# Integrating Agent-Based and Compartmental Models for Infectious Disease
Modeling: A Novel Hybrid Approach

Inan Bostanci
Zuse Institute Berlin
14195 Berlin
<EMAIL_ADDRESS>
&Tim Conrad
Zuse Institute Berlin
14195 Berlin
<EMAIL_ADDRESS>

###### Abstract

This study investigates the spatial integration of agent-based models (ABMs)
and compartmental models in infectious disease modeling, presenting a novel
hybrid approach and studying its implications. ABMs, characterized by
individual agent interactions and decision-making, offer detailed insights but
are computationally intensive for large populations. Compartmental models,
based on differential equations, provide population-level dynamics but lack
granular detail. Our hybrid model aims to balance the granularity of ABMs with
the computational efficiency of compartmental models, offering a more nuanced
understanding of disease spread in diverse scenarios, including large
populations.

We developed a custom ABM and a compartmental model, analyzing their
infectious disease dynamics separately before integrating them into a hybrid
model. This integration involved spatial coupling of discrete and continuous
populations and evaluating the consistency of disease dynamics at the macro
scale. Our key objectives were to assess the effect of model hybridization on
resulting infection dynamics, and to quantify computational cost savings of
the hybrid approach over the ABM. We show that the hybrid approach can
significantly reduce computational costs, but is sensitive to between-model
differences, highlighting that model equivalence is a crucial component of
hybrid modeling approaches. The code is available at
http://github.com/iebos/hybrid_model1.

_K_ eywords Hybrid Model $\cdot$ Coupled Models $\cdot$ Agent-Based Model
$\cdot$ Compartmental Model $\cdot$ Epidemiology $\cdot$ Infectious Disease
Modeling

##  1 Introduction

Preventing the widespread transmission of infectious diseases is a critical
objective for health systems worldwide. Preemptively addressing and mitigating
their spread is crucial for safeguarding public health, and disease modeling
plays an essential role in this effort. By analyzing the societal impacts and
developing effective intervention strategies, disease modeling has become a
necessity, as underscored by the COVID-19 pandemic [1]. Tools, such as agent-
based models (ABMs) and compartmental models, are fundamental approaches for
understanding and managing disease spread [2]. As a result, these models have
gained significant recognition in epidemiology and are invaluable to
policymakers and public health officials [3].

Compartmental models are often used for mathematical modeling of infectious
diseases [4] and provide insights into the spreading dynamics on a macro-
scale. They are often based on (deterministic) ordinary differential equations
(ODEs) and describe the rates of movement between different compartments, such
as susceptible, infected, and recovered, within a population. This modeling
approach offers valuable perspectives on overall population-level dynamics [5,
6, 7]. Although based on the simplistic assumption of a homogeneously mixed
population, their complexity and spatial resolution can be increased by
including human mobility in meta-population models [8, 9, 10] or by coupling
populations on multiple layers [11]. In contrast, ABMs simulate the behavior
of individual agents, accounting for their interactions, movements, and
decision-making processes. This enables a detailed analysis of disease
transmission dynamics under various scenarios and provides a more granular and
realistic approach to infectious disease modeling [12, 13]. This allows for
population heterogeneity, for example, by representing the population as a
network of agents [14], or by capturing complex spatial-temporal patterns
through explicit modeling of movement and interactions [15, 16, 17].

However, ABMs require significantly more computational resources compared to
(ODE-based) compartmental models, a discrepancy that grows with population
size [for a comparison, see 18, 19]. Further, ABMs often require more complex,
fine-grained data for fitting or verification [2]. This hinders ABM-based
analysis of large populations and calls for new strategies that combine the
computational efficiency of ODE-based approaches with the detailed insights of
ABM-based approaches. Such strategies could be particularly valuable to
policymakers during health emergencies such as the Covid-19 pandemic.

In this paper, we propose a hybrid modeling approach that spatially integrates
an ABM with an ODE-based compartmental model. Such a hybrid model could be
utilized to integrate multiple geographic regions with different population
characteristics and data availability, while allowing for population movement
between regions. This hybrid approach could reduce the computational costs
associated with a pure ABM while retaining analytical insights. Moreover, it
may scale up to larger populations without sacrificing significant predictive
power, as would be the case with a purely compartmental model. To facilitate
such applications, we systematically explore the proposed hybrid model and
evaluate its feasibility.

Despite their potential, hybrid models have received relatively little
attention in infectious disease modeling research. Few studies have explored
hybrid modeling approaches, most of which have not critically examined how
model coupling impacts the dynamics of infectious diseases. One of the main
goals of this paper is to analyse coupling effects on infectious disease
dynamics and the computational efficiencies of the hybrid model compared to a
pure ABM approach. For this, we study two scenarios in a simplified toy
example: In the first, movement within the ABM component is random, resulting
in a homogeneous population. In the second, agent-movement is influenced by a
landscape that encourages spatial heterogeneity, and therefore deviates from
the ODE-component.

One of the primary strengths of this study is its methodical assessment of the
effects of coupling on key indicators that reflect the infectious disease
dynamics within a spatially coupled hybrid model. We find that the spatial
location of the coupling mechanism affects resulting infectious disease
dynamics when agent movement is non-random. This finding underscores the
importance of understanding the effects of model coupling for accurate
epidemic modeling.

This study addresses the following questions: (1) How does spatial coupling of
an ABM with an ODE-based model affect the dynamics of infection spread in the
overall population? (2) How much computational cost is saved with the hybrid
model compared to a pure ABM?

The paper is structured as follows: We begin by reviewing related work on
hybrid models that couple compartmental and agent-based models, particularly
in the field of infectious disease modeling. Next, we detail the
implementation of the ABM, the compartmental model, and the hybrid model, with
an explicit description of the coupling mechanism. Following, we vary the
weight of the compartmental component within the hybrid model in two
experiments to assess the error and infection dynamics of the hybrid model.
The first experiment deals with a homogeneous population scenario, while the
second experiment deals with a hybrid model with a spatially heterogeneous
ABM-component. Finally, we discuss the results and give an outlook for future
research.

