# Building up a model family for inflammations

Cordula Reisch  Technische Universität Braunschweig, Universitätsplatz 2,
38106 Braunschweig, Germany  Corresponding author<EMAIL_ADDRESS>Sandra Nickel  Clinic for Visceral, Transplantation, Thoracic and Vascular
Surgery; Leipzig University Hospital, Liebigstrasse 20, 04103 Leipzig,
Germany<EMAIL_ADDRESS>hans-
<EMAIL_ADDRESS>Hans-Michael Tautenhahn  Clinic for
Visceral, Transplantation, Thoracic and Vascular Surgery; Leipzig University
Hospital, Liebigstrasse 20, 04103 Leipzig, Germany<EMAIL_ADDRESS>leipzig.de<EMAIL_ADDRESS>
(December 08, 2023)

###### Abstract

The paper presents an approach for overcoming modeling problems of typical
life science applications with partly unknown mechanisms and lacking
quantitative data: A model family of reaction diffusion equations is built up
on a mesoscopic scale and uses classes of feasible functions for reaction and
taxis terms. The classes are found by translating biological knowledge into
mathematical conditions and the analysis of the models further constrains the
classes. Numerical simulations allow comparing single models out of the model
family with available qualitative information on the solutions from
observations. The method provides insight into a hierarchical order of the
mechanisms. The method is applied to the clinics for liver inflammation such
as metabolic dysfunction-associated steatohepatitis (MASH) or viral hepatitis
where reasons for the chronification of disease are still unclear and time-
and space-dependent data is unavailable.

##### Keywords

Mathematical modeling, reaction diffusion equations, model hierarchy,
inflammation, hepatitis.

##### AMS subject classification

92-10, 35K57, 35Q92

##  1 Introduction

Modeling life science processes with only partly known mechanisms and actors
is challenging. The challenge becomes even harder if there is only few data or
only qualitative data available for validating the models. However,
mathematical models for not fully understood complex problems provide insight
into the relevance of mechanisms and allow testing hypotheses in-silico.

In this paper, a family of models for the dynamics of liver inflammation is
formulated, analyzed, simulated, and considered in a clinical context. Liver
inflammation like steatohepatitis or viral hepatitis often evolves into
chronic inflammations which may lead to severe secondary diseases such as
cirrhosis.

The mechanisms for the evolution of the inflammation course are not fully
understood and therefore treatment is less effective. Mathematical models
provide a scientific environment for testing hypotheses on the importance of
involved cells and mechanisms. A hierarchical order of the mechanisms is a
priori unavailable because the mechanisms are coupled and only in a few cases
isolatable for single experiments. Models on a mesoscopic scale allow an
abstraction from the unknown processes on the cell scale and allow comparing
the modeling results to qualitative data like pathological images of the
liver.

The model family describes the interactions and propagation of different
influencing variables on inflammation. In particular, the following factors
are currently considered in the model: macrophages (copper cells), viruses,
CD4+ T cells, cytotoxic T cells, and cytokines. Different approaches for known
mechanisms are used and models of different complexity are analyzed and
compared. As the focus lies on the model family, not every model will be
discussed in all detail and not all mechanisms will be used in the models. The
results for the model family provide insight into the hierarchical order of
the used mechanisms.

###  1.1 Literature overview

The literature covers three topics: First, medical and biological observations
on liver inflammations are given. Details on the known mechanism and involved
cells follow in Sec. 2.2. Second, models for inflammations are presented with
a highlight on models for viral liver infections leading to inflammation.
Third, some results on reaction diffusion equations are given.

The leading causes of liver inflammation worldwide are viruses, metabolic
diseases, autoimmune diseases, and alcohol. If left untreated over a long
period of time, all of these factors can lead to cirrhosis of the liver, which
is in fact life-threatening, [1].

The course of viral liver infections begins with the infection itself,
followed by a symptom-free incubation period. During an active phase, the
immune system fights against the virus in the liver tissue. After the active
phase, healing infection courses differ from chronic. While the amount of
virus and the activity of the immune system reduces to zero in a healing
infection course, the virus persists in the liver and the immune system
remains active during a chronic infection course, [2]. Inflammation is the
process of an active immune system reacting to infected tissue. If the
inflammation continues even for reduced viral loads, the inflammation is
called chronic.

Inflammation occurs in different tissues caused by virus, bacteria, or
autoimmune diseases. This paper focuses on inflammation caused by viral liver
infections like hepatitis B. The reasons for chronic infections are still
partly unknown, compare [3, 4], and data is rare. Qualitative data is given by
pathological images of removed tissue, [5]. The pathological images show a
partial inhomogeneous spread of T cells in the regarded tissue parts. The
amount of T cells is higher next to portal fields. During acute hepatitis B
infections, up to 95% of the liver cells are infected with the virus, [5, 2].
During chronic phases around one-third of the liver cells are infected, [5,
2]. These observations will be used as qualitative data in Sec. 3.4 for
evaluating the modeling results.

Mathematical models for inflammations are available on different length scales
and for different inflammation types. In [6] models using ordinary
differential equations are compared. The models describe the total amount of
free virus, healthy, and infected liver cells in the whole liver by
compartment models. Variations change the reaction functions or take delay
into account, compare [7]. The same components are regarded in [8] but the
diffusion of all three cell types are added. In this approach, even the liver
cells are diffusing, which is not realistic. A more realistic model that is
adapted to hepatitis B infections is presented in [9]. The modeled cells are
next to three types of liver cells, free virus and B cells and only the free
virus is diffusing. This leads to a coupled ODE-PDE system for which
stationary states are analyzed and chronic infections are regarded. The
presented models describe the dynamics of the liver cells and the free virus.
As available data is given by pathological images showing the T cells, the
results are only partly comparable with the space-dependent images.

Mathematical models need to be spatially resolved for comparison with
pathological images. Additionally, the T cells need to be modeled as these are
the only cell types next to liver cells displayed in the pathological images.
Reaction diffusion equations are one possibility for modeling interacting
cells in space and time. They allow a mathematical analysis and in some cases
a prediction of the solution behavior.

Some results on reaction diffusion equations are summarized here. Reaction
diffusion equations are used for modeling many different applications like
morphogenesis [10], the spread of populations [11] or chemical reactions [12].
Depending on the reaction functions and the diffusion parameters, the models
show different types of solutions like traveling waves, (Turing) patterns,
blow-ups, or leveling behavior, [13]. In some cases, these solutions can be
proven and predicted analytically. In the light of chronic and healing
infection courses of liver inflammations, two types of solutions are
particularly interesting: Solutions tending towards zero describe healing
infection courses and chronic infection courses are mathematically described
by solutions with a tendency to stationary and spatially inhomogeneous
distributions. As there are some results on leveling solutions, c.f. [14], the
contradiction of these results can be used for gaining requirements for
solutions not tending towards a leveled state, compare [15].

###  1.2 Overview of this paper

The model family is formulated in Sec. 2. Biological information is translated
into classes of feasible reaction functions (Sec. 2.2.1) and of taxis terms
(Sec. 2.2.2). Subsections 2.3 and 2.4 give examples of these classes. The
model family is analyzed in Sec. 3 and requirements on the classes are
formulated in Sec. 3.2.4. Sec. 3.4 presents numerical simulations and compares
them with qualitative observations. A conclusion follows in Sec. 4.

##  2 Model family

A model family consists of models describing an application using the same
modeling approach but differing in the number or complexity of included
mechanisms. First, the model type of reaction diffusion equations and boundary
conditions are discussed. Afterward, biological information on the dynamics of
liver infections is given and is translated into conditions on function
classes.

###  2.1 Reaction Diffusion Equations

The models have the general form

$\displaystyle\mathbf{q}_{,t}=\mathbf{F}(t,\mathbf{x},\mathbf{q})+\nabla\cdot\mathbf{D}(\mathbf{x},\mathbf{q},\nabla\mathbf{q}),$
(1)

where $t>0$, $\mathbf{x}\in\Omega\subset\mathbb{R}^{d}$ and $\Omega$ is a
model for the regarded part of the liver. Following, the dimension of space is
$d\in\\{1,2,3\\}$, depending on the chosen simplification.

The vector-valued function
$\mathbf{q}:[0,\infty)\times\Omega\rightarrow\mathbb{R}^{n}$ gives the time
and space-dependent amount of the acting cells and virus. The function
$\mathbf{F}$ describes the reactions between the cells and the virus, as well
as the growth or decay of any described substances. The term
$\mathbf{D}(\mathbf{x},\mathbf{q},\nabla\mathbf{q})$ models any diffusive or
chemotactic effects caused by gradients of the substances or outer influence.

The reaction diffusion in Eq. (1) is completed to an initial value problem by
$\mathbf{q}(0,\mathbf{x})=\mathbf{q}_{0}(\mathbf{x})$ for
$\mathbf{x}\in\Omega$ with $\mathbf{q}_{0}(\mathbf{x})\geq\mathbf{0}$ for all
$\mathbf{x}\in\Omega$. Additionally, we assume zero-flux boundary conditions

$\displaystyle\frac{\partial\mathbf{q}}{\partial\mathbf{n}}$
$\displaystyle=\mathbf{0}$ $\displaystyle\text{ for
}\mathbf{x}\in\partial\Omega.$ (2)

The homogeneous Neumann boundary conditions are an approximation: The liver
consists of eight different segments which are separated by infeasible tissue,
[2]. There, zero flux conditions are reasonable because the exchange with the
exterior of one segment happens via the blood vessels.

