# Gaining Profound Knowledge of Cholera Outbreak: The Significance of the Allee Effect on Bacterial Population Growth and Its Implications for Human-Environment Health

^{1}

^{2}

^{3}

^{4}

^{*}

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Related Work

#### 2.2. Model Construction

## 3. Results

#### 3.1. Sensitivity Analysis

- the parameters $\Lambda $, $\beta $, $\delta $, $\theta $, and $\rho $ should significantly affect at least one state variable of model (2);
- $\Lambda $ is not related to transmission of disease;
- influence of $\beta $ suggests to sensitize population to avoid getting in touch with bacteria;
- influences of $\theta $ and $\rho $ suggest to intensify sanitation campaigns by destroying reservoirs of V. cholera.

PRCCs and Significance | |||||
---|---|---|---|---|---|

Parameters | Range | $\mathit{S}$ | $\mathit{I}$ | $\mathit{R}$ | $\mathit{B}$ |

$\Lambda $ | [1–300] | 0.8970 ** | 0.0968 * | 0.0171 | −0.0332 |

$\beta $ | [10${}^{-6}$–0.999] | −0.9580 * | −0.2359 ** | 0.1221 ** | −0.1702 ** |

$\delta $ | [1–1000] | −0.8934 ** | −0.0545 | 0.0885 | −0.1173 * |

$\mu $ | [10${}^{-6}$–0.999] | −0.0718 | −0.0255 | 0.0270 | −0.0818 |

d | [10${}^{-6}$–0.999] | 0.0290 | 0.0432 | −0.0676 | −0.0462 |

r | [10${}^{-6}$–0.999] | −0.0282 | −0.0180 | 0.0395 | 0.0291 |

$\gamma $ | [10${}^{-6}$–0.999] | −0.0388 | 0.0147 | 0.0577 | −0.0106 |

$\alpha $ | [10${}^{-6}$–0.999] | 0.1010 | 0.0680 | 0.0530 | 0.0948 |

K | [10${}^{4}$–10${}^{17}$] | 0.0050 | 0.0335 | 0.0092 | −0.0213 |

$\rho $ | [10${}^{5}$–10${}^{20}$] | −0.0723 | −0.1149 * | 0.0561 | 0.6220 ** |

$\theta $ | [10${}^{3}$–10${}^{15}$] | 0.0210 | 0.0628 | 0.0473 | 0.5029 * |

#### 3.2. Basic Properties

#### 3.2.1. Positivity and Boundedness of Solutions

**Theorem 1.**

**Proof.**

**Step 1.**- We show that for any initial condition $({t}_{0}=0,\phantom{\rule{0.277778em}{0ex}}{X}_{0}=(S\left(0\right),I\left(0\right)$, $R\left(0\right),B\left(0\right))\in {\left({\mathbb{R}}_{+}^{*}\right)}^{4})$, the maximal solution $\left(\right[0,T[,\phantom{\rule{0.277778em}{0ex}}X=(S\left(t\right),I\left(t\right),R\left(t\right),B\left(t\right)\left)\right)$ of the Cauchy problem associated with system (2) is non-negative.Let $\tilde{T}=sup\left\{\tilde{t}\in [0;T[,\phantom{\rule{0.277778em}{0ex}}(S\left(t\right),\phantom{\rule{0.277778em}{0ex}}I\left(t\right),\phantom{\rule{0.277778em}{0ex}}R\left(t\right),\phantom{\rule{0.277778em}{0ex}}B\left(t\right))\in {\left({\mathbb{R}}_{+}^{*}\right)}^{4}\right\}$ and let us show that $\tilde{T}=T$.Suppose that $\tilde{T}<T$. At least one of the following conditions is satisfied: $S\left(\tilde{T}\right)=0$, $I\left(\tilde{T}\right)=0$, $R\left(\tilde{T}\right)=0$, or $B\left(\tilde{T}\right)=0$.Suppose $S\left(\tilde{T}\right)=0$. Then from the first equation of model (2),$$\frac{d}{dt}\left(S{e}^{{\int}_{0}^{t}\left(\lambda \left(r\right)+\mu \right)dr}\right)=(\Lambda +\gamma R){e}^{{\int}_{0}^{t}\left(\lambda \left(r\right)+\mu \right)dr},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall t\in [0;\tilde{T}[.