[1]Kejie Chen

1]School of Optoelectric Engineering and Instrumental Science, Dalian
University of Technology, Dalian, 116024, China

2]Provincial Key Lab of Digital Twin for Industrial Equipment, Dalian, 116024,
China

3]School of Energy and Power Engineering, Dalian, 116024, China

# A stochastic agent-based model to evaluate COVID-19 transmission influenced
by human mobility

<EMAIL_ADDRESS>Xiaomo Jiang  Yanqing Li  Rongxin Zhou  [  [  [

###### Abstract

The COVID-19 pandemic has created an urgent need for mathematical models that
can project epidemic trends and evaluate the effectiveness of mitigation
strategies. To forecast the transmission of COVID-19, a major challenge is the
accurate assessment of the multi-scale human mobility and how they impact the
infection through close contacts. By combining the stochastic agent-based
modeling strategy and hierarchical structures of spatial containers
corresponding to the notion of places in geography, this study proposes a
novel model, Mob-Cov, to study the impact of human traveling behaviour and
individual health conditions on the disease outbreak and the probability of
zero COVID in the population. Specifically, individuals perform power-law type
of local movements within a container and global transport between different-
level containers. Frequent long movements inside a small-level container (e.g.
a road or a county) and a small population size reduce the local crowdedness
of people and the disease infection and transmission. In contrast, travels
between large-level containers (e.g. cities and nations) facilitate global
disease spread and outbreak. Moreover, dynamic infection and recovery in the
population are able to drive the bifurcation of the system to a "zero-COVID"
state or a "live with COVID" state, depending on the mobility patterns,
population number and health conditions. Reducing total population and local
people accumulation as well as restricting global travels help achieve zero-
COVID. In summary, the Mob-Cov model considers more realistic human mobility
in a wide range of spatial scales, and has been designed with equal emphasis
on performance, low simulation cost, accuracy, ease of use and flexibility. It
is a useful tool for researchers and politicians to investigate the pandemic
dynamics and plan actions against the disease.

###### keywords:

Infectious disease, COVID-19, Agent-based modeling, Human mobility, Container
model, Bifurcation

##  1 Introduction

COVID-19 is a highly infectious disease transmitted through several ways,
including direct physical contact, inhalation of small particles and droplets
disseminated by sneezing or coughing, and virus particles or droplets landing
on eyes, nose or mouth1 . On January 30, 2020, the outbreak of COVID-19 is
declared as a Public Health Emergency of International Concern by the World
Health Organization (WHO), which has led to the global recession. Until now,
the disease is still heavily affecting the human health and economies of most
countries2 ; 3 . The COVID-19 pandemic has created an urgent need for models
to analyze and predict the interactions between the infectious disease and
human beings.

Models for studying the pandemic spread and the impact of intervention
strategies can be broadly divided into two types: partial differential
equation (PDE)-based or stochastic differential equation (SDE)-based
compartmental models4 ; 5 ; 6 ; 7 ; 8 ; 9 ; 9extra ; 9extra2  and agent-based
Monte Carlo models10 ; 11 ; 12 ; 13 ; 14 ; 15 ; 16 ; 17 ; 17extra . The
compartmental models are simple and computationally cost-effective, and thus
have been deployed to analyze the spread of COVID-19 in various places,
including Wuhan5 , Germany6  and Italy7 . For example, Dehning et al.6
combined the susceptible-infected-recovered (SIR) model with Bayesian
parameter inference to detect change points of disease spreading rate using
COVID-19 case numbers measured in Germany. On the other hand, agent-based
models consider the individual differences, microscale policies and detailed
actions, which can provide a comprehensive understanding of the transmission
dynamics as well as a flexible framework for implementing various intervention
rules10 ; 11 ; 12 ; 13 ; 14 ; 15 ; 16 ; 17 ; 18 ; 19 ; 19extra ; 19extra2 ;
19extra3 . The agent-based models have been used to study the spread of
COVID-19 in Australia10 , Singapore11  and France12  under various
interventions19extra3 . For example, Hoertel et al.12  developed an agent-
based disease transmission model on the social contact network. They examined
the impact of lockdown period and post-lockdown measures (e.g. physical
distancing, mask wearing) on the cumulative disease incidence and intensive
care unit (ICU)-bed occupancy. Meanwhile, several user-friendly, high-
performance COVID-19 microsimulation softwares, including the OpenABM-Covid
1914  and Covasim15 , have been developed for researchers and public health
officials to easily study the COVID-19 dynamics and inform policy decisions.