If $\Omega$ is a part of such a liver segment, homogeneous Neumann boundary
conditions are less intuitive. Assuming that the liver tissue in $\Omega$ is
homogeneous, periodic boundary conditions are a reasonable choice.
Biologically, liver inflammations are stronger in some parts of the liver than
in others. Therefore, the assumption of similar structures next to each other
is not perfect.

As an approximation for nearly constant amounts of acting cells close to the
boundary $\partial\Omega$, zero flux conditions are acceptable and reflect the
higher importance of local reactions compared to diffusion over the boundary.

###  2.2 Biological motivation

Viral hepatitis such as hepatitis B is named after the virus type causing an
infection and leading to inflammation of the liver tissue. Inflammation is a
consequence of the reaction of the immune system to the infection.

Hepatitis B infections show two typical infection courses. In healing
infection courses, the virus and the inflammation vanish after an acute
inflammation with a strong active phase of the immune system. Chronic
infection courses have an active phase followed by a chronic phase with mild
symptoms. In the chronic phase, virus and inflammation persist in the liver
tissue. Chronic liver infections often lead to severe secondary diseases like
liver cancer or cirrhosis.

A short summarization of the cellular immune response follows. After the
infection, the virus spreads out in the body and settles in the liver. There,
it replicates and spreads out. During this phase, the adaptive immune reaction
starts working. Dentritic cells process and opsonize virus particles to the T
cells. The T cells specify and develop to CD4+ or CD8+ T cells. Some of the
CD4+ T cells become T helper cells specializing in different types of T helper
cells. One type of T helper cell enhances the cytokine concentration, which
are signals caused by the interaction of tissue cells and virus. Cytokines
lead the way for T cells towards the virus.

The CD8+ T cells named cytotoxic T cells triggers the cell death of infected
tissue cells. Therewith, the tissue cell and the contained virus are
destroyed. Inflammation is essential to neutralize the damaging stimulus and
create the conditions for repair processes. The inflammation is caused only
indirectly by the virus.

Next, the description of infection is split up into single mechanisms. From a
model-theoretic point of view, this step is again a modeling process. We start
with naming the actors of the dynamics. Let $q_{1}=q_{1}(t,\mathbf{x})$ be the
space and time dependent amount of the virus. According,
$q_{2}=q_{2}(t,\mathbf{x})$ is the amount of T cells, which can be divided
into T helper cells $T_{h}=T_{h}(t,\mathbf{x})$ and cytotoxic T cells
$T_{c}=T_{c}(t,\mathbf{x})$. Cytokines $q_{3}=q_{3}(t,\mathbf{x})$ are the
third quantity.

The effects on each other are visualized by arrows or dependencies ($\sim$).

  1. (M1)

virus replicate and spread out: $q_{1}\rightarrow q_{1}\nearrow$

  2. (M2)

immune system reacts on virus by producing T cells: $q_{1}\rightarrow
q_{2}\nearrow$

  3. (M3)

T helper cells produce and enhance cytokines: $T_{h}\rightarrow q_{3}\nearrow$

  4. (M4)

cytokines lead cytotoxic T cells: $q_{3}\sim T_{c}$

  5. (M5)

cytotoxic T cells kill virus: $T_{c}\rightarrow q_{1}\searrow$

  6. (M6)

in absence of virus, T cells reduce (as well natural decay): $\neg
q_{1}\rightarrow q_{2}\searrow$

  7. (M7)

random spreading of T helper cells

Figure 1: Mechanisms during an inflammation. The variables are virus $q_{1}$,
cytokines $q_{3}$ and T cells $q_{2}$, divided into T helper cells $T_{h}$ and
cytotoxic T cells $T_{c}$. Positive effects are displayed as
$\footnotesize{+}$, negative effects as $\footnotesize{-}$, and movement
effects as m. The spreading of the virus and T cells is not visualized. The
dashed arrow shows the enhancement of mechanism M3 depending on $q_{1}$.

Fig. 1 visualizes the mechanisms and their effect. There is a large loop
effect: Mechanism M2 enhances the production of both T cell types according to
the virus amount $q_{1}$. The T helper cells $T_{h}$ produce, again depending
on $q_{1}$, cytokines. The cytokines $q_{3}$ control the cytotoxic T cells
$T_{c}$, which reduce the virus $q_{1}$.

Next, the mechanisms are translated into assumptions on the reaction terms.

####  2.2.1 Reactions

The next step in building up a model family is the translation of the
biological observations into mathematics. Fig. 1 is an intermediate step for
this process. There, the mechanisms are converted from a verbal description to
a logical expression. Next, the logical expressions are translated into
mathematical formulas, regarding more details of the verbal expression than in
Fig. 1.

Therefore, we specify the reaction diffusion system in (1). $F_{1}$ is the
first entry of the vector $\mathbf{F}$ and belongs to the partial differential
equation for the virus $q_{1}$. Analogously, $F_{2}$ and $F_{3}$ belong to
$q_{2}$ and $q_{3}$. If the T cell population in the model is split up into
the two sub-types $T_{h}$ and $T_{c}$, there are reaction functions
$F_{T_{h}}$ and $F_{T_{c}}$.

Every reaction function $F_{i}$ consists of different single mechanisms
$f_{i}^{Mj}$ splitting up the biological mechanisms occurring during an
inflammation. The reaction function for the change of the virus is given by

$\displaystyle
F_{1}(t,\mathbf{x},\mathbf{q})=f_{1}^{M1}(t,\mathbf{x},\mathbf{q})+f_{1}^{M5}(t,\mathbf{x},\mathbf{q}),$
(3)

where the first reaction function $f_{1}^{M1}$ describes the growth and
$f_{1}^{M5}$ describes the decay of virus. The spreading of the virus, which
is also part of mechanism M1, is described by a taxis term, which will be
discussed in Sec. 2.2.2.

If the T cells are only regarded as general T cells fulfilling all mentioned
functions of T helper cells and cytotoxic T cells, then the biological
mechanisms affecting $q_{2}$ are

$\displaystyle F_{2}(t,\mathbf{x},\mathbf{q})$
$\displaystyle=f_{2}^{M2}(t,\mathbf{x},\mathbf{q})+f_{2}^{M4}(t,\mathbf{x},\mathbf{q})+f_{2}^{M6}(t,\mathbf{x},\mathbf{q}).$

If the T cell subtypes are split up into T helper cells $T_{h}$ and cytotoxic
T cells $T_{c}$, then the mechanisms affect directly the subtypes instead of
the general T cells, so

$\displaystyle F_{T_{h}}(t,\mathbf{x},\mathbf{q})$
$\displaystyle=f_{T_{h}}^{M2}(t,\mathbf{x},\mathbf{q})+f_{T_{h}}^{M6}(t,\mathbf{x},\mathbf{q}),$
$\displaystyle F_{T_{c}}(t,\mathbf{x},\mathbf{q})$
$\displaystyle=f_{T_{c}}^{M2}(t,\mathbf{x},\mathbf{q})+f_{T_{c}}^{M4}(t,\mathbf{x},\mathbf{q})+f_{T_{c}}^{M6}(t,\mathbf{x},\mathbf{q}).$

So far, the reaction function for the cytokines consists of one mechanism,

$\displaystyle F_{3}(t,\mathbf{x},\mathbf{q})$
$\displaystyle=f_{3}^{M3}(t,\mathbf{x},\mathbf{q})$

describing the production of cytokines by T helper cells in the presence of
the virus.

The mechanisms translate into conditions on the partial derivatives of the
functions $f_{i}^{Mj}$. By considering first the translation into partial
derivatives and not directly into certain functions, the classes of reaction
functions include more variety. The deductive modeling process from biological
mechanisms to a mathematical model family is highlighted by these finer steps.

In [16] the conditions

$\displaystyle\frac{\partial F_{1}}{\partial q_{2}}<0,\quad\quad\frac{\partial
F_{2}}{\partial q_{1}}>0,\quad\quad\frac{\partial F_{2}}{\partial q_{2}}<0$

are presented, which are now regarded in more detail. We start with the
mechanisms for the virus. The growth of virus (M1) can be modeled by a
function fulfilling

$\displaystyle\frac{\partial f_{1}^{M1}(t,\mathbf{x},\mathbf{q})}{\partial
q_{1}}\geq 0.$ (4)

The equality with zero can be interpreted as a stopped growth of the virus,
for example, due to an eliminated virus or due to saturation. The second
mechanism affecting the virus is a reduction in dependency of cytotoxic T
cells. This mechanism can be expressed by the conditions on the partial
derivatives

$\displaystyle\frac{\partial f_{1}^{M5}(t,\mathbf{x},\mathbf{q})}{\partial
T_{c}}\leq 0\quad\quad\mathrm{or}\quad\quad\frac{\partial
f_{1}^{M5}(t,\mathbf{x},\mathbf{q})}{\partial q_{2}}\leq 0,$ (5)

depending on the chosen variables.

Analogously, we find for the T cells

$\displaystyle\frac{\partial f_{2}^{M2}(t,\mathbf{x},\mathbf{q})}{\partial
q_{1}}\geq 0,\quad\quad\frac{\partial
f_{2}^{M4}(t,\mathbf{x},\mathbf{q})}{\partial q_{3}}\geq
0,\quad\quad\frac{\partial f_{2}^{M6}(t,\mathbf{x},\mathbf{q})}{\partial
q_{2}}\leq 0.$ (6)

Mechanism M2 shows the effect that the immune system has on the virus by the
production of T cells. The more virus there is, the more T cells are produced.
The mechanism M4 shows that the amount of T cells increases depending on the
cytokines. In mechanism M6, the decay of T cells is modeled.

