# Computational Model Informs Effective Control Interventions against Y. enterocolitica Co-Infection

^{1}

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^{*}

## Abstract

**:**

## Simple Summary

## Abstract

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Model Description

#### 2.2. The Basic Reproduction Number ${\mathcal{R}}_{0}$

- A(1)
- ${\mathcal{F}}_{i}(0,y)=0,\phantom{\rule{0.277778em}{0ex}}{\mathcal{V}}_{i}(0,y)=0:\forall y>0\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{and}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}i=1,\dots ,n$ (no immigration of individuals into the disease compartments)
- A(2)
- ${\mathcal{F}}_{i}(x,y)\ge 0:\forall {x}_{i}\ge 0\wedge {y}_{i}\ge 0\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{and}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}i=1,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\dots ,n$ (the new infections will be represented by $\mathcal{F}$, so it cannot be negative)
- A(3)
- ${\mathcal{V}}_{i}(x,y)\le 0:\phantom{\rule{0.277778em}{0ex}}\mathrm{whenever}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{x}_{i}=0,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}i=1,\dots ,n$ (if the compartment is empty, it can only have inflow, and the net outflow from the compartment must be negative)
- A(4)
- ${\sum}_{i}{\mathcal{V}}_{i}(x,y)\ge 0:\forall {x}_{i}\ge 0\wedge {y}_{i}\ge 0$ (sum is net outflow)
- A(5)
- The system $\dot{y}=\mathcal{G}(0,y)$ has a unique asymptotically stable equilibrium, ${y}^{*}$ (all solutions with initial conditions of the form $(0,y)$ approach a point $(0,{y}^{*})\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\mathrm{as}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}t\to \infty $)

**Theorem**

**1.**

**Proof.**

#### 2.3. Center Manifold

**Theorem**

**2**(Center Manifold Theorem for Flows)

**.**

**Proposition**

**1.**

- H(1)
- In the balance equations for the infected compartments, nonlinear terms are present only in the rate of the appearance of new infections;
- H(2)
- Nonlinear terms are bilinear;
- H(3)
- There is no linear transfer from infected to uninfected compartments.

