# Structure and Hierarchy of Influenza Virus Models Revealed by Reaction Network Analysis

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## Abstract

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## 1. Introduction

## 2. Materials and Methods: Procedure for the Organizational Analysis

- Uninfected (target) cells or those resistant/refractory to infection are marked in blue, e.g., T.
- Infected cells, partially or latently infected cells, and viruses are marked in magenta, e.g., I and V.
- Bacterial co-infection species are marked in orange. These species are only occurring in Smith’s model [15].
- Text referring to any other species is marked in black, e.g., transient target cell states, passive immune system, or dead cells.

#### 2.1. Deriving the Reaction Network from the ODE System

- The term $\beta TV$ represents the reaction ${R}_{1}:\phantom{\rule{0.277778em}{0ex}}T+V\to I+V$, which in turn denotes the transformation of an uninfected target cell T to an infected cell I catalysed by the virus V.
- The terms $-\delta I$ and $-cV$ represent reactions ${R}_{2}:\phantom{\rule{0.277778em}{0ex}}I\to \varnothing $ and ${R}_{4}:\phantom{\rule{0.277778em}{0ex}}V\to \varnothing $ which are the outflow of infected cells I resp. virus V.
- The term $pI$ represents the reaction ${R}_{3}:\phantom{\rule{0.277778em}{0ex}}I\to I+V$ which is the production of viruses V catalysed by infected cells I.

- Single underline for the transformation of uninfected cells into infected ones by the action of viruses.
- $\underset{=}{\text{Double underline}}$ for kinetic terms involving interferon.

#### 2.2. Computing the Organizations from the Reaction Network

**closed**if and only if all reactions $R\in \mathcal{R}$ with $supp\left(R\right)\subseteq S$ fulfill $prod\left(R\right)\subseteq S$ too [10,18]. This means that the products of a reaction R with support in S are also in S. In other words, no species outside of S can be produced by the reactions “running on” S. As an example, we assume $S=\{T,I\}$. The reactions with support in S are ${R}_{2}$ and ${R}_{3}$. However, ${R}_{3}$ produces species V, which is not in S. Thus, S is not closed.

**self-maintaining**if and only if there exists (at least one) flux vector $\mathbf{v}\in {\mathbb{R}}^{n}$ for S that fulfills

#### 2.3. The Role Organizations Play in the Dynamics

## 3. Results and Discussion

#### 3.1. Target Cell Limited Model by Miao et al. (Miao Model, M, 2010)

#### 3.2. Target Cell Limited Model with Delayed Virus Production (Baccam II Model, Ba2, 2006)

#### 3.3. Innate and (Simple) Adaptive Immune Response (Pawelek Model, P, 2012)

- The rate term $\varphi FT$ represents the transformation of uninfected target cells to refractory cells catalysed by interferon.
- The reverse shift back from refractory to simple uninfected cells is represented by the term $\rho R$.
- Furthermore, infected cells are deleted by the action of interferon at a rate $\kappa IF$.
- Interferon is produced in the presence of infected cells at a rate $qI$.

#### 3.4. A Model Including Bacterial Co-Infection (Smith Model, Sm, 2016)

#### 3.5. Innate and Adaptive Immune Response (Handel Model, Ha, 2009)

- Infection is catalyzed by viruses V and transforms uninfected cells U to latently infected cells E and viruses V are consumed thereby. Latently infected cells E transform into infected cells I autonomously, which in turn transform into dead cells D autonomously too. Finally, the transformation of dead cells D into non-infected cells U closes the circle.
- The remaining three species V, F and X form an almost totally separate subsystem since the only interaction with the four species from the "circle" mentioned above is the catalysis of the infection by viruses V.
- The interactions within the subsystem $\{V,F,X\}$ consisting of viruses V and immune responses F and X are as follows:
- –
- Viruses V catalyze the proliferation of F and X. In the Hernandez model, proliferation of interferon F is catalyzed by infected cells instead of viruses.
- –
- There is no direct interaction between innate immune response F and adaptive immune response X.
- –
- The adaptive immune response X deletes viruses directly. Innate immune response F inhibits the self-replication of the viruses which is represented by the denominator of the fraction $\frac{p{\mathbf{I}}}{1+\kappa {\mathbf{F}}}$. We ignore the inhibition because whether the rate is zero or not is independent of F.