If the model considers a splitting up of the general T cells into T helper
cells and cytotoxic T cells, then there are conditions on the mechanism
functions for the special T cells. For the T helper cells, the mechanisms M2
and M6 are relevant and the conditions in (6) hold for $f_{T_{h}}^{M2}$ and
$f_{T_{h}}^{M6}$. The mechanisms M2, M4 and M6 are relevant for the cytotoxic
T cells and (6) hold for $f_{T_{c}}^{M2}$, $f_{T_{h}}^{M4}$ and
$f_{T_{c}}^{M6}$.

The conditions for the reaction functions of the cytokines are

$\displaystyle\frac{\partial f_{3}^{M3}(t,\mathbf{x},\mathbf{q})}{\partial
T_{h}}\geq 0\quad\quad\text{and}\quad\quad\frac{\partial
f_{3}^{M3}(t,\mathbf{x},\mathbf{q})}{\partial q_{1}}\geq 0,$ (7)

where the partial derivative with respect to the T helper cells is replaced by
the partial derivative with respect to the T cells $q_{2}$ in the case of not
splitting up into subtypes.

For all quantities, a natural decay will be discussed in Sec. 2.3. The natural
decay is a result of different biological processes and a small transport. For
the virus, the natural decay can be interpreted as a reduced growth factor.
Additionally, some processes have a natural saturation, for example, due to
restricted resources.

The general conditions on the partial derivatives of the mechanism functions
are one step of the modeling process providing the substance for building a
model family. In the next section, conditions on the taxis terms are
presented. Afterwards, particular reaction functions and taxis terms
fulfilling the conditions are given and the effects of natural decay and
saturation are discussed in more detail.

####  2.2.2 Taxis

The taxis are not fully covered by the description in Fig. 1 and therefore
here explained in more detail. The regarded domain $\Omega$ is a part of the
liver, not containing, for example, the lymph. Consequently, the details of
the transport of T cells into the liver is not part of the model. The domain
$\Omega$ contains a subdomain $\Theta$ where the inflow of T cells takes
place. If $\Omega$ is a two-dimensional domain, $\Theta$ can be seen as a cut
through a portal field of the liver structure.

The boundary conditions prevent an exchange of virus or T cells with the
exterior. In this section, the taxis of any cells in the interior of $\Omega$
are described.

The virus $q_{1}$ spreads out in the liver. The spreading is caused by two
different mechanisms, [17]. Cell-to-cell transmission leads to a spreading
with a small medium step size and the virus in one infected liver cell passes
on to a neighbor cell after replication. The second mechanism is the diffusion
of the virus through the extracellular space. The medium step size is
typically larger than for the cell-to-cell transmission. The diffusion
function in (1) for the $q_{1}$-component therefore consists of two terms

$\displaystyle\mathbf{D}_{1}(\mathbf{x},\mathbf{q},\nabla\mathbf{q})=\mathbf{D}_{1}^{\mathrm{ctc}}(\mathbf{x},\mathbf{q},\nabla\mathbf{q})+\mathbf{D}_{1}^{\mathrm{ecs}}(\mathbf{x},\mathbf{q},\nabla\mathbf{q}).$

The cell-to-cell transmission can be interpreted as a diffusion process with a
small diffusion parameter. This mechanism has the form

$\displaystyle\mathbf{D}_{1}^{\mathrm{ctc}}(\mathbf{x},\mathbf{q},\nabla\mathbf{q})=\mathbf{D}_{1}^{\mathrm{ctc}}(\nabla
q_{1}).$ (8)

The spreading through the extracellular space might depend on the blood stream
through the extracellular space and therefore on the direction and velocity of
the blood. The results term would include this extracellular spread
additionally to virus amount, for example

$\displaystyle\mathbf{D}_{1}^{\mathrm{ecs}}(\mathbf{x},\mathbf{q},\nabla\mathbf{q})=\mathbf{D}_{1}^{\mathrm{ecs}}(\mathbf{x},q_{1},\nabla
q_{1}).$ (9)

The cytotoxic T cells $T_{c}$ follow a cytokine gradient

$\displaystyle\mathbf{D}_{T_{c}}^{\mathrm{chem}}(\mathbf{x},\mathbf{q},\nabla\mathbf{q})=\mathbf{D}_{T_{c}}^{\mathrm{chem}}(T_{c},\nabla
q_{3})$

and the taxis depend on the cytotoxic T cells and the cytokine gradient.

The T helper cells are spreading out by diffusion, so
$\mathbf{D}_{T_{h}}^{\mathrm{diff}}(\mathbf{x},\mathbf{q},\nabla\mathbf{q})=\mathbf{D}_{T_{h}}^{\mathrm{diff}}(T_{h}).$

For the general T cells without a division into subtypes, both effects combine
and the dependencies on $T_{h}$ and $T_{c}$ are replaced by a dependency on
$q_{2}$.

The cytokines are signal cells without self-induced movement. Due to natural
processes, a small diffusion is possible. Additionally, the transport of the
cytokines by the bloodstream through the extracellular tissue might happen. As
a sum of those two mechanisms, the term for the cytokine $q_{3}$ is

$\displaystyle\mathbf{D}_{3}(\mathbf{x},\mathbf{q},\nabla\mathbf{q})=\mathbf{D}_{3}^{\mathrm{diff}}(\nabla
q_{3})+\mathbf{D}_{3}^{\mathrm{ecs}}(\mathbf{x},q_{3},\nabla q_{3}).$

We give examples for taxis terms in Sec. 2.4.

###  2.3 Examples of feasible reaction functions

The model in Eq. (1) is specialized for particular models of the model family.
The functions are motivated by biology, see Sec. 2.2, and still variable
enough to cover various scenarios.

Mechanism M1 describes the virus replication, and functions fulfilling Eq. (4)
are

$\displaystyle\begin{aligned}
f_{1}^{M1}(q_{1})&=a_{1}q_{1}&&(\text{unbounded}),\\\
f_{1}^{M1}(q_{1})&=a_{1}q_{1}(C_{1}-q_{1})&&(\text{bounded by }C_{1}),\\\
f_{1}^{M1}(q_{1})&=a_{1}q_{1}(C_{1}-q_{1})\frac{q_{1}-\varepsilon}{q_{1}+\kappa}&&(\text{bounded
by }C_{1}\text{ and Allee effect}).\end{aligned}$ (10)

The three examples are extensions of each other. The first function describes
an unbounded exponential growth of the virus. The second function is bounded
from above by a constant capacity $C_{1}$. The resulting growth is a logistic
growth, starting exponentially and reaching saturation which describes that
all liver cells are infected.

The third example includes a strong Allee effect, [18], modeling a small
vanishing virus amount by a negative function for $q_{1}<\varepsilon$. This
effect can be interpreted as a small immune reaction acting on very small
viral loads. A discussion of the difference due to the Allee effect can be
found in [16]. The function with the Allee effect fulfills the condition in
Eq. (4) only for values $q_{1}\geq\varepsilon$.

The mechanism M2 describes the production of T cells, both T helper cells and
cytotoxic T cells, if the virus is present. Possible mechanisms are

$\displaystyle\begin{aligned} f_{2}^{M2}(q_{1})&=a_{2}q_{1}&&(\text{local,
unbounded}),\\\
f_{2}^{M2}(q_{1},q_{2})&=a_{2}q_{1}(C_{2}-q_{2})&&(\text{local, bounded by
}C_{2}),\\\
f_{2}^{M2}(\mathbf{x},q_{1})&=a_{2}\chi_{\Theta}(\mathbf{x})\int_{\Omega}q_{1}\,\mathrm{d}\mathbf{x}&&(\text{global}),\\\
f_{2}^{M2}(\mathbf{x},q_{1})&=a_{2}\chi_{\Theta}(\mathbf{x})(C_{2}-q_{2})\int_{\Omega}q_{1}\,\mathrm{d}\mathbf{x}&&(\text{global,
with saturation}).\end{aligned}$ (11)

The first example depends only on the local virus amount. T cells appear where
the virus is. This effect is extended by an upper bound $C_{2}$ for the T
cells in the second example. The local mechanism neglects the finer liver
structure with portal fields. T cells are produced outside of $\Omega$ and are
transported via blood vessels into the liver. In the small-scale liver
structure, the influx of T cells takes place in portal fields
$\Theta\subset\Omega$.

The subdomains $\Theta$ are part of the third and fourth examples for
mechanism functions. The functions are explicitly space dependent via a
function $\chi_{\Theta}(\mathbf{x})$ fulfilling

$\displaystyle\int_{\Omega}\chi_{\Theta}(\mathbf{x})=1\quad\text{ and
}\quad\chi_{\Theta}(\mathbf{x})\begin{cases}\geq 0&\text{ for
}\mathbf{x}\in\Theta,\\\ =0&\text{ for
}\mathbf{x}\in\Omega\setminus\Theta,\end{cases}$ (12)

compare [16]. The function $\chi$ can be a characteristic function for the
subdomain $\Theta$, or, if required, a smooth function with positive values
and a local support in $\Theta$.

The integral $\int_{\Omega}q_{1}\,\mathrm{d}\mathbf{x}$ models a collection of
the total viral load in the regarded part $\Omega$. The production of T cells
is proportional to the total viral load in examples three and four in Eq.
(11).

If the T cells are distinguished into T helper cells $T_{h}$ and cytotoxic T
cells $T_{c}$, the proposed mechanisms are adapted by replacing $q_{2}$ by one
of the subtypes with individual capacities $C_{T_{h}}$, $C_{T_{c}}$. The
requirement in Eq. (6) is fulfilled for all examples.

