Optimal Control of Structured Epidemics and Public
Prevention
Bedr’Eddine AINSEBA
Bordeaux University and INRIA Bordeaux Sud Ouest
Ecole CIMPA, Pointe-à-Pitre, Guadeloupe 2009
December 28, 2008
Bedr’Eddine AINSEBA Bordeaux University
Optimal
and Control
INRIA Bordeaux
of Structured
Sud Ouest
Epidemics
Ecole
and
CIMPA,
Public
December
Pointe-à-Pitre,
Prevention
28, 2008Guadeloupe
1 / 5220
1 Control theory in population dynamics
2 Control of epidemics
3 The SIR model
4 The screening procedure
5 The classical optimal screening procedure
6 The economic framework
7 The social planner’s problem
8 A Numerical approach
9 Conclusion
10 The case of age structure
Bedr’Eddine AINSEBA Bordeaux University
Optimal
and Control
INRIA Bordeaux
of Structured
Sud Ouest
Epidemics
Ecole
and
CIMPA,
Public
December
Pointe-à-Pitre,
Prevention
28, 2008Guadeloupe
2 / 5220
Control theory in population dynamics
Motivations
-Control theory is a well-developped branch of mathematics and automatics
that identifies optimal control policies for dynamic systems. While it’s
largely used in engineering its uses in ecology is very restricted.
-The control of epidemics starded in the eighteeth century with D. Bernoulli
"... je souhaite seulement que dans une question qui regarde de si près le
bien de l’humanité, on ne décide rien qu’avec toute la connaissance de
cause qu’un peu d’analyse et de calcul peut fournir." (D. Bernoulli 1760)
-The demographical questions of control started in the end of the eighteeth
with Malthus ("An essay on the principle of population")
Bedr’Eddine AINSEBA Bordeaux University
Optimal
and Control
INRIA Bordeaux
of Structured
Sud Ouest
Epidemics
Ecole
and
CIMPA,
Public
December
Pointe-à-Pitre,
Prevention
28, 2008Guadeloupe
3 / 5220
Control theory in population dynamics
Motivations
The control is often viewed as the man action on the environment
(vaccination, screening, treatment, quarantine, health-promotion
campaigns, profilaxy, sterelisation, introduction, fishing, catching,
elimination, use of pesticides, ...)
Bedr’Eddine AINSEBA Bordeaux University
Optimal
and Control
INRIA Bordeaux
of Structured
Sud Ouest
Epidemics
Ecole
and
CIMPA,
Public
December
Pointe-à-Pitre,
Prevention
28, 2008Guadeloupe
4 / 5220
Control theory in population dynamics
Fields of use
Demography: Control the distribution of individuals within a population
-Lefkovitch (1966) (elimination of individuals on some age groups),
-Rores, Fair (1980) (Optimal control, ind. t),
-Murphy, Smith (1990), (Optimal control dep. t),
-Gurtin and Murphy (77, 81) (Non linear optimal control (the separable
case)),
-Brokate (85, 87) (Optimal control with control depending on the age),
-Keyfitz (1977) (Reach a small net growth by acting on both birth and
migrations)
Bedr’Eddine AINSEBA Bordeaux University
Optimal
and Control
INRIA Bordeaux
of Structured
Sud Ouest
Epidemics
Ecole
and
CIMPA,
Public
December
Pointe-à-Pitre,
Prevention
28, 2008Guadeloupe
5 / 5220
Control theory in population dynamics
Control of the Lotka-McKendrick model

