# The Effect of Population Flow on Epidemic Spread: Analysis and Control

Brooks Butler, Ciyuan Zhang, Ian Walter, Nishant Nair, Raphael Stern, and
Philip E. Paré *Brooks Butler, Ciyuan Zhang, Ian Walter, Nishant Nair, and
Philip. E. Paré are with the School of Electrical and Computer Engineering at
Purdue University. Emails: {brooksbutler, zhan3375, walteri, nair65,
<EMAIL_ADDRESS>Raphael Stern is with the Department of Civil,
Environmental, and Geo- Engineering at the University of Minnesota, Email:
<EMAIL_ADDRESS>This work was funded in part by the C3.ai Digital
Transformation Institute sponsored by C3.ai Inc. and the Microsoft Corporation
and in part by the National Science Foundation, grants NSF-CNS #2028738
(P.E.P.), NSF-CNS #2028946 (R.S.), and NSF-ECCS #2032258 (P.E.P).

###### Abstract

In this paper, we present a discrete-time networked SEIR model using
population flow, its derivation, and assumptions under which this model is
well defined. We identify properties of the system’s equilibria, namely the
healthy states. We show that the set of healthy states is asymptotically
stable, and that the value of the equilibria becomes equal across all sub-
populations as a result of the network flow model. Furthermore, we explore
closed-loop feedback control of the system by limiting flow between sub-
populations as a function of the current infected states. These results are
illustrated via simulation based on flight traffic between major airports in
the United States. We find that a flow restriction strategy combined with a
vaccine roll-out significantly reduces the total number of infections over the
course of an epidemic, given that the initial flow restriction response is not
delayed.

##  I Introduction

Global interconnectivity has proven to be a key factor in the propagation of
infectious diseases [ruan2006effect, tatem2006global]. Most recently, we have
seen evidence of such connectivity through the rapid spread of the COVID-19
pandemic, which propagated from its origin in Wuhan, China to every major
population center globally in a matter of weeks [rodriguez2020going]. Given
the implications of global population flow on disease spread, it becomes
important to accurately model this flow, as reliable modeling is an essential
step to developing effective and efficient mitigation strategies. Various
infection models have been proposed based on characteristics of individual
pathogens and studied in the literature, including susceptible-infected-
susceptible (SIS), susceptible-infected-removed (SIR), and susceptible-
infected-removed-susceptible (SIRS) [rock2014dynamics, mei2017dynamics]. For
this paper, we consider the recent COVID-19 pandemic as a motivating case for
the model selection and construction. Due to the delay in onset of COVID-19
symptoms [backer2020incubation, guan2020clinical, li2020early,
lauer2020incubation] and large asymptomatic populations estimated between
$17-81\%$ [byambasuren2020estimating, chang2020time, mizumoto2020estimating,
ing2020covid], we choose the susceptible-exposed-infected-removed (SEIR) model
as the foundation of our model development.

Previous work involving the incorporation of population flows in epidemic
process models include analysis of a networked SIS model with flows
[ifac_flow] as well as using a networked SIR model with flows to predict
arrival times for various epidemics using global flight data
[brockmann2013hidden], where both models are developed in continuous time.
This paper uses similar derivation techniques to define our discrete-time
epidemic model. However, we contribute to the development of such models by
including the exposed state in our model formulation, as well as provide
analysis of the discrete time dynamics. While other work has considered
capturing the effect of transportation on the spread of COVID-19 using the
SEIR model [vrabac2021capturing], the key distinction in this work is that
infection propagation over the network is modeled by the relocation of
infected individuals to other sub-populations rather than assuming direct
contact and intermingling between sub-populations. Furthermore, while previous
work assumes the likelihood of individuals traveling is independent of their
infection state, this work considers the effect of being infected on the
probability of an individual traveling.

###  I-A Contributions

In this paper, we explore a discrete-time networked SEIR model to analyze the
effect of population flows on the propagation of an infectious disease. We
summarize the contributions of this paper as follows:

  1. 1.

We derive a discrete-time networked SEIR model incorporating population flows,
including the conditional probabilities of a given state affecting the
likelihood of an individual to be traveling.

  2. 2.

We present assumptions on the parameters for the model to be well defined
given proper initial conditions.

  3. 3.

We prove model convergence and show that the equilibria (i.e., the healthy
states) are homogeneous, that is, the same value for all sub-populations.

  4. 4.

We present a feedback control law that restricts system flow globally using
the infection states, show that under the control effort the model is still
well defined, and illustrate its behavior via simulation.

In the remainder of this paper, we present our model derivation and necessary
assumptions in Section II, provide analysis of the healthy states in Section
III, propose a feedback control in Section IV, and illustrate the model and
the effectiveness of the controller through simulation in Section V. Finally,
we provide our conclusions in Section LABEL:sec:conclusion.

###  I-B Notation

We denote the set of real numbers, positive real numbers, non-negative
integers, and the positive integers as $\mathbb{R}$, $\mathbb{R}_{>0}$,
$\mathbb{Z}_{\geq 0}$, and $\mathbb{Z}_{\geq 1}$, respectively. For any
positive integer $n$, we have $[n]=\\{1,2,...,n\\}$. A diagonal matrix is
denoted as diag$(\cdot)$. The transpose of a vector $x\in\mathbb{R}^{n}$ is
$x^{\top}$. We use $\mathbf{0}$ and $\mathbf{1}$ to denote the vectors whose
entries all equal 0 and 1, respectively. We let
$\mathcal{G}=(\mathbb{V},\mathbb{E},\mathbb{W})$ denote a weighted directed
graph where $\mathbb{V}=\\{v_{1},v_{2},...,v_{n}\\}$ is the set of nodes,
$\mathbb{E}\subseteq\mathbb{V}\times\mathbb{V}$ is the set of edges, and
$\mathbb{W}:\mathbb{E}\rightarrow\mathbb{R}_{>0}$ maps to the real valued edge
weights on each edge. We denote the configuration of edges in a directed graph
at time step $k$ as $\mathcal{G}^{k}=(\mathbb{V},\mathbb{E}^{k},\mathbb{W})$,
where $\mathbb{E}^{k}$ denotes the set of edges at time step $k$. Furthermore,
we denote $\cup_{k\geq 0}\mathbb{E}^{k}$ as the union of all non-zero edge
configurations on a graph for all $k\geq 0$. We define a graph $\mathcal{G}$
as being strongly connected if there is a path from every node to every other
node in the graph.

##  II SEIR Model with Network Flows

In this section, we present a networked SEIR model incorporating the
population flow of individuals between sub-populations. First, consider a
group of $n$ sub-populations in a graph, where each sub-population $i\in[n]$
is represented by a node in the graph. We use the SEIR model to describe how
suceptible individuals in sub-population $i$ become exposed, infected, and
eventually recover as the result of an infectious disease [shu2012global]. We
begin with defining the SEIR model behavior without graph connections for each
sub-population $i\in[n]$ where $S_{i},E_{i},I_{i}$, and $R_{i}$ represent the
number of susceptible, exposed, infected, and recovered individuals in sub-
population $i$, respectively,

$\displaystyle\dot{S}_{i}(t)$
$\displaystyle=-\beta_{i}(t)\frac{I_{i}(t)}{N_{i}}S_{i}(t)$ (1a)
$\displaystyle\dot{E}_{i}(t)$
$\displaystyle=\beta_{i}(t)\frac{I_{i}(t)}{N_{i}}S_{i}(t)-\sigma_{i}(t)E_{i}(t)$
(1b) $\displaystyle\dot{I}_{i}(t)$
$\displaystyle=\sigma_{i}(t)E_{i}(t)-\delta_{i}(t)I_{i}(t)$ (1c)
$\displaystyle\dot{R}_{i}(t)$ $\displaystyle=\delta_{i}(t)I_{i}(t),$ (1d)

where $\beta_{i}$ is the infection rate, $\sigma_{i}$ is transition rate from
exposed to infected, and $\delta_{i}$ is the healing rate. We assume
$S_{i}(t)+E_{i}(t)+I_{i}(t)+R_{i}(t)=N_{i}$ for all $t$, i.e., a fixed
population size for each sub-population. We assume fixed sub-populations as
the intended time scale of the model is such that population change due to
birth/death rates and migration is negligible (e.g. rapid disease propagation
over weeks or months). While all the variables (and model parameters, except
population), will (may) continue to vary with time, we remove the time-
dependence notation for convenience and ease of reading from this point
forward.

