optimal control of an influenza model with seasonal forcing and...
TRANSCRIPT
Optimal Control of an Influenza Model with Seasonal Forcing and
Age Structure
Jeehyun Lee ∗, Jungeun Kim †, Hee-Dae Kwon ‡
Abstract
This study considers an optimal intervention strategy for influenza outbreaks. Variationsin the SEIAR model are considered to include seasonal forcing and age structure, and controlstrategies include vaccination, antiviral treatment, and social distancing such as school closures.We formulate an optimal control problem by minimizing the incidence of influenza outbreakswhile considering intervention costs. We examine the effects of delays in vaccine production,seasonal forcing, and age-dependent transmission rates on the optimal control and suggest someoptimal strategies through numerical simulations.
Keywords: Optimal control, Deterministic influenza model, Seasonal forcing, Age-dependenttransmission rate
1 Introduction
The World Health Organization (WHO) announced the emergence of Pandemic Influenza A(H1N1)on April 24, 2009 and raised the alert levels to declare pandemic on June 11, 2009 [33, 34]. FromApril 2009 to August 2010, more than 214 countries worldwide reported laboratory-confirmed casesincluding 18,449 fatal cases to the WHO [32]. A wide range of intervention strategies have beenimplemented to control the spread of influenza. In developing a response plan for a disease outbreak,it is important to predict the dynamics of the epidemic, evaluate various control strategies, anddesign the best strategy.
Mathematical models provide powerful tools for investigating the dynamics and control of in-fectious diseases. Previous studies have examined various models for predicting and assessingintervention strategies [3, 5, 11, 14, 25, 26, 35]. These recent models have been based on networksand stochastic simulations. Hethcote conducted several notable surveys of basic deterministic epi-demiological models [19, 20], and others have considered the effects of variations of these models.Brauer suggested several less traditional variations, including compartments for asymptomatic,quarantined, and isolated individuals [2, 4]. In addition, several studies have applied the optimalcontrol to specific diseases [10, 13, 21, 22, 23]. Recent years have witnessed considerable advancesin decisions on the timing and distribution of vaccination [15, 16, 27, 29, 30]. However, policy
∗Department of Computational Science and Engineering, Yonsei University, Shinchondong, Seodaemungu, Seoul120-749, Republic of Korea ([email protected]).
†Department of Computational Science and Engineering, Yonsei University, Shinchondong, Seodaemungu, Seoul120-749, Republic of Korea ([email protected]).
‡Department of Mathematics, Inha University, Yonghyundong, Namgu, Incheon 402-751, Republic of Korea([email protected]).
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optimization remains an open question because the dynamics of epidemics is complicated and de-pends on various factors. For instance, the effectiveness of an intervention strategy depends on thestructure of the population, seasonal forcing, and delays in vaccine production.
The present paper considers a deterministic model incorporating various groups and seasonalforcing. In the model, control strategies are composed of vaccination, antiviral treatment, andsocial distancing (e.g., school closures, travel restrictions, and quarantine programs). We seek anoptimal control that can minimize the incidence while considering intervention costs in varioussettings. We begin by introducing influenza models, including the SEIAR model, a SEIAR modelwith seasonal forcing, and an age-structured SEIAR model with analysis results. We then presentthe formulation of the optimal control problem with the proof of existence of an optimal controlfunction. In addition, we derive an optimality system that characterizes the optimal control.Finally, we present the results of numerical simulations for each model under various parametervalues and conclude with a summary.
2 Basic Influenza Model
The proposed influenza model is based on the SEIAR model, which is an extension of the standardSEIR model described in [2]. The SEIAR model includes two additional properties. First, infectedindividuals in the exposed stage can either develop symptoms and move to an infective stageor develop no symptoms and move to an asymptomatic infective stage. Second, some infectiveindividuals withdraw from contact with others after developing symptoms. The system of ODEsdescribing the influenza dynamics is given by
S′ = −βSΛ
E′ = βSΛ − κE
I ′ = pκE − αI
A′ = (1 − p)κE − ηA
R′ = fαI + ηA
(2.1)
with Λ = ǫE + (1 − q)I + δA and initial conditions
S(0) = S0, E(0) = E0, I(0) = I0, A(0) = A0, R(0) = R0.