$$This implies that$$\frac{d}{dt}\left(S{e}^{{\int}_{0}^{t}\left(\lambda \left(r\right)+\mu \right)dr}\right)>0,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}\forall t\in [0;\tilde{T}[.$$Integrating Equation (6) from 0 to $\tilde{T}$ yields:$$S\left(\tilde{T}\right)\ge S\left(0\right){e}^{-{\int}_{0}^{\tilde{T}}\left(\lambda \left(r\right)+\mu \right)dr}>0.$$Similarly, one can show that $I\left(\tilde{T}\right)>0$, $R\left(\tilde{T}\right)>0$, and $B\left(\tilde{T}\right)>0$, which is contradictory. Therefore, $\tilde{T}=T$ and consequently, the maximal solution $\left(S\right(t),I(t),R(t),B(t\left)\right)$ of the Cauchy problem associated to model (2) is non-negative.
**Step 2.**- We then prove that the total population of humans and bacteria satisfies the boundedness property. We first split model (2) into two parts, the human population (i.e., $S\left(t\right)$, $I\left(t\right)$ and $R\left(t\right)$) and the pathogen population (i.e., $B\left(t\right)$).Let $N=S+I+R$. Using the equation of model (2), one can deduce that$$\dot{N}=\Lambda -\mu N-dI\le \Lambda -\mu N.$$Thus,$$0\le N\left(t\right)\le \frac{\Lambda}{\mu}+\left({\displaystyle N\left(0\right)-\frac{\Lambda}{\mu}}\right){e}^{-\mu t},$$The lower limit comes naturally from the fact that the model variables are non-negative ($t\in [0,T[$) since they monitor human populations.Thus, $0\le N\left(t\right)\le \frac{\Lambda}{\mu}$ whenever $0\le N\left(0\right)\le \frac{\Lambda}{\mu}$.Suppose $0\le N\left(0\right)\le \frac{\Lambda}{\mu}$. From the last equation of model (2) and using the fact that $I\left(t\right)\le \Lambda /\mu $ for all $t\ge 0$, one has:$$\dot{B}\le f\left(B\right)+\frac{\delta \Lambda}{\mu},$$Note that $\underset{B\to +\infty}{lim}f\left(B\right)=-\infty $ and $f\left(B\right)$ is a decreasing in [$\rho ;+\infty $[. The equation of the tangent of $f\left(B\right)$ at $B=\rho $ is given by $y\left(B\right)=-r\rho (\rho -\theta )B+r{\rho}^{2}(\rho -\theta )$.It follows that, for $B>\rho $, we have:$$\dot{B}\le r{\rho}^{2}(\rho -\theta )+\frac{\delta \Lambda}{\mu}-r\rho (\rho -\theta )B.$$Integrating the above differential inequality yields:$$0\le B\left(t\right)\le \frac{r{\rho}^{2}\mu (\rho -\theta )+\delta \Lambda}{r\rho \mu (\rho -\theta )}+\left({\displaystyle B\left(0\right)-\frac{r{\rho}^{2}\mu (\rho -\theta )+\delta \Lambda}{r\rho \mu (\rho -\theta )}}\right){e}^{-r\rho (\rho -\theta )t},$$Thus, as $t\to +\infty $,$$B\left(t\right)\le \frac{r{\rho}^{2}\mu (\rho -\theta )+\delta \Lambda}{r\rho \mu (\rho -\theta )}.$$

#### 3.2.2. Positively Invariant Sets

**Lemma 1.**

**Proof.**

**Lemma 2.**

**Proof.**

#### 3.3. Existence and Stability of Equilibria

#### 3.3.1. Existence of Disease-Free Equilibria

#### 3.3.2. Stability of Equilibria and Threshold Quantities

**Proposition 1.**

- (i)
- If ${\mathcal{R}}_{0}^{0}<1$, the DFE ${Q}_{0}$ is locally stable.
- (ii)
- If ${\mathcal{R}}_{0}^{\rho}<1$, the DFE ${Q}_{\rho}$ is locally stable.
- (iii)
- The DFE ${Q}_{\theta}$ is always unstable.

