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arXiv:2111.06688v1 [math.OC] 12 Nov 2021
Model Predictive Control Tailored to Epidemic
Models
Philipp Sauerteig∗ † Willem Esterhuizen‡Mitsuru Wilson†
Tobias K. S. Ritschel§Karl Worthmann†Stefan Streif‡
Abstract
We propose a model predictive control (MPC) approach for minimising the social
distancing and quarantine measures during a pandemic while maintaining a hard in-
fection cap. To this end, we study the admissible and the maximal robust positively
invariant set (MRPI) of the standard SEIR compartmental model with control inputs.
Exploiting the fact that in the MRPI all restrictions can be lifted without violating the
infection cap, we choose a suitable subset of the MRPI to define terminal constraints
in our MPC routine and show that the number of infected people decays exponentially
within this set. Furthermore, under mild assumptions we prove existence of a uniform
bound on the time required to reach this terminal region (without violating the infection
cap) starting in the admissible set. The findings are substantiated based on a numerical
case study.
1 Introduction
As the ongoing COVID-19 pandemic demonstrates, it is important to understand how in-
fectious diseases spread and how countermeasures may affect this spread. For this reason,
evaluating the impact of countermeasures based on mathematical models is an active field of
research. In particular, there is a large body of work applying optimal control to epidemiol-
ogy. Early work on this topic includes [20] where vaccination policies are derived for an SIR
(susceptible-infective-removed) model via dynamic programming; and [32] which solves opti-
mal control problems for an SIS (susceptible-infective-susceptible) model. Many papers apply
optimal control to SIR models, [17, 24, 26, 13]; to models of HIV [21, 7]; and of malaria [1].
See also the survey paper [34].
Model-predictive control (MPC) has also been applied to epidemiology, such as in the
papers [33, 22, 35] which consider stochastic networked epidemic models. Recently, many
papers have appeared on the modelling and control of COVID-19 via MPC, see for example
[5, 23]. The papers [2, 29, 14, 27] consider MPC of COVID-19 models where the control
∗Corresponding author.
†Technische Universität Ilmenau, Ilmenau, Germany, Institute for Mathematics
([philipp.sauerteig,mitsuru.wilson,karl.worthmann]@tu-ilmenau.de).
‡Technische Universität Chemnitz, Germany, Automatic Control and System Dynamics Laboratory
([willem.esterhuizen,stefan.streif]@etit.tu-chemnitz.de).
§Max Planck Institute for Dynamics of Complex Technical Systems, Magdeburg, Germany (tobk@dtu.dk).
1
is limited to non-pharmaceutical interventions, whereas the papers [15, 28] also consider
vaccination strategies.
Set-based methods have also been applied to epidemiology with the specific goal of con-
trol design that not only eliminates a disease, but also respects hard infection caps. In [25]
where the authors describe the admissible set (also known as the viability kernel in viability
theory [3]) of a two-dimensional model of a vector-borne disease, such as dengue fever or
malaria. The paper [9] extended this study, showing that the malaria model’s admissible set
as well as a the maximal robust positively invariant set (MRPI) may be described via the
theory of barriers. In [8, 10] it is shown that parts of the boundaries of these sets are made
up of special integral curves of the system that satisfy a minimum-/maximum-like principle.
The recent paper [11] uses the theory of barriers to describe both sets for the well-known
SIR and SEIR epidemic models with and without plant-model mismatch.
The papers [25, 9, 11] argue that to maintain an infection cap, the control should be chosen
based on the location of the state with respect to these sets. If it is located in the MRPI,
intervention measures may be relaxed and economic damage may be minimised. However,
if the state is located in the admissible set then intervention measures need to be carefully
considered to avoid a breach in the infection cap.
Some other research concerned with applying set-based methods to epidemiology include
work on the set of sustainable thresholds, [4], and the paper [30], which describes the viability
kernel of an SIR model through the solution of an associated Hamilton-Jacobi equation.