## 3. Results

#### 3.1. Existence of Equilibria

- The trivial equilibrium point is as an origin equilibrium $\left(0,0,0,0,0,0,0\right)$. This solution appears when all populations are extinct. For all parameters, this point never becomes stable due to the positivity of eigenvalues in (A2).
- The first equilibrium point appears in the absence of Yersinia ${Y}_{M}^{\left(wt\right)}={Y}_{M}^{\left(mut\right)}={Y}_{L}^{\left(wt\right)}={Y}_{L}^{\left(mut\right)}=0$. System (1)–(7) has a disease-free equilibrium, which is given by$${P}^{0}=\left({B}_{M}^{0},{Y}_{M}^{0\left(wt\right)},{Y}_{M}^{0\left(mut\right)},{B}_{L}^{0},{Y}_{L}^{0\left(wt\right)},{Y}_{L}^{0\left(mut\right)},{I}^{0}\right)=\left({B}_{M}^{0},0,0,{B}_{L}^{0},0,0,{I}^{0}\right)$$$$\begin{array}{cc}\hfill {B}_{M}^{0}& ={C}_{M}\left(1-\frac{\gamma}{{\alpha}^{\left(B\right)}}\right)\hfill \end{array}$$$$\begin{array}{cc}\hfill {B}_{L}^{0}& ={C}_{L}\left(1-\frac{\beta}{{\alpha}^{\left(B\right)}}\right)\hfill \end{array}$$$$\begin{array}{cc}\hfill {I}^{0}& =1.\hfill \end{array}$$It describes a disease-free state whereby only the commensal bacteria persist. In order for the disease-free state ${P}^{0}$ to be biologically meaningful, the conditions $\gamma <{\alpha}^{\left(B\right)}$ and $\beta <{\alpha}^{\left(B\right)}$ must hold. These conditions correspond to the maximal growth rate of intestinal bacteria exceeding the rate at which intestines are charged and the maximal immunity action, which is not that strong in the absence of Yersinia strains. However, the population of the immune system is at its maximum carrying capacity (in health, not in fighting with any infection).
- A second equilibrium corresponds to the commensal bacteria’s persistence and the Yersinia mut strain in the absence of the wt strain. Without loss of generality, the commensal bacteria are supposed to be zero because they are not infective. This point is obtained by setting ${Y}_{M}^{\left(wt\right)}={Y}_{L}^{\left(wt\right)}=0$:$$\begin{array}{c}{P}^{\left(mut\right)}={P}^{1}=\left({B}_{M}^{1},{Y}_{M}^{1\left(wt\right)},{Y}_{M}^{1\left(mut\right)},{B}_{L}^{1},{Y}_{L}^{1\left(wt\right)},{Y}_{L}^{1\left(mut\right)},{I}^{1}\right)\hfill \\ \hfill =\left(0,0,{Y}_{M}^{1\left(mut\right)},0,0,{Y}_{L}^{1\left(mut\right)},{C}_{I}\right)\end{array}$$$$\begin{array}{cc}\hfill {Y}_{M}^{1\left(mut\right)}& ={C}_{M}\left(1-\frac{{C}_{I}\gamma {f}_{I}^{\left(mut\right)}}{{\alpha}^{\left(mut\right)}}\right)\hfill \end{array}$$$$\begin{array}{cc}\hfill {Y}_{L}^{1\left(mut\right)}& =\frac{1}{2}\left({C}_{L}\left(1-\frac{\beta}{{\alpha}^{\left(mut\right)}}\right)+\sqrt{4{C}_{M}{C}_{L}{\left(\frac{{C}_{I}\gamma {f}_{I}^{\left(mut\right)}}{{\alpha}^{\left(mut\right)}}-1\right)}^{2}+{C}_{L}^{2}{\left(1-\frac{\beta}{{\alpha}^{\left(mut\right)}}\right)}^{2}}\right).\hfill \end{array}$$
- The other equilibrium corresponds to the persistence of commensal bacteria and the Yersinia wt strain in the absence of the mut strain. Without loss of generality, the commensal bacteria are supposed to be zero because they are not infective. This point is obtained by setting ${Y}_{M}^{\left(mut\right)}={Y}_{L}^{\left(mut\right)}=0$:$$\begin{array}{c}\hfill {P}^{\left(wt\right)}={P}^{2}=\left({B}_{M}^{2},{Y}_{M}^{2\left(wt\right)},{Y}_{M}^{2\left(mut\right)},{B}_{L}^{2},{Y}_{L}^{2\left(wt\right)},{Y}_{L}^{2\left(mut\right)},{I}^{2}\right)=\left(0,{Y}_{M}^{2\left(wt\right)},0,0,{Y}_{L}^{2\left(wt\right)},0,{C}_{I}\right)\end{array}$$$$\begin{array}{cc}\hfill {Y}_{M}^{2\left(wt\right)}& ={C}_{M}\left(1-\frac{{C}_{I}\gamma {f}_{I}^{\left(wt\right)}}{{\alpha}^{\left(wt\right)}}\right)\hfill \end{array}$$$$\begin{array}{cc}\hfill {Y}_{L}^{2\left(wt\right)}& =\frac{1}{2}\left({C}_{L}\left(1-\frac{\beta}{{\alpha}^{\left(wt\right)}}\right)+\sqrt{4{C}_{M}{C}_{L}{\left(\frac{{C}_{I}\gamma {f}_{I}^{\left(wt\right)}}{{\alpha}^{\left(wt\right)}}-1\right)}^{2}+{C}_{L}^{2}{\left(1-\frac{\beta}{{\alpha}^{\left(wt\right)}}\right)}^{2}}\right).\hfill \end{array}$$
- Finally, the last equilibrium point corresponds to a state of the co-existence of wt and mut Yersinia strains. This point is achieved by supposing ${B}_{M}={B}_{L}=0$:$$\begin{array}{c}\hfill {P}^{\left(wt\right)\left(mut\right)}={P}^{3}=\left({B}_{M}^{3},{Y}_{M}^{3\left(wt\right)},{Y}_{M}^{3\left(mut\right)},{B}_{L}^{3},{Y}_{L}^{3\left(wt\right)},{Y}_{L}^{3\left(mut\right)},{I}^{3}\right)\\ \hfill =\left(0,{Y}_{M}^{3\left(wt\right)},Z,0,{Y}_{L}^{3\left(wt\right)},{Y}_{L}^{3\left(mut\right)},{C}_{I}\right)\end{array}$$$$\begin{array}{cc}\hfill {Y}_{M}^{3\left(wt\right)}& ={C}_{M}\left(1-\frac{{C}_{I}\gamma {f}_{I}^{\left(wt\right)}}{{\alpha}^{\left(wt\right)}}\right)-Z\hfill \end{array}$$$$\begin{array}{cc}\hfill {Y}_{L}^{3\left(wt\right)}& ={\displaystyle \frac{{\alpha}^{\left(mut\right)}\left(Z\left(1-\frac{{C}_{I}\gamma {f}_{I}^{\left(wt\right)}}{{\alpha}^{\left(wt\right)}}\right)-\gamma \phantom{\rule{0.166667em}{0ex}}{f}_{I}^{\left(wt\right)}{\left(1-\frac{{C}_{I}}{{\alpha}^{\left(wt\right)}}\right)}^{2}\right)}{\beta \left(1-\frac{{\alpha}^{\left(mut\right)}}{{\alpha}^{\left(wt\right)}}\right)}}\hfill \end{array}$$$$\begin{array}{cc}\hfill {Y}_{L}^{3\left(mut\right)}& ={\displaystyle \frac{{\alpha}^{\left(mut\right)}\left(1-\frac{{C}_{I}\gamma {f}_{I}^{\left(wt\right)}}{{\alpha}^{\left(wt\right)}}\right)\left(Z-\left(1-\frac{{C}_{I}\gamma {f}_{I}^{\left(wt\right)}}{{\alpha}^{\left(wt\right)}}\right)\right)}{\beta \left(1-\frac{{\alpha}^{\left(mut\right)}}{{\alpha}^{\left(wt\right)}}\right)}}\hfill \end{array}$$

#### 3.2. Analysis of the Disease-Free Equilibrium Point

#### 3.3. Computing and Analysis of the System through the Basic Reproduction Number

- ${\mathcal{R}}_{0}^{wt}$: the basic reproductive numbers for wt strain = $\frac{{\alpha}^{\left(wt\right)}}{{\alpha}^{\left(B\right)}{f}_{I}^{\left(wt\right)}}$
- ${\mathcal{R}}_{0}^{mut}$: the basic reproductive numbers for mut strain = $\frac{{\alpha}^{\left(mut\right)}}{{\alpha}^{\left(B\right)}{f}_{I}^{\left(mut\right)}}$.

