions for an optimal control and the corresponding states are derived using Pontryagin’s Maximum Principle is
presented in Section 3. In Section 4,we present the numerical method and the simulation results.Finally, the conclusions
are summarized in Section 5.
2
The Epidemic Model with Vaccine Dynamics
The model to be studied takes the following form

αS(t)I(t)

Ṡ(t) = rS(t)(1 − S(t)

k ) − 1+aI(t) − uS(t), S(0) = S0 ≥ 0




˙
I(t)
= αS(t)I(t)
I(0) = I0 ≥ 0
1+aI(t) − (β + γ)I(t),






Ṙ(t) = γI(t) − δR(t) + uS(t),
R(0) = R0 ≥ 0
(1)
The model has a susceptible group designated by S, an infected group I, and a recovered a group R with permanent
immunity. r is the intrinsic growth rate of susceptible, k is the carrying capacity of the susceptible in the absence of
infective, α is the maximum values of percapita reduction rate of S due to I, a is half saturation constants, u is the
vaccination coverage of susceptible, β is the death rate of infected populations, γ is the natural recover rate from infection,
δ is the death rate of recovered populations.
2.1
Stability analysis of SIR model
In this section, we discuss the local stability of the disease-free equilibria and an endemic equilibrium of system (1).
System (1) always has two disease-free equilibria
E0 = (0, 0, 0)
and
E1 = (
uK(r − u)
K(r − u)
, 0,
)
r
rδ
exists if r > u. Further, if
Kα(r − u)
> 1,
r(β + γ)
system (1) admits a unique endemic equilibrium E ∗ = (S ∗ , I ∗ , R∗ ), with
R(u) :=
S∗
=
(β+γ)(1+aI ∗ )
,
α
I∗
=
√
[aKα(r−u)−2ra(β+γ)−α2 K]+ ∆
,
2ra2 (β+γ)
R∗
=
γ ∗
δI
∆
=
[aKα(r − u) − 2ra(β + γ) − α2 K]2 + 4ra2 (β + γ)(Kα(r − u) − r(β + γ)).
+ uδ S ∗ ,
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325
Now let us(start to discuss the local
) behavior of
( the equilibria
) points
E0 , E1 =
K(r−u)
, 0, uK(r−u)
r
rδ
and E ∗ =
S ∗ , I ∗ , R∗
of the system.
Since the first and the second equation of system (1) are independent of R, we consider only these two equations for
stability analysis.
The general variational matrix M for the system (1) is given as follows:
[
]
a11 a12
M=
a21 a22
S
αI
r
αS
αI
where a11 = r(1 − K
) − 1+aI
−u− K
S, a12 = − (1+aI)
2 , a21 = 1+aI , a22 =
The variational matrix M0 corresponding to E0 is given by
[
]
−(r − u)
0
M0 =
0
−(β + γ)
αS
(1+aI)2
− (β + γ).
So, when r < u, becomes stable and in that case disease free equilibrium does not exist.
The variational matrix M1 corresponding to E1 is given by,
[
]
−αK(r−u)
−(r − u)
r
M1 =
αK(r−u)
0
− (β + γ)
r
Here the disease free equilibrium E1 is locally asymptotically stable as long as the reproduction number R(u) < 1
and unstable when R(u) > 1. So, the disease-free equilibrium is asymptotically stable in the absence of the endimic
equilibrium.
The variational matrix M2 corresponding to E ∗ is given by
[
]
∗
αS ∗
− rS
− (1+aI
∗ )2
K
M2 =
αI ∗
αS ∗
1+aI ∗
(1+aI ∗ )2 − (β + γ)
Here, using the fact that
β+γ =
we have
β+γ−
αS ∗
,
1 + aI ∗
αS ∗
αaS ∗ I ∗
=
.
∗
2
(1 + aI )
(1 + aI ∗ )2
Therefore, it’s follow that
det(M2 ) =
[
]
αS ∗
α2 S ∗ I ∗
rS ∗
(β + γ) −
+
>0
K
(1 + aI ∗ )2
(1 + aI ∗ )3
[
]
αS ∗
rS ∗
− (β + γ) −
< 0.
T race(M2 ) = −
K
(1 + aI ∗ )2
and
Hence the endemic equilibrium E ∗ is locally asymptotically stable for R(u) > 1.
In Fig.1and Fig.2, we give some examples of numerical simulation where E ∗ is locally asymptotically stable.
2.2
Vaccine induced reproduction number
Already we have defined the vaccine induced reproduction number
R(u) =
Kα(r − u)
.
r(β + γ)
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(2)
326
International Journal of Nonlinear Science, Vol.16(2013), No.4, pp. 323-333
S(t)
I(t)
2
4
6
8
10
Time (days)
12
14
16
18
20
2
Figure 1: The plot represents the populations of
susceptible and infected individuals with respect
to ’time’, where the parameters values are α =
1.49, a = 2.3, β = 0.031, γ = 0.611, K = 100,
r = 2.5 and u = 0.1.
4
6
8
10
Time (days)
12
14
16
18
20
Figure 2: The plot represents the populations of
susceptible and infected individuals with respect
to ’time’, where u = 0.9 and the other parameter
values are same as before.
−Kα
Now, R′ (u) = r(β+γ)
< 0, and hence R(u) is a decreasing function of u. This shows the impact of vaccination in
reducing the vaccine induced reproduction number. Also in the absence of vaccination i.e when, u = 0,
R(0) = R0 =
Kα
(β + γ)
(3)
From (2) it is clear that the introduction of vaccination implies R(u) ≤ R0 for 0 < u < r and consequently, if R0 < 1
then R(u) < 1. Therefore, E1 i.e. the disease-free equilibrium is asymptotically stable if R(u) < 1. Also for R0 =
1, R(u) ≤ 1 and in that case the disease-free equilibrium is asymptotically stable.
3
The Optimal Vaccination
Optimal control techniques are of great use in developing optimal strategies to control various kinds of diseases. To solve
the challenges of obtaining an optimal vaccination strategy, we use optimal control theory.
We consider the control variable u(t) ∈ Uad to be the percentage of susceptible individuals being vaccinated per unit of
time. Here
Uad = {u|u(t) is measurable, 0 ≤ u(t) ≤ umax < ∞, t ∈ [0, tend ]}
indicates an admissible control set. Now, we consider an optimal control problem to minimize the objective functional
∫ tend
1
J(u) =
[A1 S(t) + A2 I(t) + τ u2 (t)]dt
(4)
2
0
subject to system (1). Here A1 and A2 are positive constants to keep a balance in the size of S(t) and I(t), respectively.
The square of the control variable reflects the severity of the side effects of the vaccination. In the objective functional,
τ is a positive weight parameter which is associated with the control u(t). The objective of our work is to minimize the
infected and susceptible individuals and to maximize the total number of recovered individual by using possible minimal
control variables u(t).
3.1
Existence of an Optimal Control
For existence, we consider a control system (1) with initial conditions. Then, we rewrite our system (1) in the following
form:
ϕt = Bϕ + F (ϕ)
(5)
where