In the current paper, we combine the results from the paper [11] with model predictive
control (MPC) [31]. We consider the SEIR model and impose a hard cap on the proportion
of infective individuals (which might represent hospital capacity) as well as constraints on
the contact rate and removal rate (which might represent social distancing and quarantining
measures). We first show that it is possible to eliminate the disease from any initial state
located in the admissible set while always satisfying the infection cap, and that this is
always the case if the initial value is contained in the system’s MRPI. Based on our findings,
we then construct terminal conditions ensuring that the terminal costs used in our MPC
implementation are uniformly bounded and prove initial feasibility under mild assumptions.
The outline of the paper is as follows. In Section 2, we introduce the constrained epidemic
model and present important facts regarding the system’s admissible set and the MRPI.
Section 3 presents the MPC implementation that takes advantage of these sets. In this
section we also present the paper’s main result (Proposition 1) ensuring initial feasibility
and eradication of the disease if the prediction horizon is sufficiently long. A numerical case
study illustrates our findings in Section 4.
2 The SEIR model
In this section, we introduce the SEIR model and the system’s admissible set as well as the
MRPI.
2
2.1 System dynamics and constraints
We consider the SEIR model [19]
˙
S(t) = −β(t)S(t)I(t),(1a)
˙
E(t) = β(t)S(t)I(t)−ηE(t),(1b)
˙
I(t) = ηE(t)−γ(t)I(t),(1c)
˙
R(t) = γ(t)I(t),(1d)
(S, E, I , R)(0) = (S0, E0, I0, R0),(1e)
where S(t),E(t),I(t), and R(t)describe the fractions of people who are either susceptible,
exposed,infectious, or removed at time t≥0and (S0, E0, I0, R0)∈[0,1]4with S0+E0+
I0+R0= 1 denotes the initial value. In this context, the compartment Raccounts for
both recovery and death due to a fatality caused by the disease. The parameter η−1>0
denotes the average incubation time in days. Furthermore, in the standard formulation of
the SEIR model, β≡βnom >0is the rate of infectious contacts and γ≡γnom >0is the
removal rate. However, we take countermeasures into account by allowing βand γto be time-
varying control variables. In particular, a value β(t)< βnom reflects a reduction of the rate of
infectious contacts, e.g. via contact restrictions or hygiene measures, while γ(t)> γnom can
be interpreted as quarantining, and, thus, removing infectious people. These considerations
motivate our control constraints
β(t)∈[βmin, βnom ], γ(t)∈[γnom, γmax ]∀t≥0(2)
with βmin >0and γmax <∞. Moreover, we aim to maintain a hard infection cap by satisfying
the state constraint
I(t)≤Imax (3)
for some Imax ∈(0,1).
Note that Rdoes not affect the other compartments. Thus, we may ignore it and analyse
the three-dimensional system (1a)–(1c) under input and state constraints (2) and (3). To ease
our notation, we use x(t) = (x1(t), x2(t), x3(t))⊤:= (S(t), E(t), I (t))⊤∈R3to denote the
state and u(t) := (β(t), γ(t))⊤∈R2to denote the control input at time t≥0. Furthermore,
we denote the set of feasible control values by U:= [βmin, βnom]×[γnom, γmax ]⊂R2and
the set of feasible controls as U=L1
loc([0,∞), U ). Thus, for any initial value x0and control
input u∈ U there exits a unique solution of the initial value problem (1). We then write
x(·;x0, u)to denote the trajectory with respect to the three compartments S,E, and Iand
highlight the dependence on x0and u. Based on g:R3→R,x7→ x3−Imax, the set of
feasible states is given by G:= x∈R3
g(x)≤0and its interior by G−. Furthermore,
the state trajectory satisfies S(t) + E(t) + I(t)≤1for all t≥0. In other words, the system
is confined to the positively invariant set
Π := (S, E, I )∈[0,1]3
S+E+I≤1.
