Acta Mathematicae Applicatae Sinica, English Series
Vol. 20, No. 1 (2004) 25–36
Analysis of an Age Structured SEIRS Epidemic Model
with Varying Total Population Size and Vaccination
Xue-Zhi Li1 , Geni Gupur2 , Guang-Tian Zhu3
1 Department
of Mathematics, Xinyang Teachers College, Henan 464000, China (E-mail: [email protected])
2 Department
3 Academy
of Mathematics, Xinjiang University, Urumqi 830046, China
of Mathematics and Systems Science, Chinese Academy of Sciences, Beijing 100080, China
Abstract
This article focuses on the study of an age structured SEIRS epidemic model with a vaccination
program when the total population size is not kept at constant. We first give the explicit expression of the
λ) in the presence of vaccine (
λ is the exponent of growth of total population), and
reproduction number R(ψ, λ) < 1 and unstable if R(ψ, λ) > 1, then we
show that the infection-free steady state is linearly stable if R(ψ, apply the theoretical results to vaccination policies to determine the optimal age or ages at which an individual
should be vaccinated. It is shown that the optimal strategy can be either one- or two-age strategies.
Keywords
age-structured SEIRS epidemic model, vaccination; varying total population size, reproduction
number, stability, optimal vaccination strategies
2000 MR Subject Classification
1
92D30, 35L60
Introduction
Age-structured in epidemic model has been considered by many authors, because of the recognition that transmission dynamics of certain diseases can not be correctly described by the
traditional epidemic models with no age-dependence. For example, Busenberg et al [2,3], EIDoma [5], Greenhalgh [6,7], Inaba [11], Webb [15], Cha [4], etc. gave complete analyses for
fairly general SIS or SIR models with age-structure. (Here and in the following, the letters S,
E, I, R, stand for susceptible, exposed, infectious, removed, respectively). These works were
mainly concerned with finding threshold conditions for the disease to become endemic and describing the stability of steady-state solutions, often under the assumption that the population
has reached its steady state and the diseases do not affect the death rate of the population. In
fact, there are significant effects that occur when the total population is increasing, as is often
the case in underdeveloped countries and it is also inadequate to assume the population with
a steady state for disease that has a fairly rapid spread, most animal species in our world can
not satisfy this restriction. Therefore, it is necessary and also it is of practical significance to
study age structured epidemic models for time dependent population.
Studies of epidemic models that incorporate disease caused death and varying total population have become one of the important areas in the mathematical theory of epidemiology, and
they have largely been inspired by the works of Anderson and May [1, 13]. These types of models that most authors discussed are age-independent (see [12]). Researches on age-structured
Manuscript received May 9, 2001. Revised July 24, 2003.
Supported by the NSFC (No. 10371105) and the NSF of Henan Province (No. 0312002000; No. 0211044800).
X.Z. Li, G. Gupur, G.T. Zhu
26
epidemic models of SEIR or SEIRS type with varying total population size and disease-induced
death rate are scarce in the literature. This paper focuses on the study of an age-structured
SEIRS epidemic model with a vaccination program in varying total population size. We give
in the situation of varying population size
the expression of the reproductive number R(ψ, λ)
is the exponent of growth of the total
and vaccination (here ψ(a) is the vaccination rate and λ
< 1 and
population), and show that the infection-free steady state is linearly stable if R(ψ, λ)
> 1. We use the results on the dynamics of our SEIRS model to study
unstable if R(ψ, λ)
the role of vaccination on the epidemiological age-structure of a population. We consider two
optimization problem (see [9,10]): reducing R(ψ) below a certain level R∗ at minimal costs or
minimizing the reproductive number R(ψ) with a fixed resources. Following the approach used
by Hadeler and Müller [10], we show that the optimal strategies for the two problems above
have the form ‘vaccinate either at a exactly one age or at exactly two ages’.