To account for the flow of individuals between sub-populations we we expand
the model in (1):

$\displaystyle\dot{S}_{i}$
$\displaystyle=-\beta_{i}\frac{I_{i}}{N_{i}}S_{i}+\sum_{j\neq
i}\left(F_{ij}\frac{S_{j}}{N_{j}}-F_{ji}\frac{S_{i}}{N_{i}}\right)$ (2a)
$\displaystyle\dot{E}_{i}$
$\displaystyle=\beta_{i}\frac{I_{i}}{N_{i}}S_{i}-\sigma_{i}E_{i}+\sum_{j\neq
i}\left(F_{ij}\frac{E_{j}}{N_{j}}-F_{ji}\frac{E_{i}}{N_{i}}\right)$ (2b)
$\displaystyle\dot{I}_{i}$
$\displaystyle=\sigma_{i}E_{i}-\delta_{i}I_{i}+\sum_{j\neq
i}\left(F_{ij}\frac{I_{j}}{N_{j}}-F_{ji}\frac{I_{i}}{N_{i}}\right)$ (2c)
$\displaystyle\dot{R}_{i}$ $\displaystyle=\delta_{i}I_{i}+\sum_{j\neq
i}\left(F_{ij}\frac{R_{j}}{N_{j}}-F_{ji}\frac{R_{i}}{N_{i}}\right),$ (2d)

where $F_{ij}$ represents the number of individuals flowing from sub-
population $j$ to $i$, where $F_{ii}=0$. By making a substitution of variables
where
$s_{i}=S_{i}/N_{i},e_{i}=E_{i}/N_{i},x_{i}=I_{i}/N_{i},r_{i}=R_{i}/N_{i}$ we
can model the proportion of individuals as follows

$\displaystyle\dot{s}_{i}$
$\displaystyle=-\beta_{i}x_{i}s_{i}+\frac{1}{N_{i}}\sum_{j\neq
i}\left(F_{ij}s_{j}-F_{ji}s_{i}\right)$ (3a) $\displaystyle\dot{e}_{i}$
$\displaystyle=\beta_{i}x_{i}s_{i}-\sigma_{i}e_{i}+\frac{1}{N_{i}}\sum_{j\neq
i}\left(F_{ij}e_{j}-F_{ji}e_{i}\right)$ (3b) $\displaystyle\dot{x}_{i}$
$\displaystyle=\sigma_{i}e_{i}-\delta_{i}x_{i}+\frac{1}{N_{i}}\sum_{j\neq
i}\left(F_{ij}x_{j}-F_{ji}x_{i}\right)$ (3c) $\displaystyle\dot{r}_{i}$
$\displaystyle=\delta_{i}x_{i}+\frac{1}{N_{i}}\sum_{j\neq
i}\left(F_{ij}r_{j}-F_{ji}r_{i}\right),$ (3d)

where $s_{i}+e_{i}+x_{i}+r_{i}=1$. Note that both (2) and (3) assume the sub-
populations are well mixed and that the likelihood of an individual traveling
is independent of their infectious state, that is, whether they are
susceptible, exposed, infected, or recovered. We now extend our model to
include the probability that an individual is traveling, given their
infectious state.

$\displaystyle\dot{s}_{i}$
$\displaystyle=-\beta_{i}x_{i}s_{i}+\frac{1}{N_{i}}\sum_{j\neq
i}\left(F_{ij}P(s_{j}|T_{j})-F_{ji}P(s_{i}|T_{i})\right)$ (4a)
$\displaystyle\dot{e}_{i}$
$\displaystyle=\beta_{i}x_{i}s_{i}-\sigma_{i}e_{i}+\frac{1}{N_{i}}\sum_{j\neq
i}\left(F_{ij}P(e_{j}|T_{j})-F_{ji}P(e_{i}|T_{i})\right)$ (4b)
$\displaystyle\dot{x}_{i}$
$\displaystyle=\sigma_{i}e_{i}-\delta_{i}x_{i}+\frac{1}{N_{i}}\sum_{j\neq
i}\left(F_{ij}P(x_{j}|T_{j})-F_{ji}P(x_{i}|T_{i})\right)$ (4c)
$\displaystyle\dot{r}_{i}$
$\displaystyle=\delta_{i}x_{i}+\frac{1}{N_{i}}\sum_{j\neq
i}\left(F_{ij}P(r_{j}|T_{j})-F_{ji}P(r_{i}|T_{i})\right),$ (4d)

where $P(q_{i}|T_{i}),\,q_{i}\in\\{s_{i},e_{i},x_{i},r_{i}\\}$ is the
probability of an individual at sub-population $i$ being in a certain
infectious state given that they are also traveling. Note that

$P(s_{i}|T_{i})+P(e_{i}|T_{i})+P(x_{i}|T_{i})+P(r_{i}|T_{i})=1$ (5)

for all $i\in[n]$. We can compute the probability that an individual from sub-
population $i$ is traveling given that they are in state $q_{i}$ using Bayes’
rule,

$P(q_{i}|T_{i})=\frac{P(T_{i}|q_{i})P(q_{i})}{P(T_{i})},$ (6)

where $P(T_{i})$ is the probability of an individual traveling from sub-
population $i$ and $P(q_{i})=q_{i}$ is the probability they are in state
$q_{i}$. We compute the probability of an individual traveling from a given
sub-population $i\in[n]$ as

$P(T_{i})=\gamma_{i}=\frac{\sum_{j\neq i}F_{ji}}{N_{i}},$ (7)

where $\gamma_{i}$ is the proportion of the population flowing out of sub-
population $i$ and $\gamma_{i}\in[0,1]$ as it is reasonable to assume that
$\sum_{j\neq i}F_{ji}\ll N_{i}$ (as $\sum_{j\neq i}F_{ji}=N_{i}$ would imply
that the entire population is traveling at a given time). Since measuring
$P(q_{i}|T_{i})$ is practically challenging, we instead parameterize
$p_{i}^{q}:=P(T_{i}|q_{i})$, for $q_{i}\in\\{s_{i},e_{i},x_{i},r_{i}\\}$, as
follows. Using (5)-(7) we have that

$\displaystyle\frac{1}{\gamma_{i}}(p_{i}^{s}s_{i}+p_{i}^{e}e_{i}+p_{i}^{x}x_{i}+p_{i}^{r}r_{i})$
$\displaystyle=1.$ (8)

Therefore, assuming that $p_{i}^{s}=p_{i}^{e}=p_{i}^{r}=p_{i}^{T}$, we have

$\displaystyle p_{i}^{T}(s_{i}+e_{i}+r_{i})+p_{i}^{x}x_{i}$
$\displaystyle=\gamma_{i}.$

Thus, solving for $p_{i}^{T}$ yields

$p_{i}^{T}=\frac{\gamma_{i}-p_{i}^{x}x_{i}}{s_{i}+e_{i}+r_{i}},$ (9)

which allows us to use $p_{i}^{x}\in[0,1]$ as a parameter to describe how
likely an individual will be traveling given that they are infected.
Furthermore, we can compute the number of individuals flowing from sub-
population $j$ to $i$ as

$F_{ij}=\gamma_{j}w_{ij}N_{j},$ (10)

where $w_{ij}$ is the proportion of traveling individuals flowing from sub-
population $j$ to $i$ computed as

$w_{ij}=\frac{F_{ij}}{\sum_{l\neq j}F_{lj}}$ (11)

with $w_{ii}=0$. Thus, we can derive the dynamics for the susceptible
proportion at sub-population $i$ as

$\displaystyle\dot{s}_{i}$
$\displaystyle=-\beta_{i}x_{i}s_{i}+\frac{1}{N_{i}}\sum_{j\neq
i}\left(F_{ij}P(s_{j}|T_{j})-F_{ji}P(s_{i}|T_{i})\right)$
$\displaystyle=-\beta_{i}x_{i}s_{i}+\frac{1}{N_{i}}\sum_{j\neq
i}\left(\gamma_{j}w_{ij}N_{j}\frac{p_{j}^{T}s_{j}}{\gamma_{j}}-\gamma_{i}w_{ji}N_{i}\frac{p_{i}^{T}s_{i}}{\gamma_{i}}\right)$
$\displaystyle=-\beta_{i}x_{i}s_{i}+\sum_{j\neq
i}\left(\frac{N_{j}}{N_{i}}w_{ij}p_{j}^{T}s_{j}-w_{ji}p_{i}^{T}s_{i}\right).$

Using the fact that $\sum_{j\neq i}w_{ji}=1$, by (11), we have that

$\dot{s}_{i}=-(\beta_{i}x_{i}+p_{i}^{T})s_{i}+\sum_{j\neq
i}\frac{N_{j}}{N_{i}}w_{ij}p_{j}^{T}s_{j}.$