S(t)
ϕ =  I(t) 
R(t)
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327


r − u(t)
0
0
0
−(β + γ) 0 
B=
u(t)
γ
−δ


2
− rSk(t) − αS(t)I(t)
1+aI(t)


F (ϕ) =  αS(t)I(t)

1+aI(t)
0
and ϕt denote derivative of ϕ with respect to time t. Equation (5) is a non-linear system with a bounded coefficient.
We set
D(ϕ) = Bϕ + F (ϕ)
(6)
Now,
F (ϕ1 ) − F (ϕ2 )
=
)
 (
r
2 (t)I2 (t)
2
2
S
(t)
−
S
(t)
+ αS
1
1+aI2 (t) −
k 2

αS1 (t)I1 (t)

− αS2 (t)I2 (t)
1+aI1 (t)

αS1 (t)I1 (t)
1+aI1 (t) 


1+aI2 (t)
0
Therefore
(
)
F (ϕ1 ) − F (ϕ2 ) = r S22 (t) − S12 (t) + αS2 (t)I2 (t) − αS1 (t)I1 (t) + αS1 (t)I1 (t) −
1+aI2 (t)
1+aI1 (t) k
1+aI1 (t)
(
)
1 (t)I1 (t) 2 (t)I2 (t)
≤ kr S22 (t) − S12 (t) + 2α S1+aI
− S1+aI
2 (t)
1 (t) (
)
(t)I1 (t)I2 (t)−S
)( 1 (t)I1 (t)−aS
) 1 (t)I1 (t)I2 (t) ≤ kr S22 (t) − S12 (t) + 2α S2 (t)I2 (t)+aS2 (
1+aI1 (t)
≤
≤
≤
≤
≤
≤
≤
≤
≤
αS2 (t)I2 (t) 1+aI2 (t) 1+aI2 (t)
(
)
(
)
2
r
2
(t)
(t)
−
S
+
2α
S
S
(t)I
(t)
−
S
(t)I
(t)
+
aI
(t)I
(t)
S
(t)
−
S
(t)
2
1
1
1
2
2
1
1
2
k
2
(
)
(
)
r
2
2
(t)
(t)
−
S
+
2α
S
S
(t)I
(t)
−
S
(t)I
(t)
+
S
(t)I
(t)
−
S
(t)I
(t)
+
aI
(t)I
(t)
S
(t)
−
S
(t)
2
1
2
1
2
1
1
1
2
2
1
1
2
k
2
(
)
(
)
(
)
(
)
r
2
2
k S2 (t) − S1 (t) + 2αI2 (t) S2 (t) − S1 (t) + S1 (t) I2 (t) − I1 (t) + aI1 (t)I2 (t) S2 (t) − S1 (t) r
k S2 (t) − S1 (t)S2 (t) + S1 (t) + 2αS2 (t) − S1 (t)I2 (t) + aI1 (t)I2 (t) + 2αS1 (t)I2 (t) − I1 (t)
)
( S2 (t) − S1 (t) r S2 (t) + S1 (t) + 2αI2 (t) + aI1 (t)I2 (t) + 2αS1 (t)I2 (t) − I1 (t)
k
)
))
( (
(
S2 (t) − S1 (t) r S2 (t) + S1 (t) + 2α I2 (t) + aI1 (t)I2 (t)
+ 2αS1 (t)I2 (t) − I1 (t)
k (
(
))
µ
µ2
µ r
S1 (t) − S2 (t) + 2α µ (I1 (t) − I2 (t)(SeeAppendix)
2 m k + 2α m + a m2
m
[(
]
(
))
µ
µ
r
S1 (t) − S2 (t) + αI1 (t) − I2 (t)
2m
k + α 1 + am
)
(
M S1 (t) − S2 (t) + I1 (t) − I2 (t) ,
where
µ
M =2
m
Also, we get
[(
(
))]
r
µ
+α 1+a
k
m
(
)
D(ϕ1 ) − D(ϕ2 ) ≤ V ϕ1 − ϕ2 , where V = max M, ∥B∥ < ∞.
Thus, it follows that the function D is uniformly Lipschitz continuous. From the definition of the control u(t) and the
restriction on S(t), I(t) and R(t) ≥ 0, we see that a solution of the system (5) exists (Birkhoff and Rota,1989).
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328
International Journal of Nonlinear Science, Vol.16(2013), No.4, pp. 323-333