##  2 Related Work

One of the earliest hybrid model approach for infectious diseases is described
by Bobashev et al. [20]. In that study, a temporal coupling was implemented,
meaning that the model alternates between an ABM state and a compartmental
model state, with the switch being triggered when the number of infected
individuals crosses a threshold value. Aggregate population values of the ABM
are used as initial conditions for the compartmental model state. The authors
argued that insights into epidemic processes at the local level diminish in
informative value with an increasing number of infected individuals. They
evaluated several threshold values and demonstrated how such a model can be
applied to a single hypothetical city. Further, using air travel data, they
extended this model to a network of cities. [21] followed a similar approach,
applying the switching method to a meta-population setting for several regions
in Ireland. They compared computational execution times and highlighted the
potential of this model to conserve resources. However, they also noted that
the computationally most efficient configurations might yield inferior model
results due to the suboptimal performance of the compartmental model
component, measured by the total number of infected individuals and the length
of the outbreak.

Rather than using infection numbers, Nghi et al. [22] implemented a hybrid
model where the state switch is triggered when local agent density crosses a
threshold. The authors argued that dense clusters of agents can be viewed as a
homogeneous population, and therefore a compartmental model is sufficient.

Another approach, found in Bradhurst et al. [23] and Banos et al. [24],
involves using agents to couple compartmental models in a hybrid model. In
Bradhurst et al. [23], agents represent herds of livestock, and within each
agent, ordinary differential equations (ODEs) determine the infection metrics
over time. Agent-agent interactions govern the spread of diseases between
herds through different pathways. In Banos et al. [24], cities are represented
by nodes in a network. The infectious disease dynamics within the cities are
modeled with a compartmental SIR-model. A meta-population approach was
compared to a hybrid model, where the nodes are connected through agent-based
air travel: Agents can fly from one city to another, and infection events are
possible within the aircraft. The spread of a disease can be slower in the
hybrid approach if only one or a few infected agents are localized on one
node. This is because diffusion happens evenly in the meta-population model
with constant mobility rates for all compartments, while mobility is discrete
and stochastic in the hybrid approach. A similar coupling mechanism was
employed in Marilleau et al. [25], focusing on modeling the spread of a vole
population in a French region. This spatial model divides the region into grid
cells, allowing vole agents to travel between those cells. Within each cell,
the vole population size is determined by equation-based models. Once the
population of a cell reaches a critical value, the cell generates vole agents
that leave the cell and migrate to another cell with more suitable conditions.

A problem that arises when coupling an ABM with a compartmental model is the
differing population shape in each model type. In an ABM, the population is
represented by distinct entities, while an ODE model represents the population
as a continuous matter [26]. Sewall et al. [27] highlight that this is
particularly a problem when agents travel out of a compartmentally modeled
region due to information loss when converting agents to a continuous
population. Addressing this disparity is crucial in developing a successful
hybrid model, as it involves accurately transitioning the population between
discrete entities and continuous mass representations.

In conclusion, while various hybrid models have been developed to leverage the
strengths of both ABMs and compartmental models, they often face challenges
such as determining appropriate switching mechanisms and managing the
transition between discrete and continuous representations of the population.
In the following section, we will describe our suggested hybrid model and the
approaches we use to address these challenges effectively.

##  3 The Hybrid Model

This section introduces our proposed hybrid modeling approach, which combines
the strengths of a microscopic ABM with a macroscopic compartmental ODE model.
This integration aims to achieve a balance between computational efficiency
and detailed, realistic representation of infection dynamics. In the hybrid
model, infectious disease dynamics are modeled by two components in parallel:
In the first component, infections spread through agent-agent interactions in
a continuous coordinate space. The second component employs ODEs to model
these dynamics. This approach leverages the fine-grained interactions and
spatial detail of ABMs while utilizing the computational simplicity of
compartmental models to scale up to larger populations effectively.

In the following sections, we provide a detailed description of the agents,
the environment, and the coupling mechanism to illustrate how our hybrid model
addresses the challenges of combining discrete and continuous population
representations. Moreover, we demonstrate the model’s potential to reduce
computational costs while maintaining accurate and insightful predictions of
infectious disease dynamics.

###  3.1 Sub-Model M1: Agent-Based Model (ABM)

The ABM simulates agents within a 2D rectangular region in a xy-plane, defined
by the coordinates $0\leq x\leq 9$ and $0\leq y\leq 9$. Each agent occupies a
$(x,y)$-coordinate within this space and is in one of three states:
susceptible, infectious, or recovered. The probability $p_{i}$ of a
susceptible agent $i$ to become infectious within a time step is
$p_{i}=1-(1-\rho)^{k}$, where $\rho$ is the base probability and $k$ is the
number of infectious agents within a distance of $\theta$ to the susceptible
agent. This transmission model assumes a fixed transmission probability upon
contact and increases with repeated exposure, akin to a contact-duration model
[see 28, 29]. While dose-response models are more realistic [30, 31], we chose
to implement this simplified model due to its similarity to the transmission
mechanism in ODE models: In the ODE model, the transmission rate is a function
of the contact rate and the infection probability. In this version of an ABM,
the contact rate can be seen as a function of the radius size and the step
size. Infectious agents recover at a rate of $1/D$ per time step, where $D$ is
the average time an agent remains in the infectious state, becoming immune to
subsequent infections through agent-agent interactions.

####  3.1.1 Agent movement

Agent movement is implemented in two variants. In variant M1-A, agents move
randomly, simulating a homogeneous population. During a time step, an agent
can move in any direction with a maximum possible step size of $\Lambda$ under
periodic boundary conditions. Although less realistic, this resembles the
population in a compartmental model more closely.

In variant M1-B, however, we implemented a landscape that influences agent
movement [see 32, 33]. This landscape simulates non-random movement and is
modeled by a bivariate density function

$f(x,y)=-e^{\frac{\left(-(x-X)^{2}+(y-Y)^{2}\right)}{\eta}}$ (1)

In this setup, agents either move towards (attraction) or away from
(repulsion) lower elevation points. Here, the valley has its center at the
coordinates $(X,Y)$ and the width of the valley is determined by the parameter
$\eta$. Specific rules govern the transition between attraction and repulsion
states, allowing for regular movement patterns across the simulation space.
This leads to spatial heterogeneity of the agent population.

Figure 1: Visualization of the bivariate density function with X = Y = 4.5.