By similar derivations, we can rewrite (4) as

$\displaystyle\dot{s}_{i}$
$\displaystyle=-(\beta_{i}x_{i}+p_{i}^{T})s_{i}+\sum_{j\neq
i}\frac{N_{j}}{N_{i}}w_{ij}p_{j}^{T}s_{j}$ (12a) $\displaystyle\dot{e}_{i}$
$\displaystyle=\beta_{i}x_{i}s_{i}-(\sigma_{i}+p_{i}^{T})e_{i}+\sum_{j\neq
i}\frac{N_{j}}{N_{i}}w_{ij}p_{j}^{T}e_{j}$ (12b) $\displaystyle\dot{x}_{i}$
$\displaystyle=\sigma_{i}e_{i}-(\delta_{i}+p_{i}^{x})x_{i}+\sum_{j\neq
i}\frac{N_{j}}{N_{i}}w_{ij}p_{j}^{x}x_{j}$ (12c) $\displaystyle\dot{r}_{i}$
$\displaystyle=\delta_{i}x_{i}-p_{i}^{T}r_{i}+\sum_{j\neq
i}\frac{N_{j}}{N_{i}}w_{ij}p_{j}^{T}r_{j}.$ (12d)

We choose to discretize our model due to the nature of the collected data on
the spread of pandemics, where the highest resolution data is typically
recorded once per day. Using Euler’s method, we can write (12) in discrete
time as

$\displaystyle s_{i}^{k+1}$
$\displaystyle=s_{i}^{k}+h\left(-(\beta_{i}^{k}x_{i}^{k}+p_{i}^{T,k})s_{i}^{k}+\sum_{j\neq
i}\frac{N_{j}}{N_{i}}w_{ij}^{k}p_{j}^{T,k}s_{j}^{k}\right)$ (13a)
$\displaystyle e_{i}^{k+1}$
$\displaystyle=e_{i}^{k}+h\left(\beta_{i}^{k}x_{i}^{k}s_{i}^{k}-(\sigma_{i}^{k}+p_{i}^{T,k})e_{i}^{k}+\sum_{j\neq
i}\frac{N_{j}}{N_{i}}w_{ij}^{k}p_{j}^{T,k}e_{j}^{k}\right)$ (13b)
$\displaystyle x_{i}^{k+1}$
$\displaystyle=x_{i}^{k}+h\left(\sigma_{i}^{k}e_{i}^{k}-(\delta_{i}^{k}+p_{i}^{x,k})x_{i}^{k}+\sum_{j\neq
i}\frac{N_{j}}{N_{i}}w_{ij}^{k}p_{j}^{x,k}x_{j}^{k}\right)$ (13c)
$\displaystyle r_{i}^{k+1}$
$\displaystyle=r_{i}^{k}+h\left(\delta_{i}^{k}x_{i}^{k}-p_{i}^{T,k}r_{i}^{k}+\sum_{j\neq
i}\frac{N_{j}}{N_{i}}w_{ij}^{k}p_{j}^{T,k}r_{j}^{k}\right),$ (13d)

where $k\in\mathbb{Z}_{\geq 0}$ is a given time step and $h\in\mathbb{R}_{>0}$
is a sampling parameter, yielding our discrete time model.

For the model in (13) to be well-defined we require the following assumptions.

######  Assumption 1.

Let $\sum_{i\neq j}F_{ij}^{k}=\sum_{i\neq j}F_{ji}^{k}$ for all $i\in[n]$ and
$k\in\mathbb{Z}_{\geq 0}$.

This assumption requires that the total flow of individuals into a given sub-
population must be equal to the total flow out. Furthermore, we impose the
following assumption on the model parameters.

######  Assumption 2.

For all $i\in[n]$ and $k\in\mathbb{Z}_{\geq 0}$, let
$\beta_{i}^{k},\delta_{i}^{k},\sigma_{i}^{k}\in\mathbb{R}_{>0}$,
$h\beta_{i}^{k},h\delta_{i}^{k},h\sigma_{i}^{k}\in(0,1]$,
$h(\beta_{i}^{k}+p^{T,k}_{i})\leq 1,h(\sigma_{i}^{k}+p^{T,k}_{i})\leq 1,$ and
$h(\delta_{i}^{k}+p^{T,k}_{i})\leq 1$.

Under these assumptions, we can show that given proper initial conditions the
model will always remain well defined.

######  Lemma 1.

Consider the model in (13) under Assumptions 1-2. Suppose
$s_{i}^{0},e_{i}^{0},x_{i}^{0},r_{i}^{0}\in[0,1]$ and
$s_{i}^{0}+e_{i}^{0}+x_{i}^{0}+r_{i}^{0}=1$ for all $i\in[n]$. Then, for all
$k\geq 0$ and $i\in[n]$, $s_{i}^{k},e_{i}^{k},x_{i}^{k},r_{i}^{k}\in[0,1]$ and
$s_{i}^{k}+e_{i}^{k}+x_{i}^{k}+r_{i}^{k}=1$.

###### Proof.

We prove the result by induction. By assumption, it holds for the base case
$k=0$. We now show the inductive step, where given
$s_{i}^{k},e_{i}^{k},x_{i}^{k},r_{i}^{k}\in[0,1]$ and
$s_{i}^{k}+e_{i}^{k}+x_{i}^{k}+r_{i}^{k}=1$ for all $i\in[n]$, we show that
the same holds for $k+1$. By (13) and Assumption 2 we have

$\displaystyle q_{i}^{k+1}$ $\displaystyle\geq h\sum_{j\neq
i}\frac{N_{j}}{N_{i}}w_{ij}^{k}p_{j}^{T,k}q_{j}^{k}\geq 0,$ (14)

where $q_{i}^{k}\in\\{s^{k}_{i},e^{k}_{i},x^{k}_{i},r^{k}_{i}\\}$ for all
$i\in[n]$ and $k\in\mathbb{Z}_{\geq 0}$.

Furthermore, computing $s_{i}^{k+1}+e_{i}^{k+1}+x_{i}^{k+1}+r_{i}^{k+1}$
yields

$\displaystyle
s_{i}^{k+1}+e_{i}^{k+1}+x_{i}^{k+1}+r_{i}^{k+1}=s_{i}^{k}+e_{i}^{k}+x_{i}^{k}+r_{i}^{k}+h\left(-\beta_{i}^{k}x_{i}^{k}s_{i}+\beta_{i}^{k}x_{i}^{k}s_{i}-\sigma_{i}^{k}e_{i}^{k}+\sigma_{i}^{k}e_{i}^{k}-\delta_{i}^{k}x_{i}^{k}+\delta_{i}^{k}x_{i}^{k}\right)$
$\displaystyle+\frac{h}{N_{i}}\sum_{j\neq
i}\Big{(}F_{ij}^{k}(P(s_{i}|T_{i})+P(e_{i}|T_{i})+P(x_{i}|T_{i})+P(r_{i}|T_{i}))-F_{ji}^{k}(P(s_{j}|T_{j})+P(e_{j}|T_{j})+P(x_{j}|T_{j})+P(r_{j}|T_{j}))\Big{)}$
$\displaystyle\ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \ \
=s_{i}^{k}+e_{i}^{k}+x_{i}^{k}+r_{i}^{k}+\frac{h}{N_{i}}\left(F^{+}-F^{-}\right)=1,$
(15)

where $F^{+}=\sum_{j\neq i}F_{ij}^{k}$ and $F^{-}=\sum_{j\neq i}F_{ji}^{k}$
denote the total flow in and out of each sub-population, respectively, which
are equal by Assumption 1. Since each variable is non-negative by (14) and
their sum must add to one by (15), we have that
$s_{i}^{k+1},e_{i}^{k+1},x_{i}^{k+1},r_{i}^{k+1}\leq 1$. Thus, we have shown
that $s_{i}^{k+1},e_{i}^{k+1},x_{i}^{k+1},r_{i}^{k+1}\in[0,1]$, completing the
inductive step. ∎

######  Remark 1.

Assumption 2 requires that the sampling parameter be small enough in relation
to the model spread parameters such that the model remains well defined.
Furthermore, requiring $h(\beta_{i}^{k}x_{i}^{k}+p^{T,k}_{i})\leq
1,h(\sigma_{i}^{k}+p^{T,k}_{i})\leq 1,$ and $h(\delta_{i}^{k}+p^{T,k}_{i})\leq
1$ can be interpreted as requiring that no individual can both travel and
transition between infectious states during the same time step
$k\in\mathbb{Z}_{\geq 0}$, as our model does not capture infection occurring
during travel.

The following are not required for the model to remain well defined. However,
we use them in the next section to show that the set of healthy states have a
homogeneous structure.

######  Definition 1.

A graph $\mathcal{G}^{k}=(\mathbb{V},\cup_{k\geq 0}\mathbb{E}^{k},\mathbb{W})$
for $k\in\mathbb{Z}_{\geq 0}$ is _$K$ -strongly connected_ if there exist some
bound $K$ such that $(\mathbb{V},\cup_{j=k}^{k+K-1}\mathbb{E}^{j},\mathbb{W})$
is strongly connected, for all $k\in\mathbb{Z}_{\geq 0}$.