Figure 1 shows a flow diagram for model (2.1). This model includes five key compartments-S,E, I,A, and R- which denote the numbers of individuals in the susceptible, exposed, (symp-tomatic) infective, asymptomatic, and recovered compartments, respectively, and N is the totalpopulation size. A fraction p of exposed members proceed to the infective compartment at the rateκ, whereas the rest go to the asymptomatic infective compartment, (also at the rate κ). Infectiveindividuals leave the compartment at the rate α and a fraction f of members recover from thedisease when they leave I. Exposed members have their infectivity reduced by a factor of ǫ, with0 ≤ ǫ ≤ 1. The compartment E represents the latent stage when ǫ = 0 and the initial asymp-tomatic and mildly infectious stage when ǫ > 0. Asymptomatic individuals have their infectivityreduced by a factor of δ, with 0 ≤ δ ≤ 1, and go to the recovered compartment at the rate η.On average infective individuals reduce their contact rate by a factor of q. The number of contactevents sufficient for transmitting an infection per unit time per individual is βN by mass actionincidence. We refer the reader to [2], for a more detailed explanation of the SEIAR model.
2
Figure 1: SEIAR epidemic model.
We use the approach in [31] to calculate the basic reproduction number
R0 = βS0
(
ǫ
κ+
(1 − q)p
α+
δ(1 − p)
η
)
We adopt the notation g∞ for limt→∞ g(t). A similar model has been analyzed in [1], and it hasbeen shown that E∞ = I∞ = A∞ = R∞ = 0 and that the final size relation is
lnS0
S∞
= R0
(
1 −S∞
S0
)
We now extend the SEIAR model to include controls of vaccination, antiviral treatment, andsocial distancing (e.g., school closures and the cancellation of large public gatherings), which isdefined by the following system of ODEs
S′ = −β(1 − σ(t))SΛ − ν(t)S
E′ = β(1 − σ(t))SΛ − κE
I ′ = pκE − αI − τ(t)I
A′ = (1 − p)κE − ηA
R′ = fαI + τ(t)I + ηA + ν(t)S
(2.2)
with Λ = ǫE + (1 − q)I + δA and initial conditions
S(0) = S0, E(0) = E0, I(0) = I0, A(0) = A0, R(0) = R0.
The control function ν(t) measures the rate at which susceptible individuals are vaccinated duringeach time period, and the control function τ(t) measures the rate at which infectious individualsare treated during each time period. The contact rate is reduced by the control factor σ(t) becauseof social distancing.
In developing response plans for disease outbreaks, decision makers seek optimal responsesthat can minimize the incidence and disease-related mortality while considering the cost of eachmitigation strategy. The control theoretic approach assigns costs to both intervention and infectionand looks for an optimal policy that can minimize the total combined cost. Simply, we can view thecost of an infection to be proportional to the number of infected individuals during the course of the
3
epidemic. The intervention cost depends on the specific type of intervention under consideration.Here we aim to minimize the objective functional given by
J(ν, τ, σ) =
∫ T
0
PI(t) + Qν2(t) + Rτ2(t) + Wσ2(t)dt, (2.3)
where P , Q, R, and W are the weight constants of the infected individuals and control measures.They can be chosen to balance the units in the integrand and change the relative importance ofminimizing of infected individuals and intervention efforts. We assume that there are practicallimitations on the maximum rate at which individuals may be vaccinated or treated during a givenperiod of time and the maximum effect of social distancing. We seek the optimal controls ν, τ andσ in Uad such that
minimize J(ν, τ, σ) subject to (2.2) (2.4)
where Uad = {(ν, τ, σ)| ν, τ and σ are Lebesgue-integrable, 0 ≤ ν ≤ νmax, 0 ≤ τ ≤ τmax, and 0 ≤σ ≤ σmax} is the control set. The basic framework of an optimal control problem is to prove theexistence of an optimal control and then characterize it.
2.1 Existence of an optimal control
The existence of a solution to the optimal control problem can be obtained by verifying sufficientconditions. We refer to the conditions in Theorem III.4.1 and its corresponding Corollary in [12].The boundedness of solutions to the system (2.2) for the finite time interval is needed to establishthese conditions. Note that the quantities S,E, I,A, and R decrease only in proportion to theirpresent sizes, respectively, and thus, all variables remain nonnegative if the initial values are non-negative. To establish the upper bounds for the solutions, we consider an equation for the totalpopulation size N . N satisfies N ′ = −(1 − f)αI from the system (2.2) and N is bounded aboveby N(0). Because S,E, I,A, and R are all nonnegative, the upper bound for N is also the upperbound for S,E, I,A, and R. Let ~x = [S,E, I,A,R]T and ~u = [ν, τ, σ]T denote the vector of modelstates and controls, respectively.
Theorem 2.1. There exists an optimal control (ν∗, τ∗, σ∗) to the problem (2.4).
Proof. We list the requirements from the theorem in [12] as follows and verify nontrivial require-ments. Let r(t, ~x, ~u) be the right hand side of (2.2).
1. r is of class C1 and there exists a constant C such that
|r(t, 0, 0)| ≤ C, |r−→x (t,−→x ,−→u )| ≤ C(1 + |−→u |), |r−→u ((t,−→x ,−→u )| ≤ C.