Dp(a, t, x)







p(a, t, x)


p(0, t, x)





p(a, 0, x)
+µ(a, t, x; P)p(a, t, x) − k∆p(a, t, x)
= m(a)χω (x)u(a, t, x)
(a, t, x) ∈ Qa†
= 0,
(a, t, x) ∈ Σa†
Z
a†
β(a, t, x; P)p(a, t, x)da,
=
=
where D = ( ∂p
∂t +
0
p0 (a, x),
(t, x) ∈ (0, T ) × Ω
(a, x) ∈ (0, a† ) × Ω,
(1)
∂p
∂a )
Bedr’Eddine AINSEBA Bordeaux University
Optimal
and Control
INRIA Bordeaux
of Structured
Sud Ouest
Epidemics
Ecole
and
CIMPA,
Public
December
Pointe-à-Pitre,
Prevention
28, 2008Guadeloupe
6 / 5220
Control theory in population dynamics
The adjoint system

Dw (a, t, x)





w (a, t, x)



w (A, t, x)


w (a, T , x)
−µ(a, t, x; P̃)w (a, t, x) + k∆w (a, t, x)
= −β(a, t, x; P̃)w (0, t, x)
= 0,
= 0,
= g (a, x),
(a, t, x) ∈ Q
(a, t, x) ∈ Σ
(t, x) ∈ (0, T ) × Ω
(a, x) ∈ (0, A) × Ω,
(2)
Bedr’Eddine AINSEBA Bordeaux University
Optimal
and Control
INRIA Bordeaux
of Structured
Sud Ouest
Epidemics
Ecole
and
CIMPA,
Public
December
Pointe-à-Pitre,
Prevention
28, 2008Guadeloupe
7 / 5220
Control theory in population dynamics
Fields of use
Agriculture and forest management: The models used for agriculture are
relatively complete and corresponds to the real system. Models of crop
growth and of insect pests have been more and less successfully combined
and optimal strategies have been derived in many situations.
Bedr’Eddine AINSEBA Bordeaux University
Optimal
and Control
INRIA Bordeaux
of Structured
Sud Ouest
Epidemics
Ecole
and
CIMPA,
Public
December
Pointe-à-Pitre,
Prevention
28, 2008Guadeloupe
8 / 5220
Control theory in population dynamics
Estimation and control of the dynamic of Lobesia botrana
Lobesia botrana is a major grapevine pest in Europe. The larvas eat vine
buds during spring , unripe and ripe berries during summer. The grapes
become unfit for human consumption and wine making. The larvas
perforations favorize diseases (Botrytis).
Goal: Control the size of the Larvae population during spring and summer
Bedr’Eddine AINSEBA Bordeaux University
Optimal
and Control
INRIA Bordeaux
of Structured
Sud Ouest
Epidemics
Ecole
and
CIMPA,
Public
December
Pointe-à-Pitre,
Prevention
28, 2008Guadeloupe
9 / 5220
Control theory in population dynamics
Ecology
If a predator can forage on two preys, what would be its diet strategy?
Maximize the net rate of energy intake during foraging.
max
p1 λ1 e1 + p2 λ2 e2
1 + p1 λ1 h1 + +p2 λ2 h2
pi prob to attack prey i λi enconter rate with prey i ei the expected net
energy gained from prey i hi the expected handling time.
If prey 1 is more profitable he11 > he22 thus the predator will exclusively attack
prey 1 when abundance and will compose if the size of prey 1 is below a
threshold (V. Krivan, JTB).
Bedr’Eddine AINSEBA Bordeaux University
Optimal
and Control
INRIA Bordeaux
of Structured
Sud Ouest
Epidemics
Ecole
and
CIMPA,
Public
December
Pointe-à-Pitre,
Prevention
28, 2008 Guadeloupe
10 / 5220
Control theory in population dynamics
Ecology
Consider a population governed with a Mac Kendrick equation with a
mortality µ(a). Find the cheaper reproduction strategy β(a) (R0 is fixed)?
Bedr’Eddine AINSEBA Bordeaux University
Optimal
and Control
INRIA Bordeaux
of Structured
Sud Ouest
Epidemics
Ecole
and
CIMPA,
Public
December
Pointe-à-Pitre,
Prevention
28, 2008 Guadeloupe
11 / 5220
Control theory in population dynamics
Fisheries
Hard to control
Bedr’Eddine AINSEBA Bordeaux University
Optimal
and Control
INRIA Bordeaux
of Structured
Sud Ouest
Epidemics
Ecole
and
CIMPA,
Public
December
Pointe-à-Pitre,
Prevention
28, 2008 Guadeloupe
12 / 5220
Control of epidemics
Motivations
Epidemics, and notably AIDS and its estimated 38.6 million people living
with HIV worldwide in 20051 , constitute a major public health issue for
governments. Among the rich mathematical epidemiology literature, some
have used optimal control techniques to define the path of intervention
that should be followed by either governments or international
organizations in charge of public health issues. In a recent paper, Behncke
(Opt. Cont. Appl. Meth. (2000)) proposes a nice general and analytical
study of various public interventions such as vaccination, screening and
health promotion campaigns to control deterministic epidemics;
1
UNAIDS: 2006 Report on the global AIDS epidemic.
Bedr’Eddine AINSEBA Bordeaux University
Optimal
and Control