At each time step, an agent evaluates the gradient of its current position on
the xy-plane based on the underlying density function with

$\chi_{i,t}=\frac{f(x_{i,t}+\delta,y_{i,t})-f(x_{i,t}-\delta,y_{i,t})}{2\delta}-\lambda_{x,i,t}$
(2)
$\psi_{i,t}=\frac{f(x_{i,t},y_{i,t}+\delta)-f(x_{i,t},y_{i,t}-\delta)}{2\delta}-\lambda_{y,i,t}$
(3)

where $\delta$ is a fixed base step size and $\lambda$ is a random value
between $-\Lambda$ and $\Lambda$, introducing stochasticity in agent-movement.
In the attraction state, the agent’s new coordinates are

$x_{{}_{i,t+1}},y_{{}_{i,t+1}}=x_{{}_{i,t}}+\chi_{{}_{i,t}},y_{{}_{i,t}}+\psi_{{}_{i,t}}$
(4)

and in the repulsion state, they are

$x_{{}_{i,t+1}},y_{{}_{i,t+1}}=x_{{}_{i,t}}-\chi_{{}_{i,t}},y_{{}_{i,t}}-\psi_{{}_{i,t}}$
(5)

under periodic boundary conditions. Agents switch to the repulsion state once
they approached the center of the valley within a distance of 0.3 and switch
back to attraction at a rate of $1/(3\eta)$. This dynamic switching mechanism
allows for movement across the simulation space, resembling commute-like
behavior, instead of the entire population remaining around the center for the
entire simulation time. A snapshot of a realization of the ABM at $t=9$ can be
found in Figure 2, showing a dense cluster of agents at the center, surrounded
by less densely populated space.

Figure 2: Snapshot ($t=9$) of a realization ABM variant B. Each point
represents the position of a susceptible (yellow), infectious (red), or
recovered (blue) agent.

####  3.1.2 Initialization

For initialization, agents are placed randomly across the simulation space in
variant A. In variant B, initial positions and behavior states are determined
using the Monte Carlo method, based on the bivariate density function for the
coordinates, and $1/(1-1/3\eta)$ for the behavior state probabilities, and the
outbreak starts after a burn-in period.

###  3.2 Sub-Model M2: Compartmental Model

The compartmental model follows the classic SIR (Susceptible-Infectious-
Recovered) framework, describing the dynamics of disease spread in a
population with ordinary differential equations (ODEs) [34, 5]. The model
divides the population into three compartments: susceptible individuals ($S$),
infectious individuals ($I$), and recovered individuals ($R$). The dynamics of
the SIR model are governed by a system of ordinary differential equations
(ODEs). The rate of change for each compartment is given by:

$\displaystyle\frac{{dS}}{{dt}}$ $\displaystyle=-\frac{{\beta\cdot S\cdot
I}}{{N}}$ (6) $\displaystyle\frac{{dI}}{{dt}}$
$\displaystyle=\frac{{\beta\cdot S\cdot I}}{{N}}-\gamma\cdot I$
$\displaystyle\frac{{dR}}{{dt}}$ $\displaystyle=\gamma\cdot I$

where $\beta$ represents the transmission rate of the disease, $\gamma$ is the
recovery rate, and $N$ is the total population size. This model captures the
rates at which the population transitions between compartments. Given initial
conditions, step size, and time frame, the ODEs can be solved numerically,
providing compartment sizes over time - and thus simulating the dynamics of
disease spread in a population.

###  3.3 Coupling

In the hybrid model, the population-space is spatially divided into an ABM
(M1) and a compartmental region (M2), while allowing for population flow
between regions. The interface between M1 and M2 is a vertical line that can
be moved along the x-axis to adjust the proportions (or weight) of each model
component. This flexible interface allows for varying the influence of each
model within the hybrid framework.

The temporal resolution and the order of progresses can significantly impact
the results of both ABMs and compartmental models [18]. This aspect is
particularly crucial in our hybrid model due to the interaction between a
discrete model and a continuous model [26]. To maintain coherence, time is
implemented as a discrete variable in both models, and interactions within the
ABM and between models occur synchronously. Figure 3 visualizes the coupling
mechanism in the hybrid model, depicting the sequence of processes within a
single time step.

The subsequent sections will detail these processes.

Figure 3: Order of processes in hybrid model.

####  3.3.1 Population flow between model regions

Agents can transition from the M1 (ABM) area to the M2 (ODE) area through the
designated interface. Agents entering M2 are aggregated into $s_{in}$,
$i_{in}$, and $r_{in}$, representing the number of susceptible, infectious,
and recovered agents respectively. These agents are subsequently removed from
the M1 population. Movement from M2 into M1 in a given time step is governed
by the equation:

$n_{out}(t)=s_{in}(t-1)+i_{in}(t-1)+r_{in}(t-1)$ (7)

where $n_{out}$ is the number of agents leaving M2. This ensures that the flow
between the two regions remains symmetric. The lag of one time step prevents
scenarios where more agents leave M2 than are present, thus avoiding
population inconsistencies. Since M2 does not simulate spatial movement the
$y$-coordinate at which an agent leaves M2 is unknown. For variant A, this
coordinate is sampled from a uniform distribution. For variant B, the marginal
of the density function (see Equation 1) is used to derive sampling weights.

####  3.3.2 Infection dynamics within the ODE area

Due to population flow between models, the population size of M2 varies based
on the incoming and outgoing agents. Additionally, the stochastic nature of
the hybrid model requires a corrective procedure in M2 to ensure accurate
population tracking and infection dynamics. This involves several steps:

First, the compartmental model described in Section 3.2 is solved for one time
step with the current population size as shown in the following equations:

$\displaystyle\Sigma_{1}(t+1)$ $\displaystyle=S(t)-\frac{\beta\cdot S(t)\cdot
I(t)}{N(t)}dt$ (8) $\displaystyle\Phi_{1}(t+1)$
$\displaystyle=I(t)+\frac{\beta\cdot S(t)\cdot I(t)}{N(t)}-\gamma\cdot I(t)dt$
$\displaystyle\Psi_{1}(t+1)$ $\displaystyle=R(t)+\gamma\cdot I(t)dt$

Next, the state of the leaving population is determined. The state of each
leaving agent is drawn in a multinomial trial using the results from the
previous step as the class probabilities:

$s_{out}(t),i_{out}(t),r_{out}(t)\sim\textrm{Multinomial}\left(n_{out}(t),\left[\frac{\Sigma_{1}(t+1)}{N(t)},\frac{\Phi_{1}(t+1)}{N(t)},\frac{\Psi_{1}(t+1)}{N(t)}\right]\right)$
(9)

The leaving population is then subtracted from each compartment, ensuring no
negative sizes by constraining each value to zero or higher (for conciseness,
we will demonstrate the following procedures for the S-compartment, but they
also applied to the I- and R-compartment):

$\displaystyle\Sigma_{2}(t+1)$
$\displaystyle=\textrm{max}(0,\Sigma_{1}(t+1)-s_{out}(t))$ (10)