**Lemma 3.**

**Proof.**

**Hypothesis 1**

**Hypothesis 2**

**Hypothesis 3**

**Hypothesis 4**

**Hypothesis 5**

**${H}_{2}$**is satisfied.

**Theorem 2.**

**Proposition 2.**

**Proposition 3.**

**Proof.**

**Theorem 3.**

**Remark 1.**

#### 3.3.3. Endemic Equilibrium

**Lemma 4.**

- either one or three interior equilibria if ${\mathcal{R}}_{0}^{0}>1$,
- either zero or two endemic equilibria if ${\mathcal{R}}_{0}^{0}<1$.

**Lemma 5.**

**Proof.**

**Theorem 4.**

**Proposition 4.**

**Theorem 5.**

## 4. Discussion

#### 4.1. Numerical Simulations of the Proposed Model with Variation of Threshold Quantities

**Case 1:**${\mathcal{R}}_{0}\left({Q}_{\rho}\right)<1<{\mathcal{R}}_{0}^{0}$

**Case 2:**${\mathcal{R}}_{0}\left({Q}_{\rho}\right)<{\mathcal{R}}_{0}^{0}<\xi <1$

**Case 3:**$\xi <{\mathcal{R}}_{0}\left({Q}_{\rho}\right)<{\mathcal{R}}_{0}^{0}<1$

**Case 4:**$\xi <{\mathcal{R}}_{0}^{0}<1<{\mathcal{R}}_{0}\left({Q}_{\rho}\right)$

#### 4.2. Numerical Simulations of Threshold Quantities with Variation of Allee Parameters $\theta $ and $\rho $ and Bifurcation

- The curves obtained highlight the importance of emphasizing that the risk of disease outbreaks cannot be neglected for any value of $\theta $.
- Additionally, as $\theta $ approaches $\rho $, the probability of epidemic outbreaks significantly increases.

**Figure 11.**Simulation of ${\mathcal{R}}_{0}\left({Q}_{\rho}\right)$ for various values of $\theta \in [0;{10}^{6}]$ and $\rho \in [{10}^{6};{10}^{7}]$ when $\Lambda =70$, $\beta =0.001$ and the remaining parameters are consistent with those listed in Table 1.

## 5. Conclusions

- The model exhibits three disease-free equilibria related to three different real situations.
- The dynamics of the proposed model are determined by the threshold quantity ${\mathcal{R}}_{0}^{0}$.
- The phenomenon of bi-stability is observed, with backward and forward bifurcation.
- This research demonstrates that the Allee effect provides a robust framework for characterizing fluctuations in bacterial populations and the onset of cholera outbreaks

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Conflicts of Interest

## Abbreviations

DFE | Disease-Free Equilibrium |

GAS | Global Asymptotic Stability |

LHS | Latin Hypercube Sampling |

PRCC | Partial Rank Correlation Coefficients |

SIR | Susceptible Infected Recovered |

VNC | Viable but Non-Culturable |

## Appendix A. Calculation of Persistence Threshold for Disease-Free Equilibrium Qρ

## Appendix B. Proof of Lemma 4

- if ${\mathcal{R}}_{0}^{0}>1$ model (2) has:
- one endemic equilibrium if (${v}_{1}<{v}_{2}<0$, $P\left({v}_{2}\right)>0$) or if (${v}_{1}<0<{v}_{2}$, $P\left({v}_{2}\right)>0$) or if (${v}_{2}>{v}_{1}>0$, $P\left({v}_{1}\right)<0$, $P\left({v}_{2}\right)<0$) or if (${v}_{2}>{v}_{1}>0$, $P\left({v}_{1}\right)>0$, $P\left({v}_{2}\right)>0$),
- three endemic equilibria if (${v}_{2}>{v}_{1}>0$, $P\left({v}_{1}\right)<0$, $P\left({v}_{2}\right)>0$).

- if ${\mathcal{R}}_{0}^{0}<1$ model (2) has:
- no endemic equilibrium if ($0<{v}_{1}<{v}_{2}$, $P\left({v}_{1}\right)<0$, $P\left({v}_{2}\right)<0$) or if (${v}_{1}<{v}_{2}<0$, $P\left({v}_{1}\right)<0$) or if (${v}_{1}<{v}_{2}<0$, $P\left({v}_{1}\right)>0$, $P\left({v}_{2}\right)>0$) or if (${v}_{1}<0<{v}_{2}$, $P\left({v}_{1}\right)<0$, $P\left({v}_{2}\right)<0$),
- two endemic equilibria if (${v}_{1}<0<{v}_{2}$, $P\left({v}_{1}\right)<0$, $P\left({v}_{2}\right)>0$) or if ($0<{v}_{1}<{v}_{2}$, $P\left({v}_{1}\right)<0$, $P\left({v}_{2}\right)>0$).