This paper is organized as follows. Section 2 introduces an age-structured SEIRS epidemic model with vaccination in varying total population size. The reproductive numbers
R(ψ) and R0 are computed in Section 3, the linear stability of the uninfected state is
R(ψ, λ),
also studied in this section. In Section 4, we apply our results to vaccination policies and study
two optimization problems outline above.
2
The Model
This section describes the basic model we are going to analyze in this paper. This model divides
the population into classes containing susceptible, exposed, infectious and immune individuals,
(i.e., it takes into account the fact that after an individual catches the disease there is a period
during which he incubates the disease but not spread it), and describes the spread of the disease
between these classes using partial differential equations.
Let S(a, t), E(a, t), I(a, t), R(a, t) and N (a, t) denote the densities with respect to age of
the number of susceptible, exposed, infected, immune (or vaccinated) and the total number of
individuals of age a at time t, respectively. This means, for example, that the total number of
susceptible individuals between ages A1 and A2 at time t is
A2
S(a, t)da.
A1
Let µ(a) denote the age dependent mortality; b(a), the age dependent fertility; µ
(a), the mortality of infected; b(a), the fertility of infected; η(a), the differential mortality (= µ
(a) − µ(a));
α−1 (a), the average incubation period of the disease; δ(a), the recovery rate (i.e., δ −1 (a), the
average infectious period of the disease (around 1 week for measles)); γ(a), the loss of immunity
and k(a), the contact distribution.
In the present treatment of the vaccination problem the transmission coefficient is assumed
as a product of the infectivity depending on the ages of the infected present in the population
and the susceptibility depending on the age of the susceptible exposed to the infectivity.
Moreover, let β(a) denote the susceptibility of susceptibles; β(a),
the susceptibility of vacci
nated (or immune), we assume that β(a) ≤ 0 and β(a) ≤ 0; ∆(a), the differential susceptibility
(= β(a) − β(a));
ψ(a), the age dependent vaccination rate; rm , the largest age attained by the
population. The special case rm = ∞ is formally used in many demographic models.
We do not introduce a separate parameter for vaccination at birth since in practice this
effect can be seen as vaccination with high probability at a very early age.
We assume that all newborns are susceptible and that an individual may become exposed
only through contact with infectious individuals, we also assume that vaccination is partially
effective (i.e., vaccinated (or recovered) individuals can become exposed or susceptible again),
Analysis of an Age Structured SEIRS Epidemic Model
27
that only susceptible will be vaccinated, and that the disease-induced death rate can not be
neglected. Similarly to [9], we can get that the joint dynamics of the age-structured epidemiological classes are governed by the following boundary value problem which consists of four
partial differential equations:
∂S(a, t) ∂S(a, t)
+
∂t
∂a
V (t)
,
= − µ(a) + ψ(a) S(a, t) + γ(a)R(a, t) − β(a)S(a, t)
N (t)
∂E(a, t) ∂E(a, t)
+
∂t
∂a
V (t)
,
= − µ(a) + α(a) E(a, t) + β(a)S(a, t) + β(a)R(a,
t)
N (t)
∂I(a, t) ∂I(a, t)
+
=− µ
(a) + δ(a) I(a, t) + α(a)E(a, t),
∂t
∂a
∂R(a, t) ∂R(a, t)
+
∂t
∂a
V (t)
,
= − µ(a) + γ(a) R(a, t) + ψ(a)S(a, t) + δ(a)I(a, t) − β(a)R(a,
t)
N (t)
S(0, t) =
0
rm
b(a) S(a, t) + E(a, t) + R(a, t) + b(a)I(a, t) da,
E(0, t) = I(0, t) = R(0, t) = 0.
Here
N (t) =
0
rm
(2.1)
(2.2)
(2.3)
(2.4)
(2.5)
(2.6)
S(a, t) + E(a, t) + I(a, t) + R(a, t) da,
(2.7)
is the total population size and
V (t) =
0
rm
k(a)I(a, t)da
(2.8)
is the number of contacts with infecteds of one susceptible, here assumed to be independent of
the age of the susceptible.