#### Temporal Dynamics

#### 3.6. Innate Immune Response and Resistance to Infection (Hernandez Model, He, 2012)

- There is an infection reaction catalyzed by viruses like in all previous models but with one difference: during infection, healthy cells ${U}_{H}$ first transform to partially infected cells ${U}_{E}$ and only after that they transform spontaneously to infected cells ${U}_{I}$ at a rate ${k}_{e}{U}_{E}$.
- Interferon catalyzes the transformation of healthy cells to resistant cells ${U}_{R}$, like in the Pawelek Model. However, in the Pawelek Model, interferon removes infected cells. Here, interferon’s production is catalyzed by infected cells ${U}_{I}$ at a rate ${a}_{I}{U}_{I}$. There is no further influence of interferon on any other species.
- Infected cells are removed by natural killers K, which also delete partially infected cells in this model. The production of killers K is catalyzed by infected cells ${U}_{I}$ at a rate ${\mathsf{\Phi}}_{K}{U}_{I}$.
- Note that here we have an constant inflow of healthy cells ${U}_{H}$ at a rate ${S}_{H}$ (first differential equation). Thus, healthy cells cannot converge to zero.

#### 3.7. A Model with More Detailed Immune Response (Cao Model, C, 2015)

#### 3.8. Innate Immune Response and Eclipse Phase (Saenz Model, Sa, 2010)

- The “full” organization is missing here. For sure, one of the reasons is that there is no reaction producing susceptible cells T. Thus, when viruses V or interferon F are present susceptible cells T can not survive and the “full” organization neither.
- Adaptive immune response is replaced by refractory cells in the organizations here.

#### 3.9. Focusing on Innate and Adaptive Immunity (Hancioglu Model, Hcg, 2007)

- The smallest one is ${O}_{1}^{Hcg}$, which contains all the species responsible for the immune response.
- ${O}_{1}^{Hcg}$ is a subset of ${O}_{2}^{Hcg}$, which additionally contains species H and R, representing the healthy organism without infection but with the immune response turned on.
- ${O}_{3}^{Hcg}$ is the “full” organization containing all the species of the models and thus representing the organism with infection and immune response.

#### 3.10. Model with Delay Differential Equations (Bocharov Model, Bo, 1994)

#### 3.11. Complex Dual-Compartment Model (Lee Model, L, 2009)

#### 3.12. Hierarchy of Influenza A Virus Models

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## Appendix A

**Figure A1.**Reactions of Baccam model [13].

**Figure A2.**Reactions of Miao model [14].

**Figure A3.**Reactions of Baccam model [14] with delayed virus production.

**Figure A4.**Reactions of Pawelek model [23].

**Figure A5.**Reactions of Handel model [24].

**Figure A6.**Reactions of Hernandez model [26].

**Figure A7.**Reactions of Saenz model [28].

**Figure A8.**Reactions of Hancioglu model [29].

**Figure A9.**Reactions of Bocharov model [30].

**Figure A10.**Reactions of Lee model [9].

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**Figure 1.**Relation between measured data, ordinary differential equations (ODE) model, and hierarchy of organizations.

**Figure 2.**The

**Baccam Model**[13] with three variables: uninfected (susceptible) target cells (${\mathbf{T}}$), infected cells (${\mathbf{I}}$) and infectious-viral titer (${\mathbf{V}}$).

**Figure 3.**The

**Miao Model**[14] with three variables: uninfected target cells (${{\mathbf{E}}}_{{\mathbf{P}}}$), productively infected cells (${{\mathbf{E}}}_{{\mathbf{P}}}^{{*}}$) and free infectious influenza viruses (${\mathbf{V}}$).

**Figure 4.**The

**Baccam II Model**[13] with delayed virus production and four variables: uninfected (susceptible) target cells (${\mathbf{T}}$), infected cells not yet producing virus (${{\mathbf{I}}}_{{\mathbf{1}}}$), infected cells actively producing virus (${{\mathbf{I}}}_{{\mathbf{2}}}$) and infectious-viral titer (${\mathbf{V}}$).