Mechanism M3 describes the production of cytokines by the virus and the effect
of T helper cells increasing the cytokines. Examples of reaction functions are

$\displaystyle\begin{aligned} f_{3}^{M3}(q_{1})&=a_{3}q_{1}&&(\text{only
dependent on virus}),\\\ f_{3}^{M3}(T_{h})&=a_{3}T_{h}&&(\text{only dependent
on T helper cells}),\\\
f_{3}^{M3}(q_{1},T_{h})&=a_{3}q_{1}T_{h}&&(\text{unbounded}),\\\
f_{3}^{M3}(T_{h},q_{3})&=a_{3}T_{h}(C_{3}-q_{3})&&(\text{dependent on $T_{h}$
cells, bounded}),\\\
f_{3}^{M3}(q_{1},T_{h},q_{3})&=a_{3}T_{h}q_{1}(C_{3}-q_{3})&&(\text{bounded}),\end{aligned}$
(13)

where $T_{h}$ can be replaced by $q_{2}$ if there is no division into subtypes
of T cells. The examples differ in their dependencies on the involved cells
and in the boundedness of cytokines at a certain place. Dependency only on the
virus neglects the mechanism that T helper cells increase the cytokines while
functions independent of the virus neglect the mechanism that cytokines are
produced by infected liver cells.

The conditions in Eq. (7) are fulfilled. The last two examples in Eq. (7)
depend on $q_{3}$ itself and give an upper bound. By only regarding mechanism
M3, the constant $C_{3}$ gives an upper bound. Further discussion on the
boundedness of solutions follows in Sec. 3.2.

The examples for mechanism M3 in Eq. (13) can be hierarchically ranked. The
more sub-mechanisms are included, the higher the ranking is. The bottom level
contains only the first two examples. The last example contains the most
mechanisms and is therefore at the top of the ranking. Containing the most
mechanisms is not always a feature for quality as the mechanism and their
importance are unknown.

M4 describes the control of the movement of the cytotoxic T cells, c.f. Sec.
2.4.

The mechanism M5 describes the decay of the virus in dependence on cytotoxic T
cells. Possible functions for this mechanism are

$\displaystyle\begin{aligned} f_{1}^{M5}(T_{c})&=-a_{5}T_{c}&&(\text{only
dependent on cytotoxic T~{}cells}),\\\
f_{1}^{M5}(q_{1},T_{c})&=-a_{5}q_{1}T_{c}&&(\text{dependent on virus and
cytotoxic T~{}cells}).\end{aligned}$ (14)

The condition in Eq. (5) is fulfilled by both functions in (14).

Again, the dependency on the cytotoxic T cells can be replaced by a dependency
on the general T cells. The first example in Eq. (14) does not preserve the
non-negativity of $q_{1}$. The analysis of the reaction functions follows in
Sec. 3.2 in more detail.

The mechanism M6 describes that the T cell amount does not decrease if the
virus is still present. It includes different effects: First, there is a decay
of T cells. Second, this decay is smaller, if there is more virus. The
technical description of ’a lot of virus’ depends on the boundedness of the
virus $q_{1}$ by $C_{1}$. Examples are

$\displaystyle\begin{aligned} f_{2}^{M6}(q_{2})&=-a_{6}q_{2}&&(\text{natural
decay}),\\\ f_{2}^{M6}(q_{1},q_{2})&=-a_{6}q_{2}(C_{1}-q_{1})&&(\text{decay in
absence of virus}),\end{aligned}$ (15)

where the first example is included in the second because

$f_{2}^{M6}(q_{1},q_{2})=-a_{6}q_{2}(C_{1}-q_{1})=-a_{6}C_{1}q_{2}+a_{6}q_{1}q_{2}=C_{1}f_{2}^{M6}+a_{6}q_{1}q_{2}$

yields. Both examples in Eq. (15) fulfill the condition in Eq. (6) on the
partial derivative with respect to the T cells. The second example in Eq. (15)
consists of a natural decay and an increase depending on the virus and the T
cells. The increase can be interpreted as a benefit of the T cells like in a
predator-prey system or a host-pathogen system. Further details on this
comparison are discussed in [19]. If the T cells are divided into T helper
cells and cytotoxic T cells, mechanism M6 works in the same way by replacing
$q_{2}$ by $T_{h}$ or $T_{c}$ respectively.

Examples of the reaction functions of the different mechanisms are given in
Eqn. (10), (11), (13), (14) and (15). The taxis describing mechanism M4 will
be discussed in Sec. 2.4. Additionally, there is a natural decay for all
quantities.

The virus is decaying in the presence of (cytotoxic) T cells, see mechanism M5
in Eq. (14). Additionally, there is a small natural decay, which is already
regarded in the growth, covered by mechanism M1 in Eq. (10). The growth
including an Allee effect has a negative value for a very small amount of
virus. This negative growth can be interpreted as a natural decay of very
small amounts of virus. Besides, a natural decay can be expressed by a smaller
parameter $a_{1}$. Consequently, the natural decay for the virus is already
included in the example for mechanism M1 with the Allee effect. The other
examples in Eq. (10) do not include the natural decay, interpreted as a local
immune reaction, for small amounts of virus in the growth function. This
results in different analytical properties, see Sec. 3.2.

The reaction function of mechanism M6 for the T cells contains a natural
decay. In the case of a division of the T cells into the subtypes of T helper
cells and cytotoxic T cells, the mechanism M6 is used in the same way for both
subtypes.

So far, a natural decay for $q_{3}$ like

$\displaystyle f_{3}^{\mathrm{nd}}(q_{3})=-a_{\mathrm{nd}}q_{3},$ (16)

is not contained in the mechanisms. A possible constant decay is

$\displaystyle f_{3}^{\mathrm{nd}}=\begin{cases}-a_{\mathrm{nd}}&\text{ if
}q_{3}>0,\\\ 0&\text{ if }q_{3}=0,\end{cases}$ (17)

resulting in an unsteady reaction function.

This section presents examples of reaction functions fulfilling the
requirements from Sec. 2.2.1 and describing mathematically the biologically
motivated mechanisms. Next, we give examples of the movements of the different
cells.

###  2.4 Examples of feasible taxis terms

The general reaction diffusion equation in Eq. (1) has a taxis term of the
form $\nabla\cdot\mathbf{D}(t,\mathbf{x},\mathbf{q},\nabla\mathbf{q})$. This
term was divided into terms for every component in Sec. 2.2.2. In this
section, the taxis terms for every component are specified and examples are
given.

The virus $q_{1}$ spreads out using cell-to-cell transmission and diffusion
through the extracellular space. Possible dependencies for the two spreading
mechanisms are given in Eq. (8) and (9). Cell-to-cell transmission is a pure
diffusion process, so

$\displaystyle\mathbf{D}_{1}^{\mathrm{ctc}}(\nabla q_{1})$
$\displaystyle=d_{1}^{\mathrm{ctc}}\nabla q_{1}.$

The spreading of the virus through the extracellular space can be modeled as

$\displaystyle\mathbf{D}_{1}^{\mathrm{ecs}}(\nabla q_{1})$
$\displaystyle=d_{1}^{\mathrm{ecs}}\nabla q_{1}$
$\displaystyle(\text{homogeneous diffusion}),$
$\displaystyle\mathbf{D}_{1}^{\mathrm{ecs}}(\mathbf{x},\nabla q_{1})$
$\displaystyle=d_{1}^{\mathrm{ecs}}A(\mathbf{x})\nabla q_{1}$
$\displaystyle(\text{space dependent diffusion}),$

where $A(\mathbf{x})\in\mathbb{R}^{d\times d}$ is a positive definite, space-
dependent matrix modeling the flux directions in the extracellular space at
every point $\mathbf{x}$.

In more detailed modeling approaches, the mechanisms could be additionally
time-dependent and display structural deformation on cell-scale caused by
inflammation.

The taxis of T cells is type-dependent. Cytotoxic T cells are attracted by
cytokines, whereas the taxis of T helper cells is dominated by diffusion. We
start with the diffusion of the T helper cells which is analogous to the
diffusion of the virus

$\displaystyle\mathbf{D}_{T_{h}}^{\mathrm{diff}}(\nabla T_{h})$
$\displaystyle=d_{T_{h}}^{\mathrm{diff}}\nabla T_{h}.$

The motion of the cytotoxic T cells is dominated by chemotactic effects, which
means that the cytotoxic T cells follow a cytokine gradient. Examples are

$\displaystyle\begin{aligned} \mathbf{D}_{T_{c}}^{\mathrm{chem}}(\nabla
q_{3})&=d_{T_{c}}^{\mathrm{chem}}\nabla q_{3},\\\
\mathbf{D}_{T_{c}}^{\mathrm{chem}}(T_{c},\nabla
q_{3})&=d_{T_{c}}^{\mathrm{chem}}T_{c}\nabla q_{3},\end{aligned}$ (18)

where the first example in Eq. (18) does not depend on the number of cytotoxic
T cells and might lead therefore to a negative value $T_{c}$.

If the T cells are not divided up into the two subtypes, there might evolve
conflicts with the dynamic of T cells and cytokines. These problems will be
discussed later. An addition of the motion of cytotoxic T cells and T helper
cells leads to a taxis term

$\displaystyle\mathbf{D}_{2}(q_{2},\nabla q_{2},\nabla
q_{3})=d_{2}^{\mathrm{diff}}\nabla q_{2}+d_{2}^{\mathrm{chem}}q_{2}\nabla
q_{3}.$

The cytokines are spreading by some diffusion $\mathbf{D}_{3}(\nabla
q_{3})=d_{3}^{\mathrm{diff}}\nabla q_{3}.$

Next, different models are built by using the variety in the model family.