For notational convenience we introduce
a
P (a) = exp −
µ(s)ds .
(2.9)
0
The system (2.1)–(2.8) is homogeneous of degree 1. Hence the interesting objects are persistent
solution, i.e., exponential solution with constant population structure.
a quantity that must exceed
In the next section, we derive explicit expressions for R(ψ, λ),
the net reproductive
one for the disease to remain endemic (persist). In general, we call R(ψ, λ)
number, which measures the expected number of secondary infection produced by a “typical”
infected individual during its entire-death adjusted-period of infectiousness in an uninfected
population.
X.Z. Li, G. Gupur, G.T. Zhu
28
3 The Reproduction Numbers and the Stability of Uninfected State
First we consider the uninfected and non-vaccinated population which is governed by the equations (Mckendrick model [15]).
∂S(a, t) ∂S(a, t)
+
+ µ(a)S(a, t) = 0,
∂t
∂a
S(0, t) =
(3.1)
rm
b(a)S(a, t)da.
0
(3.2)
The persistent solution is obtained from the eigenvalue problem
dS(a)
= −µ(a)S(a),
da
λS(a) +
S(0) =
as
where
(3.3)
rm
0
b(a)S(a)da,
(3.4)
exp{λt},
S(a, t) = S(a)
(3.5)
S(a)
= P (a) exp{−λa},
(3.6)
is obtained from the characteristic equation
is the stable age distribution, and the exponent λ
rm
λa
b(a)P (a)e−
da = 1.
(3.7)
0
Thus the exponential solution of the system (2.1)–(2.8) describing a totally susceptible population is
λt
.
(3.8)
(S(a),
0, 0, 0)T e
Next consider the persistent solution describing a uninfected vaccinated state. It has the form
S(a), E(a), I(a), R(a)
T eλt ,
(3.9)
where
· D(a),
S(a) = P (a) exp{−λa}
(3.10)
E(a) = 0,
(3.11)
I(a) = 0,
(3.12)
− S(a) = P (a) exp{−λa}(1
R(a) = P (a) exp{−λa}
− D(a)),
(3.13)
N=
0
rm
(S(a) + R(a))da,
(3.14)
Analysis of an Age Structured SEIRS Epidemic Model
and
D(a) = 1 −
0
a
ψ(σ) exp
29
−
a
(γ(τ ) + ψ(τ ))dτ dσ.
(3.15)
σ
From (3.15) one sees that S(a, t) is an increasing function of γ(a) and a decreasing function of
ψ, as it should be according to its biological interpretation.
In order to investigate the effects of vaccination strategies on the uninfected solution, one
could try to compute the infected solution explicitly. This would provide information about
the stationary prevalence in the presence of vaccination. However, the rather complicated
calculations do not give much insight. Therefore we linearize at the vaccinated state and
determine the threshold condition as well as the reproduction number.
The following steps are applied to obtain the reproduction number. First linearize the
system (2.1–2.8) at the vaccinated state (3.9), then integrate over age to obtain an implicit
equation for V .
To get the linearized system, we need to translate the persistent solution (3.10-3.13) to the
λt
λt
origin, thus we let S(a, t) = S(a)e
+ U (a, t), R(a, t) = R(a)e
+ W (a, t). From (2.1-2.8),
the linearized system reads
∂U (a, t) ∂U (a, t)
+
∂t
∂a
V (t)
= − µ(a) + ψ(a) U (a, t) + γ(a)W (a, t) − β(a)S(a)
,
N
∂E(a, t) ∂E(a, t)
+
∂t
∂a
V (t)
= − µ(a) + α(a) E(a, t) + β(a)S(a) + β(a)R(a)
,
N
∂I(a, t) ∂I(a, t)
+
=− µ
(a) + δ(a) I(a, t) + α(a)E(a, t),
∂t
∂a
∂W (a, t) ∂W (a, t)
+
∂t
∂a
V (t)
= − µ(a) + γ(a) W (a, t) + ψ(a)U (a, t) + δ(a)I(a, t) − β(a)R(a)
,
N
U (0, t) =
rm
0
b(a) U (a, t) + E(a, t) + W (a, t) + b(a)I(a, t) da,
E(0, t) = I(0, t) = W (0, t) = 0.