## Appendix C. Proof of Theorem 4

**Theorem A1.**

- 1.
- $A={D}_{z}f(0,0)=\left(\frac{\partial {f}_{i}}{\partial {z}_{j}}(0,0)\right)$ is the linearization matrix of model (A4) around the equilibrium 0 with Φ evaluated at 0. Zero is a simple eigenvalue of A and other eigenvalues of A have negative real parts;
- 2.
- Matrix A has a right eigenvector u and a center eigenvector v (each corresponding to the zero eigenvalue). Let ${f}_{k}$ be the ${k}^{th}$ component of f and$$a=\sum _{k,i,j=1}^{n}{v}_{k}{u}_{i}{u}_{j}\frac{{\partial}^{2}{f}_{k}}{\partial {x}_{i}\partial {x}_{j}}(0,0)\phantom{\rule{1.em}{0ex}}\mathrm{and}\phantom{\rule{1.em}{0ex}}b=\sum _{k,i=1}^{n}{v}_{k}{u}_{i}\frac{{\partial}^{2}{f}_{k}}{\partial {x}_{i}\partial \Phi}(0,0),$$then, the local dynamics of the system around the equilibrium point 0 is totally determined by the signs of a and b.

- 1.
- $a>0$, $b>0$. When $\Phi <0$ with $|\Phi |\ll 1$, 0 is locally asymptotically stable and there exists a positive unstable equilibrium; when $0<\Phi \ll 1$, 0 is unstable and there exists a negative, locally asymptotically stable equilibrium;
- 2.
- $a<0$, $b<0$. When $\Phi <0$ with $|\Phi |\ll 1$, 0 is unstable; when $0<\Phi \ll 1$, 0, is locally asymptotically stable equilibrium, and there exists a positive unstable equilibrium;
- 3.
- $a>0$, $b<0$. When $\Phi <0$ with $|\Phi |\ll 1$, 0 is unstable and there exists a locally asymptotically stable negative equilibrium; when $0<\Phi \ll 1$, 0 is stable, and a positive unstable equilibrium appears;
- 4.
- $a<0$, $b>0$. When Φ changes from negative to positive, 0 changes its stability from stable to unstable. Correspondingly a negative unstable equilibrium becomes positive and locally asymptotically stable.