Let us go back to the optimal control problem, (1) and (4). In order to find an optimal solution, first we find the
Lagrangian and Hamiltonian for the optimal control problem (1) and (4). In fact, the Lagrangian of the optimal problem
is given by L(S, I, u) = A1 S(t) + A2 I(t) + 12 τ u2 (t).
We seek the minimal value of the Lagrangian. To accomplish this, we define the Hamiltonian H for the control
problem:
dS(t)
dI(t)
dR(t)
H(S, I, R, u, λ1 , λ2 , λ3 , t) = L(S, I, u) + λ1 (t)
+ λ2 (t)
+ λ3 (t)
,
(7)
dt
dt
dt
where λ1 (t), λ2 (t)and λ3 (t) are the adjoint functions to be determined suitably.
Theorem 1 There exists an optimal control u∗ (t) such that
J(u∗ (t)) = min J(u(t))
u∈Uad
subject to the control system (1) with initial conditions.
Proof. To prove the existence of an optimal control we use the result in (Lukes, 1982). Note that the control and the state
variables are nonnegative values. In this minimizing problem, the necessary convexity of the objective functional in u(t)
is satisfied.
The control space
Uad = {u|u(t) is measurable, 0 ≤ u(t) ≤ umax < ∞, t ∈ [0, tend ]}
is also convex and closed by definition. The optimal system is bounded which determines the compactness needed for the
existence of the optimal control. In addition, the integrand in the functional (4), A1 S(t) + A2 I(t) + 21 τ u2 (t) is convex
on the control u(t). Also, we can easily see that, there exist a constant ρ > 1, positive numbers ω1 and ω2 such that
ρ
J(u(t)) ≥ ω2 + ω1 (|u|2 ) 2 . We conclude that there exists an optimal control.
3.2
Characterization of the optimal control
In the previous section we showed the existence of an optimal control which maximize the functional (4) subject to system
(1). In order to derive the necessary conditions for this optimal control, we apply Pontryagin’s maximum principle to the
Hamiltonian
H(t, x(t), u(t), λ(t)) = f (t, x(t), u(t)) + λ(t)g(t, x(t), u(t)).
(8)
If (x∗ (t), u∗ (t)) is an optimal solution of an optimal control problem, then there exists a non-trivial vector function
λ(t) = (λ1 (t), λ2 (t), ..., λn (t)) satisfying the following equalities:
′
x (t) =
∂H(t,x∗ (t),u∗ (t),λ(t))
,0
∂λ
=
′
∂H(t,x∗ (t),u∗ (t),λ(t))
, λ (t)
∂u
∗
= − ∂H(t,x
(t),u∗ (t),λ(t))
,
∂x
′
where “ “ denotes the derivative with respect to time t. Which gives after derivation
 ∗
u (t) = 0
if ∂H