2. The admissible set F of all solutions to system (2.2) with corresponding control in Uad isnon-empty.
3. r(t, ~x, ~u) = a(t, ~x) + b(t, ~x)~u.
4. The control set U = [0, νmax] × [0, τmax] × [0, σmax] is closed, convex and compact.
5. The integrand of the objective functional is convex in U .
4
In order to verify these conditions, we write
r(t, ~x, ~u) =
−β(1 − σ(t))SΛ − ν(t)Sβ(1 − σ(t))SΛ − κEpκE − αI − τ(t)I(1 − p)κE − ηA
fαI + τ(t)I + ηA + ν(t)S
(2.5)
It is clear that r(t, ~x, ~u) is of class C1 and |r(t, 0, 0)| = 0. In addition, we have
|r~x(t, ~x, ~u)| =
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
−β(1 − σ)Λ − ν −β(1 − σ)Sǫ −β(1 − σ)S(1 − q) −β(1 − σ)Sδ 0β(1 − σ)Λ β(1 − σ)Sǫ − κ β(1 − σ)S(1 − q) β(1 − σ)Sδ 0
0 pκ −α − τ 0 00 (1 − p)κ 0 −η 0ν 0 fα + τ η 0
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
|r~u(t, ~x, ~u)| =
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
−S 0 βSΛ0 0 −βSΛ0 −I 00 0 0S I 0
∣
∣
∣
∣
∣
∣
∣
∣
∣
∣
Since S,E, I,A and R are bounded, there exists a constant C such that
|r(t, 0, 0)| ≤ C, |r~x(t, ~x, ~u)| ≤ C(1 + |~u|), |r~u(t, ~x, ~u)| ≤ C.
Due to Condition 1, there exists a unique solution for (2.2) for a constant control, which furtherimplies Condition 2. r is expressed as a linear function of the control variables with coefficientsdependent on time and the state variables in (2.5), which is the Condition 3. Conditions 4 and 5are obvious from the definition.
2.2 Optimality system
We characterize the optimal control functions by using Pontryagin’s Maximum Principle [28]. TheHamiltonian equation is given by
H(~x, ~u,~λ) = PI + Qν2 + Rτ2 + Wσ2
− λ1(β(1 − σ)SΛ + νS) + λ2(β(1 − σ)SΛ − κE)
+ λ3(pκE − αI − τI) + λ4((1 − p)κE − ηA) + λ5(νS + fαI + τI + ηA)
where ~λ = [λ1, λ2, λ3, λ4, λ5]T .
We find the optimal controls by solving the state equations (2.2) with initial conditions and theadjoint equations
5
λ′
1 = β(1 − σ)Λ(λ1 − λ2) + ν(λ1 − λ5)
λ′
2 = β(1 − σ)ǫS(λ1 − λ2) + κλ2 − pκλ3 − (1 − p)κλ4
λ′
3 = −P + β(1 − σ)(1 − q)S(λ1 − λ2) + (α + τ)λ3 − (fα + τ)λ5
λ′
4 = β(1 − σ)δS(λ1 − λ2) + η(λ4 − λ5)
λ′
5 = 0
with the transversality conditions
λ1(T ) = λ2(T ) = λ3(T ) = λ4(T ) = λ5(T ) = 0
The optimal control functions satisfy
ν∗ = min
(
νmax,max
(
0,(λ1 − λ5)S
2Q
))
τ∗ = min
(
σmax,max
(
0,(λ3 − λ5)I
2R
))
σ∗ = min
(
νmax,max
(
0,−(λ1 − λ2)βSΛ
2W
))
3 Variations of the SEIAR model
The SEIAR model of an epidemic can be extended to include various features of influenza thatcan have considerable influence on control strategies for epidemics. We consider two variationsto reflect seasonal force and the age-dependent structure. In this section, we provide the modelequations and an optimality system for variations of the SEIAR model. We omit the derivationsand theories that are similar to the results presented in the previous section for the SEIAR model.
3.1 Seasonal SEIAR model
An incidence of influenza exhibits strong seasonal fluctuations in temperate regions throughout theworld, which concentrate the mortality and morbidity burden of the disease in a few months eachyear. Despite eight decades of research, the causes of these fluctuations remain unclear. Thesefluctuations may be due to some mechanism inducing seasonal changes in the effective person-to-person transmission rate (often referred to as seasonal forcing). It has been shown that largefluctuations in the incidence of influenza may be caused by undetectably small seasonal changes inthe influenza transmission rate that are amplified by the dynamical resonance in [8]. The seasonalSEIAR model is derived by allowing the contact rate to vary sinusoidally according to the formula
β(t) = β0
(
1 + β1 cos
(
2πt
365
))
6
where β0 denotes the basic contact rate without seasonal forcing and β1 denotes the magnitude ofseasonal fluctuations.