**Eigenvectors of**${J}_{{\beta}^{*}}$- For the case when ${\mathcal{R}}_{0}^{0}=1$, it can be shown that the Jacobian of model (A2) has a right eigenvector (corresponding to the zero eigenvalue), given by $U={({u}_{1},{u}_{2},{u}_{3},{u}_{4})}^{T}$, where$${u}_{1}=-\left[{\displaystyle \frac{\gamma \alpha}{\mu (\mu +\gamma )}}-{\displaystyle \frac{\omega}{\mu}}\right]{u}_{2},\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{u}_{2}={u}_{2}>0,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{u}_{3}={\displaystyle \frac{\alpha}{\mu +\gamma}}{u}_{2},\phantom{\rule{0.277778em}{0ex}}\mathrm{and}\phantom{\rule{0.277778em}{0ex}}{u}_{4}={\displaystyle \frac{\delta}{r\rho \theta}}{u}_{2}.$$Similarly, the components of the center eigenvectors of ${J}_{{\beta}^{*}}$ (corresponding to the zero eigenvalue), denoted by $V={({v}_{1},{v}_{2},{v}_{3},{v}_{4})}^{T}$, are given by:$${v}_{1}=0,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{v}_{2}={v}_{2}>0,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{v}_{3}=0,\phantom{\rule{0.277778em}{0ex}}\mathrm{and}\phantom{\rule{0.277778em}{0ex}}{v}_{4}={\displaystyle \frac{\beta {S}_{0}}{Kr\rho \theta}}{v}_{2}.$$
**Computation of b**- For the sign of
**b**, it can be shown that the associated non-vanishing partial derivatives of f are:$$\frac{{\partial}^{2}{f}_{1}}{\partial {x}_{4}\partial {\beta}^{*}}(0,0)=-\frac{{S}_{0}}{K}\phantom{\rule{2.em}{0ex}}\mathrm{and}\phantom{\rule{2.em}{0ex}}\frac{{\partial}^{2}{f}_{2}}{\partial {x}_{4}\partial {\beta}^{*}}(0,0)=\frac{{S}_{0}}{K}.$$It follows that:$$b={\displaystyle \frac{\omega}{{\beta}^{*}}}{v}_{2}{u}_{2}>0.$$ **Computation of a**- For system (A2), the associated non-zero partial derivatives of f (at the DFE ${\mathsf{Q}}_{0}$) are given by:$$\phantom{\rule{-5.0pt}{0ex}}\frac{{\partial}^{2}{f}_{1}}{\partial {x}_{1}{x}_{4}}(0,0)=-\frac{{\beta}^{*}}{K},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\frac{{\partial}^{2}{f}_{1}}{\partial {x}_{4}^{2}}(0,0)=\frac{2{\beta}^{*}{S}_{0}}{K},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\frac{{\partial}^{2}{f}_{2}}{\partial {x}_{1}\partial {x}_{4}}(0,0)=\frac{{\beta}^{*}}{K},\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\frac{{\partial}^{2}{f}_{2}}{\partial {x}_{4}^{2}}(0,0)=-\frac{2{\beta}^{*}{S}_{0}}{K}$$$$\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\mathrm{and}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\frac{{\partial}^{2}{f}_{4}}{\partial {x}_{4}^{2}}(0,0)=2r(\rho +\theta ).$$Then, it follows that:$$\begin{array}{cc}a\hfill & {\displaystyle ={v}_{2}\sum _{i,j=1}^{4}{u}_{i}{u}_{j}\frac{{\partial}^{2}{f}_{2}}{\partial {x}_{i}\partial {x}_{j}}(0,0)+{v}_{4}\sum _{i,j=1}^{4}{u}_{i}{u}_{j}\frac{{\partial}^{2}{f}_{4}}{\partial {x}_{i}\partial {x}_{j}}(0,0),}\hfill \\ \phantom{\rule{1.em}{0ex}}\hfill & ={v}_{2}{u}_{2}^{2}\left[{\displaystyle \frac{\alpha \gamma \omega}{(\mu +\gamma ){S}_{0}}}+2{\left({\displaystyle \frac{\omega K}{{\beta}^{*}{S}_{0}}}\right)}^{2}{\displaystyle \frac{\omega}{\delta}}r(\rho +\theta )-{\displaystyle \frac{\omega}{{S}_{0}}}\left({\displaystyle \frac{1}{\mu}}+{\displaystyle \frac{2K}{{\beta}^{*}}}\right)\right].\hfill \end{array}$$

**a**can be positive or negative. So, if $\mathbf{b}>0$, if $\mathbf{a}>0$, model (2) undergoes the phenomenon of backward bifurcation (see Theorem A1, item (1)). Also, if $\mathbf{a}<0$ (by Theorem A1, item (4)), we have established the result about the local stability of the endemic equilibrium ${Q}^{*}$ of model (2) for ${\mathcal{R}}_{0}^{0}>1$ but close to 1.

