Mathematical Biosciences 206 (2007) 249–272
www.elsevier.com/locate/mbs
Optimal harvesting and optimal vaccination
K.P. Hadeler
a,*,1
, J. Müller
b
a
b
Department of Mathematics and Statistics, Arizona State University, Tempe, AZ 85287, USA
Zentrum Mathematik, Technical University of Munich, Boltzmannstrasse 3, 85747 Garching, Germany
Received 14 May 2005; received in revised form 3 August 2005; accepted 15 September 2005
Available online 14 November 2005
Dedicated to the memory of Ovide Arino
Abstract
Two optimization problems are considered: Harvesting from a structured population with maximal gain
subject to the condition of non-extinction, and vaccinating a population with prescribed reduction of the
reproduction number of the disease at minimal costs. It is shown that these problems have a similar structure and can be treated by the same mathematical approach. The optimal solutions have a two-window
structure: Optimal harvesting and vaccination strategies or policies are concentrated on one or two preferred age classes. The results are first shown for a linear age structure problem and for an epidemic situation at the uninfected state (minimize costs for a given reduction of the reproduction number) and then
extended to populations structured by size, to harvesting at Gurtin–MacCamy equilibria and to vaccination
at infected equilibria.
Ó 2005 Elsevier Inc. All rights reserved.
MSC classification: 92D25; 92D30; 92C60
Keywords: Structured population; Harvesting; Culling; Infectious disease; Vaccination; Optimization
*
1
Corresponding author. Tel.: +1 480 965 3779; fax: +1 480 727 7346.
E-mail address: [email protected] (K.P. Hadeler).
Funded by the Mathematical and Theoretical Biology Institute at ASU.
0025-5564/$ - see front matter Ó 2005 Elsevier Inc. All rights reserved.
doi:10.1016/j.mbs.2005.09.001
250
K.P. Hadeler, J. Müller / Mathematical Biosciences 206 (2007) 249–272
1. Introduction
Vaccination of a population against the spread of an infectious disease and harvesting from a
population have much in common. Harvesting removes individuals from a population with the
result that population growth is slowed down. The gain from the harvest is increasing with the
number of culled individuals. The possible gain and the resulting population loss must be balanced. Thus harvesting leads to several optimization problems. A vaccination campaign removes
individuals from the susceptible part of the population in order to decrease the reproduction number of the disease or the level of prevalence. Again, the expenses incurred by the vaccination program must be balanced against the possible benefits which leads to the question of an optimal
vaccination policy.
Harvesting models for structured populations are based on Leslie type matrix models [9,13], on
stage models in continuous time [14], or on the Sharpe-Lotka–McKendrick model and the Gurtin–MacCamy model [4] which has the form of a partial differential equation with boundary condition for the time dependent population distribution u(t, a). Harvesting is introduced as an
additional death term w(t, a). There is a great variety in the formulation of the problem. The problem can be time-dependent or stationary, the harvested amount w(t, a) can be absolute or proportional to population size. A distinctive feature is whether the harvesting process is connected to
population control or not. Beddington and Taylor [3] introduced the concept of maximum sustainable yield (MSY), i.e., maximizing yield without driving the population to extinction and perhaps respecting further ecological or social side conditions. Rorres and Fair [32,33] consider
harvesting absolute amounts, Gurtin and Murphy [17] study a non-linear problem where natural
mortality depends on population size and the harvesting term is w(t, a) = E(t)u(t, a), thus harvesting is independent of age, see also [31]. The goal is to maximize yield for a given initial data and a
prescribed final time and population level. Gurtin and Murphy [18,27], Murphy and Smith [28]
consider similar but more complex problems. Brokate [4] studies a (non-linear) Gurtin–MacCamy
system with harvesting over a finite time interval with uniformly bounded harvesting effort from a
given initial distribution. In [2] a periodic harvesting problem is studied which in some sense appears more general than the approach of the present paper. However, in [2] a positive immigration
rate plays a crucial role.
The variety of different vaccination models is still greater, reflecting various features of common
diseases [1]. Standard vaccination models are based on the age structured Kermack–McKendrick
SIR model. Vaccination is introduced as a term that removes individuals from the class of susceptible individuals [8,10]. Again one can consider time-dependent problems where certain goals must
be achieved within a finite time horizon with minimal effort, say, or stationary models that describe optimal policies at equilibrium. Cairns [5] studies the effect of vaccination policies on the
Jacobian at the uninfected equilibrium in multigroup epidemic models. Greenhalgh [15] considers
vaccination of fixed proportions of the population at given ages to stabilize the uninfected equilibrium. Rouderfer and Becker [34] take loss of immunity into account. Vaccination policies in
multigroup situations can lead to backward bifurcations, see [23,19,20]. The comparison of optimization concepts is of special importance since different optimization concepts do lead to different solutions. This statement sounds trivial, but it bears major problems when it comes to the
point to draw practical conclusions [11,24].
K.P. Hadeler, J. Müller / Mathematical Biosciences 206 (2007) 249–272
251
In the following we explore and exploit the analogies between harvesting and vaccination models
in an equilibrium situation. In the harvesting context we consider age-dependent proportional harvesting with maximum gain under the side-condition of non-extinction. In the vaccination context
our problem is to achieve a desired reduction of the reproduction number at minimal expenses.
We follow the approach of two previous papers [22] (see also [25] for a different mathematical
approach) where vaccination policies were applied to the uninfected equilibrium, hence to a linear
population model. In [26] this approach has been extended to non-linear situations and equilibria
where the disease is present. In the present paper we consider both approaches for the vaccination
and the harvesting problem as well.
Optimal policies turn out to be one-age or two-age policies. This fact has been shown for vaccination problems in [22,25] (see also the exposition in Section 22.4 of [36]) and for optimal harvesting problems in the early and fundamental papers [32,33]. The present problem differs from
[32,33] insofar as we consider proportional harvesting rather than culling absolute amounts.
Thus, optimal vaccination policies vaccinate a certain proportion of a single age class and possibly another proportion of some other, older, age class (in fact all of the latter). Similarly, optimal
harvesting policies select one or two age or size classes. These results are in some sense counterintuitive and it takes some effort to explain them in medical or biological terms whereafter, however, it becomes quite clear why optimal policies must have this structure. Indeed we can expect
that these sharply peaked vaccination campaigns become vaccination windows when a bound is
put on the number of possible vaccinations (or the amount of culling) per time.
The paper is organized as follows. In Section 2 we consider harvesting from a population structured by age. This problem is simpler than the vaccination problem considered in [22] and is best
suited for an interpretation of the results in biological terms. In Section 3 we consider populations
structured by size. In this case the basic tools like characteristic differential equations and renewal
equations are considerably more complicated [7]. We present them here in a form suited for the
optimization problem. In Section 4 we discuss harvesting from populations structured by size, first
for a linear model and then for a Gurtin–MacCamy situation. In Section 5 we present a vaccination problem from [22], i.e., to protect an uninfected population against the outbreak of a disease.
In Section 6 we consider the endemic case. We close with a discussion.