Given this reduction, the sum of the compartments may exceed the actual
remaining population, calculated as $N(t)-n_{out}(t)$. Therefore, a correction
term $\kappa$ is computed to balance this discrepancy:

$\displaystyle P(t+1)$
$\displaystyle=\Sigma_{2}(t+1)+\Phi_{2}(t+1)+\Psi_{2}(t+1)$ (11)
$\displaystyle\kappa(t)$ $\displaystyle=P(t+1)-(N(t)-n_{out}(t))$

This correction term is proportionally subtracted from each compartment to
ensure that their sum matches the actual population:

$\displaystyle\Sigma_{3}(t+1)$
$\displaystyle=\Sigma_{2}(t+1)-\kappa(t)\cdot\frac{\Sigma_{2}(t+1)}{P(t+1)}$
(12)

As a final step, the entering population is added to the compartments,
resulting in the final compartment sizes for the following time step:

$\displaystyle S(t+1)$ $\displaystyle=\Sigma_{3}(t+1)+s_{in}(t)$ (13)

####  3.3.3 Parameter selection

To effectively compare and couple M1 and M2, both models need to produce
similar epidemic outcomes given identical starting conditions. Following the
procedure and disease characteristics in Breitwieser et al. [29], we first
determine the parameters $\beta$ and $\gamma$ for M2, corresponding to a
measles epidemic with $R_{0}=12.9$ and an 8-day recovery period, resulting in
$\beta=1.61$ and $\gamma=1/8$. Next, we generate a time series of infectious
individuals by numerically solving the compartmental model using initial
values of $S=995,I=5,R=0$ and a step size of 0.33 days. This is achieved with
the RK45 method, as implemented in SciPy’s integrate.solve_ivp function [35].

We then fit the parameters $\Lambda$, $\rho$, and $\theta$ for M1-A to match
the time series data, with the same initial conditions and a recovery rate of
$1/24$ (to correspond to a step size of 0.33 days). This optimization is done
using the Nelder-Mead minimizer (as implemented in SciPy’s optimize.minimize
[35]) with least squares as the cost function, yielding $\Lambda=0.215$,
$\rho=0.025$, and $\theta=1.162$.

We repeat the procedure for M1-B, and additionally fit the parameter $\delta$
(the base step size), fixing $\eta$ at 3, and positioning the attraction point
at $X=Y=4.5$. This results in $\Lambda=0.268$, $\rho=0.016$, $\theta=1.01$,
and $\delta=0.01$.

Figure 4 shows that M1-A and M2 produce closely aligned SIR curves, while M1-B
exhibits a more rapid rise and a lower peak of infections. This is consistent
with the finding in Susandi et al. [36] that ABMs with heterogeneous
populations cannot produce identical results to compartmental models - which
explains this discrepancy.

Figure 4: SIR curves for model 2 (solid lines) and the average results for 100
realizations of the fitted M1 variants A (dashed lines) and B (dotted lines).

In the following section, we describe the numerical experiments conducted to
evaluate the performance and robustness of our hybrid modeling approach. The
primary objective is to understand how varying the weight of the compartmental
(ODE-based) and agent-based component affects model predictions, infection
dynamics, and computational efficiency. By systematically adjusting these
weights, we aim to uncover the nuanced behavior of the hybrid model under
different configurations. This analysis will provide valuable insights into
the potential applications and limitations of the suggested hybrid model in
infectious disease modeling.

##  4 Experimental Setup and Methodology

In this section, we describe the experimental setup and methodology used to
evaluate the fundamental dynamics of our hybrid model. By varying the weight
between its two main components - the agent-based model (M1) and the
compartmental model (M2) - we aim to understand how altering the balance
between these components and the location of the interface affects the model
outcomes in terms of the infectious wave. This setup allows us to test the
robustness of our model under various conditions. The results of these
experiments are presented in the subsequent section.

###  4.1 Model Weights

We adjust the weights of M1 ($w_{1}$) and M2 ($w_{2}$) by shifting the
interface in the hybrid model along the x-axis. For example, given $0\leq
x\leq 9$, placing the interface at x=4.5 results in $w_{1}=w_{2}=0.5$, while a
placement at x=6.75 results in $w_{1}=0.25$ and $w_{2}=0.75$. The interface
shift allows us to systematically study the impact of different model
configurations on infection dynamics.

###  4.2 Model Configurations

We compare two configurations of the hybrid model: Hybrid Model A (M1-A with
M2) and Hybrid Model B (M1-B with M2). This comparison allows us to examine
our hybrid model under two scenarios. In the first scenario (Hybrid Model A),
both model components simulate a homogeneous population, leading to very
similar results individually. In the second scenario (Hybrid Model B), the
agent-based component simulates a heterogeneous population, yielding different
epidemic dynamics compared to the M2-component. In this scenario, the
attraction point is placed at the center of the simulation space
$(x=4.5,y=4.5)$, making the distance of the attraction point to the interface
dependent on $w$. For $w_{2}>0.5$, the attraction point is located within the
borders of M2. Additionally, we examine a third scenario in the appendix,
where the location of the attraction point is always placed at the center of
M1, rather than at the center of the full simulation space. These scenarios
enable us to study the implications of spatial differences on spatial coupling
effects.

###  4.3 Evaluation Metrics

For the experiments, we evaluate the following metrics for the resulting time
series of the infectious population: (a) the root mean squared error (RMSE)
compared to the baseline from M2; (b) the peak height; (c) the time step at
which the peak occurs; (d) the Fisher-Pearson coefficient of skewness,
indicating the buildup speed of the infectious wave; (e) the full-width-at-
half-maximum (FWHM), indicating the sharpness of the peak. The RMSE quantifies
the overall deviation from the reference model, while metrics b$\ldots$e
provide detailed insights into the dynamics and shape of the epidemic.
Finally, we measure (f) the average CPU time for each model configuration. We
increase the weight of M2 from 0 to 1 in 0.02 steps, averaging the metrics
after 100 independent realizations per configuration.

##  5 Results

This section presents the results of our experiments, evaluating the
performance and robustness of the hybrid model under different configurations.
Specifically, we examine how varying the weight of the compartmental model
(M2) and the agent-based model (M1) affects various metrics (see previous
section) related to infection dynamics and computational efficiency in both
Hybrid Model A (M1-A with M2) and Hybrid Model B (M1-B with M2).