## Appendix D. Calculation of Persistence Threshold for Endemicity

## Appendix E. Proof of Theorem 5

**Eigenvectors of**${J}_{{\beta}^{*}}$- For the case when ${\mathcal{R}}_{0}\left({Q}_{\rho}\right)=1$, it can be shown that the Jacobian of model (A8) has a right eigenvector (corresponding to the zero eigenvalue), given by $U={({u}_{1},{u}_{2},{u}_{3},{u}_{4})}^{T}$, where$${u}_{1}=0,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{u}_{2}={u}_{2}>0,\phantom{\rule{0.277778em}{0ex}}\phantom{\rule{0.277778em}{0ex}}{u}_{3}={\displaystyle \frac{\alpha}{\mu +\gamma}}{u}_{2},\phantom{\rule{0.277778em}{0ex}}\mathrm{and}\phantom{\rule{0.277778em}{0ex}}{u}_{4}={\displaystyle \frac{\delta}{r\rho (\rho -\theta )}}{u}_{2}.$$Similarly, the components of the center eigenvectors of ${J}_{{\beta}^{*}}$ (corresponding to the zero eigenvalue), denoted by $V={({v}_{1},{v}_{2},{v}_{3},{v}_{4})}^{T}$, are given by:$${v}_{1}={v}_{1}>0,$$$${v}_{2}={\displaystyle \frac{[\beta \rho +\mu (\rho +K\left)\right]}{\beta \rho}}{v}_{1},$$$${v}_{3}={\displaystyle \frac{\gamma}{\gamma +\mu}}{v}_{1},$$$${v}_{4}=\left[{\displaystyle \frac{\omega [\beta \rho +\mu (\rho +K\left)\right]}{\beta \rho}}-{\displaystyle \frac{\alpha \gamma}{\gamma +\mu}}\right]{\displaystyle \frac{{v}_{1}}{\delta}}.$$
**Computation of b:**- For the sign of
**b**, it can be shown that the associated non-vanishing partial derivatives of f are:$$\frac{{\partial}^{2}{g}_{1}}{\partial {y}_{1}\partial {\beta}^{*}}(0,0)=-{\displaystyle \frac{\rho}{K+\rho}},$$$$\frac{{\partial}^{2}{g}_{1}}{\partial {y}_{1}\partial {\beta}^{*}}(0,0)=-{\displaystyle \frac{K\Lambda}{(k+\rho )[\beta \rho +\mu (\rho +K\left)\right]}},$$$$\frac{{\partial}^{2}{g}_{2}}{\partial {y}_{1}\partial {\beta}^{*}}(0,0)={\displaystyle \frac{\rho}{K+\rho}}.$$$$\mathrm{and}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\frac{{\partial}^{2}{g}_{2}}{\partial {y}_{1}\partial {\beta}^{*}}(0,0)={\displaystyle \frac{K\Lambda}{(k+\rho )[\beta \rho +\mu (\rho +K\left)\right]}}.$$It follows that:$$b=\frac{\mu (K+\rho )\omega}{{\beta}^{*}\rho}{u}_{2}>0.$$ **Computation of a:**- For system (A8), the associated non-zero partial derivatives of g (at the DFE ${\mathsf{Q}}_{\rho}$) are given by:$$\frac{{\partial}^{2}{g}_{1}}{\partial {x}_{1}{x}_{4}}(0,0)=-\frac{{\beta}^{*}K}{{(K+\rho )}^{2}},$$$$\frac{{\partial}^{2}{g}_{1}}{{\partial}^{2}{x}_{4}}(0,0)=\frac{2{\beta}^{*}K\Lambda}{{(K+\rho )}^{2}[\beta \rho +\mu (\rho +K)]},$$$$\frac{{\partial}^{2}{g}_{2}}{\partial {x}_{1}{x}_{4}}(0,0)=\frac{{\beta}^{*}K}{{(K+\rho )}^{2}}g,$$$$\mathrm{and}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\phantom{\rule{0.166667em}{0ex}}\frac{{\partial}^{2}{g}_{1}}{{\partial}^{2}{x}_{4}}(0,0)=-\frac{2{\beta}^{*}K\Lambda}{{(K+\rho )}^{2}[\beta \rho +\mu (\rho +K)]}.$$Then, it follows that:$$a=-{\left({\displaystyle \frac{\delta}{r\rho (\rho -\theta )}}\right)}^{2}\left[{\displaystyle \frac{2{\beta}^{*}K\Lambda \mu}{(K+\rho )[\beta \rho +\mu (\rho +K)]{\beta}^{*}\rho}+\left({\displaystyle \frac{\omega [\beta \rho +\mu (\rho +K\left)\right]}{\beta \rho \delta}}-{\displaystyle \frac{\alpha \gamma}{(\gamma +\mu )\delta}}\right)}\right]{v}_{1}{u}_{2}^{2}<0.$$