2. Harvesting from a population structured by age
Consider a population model of Verhulst type u_ ¼ bðuÞu lðuÞu f ðuÞ where the birth rate
b(u) decreases to zero while the death rate l(u) increases to infinity, with b(0) > l(0). Assume
f(u) is a concave function. Then there is a unique positive equilibrium u. Suppose that this population is harvested at a rate w(t) P 0 which is periodic in time with period x > 0, and ask how to
maximize the mean gain per time. The differential equation
u_ ¼ bðuÞu lðuÞu wðtÞu
has a unique positive x-periodic solution uw(t), and the mean gain at this solution is
Z x
e ðwÞ ¼ j
K
wðtÞuw ðtÞ dt
x 0
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K.P. Hadeler, J. Müller / Mathematical Biosciences 206 (2007) 249–272
where j is the gain per unit harvested. From the equation
Z
j x
e
f ðuw ðtÞÞ dt
K ðwÞ ¼
x 0
it follows that the optimal harvesting rate is a constant w that keeps f(uw) at the absolute maximum of the function f. Conversely, assume that the function f(u) assumes its maximum at
e ¼ jf ð~uÞ.
~
u 2 ð0; 1Þ, then the maximal gain is K
The situation is slightly more complicated if w is restricted to some class of functions not containing the constants, e.g., if a constant proportion is harvested at discrete intervals. Then certain
difference quotients of the solution operator must be maximized.
Now, in contrast to the Verhulst model, we consider a linear homogeneous model which, as
always in population dynamics and demography, must be seen as a short time extrapolation.
The differential equation reads u_ ¼ bu lu wu with constant b > l > 0. The gain from the harvesting policy w is jwu. We want to maximize the gain subject to the side condition w P 0 and
constant population u = w. The latter condition implies that the exponent of the exponential solution u exp kt is k = 0 and hence w = b l. Then the maximal gain is j(b l)w. The result is trivial, it says that the gain is maximized by keeping the population constant (at the prescribed level)
and harvesting the surplus.
In this one-dimensional model the answer is so simple because the only free constant is fixed by
requiring constant population size. In structured populations the state space has infinite dimension and fixing population size does not determine the harvesting rate.
Now consider a population structured by age that is governed by the classical model of SharpeLotka and McKendrick [37]. Then u(t, a) is the population density, l(a) and b(a) are the agedependent mortality and fertility. The density evolves according to
ut þ ua þ lðaÞu ¼ 0
Z 1
uðt; 0Þ ¼
bðaÞuðt; aÞ da.
ð2:1Þ
0
Under suitable conditions (saying that not all individuals die before reaching reproductive age)
there is an exponential solution (the asymptotic or persistent solution)
^
ekt uðaÞ;
ð2:2Þ
where ^
k is the exponent of exponential growth and uðaÞ is the asymptotic age distribution. The
exponent ^
k (Lotkas r) is the unique real solution of the characteristic equation
Z 1
bðaÞpðaÞ eka da ¼ 1;
ð2:3Þ
0
where
pðaÞ ¼ e
Ra
0
lðsÞ ds
is the survival function. Once ^
k is known, the asymptotic age distribution is given by
^
uðaÞ ¼ pðaÞ eka u0 ;
ð2:4Þ
K.P. Hadeler, J. Müller / Mathematical Biosciences 206 (2007) 249–272
253
where u0 is the number of newborns per time. Notice that the stationary age distribution or age
pyramid uðaÞ is monotone for growing populations ð^k P 0Þ but may be onion-shaped (or even
more irregular) for decaying populations ð^
k < 0Þ. The asymptotic solution (2.2) is stable in an
orbital sense: If a perturbation is applied, then the solution is reestablished, up to a multiplicative
factor.
To this population a harvesting or culling strategy or policy is applied: From each age class u(a)
a certain fraction w(a)u(a) (per time) is harvested. Notice that w(a) is a rate. Thus w(a) P 0, but
w(a) can be arbitrarily large. In fact it will be necessary to allow w to be a d-function peaked at a
particular age class a0. A d-function wðaÞ ¼ cda0 ðaÞ has the following interpretation: When a cohort passes through the age a0, a certain fraction 1 ec (see (2.40) below for details) is culled
instantaneously.
If the harvesting process is implemented then the dynamical equations read as follows:
ut þ ua þ lðaÞu þ wðaÞu ¼ 0
Z 1
bðaÞuðt; aÞ da.
uðt; 0Þ ¼
ð2:5Þ
0
Notice that in (2.5) harvesting is simply modelled as a process where fractions of age classes are
removed with an age-dependent rate. No decision process or reaction kinetics is incorporated. In
this respect a harvesting model is very different from a prey-predator model.
Let j(a) be the net gain from harvesting one individual of age a. Then the total gain per time
from applying the policy w is
Z 1
jðaÞwðaÞuðt; aÞ da.
ð2:6Þ
0
In the differential equation (2.5) the harvesting rate w(a) acts as an additional mortality. Similar to
Lotkas approach, we look for exponential solutions in the presence of the harvesting policy w(a).
The exponent k(w) of growth of the population subject to the policy w is the unique real root of
the equation
Z 1
Ra
wðsÞ dska
bðaÞpðaÞ e 0
da ¼ 1
ð2:7Þ
0
and the asymptotic age distribution, subject to the policy w, is
Ra
wðsÞ dskðwÞa
uðaÞ ¼ pðaÞ e 0
u0 ;
ð2:8Þ
where the free parameter u0 represents the number of births per time. If one wants to harvest
along a persistent solution without driving the population to extinction then one must require
^
k > 0 (otherwise the population would become extinct even without harvesting) and
0 6 kðwÞ 6 ^
k. If one chooses k(w) > 0 then the harvested population will still increase, and there
is some surplus that could be harvested. Hence we consider harvesting with maximal possible
intensity, k(w) = 0. Then the population is constant in time, and its age distribution is given by
(2.8) with k(w) = 0.
Now we discuss the role of the parameter u0. One needs a normalization of u0 because the problem is linear. In a first approach we assume that the number of births is a given constant u0. Hence
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K.P. Hadeler, J. Müller / Mathematical Biosciences 206 (2007) 249–272
we ask to maximize the gain subject to the side-condition of non-extinction k(w) = 0, in a persistent (stationary) situation, whereby the number of births is kept constant.
e ðwÞu0 ; where
Problem 1a. Maximize the gain functional K
Z 1
Ra
wðsÞ ds
e ðwÞ K
jðaÞpðaÞ e 0
wðaÞ da;
ð2:9Þ
0
subject to the sign condition
for a P 0
ð2:10Þ
and the balance condition
Z 1
Ra
wðsÞ ds
e
BðwÞ bðaÞpðaÞ e 0
da ¼ 1.
ð2:11Þ
wðaÞ P 0
0
Fixing u0 at a prescribed level means fixing the number of births (or of juveniles). From a bioeconomics point of view it makes more sense to ask for the maximal gain for a prescribed upper
bound w for a functional
Z 1
e
qðaÞuðaÞ da
W ¼
0
with weight q P 0 of the population u which counts total population (q(a) 1) or gives some
weight to different age classes (e.g. q(a) = 0 for a < s and q(a) = 1 for a P s, for some s > 0).
In an equilibrium situation this expression still depends on w. In this case the optimization problem assumes the following form.
e given by (2.9), subject to the sign cone ðwÞu0 , with K
Problem 2a. Maximize the gain functional K
dition (2.10) and to the balance condition (2.11) and also to the condition
Z 1
Ra
wðsÞ ds
e
W ðwÞ qðaÞpðaÞ e 0
da u0 6 w.
ð2:12Þ
0
One might think that one could solve Problem 2a by first solving Problem 1a for u0 = 1 and
then using the known w to determine u0 from equality in (2.12). But this approach does not give
the correct solution to Problem 2a.