Results are shown in Figures 5(a)-5(f) for Hybrid Model A and in Figures
6(a)-6(f) for Hybrid Model B. The x-axis represents the weight of M2
($w_{2}$). When $w_{2}=0$, the entire population is modeled by M1, and when
$w_{2}=1$, the entire population is modeled by M2.

(a)

(b)

(c)

(d)

(e)

(f)

Figure 5: Hybrid model A: Average results for 100 independent realizations per
level of model 2 weight ($w_{2}$). Bars around points represent the standard
deviation. (a) RMSE for the time series of infectious between the reference
model and the hybrid model. (b) Peak time of the time series of infectious.
(c) Peak height of the time series of infectious. (d) Fisher-Pearson
coefficient of skewness of the time series of infectious. (e) FWHM of the time
series of infectious. (f) CPU time in seconds.

###  5.1 Hybrid model A

####  5.1.1 Root Mean Squared Error (RMSE)

The RMSE for different weights behaves as expected, showing a steady drop from
25.09 to 0 as the weight of the M2-component increases. This is because the
full M2-model serves as the reference model to compute the RMSE. The standard
deviation is larger for lower values of $w_{2}$, which is due to the
stochastic nature of M1.

####  5.1.2 Peak height

The peak height of the infection curve does not change significantly with
changing $w_{2}$. It remains around 71.57% at $w_{2}=0$ and 72.46% at
$w_{2}=1$. This consistency is expected since the peak height is similar in
both individual models M1-A and M2, as shown in 4. Similar to the RMSE, the
standard deviation shrinks with increasing $w_{2}$.

####  5.1.3 Peak time

A notable coupling effect is observed in the peak time of the infectious
curve. While the peak time is slightly earlier in the reference model ($16t$)
compared to M1-A ($17.36t$), it does not gradually decrease as $w_{2}$
increases. Instead, there seems to be a quadratic relationship between the
peak time and $w_{2}$. This may be due the discrete implementation of time in
our models and the relatively small difference in peak time of 1.36 time steps
between the model components, causing the resulting peak time to align more
closely with the model component modeling the majority of the population.

####  5.1.4 Skewness

The skewness coefficient is 1.16 at $w_{2}=0$ and 1.2 at $w_{2}=1$, indicating
slightly more weight on the right tail of the infectious wave in M2 compared
to M1-A. The quadratic relationship in peak time is reflected in the skewness,
which increases more rapidly as $w_{2}$ exceeds the 0.6-mark, likely due to
the peak time influencing the skewness of the infectious wave.

####  5.1.5 Full-Width-at-Half-Maximum (FWHM)

The FWHM is 25.58 at $w_{2}=0$ and 25 at $w_{2}=1$, indicating that the
infectious wave of M2 is sharper compared to M1. However, this difference is
minor and steadily decreases as $w_{2}$ increases, suggesting minimal coupling
effects.

####  5.1.6 CPU time

The CPU time decreases from 3.18 to 0.09 seconds as $w_{2}$ increases,
following a convex pattern. This diminishing reduction in CPU time can be
attributed to the stochastic processes computed for each agent, where
calculations often involve interactions with all other agents. Thus, the
reduction in computational load is more substantial when increasing $w_{2}$
from lower values (e.g., 0 to 0.02) compared to higher values (e.g., 0.9 to
0.92), reflecting the non-linear scaling of interaction complexity.

###  5.2 Hybrid model B

(a)

(b)

(c)

(d)

(e)

(f)

Figure 6: Hybrid model B: Average results for 100 independent realizations per
level of model 2 weight ($w_{2}$). Bars around points represent the standard
deviation. (a) RMSE for the time series of infectious between the reference
model and the hybrid model. (b) Peak time of the time series of infectious.
(c) Peak height of the time series of infectious. (d) Fisher-Pearson
coefficient of skewness of the time series of infectious. (e) FWHM of the time
series of infectious. (f) CPU time in seconds.

####  5.2.1 Root Mean Squared Error (RMSE)

The RMSE at $w_{2}=0$ is 44.3, nearly twice as large as the RMSE for M1-A.
Similar to Hybrid Model A, the RMSE for Hybrid Model B drops towards zero as
$w_{2}$ increases. However, there is a notable minimum around the 0.5-mark of
$w_{2}$. The RMSE decreases for $w_{2}$ values below 0.5, then increases for
$w_{2}$ values above 0.5 before dropping again after the 0.6-mark. This
indicates that M2 has a stronger effect on model results around the 0.5 mark
than at the 0.6 mark. This can be attributed to the attraction point in the
agent-based component, resulting in a spatially heterogeneous population.
Agents form a cluster where the infection spreads more rapidly. When $w_{2}$
is set to 0.5, the attraction point lies on the interface between M1-B and M2,
attracting many agents into M2 and therefore increasing the interaction
between the models.

(a)

(b)

Figure 7: Changes in population metrics for Hybrid Model A and B for different
weights of M2. Results from 100 independent realizations. (a) Average
percentage of population entering M2 per $t$. (b) Average percentage of
overall population in M1 per $t$.

Indeed, Figure 7(a) shows that the average population flow from M1 to M2
during a time step depends on $w_{2}$. In Hybrid Model A, it is around 1.5% of
the total population for weights below 0.8, before dropping for higher
weights. In Hybrid Model B, it increases from 0.02 to 0.53, peaking at 4.5%,
before decreasing rapidly for higher weights. Additionally, Figure 7(b) shows
that the share of the overall population present in M1 drops faster around
$w=0.5$ in Hybrid Model B, while decreasing steadily in Hybrid Model A.

####  5.2.2 Peak height

The peak height of M1-B is 64.56%, considerably lower than in M2. A notable
observation is the increased standard error around the 0.5-mark of $w_{2}$,
reflecting the increased between-model interaction discussed earlier. The peak
height follows a convex pattern: For $w_{2}<0.4$, the average peak height
increases towards the higher value of M2, but decreases after the 0.4-mark
until $w_{2}=0.6$. This pattern suggests that the attraction point’s proximity
to the interface significantly influences infection dynamics.

####  5.2.3 Peak time

The peak time of a full M1-B is $16.66t$ (at $w_{2}=0$), slightly closer to
the reference model than M1-A. The attraction point appears to influence the
peak time of the hybrid model. In the full M1-B model, the peak is delayed by
0.66 compared to M2. In Hybrid Model B, the peak time is closer to the full M2
result around $w_{2}=0.4$ than around $w_{2}=0.6$, again likely due to the
increased between-model interaction around the 0.5-mark. The local minimum
around the 0.4 mark, rather than the 0.5 mark, suggests that the discrete
nature of time amplifies the interaction effect between the model components.