Given an initial state, x0∈GΠ:= G∩Π, we aim to determine a control u=u(x0)that
eliminates the disease, i.e., limt→∞ (E(t) + I(t)) = 0, while maintaining the hard infection
cap, i.e., I(t)≤Imax for all t∈[0,∞). To this end, we make use of the admissible set and
the MRPI.
3
2.2 Admissible and maximal robust positively invariant set
The admissible set contains all states, for which there exists an input such that the state
constraints are satisfied for all future time.
Definition 1. The admissible set for the system (1)–(3) is given by
A:= {x0∈GΠ| ∃u∈ U :x(t;x0, u)∈GΠ∀t∈[0,∞)}.
Moreover, we consider the set of states for which any control ensures feasibility.
Definition 2. The maximal robust positively invariant set (MRPI) contained in GΠof the
system (1)–(3) is given by
M:= {x0∈GΠ|x(t;x0, u)∈GΠ∀t∈[t0,∞)∀u∈ U } .
In [11], both sets have been characterised using the theory of barriers [8, 10]. In particular,
the sets for the system (1) are compact and never empty with M ⊆ A.
In order to eliminate the disease, we need to drive the state to the set of equilibria given
by
E:= (S, E, I )∈[0,1]3
S∈[0,1], E = 0, I = 0 .
Clearly, E ⊂ M.
The next lemma summarises some facts about the SEIR model in terms of Aand Mthat
will be useful in the sequel.
Lemma 1. The following assertions hold true.
1. For every x0∈Πand any u∈ U the epidemic will die out asymptotically, i.e., the
compartments satisfy limt→∞ E(t) = limt→∞ I(t) = 0 and limt→∞ (S(t) + R(t)) = 1.
2. For every x0∈ A there exists a u∈ U such that limt→∞ E(t) = limt→∞ I(t) = 0 and
I(t)≤Imax for all t≥0.
3. For every x0∈ M,limt→∞ E(t) = limt→∞ I(t) = 0 and I(t)≤Imax for all t≥0, for
any u∈ U.
Proof. 1) The idea is based on the proof of Theorem 5.1 in [18], where the assertion has
been shown for the SIR model. Let x0∈Πand u∈ U be arbitrary. Since Sand Rare
monotonic and bounded, the limits S∞:= limt→∞ S(t)and R∞:= limt→∞ R(t)exist with
limt→∞ ˙
S(t) = limt→∞ ˙
R(t) = 0. Hence,
0 = lim
t→∞
1
γmax
˙
R(t)≤lim
t→∞ I(t)≤lim
t→∞
1
γmin
˙
R(t) = 0
and, therefore, I∞:= limt→∞ I(t) = 0. Furthermore, E∞:= limt→∞ E(t) = 1 −(S∞+
I∞+R∞)∈[0,1] exists. Assume E∞>0. Then, there exists some t0≥0such that
˙
I(t) = ηE(t)−γ(t)I(t)>0for all t≥0, in contradiction to I∞= 0.
2) Since x0∈ A, there exists some feasible control u∈ U such that x(t;x0;u)∈GΠfor all
t≥0. Also, noting that A ⊆ Π, the assertion follows immediately from 1).
3) Since x0∈ M,x(t;x0;u)∈GΠfor all t≥0for any u∈ U. Also, noting that M ⊆ Π, the
assertion follows immediately from 1).
4
In Section 3, the set
Xf:= nx∈[0,1]3
S≤γnom
βnom , I ≤Imax, E ≤γnom
ηImax o
∪(S, 0,0)⊤
S∈[0,1]
will be used to define terminal constraints within MPC. We state the following lemma to
motivate our choice.
Lemma 2. The set Xfis contained in M. Moreover, the compartments Eand Idecay
exponentially for all x0∈Xfand all u∈ U.