As an example consider the special case Rwhere b, l, j, q are positive constants. Let w be any
R 1 la a wðsÞ ds
e
0
da. Then BðwÞ
¼ bV ¼ 1 and hence V = 1/b.
policy with k(w) = 0. Define V ¼ 0 e
e ðwÞu0 ¼ jð1 lV Þu0 ¼ jð1 l=bÞu0 which is the maximal gain for Problem 1a.
Furthermore K
e ¼ qV ¼ q=b, u0 = bw/q, and hence the maximal gain
For Problem 2a we have W
e
K u0 ¼ jðb lÞw=q. In this case there is a uniform optimal policy, w(a) = b l, but there are also
one-age policies which we shall discuss later.
As usual, we require a sufficiently high mortality (assuming a maximal possible age would work
equally well),
Z a
lðsÞ ds ¼ 1.
lim
a!1
0
K.P. Hadeler, J. Müller / Mathematical Biosciences 206 (2007) 249–272
255
Problem 1a looks highly non-linear. Therefore we apply the following substitution (see [22])
Ra
wðsÞ ds
wðaÞ
ð2:13Þ
/ðaÞ ¼ e 0
which takes the rate w into a hazard function /. Then
Z a
Ra
wðsÞ ds
/ðsÞ ds ¼ 1 e 0
.
ð2:14Þ
0
The inverse transformation is given by
wðaÞ ¼
/ðaÞ
Ra
.
1 0 /ðsÞ ds
ð2:15Þ
The transition from w to / imposes constraints on the function /:
/ðaÞ P 0 for a P 0;
Z 1
/ðaÞ da 6 1.
Qð/Þ ð2:16Þ
0
We define the kernel
LðaÞ ¼ jðaÞpðaÞ.
ð2:17Þ
e ðwÞ with
Then the gain functional (2.9) can be expressed in terms of / as Kð/Þ ¼ K
Z 1
LðaÞ/ðaÞ da u0 .
Kð/Þ ¼
ð2:18Þ
0
We use
e
BðwÞ
¼
Z
1
bðaÞpðaÞ da 0
Z
1
Z
1
bðsÞpðsÞ e
0
Rs
0
wðsÞ ds
wðsÞ ds da
ð2:19Þ
a
e
to get Bð/Þ ¼ BðwÞ
and the balance condition (2.11) in the form
Z 1
SðaÞ/ðaÞ da ¼ m 1 > 0
Bð/Þ ð2:20Þ
0
with the kernel
Z 1
bðsÞpðsÞ ds
SðaÞ ¼
ð2:21Þ
a
and the constant
Z 1
bðaÞpðaÞ da.
m¼
ð2:22Þ
0
The functionals K and B are linear in /. Notice that L(a) P 0, S(a) P 0, and S(0) = m > 1,
S 0 (a) 6 0 for 0 6 a < 1, lima!1S(a) = 0. Now Problem 1a becomes linear and it assumes the
following form.
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K.P. Hadeler, J. Müller / Mathematical Biosciences 206 (2007) 249–272
Problem 1b. Maximize K(/) under the side conditions
for 0 6 a < 1; Bð/Þ ¼ m 1; Qð/Þ 6 1.
/ðaÞ P 0
ð2:23Þ
It is somewhat astonishing that almost the same substitution works for Problem 2a. We put
vðaÞ ¼ /ðaÞu0
and define
W ðvÞ ¼
Z
ð2:24Þ
1
ð2:25Þ
RðaÞvðaÞ da
0
with
RðaÞ ¼
Z
m1 ¼
Z
1
ð2:26Þ
qðsÞpðsÞ ds;
a
1
ð2:27Þ
qðaÞpðaÞ da.
0
Then Problem 2a becomes
Problem 2b. Maximize the gain functional K(v) under the side conditions
vðaÞ P 0
for 0 6 a < 1;
u0 P 0;
BðvÞ ðm 1Þu0 ¼ 0;
QðvÞ 6 u0 ;
m1 u0 W ðvÞ 6 w.
Proposition 1. For Problem 1a there is an optimal policy w which harvests either one age class only
or at most two age classes.
Proof. We derive necessary conditions for the solutions of this problem via a Lagrangian
approach using the Kuhn–Tucker saddle point theorem. See [6,35] for an instructive exposition
of the finite-dimensional case and [29] for the infinite-dimensional situation (the problem is related
to so-called knapsack and bin packing problems [12,30]). We introduce a Lagrange function
(notice that we are looking for a maximum rather than a minimum)
H ð/; n; gÞ ¼ Kð/Þ nðBð/Þ m þ 1Þ gðQð/Þ 1Þ;
ð2:28Þ
where the parameters are n 2 R, g 2 Rþ . The Kuhn–Tucker conditions for this problem read
/ðaÞ P 0;
ð2:29Þ
Bð/Þ ¼ m 1;
ð2:30Þ
Qð/Þ 6 1;
ð2:31Þ
gðQð/Þ 1Þ ¼ 0;
ð2:32Þ
LðaÞ nSðaÞ g 1 6 0;
ð2:33Þ
Kð/Þ nBð/Þ gQð/Þ ¼ 0.
ð2:34Þ
K.P. Hadeler, J. Müller / Mathematical Biosciences 206 (2007) 249–272
257
Formally there is also the condition n(B(/) m + 1) = 0 similar to (2.32), but it follows from
(2.30) anyway. There are several cases to be distinguished.
Case 1: Q(/) < 1. Then g = 0, and from (2.33) and (2.34) we find
R1
LðaÞ/ðaÞ da
LðaÞ
6 n ¼ R01
.
SðaÞ
SðaÞ/ðaÞ da
0
ð2:35Þ
This case is impossible if b(a) = 0 for some a for which j(a) > 0, see (2.33).
Thus, n is a weighted arithmetic mean, and n is not less than the supremum of the quotients
L(a)/S(a). An arithmetic mean is not greater than this supremum. Hence the supremum n is assumed for at least one age a = A,
n¼
LðAÞ
.
SðAÞ
ð2:36Þ
Any such A is a candidate for an optimal one-age harvesting policy, i.e., the optimal policy has the
form of a delta peak w(a) = cdA(a) whereby the coefficient can be determined from the equation
e
BðwÞ
¼ 1. In a typical situation the function L(a)/S(a) assumes its maximum at a single point A. If
there are several such A then also convex combinations of the corresponding optimal policies are
optimal.
Case 2: Q(/) = 1. Then g need not vanish, from (2.33) and (2.34) we get
R1
ðLðaÞ nSðaÞÞ/ðaÞ da
R1
LðaÞ nSðaÞ 6 g ¼ 0
.
ð2:37Þ
/ðaÞ da
0
For given n, the number g is an arithmetic mean, and this mean is not less than the supremum of
the L(a) nS(a). Hence, for the n considered, the mean is identical to the supremum. There is an
a = An such that
g ¼ LðAn Þ nSðAn Þ P LðaÞ nSðaÞ
for all a. In a typical situation there are choices of the free parameter n such that there are two (or
more) such ages An, i.e., the supremum is assumed at two points, and the optimal policy w is a
specific combination of two delta peaks at ages An,1 and An,2. Such w is a candidate for an optimal
two-age policy. Hence in Case 2 the optimal policy is a one-age or a two-age policy.