- (I)
- If ${\mathcal{R}}_{0}^{wt}=1$ and ${\mathcal{R}}_{0}^{mut}=1$, then ${\alpha}^{\left(wt\right)}={\alpha}^{\left(B\right)}{f}_{I}^{\left(wt\right)}$ and ${\alpha}^{\left(mut\right)}={\alpha}^{\left(B\right)}{f}_{I}^{\left(mut\right)}$. Thus, the intersection of the transcritical curves ${\mathcal{R}}_{0}^{wt}$ and ${\mathcal{R}}_{0}^{mut}$ results in a triple transcritical bifurcation. As shown in (A9), the Jacobian has a triple zero eigenvalue at this point (${\lambda}_{2}=0,{\lambda}_{3}=0$). Kuznetsov [34] has proved that such a point would be an indicator of the onset of a non-degenerate or degenerate Bogdanov–Takens bifurcation [34,35]. The disease-free equilibrium ${P}^{0}$ loses its stability, and the wt-free and mut-free include one simple zero eigenvalue (${\lambda}_{3}=0$), meaning that the dynamics of the model change as the target parameter is within the threshold value.
- (II)
- If ${\mathcal{R}}_{0}^{wt}>1$, the wt strain equilibrium in region II will persist when ${\mathcal{R}}_{0}^{mut}<1$. The wt strain will spread and possibly persist within the host population. In general, for a strain to persist, its basic reproduction number has to be strictly greater than one. Therefore, in this region, the disease-free, mut strain, and co-existence state exchange stability: ${P}^{0}$ becomes unstable, ${P}^{2}$ becomes locally asymptotically stable, and ${P}^{1}$ and ${P}^{3}$ remain unstable. This means that the immune system could kill one of the strains more efficiently.
- (III)
- If ${\mathcal{R}}_{0}^{mut}>1$, the mut strain equilibrium in region III will persist when ${\mathcal{R}}_{0}^{wt}<1$. The mut strain will spread and possibly persist within the host population since its basic reproduction number is greater than one. Therefore, in this region, the disease-free, wt strain, and co-existence state exchange stability: ${P}^{0}$ becomes unstable, ${P}^{1}$ becomes locally asymptotically stable, and ${P}^{2}$ and ${P}^{3}$ remain unstable. This means that the immune system could defeat the wt strains. However, the risk of this situation to happen is low because the mut strains are influenced more efficiently than wt strains by immune action.
- (IV)
- If ${\mathcal{R}}_{0}^{wt}>1$ and ${\mathcal{R}}_{0}^{mut}>1$, the co-existence population spreads, and both strains persist. The overall ${\mathcal{R}}_{0}$ can be defined as ${\mathcal{R}}_{0}^{c}=d\phantom{\rule{0.166667em}{0ex}}{\mathcal{R}}_{0}^{mut}+(1-d)\phantom{\rule{0.166667em}{0ex}}{\mathcal{R}}_{0}^{wt}$. A mut with $d=1$ is thoroughly dominant, while one with $d=0$ is completely recessive; scenarios of incomplete dominance $(0<d<1)$, under-dominance $(d<0)$, and over-dominance $(d>1)$ are possible as well. For instance, a mut could achieve a higher ${\mathcal{R}}_{0}$ than the wt via a higher growth rate that increases transmission. In a co-infection, the faster-growing mut strain would outcompete the wt and reach its maximum capacity. This situation would change the co-infection to the conditions where a single infection happens. Thus, the overall ${\mathcal{R}}_{0}$ of the co-infection would be similar to that of the mut by itself, making the mut a dominant one. Furthermore, the effort for having a co-existence equilibrium and analysis of the co-infection model will fail. By contrast, let us assume a mut strain achieves a higher ${\mathcal{R}}_{0}$. Nevertheless, the virulence of the wt strain neutralizes the higher ${\mathcal{R}}_{0}$ value of the mut. This would make the mut a recessive one. In summary, virtually any two-strain co-infection model can be mapped to a set of values for d, allowing scenarios of particular interest to be explored in a context broader than the one possible with typical models.

^{−/−}) mut mice. Using the model values in Table 3, which was estimated through experimental data [10], we first examined the influence of the infection rates ${\alpha}^{\left(B\right)}$, ${\alpha}^{\left(wt\right)}$, and ${\alpha}^{\left(mut\right)}$ on the dynamics beginning by constructing one-dimensional bifurcation diagrams using ${\alpha}^{\left(B\right)}$ as the bifurcation parameter and fixing ${\alpha}^{\left(wt\right)}$ and ${\alpha}^{\left(mut\right)}$. The model values in Table 3 are the output of estimation parameters in Table 2 of the (1)–(7) by experimental data [10]. In theory, some parameter values were defined based on the biological aspect, and the rest were estimated by the optimization problem with the maximum likelihood estimation [10].