(3.16)
(3.17)
(3.18)
(3.19)
(3.20)
(3.21)
We look for exponential solutions to this problem, i.e., solutions of the form
U (a, t) = s(a)eλt ,
E(a, t) = e(a)eλt ,
I(a, t) = i(a)eλt ,
W (a, t) = r(a)eλt .
It turns out that the problem separates. The most important equations are the equations for
X.Z. Li, G. Gupur, G.T. Zhu
30
e(a) and i(a) which can be written as
V
d
e(a) = − µ(a) + α(a) + λ e(a) + β(a)S(a) + β(a)R(a)
,
da
N
(3.22)
d
i(a) = − µ
(a) + δ(a) + λ i(a) + α(a)e(a),
da
(3.23)
where
V =
rm
k(a)i(a)da.
0
(3.24)
The equation (3.23) is the key relation for the discussion of stability. From this equation we
derive a characteristic equation. For the moment we assume that V is known. Solving the
equation (3.22) and using the boundary condition e(0) = 0, we obtain an expression for e(a)
depending on V .
e(a) =
a
0
β(σ)S(σ) + β(σ)R(σ)
exp −
a
σ
V
(λ + µ(τ ) + α(τ ))dτ dσ .
N
(3.25)
Integrating Eq. (3.23) and using the boundary condition i(0) = 0, we obtain an expression for
i(a):
a
a
i(a) =
λ+µ
(τ ) + δ(τ ) dτ dσ.
(3.26)
α(σ)e(σ) exp −
0
σ
Substituting (3.25) into (3.26), we have
i(a) =
a
α(σ)
0
· exp
σ
0
−
exp −
β(ξ)S(ξ) + β(ξ)R(ξ)
V
λ+µ
(τ ) + δ(τ ) dτ dσ .
N
This equation can be more conveniently written as
a
a
β(ξ)S(ξ) + β(ξ)R(ξ)
α(σ) exp −
i(a) =
· exp
−
λ + µ(τ ) + α(τ ) dτ dξ
ξ
a
σ
0
σ
ξ
a
λ + µ(τ ) + α(τ ) dτ
ξ
σ
λ+µ
(τ ) + δ(τ ) dτ dσdξ
σ
V
.
N
(3.27)
Since V is nothing other than the weighted integral of i(a), we can multiply (3.27) from both
sides by k(a), integrate with respect to a from 0 to rm , and then divide both sides by V , we
obtain a characteristic equation for the eigenvalue λ,
Θ(λ) = 1,
(3.28)
where
rm
Θ(λ) =
0
· exp
k(a)
−
a
0
a
σ
β(ξ)S(ξ) + β(ξ)R(ξ)
a
α(σ) exp
ξ
1
λ+µ
(τ ) + δ(τ ) dτ dσdξda .
N
σ
−
λ + µ(τ ) + α(τ ) dτ
ξ
(3.29)
Analysis of an Age Structured SEIRS Epidemic Model
31
Interchanging the integral order in (3.29) and using the explicit expression for N from (3.14)
and the relations (3.10), (3.13) and (3.15), we obtain
rm
Θ(λ) =
0
λξ
β(ξ)D(ξ) + β(ξ)(1
− D(ξ)) P (ξ)e−
rm
ξ
· exp
−
σ
a
a
k(a)
α(σ)
ξ
λ + µ(τ ) + α(τ ) dτ
ξ
· exp
−
λ+µ
(τ ) + δ(τ ) dτ dσdadξ
σ
·
rm
0
−1
P (a) exp(−λa)da
.
(3.30)
is already determined by the demographic parameters and
Notice that the eigenvalue λ
equation (3.7).