Since we know that among the optimal policies there are necessarily one-age or two-age policies
we can start all over and introduce such policies in terms of rates w in the differential equations,
compute the corresponding functions / and investigate their effects on the functionals K and B. If
w is a one-age policy then
wðaÞ ¼ cdA ðaÞ;
ð2:38Þ
where A P 0 is the harvesting age and c > 0 is the intensity of harvesting. Since the integral of a
delta peak is a Heaviside jump function, we find
Z a
0 a < A;
wðsÞ ds ¼
ð2:39Þ
c a>A
0
and then taking exponentials on both sides,
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K.P. Hadeler, J. Müller / Mathematical Biosciences 206 (2007) 249–272
e
Ra
0
wðsÞ ds
¼
1
c
e
a < A;
a > A.
ð2:40Þ
The last equation says that before and after A nothing is harvested and at A exact a proportion
1 ec is harvested. Thus, harvesting as in (2.40) is the real meaning of (2.38). Using the definition (2.40) we find that the function / is again a delta peak
R
d a wðsÞ ds
¼ ð1 ec ÞdA ðaÞ
ð2:41Þ
/ðaÞ ¼ e 0
da
representing the distribution of those individuals that are actually harvested. Then the two functionals K and B assume the values
Kð/Þ ¼ LðAÞð1 ec Þ;
Bð/Þ ¼ SðAÞð1 ec Þ.
ð2:42Þ
These expressions can be used to reduce Problem 1a to a finite-dimensional optimization problem.
The case of a two-age policy
wðaÞ ¼ cdA1 ðaÞ þ c1 dA2 ðaÞ
ð2:43Þ
is somewhat more complicated. We can assume 0 6 A1 < A2 and c, c1 > 0. As before, by explicit
integration, we find
/ðaÞ ¼ dA1 ðaÞð1 ec Þ þ dA2 ðaÞðec ecc1 Þ.
ð2:44Þ
We know already that a two-age policy must satisfy Q(/) = 1, thus c1 = 1 and
/ðaÞ ¼ dA1 ðaÞð1 ec Þ þ dA2 ðaÞec
ð2:45Þ
saying that a proportion 1 ec is harvested at age A1 and everything left is harvested at age A2.
Then the functions K and B assume the values
Kð/Þ ¼ LðA1 Þð1 ec Þ þ LðA2 Þ ec ;
Bð/Þ ¼ SðA1 Þð1 ec Þ þ SðA2 Þ ec .
ð2:46Þ
These expressions can be used in Problem 1a. We have shown the following result.
Proposition 2. The one age policies form a two parameter family depending on parameters A and c,
the two age policies form a three parameter family depending on A1 < A2 and c. The functions K and
B can be expressed in terms of these parameters as in (2.42) and (2.46).
Thus, we have reduced the infinite-dimensional linear problem to a non-linear problem in two
or three dimensions. We exploit the fact that the kernel S in (2.21) is decreasing from some value
greater than 1 to zero. Since the gain rate j(a) can have any shape, the function L need not be
strictly decreasing. We define two critical ages 0 < A1 6 A2 such that S(a) > m 1 for a < A1 ,
S(a) = m 1 for a 2 ½A1 ; A2 , and S(a) < 1 for a > A2 .
Consider A 6 A2 . Then the side condition B(/) = 1 can be satisfied with a one age policy
w = cdA. In view of (2.21), (2.41) the corresponding value c is given by
1 1=SðAÞ A < A1 ;
c
e ¼
ð2:47Þ
0
A1 6 A 6 A2 .
K.P. Hadeler, J. Müller / Mathematical Biosciences 206 (2007) 249–272
259
Hence the optimal one-age policy can be found as follows. For 0 6 A 6 A2 maximize the quotient
L(a)/S(a). If the maximum is assumed at some A 2 ½A1 ; A2 then all individuals at that age are harvested, otherwise a proportion c is harvested, where c is given by (2.47). The gain for this policy is
Kð/Þ ¼
LðAÞ
.
SðAÞ
ð2:48Þ
For ages A > A2 there is no feasible one age policy. Now consider two age policies with ages
A1 < A2. From (2.46) and B(/) = 1 we find
ec ¼
SðA1 Þ 1
.
SðA1 Þ SðA2 Þ
ð2:49Þ
Since ec 2 (0,1), we have necessarily A1 2 ½0; A1 , A2 > A2 . Then the gain is
Kð/Þ ¼
1 SðA2 Þ
SðA1 Þ 1
LðA1 Þ þ
LðA2 Þ.
SðA1 Þ SðA2 Þ
SðA1 Þ SðA2 Þ
ð2:50Þ
Hence the gain of the two age policy is a convex combination of the numbers L(A1) and L(A2).
Proposition 3. The optimal harvesting policy for Problem 1a can be found as follows: Determine A1 ,
A2 from the function S. Maximize the quotient L(a)/S(a) in a P 0. Let the maximum be attained at
some A. If A 6 A2 then the optimal policy is a one-age policy at age A and the gain is L(A)/S(A). If
A > A2 then the optimal policy is a two-age policy which can be found by maximizing K(/) as given
by (2.50) over A1 2 ½0; A1 , A2 > A2 .
We return to the example above. Then L(a) = j ela, S(a) = (b/l) ela, L(a)/S(a) jl/b. We
have A1 ¼ A2 with elA1 ¼ 1 l=b. For 0 < A 6 A1 we have (1 l/b) elA 6 1. Hence we can find
c 2 (0,1] such that (1 l/b) elA = 1 ec. Then w(a) = cdA(a) is an optimal one-age policy with
e ðwÞu0 ¼ jð1 l=bÞu0 . For A ¼ A this policy says harvest all at age A. In this example the oneK
1
age policy is not unique because L(a)/S(a) does not have a unique maximum. The example shows
the biological meaning of A1 : If A > A1 then even harvest all at age A cannot keep the population
bounded.
Now we turn to Problem 2b which has the form of a linear optimization problem for the variables (v, u0) P 0 and three additional side conditions. Following up the Kuhn–Tucker machinery
for this problem as in the preceding case one can show the following.
Proposition 4. Problem 2a has a solution which harvests at most three age classes.
3. Size structure
3.1. The linear size structure model
Similar to the age structure model (2.1) we consider a population structured by a size variable
x P 0. Of course the age structure model (2.1) describes the special case where size is proportional
to age. We treat both cases separately because the age problem is easier to understand. The
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K.P. Hadeler, J. Müller / Mathematical Biosciences 206 (2007) 249–272
Rx
function u(t,x) is the time-dependent size distribution. Thus x12 uðt; xÞ dx is the number of individuals with size between x1 and x2 at time t. The model equations read
ut þ ðgðxÞuÞx þ lðxÞu ¼ 0;
Z 1
gð0Þuðt; 0Þ ¼
bðxÞuðt; xÞ dx.
ð3:1Þ
0
Again, l(x) > 0 is the size-dependent mortality, b(x) P 0 is the fertility, and g(x) > 0 is the growth
rate. The second term in the differential equation must be written as a flow and also the boundary
condition must contain the factor g(0) in order that conservation of the number of individuals
should hold in the absence of birth and death.
The characteristic equation
Z 1
R
bðxÞ x lðsÞ=gðsÞ dskrðxÞ
dx ¼ 1
ð3:2Þ
e 0
gðxÞ
0
yields the exponent ^
k and then the asymptotic size distribution
Rx
gð0Þ lðsÞ=gðsÞ ds^krðxÞ
uðxÞ ¼
u0
ð3:3Þ
e 0
gðxÞ
with
rðxÞ ¼
Z
x
0
ds
.
gðsÞ
ð3:4Þ
The characteristic equation (3.2) can be derived from (3.1) by putting u(t, x) = exp{kt}u(x). Then
ðgðxÞuðxÞÞx ¼ from where
gðxÞuðxÞ ¼ e
Rx
0
lðxÞ þ k
gðxÞuðxÞ;
gðxÞ
ððlðsÞþkÞ=gðsÞÞ ds
ð3:5Þ
gð0Þu0 .