However, the PDE/SDE-based compartmental models neglect the physical infection
process4 ; 5 ; 6 ; 7 ; 8 ; 9 ; 9extra ; 9extra2 , which loses accuracy in
reflecting real transmission dynamics. Most agent-based models and softwares
consider individual contacts using abstract networks and build disease
propagation model on these networks. There are three widely-used networks,
i.e., microscopic commute networks for work, study and other activities10 ; 11
; 13 ; 14 ; 15 , macroscopic commute networks for transport between cities and
countries20 ; 21 ; 22 ; 23 , and social contact networks among family members,
friends, workmates, etc.12 . In addition, some agent-based models consider
individual movements in a continuous physical space16 ; 17 ; 18 ; 19 ;
19extra2 . For example, Cuevas 16  studied how the combination of local random
walks and long directed movements of individuals influences COVID-19
transmission. Silva et al.17  incorporated four types of human behaviors,
including walking freely, going home, going to work and going to hospital,
into the infection model, where the four walking behaviors are modeled as
random walks of different velocities and directions.

Though the commute and contact networks can partially reflect the
characteristics of human mobility, these networks are usually biased by small
population samples used to estimate the network topology and weights of
edges20 . Furthermore, the commute networks only reflect travels in a limited
number of areas in the same spatial scale, either the microscopic scale
containing home, workplace, school and hospital10 ; 11 ; 13 ; 14 ; 15 ;
19extra2 , or in the macroscopic scale containing cities and countries21 ; 22
; 23 . The social contact networks are naturally formed in a long period of
time. But the transient contacts between strangers and acquaintances are
determined by many factors besides their social relationships24extra1 . In
addition, in the microscopic continuous space, studies have also showed that
human mobility patterns do not follow a random walk or a scale-free lévy
walk24extra2 ; 24extra3 .

Recently, Alessandretti et al.24  have showed that physical space can be
represented by nested containers with associated size, satisfying the
intuitive conception of space which is hierarchical and characterized by
typical scales. Human moving traces modeled as the combination of commuting
within the same containers and transporting between containers of different
scales are more realistic. Specifically, they proposed that the physical space
of each individual can be represented as a hierarchy of $L$ levels ordered
from the smallest to largest. The small level contains individual positions,
rooms or houses. A large level contains states or countries. At each level,
the physical space is partitioned into compact containers with certain
characteristic sizes. The individual travels among the containers at the same
or different levels based on the probabilities proportional to the container
distance and the level distance. Compared with other state-of-art
models24extra1 ; 24extra2 ; 24extra3 ; 25 ; 28 ; 29 ; 29extra1 ; 29extra2 ,
the hierarchical container model provides an unbiased performance estimate and
a significant better description of human GPS traces evaluated using several
index including the distribution of displacements, evolution of radius of
gyration and time allocation among locations and entropy. Therefore, modeling
COVID-19 infection and propagation in the nested containers can provide more
accurate and insightful results about how human mobility influences disease
transmission dynamics.

Figure 1: Schematics of the Mob-Cov model. (a) Human mobility on the three-
level nested container structures. (b) The probability of traveling from
location $j$ to $k$ is determined by two steps: selecting a level (e.g.
selecting level 3 by $P_{lv}(3)$) and selecting the nested containers level by
level (e.g. $P_{3}(k_{3})$, $P_{2}(k_{2})$, $P_{1}(k_{1})$). (c) The COVID-19
infection and recovery rules.

In this study, we present a stochastic agent-based model, Mob-Cov (mobility
influenced COVID-19 transmission model), which considers the hierarchical
geographical mobility patterns and the COVID-19 infection and recovery
processes. Specifically, in the model, each person is represented by an agent
which can move around freely and has an individual-specific health condition
represented by the infection and recovery rates. Assuming that 0.5% of the
population is infected initially. As time goes on, people surrounding the
infected individuals (i.e. these people are in the same container with the
infected ones) also get infected. Sick people travels to other containers,
which turns the local disease transmission to a global problem and eventually
induces the outbreak crisis. Next, the recovery of sick people after an
infection period is included. Instead of having an outbreak, the system is
able to achieve a dynamical equilibrium with the coexistence of the healthy
and infected people. Furthermore, the influence of population size, health
conditions, hierarchical geographical structures and population mobility on
the disease transmission dynamics are predicted. Overall, the Mob-Cov model
and the results presented in this study provide a novel and comprehensive view
of the spread of COVID-19 from a multi-scale human mobility perspective, and
can be valuable and helpful in evaluating and designing better interventions
and mitigation measures for the pandemic.