The system of state equations for the seasonal SEIAR model is given by
S′ = −β(t)(1 − σ(t))SΛ − ν(t)S
E′ = β(t)(1 − σ(t))SΛ − κE
I ′ = pκE − αI − τ(t)I
A′ = (1 − p)κE − ηA
R′ = ν(t)S + fαI + τ(t)I + ηA
(3.1)
with Λ = ǫE + (1 − q)I + δA and initial conditions
S(0) = S0, E(0) = E0, I(0) = I0, A(0) = A0, R(0) = R0.
We define the same objective functional given in equation (2.3)
J(ν, τ, σ) =
∫ T
0
PI(t) + Qν2(t) + Rτ2(t) + Wσ2(t)dt
and determine the optimal controls ν, τ and σ in Uad such that
minimize J(ν, τ, σ) subject to (3.1) (3.2)
where Uad = {(ν, τ, σ)|ν, τ and σ are Lebesgue-integrable, 0 ≤ ν ≤ νmax, 0 ≤ τ ≤ τmax, and 0 ≤σ ≤ σmax}.
The Hamiltonian equation is given by
H(~x, ~u,~λ) = PI + Qν2 + Rτ2 + Wσ2 − λ1(β0(1 + β1 cos(2πt/365))(1 − σ)SΛ + νS)
+ λ2(β0(1 + β1 cos(2πt/365))(1 − σ)SΛ − κE) + λ3(pκE − αI − τI)
+ λ4((1 − p)κE − ηA) + λ5(νS + fαI + τI + ηA).
We find the optimal controls by solving state equations (3.1) with initial conditions and theadjoint equations
λ′
1 = β0(1 + β1 cos(2πt/365))(1 − σ)Λ(λ1 − λ2) + ν(λ1 − λ5)
λ′
2 = β0(1 + β1 cos(2πt/365))(1 − σ)ǫS(λ1 − λ2) + κλ2 − pκλ3 − (1 − p)κλ4
λ′
3 = −P + β0(1 + β1 cos(2πt/365))(1 − σ)(1 − q)S(λ1 − λ2) + (α + τ)λ3 − (fα + τ)λ5
λ′
4 = β0(1 + β1 cos(2πt/365))(1 − σ)δS(λ1 − λ2) + η(λ4 − λ5)
λ′
5 = 0
with the transversality conditions
λ1(T ) = λ2(T ) = λ3(T ) = λ4(T ) = λ5(T ) = 0. (3.3)
7
The optimal control functions satisfy
ν∗ = min
(
νmax,max
(
0,(λ1 − λ5)S
2Q
))
τ∗ = min
(
σmax,max
(
0,(λ3 − λ5)I
2R
))
σ∗ = min
(
νmax,max
(
0,−(λ1 − λ2)β0(1 + β1 cos(2πt/365))SΛ
2W
))
.
3.2 Age-structured SEIAR model
Vaccination and social distancing are among the most important control measures for reducing thespread of many infectious diseases. Thus, it is of great interest to evaluate various policies and/orseek the best strategies to better allocate resources and minimize disease burdens. When thereare restrictions on resources, such as the limited availability of vaccines, the optimal allocation ofresources is critical. The degree of protection conferred by an influenza vaccine tends to be lower forthe elderly than for the general population, and it has been suggested that an immunization strategybased on reducing the transmission rate should substantially reduce the overall disease burden[6, 24]. In particular, the potential benefit of preferentially vaccinating school-aged children hasbeen discussed because this age group is disproportionately responsible for influenza transmission[7, 9, 18]. Thus, it is necessary to take into account the transmission dynamics of different agegroups to determine the optimal control policy. In this regard, we consider an age-structuredmodel.
S′i = −SiΛi − νi(t)Si
E′i = SiΛi − κiEi
I ′i = piκiEi − αiIi − τi(t)Ii
A′i = (1 − pi)κiEi − ηiAi
R′i = νi(t)Si + fiαiIi + τi(t)Ii + ηiAi
(3.4)
with Λ1 = β0C11(1− σ11(t))[ǫ1E1 + (1− q1)I1 + δ1A1] + β0C12(1− σ12(t))[ǫ2E2 + (1− q2)I2 + δ2A2]
and Λ2 = β0C21(1− σ21(t))[ǫ1E1 + (1− q1)I1 + δ1A1] + β0C22(1− σ22(t))[ǫ2E2 + (1− q2)I2 + δ2A2]
and the initial conditions
Si(0) = Si0, Ei(0) = Ei0, Ii(0) = Ii0, Ai(0) = Ai0, Ri(0) = Ri0, i = 1, 2
where Cij is the frequency of contact between an individual in the jth age-group and members ofthe ith age-group.