We introduce this expression into the recruitment law (3.1), divide by g(0)u0, and find (3.2).
3.2. Size structured Gurtin–MacCamy system
If the birth and death rates in (3.1) depend on total population size then we arrive at models of
Gurtin–MacCamy type for size-structured populations. Here we assume that individuals compete
via a death rate depending on the weighted population size W, the birth rate is independent of W,
ut þ ðgðxÞuÞx þ lðx; W Þu ¼ 0;
Z 1
gð0Þuðt; 0Þ ¼
bðxÞuðt; xÞ dx;
0
Z 1
qðxÞuðt; xÞ dx.
W ðtÞ ¼
0
Here, the weight q reflects the impact on the ecosystem of the different size classes.
ð3:6Þ
K.P. Hadeler, J. Müller / Mathematical Biosciences 206 (2007) 249–272
261
We assume that l(x, W) is strictly increasing as a function of W and l(x, W) ! 1 for W ! 1.
These assumptions can be weakened, but then we have to introduce additional technical
conditions that do not lead to further insight; hence we stay with the most simple case. We expect,
under suitable conditions, a non-trivial stationary solutions to exist.
Similar reasoning as above yields for a stationary solution first an equation for W, then a
formula for u0 and then a formula for u(x),
Z 1
R
Þ
bðxÞ x lðs;W
ds
RðW Þ e 0 gðsÞ dx ¼ 1;
gðxÞ
0
Z 1
R
Þ
qðxÞ x lðs;W
ds
e 0 gðsÞ dx u0 ;
W ¼ gð0Þ
ð3:7Þ
gðxÞ
0
R
Þ
gð0Þ x lðs;W
ds
e 0 gðsÞ u0 .
uðxÞ ¼
gðxÞ
From these equations, we find immediately the following conclusion.
Conclusion: If R(0) > 1 then there is a unique non-trivial equilibrium, i.e., an equilibrium with
W > 0.
Uniqueness is a consequence of the monotonicity of l(x, W) with respect to W that implies that
R(W) is strictly decreasing in W.
It is well known, that this equilibrium may undergo a Hopf bifurcation and become unstable
[16]. However, we will focus on a static analysis (we aim at an optimal harvesting rate for the equilibrium), implicitly assuming that for the cases considered the non-trivial equilibrium is locally
stable.
4. Harvesting a size-structured population
4.1. The linear problem with harvesting
Applying a harvesting policy w to the system (3.1) leads to
ut þ ðgðxÞuÞx þ lðxÞu þ wðxÞu ¼ 0;
Z 1
gð0Þuðt; 0Þ ¼
bðxÞuðt; xÞ dx.
ð4:1Þ
0
As in Section 2, the function w enters formally as an additional mortality.
From Eq. (3.2), with k = 0, we obtain the condition for marginal harvesting as
Z 1
R
R
bðxÞ x lðsÞ=gðsÞ ds x wðsÞ=gðsÞ ds
e
0
0
e
BðwÞ dx ¼ 1.
gðxÞ
0
ð4:2Þ
Let j(x) be the net gain from harvesting one individual of size x. Then the gain functional is
e ðwÞu0 with
K
Z 1
R
R
gð0Þ x lðsÞ=gðsÞ ds x wðsÞ=gðsÞ ds
e
0
0
K ðwÞ jðxÞ
wðxÞ dx.
ð4:3Þ
e
gðxÞ
0
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K.P. Hadeler, J. Müller / Mathematical Biosciences 206 (2007) 249–272
In the special case g(x) 1 we recover (2.9) and (2.11). Now we apply a substitution similar to
(2.13)
Rx
1
wðsÞ=gðsÞ ds
Rx
wðxÞ=gðxÞ; wðxÞ ¼ gðxÞ
/ðxÞ ¼ e 0
/ðxÞ.
ð4:4Þ
1 0 /ðyÞ dy
e ðwÞ ¼ Kð/Þ, and the condition BðwÞ
e
Then K
¼ 1 becomes B(/) = 1, where
Z 1
LðxÞ/ðxÞ dx;
Kð/Þ ¼
Bð/Þ ¼
Z
0
ð4:5Þ
1
SðxÞ/ðxÞ dx ¼ m 1;
0
with
m¼
Z
1
bðxÞ e
gðxÞ
0
Rx
0
ðlðsÞ=gðsÞÞ ds
dx;
Rx
lðsÞ=gðsÞ ds
LðxÞ ¼ jðxÞgð0Þe 0
;
Z 1
Rs
bðsÞ lðsÞ=gðsÞ ds
SðxÞ ¼
ds.
e 0
gðsÞ
x
ð4:6Þ
ð4:7Þ
We can formulate Problem 1b as in Section 2. Propositions 1–3, are verbally the same, with L, S
defined by (4.7).
4.2. Harvesting at equilibrium
We now consider the Gurtin–MacCamy system (3.6) with a linear harvesting policy,
ut þ ðgðxÞuÞx þ lðx; W Þu þ wðxÞu ¼ 0;
Z 1
bðxÞuðt; xÞ dx;
gð0Þuðt; 0Þ ¼
W ðtÞ ¼
0
Z
ð4:8Þ
1
qðxÞuðt; xÞ dx.
0
We assume that there is a non-trivial equilibrium for w = 0 (otherwise the population dies out
even without harvesting). A policy w with u(x) = 0 cannot be optimal since very small harvesting
rates yield a positive gain. Hence in the following we may restrict our consideration to those w(x)
for which u(x) 5 0. For practical purposes we put u(x) 0 if there is only the trivial equilibrium.
Then the optimization problem reads:
Problem 3a. For a given function w(x), define u(x) to be the non-trivial equilibrium (if it exists),
and u(x) = 0 otherwise. Maximize the gain
Z 1
e
jðxÞwðxÞuðxÞ dx.
K ðwÞ ¼
0
K.P. Hadeler, J. Müller / Mathematical Biosciences 206 (2007) 249–272
263
For a given rate w, the equilibrium conditions are given by Eqs. (3.7), with l(s,W) replaced by
l(s, W) + w(s). We use the abbreviation
Rx
e W Þ ¼ e 0 lðs;W Þ=gðsÞ ds
Eðx;
and the transformation (4.4) to formulate these conditions in terms of the function /. We define
the numbers
Z 1
Z 1
bðxÞ e
1 e
~ 1 ðW Þ ¼
~ Þ¼
Eðx; W Þ dx; m
Eðx; W Þ dx
mðW
gðxÞ
gðxÞ
0
0
and the kernels
e
e W Þ u0 ;
Lðx; W Þ ¼ jðxÞ Eðx;
Z 1
bðyÞ e
e
Eðy; W Þ dy;
S ðx; W Þ ¼
gðyÞ
x
Z 1
qðyÞ e
Te ðx; W Þ ¼
Eðy; W Þ dy.
gðyÞ
x
Then the conditions for an equilibrium assume the form
Z 1
e
~ Þ 1;
S ðx; W Þ/ðxÞ dx ¼ mðW
0
Z
~ 1 ðW Þ W ¼ gð0Þ m
1
Te ðx; W Þ/ðxÞ dx u0 ;
ð4:9Þ
ð4:10Þ
ð4:11Þ
0
uðxÞ ¼
Z x
gð0Þ e
Eðx; W Þ 1 /ðyÞ dy u0 ;
gðxÞ
0
e ðwÞ ¼ Kð/Þ becomes
and the gain functional K
Z 1
e
Kð/Þ ¼
Lðx; W Þ/ðxÞ dx.