^{−/−}mice. As the responses of the immune system in these mice are different, we conclude different behaviors. Let us denote the numerical simulation:

- The effect of the maximum rate of immune growth $\kappa $ on wt Yersinia strain in the mucosa, Figure 7;
- The effect of the maximum rate of immune growth $\kappa $ on mut Yersinia strain in the mucosa, Figure 8;
- The effect of the maximum rate of immune growth $\kappa $ on wt Yersinia strain in the lumen, Figure 9;
- The effect of the maximum rate of immune growth $\kappa $ on mut Yersinia strain in the lumen, Figure 10,

## 4. Discussion and Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## Abbreviations

AMP | antimicrobial peptide |

BMBF | Federal Ministry of Education and Research |

BMBF-DZG | Deutsche Zentren der Gesundheitsforschung |

CFU | colony-forming unit |

DFE | disease-free equilibrium |

DFG | Deutsche Forschungsgemeinschaft |

DZIF | German Center for Infection Research |

GF | germ-free |

GIT | gastrointestinal tract |

mut | mutant |

MyD88^{−/−} | MyD88-deficient mice |

ODE | Ordinary Differential Equation |

PD | Process Description |

RKI | Robert Koch Institute |

SBGN | Systems Biology Graphical Notation |

SI | small intestine |

SPF | specific-pathogen-free |

T3SS | Type [email protected] secretion system |

wt | wild-type |

YadA | Yersinia adhesin A |

Ye | Yersinia entercolitica |

Yop | Yersinia outer protein |

## Appendix A. Mathematical Calculation

#### Appendix A.1. Jacobian

#### Appendix A.2. The Eigenvalues of Trivial Equilibrium Point

#### Appendix A.3. The Eigenvalues and Eigenvectors of the Disease-Free Equilibrium point

#### Appendix A.4. The Eigenvalues wt Strain Equilibrium

#### Appendix A.5. The Eigenvalues mut Strain Equilibrium

#### Appendix A.6. The Basic Reproduction Numbers

- Based on Equations (14) and (15), we split (22)–(28) into two compartments: one disease compartment with populations $\left({Y}_{M}^{\left(wt\right)},{Y}_{M}^{\left(mut\right)},{Y}_{L}^{\left(wt\right)},{Y}_{L}^{\left(mut\right)}\right)$ and one non-disease compartment with populations $({B}_{M},{B}_{L},I)$.
- ${\mathcal{F}}_{i}\left(x\right)$ corresponds to the rate of the appearance of new infections, and ${\mathcal{V}}_{i}$ corresponds to the rate of transfer assumed as$$\begin{array}{cc}\hfill {\mathcal{F}}_{1}& =\left({\alpha}^{\left(wt\right)}-{\sigma}_{M\u27f6L}^{\left(wt\right)}\right)\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{Y}_{M}^{\left(wt\right)}\hfill \end{array}$$$$\begin{array}{cc}\hfill {\mathcal{F}}_{2}& =\left({\alpha}^{\left(mut\right)}-{\sigma}_{M\u27f6L}^{\left(mut\right)}\right)\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{Y}_{M}^{\left(mut\right)}\hfill \end{array}$$$$\begin{array}{cc}\hfill {\mathcal{F}}_{3}& =\left({\alpha}_{L}^{\left(wt\right)}\right)\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{Y}_{L}^{\left(wt\right)}\hfill \end{array}$$$$\begin{array}{cc}\hfill {\mathcal{F}}_{4}& =\left({\alpha}_{L}^{\left(mut\right)}\right)\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{Y}_{L}^{\left(mut\right)}\hfill \end{array}$$$$\begin{array}{cc}\hfill {\mathcal{V}}_{1}& =\left(\gamma \phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{f}_{I}^{\left(wt\right)}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}I\right)\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{Y}_{M}^{\left(wt\right)}\hfill \end{array}$$$$\begin{array}{cc}\hfill {\mathcal{V}}_{2}& =\left(\gamma \phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{f}_{I}^{\left(mut\right)}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}I\right)\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{Y}_{M}^{\left(mut\right)}\hfill \end{array}$$$$\begin{array}{cc}\hfill {\mathcal{V}}_{3}& =\left(\beta \right)\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{Y}_{L}^{\left(wt\right)}-{\sigma}_{M\u27f6L}^{\left(wt\right)}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{Y}_{M}^{\left(wt\right)}\hfill \end{array}$$$$\begin{array}{cc}\hfill {\mathcal{V}}_{4}& =\left(\beta \right)\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{Y}_{L}^{\left(mut\right)}-{\sigma}_{M\u27f6L}^{\left(mut\right)}\phantom{\rule{0.166667em}{0ex}}\xb7\phantom{\rule{0.166667em}{0ex}}{Y}_{M}^{\left(mut\right)}\hfill \end{array}$$
- The basic reproduction number is achieved as the spectral radius of the matrix $\left(F{V}^{-1}\right)$;as $\mathcal{R}=\rho \left(F{V}^{-1}\right)$ where $F=\frac{\partial {\mathcal{F}}_{i}\left({P}^{0}\right)}{\partial {x}_{j}}$, $V=\frac{\partial {\mathcal{V}}_{i}\left({P}^{0}\right)}{\partial {x}_{j}}$, and ${x}_{j}$ are as follows:$$\begin{array}{c}\hfill {x}_{j}=\left({Y}_{M}^{\left(wt\right)},{Y}_{M}^{\left(mut\right)},{Y}_{L}^{\left(wt\right)},{Y}_{L}^{\left(mut\right)}\right)\end{array}$$$$\begin{array}{c}\hfill F=\left(\begin{array}{cccc}{\alpha}^{\left(wt\right)}(\frac{\gamma}{{\alpha}^{\left(B\right)})}& 0& 0& 0\\ 0& {\alpha}^{\left(mut\right)}(\frac{\gamma}{{\alpha}^{\left(B\right)})}& 0& 0\\ 0& 0& {\alpha}^{\left(wt\right)}(\frac{\beta}{{\alpha}^{\left(B\right)})}& 0\\ 0& 0& 0& {\alpha}^{\left(mut\right)}(\frac{\beta}{{\alpha}^{\left(B\right)})}\end{array}\right)\end{array}$$$$\begin{array}{c}\hfill V=\left(\begin{array}{cccc}\gamma \phantom{\rule{0.166667em}{0ex}}{{f}_{I}}^{\left(wt\right)}& 0& 0& 0\\ 0& \gamma \phantom{\rule{0.166667em}{0ex}}{{f}_{I}}^{\left(mut\right)}& 0& 0\\ -\frac{{\alpha}^{\left(wt\right)}\left(-\gamma +{\alpha}^{\left(B\right)}\right)}{{\alpha}^{\left(B\right)}}& 0& \beta & 0\\ 0& -\frac{{\alpha}^{\left(mut\right)}\left(-\gamma +{\alpha}^{\left(B\right)}\right)}{{\alpha}^{\left(B\right)}}& 0& \beta \end{array}\right).\end{array}$$Then$$\begin{array}{cc}\hfill \mathcal{R}& =\rho \left(F{V}^{-1}\right)\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& =\mathrm{max}\left\{\left|\frac{{\alpha}^{\left(wt\right)}}{{\alpha}^{\left(B\right)}}\right|,\left|\frac{{\alpha}^{\left(mut\right)}}{{\alpha}^{\left(B\right)}}\right|,\left|\frac{{\alpha}^{\left(wt\right)}}{{\alpha}^{\left(B\right)}{{f}_{I}}^{\left(wt\right)}}\right|,\left|\frac{{\alpha}^{\left(mut\right)}}{{\alpha}^{\left(B\right)}{{f}_{I}}^{\left(mut\right)}}\right|\right\}\hfill \end{array}$$