∂u < 0




u∗ (t) ∈ [0, umax ] if ∂H
∂u = 0




 ∗
u (t) = umax
if ∂H
∂u > 0
Now, we apply the necessary conditions to the Hamiltonian H (7).
Theorem 2 Let S ∗ (t), I ∗ (t) and R∗ (t) be optimal state solutions with associated optimal control variable u∗ (t) for the
optimal control problem (1) and (4). Then, there exist adjoint variables λ1 , λ2 and λ3 that satisfy
(
)
(
)

(
)

∗
∗
∗
dλ
(t)

αI
αI
1
∗

= −A1 − λ1 (t) r 1 − 2Sk − 1+aI
− λ2 (t) 1+aI
− λ3 (t)u∗ (t)

∗ − u (t)
∗
dt



)
)
(
(
(
)
∗
∗
dλ
(t)
αS
αS
2

− λ3 (t)γ
= −A2 + λ1 (t) (1+aI ∗ )2 − λ2 (t) (1+aI ∗ )2 − β + γ

dt




 dλ3 (t)
= λ3 (t)δ
dt
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329
with transversality conditions
λi (tend ) = 0, i = 1, 2, 3.
(10)
Furthermore, the optimal control u∗ (t) is given by
(
(
) )
(λ1 (t) − λ3 (t))S ∗ (t)
u (t) = max min
, umax , 0
τ
∗
(11)
Proof. We use the Hamiltonian (7) in order to determine the adjoint equations and the transversality conditions. From
setting S(t) = S ∗ (t), I(t) = I ∗ (t) and R(t) = R∗ (t), and differentiating the Hamiltonian with respect to S, I and R,
respectively, we obtain (2). And by using the optimality conditions we find
∂H
= τ u∗ (t) − λ1 (t)S ∗ + λ3 (t)S ∗ = 0, at u = u∗ (t)
∂u
which gives
(
u∗ (t) =
)
λ1 (t) − λ3 (t) S ∗ (t)
τ
Using the property of the control space, we obtain