##  2 The proposed Mob-Cov model

###  2.1 Agent-based model

To investigate the impact of human mobility on the transmission of COVID-19,
we develop a stochastic agent-based Mob-Cov model. The model includes three
key components: (1) a synthetic population generated with the hierarchical
geographical characteristics (Figure 1a); (2) realistic human travel rules,
taking into account the power law behaviors of travelling within a container
and transporting between containers (Figure 1b); (3) the infection and
recovery with probabilities influenced by individual health conditions (Figure
1c). The first two components about human mobility are discussed in
subsections II B and II C. The last component about disease infection and
recovery is discussed in the subsection II D. The flow chart (framework) of
the Mob-Cov model is shown in Figure 2.

This model framework includes a flexible individual-based approach that can
capture not only the local interactions between individuals and microscopic
disease transmission, but also the global spread of the disease influenced by
the really long-distance cross-district travels. The framework allows
policymakers to define measures at various spatial-temporal scales and
evaluate the effective length scale of the interventions.

In the following sections, the three key components of the model and their
mathematical foundations are first introduced. The selection of model
parameters and the evaluation methods are then presented. Lastly, the pseudo
code of the model and the computational procedures are given.

###  2.2 Human mobility in the nested containers

The hidden geographical structure of the population is modeled using nested
containers proposed by Alessandretti et al.24 . Specifically, the physical
space is assumed to contain hierarchy of $L$ levels. Level 1 is the smallest
and level $L$ is the largest. At any level $l$ ($l=1,2,...,L$), the space is
partitioned into $n_{l}$ topologically compact containers. For $l<L$, a
container is fully included within a single parent container. Thus, each
geographical location $k$ can be uniquely identified as a sequence of
containers ($k=(k_{1},k_{2},...,k_{L})$), where the child container at $l$
level $k_{l}$ is fully included in the mother container at $l+1$ level
$k_{l+1}$. For example, Figure 1a shows the schematics of a three-level
hierarchical structure. At the largest level (level 3), there are 3
containers, as shown by the light orange polygons. Level 2 has 9 containers,
as shown by the orange polygons. And level 3 contains 21 individual positions,
represented by the black dots. The location of agent $j$ is defined as
$j=(j_{1},j_{2},j_{3})$, where $j_{3}$ is the ID of the left large polygons in
level 3, $j_{2}$ represents the middle polygon in $j_{3}$ and $j_{1}$ is the
ID of one black dot in $j_{2}$.

Figure 2: The flowchart of the Mob-Cov model. $rd$ is a random number
uniformly distributed between 0 and 1. $t$ is the current simulation time.
$\tilde{t}_{i}$ is the infection time of agent $i$. $P_{mi}$, $P_{lv}(l)$,
$P_{j}(k_{j})$, $P_{i}$, $R_{i}$ and $T_{inf}$ are model parameters discussed
in section II.

Travelling in the nested containers follows a two-stage process: first
deciding which level to travel at, second selecting a container at that level,
and then a container within the chosen container and so on down to the lowest
level. Specifically, suppose an agent is at position
$j=(j_{1},j_{2},...,j_{L})$, the probability for the individual to travel to
position $k=(k_{1},k_{2},...,k_{L})$ can be expressed as (Figure 1b)

$P(j\rightarrow k)=P_{lv}(l)\prod_{i\leq l}P_{i}(k_{i})$ (1)

where $P_{lv}(l)$ is the probability of the agent choosing level $l$. The
agent stays in the same container if the container level is larger than $l$.
In $1,2,...,l$ levels, the agent switches to different containers. Thus,
$k_{l+1}=j_{l+1}$, $k_{l+2}=j_{l+2}$, …, $k_{L}=j_{L}$, but $k_{0}\neq j_{0}$,
$k_{1}\neq j_{1}$, …, $k_{l}\neq j_{l}$. $P_{lv}(l)$ follows an exponential
distribution with a parameter $d$, which is expressed as

$P_{lv}(l)=e^{-d\cdot l},\,\,\,\,\forall\,\,\,l\in[1,L]$ (2)

where $l$ is an integer representing the level. Eq. (2) indicates that
switching between large-level containers (e.g. between countries) is less
frequent than switching between small-level containers (e.g. between houses or
streets).

$P_{i}(k_{i})$ is the probability of choose container $k_{i}$ at level $i$.
$P_{i}(k_{i})$ follows a long-tail distribution with parameters $c_{0}$ and
$c_{1}$, which is expressed as

$P_{i}(k)=k^{-(c_{0}+c_{1}\cdot i)},\,\,\,\,\forall\,\,\,i\in[1,l]$ (3)

where $i$ is the level. $k$ is the container position. Eq. (3) suggests that
the travel probability decreases over the distance, especially at a large
level $i$. This indicates that, for example, traveling to a distant country is
more scarcer than going to a remote road. But compared with the widely-used
exponential distribution, the long-tail distribution allows the happening of
long-distance movements at all levels.