Since 0 ≤ D(s) ≤ 1, the kernel in (3.30) is nonnegative for real λ. As a function for real
λ, it is strictly decreasing. It goes to zero for λ going to ∞, and it goes to ∞ for λ goes to
−∞. Therefore the characteristic equation (3.28) has a unique real root λ0 . Let λ = x + iy be
another root. From
1 = Θ(λ) = Θ(x + iy) ≤ Θ(x),
it follows that Reλ ≤ λ0 . Hence λ0 is the leading root. From the general theory of homogeneous
equations (sees Hadeler [14] for a review of the finite dimensional case) it follows that λ0 has
The persistent solution is linearly stable if
to be compared with the exponent of growth λ.
(with λ = λ,
multiplicities counted) have negative real parts. Comparing
the differences λ − λ
instead of 0 can be understood as taking into account the ’dilution’ of the
the real parts to λ
infected in the growing population. Therefore we have the following stability criterion.
and it is linearly
Theorem 1. The uninfected state (3.10)–(3.13) is linearly stable if λ0 < λ
unstable if λ0 > λ.
Note that the right hand side of (3.30) can also be written in the form
rm
Θ(λ) =
0
β(ξ)D(ξ) + β(ξ)
1 − D(ξ) P (ξ)
rm
λ
λ)
k(a)e−a
· e−(a−ξ)(λ−
ξ
·
a
α(σ) exp
ξ
·
−
σ
µ(τ ) + α(τ ) dτ · exp −
ξ
0
rm
−1
P (a) exp(−λa)da
.
a
µ
(τ ) + δ(τ ) dτ dσdadξ
σ
(3.31)
X.Z. Li, G. Gupur, G.T. Zhu
32
We define
= Θ(λ)
=
R(ψ, λ)
rm
0
β(ξ)D(ξ) + β(ξ)(1
− D(ξ)) P (ξ)
rm
−
σ
µ(τ ) + α(τ ) dτ · exp −
ξ
rm
0
a
α(σ)
ξ
ξ
· exp
·
λ
k(a)e−a
a
µ
(τ ) + δ(τ ) dτ dσdadξ
σ
−1
da
P (a) exp − λa
.
(3.32)
< 1, i.e.,
According to the monotonicity of Θ(λ) on real λ, it is easy to see that if R(ψ, λ)
then λ0 < λ;
If R(ψ, λ)
> 1 , i.e., Θ(λ0 ) = 1 < Θ(λ),
then λ0 > λ.
Thus by
Θ(λ0 ) = 1 > Θ(λ),
Theorem 1 we get
< 1 and it is
Theorem 2. The uninfected state (3.10)–(3.13) is linearly stable if R(ψ, λ)
> 1.
linearly unstable if R(ψ, λ)
The number R(ψ, λ) is called the the reproductive number in the case of varying total
we find that the growth
population size and vaccination. From the expression (3.32) of R(ψ, λ)
of the total population size affects the reproductive number.
exponent λ
From the explicit formulas (3.30)–(3.31), we observe the following features (which are biologically evident but lend some credibility to the model).
increases;
λ0 increases if one of the following cases holds: (1) k(a) increases; (2) β(a) or β(a)
(3) γ(a) increases; (4) α(a) increases.
(a) increases;
λ0 decreases if one of the following cases holds: (1) δ(a) increases; (2) µ
(3) ψ(a) increases (this is the most important case).
Notice that the coefficient b(a) does not influence λ0 . In fact, it should be emphasized that
λ0 is not directly related to the exponent of exponential growth of an infected solution.
= 0. This assumption says that
Now we consider the case of constant population size λ
in the uninfected state there is balanced population growth. The uninfected population is in
static equilibrium,
rm
b(a)P (a)da = 1.