## References

- World Health Organization, Cholera. Available online: https://www.who.int/news-room/fact-sheets/detail/cholera. (accessed on 30 March 2023).
- Lipp, E.K.; Huq, A.; Colwell, R.R. Effects of global climate on infectious disease: The cholera model. Clin. Microbiol. Rev.
**2002**, 15, 757–770. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Islam, M.S.; Drasar, B.S.; Sack, R.B. The aquatic environment as a reservoir of Vibrio cholerae: A review. J. Diarrhoeal Dis. Res.
**1993**, 11, 197–206. [Google Scholar] [PubMed] - Kolaye, G.; Bowong, S.; Houe, R.; Aziz-Alaoui, M.A.; Cadivel, M. Mathematical assessment of the role of environmental factors on the dynamical transmission of cholera. Commun. Nonlinear Sci. Numer. Simul.
**2019**, 67, 203–222. [Google Scholar] [CrossRef] [Green Version] - Codeço, C.T. Endemic and epidemic dynamics of cholera: The role of the aquatic reservoir. BMC Infect. Dis.
**2001**, 1, 1. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Hartley, D.M.; Morris, J.G., Jr.; Smith, D.L. Hyperinfectivity: A critical element in the ability of V. cholerae to cause epidemics? PLoS Med.
**2006**, 3, e7. [Google Scholar] [CrossRef] [PubMed] - Kolaye, G.; Damakoa, I.; Bowong, S.; Houe, R.; Békollè, D. Theoretical assessment of the impact of climatic factors in a Vibrio cholerae model. Acta Biotheor.
**2018**, 66, 279–291. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kolaye, G.; Damakoa, I.; Bowong, S.; Houe, R.; Békollè, D. A mathematical model of cholera in a periodic environment with control actions. Int. J. Biomath.
**2020**, 13, 2050025. [Google Scholar] [CrossRef] - Colwell, R.; Brayton, P.; Grimes, D.; Roszak, D.; Huq, S.; Palmer, L. Viable but non-culturable Vibrio cholerae and related pathogens in the environment: Implications for release of genetically engineered microorganisms. Nat. Biotechnol.
**1985**, 3, 817–820. [Google Scholar] [CrossRef] - Roszak, D.; Colwell, R. Survival strategies of bacteria in the natural environment. Microbiol. Rev.
**1987**, 51, 365–379. [Google Scholar] [CrossRef] [PubMed] - Campus de Microbiologie Medicale, VIBRIO. Available online: http://www.microbes-edu.org/etudiant/vibrio.html (accessed on 18 December 2022).
- Cuzin, L.; Delpierre, C. Épidémiologie des maladies infectieuses. EMC Mal. Infect.
**2005**, 2, 157–162. [Google Scholar] [CrossRef] - Statistiques Mondiales. Available online: http://www.statistiques-mondiales.com/cameroun.htm (accessed on 18 December 2022).
- Bayleyegn, Y.N. Mathematical Analysis of a Model of Cholera Transmission Dynamics. Master’s Thesis, African Institute for Mathematical Sciences (AIMS), Cape Town, South Africa, 2009. [Google Scholar]
- King, A.A.; Ionides, E.L.; Pascual, M.; Bouma, M.J. Inapparent infections and cholera dynamics. Nature
**2008**, 454, 877–880. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Chitnis, N.; Hyman, J.M.; Cushing, J.M. Determining important parameters in the spread of malaria through the sensitivity analysis of a mathematical model. Bull. Math. Biol.
**2008**, 70, 1272–1296. [Google Scholar] [CrossRef] [PubMed] - Blower, S.M.; Dowlatabadi, H. Sensitivity and uncertainty analysis of complex models of disease transmission: An HIV model, as an example. Int. Stat. Rev./Rev. Int. Stat.
**1994**, 62, 229–243. [Google Scholar] [CrossRef] - Marino, S.; Hogue, I.B.; Ray, C.J.; Kirschner, D.E. A methodology for performing global uncertainty and sensitivity analysis in systems biology. J. Theor. Biol.
**2008**, 254, 178–196. [Google Scholar] [CrossRef] [PubMed] [Green Version] - Kamgang, J.C.; Sallet, G. Global asymptotic stability for the disease free equilibrium for epidemiological models. C. R. Math.
**2005**, 341, 433–438. [Google Scholar] [CrossRef] - Kamgang, J.C. Contribution à la stabilisation des systèmes mécaniques: Contribution à l’étude de la stabilité des modèles épidémiologiques. Ph.D. Thesis, Université Paul Verlaine, Metz, France, 2003. [Google Scholar]
- 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] [PubMed] - Castillo-Chavez, C.; Song, B. Dynamical models of tuberculosis and their applications. Math. Biosci. Eng
**2004**, 1, 361–404. [Google Scholar] [CrossRef] [PubMed] - Bouma, M.J.; Pascual, M. Seasonal and interannual cycles of endemic cholera in Bengal 1891–1940 in relation to climate and geography. In Proceedings of the The Ecology and Etiology of Newly Emerging Marine Diseases; Porter, J., Ed.; Springer: Dordrecht, The Netherlands, 2001; pp. 147–156. [Google Scholar] [CrossRef]
- Carr, J. Applications of Centre Manifold Theory; Springer Science & Business Media: Berlin/Heidelberg, Germany, 2012; Volume 35. [Google Scholar]