##  3 Analysis of the model family

The reaction functions and taxis terms in Sec. 2.3 and 2.4 are combined to
models of the model family. Afterward, analytical results for the models are
discussed. This analysis gives further requirements on the variety of possible
reaction and taxis functions. Numerical simulations of acceptable models
follow in Sec. 3.4.

###  3.1 Examples for models

The model family includes models for different sets of cell types. The largest
set is the combination $(q_{1},T_{h},T_{c},q_{3})$. Another possible set of
participating cells is $(q_{1},q_{2},q_{3})$. The smallest model with
interactions consists of virus and T cells, $(q_{1},q_{2})$ and neglects the
cytokines. For certain analysis steps, different combinations or one-component
models are meaningful as well, see [16, 15].

######  Remark 1 (Model 1).

A large model in our model family for the four components virus $q_{1}$, T
helper cells $T_{h}$, cytotoxic T cells $T_{c}$, and cytokines $q_{3}$ is

$\displaystyle q_{1,t}$
$\displaystyle=a_{1}q_{1}(C_{1}-q_{1})\frac{q_{1}-\varepsilon}{q_{1}+\kappa}-a_{5}q_{1}T_{c}+\nabla\cdot[d_{1}^{\mathrm{ctc}}\nabla
q_{1}+d_{1}^{\mathrm{ecs}}A(\mathbf{x})\nabla q_{1}],$ (19) $\displaystyle
T_{h,t}$
$\displaystyle=a_{2,h}\chi_{\Theta}(\mathbf{x})(C_{T_{h}}-T_{h})\int_{\Omega}q_{1}\,\mathrm{d}\mathbf{x}-a_{6,h}T_{h}(C_{1}-q_{1})+\nabla\cdot[d_{T_{h}}^{\mathrm{diff}}\nabla
T_{h}],$ (20) $\displaystyle T_{c,t}$
$\displaystyle=a_{2,c}\chi_{\Theta}(\mathbf{x})(C_{T_{c}}-T_{c})\int_{\Omega}q_{1}\,\mathrm{d}\mathbf{x}-a_{6,c}T_{c}(C_{1}-q_{1})-\nabla\cdot[d_{T_{c}}^{\mathrm{chem}}T_{c}\nabla
q_{3}],$ (21) $\displaystyle q_{3,t}$
$\displaystyle=a_{3}T_{h}q_{1}-a_{\mathrm{nd}}q_{3}+\nabla\cdot
d_{3}^{\mathrm{diff}}\nabla q_{3},$ (22)

for $\mathbf{x}\in\Omega$ and $t>0$. Let
$\mathbf{q}=(q_{1},T_{h},T_{c},q_{3})^{\mathrm{T}}$. Boundary conditions are
like in Eq. 2 zero flux conditions. Non-negative initial values complete the
problem. All parameters have positive values. The matrix $A$ is positive
definite and $\chi_{\Theta}$ fulfills Eq. (12).

Model 1 in Rem. 1 uses all mechanisms and for every mechanism the example with
the highest complexity in comparison to the other examples. Of course, there
are many more detailed functions and models with many more components
thinkable. Next, a model with three interacting cell types is presented,
compare [16].

######  Remark 2 (Model 2).

A second model for inflammation contains the three components virus $q_{1}$, T
cells $q_{2}$ and cytokines $q_{3}$. The dynamics are given by

$\displaystyle q_{1,t}$
$\displaystyle=a_{1}q_{1}(C_{1}-q_{1})\frac{q_{1}-\varepsilon}{q_{1}+\kappa}-a_{5}q_{1}q_{2}+\nabla\cdot[d_{1}^{\mathrm{ctc}}\nabla
q_{1}],$ (23) $\displaystyle q_{2,t}$
$\displaystyle=a_{2}\chi_{\Theta}(\mathbf{x})\int_{\Omega}q_{1}\,\mathrm{d}\mathbf{x}-a_{6}q_{2}(C_{1}-q_{1})+\nabla\cdot[d_{2}^{\mathrm{diff}}\nabla
q_{2}-d_{2}^{\mathrm{chem}}q_{2}\nabla q_{3}],$ (24) $\displaystyle q_{3,t}$
$\displaystyle=a_{3}q_{1}-a_{\mathrm{nd}}q_{3}+\nabla\cdot[d_{3}^{\mathrm{diff}}\nabla
q_{3}],$ (25)

again completed with zero flux boundary conditions Eq. (2) and non-negative
initial values. The parameters are positive and $\chi_{\Theta}$ fulfills Eq.
(12).

As a further reduction, the cytokines are not regarded. The right circle in
Fig. 1 containing the mechanisms of the cytokines is neglected. The resulting
model only contains the dynamics of virus and T cells, compare [20, 16].

######  Remark 3 (Model 3).

A two-component model of virus $q_{1}$ and T cells $q_{2}$ is

$\displaystyle q_{1,t}$
$\displaystyle=a_{1}q_{1}(C_{1}-q_{1})\frac{q_{1}-\varepsilon}{q_{1}+\kappa}-a_{5}q_{1}q_{2}+\nabla\cdot[d_{1}^{\mathrm{ctc}}\nabla
q_{1}],$ (26) $\displaystyle q_{2,t}$
$\displaystyle=a_{2}\chi_{\Theta}(\mathbf{x})\int_{\Omega}q_{1}\,\mathrm{d}\mathbf{x}-a_{6}q_{2}(C_{1}-q_{1})+\nabla\cdot[d_{2}^{\mathrm{diff}}\nabla
q_{2}].$ (27)

Again, the parameters are positive, $\chi_{\Theta}$ fulfills Eq. (12), the
initial conditions are non-negative and there are zero flux boundary
conditions.

Model 3 in Rem. 3 uses different examples of reaction functions including
fewer effects than model 1 in Rem. 1. This is not only a consequence of the
reduction of modeled cell types but also depending on the used reaction
functions.

It is not possible to decide a priori, that means before an analytical
analysis and some simulations, for the best model. This is caused by the
uncertainty in choosing mechanisms and concrete reaction and taxis functions.
Further, it is a consequence of the missing definition for a ’best’ model.

With these three different models in mind, we start the analysis of the models
with as general reaction functions as possible.

###  3.2 Analytical results

The analysis of the reaction diffusion equations covers some basic properties.

####  3.2.1 Non-negativity of solutions

The first property which will be discussed is the non-negativity of the
solutions. As the solutions are interpreted as an amount of a certain cell
type, negative values are meaningless in the light of application.

The reaction diffusion equations consist of some reaction functions and
divergence terms. If the divergence term considers only diffusive effects, the
regarded system quantity will not become negative due to the diffusive term.
In those cases, it is sufficient to regard the reaction terms and check
whether it allows a negative quantity. We start with the discussion of the
reaction functions and come afterward to the taxis.

The reaction function for the virus $q_{1}$ consists of two mechanism
functions for M1 and M5, see Eq. (3). The mechanism M1 depends on the virus,
so $f_{1}^{M1}=f_{1}^{M1}(t,\mathbf{x},q_{1})$. The explicit time and space
dependency is only theoretical and not discussed.

The mechanism M5 depends on the amount of (cytotoxic) T cells, $T_{c}$ or
$q_{2}$, and the amount of virus. The dependency reads
$f_{1}^{M5}=f_{1}^{M5}(t,\mathbf{x},q_{1},T_{c})$, where $T_{c}$ can be
replaced by $q_{2}$. Together, the reaction function is called
$F_{1}(t,\mathbf{x},q_{1},T_{c})$, and it specifies Eq. (3). A necessary
condition for the non-negativity of $q_{1}$ is

$\displaystyle F_{1}(t,\mathbf{x},0,T_{c})\geq 0.$ (28)

A function modeling the growth of a virus should fulfill

$\displaystyle f_{1}^{M1}(t,\mathbf{x},0)=0.$ (29)

All examples in Eq. (10) fulfill Eq. (29), so there is no increase of virus if
no virus is present.

The mechanism M5 depends not only on the virus itself but also on the amount
of (cytotoxic) T cells. Additionally, the requirement Eq. 5 is a non-positive
partial derivative with respect to the T cells. Therefore, $f_{1}^{M5}$ has
negative values, and should be zero if $q_{1}=0$. Otherwise, there would be an
automated cell death without any virus. Again, the requirement in Eq. 28
transfers to a condition on $f_{1}^{M5}(t,\mathbf{x},q_{1},q_{2})$ as

$\displaystyle f_{1}^{M5}(t,\mathbf{x},0,q_{2})=0.$ (30)

Regarding the examples of reaction functions for mechanism M5 in Eq. (14),
only the second example fulfills these requirements. The first example depends
only on the T cells and can lead to a negative amount of virus if $q_{1}=0$.

Next, we regard the reaction functions of the T cells $q_{2}$. The discussion
holds as well for the subtypes T helper cells $T_{h}$ and cytotoxic T cells
$T_{c}$. The reaction function consists of two mechanisms, M2 for the
production and M6 for the decrease. The production depends on the virus and,
if a saturation effect is considered, on the amount of T cells. So,
$f_{2}^{M2}=f_{2}^{M2}(t,\mathbf{x},q_{1},q_{2})$. The decay of T cells
depends as well on the virus and on the amount of T cells. While the
dependency on $q_{1}$ is a fundamental requirement in mechanism M2, the
dependency on $q_{1}$ in M6 is optional, compare Eqn. (11) and (15). In total,
the reaction function for the T cells and the subtypes is a function
$F_{2}(t,\mathbf{x},q_{1},q_{2})$. As the function of M2 is non-negative, the
requirement

$\displaystyle F_{2}(t,\mathbf{x},q_{1},0)\geq 0$ (31)

transfers directly to a requirement on the reaction function of mechanism M6,
so

$\displaystyle f_{2}^{M6}(t,\mathbf{x},q_{1},0)\geq 0.$ (32)

All examples in (11) and (15) fulfill Eq. (32) and are therefore, concerning
the non-negativity of solutions, suitable functions. Of course, other
functions are thinkable as long as the non-negativity requirements are
fulfilled.