Eqs. (1)-(3) calculate the probability of switching from location $j$ to
location $k$ if the agent takes a movement. We consider the fact that people
do not travel at every time moment but need to take a rest or stay at a place
for a while. Thus, the probability for an agent to take a movement is modeled
with a probability $P_{m}$, which is uniformly distributed between 0.2 and
0.4. A portion of people with low movement tendency (i.e. low mobility
population) is further considered. The ratio of the low mobility people in the
population $r_{lm}$ is a model parameter. The probability for the low mobility
people to take a movement is only $P_{m}=0.03$.

###  2.3 Infection and recovery

If two agents $i$ and $j$ ($i$ is infected and $j$ is healthy) are in the same
level-2 container (level-1 containers are the individual positions), infection
happens with a probability $P_{ij}$ (Figure 1c). If there are more than one
infected agents, the infection probability of a healthy agent $j$ is
calculated as $P_{j}=1-\prod_{i}(1-P_{ij})$, for all infected agents $i$ in
the same level-2 container with $j$. For the high risk population, $P_{ij}$
follows a uniform distribution between 0.1 and 0.3. For the low risk
population, $P_{ij}=0.02$. The low risk people ratio in the population
$r_{li}$ is a model parameter. Note that, $P_{ij}$ is chosen according to
previous studies 15 ; 16 . Kerr et al. 15  set $P_{ij}=0.05$ for household-
level transmission. Cuevas 16  set $P_{ij}=0.2$.

After being infected for a time period $T_{inf}$, the individual $i$ is able
to recover to a healthy state with a probability $R_{i}$ (Figure 1c).

###  2.4 Outbreak endpoint

To evaluate the spread speed of the disease in the population, the outbreak
endpoint is calculated. Using the outbreak endpoint instead of the time when
all the agents are infected is because there are always few individuals having
really low infectious probability $P_{ij}$. The time it takes to get the whole
population infected is heavily biased by the last few individuals, which
cannot directly and accurately reflect the system property. Therefore, an
outbreak endpoint which considers the effects of both the infected population
ratio and the infection rate is used in this study13 ; 30 .

The detection of the outbreak endpoint is similar to find the knee point in
system engineering. The knee point usually represents the "right decision
point", after which the system properties are no longer significantly
influenced by the variation of the variable30 . Thus, in this study, we follow
the previously proposed methods and calculate the maximum point of the cost
function. The maximum point is defined as the outbreak endpoint of the system.
The cost function is expressed as13

$J(t)=\frac{N_{inf}(t)}{N_{p}}-\frac{t}{T_{max}}$ (4)

where $N_{inf}$ is the number of infected agents at time step (iteration step)
$t$, $N_{p}$ is the total population number, $T_{max}$ is the total time
(maximum iteration number), which is defined as the time when 90% of the
population got infected plus 100 extra time steps.

It has been proved that the cost function $J$ can only have one global maximum
point, which corresponds to the outbreak endpoint13 . Therefore, the outbreak
time can be calculated as

$T_{b}=\arg_{t\in[1,T_{max}]}\,max\,J(t)$ (5)

Figure 3 shows a simulation example of the infection rate over time and the
determination of the outbreak endpoint. Figure 3a shows that the number of
newly infected people is the largest during the first 70 time steps. After 70
time steps, the infection rates gradually decrease to values close to zero. As
expected, the proportion of infected people keeps increasing over time, but
the increasing speed slows down (Figure 3b). By finding the maximum point of
the cost function (Figure 3c), the outbreak point can be identified when $t$
is around 70, which corresponds to an infection ratio around 80%. The results
suggest that the first 70 time steps are the most critical period when the
number of infected people increases rapidly. After around 80% of the
population gets infected, the infection dynamics slows down. And it can take a
really long time (can be longer than 2000 time steps) for the whole population
to get infected.

It should also be noted that the outbreak happens when only infection of
individuals is included in the model. After considering the recovery of sick
people, the system is likely to reach a dynamical equilibrium with both
infected and healthy population, instead of reaching the outbreak endpoint.

Figure 3: The evolution of the infection process (a) The rate of infection
(the number of new infected people) over time. (b) The infection ratio of the
population over time. (c) The cost function $J(t)$ over time. The outbreak
endpoint is the global maximum point of the cost function, labeled in red.