INRIA Bordeaux
of Structured
Sud Ouest
Epidemics
Ecole
and
CIMPA,
Public
December
Pointe-à-Pitre,
Prevention
28, 2008 Guadeloupe
13 / 5220
Control of epidemics
Motivations
Behncke shows that the effort control should be maximal on some initial
time interval. This quite intuitive result is however not observed
empirically: for most epidemics public intervention has been delayed and,
once set up, has grown with the importance of the epidemics. The case of
AIDS is impressive: from UNAIDS’ launch in 1996 until 2005, available
annual funding for low and middle income countries increased 28-fold1 .
Moreover, all around the world, local government reactions to the epidemic
have been highly heterogeneous, and strongly related to national wealth.
Bedr’Eddine AINSEBA Bordeaux University
Optimal
and Control
INRIA Bordeaux
of Structured
Sud Ouest
Epidemics
Ecole
and
CIMPA,
Public
December
Pointe-à-Pitre,
Prevention
28, 2008 Guadeloupe
14 / 5220
Control of epidemics
Motivations
In this talk, we argue that introducing some economic constraints -and the
level of economic development- in an epidemic model, may yield to a more
realistic path of public intervention. The idea is simple once admitted that
public health policies are not only an immediate cost but also affect the
future wealth of the economy. Resources engaged in health expenses are
indeed not devoted to investment. However, public health intervention
reduces the epidemic whose propagation increases with the prevalence rate.
The optimal path of intervention should hence integrate this intertemporal
trade-off.
Bedr’Eddine AINSEBA Bordeaux University
Optimal
and Control
INRIA Bordeaux
of Structured
Sud Ouest
Epidemics
Ecole
and
CIMPA,
Public
December
Pointe-à-Pitre,
Prevention
28, 2008 Guadeloupe
15 / 5220
Control of epidemics
Motivations
We study a standard model of optimal economic growth (a ‘Ramsey’
model) in which the whole population may be affected by an epidemic
similar to Naresh, Tripathi and Omar (2006). The public health control is a
screening procedure, in the tradition of the pioneer work of Wickwire (75),
that identifies infected individuals to heal them and to remove them from
the infectious population. As one may imagine, the key assumption relies
on the choice of the social welfare function. We have chosen a general
criterion that combine standard objectives used in mathematical
epidemiology and in economics. The first one is the minimization of the
number of infected people at the end of the planned period while the
second is the present discount value of the product of individual utility and
the size of the population.
Bedr’Eddine AINSEBA Bordeaux University
Optimal
and Control
INRIA Bordeaux
of Structured
Sud Ouest
Epidemics
Ecole
and
CIMPA,
Public
December
Pointe-à-Pitre,
Prevention
28, 2008 Guadeloupe
16 / 5220
The SIR model
The dynamics of the epidemic
The population, denoted P (t), is initially divided into three classes SIR. It
is assumed that only individuals in the class I are active in the transmission
of the disease (unfortunately data shows that this is not the case) through
two distinct channels: horizontal contamination, modelled using a law of
action in frequency with parameter λ > 0 and vertical contamination
assuming that the proportion of infected child born from infected parents is
a constant denoted ι ∈ [0, 1]. Infected individuals develop the illness at rate
δ1 > 0 and consequently 1/δ1 is the average duration of the incubation
period. Finally, the per capita birth rate is β > 0 for susceptible and
infected individuals while it is zero for those who belongs to class R (t); the
basic per capita death rate is µ > 0 and is augmented by µ1 > 0 and
µ2 > µ1 for respectively the infected and the ill individuals.
Bedr’Eddine AINSEBA Bordeaux University
Optimal
and Control
INRIA Bordeaux
of Structured
Sud Ouest
Epidemics
Ecole
and
CIMPA,
Public
December
Pointe-à-Pitre,
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28, 2008 Guadeloupe
17 / 5220
The SIR model
The SIR Model
Hence, the dynamics of the population writes:
dS (t)
dt
dI (t)
dt
dR (t)
dt
S (t) I (t)
,
S (t) + I (t)
S (t) I (t)
= (ιβ − µ − µ1 ) I (t) + λ
− δ1 I (t) ,
S (t) + I (t)
= (β − µ) S (t) + (1 − ι) βI (t) − λ
= δ1 I (t) − (µ + µ2 ) R (t) ,