We use the approach in [31] to calculate the basic reproduction number
R0 =1
2β0
[
S1C11L1 + S2C22L2 +√
(S1C11L1 − S2C22L2)2 + 4S1C21L1S2C12L2
]
where
L1 =ǫ1
κ1
+(1 − q1)p1
α1
+δ1(1 − p1)
η1
8
and
L2 =ǫ2
κ2
+(1 − q2)p2
α2
+δ2(1 − p2)
η2
and analyze the model to obtain the final size relation
lnS10
S1∞= β0
[
S10C11L1
(
1 −S1∞
S10
)
+ S20C12L2
(
1 −S2∞
S20
)]
and
lnS20
S2∞= β0
[
S10C21L1
(
1 −S1∞
S10
)
+ S20C22L2
(
1 −S2∞
S20
)]
.
Now we define an objective functional as
J(ν1, ν2, τ1, τ2, σ11, σ12, σ21, σ22)
=2
∑
i=1
∫ T
0
PiIi(t) + Qiν2
i (t) + Riτ2
i (t) + Wi1σ2
i1(t) + Wi2σ2
i2(t) dt (3.5)
where Pi, Qi, Ri and Wij are the weight constants of three controls and infective individuals.We seek the optimal controls νi, τi (i = 1, 2) and σij (i, j = 1, 2) in Uad such that
minimize J(ν1, ν2, τ1, τ2, σ11, σ12, σ21, σ22) subject to (3.4) (3.6)
where Uad = {(ν1, ν2, τ1, τ2, σ11, σ12, σ21, σ22)| νi, τi and σii are Lebesgue-integrable, 0 ≤ νi ≤ νimax,0 ≤ τi ≤ τimax, (i = 1, 2) and 0 ≤ σij ≤ σijmax, (i, j = 1, 2)} is the control set.
The Hamiltonian equation is given by
H(~x, ~u,~λ) =2
∑
i=1
[PiIi + Qiν2
i + Riτ2
i + Wi1σ2
i1 + Wi2σ2
i2]
− λ1(Λ1 + ν1)S1 + λ2(S1Λ1 − κ1E1) + λ3(p1κ1E1 − (α1 + τ1)I1)
+ λ4((1 − p1)κ1E1 − η1A1) + λ5(ν1S1 + (f1α1 + τ1)I1 + η1A1)
− λ6(Λ2 + ν2)S2 + λ7(S2Λ2 − κ2E2) + λ8(p2κ2E2 − (α2 + τ2)I2)
+ λ9((1 − p2)κ2E2 − η2A2) + λ10(ν2S2 + (f2α2 + τ2)I2 + η2A2)
where ~x = [S1, S2, E1, E2, I1, I2, A1, A2, R1, R2]T , ~u = [ν1, ν2, τ1, τ2, σ1, σ2]
T and ~λ = [λ1, λ2, λ3, λ4,λ5, λ6, λ7, λ8, λ9, λ10]
T .We find the optimal controls by solving the state equations (3.1) with initial conditions and the
adjoint equations
9
λ′
1 = Λ1(λ1 − λ2) + ν1(λ1 − λ5)
λ′
2 = β0C11(1 − σ11)ǫ1S1(λ1 − λ2) + λ2κ1 − λ3p1κ1 − λ4(1 − p1)κ1 + β0C21(1 − σ21)ǫ1S2(λ6 − λ7)
λ′
3 = −P1 + β0C11(1 − σ11)(1 − q1)S1(λ1 − λ2) + λ3(α1 + τ1) − λ5(f1α1 + τ1)
+ β0C21(1 − σ21)(1 − q1)S2(λ6 − λ7)
λ′
4 = β0C11(1 − σ11)δ1S1(λ1 − λ2) + η1(λ4 − λ5) + β0C21(1 − σ21)δ1S2(λ6 − λ7)
λ′
5 = 0
λ′
6 = Λ2(λ6 − λ7) + ν2(λ6 − λ10)
λ′
7 = β0C12(1 − σ12)ǫ2S1(λ1 − λ2) + β0C22(1 − σ22)ǫ2S2(λ6 − λ7) + λ7κ2 − λ8p2κ2 − λ9(1 − p2)κ2
λ′
8 = −P2 + β0C12(1 − σ12)(1 − q2)S1(λ1 − λ2) + β0C22(1 − σ22)(1 − q2)S2(λ6 − λ7) + λ8(α2 + τ2)
− λ10(f2α2 + τ2)
λ′
9 = β0C12(1 − σ12)δ2S1(λ1 − λ2) + β0C22(1 − σ22)δ2S2(λ6 − λ7) + η2(λ9 − λ10)
λ′
10 = 0
with the transversality conditions
λ1(T ) = λ2(T ) = λ3(T ) = λ4(T ) = λ5(T ) = λ6(T ) = λ7(T ) = λ8(T ) = λ9(T ) = λ10(T ) = 0.