###  2.5 Computational procedure

The Mob-Cov model is implemented as an iterative scheme. The flowchart is
shown in Figure 2 and the pseudo-code is shown below. In the pseudo-code, $L$
is the number of levels. $N_{cont}=(n_{1},n_{2},...,n_{L})$ is the number of
containers in each level. $c_{0}$, $c_{1}$ and $d$ are parameters that
determine the mobility, as discussed in section II2. $N_{p}$ is the number of
agents. $N_{inf0}$ is the number of initially infected agents. $T_{inf}$ is
the infection time period before the agent having a probability to recovery.
$r_{lm}$ is the ratio of low mobility population. $r_{li}$ is the ratio of low
infection risk population. NestedCont is the nested container structure, which
is a nested random dictionary implemented in Python. $P_{mi}$ is the
probability of taking a movement for agent $i$. $P_{ij}$ is the probability of
agent $j$ getting infected by agent $i$. $R_{i}$ is the recovery rate of agent
$i$. If agents $i$ and $j$ are in the same level-2 container, $j$ is defined
as the neighboring agent of $i$.

Input: $L$, $N_{cont}$, $c_{0}$, $c_{1}$, $d$, $N_{p}$, $N_{inf0}$, $T_{inf}$

Initialise:

NestedCont (nested container structure)

agent position $\mathbf{X}_{i}$ ($i=1,2,..,N_{p}$)

infected agent IDs

$P_{mi}$ (mobility) for each agent $i$

$P_{ij}$ (infection) for agents $i$ and $j$ ($i\neq j$)

$R_{i}$ (recovery) for each agent $i$

while  _(iter $<$ MaxIteration) or (stop criterion)_ do

for  _each agent $i$_ do

if  _(rand $\leq P_{mi}$)_ then

Update $\mathbf{X}_{i}$ using Eqs. (1)-(3)

end if

end for

for  _each infected agent $i$_ do

for  _each neighboring healthy agent $j$_ do

if  _(rand $\leq P_{ij}$)_ then

Update agent $j$ as infected

end if

end for

if  _(Infection time $>T_{inf}$) & (rand $\leq R_{i}$)_ then

Update agent $i$ as healthy

end if

end for

end while

Post process results and visualization

Algorithm  Mob-Cov model

In the simulation, the maximum iteration number is set as 2000, which is long
enough for the system to reach the equilibrium.The number of agents is 1000.
The number of initially infected people is 5. For each experiment, 50
simulations are performed, and the mean and variance of the results (including
the outbreak time, infected population ratio when outbreak, infected ratio,
zero-COVID ratio) are calculated.

##  3 Results

###  3.1 The spread dynamics of COVID-19

Figure 4a shows a simulation result of the population distribution and the
spread of COVID-19 disease. In the simulation, the number of levels is $L=5$.
Parameters of mobility are $d=1.5$, $c_{0}=0.8$, $c_{1}=0.35$. The number of
containers in each level is $N_{cont}=(8,4,9,9,250)$. Containers in level 5
are represented by colorful and transparent polygons. Individuals in different
level-5 containers are represented by dots labeled with different colors.
Figure 4b is the probability of selecting a container level, $P_{lv}(l)$. The
grey bars are simulation results, which fit well with the analytical
expression in Eq. (2). Figure 4c is the probability of selecting a container
in level 2, $p_{2}(k)$. The simulation results represented by the black dots
fit well to the analytical expression in Eq. (3).

Figure 4: Visual results of the population mobility and the spread of the
disease. (a) Distribution of the health and infected individuals at four time
steps. In the simulation, $L=5$, and there are 8 containers at level 5, which
are represented by the colorful polygons. Individuals in the 8 containers are
represented by the colorful dots. Infected individuals are labeled as red. (b)
Probability of switching to level $l$ ($l\in[1,5]$). The grey bars are the
simulation results. The dotted line is calculated using Eq. (2). (c)
Probability of moving to container $k$ at level 2. The black dots are the
simulation results, and the black line is calculated using Eq. (3).

At the beginning, only five individuals (four in the blue container and one in
the red container) are infected. Due to the high population density in the
blue container, the disease quickly spreads out in one cluster at time
$t_{0}$. Many infected people then travels to the nearby cluster in the same
blue container, accelerating the local transmission. Meanwhile, the spread of
the disease is relatively slow in the red container due to the low population
density. Afterwards, one or two infected people travel between the level-5
containers and reach the grey and brown areas, indicating that the disease is
transmitted globally. After a further local spread at time $t_{2}$, a disease
outbreak happens at $t_{3}$.

Overall, the spread of the disease is a multi-scale dynamic process influenced
by the probability of switching between levels (Figure 4b) and the movement
distance (Figure 4c). The results indicate that the total population density
and the initial infected people number does not need to be really large. When
the disease starts from a local population cluster with a high people density
(e.g. a crowded market), a small probability of traveling between large-level
containers (e.g. traveling cross cities or countries) can lead to a rapid
global spread and outbreak. The local-to-global spread of the disease heavily
relies on the long movements between containers. Thus, accurately modeling
both the short walks and long sojourns using the hierarchical container
structure in the Mob-Cov model ensures the precise prediction of the disease
dynamics.