Proof. Clearly, any point x0with S0∈[0,1] and E0=I0= 0 is in M. Let x0∈Xfwith
E0+I0>0be given and consider u=unom. First, note that if I0=Imax , then
˙
I(0) = ηE(0) −γnomImax ≤0
by definition of Xf. Moreover, if I0= 0, then ˙
I(0) >0and ˙
E(0) <0. Due to continuity,
there exists some (sufficiently small) time δ > 0such that I(δ)>0and x(δ)∈Xf. Since S
decreases monotonically, the compartments Eand Iare bounded from above by the solution
of the initial value problem
˙e(t) = βS(δ)i(t)−ηe(t), e(0) = E(δ),(4a)
˙
i(t) = ηe(t)−γi(t), i(0) = I(δ).(4b)
The solution of (4) is given by
(e(t), i(t))⊤= eAt(E(δ), I (δ))⊤
with matrix
A=−η βS(δ)
η−γ.
The eigenvalues of Aare given by
λ1,2=−η+γ
2±r(η+γ)2
4−ηγ +ηβS(δ).
Hence, S(δ)< S0≤γ/β yields exponential stability of A. Consequently, eand iand, thus,
Eand Idecay exponentially.
Remark 1. As in the SIR model [18], S(t) = γ(t)/β(t)is a threshold in the following sense
d
dt(E(t) + I(t)) = β(t)S(t)I(t)−γ(t)I(t)
>0if S(t)> γ(t)/β(t),
= 0 if S(t) = γ(t)/β(t),
<0if S(t)< γ(t)/β(t),
i.e., as long as Sis sufficiently big, the total number of infected (exposed or infectious)
people increases monotonically and decays otherwise. In the following, we use the notation
¯
S=γnom/βnom to denote the threshold for herd immunity.
5
3 MPC with terminal cost and set
We propose an MPC scheme for determining a feedback control that aims to minimise the
required contact restrictions and quarantine measures while maintaining a hard infection cap,
i.e., given an initial value x0∈GΠwe are interested in solving the optimal control problem
inf
u∈U J(x0, u) := Z∞
0
ℓ(x(t;x0, u), u(t)) dt(5a)
s.t.˙x(t) = f(x(t), u(t)), x(0) = x0(5b)
with stage costs ℓ:R3×R2→R,
ℓ(x, u) := E2+I2+ (β−βnom)2+ (γ−γnom )2.(5c)
For an introduction to MPC we refer to [31].
3.1 MPC formulation
We present a solution to the stated problem using continuous-time MPC. Thus, we consider
the following finite-horizon optimal control problem,
OCPT: inf
u∈U
Xf
T(x0)
JT(x0, u),
with JT(x0, u) : R3× L1
loc([0, T ),R2)→R≥0, the cost functional, specified as
JT(x0, u) := ZT
0
ℓ(x(t;x0, u), u(t)) dt+Jf(x(T;x0, u)),
with terminal cost
Jf(ˆx) := Z∞
0
x2
2(t; ˆx, unom) + x2
3(t; ˆx, unom) dt
=Z∞
0
E2
nom(t) + I2
nom(t) dt.
Here, Enom and Inom denote the exposed and infective compartments of the state trajectory
obtained with the nominal input unom ≡(βnom , γnom )⊤. Given an initial state x0∈GΠ
and a horizon length T∈[0,∞), the set UXf
T(x0)denotes all control functions u∈ U for
which x(t;x0, u)∈GΠfor all t∈[0, T ], and x(T;x0, u)∈Xf. The optimal value function,
VT(x0) : R3→R≥0∪ {+∞}, is defined as
VT(x0) := inf
u∈U
Xf
T(x0)
JT(x0, u).
If there does not exist a solution to OCPT, i.e., if UXf
T(x0) = ∅, we take the convention that
VT(x0) = ∞. The continuous-time MPC algorithm is outlined in Algorithm 1.
1Since the system is time-invariant we conveniently redefine the initial time to 0 on each iteration.
6
Algorithm 1 Continuous-time MPC with target set
Input: time shift δ∈R>0,N∈N, initial state x0∈ A,Xf⊂ M
Set prediction horizon T←N δ,ˆx←x0.