0
Thus zero is the critical value for λ0 . Hence the critical quantity is
R(ψ) := R(ψ, 0) =
rm
0
· exp
·
β(ξ)D(ξ) + β(ξ)(1
− D(ξ)) P (ξ)
0
−
ξ
rm
σ
µ(τ ) + α(τ ) dτ exp −
−1
P (a)da
.
rm
a
k(a)
α(σ)
ξ
ξ
a
µ
(τ ) + δ(τ ) dτ dσdadξ
σ
(3.33)
In the case of constant population size, we define the number R(ψ) as the reproduction number
in the presence of the vaccination strategy ψ.
If no vaccination is applied (i.e., ψ ≡ 0), then R(0) = R0 , where R0 is the basic reproduction
number (when a purely susceptible population is considered). By what has been said earlier,
vaccination leads to a decrease of R. In particular R(ψ) ≤ R0 for any ψ.
Analysis of an Age Structured SEIRS Epidemic Model
33
For subsequent applications it is convenient to write R(ψ) in a different form. Since ψ(a) ≡ 0
implied D(a) ≡ 1, we find the basic reproduction number R0 = R(0) as
σ
a
rm
a
rm
−
(µ(τ )+α(τ ))dτ −
(
µ(τ )+δ(τ ))dτ
ξ
σ
β(ξ)P
(ξ)
k(a)
α(σ)e
e
dσdadξ
0
ξ
ξ
rm
R0 =
.
(3.34)
P (a)da
0
and the representation
R(ψ) = R0 −
F (ψ)
,
N
(3.35)
where
F (ψ) =
(β(ξ) − β(ξ))(1
− D(ξ) P (ξ)
rm
0
· exp
−
a
k(a)
ξ
µ(τ ) + α(τ ) dτ exp −
σ
rm
ξ
a
α(σ)
ξ
µ
(τ ) + δ(τ ) dτ dσdadξ,
σ
and
N=
(3.36)
rm
0
P (a)da.
(3.37)
Using the previously defined differential susceptibility and (3.15), F (ψ) can be rewritten as
F (ψ) =
rm
k(a)P (a)
0
·
0
rm
∆(ξ)
−
ψ(ζ)e
ξ
ζ
−
α(σ)e
σ
ξ
α(τ )dτ
−
·e
a
σ
(η(τ )+δ(τ ))dτ
dσda
ξ
ξ
ξ
a
(γ(τ )+ψ(τ ))dτ
dζ dξ,
(3.38)
From the above discussions we observe that the quantity F (ψ) is (up to the factor N ) the
reduction in the reproduction number that can be achieved by applying the vaccination strategy
ψ.
It is interesting to observe that this quantity depends on the differential susceptibility ∆(a)
and not on β(a), β(a)
separately.
Any actual vaccination campaign must aim at making F (ψ) large and thus possibly reducing
R(ψ) to values below 1. Whether this is possible depends, apart from economic and social side
conditions (so far not incorporated into the model), on the differential susceptibility ∆(a) and
the loss of immunity γ(a).
There are three rather different situations. In the first situation we have R0 < 1. Then R
can be further reduced (e.g., to make it considerably less than one), but in principle vaccination
is not necessary. In the second situation we have R0 > 1, where
σ
a
a
rm
rm
−
(µ(τ )+α(τ ))dτ −
(
µ(τ )+δ(τ ))dτ
ξ
σ
k(a)
α(σ)
·
e
e
dσdadξ
β(ξ)P
(ξ)
ξ
0
ξ
,
(3.39)
R0 =
N
is the reproduction rate in a totally vaccinated population. Then the disease can spread in a
totally vaccinated population, and this vaccination seems useless. Nevertheless vaccination can
be used to give partial protection to individuals. The third, and most interesting situation,
occurs when R0 < 1 < R0 . In that case certain strategies will reduce R(ψ) to values below
one and thus will lead to the elimination of the disease. The problem of how to choose optimal
strategies will be covered in the next section
X.Z. Li, G. Gupur, G.T. Zhu
34
4
Optimal Vaccination Strategies
In this section, we consider the situation in which the population is in demographic equilibrium.