######  Assumption 3.

Let the graph $\mathcal{G}^{k}=(\mathbb{V},\cup_{k\geq
0}\mathbb{E}^{k},\mathbb{W})$, where
$\mathbb{W}:\mathbb{E}^{k}\rightarrow\mathbb{R}_{>0}$ is defined by
$w_{ij}^{k}$, be $K$-strongly connected.

##  III Model Analysis

In this section we analyze the equilibria of the model in (13), i.e., the
healthy states of the system, which we define as
$q_{i}^{*}=\lim_{k\rightarrow\infty}q_{i}^{k}$ for all $i\in[n]$ where
$q_{i}^{*}\in\\{s_{i}^{*},e_{i}^{*},x_{i}^{*},r_{i}^{*}\\}$. We use the
following result given in [blondel2005convergence] on the conditions for
discrete-time consensus models.

######  Lemma 2.

Let a discrete-time system defined by the transition matrix $L^{k}$ satisfy
the following properties, where $l_{ij}^{k}$ is the corresponding entry in the
$i$th row and $j$th column at time step $k\in\mathbb{Z}_{\geq 0}$:

  1. (i)

The graph $G=(\mathbb{V},\cup_{k\geq 0}\mathbb{E}^{k})$, where the edge
weights at time step $k$ are given by $L^{k}$, is $K$-strongly connected.

  2. (ii)

There exists a positive constant $y\in\mathbb{R}_{>0}$ such that for all
$i,j\in[n]$ and $k\in\mathbb{Z}_{\geq 0}$ we have

    1. (a)

$l_{ii}^{k}\geq y$

    2. (b)

$l_{ij}^{k}\in\\{0\\}\cup[y,1]$

    3. (c)

$\sum_{j=1}^{n}l_{ij}^{k}=1$.

Then, the system dynamics defined by $L^{k}$ guarantee asymptotic consensus.

We now present results on the asymptotic convergence of the healthy states for
the system in (13).

######  Theorem 1.

Consider the model in (13) under Assumptions 1-3. Given that there exists some
$i\in[n]$ such that $x_{i}^{0}\in(0,1]$ or $e_{i}^{0}\in(0,1]$, then there
exists a set of asymptotically stable equilibria of the form
$(\mathbf{s}^{*},\mathbf{0},\mathbf{0},1-\mathbf{s}^{*})$, where
$\mathbf{s}^{*}=\alpha\mathbf{1}$, $\alpha\in[0,1]$.

###### Proof.

We prove the result by splitting it into two parts, namely,

  1. 1.

$e_{i}^{*}=0$ and $x_{i}^{*}=0$, for all $i\in[n]$

  2. 2.

$s_{i}^{*}=\alpha$ for all $i\in[n]$.

The parts are presented sequentially:

1) We denote $\mathbf{S}^{k}=\sum_{i=1}^{n}s_{i}^{k}\in[0,n]$,
$\mathbf{E}^{k}=\sum_{i=1}^{n}e_{i}^{k}\in[0,n]$ and
$\mathbf{X}^{k}=\sum_{i=1}^{n}x_{i}^{k}\in[0,n]$ as the sum of the
susceptible, exposed, and infected states of all the sub-populations,
respectively. Hence, based on the dynamics of the system states in (13) and by
Assumption 1, we have

$\displaystyle\mathbf{S}^{k+1}$
$\displaystyle=\mathbf{S}^{k}-h\sum_{i=1}^{n}\beta_{i}^{k}x_{i}^{k}s_{i}^{k}$
(16a) $\displaystyle\mathbf{E}^{k+1}$
$\displaystyle=\mathbf{E}^{k}+h\left(\sum_{i=1}^{n}\beta_{i}^{k}x_{i}^{k}s_{i}^{k}-\sum_{i=1}^{n}\sigma_{i}^{k}e_{i}^{k}\right)$
(16b) $\displaystyle\mathbf{X}^{k+1}$
$\displaystyle=\mathbf{X}^{k}+h\left(\sum_{i=1}^{n}\sigma_{i}^{k}e_{i}^{k}-\sum_{i=1}^{n}\delta_{i}^{k}x_{i}^{k}\right).$
(16c)

By Assumption 2 and Lemma 1, we have that the rate of change of
$\mathbf{S}^{k}$, $-h\sum_{i=1}^{n}\beta_{i}^{k}x_{i}^{k}s_{i}^{k}$, is non-
positive for all $k\geq 0$ and $\mathbf{S}^{k}$ is lower bounded by zero.
Hence, by Lemma 1, we obtain that $\lim_{k\rightarrow\infty}\mathbf{S}^{k}$
exists, and

$\lim_{k\rightarrow\infty}-h\sum_{i=1}^{n}\beta_{i}^{k}x_{i}^{k}s_{i}^{k}=0.$
(17)

Accordingly, we can write that
$\lim_{k\rightarrow\infty}(\mathbf{E}^{k+1}-\mathbf{E}^{k})=\lim_{k\rightarrow\infty}-h\sum_{i=1}^{n}\sigma_{i}^{k}e_{i}^{k}$.
From Assumption 2, we know that $h\sigma_{i}^{k}>0$ for all $i\in[n]$ and
$k\in\mathbb{Z}_{\geq 0}$. Therefore, by Lemma 1, we conclude that
$\lim_{k\rightarrow\infty}\mathbf{E}^{k}=0$ and thus
$\lim_{k\rightarrow\infty}e_{i}^{k}=0$ for all $i\in[n]$.

Similarly, we acquire that
$\lim_{k\rightarrow\infty}(\mathbf{X}^{k+1}-\mathbf{X}^{k})=\lim_{k\rightarrow\infty}-h\sum_{i=1}^{n}\delta_{i}^{k}x_{i}^{k}$,
since $\lim_{k\rightarrow\infty}e_{i}^{k}=0$ for all $i\in[n]$. By Assumption
2 and Lemma 1, $h\delta_{i}^{k}>0$ and $x_{i}^{k}$ is well-defined, we acquire
that $\lim_{k\rightarrow\infty}\mathbf{X}^{k}=0$ and thus
$\lim_{k\rightarrow\infty}x_{i}^{k}=0$ for all $i\in[n]$.

2) In order to show that the susceptible states become equally mixed as
$k\rightarrow\infty$, where $k\in\mathbb{Z}_{\geq 0}$, we leverage Lemma 2.
First, we construct dynamics for the susceptible states in (13a) when
$x_{i}^{*}=e_{i}^{*}=0$ for all $i\in[n]$, as proven in 1), which yields

$s_{i}^{k+1}=s_{i}^{k}+h\left(-\gamma_{i}^{k}s_{i}^{k}+\sum_{j\neq
i}\frac{N_{j}}{N_{i}}w_{ij}^{k}\gamma_{i}^{k}s_{j}^{k}\right),$ (18)

where $p^{T,k}_{i}=\gamma_{i}$ by (9) and Lemma 1. We can express these
dynamics in matrix form as

$s^{k+1}=\underbrace{\left(I+h\left(-\Gamma^{k}+N^{-1}W^{k}\Gamma^{k}N\right)\right)}_{L^{k}}s^{k},$
(19)

where $s^{k}=[s_{1}^{k},\dots,s_{n}^{k}]^{\top}$,
$\Gamma^{k}=\text{diag}(\gamma_{1}^{k},\dots,\gamma_{n}^{k})$,
$N=\text{diag}(N_{1},\dots,N_{n})$, and $W^{k}$ is the matrix defined by
entries $w_{ij}^{k}$ on the $i$th row and $j$th column. In order to guarantee
that $s_{i}^{*}$ is homogeneous for all $i\in[n]$, we will apply Lemma 2 to
(19).