The reaction function $F_{3}$ of the cytokines $q_{3}$ consists of functions
for the mechanism M3 and on the natural decay, compare Eqn. (13), (16) and
(17). Mechanism M3 describes the increase of cytokines and can depend on the
virus, the T (helper) cells, and the amount of cytokines, so
$f_{3}^{M3}=f_{3}^{M3}(t,\mathbf{x},q_{1},q_{2},q_{3})$. The natural decay
depends only on the amount of cytokines, either directly like in Eq. (16), or
indirectly like in Eq. (17). In total, the reaction function $F_{3}$ of the
cytokines depends on all cells, so the non-negativity requirement reads

$\displaystyle F_{3}(t,\mathbf{x},q_{1},q_{2},0)\geq 0.$ (33)

The function $f_{3}^{M3}$ describes an increase and therefore is non-negative.
A function modeling the natural decay of cytokines should fulfill the
requirement

$\displaystyle f_{3}^{\mathrm{nd}}(t,\mathbf{x},0)=0$ (34)

to model the natural decay. Both examples in Eq. (16) and (17) fulfill the
condition.

As seen, most of the conditions on the reaction functions for preserving non-
negativity transfer directly into conditions on single reaction functions.
Most of the introduced examples for reaction functions fulfill these
conditions.

As a next step, the taxis terms are considered. We start with the diffusive
movement, described by a term $\nabla\cdot[d_{i}^{\mathrm{diff}}\nabla
q_{i}]=d_{i}^{\mathrm{diff}}\Delta q_{i}$.

The diffusive spreading is well known, for example from the heat equation. It
can not lead to a negative amount of $q_{i}$ if $q_{i}$ is non-negative for
all $\mathbf{x}\in\Omega$. This is a consequence of the leveling nature of
diffusion and can be shown by using the fundamental solution of the heat
equation after estimating the reaction function $f$ as a non-negative value,
compare [21]. The same follows in case of a diffusion term
$\nabla\cdot[d_{i}^{\mathrm{diff}}A(\mathbf{x})\nabla q_{i}]$, where $A$ is
for all $\mathbf{x}\in\Omega$ positive definite. Consequently, the non-
negativity of the solutions for the cells with only diffusive spreading is
assured.

In the case of chemotactic effects, like for the cytotoxic T cells $T_{c}$, we
show the non-negativity as well. An example of a reaction diffusion equation
for $T_{c}$ is given by

$\displaystyle
T_{c,t}=F_{T_{c}}(t,\mathbf{x},q_{1},T_{c},q_{3})+\nabla\cdot[d_{T_{c}}^{\mathrm{chem}}T_{c}\nabla
q_{3}],$

analogously for $q_{2}$. As the reaction function fulfills
$F_{T_{c}}(t,\mathbf{x},q_{1},0,q_{3})\geq 0$, the non-negativity of $T_{c}$
is proven if the divergence term does not lead to $T_{c}(t,\mathbf{x})<0$ for
any $t>0$ and any $\mathbf{x}\in\Omega$. The chemotactical term is the second
example in Eq. (18).

Under the assumption of a continuous function, $T_{c}(t,\mathbf{x})$ is zero
before it might become negative. If $T_{c}$ is negative at a time $t$ and a
place $\mathbf{x}$, then the term for the chemotactic effects is negative as
well, due to the linear dependency on $T_{c}$. So, $T_{c}$ has non-negative
values if the taxis term depends explicitly on $T_{c}$, analogously for
$q_{2}$.

The first example in Eq. (18) does not depend on $T_{c}$. This term might lead
to a negative value of $T_{c}$, for example, if $T_{c}(t,\mathbf{x})\equiv 0$
for a certain time $t$ and all $\mathbf{x}\in\Omega$. Then, depending on the
distribution of $q_{3}$, a negative diffusion $\Delta q_{3}$ is possible for a
point $\mathbf{x}$. The reaction function can be zero at this point, for
example, if $q_{1}(t,\mathbf{x})\equiv 0$ as well. It is required, that the
taxis term depends on the T cells

$\displaystyle\mathbf{D}_{T_{c}}^{\mathrm{chem}}=\mathbf{D}_{T_{c}}^{\mathrm{chem}}(T_{c},\nabla
q_{3}).$

Altogether, systems fulfilling the requirements on the reaction functions and
on the taxis terms are candidates for well-suited models w.r.t. non-negative
solutions. Of course, the requirements are only necessary and not sufficient
in all cases.

####  3.2.2 Boundedness of the solutions

The non-negativity of solutions, which is a lower bound of the solutions, was
discussed in Sec. 3.2.1. Now, an upper bound for the solutions is discussed.
The effort to show the existence of an upper bound of the solutions depends
strongly on the chosen reaction functions. Of course, mechanisms with a
negative impact on the variable do not increase an upper bound. If the
mechanisms with a positive impact on the variable are already bounded, like
some examples in Eqn. (10), (11) and (13), then the solution is bounded. If
one mechanism is unbounded, proving an upper bound requires more effort.

From a biological point of view, the existence of upper bounds is a desired
property. As the space in the liver is finite, there cannot be infinitely many
particles.

We discuss the boundedness of $q_{1}$, $q_{2}$, and $q_{3}$ separately. The
existence of upper bounds for $T_{h}$ and $T_{c}$ follows along the discussion
of $q_{2}$. In some cases, an upper bound of one variable is already required
for formulating some other mechanisms.

Again, we start with the boundedness of the virus $q_{1}$. As the virus needs
liver cells for reproduction, the reproduction is limited by the number of
liver cells in the domain $\Omega$. Additionally, there can be free virus in
the extracellular space, but again, this space is limited as well.
Consequently, a growth function of mechanism M1
$f_{1}^{M1}(t,\mathbf{x},q_{1})$ should fulfill the requirement that

$\displaystyle f_{1}^{M1}(t,\mathbf{x},C_{1})=0$ (35)

for a capacity $C_{1}>0$. Only the first example in Eq. (10) does not fulfill
this requirement. This example models an unbounded growth, which is not
biologically realistic.

The mechanism M6 depends on the existence of an upper bound for the virus.

Next, we regard the cytokines $q_{3}$. The reaction function consists of a
growth term and a natural decay. The growth function $f_{3}^{M3}$ may depend
on the virus, the T (helper) cells, and the cytokines themselves.

We start discussing the cases of $f_{3}^{M3}=f_{3}^{M3}(q_{1})$. If, as
discussed before, the virus is bounded by $C_{1}$, then the influence of
$q_{1}$ on the increase of $q_{3}$ is bounded as well. Together with the
natural decay, there will be a value $q_{3}(t,\mathbf{x})=C_{3}$ where the
natural decay equalizes the production of cytokines.

For functions $f_{3}^{M3}=f_{3}^{M3}(q_{1},q_{2},q_{3})$, the influence of
$q_{3}$ gives an upper bound through a bounded increase. In the case of
$f_{3}^{M3}=f_{3}^{M3}(q_{1},q_{3})$, this upper bound is not required due to
the interplay with the natural decay.

So far, we did not discuss the boundedness of $q_{2}$. If
$q_{2}(t,\mathbf{x})\leq C_{2}$ for all $t>0$ and all $\mathbf{x}\in\Omega$,
then there is an upper bound for $q_{3}$ as well. This follows by analogous
arguments as for the dependency of the increase only on the virus. If $q_{2}$
is unbounded, the increase of cytokines $q_{3}$ is unbounded as well.

There are two cases of increasing T cells regarding the boundedness, compare
Eq. (11). Either, the mechanism for the increase already includes a point-wise
upper bound $C_{2}$ or not. If the mechanism does not include such as
boundedness, then the discussion needs more effort, compare [22]. We refer to
this longer discussion of a maximal value and formulate as a requirement that
there is a value $\tilde{C}_{2}$ such that the time derivative
$q_{2,t}(t,\mathbf{x})$ becomes negative for
$q_{2}(t,\mathbf{x})>\tilde{C}_{2}$.

This requirement is for example fulfilled for the model in Rem. 3, see [22].

Altogether, for gaining a realistic model, all variables should be in
$L^{\infty}(\Omega_{T})$. If the boundedness of the virus is included in its
growth function, the boundedness of the T cells $q_{2}$ and the cytokines
$q_{3}$ follow in some cases.

####  3.2.3 Longtime behavior

As the solution $\mathbf{q}=\mathbf{q}(t,\mathbf{x})$ gives the amount of
virus, T cells and cytokines at a time $t$ and a space $\mathbf{x}$, the time
evolution of the solution can be interpreted as an infection course. The
question answered in this section is what types of solutions can be expected
and whether these can be predicted a priori. Solutions of reaction diffusion
equations show different behavior like a tendency towards steady states,
traveling waves, or blow-ups, compare [13].

In the light of modeling inflammations, requirements for the boundedness of
solutions were formulated. These requirements and a bounded domain $\Omega$
lead to the boundedness of solutions in any $L^{p}$-norm. Consequently, blow-
ups are neither desirable for modeling inflammations nor occurring. Solutions
of traveling waves are common for modeling atherogenesis, a certain form of
inflammation, compare [23, 24]. This type of solution is not suitable for
modeling liver infections with a chronic infection course, due to the
biological observations of more stationary inflammations, compare [2]. The
observed spread of T cells can be interpreted as a spatially inhomogeneous
stationary solution.