The optimal control functions satisfy
ν∗1 = min
(
ν1max,max
(
0,(λ1 − λ5)S1
2Q1
))
ν∗2 = min
(
ν2max,max
(
0,(λ6 − λ10)S2
2Q2
))
τ∗1 = min
(
τ1max,max
(
0,(λ3 − λ5)I1
2R1
))
τ∗2 = min
(
τ2max,max
(
0,(λ8 − λ10)I2
2R2
))
σ∗11 = min
(
σ11max,max
(
0,−(λ1 − λ2)S1β0C11[ǫ1E1 + (1 − q1)I1 + δ1A1]
2W11
))
σ∗12 = min
(
σ12max,max
(
0,−(λ1 − λ2)S1β0C12[ǫ2E2 + (1 − q2)I2 + δ2A2]
2W12
))
σ∗21 = min
(
σ21max,max
(
0,−(λ6 − λ7)S2β0C21[ǫ1E1 + (1 − q1)I1 + δ1A1]
2W21
))
σ∗22 = min
(
σ22max,max
(
0,−(λ6 − λ7)S2β0C22[ǫ2E2 + (1 − q2)I2 + δ2A2]
2W22
))
.
10
4 Numerical simulations
We solved the optimal control problem by using a gradient-type iterative scheme [17]. With aninitial guess for the control variables, we solved the state system forward in time and then the ad-joint system backward in time. We updated the values of control variables by using the optimalitycondition and repeated the process until the system converged. In the numerical simulations, wecompared the optimal intervention strategies in settings that varied according to the basic repro-duction number, delays in vaccine production, seasonal forcing, and the age-dependent incidencerate.
Parameter Description Value
κ transition rate for the exposed 0.526α recovery rate for the (symptomatic) infective 0.244η recovery rate for the asymptomatic 0.244p fraction of developing symptoms 0.667f 1-fatality rate (one minus fatality rate) 0.98ǫ infectivity reduction factor for the exposed 0δ infectivity reduction factor for the asymptomatic 1q contact reduction by isolation 0.5
R0 basic reproduction number 1.9847νmax maximum vaccination rate 0.01τmax maximum treatment rate 0.05σmax maximum rate for reduction in contact 0.01S0 initial susceptible population 1000000E0 initial exposed population 0I0 initial (symptomatic) infective population 1A0 initial asymptomatic population 0
Table 1: Parameter values in simulations
The proposed model contains a number of parameters that must be assigned values beforeany simulation. In specifying model parameters, we employed values similar to those reported orjustified in previous studies whenever possible. Table 1 provides a summary of the definitions andvalues of the parameters in the numerical computation. We obtained the parameters mainly fromthe Longini et al. [25] and Arino et al. [2].
4.1 Basic SEIAR Model
For the basic SEIAR model, we compared the optimal intervention strategies by varying the basicreproduction number and delays in vaccine production. We first stimulated the SEIAR modelwithout controls for reference purposes. Figure 2 shows the effects of the basic reproduction numberR0, which R0 ranged from 1.4 to 2.6. As expected, a decrease in the incidence rate delayed thepeak time and reduced the number of infected individuals.
The optimal intervention strategies and the corresponding states for various R0 are presentedin Figure 3. Since the magnitude of the number of infected individuals far exceeded that of the costof controls in the objective functionals, we balanced this difference in units by the choice of weights
11
0 50 100 150 200 250 300−1
0
1
2
3
4
5
time
infe
ctiv
e
R0 = 2.6
R0 = 2.2
R0 = 1.8
R0 = 1.4
Figure 2: Disease dynamics by various basic reproduction numbers: The graph uses the log scaleto display the number of infective individuals to compare with the results using controls.
P = 1, Q = 106, R = 106, and W = 106. In addition, in numerical simulations, one may change theweights to place greater emphasis on the minimization of infected individuals or on control efforts.As shown in Table1, we assumed a maximum vaccination rate of 0.01, a maximum treatment rateof 0.05, and a maximum contact rate reduction of 0.01. These maximum values were chosen ratherarbitrarily, but the qualitative results were not sensitive to changes in these parameters.
As shown in the graphs, regardless of the basic reproduction number, it was essential for theeffective control of an epidemic to vaccinate at the highest possible rate as early as possible tominimize the number of infected individuals as well as the amount of effort required for controllingthe epidemic. For treatment and social distancing, however, the results indicate that the vaccinationrate should be changed according to the dynamics of the infected population.