**Figure 5.**The

**Pawelek Model**[23] with five variables: (uninfected) target cells (${\mathbf{T}}$), productively infected cells (${\mathbf{I}}$), uninfected cells refractory to infections (${\mathbf{R}}$), free viruses (${\mathbf{V}}$) and interferon (${\mathbf{F}}$).

**Figure 6.**The

**Smith Model**[15] with five variables: susceptible target cells (${\mathbf{T}}$), two classes of infected cells (${{\mathbf{I}}}_{{\mathbf{1}}}$ and ${{\mathbf{I}}}_{{\mathbf{2}}}$), free viruses (${\mathbf{V}}$), and bacteria (${\mathbf{P}}$).

**Figure 7.**The

**Handel model**[24] with seven variables: uninfected cells (${\mathbf{U}}$), latently infected cells (${\mathbf{E}}$), productively infected cells (${\mathbf{I}}$), dead cells (D), free viruses (${\mathbf{V}}$), innate immune response (${\mathbf{F}}$) and adaptive immune response (${\mathbf{X}}$). The dotted arrows denote the projection of the dynamics shown in Figure 8.

**Figure 8.**Temporal dynamics of the Handel model. By projecting the seven-dimensional trajectory to organizations (dotted arrows in Figure 7b) we find three phases: (Phase 1) Until day number 0, there are solely $7\times {10}^{9}$ uninfected cells ${\mathbf{U}}$ in the system represented by the organization ${O}_{2}^{Ha}=\left\{{\mathbf{U}}\right\}$. (Phase 2) At day 0,

**infection**is simulated by adding $V\left(0\right)={10}^{4}$ virus particles to the system. The resulting state $\{{\mathbf{U}},{\mathbf{V}}\}$ is projected to organization ${O}_{5}^{Ha}$ (all species). (Phase 3) Lastly, at day t = 37d past infection the system settles in the final organization, namely ${O}_{4}^{Ha}=\{{\mathbf{U}},{\mathbf{X}}\}$, which is generated by the set $\{{\mathbf{U}},{\mathbf{X}},D\}$ (see text). The values of the model parameters are (from [24]): $\lambda =0.25$, $b=2.1\times {10}^{-7}$, $g=4$, $d=2$, $p=5\times {10}^{-2}$, $\kappa =1.8\times {10}^{-2}$, $c=10$, $\gamma =7.5\times {10}^{-4}$, $k=1.8$, $w=1$, $\delta =0.4$, $f=2.7\times {10}^{-6}$, and $r=0.3$. Note that the number of uninfected cells ${\mathbf{U}}$ is not constant after infection as it may seem from the figure. In fact, after infection, the number of uninfected cells first decreases and than rises again [24].

**Figure 9.**The

**Hernandez Model**[26] with seven variables: healthy cells (${{\mathbf{U}}}_{{\mathbf{H}}}$), partially infected cell (${{\mathbf{U}}}_{{\mathbf{E}}}$), infected cells (${{\mathbf{U}}}_{{\mathbf{I}}}$), cells resistant to infection (${{\mathbf{U}}}_{{\mathbf{R}}}$), virus particles (${\mathbf{V}}$), interferon (${\mathbf{F}}$) and natural killers (${\mathbf{K}}$).

**Figure 10.**The

**Cao Model**[27] with seven variables: target cells (${\mathbf{T}}$), infected cells (${\mathbf{I}}$), viruses (${\mathbf{V}}$), resistant cells (${\mathbf{R}}$), interferon (${\mathbf{F}}$), B cells (${\mathbf{B}}$), and antibodies (${\mathbf{A}}$).

**Figure 11.**The

**Saenz Model**[28] with eight variables: Epithelial cells in one of the states: susceptible (${\mathbf{T}}$), eclipse phase (${{\mathbf{E}}}_{{\mathbf{1}}}$ and ${E}_{2}$), prerefractory (W), refractory (${\mathbf{R}}$) and infectious (${\mathbf{I}}$). The further variables are: virus cells (${\mathbf{V}}$) and interferon (${\mathbf{F}}$).