A vaccination strategy ψ converts susceptibles of age a into (partly) immunes of age a. By (3.35),
the aim of the vaccination strategy ψ is to make F (ψ) large and thus decrease R(ψ). In practice,
the application of vaccination strategies is limited by costs. The costs include expenses for the
application of the vaccine, compliance of the different age classes and necessary advertisement
campaign, etc. We assume that the expense of one vaccination at age a are given by a positive
number κ(a), and that the total costs depend linearly on the number of vaccinations.
We assume that only susceptibles are vaccinated and that the costs of the vaccination
strategy ψ are C(ψ). Then the total cost of the vaccination strategy is
C(ψ) =
By (3.10) we have
C(ψ) =
rm
κ(a)ψ(a)S(a)da.
(4.1)
κ(a)P (a)ψ(a)D(a)da
(4.2)
0
rm
0
where D(a) is given by (3.15). For simplicity, we consider the case in which immunity lasts,
i.e., γ(a) ≡ 0. Then the functionals F (ψ), C(ψ) are simplified to
F (ψ) =
rm
·
C(ψ) =
0
rm
−
ξ
ψ(θ)e
θ
σ
ξ
α(τ )dτ
−
·e
a
σ
(η(τ )+δ(τ ))dτ
dσda
ξ
ψ(τ )dτ
dθdξ,
a
−
κ(a)P (a)ψ(a)e
0
−
α(σ)e
ξ
ξ
a
κ(a)P (a)
0
rm
∆(ξ)
0
(4.3)
ψ(τ )dτ
da.
(4.4)
Two optimization problems can be defined as follows. Let R∗ and C∗ be two constants.
(1) Find a vaccination strategy ψ(a) that minimizes C(ξ) constrained by R(ψ) ≤ R∗ .
(2) Find a vaccination strategy ψ(a) that minimizes R(ψ) constrained by C(ψ) ≤ C∗ .
The difficulty associated with these optimization problems is due to the fact that C(ψ) and
F (ψ) are nonlinear functionals of ψ. Hadelar and Müller [13] showed how to overcome the
difficulty. In order to make both C(ψ) and F (ψ) linear functionals we apply the transformation
φ(a) = −
d
exp −
da
0
a
ψ(τ )dτ
= ψ(a) exp
−
a
0
ψ(τ )dτ ,
(4.5)
or, conversely
ψ(a) =
1−
φ(a)
a
.
φ(τ )dτ
0
We denote F (φ) = F (ψ), C(φ) = C(ψ). Then, the substitution of (4.5) into (4.4) leads to
F (φ) =
rm
∆(ξ)
0
·
0
rm
κ(a)P (a)
ξ
a
−
α(σ)e
σ
ξ
α(τ )dτ
−
·e
a
σ
(η(τ )+δ(τ ))dτ
dσda
ξ
ξ
φ(ζ)dζdξ.
(4.6)
Analysis of an Age Structured SEIRS Epidemic Model
35
If we change the order of integral in (4.6), we arrive at
F (φ) =
rm
φ(ζ)
0
rm
κ(a)P (a)
ζ
a
rm
∆(ξ)
−
σ
α(σ)e
ξ
ξ
α(τ )dτ
−
·e
a
σ
(η(τ )+δ(τ ))dτ
dσdadξdζ.
ξ
Therefore, we can write F (φ), C(φ) as follows
F (φ) =
rm
0
C(φ) =
where the kernels are given by
rm
rm
K(ζ) =
∆(ξ)
κ(a)P (a)
ζ
ξ
K(ζ)φ(ζ)dζ,
(4.7)
B(a)φ(a)da,
(4.8)
rm
0
a
−
σ
α(σ)e
ξ
α(τ )dτ
−
·e
a
ξ
σ
(η(τ )+δ(τ ))dτ
dσdadξ,
(4.9)
B(a) = κ(a)P (a).
(4.10)
Hence, we have replaced two nonlinear functionals with the linear functional given by F (φ) and
C(φ). If we let
rm
Q(φ) =
φ(a)da.
0
Then it is easy to see that Q(φ) ≤ 1.