ð4:12Þ
ð4:13Þ
0
Now the optimization problem assumes the form
Problem 3b. Maximize theR gain K(/) with W and u0 given by (4.10) and (4.11) under the side con1
ditions /(x) P 0, Qð/Þ ¼ 0 /ðxÞ dx 6 1.
Here, the population size W plays the role of the exponent k in the linear setting. In that case, it
has been clear that k = 0 for the optimal solution. In the present case, we do not have an a priori
guess how to choose W. Also, due to the non-linear form of the model, there is no direct transformation into a linear problem. Hence we assume that there is an optimal solution with corresponding population size W* and optimal population u*(x).
W = W* is introduced as an additional, artificial side condition. As a consequence, the population size for x = 0, i.e., u0, is given by a linear functional of /. However, even if we fix W = W*,
K(Æ) is still not linear in /. In order to get rid of this problem, we not only demand W = W*, but
also u0 ¼ u0 . As a consequence, we find that K( Æ ) becomes linear.
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K.P. Hadeler, J. Müller / Mathematical Biosciences 206 (2007) 249–272
We will see, that this somewhat barefaced strategy works. We cannot prove the existence of an
optimal strategy in the present setting, since the Theorem of Kuhn and Tucker only yields necessary conditions for optimal solutions. Using compactness arguments, also existence can be ensured [26].
Assumption. A solution /* of the optimization problem exists.
~ Þ, m1 ¼ m
~ 1 ðW Þ,
For this solution, denote W, u, u0 by W*, u*, u0 Define m ¼ mðW
e
EðxÞ ¼ Eðx; W Þ,
LðxÞ ¼ e
Lðx; W Þ;
and
Bð/Þ ¼
Z
SðxÞ ¼ e
S ðx; W Þ;
1
SðxÞ/ðxÞ dx;
Dð/Þ ¼
0
T ðxÞ ¼ Te ðx; W Þ
Z
1
T ðxÞ/ðxÞ dx.
0
Then we formulate a linear optimization problem.
Problem 3c. Maximize the gain
Z 1
Kð/Þ ¼
LðxÞ/ðxÞ dx u0
0
under the side conditions
/ðxÞ P 0;
Qð/Þ ¼
Z
1
/ðxÞ dx 6 1;
Bð/Þ ¼ 1;
0
W ¼ gð0Þ½m1 Dð/Þu0 .
Proposition 5. A solution / of the linear Problem 3c is a solution of the full Problem 3b.
Proof. We first show that a solution of Problem 3c is a feasible policy also for Problem 3b. In the
second step, we show that the solution of 3c performs (in the setting of Problem 3b) at least as well
as the solution /* of Problem 3b.
Let /^ be a solution of Problem 3c. Then Eq. (4.10) is satisfied with for / ¼ /^ with W = W* in
^ ¼ 1 and Dð/Þ ¼ u says that Eq. (4.11) is satisfied with W = W*, uð0Þ ¼ u .
view of Bð/Þ
0
0
Hence we can define
Z x
gð0Þ
^
^
EðxÞ 1 /ðyÞ dy u0
uðxÞ ¼
gðxÞ
0
and see that /^ is feasible for Problem 3b.
Since /* is also an admissible function for Problem 3c, we have
^ P Kð/ Þ ¼ Kð/ Þ.
^ ¼ Kð/Þ
Kð/Þ
^ is also solution
Thus, /^ performs as least as good as /* (with respect to Problem 3b). Thus, /ðxÞ
of Problem 3b.
The structure of Problem 3c is the same as in the linear case with one additional linear side
condition. This additional side condition may cause the optimal solution to consist of three point
K.P. Hadeler, J. Müller / Mathematical Biosciences 206 (2007) 249–272
265
masses instead of two, but the arguments parallel the arguments for Proposition 1. Thus, we skip
the proof but only state the result.
Proposition 6. For Problem 3a there is an optimal policy which harvest either one or two or at most
three age classes.
5. Epidemic spread and vaccination
We consider an epidemic model [22] based on the McKendrick system (2.1) for susceptible u,
vaccinated w, and infectious v individuals in the form of three partial differential equations.
~ðaÞ for susceptibles/vaccinated
The parameters are the fertility bðaÞ P ~
bðaÞ, the mortality lðaÞ 6 l
and infectious, respectively, and the recovery rate a (assumed independent of age). Vaccination
~
may not lead to complete protection. The individual susceptibility is bðaÞ P bðaÞ
for susceptibles
~
and vaccinated, respectively. The case b ¼ 0 corresponds to complete protection. The infectivity
k(a) also depends on age. Thus, we consider a separable model: The rate of transmission from an
infected individual of age b to a susceptible or vaccinated individual of age a is b(a)k(b) or
~
bðaÞkðbÞ,
respectively.
The vaccination policy is given by the rate w(a) at which susceptible individuals are moved into
the vaccinated class. This scenario is described by the following equations:
ut þ ua ¼ lðaÞu wðaÞu bðaÞuV =N ;
~
=N ;
wt þ wa ¼ lðaÞw þ wðaÞu þ av bðaÞwV
ð5:1Þ
~
vt þ va ¼ ~
lðaÞv av þ ðbðaÞu þ bðaÞwÞV
=N
with boundary conditions (vertical transmission is not considered)
Z 1
uðt; 0Þ ¼
½bðaÞðuðt; aÞ þ wðt; aÞÞ þ ~
bðaÞvðt; aÞ da;
0
wðt; 0Þ ¼ 0;
where
NðtÞ ¼
Z
ð5:2Þ
vðt; 0Þ ¼ 0;
1
½uðt; aÞ þ wðt; aÞ þ vðt; aÞ da
ð5:3Þ
0
is the total population size and
Z 1
kðaÞvðt; aÞ da
V ðtÞ ¼
ð5:4Þ
0
is the number of contacts per time with infected individuals
for a given susceptible individual.
Ra
As before we use the survival function P ðaÞ ¼ expð 0 lðsÞ dsÞ, in addition we define the function D(a) describing the escape from vaccination
Z a Ra
Ra
wðsÞ ds
wðsÞ ds
¼1
e s
wðsÞ ds.
ð5:5Þ
DðaÞ ¼ e 0
0
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K.P. Hadeler, J. Müller / Mathematical Biosciences 206 (2007) 249–272
^
In the absence of vaccination and infection there is a persistent solution ðu; 0; 0Þekt where ^k, u are
given by (2.3) and (2.4). If a vaccination policy w is applied then this solution changes to
^
; 0Þekt
ð
u; w
ð5:6Þ
(with the same exponent, here w does not act like a mortality) where
^
^
uðaÞ ¼ P ðaÞDðaÞeka ;
ðaÞ ¼ P ðaÞeka uðaÞ.
w
ð5:7Þ
describes the remaining non-vaccinated individuals. It depends on the
Notice that the function u
^
^
; 0Þekt is not a bifurcation, the persisvaccination policy w. The transition from ð
u; 0; 0Þekt to ðu; w
tent solution depends on the parameter w in a smooth manner.