## Appendix B. Data Availability

- The Matlab script
- The Maple workbook
- The described model is available in SBML format [26] (Level 3 Version 2 [27]) from BioModels database [36] under model identifier
`MODEL2002070001`.

## References

- El Tahir, Y.; Skurnik, M. YadA, the multifaceted Yersinia Adhes. Int. J. Med. Microbiol.
**2001**, 291, 209–218. [Google Scholar] [CrossRef] [PubMed] - Handley, A.J.; Koster, R.; Monsieurs, K.; Perkins, G.D.; Davies, S.; Bossaert, L. European Resuscitation Council Guidelines for Resuscitation 2005: Section 2. Adult basic life support and use of automated external defibrillators. Resuscitation
**2005**, 67, S7–S23. [Google Scholar] [CrossRef] [PubMed] - Mühlenkamp, M.; Oberhettinger, P.; Leo, J.C.; Linke, D.; Schütz, M.S. Yersinia Adhes. A (YadA)- Beast. Int. J. Med. Microbiol.
**2015**, 305, 252–258. [Google Scholar] [CrossRef] [PubMed] - Young, G.A.; LaVon, G.D.; Taylor, G.W. High Efficiency Absorbent Articles for Incontinence Management. US Patent 5,147,345, 15 September 1992. [Google Scholar]
- Pepe, G.J.; Albrecht, E.D. Actions of placental and fetal adrenal steroid hormones in primate pregnancy. Endocr. Rev.
**1995**, 16, 608–648. [Google Scholar] [PubMed] - Cornelis, G.R. Yersinia Type III Secretion: Send Eff. J. Cell Biol.
**2002**, 158, 401. [Google Scholar] [CrossRef][Green Version] - Ruckdeschel, K.; Roggenkamp, A.; Schubert, S.; Heesemann, J. Differential contribution of Yersinia Enterocolitica Virulence Factors Evas. Microbicidal Action Neutrophils. Infect. Immun.
**1996**, 64, 724–733. [Google Scholar] [CrossRef][Green Version] - Lupp, C.; Robertson, M.L.; Wickham, M.E.; Sekirov, I.; Champion, O.L.; Gaynor, E.C.; Finlay, B.B. Host-mediated inflammation disrupts the intestinal microbiota and promotes the overgrowth of Enterobacteriaceae. Cell Host Microbe
**2007**, 2, 119–129. [Google Scholar] [CrossRef][Green Version] - Stecher, B.; Robbiani, R.; Walker, A.W.; Westendorf, A.M.; Barthel, M.; Kremer, M.; Chaffron, S.; Macpherson, A.J.; Buer, J.; Parkhill, J.; et al. Salmonella enterica serovar typhimurium exploits inflammation to compete with the intestinal microbiota. PLoS Biol.
**2007**, 5, e244. [Google Scholar] [CrossRef][Green Version] - Geißert, J.; Bohn, E.; Mostolizadeh, R.; Dräger, A.; Autenrieth, I.; Beier, S.; Deusch, O.; Eichner, M.; Schütz, M. Model-based prediction of bacterial population dynamics in gastrointestinal infection. bioRxiv
**2020**. [Google Scholar] [CrossRef] - Nowak, M.; Robert, M. May. Superinfection and the evolution of parasite virulence. Proc. R. Soc. B
**1994**, 255, 81n89. [Google Scholar] - Castillo-Chavez, C.; Velasco-Hernández, J.X. On the relationship between evolution of virulence and host demography. J. Theor. Biol.
**1998**, 192, 437–444. [Google Scholar] [CrossRef] [PubMed] - Dye, C.; Williams, B.G.; Espinal, M.A.; Raviglione, M.C. Erasing the world’s slow stain: Strategies to beat multidrug-resistant tuberculosis. Science
**2002**, 295, 2042–2046. [Google Scholar] [CrossRef] [PubMed][Green Version] - Boldin, B.; Geritz, S.A.; Kisdi, É. Superinfections and adaptive dynamics of pathogen virulence revisited: A critical function analysis. Evol. Ecol. Res.
**2009**, 11, 153–175. [Google Scholar] - Liu, W.m.; Hethcote, H.W.; Levin, S.A. Dynamical behavior of epidemiological models with nonlinear incidence rates. J. Math. Biol.
**1987**, 25, 359–380. [Google Scholar] [CrossRef] [PubMed] - Madden, L.V.; Hughes, G.; Van Den Bosch, F. The Study of Plant Disease Epidemics; Am Phytopath Society: St. Paul, MN, USA, 2007. [Google Scholar]
- Gross, K.L.; Porco, T.C.; Grant, R.M. HIV-1 superinfection and viral diversity. Aids