**Figure 12.**The

**Hancioglu Model**[29] with 10 variables: viral load (${\mathbf{V}}$), healthy cells (${\mathbf{H}}$), infected cells (${\mathbf{I}}$), antigen presenting cells (M), interferon (${\mathbf{F}}$), resistant cells (${\mathbf{R}}$), effector cells (${\mathbf{E}}$), plasma cells (${\mathbf{P}}$), antibodies (${\mathbf{A}}$) and antigenic distance (S).

**Figure 13.**The

**Bocharov Model**[30] with 10 variables: infective IAV particles (${{\mathbf{V}}}_{{\mathbf{f}}}$), IAV-infected cells (${\mathbf{C}}$), destroyed epithelial cells (m), stimulated macrophages (${{\mathbf{M}}}_{{\mathbf{V}}}$), activated helper T cells providing proliferation of cytotoxic T cells (${{\mathbf{H}}}_{{\mathbf{E}}}$), activated helper T cells providing proliferation and differentiation of B cells B (${{\mathbf{H}}}_{{\mathbf{B}}}$), activated CTL (${\mathbf{E}}$), B cells (${\mathbf{B}}$), plasma cells (${\mathbf{P}}$), antibodies to IAV (${\mathbf{F}}$), and uninfected epithelial cells (${\mathbf{U}}$). Note that, for clarity, we have added ${\mathbf{U}}$ as a state variable, which is only implicitly represented as ${\mathbf{U}}={C}^{*}-{\mathbf{C}}-m$ in the original model by Bocharov et al.

**Figure 14.**The

**Lee model**[9] which contains 15 variables: uninfected epithelial cells (${{\mathbf{E}}}_{{\mathbf{P}}}$), infected epithelial cells (${{\mathbf{E}}}_{{\mathbf{P}}}^{{*}}$), virus titer ($EI{D}_{50}/ml$) (${\mathbf{V}}$), immature dendritic cells (${\mathbf{D}}$), virus-loaded dendritic cells (${{\mathbf{D}}}^{{*}}$), mature dendritic cells (${{\mathbf{D}}}_{{\mathbf{M}}}$), naive CD4+ T cells (${H}_{N}$), effector CD4+ T cells (${{\mathbf{H}}}_{{\mathbf{E}}}$), naive CD8+ T cells (${T}_{N}$), effector CD8+ T cells (${{\mathbf{T}}}_{{\mathbf{E}}}$), naive B cells (${B}_{N}$), activated B cells (${{\mathbf{B}}}_{{\mathbf{A}}}$), short-lived plasma (antibody-secreting) B cells (${{\mathbf{P}}}_{{\mathbf{S}}}$), long-lived plasma (antibody-secreting) B cells (${{\mathbf{P}}}_{{\mathbf{L}}}$) and antiviral antibody titer (${\mathbf{A}}$). Note that here we have colored green only those species representing the immune system when activated.

**Figure 15.**Hasse-diagram of the hierarchy of IAV models with respect to their long-term behaviour. In brackets (), we added the number of species of each model. Underneath (marked by colors) the kinds of species contained in the organizations belonging to each model. The meaning of the four colors is as follows: Species belonging to the healthy state of the organism are colored blue, those belonging to the immune response are colored green, those belonging to infection like infected cells and viruses are colored magenta, and bacteria from bacterial co-infection are colored orange. Horizontally, the diagram consists of four lines. The models in the lowest line contain organizations with exactly two different kinds of species (colors) (including the empty set). In the second line above, there are three different combinations of species (colors) to be found in each model. There is only one model in each of the highest two lines: The Smith model [4] is the only one with bacteria and contains four different combinations of colors. In the Handel Model, there are even five different combinations of colors out of ${2}^{4}=16$ possible combinations.

**Table 1.**Overview of all models and organization types contained. An organization type like ${\mathbf{X}}{\mathbf{X}}$ denotes the type of species contained in an organization, according to our coloring scheme. The set of organization types of a model is called its signature.