Property (i) of Lemma 2 holds by Assumption 3. Furthermore, (ii.a) and (ii.b)
are true by construction. We now show (ii.c) is true by computing
$L^{k}\mathbf{1}$, which yields

$\displaystyle
L^{k}\mathbf{1}=\mathbf{1}+h\left(-\bm{\gamma}^{k}+N^{-1}W^{k}\Gamma^{k}N\mathbf{1}\right),$
(20)

where $\bm{\gamma}^{k}=[\gamma_{i}^{k},\dots,\gamma_{n}^{k}]^{\top}$.
Computing $N^{-1}W^{k}\Gamma^{k}N\mathbf{1}$ yields

$\displaystyle
N^{-1}W^{k}\Gamma^{k}N\mathbf{1}=N^{-1}\begin{bmatrix}\sum_{j\neq
1}\gamma_{j}^{k}w_{1j}^{k}N_{j}\\\ \vdots\\\ \sum_{j\neq
n}\gamma_{j}^{k}w_{nj}^{k}N_{j}\end{bmatrix}.$ (21)

Using (10), we can write (21) as

$\displaystyle N^{-1}\begin{bmatrix}\sum_{j\neq 1}F_{1j}^{k}\\\ \vdots\\\
\sum_{j\neq n}F_{nj}^{k}\end{bmatrix}=\begin{bmatrix}\sum_{j\neq
1}\frac{F_{1j}^{k}}{N_{1}}\\\ \vdots\\\ \sum_{j\neq
n}\frac{F_{nj}^{k}}{N_{n}}\end{bmatrix},$ (22)

which, by Assumption 1 and (7), becomes

$\displaystyle\begin{bmatrix}\sum_{i\neq 1}\frac{F_{i1}^{k}}{N_{1}}\\\
\vdots\\\ \sum_{i\neq
n}\frac{F_{in}^{k}}{N_{n}}\end{bmatrix}=\begin{bmatrix}\gamma_{1}^{k}\\\
\vdots\\\ \gamma_{n}^{k}\end{bmatrix}=\bm{\gamma}^{k}.$

Thus, $L^{k}\mathbf{1}=\mathbf{1}$, showing that $L^{k}$ meets the
requirements to guarantee consensus of the susceptible states as stated in
Lemma 2. Therefore, $s_{i}^{*}=\alpha$ for all $i\in[n]$, where by Lemma 1 we
have that $\alpha\in[0,1]$. ∎

##  IV Feedback Control

We now propose a feedback control strategy for the model in (13) derived from
the current infection states. Given that $\gamma_{i}^{k}$ represents the
unimpeded flow out of sub-population $i$ at time step $k\in\mathbb{Z}_{\geq
0}$, we can implement a scheme that restricts travel between all sub-
populations proportionally:

$\tilde{\gamma}_{i}^{k}=\theta^{k}\gamma_{i}^{k},$ (23)

where $\theta^{k}\in[0,1]$ is the flow restriction penalty. We propose a flow
restriction penalty that is a function of the average infection level:

$\theta^{k}=1-(\bar{x}^{k})^{\frac{1}{\eta}},$ (24)

where $\bar{x}^{k}=\frac{1}{n}\sum_{i\in[n]}x_{i}^{k}$ is the average
proportion of infected individuals across all sub-populations and
$\eta\in\mathbb{R}_{>0}$ can be viewed as a sensitivity parameter, where
$\eta>1$ denotes a higher sensitivity and $\eta<1$ denotes a lower sensitivity
to the average infection level in the network. The magnitude of $\eta$ can
also be viewed as the strength of the controller in reaction to the overall
infection. We now show that applying the strategy in (23) still maintains the
assumptions imposed in Section II, enforcing that the model remains well
defined. We define $\widetilde{\eqref{eq:flows_ind_node_disc}}$ as the system
with dynamics in (13) including the control strategy in (23).

######  Proposition 1.

Consider $\widetilde{\eqref{eq:flows_ind_node_disc}}$ under Assumption 2 and
with $\theta^{k}\in[0,1]$ for all $k\geq 0$. If with
$s_{i}^{0},e_{i}^{0},x_{i}^{0},r_{i}^{0}\in[0,1]$ and
$s_{i}^{0}+e_{i}^{0}+x_{i}^{0}+r_{i}^{0}=1$ for all $i\in[n]$, then
$s_{i}^{k},e_{i}^{k},x_{i}^{k},r_{i}^{k}\in[0,1]$ and
$s_{i}^{k}+e_{i}^{k}+x_{i}^{k}+r_{i}^{k}=1$, for all $k\geq 0$ and $i\in[n]$.

###### Proof.

As control is only applied to $\gamma_{i}^{k}$ for all $i\in[n]$, we must only
verify that Assumption 1 is met under the proposed control strategy and the
rest will follow from Lemma 1. Since $\theta^{k}\in[0,1]$ we have by (7) that
$\tilde{\gamma}_{i}^{k}\in[0,1]$. Furthermore, by (10) we have

$\tilde{F}_{ij}^{k}=\tilde{\gamma_{j}}^{k}w_{ij}^{k}N_{j}.$ (25)

Computing the flows in versus the flows out, $\tilde{F}^{+}-\tilde{F}^{-}$,
yields

$\displaystyle\tilde{F}^{+}-\tilde{F}^{-}$ $\displaystyle=\sum_{j\neq
i}\tilde{F}_{ij}^{k}-\sum_{j\neq i}\tilde{F}_{ji}^{k}$
$\displaystyle=\sum_{j\neq i}\tilde{\gamma_{j}}^{k}w_{ij}^{k}N_{j}-\sum_{j\neq
i}\tilde{\gamma_{i}}^{k}w_{ji}^{k}N_{i}$
$\displaystyle=\theta^{k}\left(\sum_{j\neq
i}\gamma_{j}^{k}w_{ij}^{k}N_{j}-\sum_{j\neq
i}\gamma_{i}^{k}w_{ji}^{k}N_{i}\right)$
$\displaystyle=\theta^{k}(F^{+}-F^{-})=0.$

Thus, Assumption 1 is maintained. Therefore, by Lemma 1
$s_{i}^{k},e_{i}^{k},x_{i}^{k},r_{i}^{k}\in[0,1]$ and
$s_{i}^{k}+e_{i}^{k}+x_{i}^{k}+r_{i}^{k}=1$, for all $k\geq 0$ and $i\in[n]$.
∎

By restricting $\gamma_{i}^{k}$, we directly reduce the flow of both infected
and non-infected individuals according to (9) as

$\small\theta^{k}\left(p_{i}^{T,k}(s_{i}^{k}+e_{i}^{k}+r_{i}^{k})+p_{i}^{x,k}x_{i}^{k}\right)=\theta^{k}\gamma_{i}^{k}.$
(26)

Thus, our controlled flow rates, with respect to the conditional probability
parameters, are given by

$\tilde{p}_{i}^{q,k}=\theta^{k}{p}_{i}^{q,k}$ (27)

where $p_{i}^{q,k}\in\\{p_{i}^{s,k},p_{i}^{e,k},p_{i}^{x,k},p_{i}^{r,k}\\}$.

The control strategy in (23)-(24) effectively restricts flow between all sub-
populations proportionally in the presence of any amount of system infection.
Note that by construction
$\theta^{k}=1-(\bar{x}^{k})^{\frac{1}{\eta}}\in[0,1]$ for all $k\geq 0$. Thus,
by Proposition 1, the system is well defined.

It should be noted that while proportional restrictions to all flow are not
the most precise form of control that can be applied to this model, this
approach is not dissimilar to the travel policies on global and regional
flights during the height of the COVID-19 pandemic [suzumura2020impact]. In
the following section we apply this control strategy to a simplified model of
a travel network between populous cities based on median flight data, and
evaluate its effectiveness on mitigating disease spread.

##  V Simulations

In this section, we detail the methods and parameters used to simulate our
model, its limiting behavior, and our proposed control strategy as described
in Sections II-IV. We construct our simulations using population data from the
US cities of Atlanta, Los Angeles, Chicago, and Dallas, and the flights
between each city’s primary airport (ATL, LAX, ORD, DFW). The infection starts
in Los Angeles and propagates through the network, reaching an equilibrium
where $x^{k}_{i}=0$ for all $i\in[n]$.

To simulate the states for the SEIR model we use (13) with fixed homogeneous
spread parameters (i.e., the same for every sub-population and static),
$(\beta,\delta,\sigma,h,p^{x})=(0.5,0.34,0.19,0.14,0.005)$. The population of
each city is given by
$(N_{\text{ATL}},N_{\text{LAX}},N_{\text{ORD}},N_{\text{DFW}})\approx(0.5,4,2.7,1.3)*10^{6}$,
where the population sizes are approximated from [census2019] and
[datacommons]. The population traveling between the cities is approximated by
the median number of daily flights between the airports in March
2021[flightsfrom]:

$F=\xi\begin{bmatrix}0&15&23&19\\\ 15&0&22&21\\\ 23&22&0&23\\\
19&21&23&0\end{bmatrix},$

where $\xi\in\mathbb{R}_{\geq 0}$ is a scaling factor, which is used to
increase or decrease the total volume of population flow. The initial
conditions for the sub-populations are $s^{0}=[1,0.99,1,1],\
e^{0}=[0,0.005,0,0],\ x^{0}=[0,0.005,0,0]$, and $r^{0}=[0,0,0,0]$.