Figure 5: The influence of human mobility and health conditions on the disease
spread and outbreak. The influence of moving distance regulated by parameter
$c_{1}$ on (a) the infected people ratio at outbreak and (b) the outbreak
time. The influence of travelling between levels regulated by parameter $d$ on
(c) the infected ratio and (d) the outbreak time. The influence of the
percentage of low mobility population on (e) the infected ratio and (f) the
outbreak time. The influence of the percentage of low risk population on (g)
the infected ratio and (h) the outbreak time. The influence of population size
on (i) the infected ratio and (j) the outbreak time. The influence of the
number of hierarchical levels on (k) the infected ratio and (l) the outbreak
time.

###  3.2 Influential factors on disease outbreak

The spread and outbreak of COVID-19 is influenced by various complex factors,
such as biological characteristics of the virus, health and immune conditions,
population density, environmental conditions and policies. Using the Mob-Cov
model, we mainly decipher the impact of microscopic and universal human
mobility and individual health conditions on the disease spread and outbreak
(Figure 5).

For the mobility, the results show that the moving distance and the
probability of traveling between large-level containers have significant
influence on the disease spread and the onset of the outbreak. When long
movements in the same level happen more frequently, which corresponds to a
smaller value of $c_{1}$, the local crowdedness is reduced. Thus, it takes a
longer time to reach outbreak (Figure 5b). The infected people are likely to
be distributed in every containers and the ratio of the infected population
can be relatively small when the outbreak happens (Figure 5a). When
individuals are likely to travel between large-level containers (e.g. cities
and countries), which corresponds to a small value of $d$, the global
transmission of the disease is enhanced. The time it takes for the onset of
the outbreak becomes shorter. At the outbreak endpoint, more than 80% of the
population have been infected. Restricting travels between large-level
containers helps slow down the spread of the disease and reduce the size of
the infected population (Figure 5c and Figure 5d). The ratio of the low
mobility population over the total population does not have a significant
influence on the infected people number when the outbreak happens (Figure 5e).
But if most people prefer to stay at the same place than move, the time it
takes for reaching the outbreak endpoint increases from 100 time steps to more
than 600 time steps. In addition, the variation of the outbreak time is large
when the mobility of most people is low, because the outbreak time now heavily
relies on the random initial distribution of the population (Figure 5f).

For the individual health conditions, if a large percentage of the population
have low infection risk (e.g. most people get a COVID-19 vaccination), the
spread speed of the disease is slowed down (Figure 5h). The low risk people
ratio does not have significant influence on the infected population size at
the outbreak endpoint. Around 70% of the people are infected at the outbreak
(Figure 5g). The total population size also does not have a significant impact
on the infected population ratio (Figure 5i). But when the population number
is large, the local crowdedness increases the infection probability of
individuals and accelerates the spread of the disease (Figure 5i).

Lastly, we find that the hierarchical structure of the containers has a large
impact on the infected population ratio and the outbreak time. If the hidden
network of human travels contains many levels, the spread speed of the disease
is slower and the outbreak time is larger (Figure 5l). In addition, less
people get infected at the outbreak endpoint (Figure 5k). The result suggests
that multi-layer geographical barriers and borders that restrict human
mobility can help reduce the spread of the disease. For example, in the areas
of high population density, the spread of COVID-19 tends to be fast. By zoning
and adding multi-level physical borders, the transmission of the disease may
be effectively reduced.

Figure 6: Bifurcation diagrams and the transition between "living with COVID"
and "zero-COVID" states. (a) The bifurcation diagram of infected population
ratio influenced by mobility parameter $c_{1}$. Black dots and line are for
"zero-COVID" state. Orange dots and line are for "living with COVID" state.
Each dot is the mean result of 50 simulations. (b) The percentage of achieving
"zero-COVID" influenced by $c_{1}$. (c) Variation of the infected population
ratio over time in the two states when $c_{1}=0.25$. (d) Bifurcation diagram
of infected ratio influenced by the percentage of low mobility people. Grey
and light orange dots represent rare cases (1 or 2 out of 50 simulation
cases). (e) The percentage of achieving "zero-COVID" influenced by the low
mobility people ratio. (f) Variation of the infected ratio over time when the
low mobility people ratio is 0.7. (g) Bifurcation diagram of the infected
ratio influenced by the population number. (h) The percentage of achieving
"zero-COVID" influenced by the population number. (i) Variation of the
infected ratio over time when the population number is 400.