**2004**, 18, 1513–1520. [Google Scholar] [CrossRef] [PubMed] - Nurtay, A.; Hennessy, M.G.; Sardanyés, J.; Alsedà, L.; Elena, S.F. Theoretical conditions for the coexistence of viral strains with differences in phenotypic traits: A bifurcation analysis. R. Soc. Open Sci.
**2019**, 6, 181179. [Google Scholar] [CrossRef][Green Version] - Mostolizadeh, R.; Shamsara, E.; Afsharnezhad, Z. Hopf Bifurcation and Chaos in a model for HTLV-I infection of CD4
^{+}T- cells. Ann. Tiberiu Popoviciu Semin. Funct. Equ. Approx. Convexity Ser. B Math. Interdiscip. Res.**2017**, 15, 59–79. [Google Scholar] - Mostolizadeh, R.; Afsharnezhad, Z. Hopf bifurcation in a model for adult T-cell leukemia. Math. Methods Appl. Sci.
**2018**, 41, 6210–6225. [Google Scholar] [CrossRef] - Shamsara, E.; Mostolizadeh, R.; Afsharnezhad, Z. Transcritical bifurcation of an immunosuppressive infection model. Iran. J. Numer. Anal. Optim.
**2016**, 6, 1–16. [Google Scholar] - Hethcote, H.W. The mathematics of infectious diseases. SIAM Rev.
**2000**, 42, 599–653. [Google Scholar] [CrossRef][Green Version] - Anderson, R.M.; May, R. Infectious Diseases of Humans; Oxford Science Publication: New York, NY, USA, 1991. [Google Scholar]
- Touré, V.; Dräger, A.; Luna, A.; Dogrusoz, U.; Rougny, A. The Systems Biology Graphical Notation: Current Status and Applications in Systems Medicine. In Systems Medicine; Wolkenhauer, O., Ed.; Academic Press: Oxford, UK, 2020; Volume 3, pp. 372–381. [Google Scholar]
- Rougny, A.; Touré, V.; Moodie, S.; Balaur, I.; Czauderna, T.; Borlinghaus, H.; Dogrusoz, U.; Mazein, A.; Dräger, A.; Blinov, M.L.; et al. Systems Biology Graphical Notation: Process Description language Level 1 Version 2.0. J. Integr. Bioinform.
**2019**, 16. [Google Scholar] [CrossRef] [PubMed] - Keating, S.M.; Waltemath, D.; König, M.; Zhang, F.; Dräger, A.; Chaouiya, C.; Bergmann, F.T.; Finney, A.; Gillespie, C.S.; Helikar, T.; et al. SBML Level 3: An extensible format for the exchange and reuse of biological models. Mol. Syst. Biol.
**2020**, 16, e9110. [Google Scholar] [CrossRef] [PubMed] - Hucka, M.; Bergmann, F.T.; Chaouiya, C.; Dräger, A.; Hoops, S.; Keating, S.M.; König, M.; Le Novère, N.; Myers, C.J.; Olivier, B.G.; et al. Systems Biology Markup Language (SBML) Level 3 Version 2 Core Release 2. J. Integr. Bioinform.
**2019**, 16, 1. [Google Scholar] [CrossRef] [PubMed] - Diekmann, O.; Heesterbeek, J.A.P.; Metz, J.A. On the definition and the computation of the basic reproduction ratio R 0 in models for infectious diseases in heterogeneous populations. J. Math. Biol.
**1990**, 28, 365–382. [Google Scholar] [CrossRef][Green Version] - Van den Driessche, P.; Watmough, J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci.
**2002**, 180, 29–48. [Google Scholar] [CrossRef] - Van den Driessche, P.; Watmough, J. Further notes on the basic reproduction number. In Mathematical Epidemiology; Springer: Berlin/Heidelberg, Germany, 2008; pp. 159–178. [Google Scholar]
- Buonomo, B. A note on the direction of the transcritical bifurcation in epidemic models. Nonlinear Anal. Model Control
**2015**, 20, 38–55. [Google Scholar] [CrossRef][Green Version] - Guckenheimer, J.; Holmes, P. Nonlinear Oscillations, Dynamical Systems, and Bifurcations of Vector Fields; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2013; Volume 42. [Google Scholar]
- Wiggins, S. Introduction to Applied Nonlinear Dynamical System and Chaos; Spring: New York, NY, USA, 1990. [Google Scholar]
- Kuznetsov, Y.A. Elements of Applied Bifurcation Theory; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2013; Volume 112. [Google Scholar]
- Kuznetsov, Y.A. Practical computation of normal forms on center manifolds at degenerate Bogdanov–Takens bifurcations. Int. J. Bifurc. Chaos
**2005**, 15, 3535–3546. [Google Scholar] [CrossRef] - Malik-Sheriff, R.S.; Glont, M.; Nguyen, T.V.N.; Tiwari, K.; Roberts, M.G.; Xavier, A.; Vu, M.T.; Men, J.; Maire, M.; Kananathan, S.; et al. BioModels—15 years of sharing computational models in life science. Nucleic Acids Res.
**2020**, 48, D407–D415. [Google Scholar] [CrossRef][Green Version]