Letting ρ = (R0 − R∗ )N . We are able to replace Problem (1) by the following linear
optimization problem:
Minimize
C(φ),
Subject to f (φ) ≤ 0,
(4.11)
φ ≥ 0,
where
f (φ) =
f1 (φ)
f2 (φ)
=
ρ − F (φ)
Q(φ) − 1
and f (φ) ≤ 0 is equivalent to fi (φ) ≤ 0 (i = 1, 2). After using (formally) the Saddle Point
Theorem of Kuhn and Tucker for the convex optimization problem (see [14]) we can show that
(4.11) is mathematically equivalent to P1 in [10]. Hence, using the same arguments we arrive
at the following conclusion.
Theorem 4.1.
There are two possible optimal vaccination strategies in Problem (1):
(i) one-age strategy: vaccinate the susceptible population at exactly age A;
(ii) two-age strategy: vaccinate part of the susceptible population at age A1 , and the remaining susceptibles at a latter age A2 .
For Problem (2), a similar conclusion to Theorem 4.1 can be obtained, i.e., the optimal
vaccination strategy is either one- or two-age, and the optimal ages can be determined in the
same manner as that in [13].
36
X.Z. Li, G. Gupur, G.T. Zhu
References
[1] Anderson, R.M., May, R.M. Population biology of infectious disease I, Nature, 180: 361 (1979)
[2] Busenberg, S., Iannelli, M., Thieme, H. Dynamics of an age-structured epidemic model, in Proceedings
on Dynamical Systems, Lecture Notes in Math. (Nankai Subseries) (Edited by S.T.Liao, Y.Q. Ye and
T.R.Ding), World scientific, 1–19 (1993)
[3] Busenberg, S., Iannelli, M., Thieme, H. Global behavior of an age-structured epidemic model. SIAM J.
Math. Anal., 22: 1065–1080 (1991)
[4] Cha, Y., Iannelli, M., Milner, E. Existence and uniqueness of endemic states for the age-structured SIR
epidemic model. Math. Biosci., 150: 177–190 (1998)
[5] El-Doma, M. Analysis of an age-dependent SIS epidemic model with vertical transmission and proportionate
mixing assumption. Math. Comput. Model. 29: 31–43 (1999)
[6] Greenhalgh, D. Analytical threshold and stability results on age-structured epidemic models with vaccination. Theor. Popul. Biol., 33: 266–290 (1988)
[7] Greenhalgh, D. Threshold and stability results for an epidemic model with an age-structured meeting rate.
IMA J. Math. Appl. Med. Biol., 5: 81–100 (1988)
[8] Hadeler, K.P. Periodic solutions of homogeneous equations. J. Diff. Equ., 95: 183–202 (1992)
[9] Hadeler, K.P., Müller, J. Vaccination in age-structured populations I: The reproduction number, in Models for Infectious Human Diseases: Their Structured Relation to Data. V.Isham and G. Medley, eds.,
Cambridge University Press. Cambridge, 90–101, 1993
[10] Hadeler, K.P., Müller, J. Vaccination in age-structured populations II: Optimal strategies, in Models for
Infectious Human Diseases: Their Structured Relation to Data. V.Isham and G. Medley, eds. Cambridge
University Press. Cambridge, 90–101 (1993)
[11] Inaba, H. Threshold and stability results for an age-structured epidemic model. J. Math. Biol., 28:
149–175 (1990)
[12] Li, M.Y., J.R.Graef, Liancheng Wang and J.Karsai, Global dynamics of a SEIR model with varying total
population size. Math. Biosci. 160: 791–213 (1999)
[13] May, R.M., Anderson, R.N. Population biology of infectious disease II, Nature, 180: 455 (1979)
[14] Stoer, J., Witzgall, Chr. Convexity and Optimization in Finite Dimension I. Springer Verlag, 1970
[15] Webb, G.F. Theory of Nonlinear-Dependent Population Dynamics. Marcel Dekker, New York and Basel,
1985