###  3.3 Bifurcation and state transition

When the recovery from infection is considered, the system is no longer going
to reach the outbreak endpoint. In contrast, depending on the infection rate,
recovery speed and rate, and many other factors, two possible states may
happen: (1) "zero-COVID" state, which means all the infected people have
recovered; (2) "living with COVID" state, during which the system reaching an
equilibrium state and the number of newly-infected people is equal to the
number of newly-recovered people. In this section, the influential factors on
the transition between "zero-COVID" state and "living with COVID" state are
investigated.

First, the influence of moving distance determined by parameter $c_{1}$ is
studied. When individuals tend to take long movements in same-level containers
(i.e. mobility parameter $c_{1}$ is smaller than 0.25), the system is able to
achieve "zero-COVID". Increasing $c_{1}$ leads to the local accumulation of
people and facilitates the spread of the disease. Bifurcation happens when
$c_{1}$ becomes larger than 0.25. The transition appears to be discontinuous,
and a hysteresis loop appears in the bi-stable region where both "zero-COVID"
state (black dots) and "living with COVID" state (orange dots) coexist. When
$c_{1}$ is larger than 0.4, the percentage of infected people in the
population increases to more than 30% (Figure 6a). Figure 6b shows that the
probability of zero-COVID after a long enough time rapidly decreases to zero
when $c_{1}$ is larger than 0.25. We further show the variation of the
infected people ratio over time (Figure 6c). If the system eventually reaches
a "zero-COVID" state, the fluctuation curves are labeled in pink colors.
Otherwise, the curves are labeled in grey colors. Figure 6c shows that there
is a peak of the infected ratio at the beginning, because healthy people keeps
being infected but infected people have not recovered yet. Afterwards, the
infected ratio decreases and then fluctuates between 3% to 5%. In some cases,
the infected ratio decreases to zero, indicating that all the sick people
become recovered.

Next, effects of the percentage of low mobility people in the population is
evaluated. Increasing the percentage of low mobility people reduces the spread
of the disease and the infected ratio. Thus, the system transitions from the
"living with COVID" state to a bistable region, and then to the "zero-COVID"
state (Figure 6d). The probability of achieving zero-COVID increases rapidly
when the low mobility people ratio is larger than 60% (Figure 6e). In
addition, the variation of the infected ratio is large and the time for the
system to reach the "zero-COVID" state is longer when the low mobility people
ratio is 70% (Figure 6f). This phenomenon may be caused by the stochastic
locations and movements of infected people. When the infected people stay in
the crowds for a while, it facilitates the disease transmission and increases
the infected ratio. But if the infected people are located in a remote area,
the low mobility characteristics helps reducing the infection.

Lastly, we also found that varying the population number leads to the
bifurcation, and the bistable region with both "zero-COVID" state and "living
with COVID" state appears at an intermediate population number (i.e. around
400 to 500 people) (Figure 6g). The probability of zero COVID decreases in the
bistable region (Figure 6h). When the population size is larger than 600, the
system is likely to reach an equilibrium and has more than 20% of the
population got infected (Figure 6g). When the population number is less than
400, all the infected people can recover from the disease at the first 1000
time steps (Figure 6i) in most simulation cases.

In addition, effects of infection and recovery probabilities on the disease
spread and the infected ratio have also been studied. As expected, increasing
the infection probability leads to the increase of infected ratio, while
increasing the recovery probability reduces infected ratio. These results are
trivial and are not shown here.

Figure 7: Phase diagram of the transient zero-COVID ratio over time influenced
by human mobility and population size. (a) The effect of mobility parameter
$c_{1}$ which represents the traveling distance in the containers. (b) The
effect of low mobility people ratio in the population. (c) The effect of the
population number.

###  3.4 Phase diagram and transient dynamics

When people take frequent and short local movements, or the population size is
large, the spread of the disease is easier and faster. The infected people
ratio is larger than 10%. Reducing local crowdedness helps decrease the
infected ratio to around 5%. Then, fluctuations in the system originated from
the random human travel, infection and recovery behaviors may lead to the
disappearing of COVID-19 infection (Figures 6c, f and i). The probability to
zero COVID-19 and the time it takes are influenced by many factors. In this
section, we analyzed the phase diagrams of the infected ratio over time
influenced by mobility parameter $c_{1}$, low mobility people ratio and
population number.