(3)
(4)
(5)
with S (0) , I (0) and R (0) given.
Bedr’Eddine AINSEBA Bordeaux University
Optimal
and Control
INRIA Bordeaux
of Structured
Sud Ouest
Epidemics
Ecole
and
CIMPA,
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December
Pointe-à-Pitre,
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28, 2008 Guadeloupe
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The SIR model
The SIR Model
This system is easy to solve introducing variable a (t) = I (t) /S (t). Using
(3) and (4) yields the following logistic equation:
da (t)
= νa (t) − πa2 (t) ,
dt
(6)
where ν = λ − (µ + µ1 − ιβ + δ1 ) − (β − µ) and π = (1 − ι) β. The
dynamics of a (t) is therefore given by:
a (t) =
a (0) +
ν
π a (0)
,
ν
−νt
π − a (0) e
(7)
and admits two steady-states if ν > 0: a∗ = 0, which is unstable and
a∗∗ = ν/π, which is stable and one steady-state if ν ≤ 0: a∗ = 0, which is
stable.
Bedr’Eddine AINSEBA Bordeaux University
Optimal
and Control
INRIA Bordeaux
of Structured
Sud Ouest
Epidemics
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and
CIMPA,
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December
Pointe-à-Pitre,
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The screening procedure
Modeling
The epidemic can be controlled with screening. This procedure is imposed
on a fraction of the population which is not ill and allows distinguishing the
susceptible from the infected individuals. Once screened, an infected
individual belongs to a new subclass of the population, denoted D (t),
which is not active in the transmission of the epidemics (neither by
horizontal nor vertical contamination). He is supposed to have the same
mortality rate as any infected individual but he benefits from public
medication and hence has a lower probability to develop the illness:
δ2 ∈ [0, δ1 ].
Bedr’Eddine AINSEBA Bordeaux University
Optimal
and Control
INRIA Bordeaux
of Structured
Sud Ouest
Epidemics
Ecole
and
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December
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The screening procedure
The model
The dynamic system is now composed of (3) and:
dI (t)
dt
dD (t)
dt
dR (t)
dt
S (t) I (t)
S (t) + I (t)
−δ1 I (t) − σ (S (t) + I (t)) v (t) I (t) ,
= (ιβ − µ − µ1 ) I (t) + λ
(8)
= σ (S (t) + I (t)) v (t) I (t) − (µ + µ1 ) D (t) − δ2 D (t) , (9)
= δ1 I (t) + δ2 D (t) − (µ + µ2 ) R (t) .
(10)
Bedr’Eddine AINSEBA Bordeaux University
Optimal
and Control
INRIA Bordeaux
of Structured
Sud Ouest
Epidemics
Ecole
and
CIMPA,
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December
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The screening procedure
The functional response to the screening offer
σ (S (t) + I (t)) v (t) is the rate at which individuals are screened and
removed from the infected population. The interpretation is the following:
I (t) / (S (t) + I (t)) dt is the probability that a screened individual is
actually infected, σ (S (t) + I (t)) (S (t) + I (t)) is the functional response
to the screening offer and v (t) represents the screening strategy, i.e. the
number of individual who are planned to be screened in the unit of time.
Bedr’Eddine AINSEBA Bordeaux University
Optimal
and Control
INRIA Bordeaux
of Structured
Sud Ouest
Epidemics
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and
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December
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The classical optimal screening procedure
The cost of the disease
Cost of the screening compagn
Z
T
pσ(S + I )(t)v (t)(I + S)(t)dt
0
Cost of the treatment
Z
T
qσ(S + I )(t)v (t)I (t)dt
0
Social cost
Z
T
I (t)dt
0
Terminal cost
cI (T )
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The economic framework
The Ramsey Model
The economic part of the model is based on the Ramsey model (see Cass
(Rev. Eco. Studies (65)) and Koopmans (65). It supposes the existence of
a social planner whose goal is to choose an optimal consumption path in
order to maximize the welfare of the population under some economic
constraints. There exists one good which is produced using a neoclassic
production function F (., .) where the inputs are capital, K (t), and labor,
L (t).
Bedr’Eddine AINSEBA Bordeaux University
Optimal
and Control
INRIA Bordeaux