1. Find a minimizer u⋆∈arg inf u∈UXf
T(ˆx)JT(ˆx, u).
2. Implement u⋆(·), for t∈[0, δ).1
3. Set ˆx←x(δ; ˆx, u⋆)and go to step 1).
We assume that if UXf
T(ˆx)6=∅in step 1), then a minimizer is an element of UXf
T(ˆx),
see [16, p. 56] for a discussion on this issue. Thus, given an initial state x0∈ A,OCPT
is solved with the finite horizon T=N δ, the first portion of the minimizer is applied to
the system, the state is measured, and the process is iterated indefinitely. The algorithm
implicitly produces the time-varying sampled-data MPC feedback,µT,δ : [0, δ)×GΠ→U,
which results in the MPC closed-loop solution, denoted xµT,δ (·;x0).
Remark 2. Initial feasibility yields recursive feasibility.
The following result is essential to prove that the MPC implementation works.
Lemma 3. If no countermeasures are implemented, i.e., u=unom , the infinite horizon cost
functional is uniformly bounded on the MRPI, i.e.,
∃C∈R>0∀x0∈ M :J∞(x0, unom)< C.
Consequently, V∞(x0)< C for all x0∈ M.
Proof. Let x0∈ M be arbitrary. Then, any control u∈ U yields x(t;x0, u)∈GΠfor all
t≥0. Moreover, I∞:= limt→∞ I(t) = limt→∞ E(t) = 0 and S∞:= limt→∞ S(t)and
R∞:= limt→∞ R(t)exist. Hence,
Z∞
0
β(t)S(t)I(t) dt=−Z∞
0
˙
S(t) dt=S0−S∞∈[0,1]
and, therefore,
Z∞
0
E(t) dt=1
ηZ∞
0
β(t)S(t)I(t) dt−Z∞
0
˙
E(t) dt
=1
η(S0−S∞+E0).
Moreover, since E(t)∈[0,1], we get E(t)2≤E(t)for all t≥0and, thus,
Z∞
0
E(t)2dt≤Z∞
0
E(t) dt=1
η(S0−S∞+E0)<∞.
Analogously, one can show that
Z∞
0
I(t)2dt≤Z∞
0
I(t) dt=1
γnom (1 −R0−S∞)<∞.
7
In conclusion, we get
V∞(x0)≤J∞(x0, unom)
=Z∞
0
ℓ(x(t;x0, unom), unom(t)) dt
=Z∞
0
E(t)2+I(t)2dt
≤1
η(S0−S∞+E0) + 1
γnom (1 −R0−S∞)
≤2
η+1
γnom <∞,
which completes the proof.
3.2 Initial feasibility
Next we show initial (and, thus, recursive) feasibility if the following mild assumption holds.
Assumption 1. If there are infected people, i.e., E0+I0>0, then we assume the initial
values to be bounded from below, i.e., there exists some ε0>0such that
ε0≤min{E0, I0}.(6)
Furthermore, we assume there exists some K∈R>0such that
I(t)
I(t) + E(t)≥K∀t≥0(7)
or, equivalently, I(t)≥K(E(t) + I(t)), i.e., the fraction of infected people who are infectious
is bounded from below. In order to facilitate the proof of Lemma 3, we further assume that
βmin ≤γmax. Note that this can always be achieved by allowing stricter social distancing or
quarantine measures.
Our numerical simulations in Section 4 show that Assumption (6) is essential to reach
the terminal set in finite time, see Figure 3. However, from a practical point of view, we are
only interested in a situation where there already are infected people.
Based on Assumption 1, we are able to show that the terminal set is reached in finite
time.
Lemma 4. The time required to reach the terminal set is uniformly bounded on A′:=
{x0∈ A | (6) and (7) hold }, i.e.
∃T∈R>0∀x0∈ A′∃u∈ U :x(T;x0, u)∈Xf.