(λ1 (t)−λ3 (t)S ∗ (t)
∗

≤0
 u (t) = 0, if
τ
∗
(λ1 (t)−λ3 (t)S ∗ (t)
∗
u (t) =
, if 0 < λ1 (t)−λτ3 (t)S (t) < umax
τ

∗
 ∗
u (t) = umax , if λ1 (t)−λτ3 (t)S (t) ≥ umax
So the optimal control is characterized as
(
∗
u (t) = max
)
((
) )
λ1 (t) − λ3 (t) S ∗ (t)
min
, umax , 0 .
τ
The optimal control and the state are found by solving the optimality system, which consists of the state system (1),
the adjoint system (2), initial conditions at t = 0, boundary conditions (10), and the characterization of the optimal control
(11).
So the optimality system is given by

























(
∗
Ṡ (t)
=
rS (t)(1 −
I˙∗ (t)
=
αS ∗ (t)I ∗ (t)
1+aI ∗ (t)
∗
Ṙ (t) =
λ1˙(t)



















λ2˙(t)




 ˙
λ3 (t)
=
∗
∗
S ∗ (t)
k )
−
αS ∗ (t)I ∗ (t)
1+aI ∗ (t)
− (β + γ)I ∗ (t),
(
− max
((
∗
((
min
)
)
λ1 (t)−λ3 (t) S ∗ (t)
, umax
τ
λ1 (t)−λ3 (t) S ∗ (t)
, umax
τ
) )
, 0 S ∗ (t),
) )
, 0 S ∗ (t),
γI (t) − δR (t) + max min
(
(
)
((
) ))
(
λ1 (t)−λ3 (t) S ∗ (t)
2S ∗ (t) )
αI ∗ (t)
−A1 − λ1 (t) r 1 − k
, umax , 0
− 1+aI ∗ (t) − max min
τ
(
)
(
)
)
((
)
−λ2 (t)
αI ∗ (t)
1+aI ∗ (t)
(
=
−A2 + λ1 (t)
=
λ3 (t)δ
− λ3 (t) max
)
αS ∗ (t)
(1+aI ∗ (t))2
min
(
− λ2 (t)
λ1 (t)−λ3 (t) S ∗ (t)
, umax
τ
αS ∗ (t)
(1+aI ∗ (t))2
(
)
− β+γ
,0
)
− λ3 (t)γ
with λ1 (tend ) = 0, λ2 (tend ) = 0, λ3 (tend ) = 0, S(0) = S0 , I(0) = I0 , R(0) = R0
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(12)
330
International Journal of Nonlinear Science, Vol.16(2013), No.4, pp. 323-333
4
Numerical illustration
4.1
The improved GSS1 method
The resolution( of the optimality system
is created improving the Gauss Seidel − like implicit finite-difference method
)
developed by Gumel and al. 2001 and denoted GSS1 method. It consists on discretizing the interval [t0 , tend ] by the
points tk = kl + t0 (k = 0, 1, ..., n), where l is the time step. Next, we define the state and adjoint variables S(t), I(t),
R(t) , λ1 (t), λ2 (t), λ3 (t) and the control u(t) in terms of nodal points S k , I k , Rk , λk1 , λk2 , λk3 and uk with S 0 , I 0 , R0 ,
λ01 , λ02 , λ03 and u0 as the state and adjoint variables and the controls at initial time t0 . S n , I n , Rn , λn1 , λn2 , λn3 and un as
the state and adjoint variables and the controls at final time tend .
As it is well known, the approximation of the time derivative by its first-order forward-difference is given, for the first
S(t+l)−S(t)
.
state variable S(t), by dS(t)
dt = lim
l
l→0
We use the scheme developed by Gumel et al. to adapt it to our case as following:
S k+1 − S k
l
I k+1 − I k
l
Rk+1 − Rk
l
(
)
S k+1
αS k+1 I k
= rS k+1 1 −
−
− uk S k+1
k
1 + aI k
=
(13)
αS k+1 I k+1
− (β + γ)I k+1
1 + aI k+1
(14)
= γI k+1 − δRk+1 + uk S k+1
(15)
By applying an analogous technology, we approximate the time derivative of the adjoint variables by their first-order
backward-difference and we use the appropriated scheme as follows:
λn−k
− λn−k−1
1
1
l
=
λn−k
− λn−k−1
2
2
l
=
λn−k
− λn−k−1
3
3
l
=
4.2
)
)
(
(
(
αI k+1
2S k+1 )
αI k+1
n−k
k
−
−A1 − λn−k−1
−
u
−
λ
− λn−k
uk (16)
r
1
−
1
2
3
k
1 + aI k+1
1 + aI k+1
)
)
(
(
αS k+1
αS k+1
n−k−1
n−k−1
−A2 + λ1
− λ2
− (β + γ) − λn−k
(17)
γ
3
(1 + aI k+1 )2
(1 + aI k+1 )2
λn−k−1