####  5.2.4 Skewness

The skewness of the resulting infectious data is considerably lower in M1-B
than in M2, indicating a more symmetrical infection wave in M1-B. Although
skewness increases with an increasing $w_{2}$, a local maximum at $w_{2}=0.38$
is followed by a minor drop until $w_{2}=0.64$. This reflects the shift
observed in the previous metrics for Hybrid Model B.

####  5.2.5 Full-Width-at-Half-Maximum (FWHM)

At $w_{2}=0$, the FWHM is considerably higher (29.8) than at $w_{2}=1$ (25.0),
indicating that the infectious peak of M1-B is more truncated than in M1-A or
M2. Similar to the previous metric, a regression can be observed as $w_{2}$
increases with a temporary shift after the 0.38-mark. Both observations are
consistent with the increased between-model exchange around this value.

####  5.2.6 CPU time

For lower $w_{2}$-values, the average CPU time of Hybrid Model B is higher
than in the case of Hybrid Model A, at 3.37 seconds when $w_{2}=0$. This is
due to the increased complexity introduced by the attraction points.
Additionally, there is an increase from $w_{2}=0$ to $w_{2}=0.06$, possibly
due to the further complexity introduced by population flow from M2 to M1,
influenced by the landscape in M1. For Hybrid Model B, the drop in CPU time
follows the drop in population size of the M1-B component, as visualized in
Figure 7(b). The computational demands of M1 stem from simulating each
individual agent, making the population size a key factor in determining CPU
time.

##  6 Discussion

The results for Hybrid Model A demonstrate that the outcomes are directly
proportional to the weighting of each model component. Specifically, as the
weighting shifts, the model’s outcomes increasingly resemble the results of
the predominant component. This proportional relationship indicates that the
hybrid model effectively balances the computational efficiency of the
compartmental model (M2) with the detailed interactions of the agent-based
model (M1-A). By adjusting the weight between these components, the hybrid
model can be fine-tuned to optimize either efficiency or detail, providing a
versatile and scalable approach to infectious disease modeling.

In contrast, the results for Hybrid Model B reveal a more complex interaction
involving the implemented attraction point and the between-model interface.
This interaction triggers switches in the model outcomes, favoring one
component over the other under different conditions. The direction-switching
effect observed in the peak time and peak height for Hybrid Model B can be
explained by the presence of the attraction point. Without the attraction
point, these metrics would likely follow a trajectory similar to Hybrid Model
A. However, the increased between-model interaction around $w_{2}=0.5$ leads
to a local change, emphasizing the sensitivity of the hybrid model to the
spatial configuration of the attraction point.

###  6.1 Impact on Infection Spread Dynamics

The study reveals that spatial coupling of an ABM with an ODE-based model can
effectively capture the dynamics of infection spread in the overall
population. In Hybrid Model A, where both components simulate a homogeneous
population, the infection dynamics transition smoothly as the weight shifts
between M1 and M2. This smooth transition reflects a successful integration of
the detailed agent interactions in M1 with the aggregate population-level
processes in M2, maintaining consistency in the model’s outcomes.

In Hybrid Model B, where the agent-based component includes spatial
heterogeneity, the infection dynamics reveal a deeper complexity. The presence
of an attraction point introduces a non-uniform distribution of agents,
leading to varied infection dynamics depending on the location of the
interface. This demonstrates that spatial factors and agent clustering can
significantly influence the spread of infection, underscoring the model’s
ability to adapt to complex scenarios by accurately reflecting the impact of
spatial configurations and agent movements.

These findings highlight that the location of the coupling mechanism in a
hybrid model can amplify the differences between model components. Differences
between models and sub-populations, as well as the level of population flow,
should be carefully considered in real-world applications. The increased
between-model interaction observed in Hybrid Model B underscores the need for
a nuanced approach to spatial coupling, especially when dealing with
heterogeneous populations, thereby enhancing the model’s applicability and
precision in various scenarios.

###  6.2 Computational Efficiency

Our experiments also provide insights into the computational efficiency of the
hybrid model compared to a pure ABM. For Hybrid Model A, the hybrid approach
achieves a substantial reduction in CPU time while maintaining accuracy in the
infection dynamics. The proportional relationship between the weight of the
compartmental model and the CPU time underscores the potential for significant
computational savings. This is particularly beneficial in large-scale
simulations where computational resources are a critical constraint.

In Hybrid Model B, while the overall trend shows reduced CPU time with
increasing weight of the compartmental model, the relationship is more
complex. The initial increase in CPU time at low values of $w_{2}$ reflects
the added complexity due to the attraction point and the increased interaction
between the models. Despite this complexity, the hybrid model still
demonstrates a net reduction in computational cost compared to a pure ABM,
especially at higher values of $w_{2}$. This indicates that even with added
spatial heterogeneity, the hybrid model can offer computational advantages,
though the exact savings depend on the specific configuration and interaction
dynamics.

###  6.3 Limitations and Future Directions

One limitation of the hybrid model, compared to a pure ABM, is the loss of
information during each exchange between the two model components. Whenever
population flow from M2 to M1 is modeled, new agents are created and their
individual statuses are randomly distributed based on the population’s current
state. Consequently, an agent that recently entered M2 as recovered may
effectively leave as susceptible or infectious. Exploring alternative
strategies for information exchange between the models and refining the
coupling mechanism could potentially address this limitation and improve the
accuracy of the hybrid model in capturing individual-level dynamics.

Overall, these findings provide valuable insights into the strengths and
limitations of hybrid modeling approaches. They emphasize the importance of
carefully considering the spatial configuration and interaction dynamics
between model components to optimize both accuracy and computational
efficiency. Future research could focus on developing more sophisticated
coupling mechanisms and exploring the application of hybrid models to a
broader range of infectious diseases and real-world scenarios.

##  7 Conclusion

In this paper, we addressed a significant research gap by introducing a novel
spatial hybrid modeling approach that integrates a microscopic agent-based
model (ABM) with a macroscopic compartmental ordinary differential equation
(ODE) model. This topic has been scarcely explored in the existing literature
on hybrid models. Our primary aim was to explore the potential of this hybrid
model in reducing computational costs while maintaining accurate and detailed
infection dynamics. We demonstrated how such coupling can be effectively
achieved, highlighting the potential of spatial coupling in real-world
epidemic modeling, advancing the understanding of hybrid systems, and setting
a foundation for future research in this domain.