Hence,
∃C∈R>0∀x0∈ A′:V∞(x0)< C.
Proof. For x0∈ M, the assertion follows from Lemma 3. Consider an arbitrary x0∈ A′\ M.
Then, x0/∈Xfaccording to Lemma 2. We construct a control ˜u∈ U such that x(t;x0,˜u)∈
GΠfor all t≥0and x(T;x0,˜u)∈Xffor some finite T > 0. To this end, we ensure a decay
of Sas follows.
8
Note that since x0∈ A′\ M and limt→∞ I(t) = 0 there exists some control u1∈ U
satisfying x(t;x0, u1)∈GΠfor all t≥0and some time t1≥0such that I(t1;x0, u1) =
1/2·Imax.
Moreover, based on Assumption 1 we may assume that
γnom
βnom
<1<γmax
βmin
since otherwise, the pandemic would simply abate without control, see Remark 1. Therefore,
there exists some time t3∈(t1,∞]and a control u2= (β2, γ2)⊤∈ U such that
γ2(t)
β2(t)=S(t)∈γnom
βnom
,γmax
βmin ∀t∈[t1, t3]
and, hence, the number of infected E+Iis constant on the interval [t1, t3]. Since limt→∞(E(t)+
I(t)) = 0, this implies t3<∞.
Next, we show that there exists some t2∈[t1, t3]and a control u3= (β3, γ3)⊤∈ U such
that E+Iis constant on t∈[t2, t3]and, in addition,
−˙
E(t) = ˙
I(t) = ηE(t)−γ3(t)I(t) = 0 ∀t∈[t2, t3].
Suppose ηE(t)/I(t)< γnom for all t > t1. Then, ˙
E(t)>0for all t > t1, in contradiction to
limt→∞ E(t) = 0. On the other hand, if ηE(t)/I(t)> γmax for all t > t1, then ˙
I(t)>0for
all t > t1, in contradiction to limt→∞ I(t) = 0. Therefore, t2and u3as above exist with
γ3(t) = ηE(t)
I(t)and β3(t) = ηE(t2)
S(t)I(t2)
for all t∈[t2, t3]. (The feasibility of β3can be argued analogously to the one of γ3.)
Thus, we have
γ3(t)
β3(t)=S(t) = S(t2)−I(t2)Zt
t2
γ3(s)ds
≤S(t2)−I(t2)γnom(t−t2)
for all t∈[t2, t3]. With the maximal t3we arrive at
S(t3) = ¯
S=γnom
βnom
and, thus, x(T, x0,˜u)∈Xfwith T=t3and
˜u(t) =
u1(t)if t < t1,
u2(t)if t∈[t1, t2),
u3(t)if t∈[t2, t3).
Due to Assumption 6, the value
sup
x0∈A′
inf t1≥0
∃u∈ U :x(t;x0, u)∈GΠ∀t≥0
I(t1;x0, u) = 1/2·Imax
9
is bounded uniformly since
˙
S(t)≤ −βminK(E0+I0)S(t)≤ −2βminK ε0S(t)
for all t≥0with S(t)> γnom/βnom .
As argued in the proof of Lemma 4, assumption (7) is not too restrictive. Whenever I(t)
is small while E(t)is big at the same time, then ˙
I(t)>0, meaning that Iis going to increase.
Thus, after some time into the pandemic, there will always be a certain fraction of infected
people who are infectious.
Our main result states that the MPC feedback generated by Algorithm 1 approximates
the solution of (5).
Proposition 1. There exists a finite prediction horizon T > 0such that OCPT(5) is
initially and, thus, recursively feasible for every x0∈ A′with limt→∞ xµT,δ (t, x0)→ E, and
g(xµT,δ(t, x0)) ≤0for all t≥0under the MPC feedback µT,δ produced by Algorithm 1.
In other words, from any initial state in Athere exists a sufficiently long finite prediction
horizon such that the resulting MPC controller eliminates the disease while always satisfying
the infection cap.