of Structured
Sud Ouest
Epidemics
Ecole
and
CIMPA,
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December
Pointe-à-Pitre,
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The economic framework
Hypotheses
We assume that F is C ∞ (R+ , R+ ), homogenous of degree 1, and satisfies
F (0, L) = F (K , 0) = 0 and ∂F /∂K > 0, ∂F /∂L > 0, ∂ 2 F /∂K 2 < 0,
∂ 2 F /∂L2 < 0. Moreover, limK →0 F (K , L) = limL→0 F (K , L) = 0 and
limK →∞ F (K , L) = limL→∞ F (K , L) = ∞. Production can be used for
consumption C (t), health expenses H (t), or for adding to the capital
stock dK (t) /dt.
Bedr’Eddine AINSEBA Bordeaux University
Optimal
and Control
INRIA Bordeaux
of Structured
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The economic framework
Disease cost
Health expenses share into preventive and treatment cost: screening cost is
p1 σ (S (t) + I (t)) (S (t) + I (t)) v (t), where p1 > 0 denotes the unit cost
of screening. We assume that healing costs of screened infected individuals
and ill individuals are proportional to capital: k (t) p2 D (t) and k(t) p3 R (t)
with p3 > p2 > 0.
Bedr’Eddine AINSEBA Bordeaux University
Optimal
and Control
INRIA Bordeaux
of Structured
Sud Ouest
Epidemics
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and
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December
Pointe-à-Pitre,
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The economic framework
Modeling
Using variables per capita (k = K /P, c = C /P), the resources constraint
of the economy writes:
dk (t)
L (t)
k (t) dP (t)
D (t)
R (t)
= F k (t) ,
− c (t) −
− p2
− p3
dt
P (t)
P (t) dt
P (t)
P (t)
σ (S (t) + I (t)) (S (t) + I (t)) v (t)
−p1
.
(11)
P (t)
where P (t) is the total population. The second constraint is full
employment; since it is assumed that ill individuals are not able to work,
the constraint writes:
L (t) = S (t) + I (t) + D (t) .
(12)
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The social planner’s problem
The cost/gain function
The objective is supposed to be a combination of two objects that
constitutes standard objectives in respectively epidemiological and
economic framework. The first one is the minimization of the number of
infected individuals at time T (social cost). The economic one is the
maximization of the discounted sum of instantaneous utility.
Bedr’Eddine AINSEBA Bordeaux University
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The social planner’s problem
The cost/gain function
It is assumed that individuals derive utility from consumption, which can be
measured using function u (c (t)) with u 0 (.) > 0, u 00 (.) < 0 and
limc→0 u 0 (c) = ∞. The economic objective is:
Z
0
T
e −ρt P (t) u (c (t)) dt + e −ρT P (T )
u (f (k (T )) − nk (T ))
,
ρ−n
(13)
where ρ is the discount rate, n is a bound on the population increase, and
ρ > n to ensure the convergence for large T . The terminal cost
e −ρT G (k (T )) may be choosen as in (Mercenier (94)) to statisfy the
invariance property.
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The social planner’s problem
The cost/gain function
By adding the economic and the epidemiological objectives, the social
planner’s objective writes:
T
e −ρT P (T ) u (f (k (T )) − nk (T ))
−ε2 I 2 (T ) .
ρ
−
n
0
(14)
where εi is the relative weight of the corresponding objective. The problem
of the social planner is then to choose c (t) and v (t) that maximize (14)
subject to (3), (8), (9), (10), (11) and
Z
ε1
e −ρt P (t) u (c (t)) dt+
k (t) ≥ 0, c (t) ≥ 0, 0 ≤ v (t) ≤ ζ,
k0 > 0, S0 > 0, I0 > 0, D0 ≥ 0, R0 > 0 given.
(15)
(16)
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Optimal
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The social planner’s problem
The optimality system
The optimal solution is the solution of the system composed by five state
equations (3), (8), (9), (10), (11) and by the following five adjoint
equations (where the dependence of the variables to time has been
omitted):
dq1
dt
λI (P − S)
= ε1 u (c) + q1 [ρ − (β − µ)] + [q1 − q2 ]
2
P
q5
L
+ [q2 − q4 ] σ 0 ((S + I )) vI − 2 RF20 k,
P
P
q5
+ 2 {k [β (R + D) + µ1 (I + D) + µ2 R] − p2 kD − p3 kR}
P
q5 p1 0
+ 2 σ ((S + I )) (S + I ) Pv + σ ((S + I )) (D + R) v ,(17)
P
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The social planner’s problem