**Figure 1.**An overview of the Yersinia enterocolitica population dynamics model. Filled circles represent the entity pool nodes for the populations of Ye in their respective compartments as well as the strength of the immune reaction I. Table 1 explains the notations used in the model and this figure. The black arrows represent processes with an impact on the population dynamics of the entity pools. Arrows pointing from empty set symbols to pool nodes denote an increase in the population size or a decrease if the process arcs point from entity pools to empty sets. Migration across compartments of the respective populations appears as vertical process arcs. Some of these processes receive stimulating effects from the immune reaction or from the size of the Ye populations within the mucosa, as colored arcs indicate. Reference [10] provides a more detailed description of the model’s structure.

**Figure 2.**The diagram displays three regions with different qualitative behaviors in terms of the basic reproduction numbers. Region I: infection-free state; Region II: mut-free state; Region III: wt-free state; Region IV: co-existence of all strains.

**Figure 3.**The diagrams display the role of ${\mathcal{R}}_{0}^{com}$ in appearance/non-appearance of the trivial solution and disease-free equilibrium. (

**a**) By fixing the parameter values Table 3 in model (14), the commensal bacteria in the mucosa appear when ${\mathcal{R}}_{0}^{com}>1$. Therefore, as long as ${\mathcal{R}}_{0}^{com}<1$, only the trivial solution for the model exists. Since the trivial solution is always unstable, the extinction of all populations is never achieved. (

**b**) By fixing the parameter values Table 3 in model (14), the commensal bacteria in the lumen appear when ${\mathcal{R}}_{0}^{com}>1$. Therefore, as long as ${\mathcal{R}}_{0}^{com}<1$, only the trivial solution for the model exists. Since the trivial solution is always unstable, the extinction of all populations is never achieved.

**Figure 5.**The sensitivity of parameter ${\alpha}^{\left(B\right)}$ with respect to commensal bacteria, wild-type, and mutant Yersinia in the mucosa and lumen for 336 h. $0<{\alpha}^{\left(B\right)}<1$, no biologically meaningful region (out of our interest). ${\alpha}^{\left(B\right)}>1$, all populations appear. When $1<{\alpha}^{\left(B\right)}<2.28$, the region is a region for the appearance of the co-existence equilibrium, but the hypothesis of the co-existence equilibrium is not satisfied. Therefore, wild-type strain does not grow, and the mutant strain is going down slowly. When ${\alpha}^{\left(B\right)}>2.28$, this is a region of the appearance of a wt equilibrium ${\mathcal{R}}_{0}^{wt}>1$. Thus, (

**a**,

**e**) are increasing fast and stay at the maximum level as (

**a**,

**d**) are going back to zero. Additionally, (

**c**,

**d**) do not grow anymore ${\mathcal{R}}_{0}^{mut}<1$.

**Figure 6.**The diagram displays the appearance of the co-existence of wt and mut Yersinia strains as ${\alpha}^{\left(B\right)}$ is changing in a region where ${\mathcal{R}}_{0}^{wt}>1$ and ${\mathcal{R}}_{0}^{mut}>1$. (

**a**) An immune reaction influences the wt and mut Yersinia strains in the mucosa. Additionally, another part is spilled over and moves to the lumen compartment. Therefore, they simultaneously increase or decrease to reach their maximum values. (

**b**) The wt and mut Yersinia strains in the lumen simultaneously increase to reach the carrying capacity of the lumen compartment. In both (

**a**,

**b**), when ${\alpha}^{\left(B\right)}$ reaches 4.90, the co-existence of wt and mut Yersinia strains disappears.

**Figure 7.**The sensitivity of parameter κ in the elimination of wt Yersinia strain in the mucosa. (

**a**–

**c**) The effect of the parameter κ on the different types of mice. (

**d**–

**f**) The projection of κ for a different types of mice.

**Figure 8.**The sensitivity of parameter κ in the elimination of mut Yersinia strain in the mucosa. (

**a**–

**c**) The effect of the parameter κ on the different types of mice. (

**d**–

**f**) The projection of κ for different types of mice.

**Figure 9.**The sensitivity of parameter κ in the elimination of wt Yersinia strain in the lumen. (

**a**–

**c**) The effect of the parameter κ on the different types of mice. (

**d**–

**f**) The projection of κ for different types of mice.

**Figure 10.**The sensitivity of the parameter κ in the elimination of the mut Yersinia strain in the lumen. (

**a**–

**c**) The effect of the parameter κ on the different types of mice. (

**d**–

**f**) The projection of κ for different types of mice.

**Figure 11.**Diagram displaying the disease-free state by fixing the parameter values Ye SPF wt/A0 from Table 3 in Model (14). As long as ${\alpha}^{\left(B\right)}$ is large enough, the disease-free state persists. However, by reducing ${\alpha}^{\left(B\right)}$, the basic reproduction number ${\mathcal{R}}_{0}^{wt}$ reaches its threshold value (${\mathcal{R}}_{0}^{wt}=1$). This causes changes in the dynamic behavior of the model (14), as shown in Figure 4.