Proof. From the proof of Lemma 4, the target set Xfis reachable from any x0∈ A′in finite
time. Thus, there exists a finite T > 0such that for every x0∈ A′the problem OCPT
is initially feasible. Hence, OCPTis recursively feasible, implying that the infection cap is
respected for all t≥0. After the state reaches Xf⊂ M, it approaches Easymptotically
according to Lemma 1.
Remark 3. For the SEIR model, the final cost Jf(x)is in fact not needed to ensure stability,
as is usual in MPC. However, it does improve the closed-loop transient performance.
4 Numerical case study
We now illustrate the paper’s theoretical results with simulations in Matlab. We consider
the constrained SEIR model (1a)–(1c) under constraints (2) and (3). We take βnom = 0.44,
γnom = 1/6.5,η= 1/4.6and Imax = 0.05 [15]. Furthermore, we take γmax = 0.5and
βmin = 0.22. The boundaries of Mand Afor these parameters are indicated in Figure 1,
along with a typical run of the MPC closed loop as produced by Algorithm 1 with δ= 1.
The prediction horizon is set to T= 25 days and the initial value x0= (0.50,0.18,0.01)⊤is
close to the boundary of A.
In Figure 2 the evolution of Isubject to the MPC feedback µTis depicted. The trajectory
follows the construction in the proof of Lemma 4. First, Iincreases until it reaches some
controlled equilibrium ˙
I(t2) = 0 until herd immunity is achieved with S(t3) = ¯
S. Then, it
decays asymptotically towards zero. Note that the optimal equilibrium in this context is the
infection cap, i.e., I≡Imax since it maximises the descent of S. Furthermore, the cost term
with respect to γTis much bigger than the one associated with βT, meaning that in our
setting quarantine is more efficient than social distancing. However, keep in mind that in our
model, only infectious people are put into quarantine and we do not consider re-infections,
i.e., they are completely removed from the system dynamics.
10
Figure 1: The point clouds represent the boundaries of the admissible set A(blue) and the
MRPI set M(red) while the planes depict the boundaries of the terminal set Xf(grey)
and Π(blue). The trajectory obtained by the MPC feedback law starts in A(black circle),
approaches its boundary, stays there until it reaches M, and converges asymptotically to a
disease-free equilibrium (black cross), see also Figure 2.
Figure 2: Evolution of the infectious compartment Isubject to the MPC feedback µT=
(βT, γT)⊤over time. First, Iincreases until I(t2) = Imax, then it stays there until S(t3) = ¯
S,
and decays monotonically afterwards.
Figure 3 motivates assumption (6). We observe that starting close to S0=¯
Sand decreas-
ing E0and I0, the time for a pandemic to break out explodes. Moreover, the overall cost
J∞(x0, unom)seems to be independent of the choice of E0and I0. One reason for that is the
fact that E+Iincreases as long as S(t)>¯
S. Hence, for the pandemic to die out, S0−¯
Smany
11
Figure 3: Impact of initial value for small x0= ( ¯
S+ 0.01, ε0, ε0)⊤without control, i.e.,
u≡unom. Here, ε= 2.2204 ·10−16 denotes the machine precision.
people have to get infected first. Consequently, we do not have cost controllability on A.2
5 Conclusions
In this paper, we considered the SEIR compartmental model with control inputs representing
social distancing and quarantine measures. Based on a hard infection cap, we determined a
subset in the state space, where the pandemic is contained without enforcing countermea-
sures. We used this set to define terminal constraints for our MPC algorithm and showed
initial and, thus, recursive feasibility under mild assumptions. Our numerical simulations
show that the approach is suitable to maintain the infection cap while keeping the total
amount of time short where countermeasures have to be enforced. We found that in order for
the pandemic to abate, sufficiently many people have to be infected first, i.e., herd immunity
needs to be established. Furthermore, in our model, quarantine is more efficient than social
distancing.
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