The optimality system
dq2
dt
λS (P − I )
= ε1 u (c) + [q2 − q1 ] (1 − ι) β −
− δ1 q3
P2
+q2 [ρ + µ + µ1 − β + δ1 ]
+ [q2 − q4 ] σ 0 (S + I ) vI + σ (S + I ) v
q5
L
0
+ 2 −RF2 k,
+ k [β (R + D) − µ1 (S + R) + µ2 R]
P
P
q5
+ 2 p1 Pv σ 0 (S + I ) (S + I ) + σ (S + I ) − p2 kD − p3 kR(18)
,
P
Bedr’Eddine AINSEBA Bordeaux University
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The social planner’s problem
The optimality system
dq3
dt
λSI
= ε1 u (c) + q3 (ρ + µ + µ2 ) − (q1 − q2 ) 2
P
q5
L
+ 2 LF20 k,
+ k (−β (S + I ) + µ1 (I + D) − µ2 (P − R))
P
P
q5
(19)
+ 2 [−p1 σ (S + I ) (S + I ) v − p2 kD + p3 k (P − R)] ,
P
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The social planner’s problem
The optimality system
dq4
dt
λSI
= ε1 u (c) + q4 (ρ + µ + µ1 + δ2 ) − δ2 q3 − (q1 − q2 ) 2
P
q5
L
0
+ 2 −RF2 k,
+ k (−β (S + I ) − µ1 (S + R) + µ2 R)
P
P
q5
+ 2 [−p1 σ (S + I ) (S + I ) v − p2 k (P − D) + p3 kR] ,
(20)
P
and
dq5
L
dP (t)
D (t)
R (t)
0
= q5 ρ − F1 k,
+
+ p2
+ p3
dt
P
P (t) dt
P (t)
P (t)
(21)
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The social planner’s problem
The optimality system
Finally, the system shall satisfy the following transversality conditions:
q1 (T ) = 0, q2 (T ) = 2ε2 I (T ) , q3 (T ) = 0, q4 (T ) = 0,
0
q5 (T ) = −G (k (T )) .
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The social planner’s problem
The optimality system
Moreover, the consumption per capita c (t) solves:
ε1 P (t) u 0 (c (t)) = q5 ,
(22)
while the control v (t), which is linear in the objective, satisfies:
0 if [q2 (t) − q4 (t)] I (t) + q5 (t) [p1 (S (t) + I (t)) /P (t)] < 0
v (t) = .
ζ if [q2 (t) − q4 (t)] I (t) + q5 (t) [p1 (S (t) + I (t)) /P (t)] > 0
(23)
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The social planner’s problem
Remark that differentiating (22) with respect to time and rearranging using
(21) yields the standard dynamics for consumption:
dc (t)
u 0 (c (t))
L (t)
D (t)
R (t)
0
= 00
ρ − F1 k (t) ,
+ p2
+ p3
. (24)
dt
u (c (t))
P (t)
P (t)
P (t)
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A Numerical approach
Numerical simulations are needed to solve the optimal dynamics. An
iterative method is used as follows: first, the state equations are solved
with a guess for the controls over the simulated time using a semiimplicit
scheme. Then, using the transversality conditions and the current iteration
solution for the state equations, the adjoint equations are solved with a
backward semiimplicit scheme. Then, the controls are updated using
gradients methods. This procedure is repeated till the values of unknowns
obtained at a given iteration are very close to the ones obtained at the
previous iteration. Figures are obtained with function bcong from IMSL
module. In order to cope with the problem which is initialy linear
R T in the
effort command, we introduce in the objective an extra term ε 0 v (t)2 dt,
and repeat the previous algorithm with smaller and smaller ε, in order to
mimic the linear objective.
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A Numerical approach
Functional forms that have been chosen are:
√
c,
u (c) =
F (K , L) = AK
1
σ (x) =
.
x
(25)
0.3 0.7
L
,
(26)
(27)
The functional for the instantaneous utility implies a constant utility of
substitution with substitution effects dominating the revenue effects. The
production is Cobb-Douglas with a share of capital in GDP that is equal to
0.3. Finally, the specification for function σ (.), which is the one used by
Hseih [?] and Velasco and al [?], implies that the functional response to the
screening offer is normalized to 1.
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A Numerical approach
Data
The values of the demographic parameters are:
µ
µ 1 µ2
β
ι
λ
δ1
δ2
0.01 0.03 0.6 0.03 1/3 0.5 0.125 0.07
Table 1: Demographic parameters
which, given the chosen discretization path, correspond to annual
population growth rate that is lower than 1%.
We can refer to Hyman and all (88,00) for parameters relative to HIV.
Duration before developping AIDS after contacting HIV is 8 years.
Moreover, the contamination coefficient is fairly high implying that without
the set up of the screening procedure, the epidemic would last forever.
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A Numerical approach