######  Remark 4.

We interpret solutions tending towards a stationary state which is spatially
inhomogeneous as chronic liver infections, compare [20]. Solutions tending
towards zero are interpreted as healing infection courses.

Suitable models for inflammations should show both relevant solution types,
depending on the used parameters in the model or on the domain $\Omega$.
Analytically, it is desirable to have a priori knowledge about the long-term
behavior of the solution. As the tendency towards a spatially homogeneous
steady state is a well-analyzed behavior of reaction diffusion equations,
compare for example [14], there is some hope to predict those solutions. The
theory of reaction diffusion equations with solutions tending towards a
spatially inhomogeneous steady-state solution is still an open research field.

We discuss two approaches for a priori statements on the occurrence of
decaying solutions. The first approach is based on [14, Thm. 14.17] and was as
well used in [20] in the context of a selected model for inflammations.

The theorem in [14] yields for reaction diffusion equations
${\mathbf{q}_{,t}=\mathbf{F}(\mathbf{q})+D\Delta\mathbf{q}}.$ Models with
chemotactic effects are not covered directly by this theorem. The statement of
the theorem requires some definitions.

######  Definition 5 (compare [14]).

Let $\lambda$ be the first non-zero eigenvalue to the eigenfunctions of the
negative Laplacian on $\Omega$ with zero flux boundary conditions. Further,
let $d$ be the smallest eigenvalue of the diffusion matrix $D$. Let $\Sigma$
be the invariant domain of the solutions $\mathbf{q}$,
$M=\max_{\mathbf{q}\in\Sigma}\lVert\frac{\partial\mathbf{F}}{\partial\mathbf{q}}\rVert_{\mathbb{R}^{n\times
n}}$ and $\sigma=\lambda d-M$.

######  Remark 6 (compare [14]).

The reaction diffusion system
$\mathbf{q}_{,t}=\mathbf{F}(\mathbf{q})+D\Delta\mathbf{q}$ has an invariant
domain of the solutions $\Sigma$, $\mathbf{F}$ is smooth and $D$ is positive
definite. If $\sigma>0$ yields, then

$\displaystyle\lVert\mathbf{q}(t,\mathbf{x})-\bar{\mathbf{q}}(t)\rVert_{L^{\infty}(\Omega)}\leq
c\mathrm{e}^{-\sigma t}$

where $\bar{\mathbf{q}}(t)$ is the solution of
$\bar{\mathbf{q}}_{,t}=\mathbf{F}(\bar{\mathbf{q}})+\mathbf{g}(t)$ with
$\mathbf{g}$ decaying exponentially.

Further, if $D$ is diagonal, then $\mathbf{q}$ tends in $L^{\infty}(\Omega)$
towards $\bar{\mathbf{q}}$.

Similar results with slightly different requirements can be found in [13].

Next, the requirements of Rem. 6 are checked for the models for inflammations.
As the structure of the reaction diffusion equations only considers diffusion,
models with chemotactic effects are not covered by the theorem. This is a
problem for all models with cytokines. Models with mechanisms like in the
first example of Eq. (18) can be written in the form of Rem. 6 but those do
not fulfill the conditions for non-negative solutions.

Solutions interpreted as chronic infections may occur if $\sigma<0$. This is a
necessary condition for the occurrence of solutions tending toward a spatially
inhomogeneous steady-state distribution. In [20] this is checked for a model
similar to the one in Rem. 3. In [16] some simplified models with chemotactic
effects are analyzed.

For the wide class of possible models in this paper, the theorem of Rem. 6 is
not well suited and fits only for models without chemotactic effects.

A different approach for predicting the longtime behavior are entropy methods,
compare [25]. As those methods require the definition of an entropy functional
depending on the reaction and diffusion terms, a general statement on the use
of entropy functionals is not possible. In [15] a simplified model for liver
inflammations consisting only of one variable was analyzed using entropy
methods. The main analytical result for the reduced model is that if the decay
terms of the virus overrule the growth of the virus, then only solutions
interpreted as healing infection courses occur. This result is not directly
transferable to the analysis of a two or more component model as the interplay
between the different components dominates the system behavior.

Altogether, only a few conclusions can be drawn from the existing theorems on
the longtime behavior of very general reaction diffusion systems.

The next section gives an overview of the requirements of the reaction and
taxis functions. Additionally, the requirements are checked for the three
presented models.

####  3.2.4 Summarize of the requirements

The requirements are summarized according to the cell types and the introduced
mechanisms. First, the requirements R.1 belonging to the virus are presented.

######  Remark 7.

The dynamics of the virus are described by

$\displaystyle q_{1,t}$
$\displaystyle=f_{1}^{M1}(t,\mathbf{x},q_{1})+f_{1}^{M5}(t,\mathbf{x},q_{1},q_{2})+\nabla\cdot\mathbf{D}_{1}(\mathbf{x},\nabla
q_{1})$ (36)

where $f_{1}^{M1}$, $f_{1}^{M5}$ and $\mathbf{D}_{1}$ fulfill the requirements

  1. (R.1.1)

$f_{1,q_{1}}^{M1}\geq 0$ for $q_{1}>\varepsilon>0$, compare Eq. (4),

  2. (R.1.2)

$f_{1}^{M1}(t,\mathbf{x},0)=0$, compare Eq. (29),

  3. (R.1.3)

$f_{1}^{M1}(t,\mathbf{x},C_{1})=0$ for a capacity $C_{1}>\varepsilon>0$,
compare Eq. (35)

  4. (R.1.4)

$f_{1}^{M5}(t,\mathbf{x},0,q_{2})\geq 0$, compare Eq. (5),

  5. (R.1.5)

$f_{1,q_{2}}^{M5}\leq 0$, compare Eq. (29),

  6. (R.1.6)

if $\mathbf{D}$ has the form $A\nabla q_{1}$, then $A$ is positive definite.

If the general T cells are divided into T helper cells and cytotoxic T cells,
the dependency of $f_{2}^{M5}$ on $q_{2}$ is replaced by a dependency on the
cytotoxic T cells $T_{c}$.

######  Remark 8.

The dynamics of the T cells are described by

$\displaystyle q_{2,t}$
$\displaystyle=f_{2}^{M2}(t,\mathbf{x},q_{1},q_{2})+f_{2}^{M6}(t,\mathbf{x},q_{1},q_{2})+\nabla\cdot\mathbf{D}_{2}(\mathbf{x},q_{2},\nabla
q_{2},\nabla q_{3})$ (37)

which can be adapted for only T helper cells ($T_{h}$ instead of 2 as index)
or cytotoxic T cells ($T_{c}$ instead of 2 as index). In any case, the
functions $f_{2}^{M2}$, $f_{2}^{M6}$ and $\mathbf{D}_{2}$ fulfill the
requirements

  1. (R.2.1)

$f_{2,q_{1}}^{M2}\geq 0$, compare Eq. (6),

  2. (R.2.2)

$f_{2}^{M2}(t,\mathbf{x},q_{1},q_{2})\geq 0$,

  3. (R.2.3)

$f_{2,q_{2}}^{M6}(t,\mathbf{x},q_{1},q_{2})\leq 0$, compare Eq. (6)

  4. (R.2.4)

$f_{2}^{M6}(t,\mathbf{x},q_{1},0)\geq 0$, compare Eq. (32),

  5. (R.2.5)

there exists a value $C_{2}$ with $q_{2,t}\leq 0$ for all $q_{2}>C_{2}$,

  6. (R.2.6)

if $\mathbf{D}_{2}$ has a function
$\mathbf{D}_{2}^{\mathrm{ecs}}(t,\mathbf{x},\nabla q_{2})$ with $A\nabla
q_{2}$, then $A$ is positive definite,

  7. (R.2.7)

if $\mathbf{D}_{2}$ has a function
$\mathbf{D}_{2}^{\mathrm{chem}}(q_{2},\nabla q_{3})$, then the dependency on
$q_{2}$ is obligatory.

As a last quantity, the cytokines are regarded.

######  Remark 9.

The dynamics of the cytokines $q_{3}$ are described by

$\displaystyle q_{3,t}$
$\displaystyle=f_{3}^{M3}(t,\mathbf{x},q_{1},q_{2},q_{3})+f_{3}^{\mathrm{nd}}(t,\mathbf{x},q_{3})+\nabla\cdot\mathbf{D}_{3}(\mathbf{x},\nabla
q_{3})$ (38)

where $f_{3}^{M3}$, $f_{3}^{\mathrm{nd}}$ and $\mathbf{D}_{3}$ fulfill the
requirements

  1. (R.3.1)

$f_{3,q_{1}}^{M3}\geq 0$ and $f_{3,T_{h}}^{M3}\geq 0$, compare Eq. (7),

  2. (R.3.2)

$f_{3}^{M3}(t,\mathbf{x},q_{1},q_{2},q_{3})\geq 0$,

  3. (R.3.3)

$f_{3}^{\mathrm{nd}}(t,\mathbf{x},q_{3})\leq 0$ for all $q_{3}\geq 0$,

  4. (R.3.4)

$f_{3}^{\mathrm{nd}}(t,\mathbf{x},0)=0$, compare Eq. (34),

  5. (R.3.5)

there exists a value $C_{3}$ with $q_{3,t}\leq 0$ for all $q_{3}>C_{3}$,

  6. (R.3.6)

if $\mathbf{D}_{3}$ has the form $A\nabla q_{3}$, then $A$ is positive
definite.