If some infectious individuals are introduced then they can interact with the remaining susceptible population, i.e., with susceptible individuals and with vaccinated individuals with reduced
~ 6 0. The (orbital) stability of the uninfected solution (5.6) is determined
susceptibility in case b
by the reproduction number R(w). The number R(w) is the average number of secondary cases
produced by one infective individual in a population of susceptible and vaccinated individuals
subject to the vaccination policy w. If R(w) > 1 then the uninfected solution is unstable, otherwise
it is stable. The reproduction number for this model has been computed in [22] (see also [21]). Various (equivalent) expressions for R(w) can be given. We present two of these expressions here. We
introduce the differential mortality and the differential susceptibility as
~ðaÞ lðaÞ;
dðaÞ ¼ l
~
DðaÞ ¼ bðaÞ bðaÞ.
Then
Z
RðwÞ ¼
1
^ka
kðaÞP ðaÞe
0
where
N¼
Z
ð5:8Þ
Z
a
e
Ra
s
ðdþaÞ ds
~
½bðsÞDðsÞ þ bðsÞð1
DðsÞÞ ds da=N ;
ð5:9Þ
0
1
^
P ðaÞeka da.
0
The following expression is obtained from (5.9) by applying partial integration,
Z 1
Z 1
Rs
^ka a ðdðsÞþaÞ ds
~
½DðaÞbðaÞ þ ð1 DðaÞÞbðaÞ
kðsÞP ðsÞe e
ds da=N .
RðwÞ ¼
0
ð5:10Þ
a
From now on we assume a stationary demography, i.e., ^k ¼ 0. In the absence of vaccination, for
w = 0, the number R(0) = R0 is the basic reproduction number
Z 1
Z 1
Rs
ðdðsÞþaÞ ds
R0 ¼
bðaÞ
kðsÞP ðsÞe a
ds da=N .
ð5:11Þ
0
a
Using R0 the reproduction number R(w) can be represented in the form
RðwÞ ¼ R0 F ðwÞ=N ;
where
F ðwÞ ¼
Z
Z
1
DðaÞ
0
a
1
kðsÞP ðsÞe
Rs
a
ðdðsÞþaÞ ds
Z
a
ds
e
0
Ra
r
wðsÞ ds
wðrÞ dr da
ð5:12Þ
ð5:13Þ
K.P. Hadeler, J. Müller / Mathematical Biosciences 206 (2007) 249–272
267
is (up to the factor 1/N) the reduction of the reproduction number achieved by applying the policy
w, see (3.25) in [22].
We consider a situation where vaccination is applied only to susceptible individuals. This
assumption requires in practice that vaccinated individuals are marked in such a way that they
can be recognized by the medical service. Suppose the costs of moving one individual of age a
from the susceptible class to the vaccinated class are j(a). Then the total costs incurred by the policy w are
Z 1
e
jðaÞP ðaÞwðaÞDðaÞ da.
ð5:14Þ
CðwÞ ¼
0
We fix a level R* < 1 to which the reproduction number shall be reduced. Let q = (R0 R*)N.
Then we have the following optimization problem (Problem P1 of [22]).
e
Problem 4a. Minimize the functional CðwÞ
under the sign condition w(a) P 0 for a P 0 and the
condition F(w) P q.
We define the kernels
Z 1
Z 1
Rs
ðdþaÞ ds
DðrÞ
kðsÞP ðsÞe r
ds dr;
SðaÞ ¼
ð5:15Þ
a
r
LðaÞ ¼ jðaÞP ðaÞ.
Using the transformation (2.13) this problem can be transformed into a linear problem. We
define the functionals Q, K and B as in (2.16), (2.18) and (2.20).
Problem 4b. Minimize C(/) under the side conditions
/ðaÞ P 0
for
0 6 a < 1;
Bð/Þ P q;
ð5:16Þ
Qð/Þ 6 1.
This problem is similar to Problem 1b. Instead of a maximum we are looking for a minimum,
the kernel S(a) is monotone, the side conditions are both inequalities. As before we define a
Lagrange function
H ð/; n; gÞ ¼ Cð/Þ nðBð/Þ qÞ gðQð/Þ 1Þ
ð5:17Þ
with n; g 2 Rþ . Then we get the Kuhn–Tucker conditions
/ðaÞ P 0;
Bð/Þ P q;
Qð/Þ 6 1;
nðBð/Þ qÞ ¼ 0;
gðQð/Þ 1Þ ¼ 0;
LðaÞ nSðaÞ g 1 6 0;
Cð/Þ nBð/Þ gQð/Þ ¼ 0.
ð5:18Þ
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K.P. Hadeler, J. Müller / Mathematical Biosciences 206 (2007) 249–272
Again we distinguish the cases Q(/) < 1 (one-age policies) and Q(/) = 1 (two-age policies). We
reduced the infinite dimensional optimization problem to a two-dimensional problem (in case
Q(/) < 1) or a three-dimensional problem (in case Q(/) = 1) problem, which can be solved
numerically. The details of the reduced problem, which go parallel with the derivation of Propositions 2 and 3, have been presented in [22], II.
6. Endemic case
In this section we consider the endemic scenario [26]. The problem in the endemic case is that
we do not know who already had the disease and who had not. We do only know who has been
vaccinated. Hence we slightly modify the model of the previous section: the vaccination strategy
distinguishes only between vaccinated and non-vaccinated individuals but does not take into account immunity acquired by passing through the disease.
There are three classes of non-vaccinated individuals: susceptible individuals u, infected individuals v and immune individuals w. The vaccination strategy is applied to all three classes. However,
vaccination means different things for the different classes. Vaccination carries a susceptible individual into the class of vaccinated former susceptible individuals uv. Both the individuals in classes
uv and w are immune, but only those in uv are known to be immune. Vaccination of an individual
from class w does not change his/her immunological status, but vaccination nevertheless causes a
transition to another class of individuals known to be immune, namely to vaccinated former immune individuals wv. In field work one tries to avoid vaccination of infected individuals. However,
avoiding vaccination of infected individuals requires that infected individuals can be recognized.
We assume that there is a class of infected individuals so far not having been recognized v and that
for an individual in this class being summoned to vaccination makes the infected status obvious,
and consequently the individual is transferred to the class of recognized infected vi. After recovery
these individuals enter the class uv. Summing up, within the framework of this model, information
(available to the authority applying the vaccination policy) about the health status of an individual is as essential as the health status itself.
In order to keep the model simple we assume that the disease has no effect on the birth rate or
death rate. In contrast to the previous sections, immune individuals are assumed to be fully pro~ ¼ 0. Moreover, the population is assumed to be in demographic equilibrium, i.e.,
tected, i.e.,R b
a
R1
lðsÞ ds
bðaÞe 0
da ¼ 1. The model reads (using the same notations as before)
0
ut þ ua ¼ lðaÞu wðaÞu bðaÞuV 0 =N ;
vt þ va ¼ lðaÞv wðaÞv av þ bðaÞuV 0 =N;
wt þ wa ¼ lðaÞw wðaÞw þ av;
ðuv Þt þ ðuv Þa ¼ lðaÞuv þ wðaÞu;
ðvi Þt þ ðvi Þa ¼ lðaÞvi þ wðaÞv avi ;
ðvv Þt þ ðvv Þa ¼ lðaÞvv þ avi ;
ðwv Þt þ ðwv Þa ¼ lðaÞwv þ wðaÞw;
K.P. Hadeler, J. Müller / Mathematical Biosciences 206 (2007) 249–272
Z
uðt; 0Þ ¼
269
1
bðaÞ½u þ v þ w þ uv þ vi þ vv þ wv da;
0
vðt; 0Þ ¼ wðt; 0Þ ¼ uv ðt; 0Þ ¼ vi ðt; 0Þ ¼ wv ðt; 0Þ ¼ 0
ð6:1Þ
with
V0 ¼
Z
1
kðaÞ½vðt; aÞ þ vi ðt; aÞ da;
ð6:2Þ
0
NðtÞ ¼
Z
1
½u þ v þ w þ uv þ vi þ vv þ wv da.