The economic parameters are chosen such that:
ρ p1 p2 p3
0 0.5 0.5 1
Table 2: Economic parameters
Hence, there is no discount rate, meaning that each generation is treated
equally by the social planner.
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A Numerical approach
We compare two situations, the one of country with a very high level of
prevalence, such as botswana (Almeder (04)), and the one with low
prevalence. In this last case, the country for which we take initial
conditions (according to data from INVS (2005)) is France. Figures (1)
and (2) give prevalence from 2000 to 2050 for these two countries without
screening procedure.
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A Numerical approach
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A Numerical approach
The simulations compute the optimal screening v (t) for two economies
that have distinct initial level of capital per capita: it is supposed 15-fold
higher in the ‘rich’ economy than in the ‘poor’ one (NGP for Botswana is
5.4 milliard dollars compare to. 1.423 for France in 2005). Remark
nevertheless that the terminal level of capital per capita is the same in the
two economies. Then, two cases are studied depending on the initial level
of prevalence of the epidemic in the economy. In case 1, a relatively high
level of prevalence is considered:
S (0) /P (0) I (0) /P (0) R (0) /P (0) D (0) /P (0)
0.89
0.1
0.01
0
Table 3: Initial conditions for case 1
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A Numerical approach
The optimal paths are then plotted in the following figure:
Figure 3: high prevalence with ε2 = 0.9, ε1 = 0.1
Population dynamics
Figure 3 shows that the screening control is active along all the period,
that enable to reduce the number of new cases. It can be shown that with
an other policy, which takes at the same weight economics and wealth by
taking (ε2 = 0.5, ε1 = 0.5), the same kind of results can be obtain. The
only difference between these two policies is the reduction of the number of
infectives.
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A Numerical approach
Moreover, we can see that on the contrary to the static case, the
preventive effort is not necessary higher at the beginning of the period,
than at the end of the period, as it can be shown
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A Numerical approach
In case 2, a :low level of prevalence is considered:
S (0) /P (0) I (0) /P (0) R (0) /P (0) D (0) /P (0)
0.9855
0.012
0.0025
0
Table 4: Initial conditions for case 2
The optimal paths are plotted in the following figure:
Figure 4: Low prevalence
Population dynamics
Figure 4 shows that when a mixed policy is taken (ε2 = 0.5, ε1 = 0.5), no
screening is done, even with a high level of capital.
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A Numerical approach
On the contrary, when a policy taking more into account health is followed
(figure 5), screening enable to reduce the prevalence. More over, figure 6
shows that screening effort is higher at the begining of the plan.
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Conclusion
1. Public prevention of epidemics within an economical context
Joint work with E. Augeraud (La Rochelle University), H. D’Albis
(Toulouse University)
2. Contact Tracing program
3. The screened individual are active in the infection transmission.
4. Age since infection
Joint work with M. Iannelli (Trento University)
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The case of age structure
Let S(t), i(θ, t), R(t) describe the status of the population in the usual
epidemic classes of susceptibles, infected, removed individuals. Namely
S(t) and R(t) denote the number of individuals in the corresponding class,
while i(θ, t) denotes the density of individuals with respect to θ which is
the age of infection, i.e. the time elapsed since the individual has been
infected. We assume a simple demographic process summarized by a
recruitment rate of susceptibles Λ and a mortality rate µ, so that the
classical S-I-R model structured by the age of infection and including a
screening strategy is governed by the following system
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The case of age structure