If the T cells are not split up, the partial derivative in (R.3.1) with
respect to $T_{h}$ is replaced by the partial derivative with respect to
$q_{2}$.

The requirements in Rem. 7, 8, 9 are reasonable conditions on the function of
a thinkable model for inflammations. Of course, they are neither sufficient
nor necessary for formulating a model describing inflammation in a general
context.

For the first application, the requirements are checked for the models 1,2 and
3.

######  Remark 10.

Model 1 (Rem. 1) fulfills all requirements (R.1), (R.2), (R.3).

######  Remark 11.

Model 2 (Rem. 2) fulfills all requirements (R.1), (R.2) and (R.3) under the
assumption that $q_{1}\leq C_{1}$. Due to the requirements, the boundedness of
the virus is preserved by the dynamics if the initial conditions are locally
bounded by $C_{1}$. The boundedness of the T cells $q_{2}$ is a consequence of
the observations on the two-component model in [22].

######  Remark 12.

Model 3 in Rem. 3 fulfills the requirements (R.1) and (R.2). The boundedness
of the T cells $q_{2}$ was shown in [22].

In total, all three models fulfill the requirements, even though, in some
cases, the proof of the boundedness is more complicated and a priori
predictions are not always possible.

###  3.3 Numerical results

Simulations provide a visual understanding of models and possible solutions.
In this section, simulations for the models 1-3 are presented.

The simulations use a semi-discretization of the space coordinates for
$\Omega=(0,1)\times(0,1)\subset\mathbb{R}^{2}$. Afterward, standard solvers
for ordinary differential equations are used. As the motion terms include
Laplacians, the resulting ordinary differential equations of the space
discretization may become stiff due to the growing quotient of the smallest
and the largest eigenvalue of the discretized Laplacian. The number of
discretization points is fixed by $\Delta x=0.05$ and, in case of long
calculation time for solving the ordinary differential equations, a solver for
stiff equations is used.

This section focuses on two points. First, the two solution types interpreted
as chronic and as healing infection courses are presented. Second, there is a
comparison of the three models using two to four different cell types. A
leading question is whether even a model covering only two of four cell types
can provide any insight.

The initial conditions are chosen as $q_{1}(0,\mathbf{x})\equiv 1$,
$q_{2}(0,\mathbf{x})=T_{c}(0,\mathbf{x})=T_{h}(0,\mathbf{x})\equiv 0$ and
$q_{3}(0,\mathbf{x})\equiv 0.1$ in all simulations. The function
$\chi_{\Theta}(\mathbf{x})$ is the normalized characteristic function for
$\Theta$. Most of the parameters are fixed for all simulations, see Table 1.

Table 1: Parameter values for all three models which are chosen as constant if the mechanism is included.  $a_{1}$ | $C_{1}$ | $\varepsilon$ | $\kappa$ | $d_{1}^{\mathrm{ctc}}$ | $a_{2,h}$ | $C_{T_{h}}$ | $a_{6}$ | $d_{T_{h}}^{\mathrm{diff}}$ | $C_{T_{c}}$ | $a_{3}$ | $a_{\mathrm{nd}}$ | $d_{3}^{\mathrm{diff}}$
---|---|---|---|---|---|---|---|---|---|---|---|---
1 | 1 | 0.05 | 0.01 | 0.6 | 2 | 8 | 0.2 | 0.9 | 15 | 0.8 | 0.6 | 0.5

Model 1 shows solutions with a tendency towards zero and solutions with a
tendency towards a spatial inhomogeneous steady state, depending on one
parameter. In this case, the parameter $d_{T_{c}}^{\mathrm{diff}}$ changes the
behavior, see Fig. 2 and Table 2 for the used parameter values. The spread of
the virus is modeled by diffusive cell-to-cell transmission. Extracellular
movement through a blood flow is not regarded, $d_{1}^{\mathrm{ecs}}=0$.

Figure 2: Model 1 shows solutions interpreted as healing infection courses (upper row) and as chronic infections (lower row). Different time steps are in different shadings from $t_{0}=0$ in light over $t_{1}=16$ to $t_{2}=40$ in dark.  Table 2: Parameter values for the changed parameters in bold.  | model 1 | model 2 | model 3
---|---|---|---
| $a_{5}$ | $a_{2,c}$ | $d_{T_{c}}^{\mathrm{chem}}$ | $a_{5}$ | $a_{2}$ | $d_{2}^{\mathrm{chem}}$ | $a_{5}$ | $a_{2}$
healing | 2 | 2 | 1 | 1 | 2 | 1 | 0.5 | 2
chronic | 2 | 2 | 8 | 0.5 | 2 | 1 | 0.5 | 0.7

The simulations of model 1 show that there is a remarkable difference in the
modeled spreading of T helper cells $T_{h}$ and cytotoxic T cells $T_{c}$.
While the spread of T helper cells is modeled by diffusion in model 1, compare
Rem. 1, the motion of cytotoxic T cells is modeled by chemotaxis.
Consequently, the cytotoxic T cells $T_{c}$ in Fig. 2(h) spread out less and
the immune reaction is less effective in reducing the viral load. In the
chronic case, the number of T helper cells $T_{h}$ reduces after an active
phase but remains high for all time.

In model 2, there is only one type of T cells spreading out by diffusion and
following the gradient of the cytokines. The parameter under variation in this
case is $a_{5}$, see Table 2, which describes the effectiveness of the T
cells. Fig. 3 shows a solution interpreted as chronic infection course. The
virus remains in the whole liver with a spatial inhomogeneous spreading. The
cytokines remain at a high level everywhere.

Figure 3: Model 2 shows solutions interpreted as chronic infection courses.
Different time steps are in different shadings from $t_{0}=0$ in light over
$t_{1}=32$ to $t_{2}=80$ in dark. The portal field $\Theta$ is in the right
corner in (b).

The changed parameter for model 3 is $a_{2}$, see Table 2, which regulates the
inflow of T cells through the portal field in dependency on the total virus at
a time $t$. The results of the simulation of model 3 are comparable to the
simulation results of models 1 and 2. A comparison of all three models for a
chronic course is in Fig. 4. As the T cells $q_{2}$ are a sum of the T helper
cells $T_{h}$ and the cytotoxic T cells $T_{c}$, Fig. 4 compares the $L_{1}$
norms of $q_{1}$ in (b) of $q_{2}$ and $T_{h}+T_{c}$.

Figure 4: $L_{1}$ norms of the solutions of model 1, 2 and 3 for the parameter
sets in Table 2 interpreted as chronic infection course. (a) total virus (b)
total T cells.

The amount of virus tends in all three models to comparable values due to the
small changes of parameters. The total amount of T cells is comparable for
models 2 and 3 but different for model 1 due to the differentiation between T
helper cells and cytotoxic T cells with different maximal values. The curves
of models 2 and 3 show small oscillations in the first time but a damping
towards a constant value later on.

###  3.4 Clinical impact

The mathematical model for liver inflammation developed and described in this
thesis may have significant clinical implications as it provides valuable
insights into the underlying dynamics and mechanisms of liver inflammation
caused by viruses. Some potential clinical improvements include a deeper
understanding of the disease and its progression, supporting clinical
decisions through early detection and diagnosis, and optimizing treatment. All
of this could contribute to personalized diagnosis and treatment of viral
hepatitis in the future. More specifically, this mathematical model can help
to detect subtle changes in influencing variables or physiological parameters
associated with liver inflammation at an early stage. This early detection
allows for rapid intervention and treatment that can potentially prevent the
progression of liver inflammation to cirrhosis. Such predictive models can
help physicians use their resources more efficiently by identifying patients
who are likely to require more intensive monitoring or intervention. This is
of paramount importance in healthcare systems with limited resources.
Mathematical models provide a platform for simulating the dynamic processes
involved in liver inflammation. Understanding disease progression at a
mechanistic level can contribute to the development of new therapeutic targets
and strategies or prevent inflammation from becoming chronic. More and more
patients are demanding shared decision making, where the visualization and
explanation derived from the mathematical model can be used to educate
patients about their disease. Better patient understanding can improve
treatment adherence and lifestyle change, which has a positive impact on long-
term outcomes. In summary, a mathematical model for liver inflammation can
provide clinicians with valuable tools to improve diagnosis, treatment
planning, and patient outcomes. Its integration into clinical practice has the
potential to usher in a new era of precision medicine for liver disease.

##  4 Conclusions

A deductive modeling approach was presented for the application of liver
infections leading to inflammations. This life science application has unknown
mechanisms leading to chronic infection courses for which only qualitative
data is available. Building up a model family allows to gain a deeper
understanding of the involved mechanisms. The model family consists of
reaction diffusion equations and the reaction or taxis functions can be chosen
from a class of feasible functions. The feasible classes fulfill different
biologically motivated properties. Analytical investigations restricted the
function classes further.

In the case of inflammation modeling, three models were chosen out of the
model family and all three models reproduce both different infection courses
depending on some parameter values. Consequently, already the smallest of the
three models is able to cover basic observations even if less information is
used than in the more complex models of the model family. Depending on the
modeling purpose, a suitable model from the model family can be chosen.

##  5 Acknowledgments

HM Tautenhahn disclosed receipt of the following financial support for the
authorship of this article: This work was supported by:

Federal Ministry of Education and Research; ATLAS [031L0304C] HM Tautenhahn

German Research Foundation SIMLIVA [465194077] HM Tautenhahn

German Research Foundation [428832822] HM Tautenhahn

This research did not contain any studies involving animal or human
participants, nor did it take place on any private or protected areas. No
specific permissions were required for corresponding locations.

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