To evaluate the effectiveness of our approach, we utilized several performance
metrics to detect coupling effects on the resulting infectious disease
dynamics. Our experiment with two toy scenarios revealed that the outcomes of
the hybrid model depend on the similarity between the integrated models. In
both scenarios, the ODE component lacked the spatial resolution of the ABM-
component. However, in the first scenario, where both model components
described a homogeneous population, the hybrid model produced the same
infectious disease dynamics as the reference model. In the second scenario, we
introduced spatial heterogeneity within the ABM by enabling non-random agent
movement and localized clustering around an attraction point. The
corresponding hybrid model showed that disease dynamics were influenced by the
location of the between-model interface relative to the attraction point.

To address this, we suggest a meta-population approach for the ODE-component,
which could provide spatial resolution and heterogeneity without significantly
increasing computational demands. We also noted that the extreme variations in
agent density in our toy examples may not be as pronounced in real-world
applications, potentially mitigating concerns about model discrepancies.

To focus our analysis on the coupling effects, we intentionally simplified the
model components, maintaining the standard Susceptible, Infectious, and
Recovered (S, I, R) categories in both model components. This simplification
allowed us to evaluate the impact of spatial interactions without the added
complexity of more elaborate disease states or interactions. However,
increasing the number of categories could further highlight the computational
advantages of the hybrid approach compared to a purely microscopic approach.
Future research will explore more realistic scenarios to further validate and
enhance our proposed hybrid model.

In conclusion, our spatial hybrid modeling approach presents a promising
avenue for studying infectious diseases and complex systems. It offers
opportunities to refine model accuracy and optimize computational efficiency,
paving the way for more sophisticated and practical applications in future
research endeavors.

##  8 Implementation

The model is implemented in Python v3.9 [37] and the code is available on
GitHub (http://github.com/iebos/hybrid_model1). The model includes three
primary class objects: The Person class which creates an agent, the Model_ODE
class which solves the ODEs and the Model_hybrid_realistic_continuous which
creates the ABM and couples it to the ODE.

##  9 Acknowledgements

The work on the paper was funded by the German Ministry of research and
education (BMBF) (project MODUS-Covid, grant number 031L0302C), and under
Germany’s Excellence Strategy (MATH+ -The Berlin Mathematics Research Center,
grant number EXC-2046/1, project no. 390685689, subproject EF45-4). The
authors extend their gratitude to Kai Nagel and Kristina Maier for their
valuable insights and comments on an early draft of this manuscript.

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##  Appendix A Appendix 1

(a)

(b)

(c)

(d)

(e)

(f)

Figure 8: Hybrid model C: Average results for 100 independent realizations per
level of model 2 weight ($w_{2}$). Bars around points represent the standard
deviation. (a) RMSE for the time series of infectious between the reference
model and the hybrid model. (b) Peak time of the time series of infectious.
(c) Peak height of the time series of infectious. (d) Fisher-Pearson
coefficient of skewness of the time series of infectious. (e) FWHM of the time
series of infectious. (f) CPU time in seconds.

Figures 8(a) \- 8(f) show results for Hybrid Model C, where the attraction
point is placed at the center of M1. For example, at the pure ABM at
$w_{2}=0$, it is at the same location as in Hybrid Model B at $(x=4.5,y=4.5)$.
However, at $w_{2}=0.5$, it is placed at $(x=2.25,y=4.5)$. Most metrics behave
differently than in results for Hybrid Model A and B. Rather than a steady or
irregular trend towards the value of the reference model, most metrics rapidly
increase or decrease towards the direction of the reference model and exceed
or precede the reference value around $w_{2}=0.8$, before regressing towards
it again. Most notably, the RMSE increases from $w_{2}=0.2$ to $w_{2}=0.74$
before sharply dropping. Given that the location of the attraction point can
affect the population density within M1 and also the overall population
distribution between the model components, it is likely that the fitted
parameters of the pure ABM are not appropriate under these different
conditions. Further, while the attraction point remains within M1, the
distance to the interface grows smaller with increasing $w_{2}$. Figure 9(a)
shows that as a result, between-model interaction increases continuously
rather than peaking around the middle as is the case in Hybrid Model B.
Additionally, Figure 9(b) shows that the majority of the overall population is
located within M1 for $w_{2}<0.85$. Therefore, for Hybrid Model C, the weight
of M2 is not a sufficient indicator of the influence of M2 on overall
population dynamics.

(a)

(b)

Figure 9: Changes in population metrics for Hybrid Model A, B, and C for
different weights of M2. Results from 100 independent realizations. (a)
Average percentage of population entering M2 per $t$. (b) Average percentage
of overall population in M1 per $t$.


MIDLMedical Imaging with Deep Learning 2019 MIDL 2019 – Extended Abstract
Track Rune Wetteland1<EMAIL_ADDRESS>
Kjersti Engan1<EMAIL_ADDRESS>
Trygve Eftestøl1<EMAIL_ADDRESS>
Vebjørn Kvikstad2<EMAIL_ADDRESS>
Emilius A.M. Janssen2,3<EMAIL_ADDRESS>
1Department of Electrical Engineering and Computer Science, University of
Stavanger, Norway
2Department of Pathology, Stavanger University Hospital, Norway
3Department of Mathematics and Natural Sciences, University of Stavanger,
Norway

# Multiscale Deep Neural Networks for Multiclass Tissue Classification of
Histological Whole-Slide Images

###### Abstract

Correct treatment of urothelial carcinoma patients is dependent on accurate
grading and staging of the cancer tumour. This is determined manually by a
pathologist by examining the histological whole-slide images (WSI). The large
size of these images makes this a time-consuming and challenging task. The WSI
contain a variety of tissue types, and a method for defining diagnostic
relevant regions would have several advantages for visualization as well as
further input to automated diagnosis systems. We propose an automatic
multiscale method for classification of tiles from WSI of urothelial carcinoma
patients into six classes. Three architectures based on convolutional neural
network (CNN) were tested: MONO-CNN (400x), DI-CNN (100x/400x) and TRI-CNN
(25x/100x/400x). The preliminary results show that the two multiscale models
performed significantly better than the mono-scale model, achieving an
F1-score of 0.986, substantiating that utilising multiple scales in the model
aids the classification accuracy.