i)










ii)





S 0 (t) = Λ − λ(t)S(t) − µS(t)
∂i
∂i
+
+ [µ + ν(θ) + γ(θ) + v (t)Ψ(N(t))φ(θ)] i(θ, t) = 0
∂t
∂θ
Z θm

0

iii)
R
(t)
=
[γ(θ) + v (t)Ψ(N(t))φ(θ)] i(θ, t)d θ − µR(t)



0





Z θm




i(θ, t)d θ + R(t)
 iv ) N(t) = S(t) +
0
(28)
endowed with the following incidence condition
i(0, t) = λ(t)S(t)
(29)
and the initial conditions
S(0) = S0 ,
i(θ, 0) = i0 (θ),
R(0) = R0
(30)
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The case of age structure
In (28) and (29) λ(t) is the force of infection and is assumed to have the
following constitutive form
λ(t) =
c
N(t)
Z
θm
χ(θ)i(θ, t)d θ
(31)
0
where c is the contact rate, χ(θ) the infectiveness of a contact (depending
of the disease advancement). We note that in (28, iv), N(t) denotes the
total-active population. This problem will be studied and a functional, to
be minimized, composed by social costs, screening costs and treatments
cost will be build. We will show how we can obtain the existence of optimal
screening strategies compatible with the program purposes.
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