**Table 1.**The variable symbols and their meaning. The lumen is abbreviated with L, the mucosa as M. We also indicate each variable with mut or wt to denote to which population they refer. An upper-case I refers to the immune system. SBML [26,27] defines the units item and dimensionless to indicate that a quantity occurs in a piece number or that its unit originates from the cancellation of other units.

Variable Symbol | Meaning | Units |
---|---|---|

${B}_{M}$ | Commensal bacteria in the mucosa | item |

${Y}_{M}^{\left(wt\right)}$ | wt Yersinia in the mucosa | item |

${Y}_{M}^{\left(mut\right)}$ | mut Yersinia in the mucosa | item |

${B}_{L}$ | Commensal bacteria in the lumen | item |

${Y}_{L}^{\left(wt\right)}$ | wt Yersinia in the lumen | item |

${Y}_{L}^{\left(mut\right)}$ | mut Yersinia in the lumen | item |

I | Strength of immune reaction | dimensionless |

Parameter | Definition | Unit |
---|---|---|

${\alpha}^{\left(B\right)}$ | Maximal growth rate of intestinal bacteria | 1/$\mathrm{h}$ |

${\alpha}^{\left(wt\right)}$ | Maximal growth rate of wt Yersinia | 1/$\mathrm{h}$ |

${\alpha}^{\left(mut\right)}$ | Maximal growth rate of mut Yersinia | 1/$\mathrm{h}$ |

${f}_{I}^{\left(wt\right)}$ | Immunity adjustment factor for wt Yersinia | dimensionless |

${f}_{I}^{\left(mut\right)}$ | Immunity adjustment factor for mut Yersinia | dimensionless |

${C}_{M}$ | Carrying capacity of the mucosa | item |

${C}_{L}$ | Carrying capacity of the lumen | item |

${C}_{I}$ | Carrying capacity of the immune system | item |

$\gamma $ | Maximal immunity action | 1/$\mathrm{h}$ |

$\kappa $ | Maximal rate of immune growth | 1/$\mathrm{h}$ |

$\beta $ | Rate at which intestines are discharged | 1/$\mathrm{h}$ |

Parameter | Values in Ye SPF wt/A0 | Values in Ye SPF wt/T3S0 | Values in Ye GF wt/A0 | Values in Ye MyD88^{−/−} wt/A0 |
---|---|---|---|---|

${\alpha}^{\left(B\right)}$ | 4.89 × 10^{−1} | 2.00 | 1.99 | 5.40 × 10^{−1} |

${\alpha}^{\left(wt\right)}$ | 4.44 × 10^{−1} | 1.86 | 1.60 | 5.78 × 10^{−1} |

${\alpha}^{\left(mut\right)}$ | 4.44 × 10^{−1} | 1.86 | 1.60 | 5.78 × 10^{−1} |

${f}_{I}^{\left(wt\right)}$ | 3.96 × 10^{−1} | 9.48 × 10^{−3} | 1.10 × 10^{−1} | 6.23 × 10^{−2} |

${f}_{I}^{\left(mut\right)}$ | 1.95 × 10^{−1} | 3.73 × 10^{−1} | 1.19 × 10^{−1} | 1.28 × 10^{−1} |

${C}_{M}$ | 1.76 × 10^{5} | 6.27 × 10^{3} | 1.3 × 10^{6} | 1.28 × 10^{5} |

${C}_{L}$ | 2.14 × 10^{7} | 6.13 × 10^{6} | 4.99 × 10^{9} | 9.98 × 10^{9} |

${C}_{I}$ | 1.00 | 1.00 | 1.00 | 1.00 |

$\gamma $ | 1.00 | 1.00 | 9.97 × 10^{−1} | 1.00 × 10^{−1} |

$\kappa $ | 7.83 × 10^{−1} | 4.28 × 10^{−1} | 6.50 × 10^{−1} | 4.37 × 10^{−1} |

$\beta $ | 2.50 × 10^{−1} | 2.50 × 10^{−1} | 8.33 × 10^{−2} | 1.82 × 10^{−1} |

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**MDPI and ACS Style**

Mostolizadeh, R.; Dräger, A. Computational Model Informs Effective Control Interventions against *Y. enterocolitica* Co-Infection. *Biology* **2020**, *9*, 431.
https://doi.org/10.3390/biology9120431

**AMA Style**

Mostolizadeh R, Dräger A. Computational Model Informs Effective Control Interventions against *Y. enterocolitica* Co-Infection. *Biology*. 2020; 9(12):431.
https://doi.org/10.3390/biology9120431

**Chicago/Turabian Style**

Mostolizadeh, Reihaneh, and Andreas Dräger. 2020. "Computational Model Informs Effective Control Interventions against *Y. enterocolitica* Co-Infection" *Biology* 9, no. 12: 431.
https://doi.org/10.3390/biology9120431