ð6:3Þ
0
The costs of a vaccination policy w are defined as the number of vaccinations times the costs for
one vaccination. Since vaccination means different things for susceptible/immune and for infected individuals, the costs will be different for these classes. Hence we get the cost functional
Z 1
e
CðwÞ
¼
wðaÞ½j1 ðaÞðuðt; aÞ þ wðt; aÞÞ þ j2 ðaÞvðt; aÞ da.
ð6:4Þ
0
In the endemic case the reproduction number is not an appropriate measure for the quality of a
vaccination strategy. Instead we count infected individuals and weigh these with a factor g(a) measuring the social impact of one diseased case (e.g. in rubella pregnant women would get a high
factor). Hence we consider the weighted prevalence
Z 1
~IðwÞ ¼
gðaÞ½vðt; aÞ þ vi ðt; aÞ da.
ð6:5Þ
0
We investigate an endemic equilibrium. In order to guarantee the existence of such an equilibrium
we demand that R(w)jw=0 > 1. We prescribe an upper level for the weighted prevalence eI ðwÞ 6 i0
e
and we minimize the costs CðwÞ
necessary to stay below that level. The equations for the endemic
equilibrium are non-linear due to the factor V0 which depends on v and vi. We remove this nonlinearity in the following manner. In the stationary equations of (6.1) we assume that V0 is another
unknown constant. This constant is determined by the linear side condition Ve ðwÞ ¼ V 0 with
Z 1
e
V ðwÞ ¼
kðaÞ½vðaÞ þ vi ðaÞ da.
ð6:6Þ
0
The optimization problem reads
Problem 5a. Minimize the functional C(w) under the side conditions w(a) P 0 for a P 0,
eI ðwÞ 6 i0 , and Ve ðwÞ ¼ V 0 .
Now we compute some components of the stationary endemic solution in terms of w and V0,
Ra
bV =N ds
uðaÞ ¼ P ðaÞDðaÞe 0 0
;
Z a Rs
V0
bV =N drðasÞa
ds;
vðaÞ ¼ P ðaÞDðaÞ
e 0 0
bðsÞ
N
0
Z a Z s Rr
V0
bV =N dr0 ðsrÞa
wðaÞ ¼ P ðaÞDðaÞa
e 0 0
bðrÞ
dr ds;
N
0
0
270
K.P. Hadeler, J. Müller / Mathematical Biosciences 206 (2007) 249–272
vðaÞ þ vi ðaÞ ¼ P ðaÞ
Z
a
e
Rs
0
bV 0 =N drðasÞa
0
DðsÞbðsÞ
V0
ds.
N
~ and Ve as
Hence we can express the functions ~I, C
Z 1
Z 1
R
V 0 s bV 0 =N drðasÞa
eI ðwÞ ¼
DðsÞ
gðaÞP ðaÞbðsÞ e 0
da ds;
N
0
s
Z 1
Ra
h
bV =N ds
~
CðwÞ
¼
wðaÞDðaÞP ðaÞ j1 ðaÞe 0 0
0
Z a Z s Rr
V0
bV =N dr0 ðsrÞa
þ j1 ðaÞa
e 0 0
bðrÞ
dr ds
N
0
0
Z a Rs
V0
bV 0 =N drðasÞa
0
ds da;
e
bðsÞ
þ j2 ðaÞ
N
0
Z 1
Z 1
R
V 0 s bV =N drðasÞa
Ve ðwÞ ¼
DðsÞ
kðaÞP ðaÞbðsÞ e 0 0
da ds.
N
0
s
ð6:8Þ
Using the transformation (2.13), this problem becomes linear. Similar to the previous sections we
define the kernels
Z
Z 1
R
V 0 s bV 0 =N dr 1
0
bðsÞ e
gðsÞP ðsÞeaðssÞ ds ds;
SðaÞ ¼
N
a
s
Z a Z s Rr
Ra
V0
bV 0 =N ds
bV =N dr0 ðsrÞa
dr ds
LðaÞ ¼ P ðaÞ j1 ðaÞe 0
þ j1 ðaÞa
e 0 0
bðrÞ
N
0
0
ð6:9Þ
Z a Rs
V0
bV 0 =N drðasÞa
0
þ j2 ðaÞ
e
bðsÞ
ds ;
N
0
Z 1
Z
R
V 0 s bV =N dr 1
MðaÞ ¼
bðsÞ e 0 0
kðsÞP ðsÞeaðssÞ ds ds.
N
a
s
Then we obtain
Z 1
/ðaÞSðaÞ da;
Ið/Þ ¼
0
Z 1
e
CðwÞ ¼ Cð/Þ; Cð/Þ ¼
/ðaÞLðaÞ da;
0
Z 1
e
e
V ðwÞ ¼ V ð0Þ V ð/Þ; V ð/Þ ¼
/ðaÞMðaÞ da.
eI ðwÞ ¼ eI ð0Þ Ið/Þ;
0
With these definitions Problem 5a becomes
Problem 5b. Minimize C(/) under the side conditions
/ðaÞ P 0 for 0 6 a < 1;
Ið/Þ P eI ð0Þ i0 ;
Qð/Þ 6 1;
V ð/Þ ¼ Ve ð0Þ V 0 .
ð6:10Þ
ð6:11Þ
ð6:12Þ
K.P. Hadeler, J. Müller / Mathematical Biosciences 206 (2007) 249–272
271
In contrast to the previous problems there is one more side condition. This additional side condition may cause three instead of two delta peaks in optimal vaccination strategies. Since the argument parallels that of the previous sections, we do not give full details of the proof but state
merely the result.
Proposition 6. There is an optimal strategy that vaccinates in at most three age classes,
wðaÞ ¼ c1 dA1 ðaÞ þ c2 dA2 ðaÞ þ c3 dA3 ðaÞ;
0 6 A1 < A2 < A3 < 1.
If the weights c1, . . . , c3 are all different from zero then at age A3 all remaining individuals are vaccinated, i.e., those who have not been vaccinated at ages A1 or A2.
7. Discussion
We have shown that there is a striking similarity between harvesting populations structured by
age or by size on the one hand and vaccination against contagious diseases in populations structured by age on the other hand. The common feature is, of course, that harvesting or culling and
vaccination moves individuals from one class to another. In either case we have considered the
situation of continuous harvesting or vaccination at equilibrium, in contrast to most other papers
on harvesting which consider a given initial data and a fixed time horizon. In the linear situation
we start from a persistent solution, in the non-linear model we consider a non-trivial equilibrium.
Whereas in the harvesting situation the side condition for the maximum yield problem is nonextinction, in the vaccination situation it is a given level for the reduction of the infection. In
all cases we find that a small finite number of age or size classes should be harvested or vaccinated
to get an optimal solution. We did not impose any bounds on the harvesting or vaccination rates.
We expect that, if such further conditions are imposed, the finite number of age or size classes
become age or size intervals or windows.
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