δ
3
(18)
Numerical results
We use the following data:
S0 = 50; I0 = 35; R0 = 15; a = 2.3; α = 1.49; τ = 50; r = 2.5
A1 = 0.1; A2 = 0.5; γ = 0.611; β = 0.031; δ = 0.01; k = 100
We obtain the following figures
75
75
S without control
S with control
65
65
60
60
55
55
50
50
45
45
40
40
35
0
20
40
60
Time (days)
80
100
I without control
I with control
70
I(t)
S(t)
70
35
0
20
40
60
80
100
Time (days)
Figure 3: The plot represents the susceptible popula- Figure 4: The plot represents the infected populations
tions both with control and without control.
both with control and without control.
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Hassan Laarabi, Abdelhadi Abta.et.al:Stability Analysis and Optimal Vaccination Strategies for · · ·
4000
331
1
R without control
R with control
3500
0.9
0.8
3000
0.7
Control u
R(t)
2500
2000
0.6
0.5
0.4
1500
0.3
1000
0.2
500
0
0.1
0
20
40
60
80
100
0
0
20
40
Time (days)
60
80
100
Time (days)
Figure 5: The plot represents the recovered populations Figure 6: The Plot represents the control variable u(t)
both with control and without control.
with respect to time.
When viewing the graphs, remember that each of the individuals without control is marked by dashed black lines.
The control individuals are marked by blue lines. The recovered individuals, with control, increase rapidly while the
susceptible and infected individuals, with control, rapidly decrease. This is due to the vaccination strategy.
5
Conclusion
In this paper we do not consider any special disease but our aim is to set up an optimal control problem relative to epidemic
model with saturated incidence rate and saturated treatment function, so it is to minimize the infected and susceptible
populations and to maximize recovered populations.
A comparison between optimal control and no control is presented. It is easy to see that the optimal vaccination
is much more effective for reducing the number of infected and susceptible individuals and increasing the number of
recovered individuals. In order to illustrate the overall picture of the epidemic, the numbers of infected, susceptible and
recovered individuals under the optimal control and no control are shown in figures.
Appendix
Let us consider the function N (t) = S(t) + I(t) + R(t). Now the time derivative of N(t) along solution of (1) is
(
)
dN (t)
dS(t) dI(t) dR(t)
S(t)
=
+
+
= rS(t) 1 −
− δR(t) − βI(t)
dt
dt
dt
dt
k
Therefore,
[ (
]
)
dN (t)
S(t)
+ cN (t) = r 1 −
+ c S(t) + (c − δ)R(t) + (c − β)I(t).
dt
k
Let m = min(β, δ) and c = m
therefore
[ (
]
)
dN (t)
S(t)
+ mN (t) ≤ r 1 −
+ m S(t).
dt
k
or,
k(r + m)2
dN (t)
+ mN (t) ≤ µ, where µ =
.
dt
4r
Applying the theory of differential inequality (Birkhoff and Rota, 1982) we obtain
µ
0 < N (t) < (1 − e−mt ) + N (0)e−mt
m
µ
Which, upon letting t → +∞, yield 0 < N (t) < m
So, we have that all the solutions of the systems (1) that starts in R3+ are confined to the region
µ
Ω = {(S(t), I(t), R(t)) ∈ R3+ /S(t) + I(t) + R(t) ≤ }
m
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International Journal of Nonlinear Science, Vol.16(2013), No.4, pp. 323-333
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