Model | Number of Variables | Number of Reactions | Number of Organizations | Organizations & Signature | |||
---|---|---|---|---|---|---|---|

Baccam [13] 2006 | 3 | 4 | 2 | ${O}_{1}^{Ba1}=\varnothing $ | |||

${O}_{2}^{Ba1}$ | X | ||||||

Miao [14] 2010 | 3 | 5 | 3 | ${O}_{1}^{M}=\varnothing $ | |||

${O}_{2}^{M}$ | X | ||||||

${O}_{3}^{M}$ | X | X | |||||

Baccam II [13] 2006 | 4 | 5 | 2 | ${O}_{1}^{Ba2}=\varnothing $ | |||

${O}_{2}^{Ba2}$ | X | ||||||

Pawelek [23] 2012 | 5 | 9 | 2 | ${O}_{1}^{P}=\varnothing $ | |||

${O}_{2}^{P}$ | X | ||||||

Smith [15] 2016 | 5 | 12 | 4 | ${O}_{1}^{Sm}=\varnothing $ | |||

${O}_{2}^{Sm}$ | X | ||||||

${O}_{3}^{Sm}$ | X | ||||||

${O}_{4}^{Sm}$ | X | X | |||||

Handel [24] 2010 | 7 | 12 | 5 | ${O}_{1}^{Ha}=\varnothing $ | |||

${O}_{2}^{Ha}$ | X | ||||||

${O}_{3}^{Ha}$ | X | ||||||

${O}_{4}^{Ha}$ | X | X | |||||

${O}_{5}^{Ha}=\left\{all\right\}$ | X | X | X | ||||

Hernandez [26] 2012 | 7 | 16 | 2 | ${O}_{1}^{He}$ | X | X | |

${O}_{2}^{He}=\left\{all\right\}$ | X | X | X | ||||

Cao [27] 2015 | 7 | 26 | 3 | ${O}_{1}^{C}=\varnothing $ | |||

${O}_{2}^{C}$ | X | ||||||

${O}_{3}^{C}=\left\{all\right\}$ | X | X | X | ||||

Saenz [28] 2010 | 8 | 12 | 4 | ${O}_{1}^{Sa}=\varnothing $ | |||

${O}_{2}^{Sa}$, ${O}_{3}^{Sa}$, ${O}_{4}^{Sa}$ | X | ||||||

Hancioglu [29] 2007 | 10 | 44 | 3 | ${O}_{1}^{Hcg}$ | X | ||

${O}_{2}^{Hcg}$ | X | X | |||||

${O}_{3}^{Hcg}=\left\{all\right\}$ | X | X | X | ||||

Bocharov [30] 1994 | 10 | 45 | 2 | ${O}_{1}^{Bo}$ | X | ||

${O}_{2}^{Bo}$ | X | X | |||||

Lee [9] 2009 | 15 | 37 | 8 | ${O}_{1}^{L}$ | X | ||

${O}_{2}^{L}$, ${O}_{3}^{L}$, ${O}_{4}^{L}$, ${O}_{5}^{L}$, ${O}_{6}^{L}$ | X | X | |||||

${O}_{7}^{L}$, ${O}_{8}^{L}=\left\{all\right\}$ | X | X | X |

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**MDPI and ACS Style**

Peter, S.; Hölzer, M.; Lamkiewicz, K.; di Fenizio, P.S.; Al Hwaeer, H.; Marz, M.; Schuster, S.; Dittrich, P.; Ibrahim, B.
Structure and Hierarchy of Influenza Virus Models Revealed by Reaction Network Analysis. *Viruses* **2019**, *11*, 449.
https://doi.org/10.3390/v11050449

**AMA Style**

Peter S, Hölzer M, Lamkiewicz K, di Fenizio PS, Al Hwaeer H, Marz M, Schuster S, Dittrich P, Ibrahim B.
Structure and Hierarchy of Influenza Virus Models Revealed by Reaction Network Analysis. *Viruses*. 2019; 11(5):449.
https://doi.org/10.3390/v11050449

**Chicago/Turabian Style**

Peter, Stephan, Martin Hölzer, Kevin Lamkiewicz, Pietro Speroni di Fenizio, Hassan Al Hwaeer, Manja Marz, Stefan Schuster, Peter Dittrich, and Bashar Ibrahim.
2019. "Structure and Hierarchy of Influenza Virus Models Revealed by Reaction Network Analysis" *Viruses* 11, no. 5: 449.
https://doi.org/10.3390/v11050449