For the 2009 A(H1N1) influenza outbreak, vaccine production started in the early summer of2009, and it was expected that the first production batch would be ready in the early stages of theepidemic. However, for most countries, the vaccine arrived much later than expected. The influenzabegan spreading in April 2009, and the vaccination started in October 2009. To investigate theeffects of such delays in vaccine production on optimal controls, we varied the delays until thevaccine was available from 0 to 180 days. Figure 4 shows the results for three basic reproductionnumbers: 1.4, 2.0, and 2.6. Consistent with the results of previous simulations, vaccinating at thehighest possible rate as soon as the vaccine was available led to the best outcomes. This suggeststhat if the vaccine is available earlier in the epidemic, then the control strategies should be moreeffective and require less effort. However, once a large portion of the population is infected and laterbecomes immune, then the vaccine may have little effect. Therefore, it is crucial to get vaccinesready early enough, particularly for those diseases with high incidence rates.
4.2 Seasonal SEIAR model
The incidence of influenza shows a seasonal pattern in temperate areas, occurring in the northernand southern hemispheres during their respective winters. This raises the important question of howthis seasonality influences intervention strategies such as the timing of vaccination. To understandthe effects of seasonal forcing on control strategies for epidemics, we computed optimal controls by
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0 50 100 150 200 250 3000
0.002
0.004
0.006
0.008
0.01
time
vaccination
2.61.8
2.21.4
0 50 100 150 200 250 300−1
0
1
2
3
4
5
time
infective
2.2 2.6
1.8
1.4
0 50 100 150 200 250 3000
0.01
0.02
0.03
0.04
0.05
0.06
time
treatment
2.6
2.2 1.4
1.8
0 50 100 150 200 250 300−1
0
1
2
3
4
5
time
infective
2.6
1.8
2.21.4
0 50 100 150 200 250 3000
0.002
0.004
0.006
0.008
0.01
time
social distancing
1.4
2.6
1.8
2.2
0 50 100 150 200 250 300−1
0
1
2
3
4
5
time
infective
1.4
2.2
2.6
1.8
Figure 3: Optimal strategies with various basic reproduction numbers R0. The graphs in the leftcolumn are optimal control functions (from the top, vaccination, treatment and social distancing).The right column uses the log scale to display the corresponding number of infective individualswith highly disparate scales due to different values of R0.
using the time-dependent contact rate discussed in Section 3.1.
β(t) = β0
(
1 + β1 cos
(
2π
(
t
365− s
)))
where s is the shift in seasonal forcing. In Figure 5, the graph on the left shows the time dependenttransmission rates according to shifts in the seasonal forcing function β(t). These changes in theincidence influenced the dynamics of the epidemic (see the graph on the right).
The results in Figure 6 are consistent with those of simulations using the basic SEIAR model.Regardless of the starting season, it was optimal to vaccinate at the highest possible rate as early aspossible to minimize the number of infective individuals as well as the amount of effort. However,the amount of effort required for controlling the spread of influenza was closely related to the timeof its emergence. If influenza starts in the season when transmission rate is low or decreasing, thecontrol was more effective and required less vaccination effort. The results for treatment and social
13
time
dela
ys
vaccination
0 50 100 150 200 250 300
0
30
60
90
120
150
180
0
2
4
6
8
x 10−3
time
dela
ys
infective
0 50 100 150 200 250 300
0
30
60
90
120
150
180
−1
0
1
2
3
4
time
dela
ys
vaccination
0 50 100 150 200 250 300
0
30
60
90
120
150
180
0
2
4
6
8
x 10−3
time
dela
ys
infective
0 50 100 150 200 250 300
0
30
60
90
120
150
180
−1
0
1
2
3
4
time
dela
ys
vaccination
0 50 100 150 200 250 300
0
30
60
90
120
150
180
0
2
4
6
8
x 10−3
time
dela
ysinfective
0 50 100 150 200 250 3000
30
60
90
120
150
180
−1
0
1
2
3
4
5
Figure 4: Optimal vaccination under delays in vaccine production. The graphs in the left columnindicate optimal vaccination and those in the right column show the corresponding number ofinfective individuals in the log scale. The first, second, third rows show the results for R0 = 1.4,R0 = 2.0, and R0 = 2.6, respectively.
distancing are consistent with those for the basic SEIAR model, that is, the rates followed thepattern of the epidemic curve. In other words, efforts for treatment and social distancing shouldbe concentrated around the peak time for the infected population.