##  1 Introduction

Bladder cancer is the 10th most common cancer type worldwide [Bray et al.
(2018)]. More than 90% of bladder cancer cases are urothelial carcinomas which
has a particular high recurrence (50-70%) and progression rate (10-30%),
making correct treatment and follow-up vital for survivability. Treatment is
dependent on the cancer grade and stage, determined manually by an expert
pathologist examining the histological whole-slide images (WSI). This is a
time-consuming and challenging task, and studies have shown that it may have a
low reproducibility in some cases, such as grading of urothelial carcinoma
[Mangrud (2014)].

Examination of the WSI is challenging because of the large size of the image,
which contains several different tissue types, where only some are useful for
diagnostic information. An automatic tool for identification of such regions
would be beneficial for both guiding a pathologist to the useful areas of the
large WSI during examination, and for ROI extraction of useful tissue for a
computer aided diagnostic solution. In this paper we present an automatic
method for classification of tiles from WSI of urothelial carcinoma patients
into the classes: urothelium, stroma, muscle, damaged tissue, blood and
background. The tiles are extracted at different magnification levels, to
combine and utilise information at different scales in a similar fashion to
that of a pathologist.

Multiscale approaches to tile-based classification have previously been done
on other cancer types. In the work of Li et al. (2017) a multiscale U-Net was
proposed for segmentation of histological images from radical prostatectomies
to classify tiles into four classes. Tiles of size 100x100, 200x200 and
400x400 pixels were all extracted from histological images at 200x
magnification. Features from the different tiles were then concatenated and
used as input to the multiscale U-net. The model achieved a mean Jaccard index
of 65.8% over the four classes. In Sirinukunwattana et al. (2018) a comparison
of five single-scale and five multiscale architectures were tested on two
datasets. Their best model (G) was a multiscale model which achieved an
average F1-score of 0.782$\pm$0.07 across four classes of prostate cancer and
0.538$\pm$0.08 across four classes of breast cancer. Their result supports the
claim that incorporating a larger visual context improves the results. In
Wetteland et al. (2019) we presented a method based on deep convolutional
neural networks (CNN) for classifying tiles of urothelial carcinoma WSI into
the six classes mentioned above. This was a single-scale approach where all
tiles were extracted from the full resolution image of 400x magnification. The
method got an F1-score of 0.934$\pm$0.061.

##  2 Data Material

The data material consists of Hematoxylin Eosin Saffron (HES) stained WSI from
patients diagnosed with primary papillary urothelial carcinoma, collected at
the University Hospital of Stavanger, Norway. An expert pathologist has
carefully annotated 239 selected regions from 50 WSI from 32 unique patients,
where each region includes one of the five foreground classes. Regions
belonging to the background class was annotated on seven randomly selected
patients.

Tiles were extracted from these regions at 25x, 100x and 400x magnification in
such a manner that the centre pixel is the same in all three tiles. All tiles
have the same size of 128x128x3 pixels. Tiles belonging to the test set was
extracted from patients not present in the training data. The remaining data
was augmented to balance the dataset and was further randomly shuffled and
split into 85% training and 15% validation data. A random seed was set to
ensure that the shuffling was the same for each model. The final datasets
consist of 128K training tiles, 23K validation tiles and 11K test tiles.

##  3 Method and Results

This paper compares three architectures referred to as the MONO-, DI- and TRI-
CNN model. The three architectures have one (400x), two (100x, 400x) and three
(25x, 100x, 400x) inputs, respectively. Each input is fed into a pre-trained
VGG16 network Simonyan and Zisserman (2014) which acts as a feature extractor.
The fully-connected (FC) layers of VGG16 are replaced with a classification
network consisting of two FC-layers, each followed by a dropout layer, and a
final softmax layer with one output node for each of the six classes. The DI-
CNN and TRI-CNN models have two and three parallel VGG16 branches,
respectively, which are concatenated before entering the classification
network.

The FC-layers were tested with 512, 1024, 1536, 2048 and 4096 neurons, and
dropout rates of 0, 0.3 and 0.5. This 15-model hyperparameter search was
conducted on each of the three architectures, resulting in 45 models. These 45
models were run three consecutive times and averaged together for a more
accurate result. Each model was trained using early stopping, stopping the
model if validation loss did not decrease within 30 epochs. All model
selections were based on the validation set performance. After training, the
weight parameters from the epoch which performed best on the validation
dataset were restored, and a final evaluation of the model was performed on
the test dataset. The VGG16 networks had their weight parameters frozen during
training. The model was written in Python 3.5 using the Keras machine learning
library Chollet et al. (2015).

Table LABEL:tab:result shows the hyperparameters for the best performing
models and their average result from the three consecutive runs. The MONO-CNN
model achieves a result similar to that of the autoencoder. The two multiscale
models perform equally and significantly better than the mono-scale models.
The multiscale models also have a lower standard deviation on all metrics.
Since both multiscale models achieve the same result, one could argue that the
simplest model of the two should be chosen. In that case, DI-CNN with its 36M
parameters is a simpler model than TRI-CNN which has 47M parameters in total.
DI-CNN also have a marginally lower standard deviation than TRI-CNN.

tab:result  Model Input Scale Dropout FC-Neurons # Parameters F1-Score
Autoencoder111Model trained and evaluated on the same dataset Wetteland et al.
(2019). 400x 0.1 256/512 89M/89M 0.934 $\pm$ 0.061 MONO 400x 0.3 2048 5.3M/20M
0.944 $\pm$ 0.007 DI 100x/400x 0.0 2048 6.3M/36M 0.986 $\pm$ 0.002 TRI
25x/100x/400x 0.5 1024 2.6M/47M 0.986 $\pm$ 0.003

Table 1: Models evaluated on the test set. F1-Score is presented as the total
average and standard deviation calculated across all six classes over three
consecutive runs. Parameters are shown as no. of trainable parameters / no. of
total parameters.

##  4 Conclusion

In this paper, we present preliminary results from a multiscale tile-based
classification model. Tiles from six classes were extracted at multiple scales
from WSI of patients diagnosed with urothelial carcinoma. Three model
architectures were compared: MONO-CNN (400x), DI-CNN (100x, 400x) and TRI-CNN
(25x, 100x, 400x). Results for an autoencoder model from previous work was
also included for reference. Both multiscale models outperform the two single-
scale models and achieve a very good result indicating the advantage of
utilising multiple scales. The model can be used as an ROI extraction method
for relevant tissue areas in the large WSI, useful for both pathologist and
computer-aided diagnostic systems. Some more experiments should be performed
to clarify if the behaviour stems from the multiscale approach or the extended
field-of-view.

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