\begin{overpic}[trim=19.63246pt 22.76228pt 0.0pt
0.0pt,clip,width=216.81pt]{S1.png}
\put(-6.5,25.0){\large{\parbox{325.215pt}{90}{$s_{i}^{k}$}}}
\put(55.0,-2.0){\large{\parbox{325.215pt}{}\normalsize\put(86.5,35.0){\hbox{\pagecolor{white}\parbox{13.00806pt}{\footnotesize$s_{0}^{k}$
\par\vspace{.45ex} \par$s_{1}^{k}$ \par\vspace{.45ex} \par$s_{2}^{k}$
\par\vspace{.45ex} \par$s_{3}^{k}$}}} \end{overpic} \@@toccaption{{\lx@tag[
]{{1}}{ An illustration of the model in \eqref{eq:flows_ind_node_disc}
reaching consensus after the infection dies out in the system, indicated by
the vertical black dotted line where $\bar{x}\leq
10^{-4}$.}}}\@@caption{{\lx@tag[: ]{{Figure 1}}{ An illustration of the model
in \eqref{eq:flows_ind_node_disc} reaching consensus after the infection dies
out in the system, indicated by the vertical black dotted line where
$\bar{x}\leq 10^{-4}$.}}} \end{figure} \par First, we show the consensus
behavior from Theorem\leavevmode\nobreak\ \ref{thm:equilibrium} by simulating
\eqref{eq:flows_ind_node_disc} using the given initial conditions and model
parameters. In Figure\leavevmode\nobreak\ \ref{fig:consensus_plot}, we see
that the susceptible proportions of sub-populations reach consensus. The
dashed vertical line in the graph indicates the time step at which the average
infected proportion of the population reaches near zero ($\bar{x}^{k}\leq
10^{-4}$). These results suggest that, by the end of the epidemic process, the
flow dynamics dominate the behavior of the model and eventually lead to an
equal mixing of susceptible and recovered populations given enough time, as
suggested by Theorem\leavevmode\nobreak\ \ref{thm:equilibrium}.
\par\begin{figure}\centering\begin{overpic}[trim=56.9055pt 62.59596pt
59.75095pt 1.00374pt,clip,width=208.13574pt]{Images/infectedplot.png}
\put(-4.0,38.0){{\parbox{325.215pt}{90}{$\bar{x}^{k}$}}}
\put(50.0,-3.0){\large{\parbox{325.215pt}{}
\put(100.0,38.0){\normalsize{\parbox{325.215pt}{90}{$\theta^{k}$}}}\normalsize\end{overpic}
\hfill\begin{overpic}[trim=1.00374pt 56.9055pt 1.00374pt
-28.45274pt,clip,width=203.80193pt]{Images/recoveredcontour.png}
\put(-2.0,34.0){{\parbox{325.215pt}{90}{$\gamma$}
}}\normalsize\put(44.0,-1.0){{\parbox{325.215pt}{}
\put(100.0,34.0){{\parbox{325.215pt}{90}{$\bar{r}^{*}$}}}
\normalsize\end{overpic} \@@toccaption{{\lx@tag[ ]{{2}}{System response to the
controller in \eqref{eq:control}-\eqref{eq:theta} with different strength
values ($\eta$) and flow rates ($\gamma$). (Top) Plot of $\bar{x}^{k}$ with
$\eta$ ranging from $0$ to $1000$ and $\xi=100$. An $\eta=0$ means the control
strategy is not used. The dotted lines of corresponding color denote the
control penalty applied to the flow rates in the system with respect to the
total infection level. (Bottom) The average proportion of recovered
individuals $\bar{r}^{*}$ for a spectrum of equilibria. The baseline
$\gamma=1.0$ is when $\xi=100$.}}}\@@caption{{\lx@tag[: ]{{Figure 2}}{System
response to the controller in \eqref{eq:control}-\eqref{eq:theta} with
different strength values ($\eta$) and flow rates ($\gamma$). (Top) Plot of
$\bar{x}^{k}$ with $\eta$ ranging from $0$ to $1000$ and $\xi=100$. An
$\eta=0$ means the control strategy is not used. The dotted lines of
corresponding color denote the control penalty applied to the flow rates in
the system with respect to the total infection level. (Bottom) The average
proportion of recovered individuals $\bar{r}^{*}$ for a spectrum of
equilibria. The baseline $\gamma=1.0$ is when $\xi=100$.}}} \end{figure}
\par\begin{figure}\centering\begin{overpic}[trim=56.9055pt 62.59596pt
59.75095pt 1.00374pt,clip,width=203.80193pt]{Images/infectedplot_vaccine.png}
\put(-4.0,38.0){{\parbox{325.215pt}{90}{$\bar{x}^{k}$}}}
\put(50.0,-3.0){\large{\parbox{325.215pt}{}
\put(100.0,38.0){\normalsize{\parbox{325.215pt}{90}{$\theta^{k}$}}}\normalsize\end{overpic}
\hfill\begin{overpic}[trim=1.00374pt 56.9055pt 1.00374pt
-28.45274pt,clip,width=203.80193pt]{Images/recoveredcontour_vaccine.png}
\put(-2.0,34.0){{\parbox{325.215pt}{90}{$\gamma$}
}}\normalsize\put(44.0,-1.0){{\parbox{325.215pt}{}
\put(100.0,30.0){{\parbox{325.215pt}{90}{$\delta\sum_{k}\bar{x}^{k}$}}}
\normalsize\end{overpic} \@@toccaption{{\lx@tag[ ]{{3}}{Implementation of a
vaccine roll-out. Plots mirror those in Figure\leavevmode\nobreak\
\ref{fig:control_no_vaccine}, with the roll-out starting at $k=500$, moving
$0.1\%$ of $s_{i}^{k}\rightarrow r_{i}^{k}$ until $\bar{s}^{k}=0.01$. (Top)
Plot of $\bar{x}^{k}$ with $\eta$ ranging from $0$ to $1000$ and $\xi=100$. An
$\eta=0$ means the control strategy is not used. The dotted lines of
corresponding color denote the control penalty applied to the flow rates in
the system with respect to the total infection level. (Bottom) The average
proportion of recovered individuals due to infection
$\delta\sum_{k}\bar{x}^{k}$, which is equivalent to $\bar{r}^{*}$ when there
is no vaccine, for a spectrum of equilibria. The baseline $\gamma=1.0$ is when
$\xi=100$. }}}\@@caption{{\lx@tag[: ]{{Figure 3}}{Implementation of a vaccine
roll-out. Plots mirror those in Figure\leavevmode\nobreak\
\ref{fig:control_no_vaccine}, with the roll-out starting at $k=500$, moving
$0.1\%$ of $s_{i}^{k}\rightarrow r_{i}^{k}$ until $\bar{s}^{k}=0.01$. (Top)
Plot of $\bar{x}^{k}$ with $\eta$ ranging from $0$ to $1000$ and $\xi=100$. An
$\eta=0$ means the control strategy is not used. The dotted lines of
corresponding color denote the control penalty applied to the flow rates in
the system with respect to the total infection level. (Bottom) The average
proportion of recovered individuals due to infection
$\delta\sum_{k}\bar{x}^{k}$, which is equivalent to $\bar{r}^{*}$ when there
is no vaccine, for a spectrum of equilibria. The baseline $\gamma=1.0$ is when
$\xi=100$. }}} \end{figure}
\par\begin{figure}\centering\begin{overpic}[trim=56.9055pt 65.44142pt
62.59596pt 1.00374pt,clip,width=216.81pt]{Images/binarycontroller.png}
\put(-4.0,38.0){{\parbox{325.215pt}{90}{$\bar{x}^{k}$}}}
\put(50.0,-3.0){\large{\parbox{325.215pt}{}
\put(100.0,38.0){{\parbox{325.215pt}{90}{$\theta^{k}$}}}
\normalsize\end{overpic} \@@toccaption{{\lx@tag[ ]{{4}}{ System response
(solid lines) to the controller in \eqref{eq:control} with an on-off flow
penalty (dotted lines). The case $\eta=0$ has $\theta^{k}=1$ for all $k\geq
0$, the cases $\eta=1,2$ have $\theta^{k}=1$ for all $k\in\\{0,\dots,49\\}$
and set $\theta^{k}=0$ starting at $k=50$, the cases $\eta=3,4$ have
$\theta^{k}=0$ starting from $k=0$, cases $\eta=1,3$ re-open travel (set
$\theta^{k}=1$) for all $k$ after $\bar{x}^{k}<0.001$, and cases $\eta=2,4$
only re-open travel after $\bar{x}^{k}\approx 0$. }}}\@@caption{{\lx@tag[:
]{{Figure 4}}{ System response (solid lines) to the controller in
\eqref{eq:control} with an on-off flow penalty (dotted lines). The case
$\eta=0$ has $\theta^{k}=1$ for all $k\geq 0$, the cases $\eta=1,2$ have
$\theta^{k}=1$ for all $k\in\\{0,\dots,49\\}$ and set $\theta^{k}=0$ starting
at $k=50$, the cases $\eta=3,4$ have $\theta^{k}=0$ starting from $k=0$, cases
$\eta=1,3$ re-open travel (set $\theta^{k}=1$) for all $k$ after
$\bar{x}^{k}<0.001$, and cases $\eta=2,4$ only re-open travel after
$\bar{x}^{k}\approx 0$. }}} \end{figure}
\par\par\begin{figure}\centering\begin{overpic}[trim=56.9055pt 65.44142pt