4.3 Age-structured SEIAR Model
Vaccination and social distancing are among the most important control measures for reducingthe spread of many infectious diseases. Thus, it is great interest to find the best strategies forallocating resources and minimizing disease burdens. This problem is complex and depends ondiverse factors. For example, the optimal use of vaccines depends on the population structure. Inaddition, it is necessary to take into account the transmission dynamics of various age groups todetermine optimal control policies. Figure 7 shows the number of infective individuals for group1 (a high contact rate) and group 2 (a low contact rate) as we vary the relative contact rates c of
14
0 50 100 150 200 250 3002
3
4
5
6
7
8
9x 10−7
time
cont
act r
ates
s = 0 s = 1/4 s = 1/2 s = 3/4
0 50 100 150 200 250 300−1
0
1
2
3
4
5
time
infe
ctiv
e
s = 0
s = 1/2 s = 3/4
s = 1/4
Figure 5: Disease dynamics with seasonal forcing. The graph on the left shows the time dependentcontact rate β(t) for various shifts and the one on the right shows the disease dynamics accordingto the seasonal forcing.
contact matrix in the form of [c 1;1 1].The optimal intervention strategies for vaccination and social distancing varying contact matrix
with a fixed basic reproduction number are presented in Figure 8. The results suggest that focusingvaccination on high-transmission groups such as students can help prevent the spread of diseases.Also, as the contact rate ratio of high-transmission group to low-transmission group increases, thevaccination program can place greater emphasis on the former. Although this strategy makes sensein the early stages of an epidemic, its effectiveness decreases once the epidemic begins. As shown inFigure 8, it may be beneficial to place greater emphasis on social distancing for high-transmissiongroups.
5 Conclusions
This study employs techniques and ideas from control theory to derive optimal intervention strate-gies for influenza outbreaks. In particular, the study considers vaccination, antiviral treatment, andsocial distancing (e.g., school closures and the cancellation of large public gatherings) and employsthe SEIAR model, a SEIAR model with seasonal forcing, and an age-structured SEIAR model asthe mathematical models of influenza. The purpose of this study is to minimize the number ofinfected individuals while considering the cost of each mitigation strategy. The results of numericalsimulations suggest that to effectively control an epidemic, vaccination should occur at the highestpossible rate as early as possible. For treatment and social distancing, however, the strategy shouldbe tailored to the dynamics of the infected population. Finally, focusing the vaccination strategyon high-transmission groups such as students should help prevent the spread of infectious diseases.
Acknowledgements
The work of Jeehyun Lee was supported by the Korea Research Foundation Grant funded by theKorean Government (KRF-2008-531-C00012). The work of Hee-Dae Kwon was supported in part bythe National Research Foundation of Korea(NRF) Grant funded by the Korean government(MEST)(2009-0065241) and in part by the Korea Research Foundation Grant funded by the Korean Government(KRF-2008-314-C00043).
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0 50 100 150 200 250 3000
0.002
0.004
0.006
0.008
0.01
time
vaccination
1/21/4
0
3/4
0 50 100 150 200 250 300−1
0
1
2
3
4
5
time
infective
3/4 0
1/4
1/2
0 50 100 150 200 250 3000
0.01
0.02
0.03
0.04
0.05
0.06
time
treatment
1/4
0 1/2 3/4
0 50 100 150 200 250 300−1
0
1
2
3
4
5
time
infective
3/4
1/2
1/40
0 50 100 150 200 250 3000
0.002
0.004
0.006
0.008
0.01
time
social distancing
0
1/4 1/2 3/4
0 50 100 150 200 250 300−1
0
1
2
3
4
5
time
infective
0
1/4 3/4
1/2
Figure 6: Optimal strategies with seasonal forcing. The graphs in the left column indicate threetypes of optimal control functions - vaccination, treatment, and social distancing - with shifts inseasonal forcing, whereas those in the right column indicate the corresponding number of infectiveindividuals in a log scale.
0 100 200 300 4000
1000
2000
3000
4000
5000
6000
7000
8000[1111]
[7111]
0 100 200 300 4000
1000
2000
3000
4000
5000
6000
7000
[1111]
[7111]
[5111]
[3111]
Figure 7: Disease dynamics of the age-structured model with various c of contact matrix in theform of [c 1;1 1]. The left graph shows the disease dynamics for group 1 (a high contact rate), andthe right graph shows the disease dynamics for group 2 (a low contact rate).
16
0 50 100 150 200 250 3000
0.002
0.004
0.006
0.008
0.01
time
vaccination1
17
0 50 100 150 200 250 3000
0.002
0.004
0.006
0.008
0.01
time
vaccination2
1
5
3
7
0 50 100 150 200 250 3000
0.002
0.004
0.006
0.008
0.01
time
vaccination1
0 50 100 150 200 250 3000
0.002
0.004
0.006
0.008
0.01
time
vaccination2
7
1
0 50 100 150 200 250 3000
0.002
0.004
0.006
0.008
0.01
time
social distancing11
1
7
0 50 100 150 200 250 3000
0.002
0.004
0.006
0.008
0.01
time
social distancing22
1
7
5
3
Figure 8: Optimal strategies for the age-structured model. The graphs in the first row show theresults for vaccination without delay; those in the second row, vaccination with a 30-day delay; andthose in the third row, social distancing with various c of the contact matrix in the form of [c 1;11]. The first column shows the optimal controls for group 1 (a high contact rate), and the secondcolumn shows them for group 2 (a low contact rate).
17
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