Figure 7a shows that the system reaches equilibrium state quickly (i.e. either
the "zero-COVID" state or the "living with COVID" state) when $c_{1}$ is
larger than 0.4 or smaller than 0.25. When $c_{1}\in[0.25,0.4]$, the system is
bistable and the probability of zero-COVID drops significantly with a small
increase of $c_{1}$. The time for a thorough recovery from COVID-19 becomes
longer. From Figure 7b, it can be found that having a large portion of low
mobility people in the population (i.e. more than 30%) increases the time
needed for zero-COVID. Though the recovery time is longer, the probability of
reaching "zero-COVID" state is higher. When the low mobility people ratio is
small, the system is likely to be in the "living with COVID" equilibrium
state. Moreover, the population number also influences the transient COVID-19
transmission dynamics (Figure 7c). When the number of people in the population
is small (i.e. less than 400), all the people can recover from the disease
within the first 500 time steps. When the people number increases, the
probability of reaching the zero-COVID state is lower and the population
recovery takes a longer time. When the population contains more than 800
people, the newly infected people and recovered people achieves a dynamical
equilibrium. The system is in the "live with COVID" state, and more than 20%
of people in the population are infected.

##  4 Discussion

The COVID-19 pandemic has presented an unprecedented challenge and an urgent
need for accurate understanding and rapid prediction of the disease dynamics
in a wide range of scales. As an airborne disease which transmits mainly
through close-contact meetings and interactions, peoples’ physical locations
and movements are most fundamental sources that drive the spread of COVID-19.
However, due to the limited understanding of human mobility, until now, how
the intertwined long-distance travels and local movements lead to contagion
events and virus spread remains illusive.

In this study, we developed a mobility-influenced COVID-19 transmission model
(Mob-Cov) using the agent-based Monte Carlo modeling strategy and the multi-
level nested spatial containers. Human mobility behavior is restricted based
on a hierarchical structure of spatial containers, corresponding to the
geographical places from buildings, via streets and cities, to nations and
continents. By merging the power-law movements within containers and the
macroscopic transport between containers, the generated human travel patterns
are highly realistic, which ensures an accurate study of the transmission
dynamics of COVID-19 influenced by mobility. In addition, due to the
flexibility of agent-based modeling strategy, different individual moving
actions and health conditions are considered in the model. We found that long
travels within a small-level container (e.g. a street or a county) helps
alleviate local crowdedness of people and reduces infection. But traveling
between levels and large-level containers (e.g. traveling between cities and
nations) converts the local transmission to global disease spread,
facilitating the outbreak of the disease. Moreover, the infection and recovery
of an individual leads to the bifurcation and the appearance of two dynamic
states, "zero-COVID" state and "live with COVID" state. We conducted
bifurcation analysis and found that the ability of zero COVID, which means all
the infected people are recovered, is largely influenced by human mobility and
population number. The Mob-Cov model might be further used to quantitatively
measure how much local accumulation of people and global travels need to be
reduced in order to achieve successfully removing the COVID-19 infection from
the population.

Overall, the Mob-Cov model can be extended in four major directions in future
works. (1) Infection in a wide range of spatial scales can be considered.
Specifically, compared with previous works, infection on a contact network
estimated from human GPS data mainly captures the transmission between cities
and nations21 ; 22 ; 23 . Agent-based models using random walks to describe
human movements can only study the microscopic actions within a building or a
district16 ; 17 ; 18 ; 19 . The multi-level hierarchical container structures
in the Mob-Cov provides a more realistic and accurate description of
individuals moving cross multiple length scales while maintaining a reasonable
computational cost. (2) The nested container structures of human mobility can
be estimated from the real-world human GPS data using maximum likelihood
estimator24 . Thus, the model is capable of predicting the distribution of
COVID-19 infection in reality and providing the early warning of disease
outbreak based on the past infection data and the geographical information.
(3) Various interventions can also be easily implemented and used to study the
impact of policies on the control of the COVID-19 pandemic, especially the
effective length scales of these policies. (4) In this work, we focused on
modeling the multi-level human mobility while implementing only simple rules
of disease infection. It would be interesting to further extend the Mob-Cov
model to complex infection scenarios, including the susceptible, exposed,
infectious and recovered states. In particular, the susceptible and exposed
states can be critical, since the latent period between being infected and
becoming infectious has made the disease more contagious. Moreover, whether
the epidemic wave changes on the nested mobility networks, and how does it be
influenced by population characteristics (e.g. age, gender, level of
urbanization) should be further investigated using our Mob-Cov model.

## Acknowledgments

We acknowledge support from the National Natural Science Foundation of China
(Grant No. 12102081) and Fundamental Research Funds for the Central
Universities in China (Grant No. DUT21RC(3)044).

## Data Availability

The data that support the findings of this study are available from the
corresponding author upon reasonable request.

## Conflict of interest

The authors declare that they have no conflict of interest.

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