62.59596pt 1.00374pt,clip,width=216.81pt]{Images/binarycontroller_vaccine.png}
\put(-4.0,38.0){{\parbox{325.215pt}{90}{$\bar{x}^{k}$}}}
\put(50.0,-3.0){\large{\parbox{325.215pt}{}
\put(100.0,38.0){{\parbox{325.215pt}{90}{$\theta^{k}$}}}
\normalsize\end{overpic} \@@toccaption{{\lx@tag[ ]{{5}}{ System response to
the controller in \eqref{eq:control} with an on-off flow penalty combined with
a vaccine roll-out. The on-off controller is identical to
Figure\leavevmode\nobreak\ \ref{fig:binary_controller_no_vaccine} and the
vaccine roll-out is identical Figure \ref{fig:control_vaccine}.
}}}\@@caption{{\lx@tag[: ]{{Figure 5}}{ System response to the controller in
\eqref{eq:control} with an on-off flow penalty combined with a vaccine roll-
out. The on-off controller is identical to Figure\leavevmode\nobreak\
\ref{fig:binary_controller_no_vaccine} and the vaccine roll-out is identical
Figure \ref{fig:control_vaccine}. }}} \end{figure} \par Second, we simulate
the control strategy proposed in Section \ref{sec:control} with and without an
additional heuristic for vaccine distribution. With no vaccine strategy
applied, these simulations show that exclusively implementing a control law
that uniformly reduces travel based on $\bar{x}^{k}$, namely using
\eqref{eq:control}-\eqref{eq:theta}, does not have a significant impact on the
total number of people infected. Furthermore, in many cases increasing the
strength of the controller will cause the total number of recovered
individuals to increase. This phenomenon is illustrated in the plot at the
bottom of Figure \ref{fig:control_no_vaccine} by the fact that as the x-axis
increases (recall that the $\eta$ value corresponds to the strength of the
controller), so does $\bar{r}^{*}$ (except for very small $\gamma$ values).
This behavior is the consequence of at least two reasons: 1) the controller
does not prevent infections from occurring inside the sub-populations (i.e. no
lock downs), and 2) the controller in some cases `flattens the curve' which
prolongs the outbreak and increases the integral under the curve (i.e. a
higher number of total infections). \par On the other hand, a reduction in the
peak infected population reduces strain on healthcare systems, improving
medical outcomes and decreasing fatality rates\leavevmode\nobreak\
\cite[cite]{[\@@bibref{}{kenyon2020flattening}{}{}]}, and creates an
opportunity for a vaccine to be more impactful. In Figure\leavevmode\nobreak\
\ref{fig:control_vaccine} we simulate the network with the same initial
conditions and parameters as Figure\leavevmode\nobreak\
\ref{fig:control_no_vaccine}, adding the distribution of a vaccine, starting
at time step $k=500$, which, for each sub-population, moves $0.1\%$ percent of
the susceptible proportion directly to the recovered proportion at each time
step. In the bottom of Figure\leavevmode\nobreak\ \ref{fig:control_vaccine},
we show the mean infected proportion of the system for increasing levels of
sensitivity as well as the total proportion of the recovered population who
were infected by the disease\leavevmode\nobreak\ {($\delta\sum_{k\geq
0}\bar{x}^{k}$)}. We see that a combined strategy of restricting flow and
vaccine distribution can have a marked effect on the number of individuals
infected while simultaneously reducing the peak infection level of the system.
\par Lastly, we implement the controller in \eqref{eq:control} with
$\theta^{k}$ being strictly binary, that is, $\theta^{k}$ is either 0 or 1. In
Figure\leavevmode\nobreak\ \ref{fig:binary_controller_no_vaccine}, control
strategies $\eta=1,2$ shut down all travel after $50$ time steps, imitating a
delay in decision making from policymakers while strategies $\eta=3,4$ shut
down travel immediately when the infection is first detected in any sub-
population. A critical note is that even when travel is eventually completely
closed, if it does not happen quickly enough then there will be almost no
discernible impact on infection levels, as seen by comparing control
strategies $\eta=1,2$. These extreme cases illustrate that unless the
infection is completely eradicated ($\bar{x}^{k}=0$) prior to reopening
travel, it will always spread throughout the network after travel is reopened.
When we tested the binary controller case $\eta=3$ reopening travel after
$\bar{x}^{k}<10^{-9}$, there was a second $\bar{x}^{k}$ peak around $k=7000$.
When a vaccine is distributed, this delayed second wave can be significantly
mitigated as shown in Figure\leavevmode\nobreak\
\ref{fig:binary_controller_vaccine}. While costly, completely closing all
travel between sub-populations can enable a vaccine to have a tremendous
impact but only if the initial response is not delayed.
\par\par\par\par\par\par\par\par\par\par\@@numbered@section{section}{toc}{Conclusion}
\par In this paper, we have constructed a networked discrete-time SEIR
epidemic model that incorporates population flows, presented conditions under
which the model is well-defined, and shown asymptotic convergence to the
healthy states, the set of equilibria. Additionally, we have proposed a
control policy for restricting population flow which can be interpreted as
implementing travel restrictions/bans, showed it is well defined, and
illustrated its behavior via simulation. We have found that only restricting
the flow of the population is typically insufficient to reduce the total
number of infections over the course of an epidemic. More severe restrictions
on the population flow can decrease the peak infection level, which can
alleviate stress on healthcare facilities. Further, applying population flow
restrictions together with a vaccination strategy can significantly reduce the
total number of infections. \par For future work we plan to incorporate the
possibility of infections occurring while individuals are traveling (i.e.,
infections occurring on the edges of the graph) as well as using real travel
and infection data from the COVID-19 pandemic to learn the model parameters.
Finally, note that our model does not capture asymptomatic transmission of the
virus, a key component of the infectious behavior of COVID-19, therefore,
developing a similar SAIR formulation remains open to future work.
\par\par\par\par\par\bibstyle{IEEEtran} \thebibliography
\lx@bibitem{ruan2006effect} S.\leavevmode\nobreak\ Ruan,
W.\leavevmode\nobreak\ Wang, and S.\leavevmode\nobreak\ A. Levin, ``The effect
of global travel on the spread of {SARS},'' \emph{Mathematical Biosciences \&
Engineering}, vol.\leavevmode\nobreak\ 3, no.\leavevmode\nobreak\ 1, p. 205,
2006. \par\lx@bibitem{tatem2006global} A.\leavevmode\nobreak\ J. Tatem,
D.\leavevmode\nobreak\ J. Rogers, and S.\leavevmode\nobreak\ I. Hay, ``Global
transport networks and infectious disease spread,'' \emph{Advances in
Parasitology}, vol.\leavevmode\nobreak\ 62, pp. 293--343, 2006.
\par\lx@bibitem{rodriguez2020going} A.\leavevmode\nobreak\ J.
Rodr{\'{\i}}guez-Morales, K.\leavevmode\nobreak\ MacGregor,
S.\leavevmode\nobreak\ Kanagarajah, D.\leavevmode\nobreak\ Patel, and
P.\leavevmode\nobreak\ Schlagenhauf, ``Going global--travel and the 2019 novel
coronavirus,'' \emph{Travel Medicine and Infectious Disease},
vol.\leavevmode\nobreak\ 33, p. 101578, 2020.
\par\lx@bibitem{rock2014dynamics} K.\leavevmode\nobreak\ Rock,
S.\leavevmode\nobreak\ Brand, J.\leavevmode\nobreak\ Moir, and
M.\leavevmode\nobreak\ J. Keeling, ``Dynamics of infectious diseases,''
\emph{Reports on Progress in Physics}, vol.\leavevmode\nobreak\ 77,
no.\leavevmode\nobreak\ 2, p. 026602, 2014\. \par\lx@bibitem{mei2017dynamics}
W.\leavevmode\nobreak\ Mei, S.\leavevmode\nobreak\ Mohagheghi,
S.\leavevmode\nobreak\ Zampieri, and F.\leavevmode\nobreak\ Bullo, ``On the
dynamics of deterministic epidemic propagation over networks,'' \emph{Annual
Reviews in Control}, vol.\leavevmode\nobreak\ 44, pp. 116--128, 2017.
\par\lx@bibitem{backer2020incubation} J.\leavevmode\nobreak\ A. Backer,
D.\leavevmode\nobreak\ Klinkenberg, and J.\leavevmode\nobreak\ Wallinga,
``Incubation period of 2019 novel coronavirus (2019-{nCoV}) infections among
travellers from {Wuhan}, {China}, 20--28 {January} 2020,''
\emph{Eurosurveillance}, vol.\leavevmode\nobreak\ 25, no.\leavevmode\nobreak\
5, p. 2000062, 2020. \par\lx@bibitem{guan2020clinical} W.\leavevmode\nobreak\
Guan, Z.\leavevmode\nobreak\ Ni, Y.\leavevmode\nobreak\ Hu,
W.\leavevmode\nobreak\ Liang, C.\leavevmode\nobreak\ Ou,
J.\leavevmode\nobreak\ He, L.\leavevmode\nobreak\ Liu, H.\leavevmode\nobreak\
Shan, C.\leavevmode\nobreak\ Lei, D.\leavevmode\nobreak\ S. Hui
\emph{et\leavevmode\nobreak\ al.}, ``Clinical {Characteristics} of
{Coronavirus} {Disease} 2019 in {China},'' \emph{New England Journal of
Medicine}, vol. 382, no.\leavevmode\nobreak\ 18, pp. 1708--1720, 2020.
\par\lx@bibitem{li2020early} Q.\leavevmode\nobreak\ Li, X.\leavevmode\nobreak\
Guan, P.\leavevmode\nobreak\ Wu, X.\leavevmode\nobreak\ Wang,
L.\leavevmode\nobreak\ Zhou, Y.\leavevmode\nobreak\ Tong,
R.\leavevmode\nobreak\ Ren, K.\leavevmode\nobreak\ S. Leung,
E.\leavevmode\nobreak\ H. Lau, J.\leavevmode\nobreak\ Y. Wong
\emph{et\leavevmode\nobreak\ al.}, ``Early transmission dynamics in {Wuhan,
China}, of novel coronavirus--infected pneumonia,'' \emph{New England Journal
of Medicine}, 2020. \par\lx@bibitem{lauer2020incubation}
S.\leavevmode\nobreak\ A. Lauer, K.\leavevmode\nobreak\ H. Grantz,
Q.\leavevmode\nobreak\ Bi, F.\leavevmode\nobreak\ K. Jones,
Q.\leavevmode\nobreak\ Zheng, H.\leavevmode\nobreak\ R. Meredith,
A.\leavevmode\nobreak\ S. Azman, N.\leavevmode\nobreak\ G. Reich, and
J.\leavevmode\nobreak\ Lessler, ``The incubation period of coronavirus disease
2019 ({COVID-19}) from publicly reported confirmed cases: Estimation and
application,'' \emph{Annals of Internal Medicine}, vol. 172,
no.\leavevmode\nobreak\ 9, pp. 577--582, 2020.
\par\lx@bibitem{byambasuren2020estimating} O.\leavevmode\nobreak\ Byambasuren,
M.\leavevmode\nobreak\ Cardona, K.\leavevmode\nobreak\ Bell,
J.\leavevmode\nobreak\ Clark, M.-L. McLaws, and P.\leavevmode\nobreak\
Glasziou, ``Estimating the extent of true asymptomatic {COVID-19} and its
potential for community transmission: Systematic review and meta-analysis,''
\emph{Official Journal of the Association of Medical Microbiology and
Infectious Disease Canada}, vol.\leavevmode\nobreak\ 5,
no.\leavevmode\nobreak\ 4, pp. 223--234, 2020. \par\lx@bibitem{chang2020time}
D.\leavevmode\nobreak\ Chang, G.\leavevmode\nobreak\ Mo,
X.\leavevmode\nobreak\ Yuan, Y.\leavevmode\nobreak\ Tao,
X.\leavevmode\nobreak\ Peng, F.-S. Wang, L.\leavevmode\nobreak\ Xie,
L.\leavevmode\nobreak\ Sharma, C.\leavevmode\nobreak\ S.
Dela\leavevmode\nobreak\ Cruz, and E.\leavevmode\nobreak\ Qin, ``Time kinetics
of viral clearance and resolution of symptoms in novel coronavirus
infection,'' \emph{American Journal of Respiratory and Critical Care
Medicine}, vol. 201, no.\leavevmode\nobreak\ 9, pp. 1150--1152, 2020\.
\par\lx@bibitem{mizumoto2020estimating} K.\leavevmode\nobreak\ Mizumoto,
K.\leavevmode\nobreak\ Kagaya, A.\leavevmode\nobreak\ Zarebski, and
G.\leavevmode\nobreak\ Chowell, ``Estimating the asymptomatic proportion of
coronavirus disease 2019 ({COVID-19}) cases on board the {Diamond Princess}
cruise ship, {Y}okohama, {J}apan, 2020,'' \emph{Eurosurveillance},
vol.\leavevmode\nobreak\ 25, no.\leavevmode\nobreak\ 10, p. 2000180, 2020.
\par\lx@bibitem{ing2020covid} A.\leavevmode\nobreak\ J. Ing,
C.\leavevmode\nobreak\ Cocks, and J.\leavevmode\nobreak\ P. Green,
``{COVID-19}: In the footsteps of {E}rnest {S}hackleton,'' \emph{Thorax},
2020. \par\lx@bibitem{ifac_flow} M.\leavevmode\nobreak\ Ye,
J.\leavevmode\nobreak\ Liu, C.\leavevmode\nobreak\ Cenedese,
Z.\leavevmode\nobreak\ Sun, and M.\leavevmode\nobreak\ Cao, ``A network {SIS}
meta-population model with transportation flow,'' in \emph{Proceedings of the
IFAC World Congress}, 2020. \par\lx@bibitem{brockmann2013hidden}
D.\leavevmode\nobreak\ Brockmann and D.\leavevmode\nobreak\ Helbing, ``The
hidden geometry of complex, network-driven contagion phenomena,''
\emph{Science}, vol. 342, no. 6164, pp. 1337--1342, 2013\.
\par\lx@bibitem{vrabac2021capturing} D.\leavevmode\nobreak\ Vrabac,
M.\leavevmode\nobreak\ Shang, B.\leavevmode\nobreak\ Butler,
J.\leavevmode\nobreak\ Pham, R.\leavevmode\nobreak\ Stern, and
P.\leavevmode\nobreak\ E. Par{\'{e}}, ``Capturing the effects of
transportation on the spread of {COVID-19} with a multi-networked {SEIR}
model,'' \emph{IEEE Control Systems Letters}, 2021.
\par\lx@bibitem{shu2012global} H.\leavevmode\nobreak\ Shu,
D.\leavevmode\nobreak\ Fan, and J.\leavevmode\nobreak\ Wei, ``Global stability
of multi-group {SEIR} epidemic models with distributed delays and nonlinear
transmission,'' \emph{Nonlinear Analysis: Real World Applications},
vol.\leavevmode\nobreak\ 13, no.\leavevmode\nobreak\ 4, pp. 1581--1592, 2012.
\par\lx@bibitem{blondel2005convergence} V.\leavevmode\nobreak\ D. Blondel,
J.\leavevmode\nobreak\ M. Hendrickx, A.\leavevmode\nobreak\ Olshevsky, and
J.\leavevmode\nobreak\ N. Tsitsiklis, ``Convergence in multiagent
coordination, consensus, and flocking,'' in \emph{Proceedings of the 44th IEEE
Conference on Decision and Control}.\hskip 12.00003pt plus 6.00002pt minus
4.79993ptIEEE, 2005, pp. 2996--3000. \par\lx@bibitem{suzumura2020impact}
T.\leavevmode\nobreak\ Suzumura, H.\leavevmode\nobreak\ Kanezashi,
M.\leavevmode\nobreak\ Dholakia, E.\leavevmode\nobreak\ Ishii,
S.\leavevmode\nobreak\ A. Napagao, R.\leavevmode\nobreak\ P{\'{e}}rez-Arnal,
and D.\leavevmode\nobreak\ Garcia-Gasulla, ``The impact of {COVID-19} on
flight networks,'' \emph{arXiv preprint arXiv:2006.02950}, 2020.
\par\lx@bibitem{census2019} ``Annual estimates of the resident population for
incorporated places of 50,000 or more,'' 2019. [Online]. Available:
\url{https://www.census.gov/data/tables/time-series/demo/popest/2010s-total-
cities-and-towns.html} \par\lx@bibitem{datacommons} ``Place explorer.''
[Online]. Available: \url{https://datacommons.org/place}
\par\lx@bibitem{flightsfrom} ``All routes and scheduled flights from every
airport.'' [Online]. Available: \url{https://www.flightsfrom.com/}
\par\lx@bibitem{kenyon2020flattening} C.\leavevmode\nobreak\ Kenyon,
``Flattening-the-curve associated with reduced {COVID-19} case fatality rates-
an ecological analysis of 65 countries,'' \emph{Journal of Infection},
vol.\leavevmode\nobreak\ 81, no.\leavevmode\nobreak\ 1, pp. e98--e99, 2020.
\par\endthebibliography
\par\par\par\par\par\par\par\par\par\par\par\par\par\par\par\end{document}}}}}}}}}}}}}}}