**Figure 2.**Simulation of model (2) when $\Lambda =10$, $\beta =0.0001$ and $\u03f5=50,000$ (so that $\theta -\u03f5=9.5\times {10}^{5}$, ${\mathcal{R}}_{0}^{0}=0.0103$, and $\xi =0.0495$) using various initial conditions chosen in attraction domain of ${Q}_{0}$. The remaining parameters are consistent with those listed in Table 1. Each subfigure corresponds to each state of model (2). (

**a**) Susceptible. (

**b**) Infected. (

**c**) Recovered. (

**d**) Bacteria.

**Figure 3.**Simulation of model (2) when $\Lambda =50$, $\beta =0.0015$, $\theta ={10}^{6}$ and $\u03f5=50,000$ (so that $\theta -\u03f5=9.5\times {10}^{5}$, $\xi =0.0495$, and ${\mathcal{R}}_{0}^{0}=0.7702$) using various initial conditions $\left(S\right(0),I(0),R(0),B(0\left)\right)$ chosen in the domains ${\mathcal{D}}_{1}=]0;50,000]\times ]0;10]\times ]0;50]\times \left\{2\times {10}^{5},3\times {10}^{5},4\times {10}^{5},5\times {10}^{5}\right\}$ and ${\mathcal{D}}_{2}=]0;50,000]\times ]0;10]\times ]0;50]\times \left\{7.5\times {10}^{5},8\times {10}^{5},9\times {10}^{5},9.5\times {10}^{5}\right\}$. The remaining parameters are consistent with those listed in Table 1. Each subfigure corresponds to each state of model (2). (

**a**) Susceptible. (

**b**) Infected. (

**c**) Recovered. (

**d**) Bacteria.

**Figure 4.**Simulation of model (2) when $d=0.7$, $\gamma =0.5$, $\alpha =0.45$ (so that ${\mathcal{R}}_{0}^{0}=0.0058$, $\xi =0.0505$ and ${\mathcal{R}}_{0}\left({Q}_{\rho}\right)=0.0033$) and initial conditions are chosen in ${\mathcal{D}}_{3}=]0;50,000]\times ]0;10]\times ]0;50]\times \left[0.9\times {10}^{6},1.2\times {10}^{8}\right]$. The remaining parameters are consistent with those listed in Table 1. Each subfigure corresponds to each state of model (2). (

**a**) Susceptible. (

**b**) Infected. (

**c**) Recovered. (

**d**) Bacteria. (

**e**) Zoom of bacteria graph.

**Figure 5.**Simulation of model (2) when $\Lambda =30$, $\beta =0.01$ (so that ${\mathcal{R}}_{0}^{0}=3.0809>1$) and various initial conditions have been taken in ${\mathcal{D}}_{3}=]0;7.5\times {10}^{6}]\times ]0;1.5\times {10}^{3}]\times ]0;7.5\times {10}^{3}]\times \left[0,1.5\times {10}^{8}\right]$. The remaining parameters are consistent with those listed in Table 1. Each subfigure corresponds to each state of model (2). (

**a**) Susceptible. (

**b**) Infected. (

**c**) Recovered. (

**d**) Bacteria. (

**e**) Zoom of bacteria graph.

**Figure 6.**Simulation of model (2) when $\Lambda =50$, $\beta =0.02$ and $r={10}^{-20}$ (so that ${\mathcal{R}}_{0}\left({Q}_{\rho}\right)=0.0526$ and ${\mathcal{R}}_{0}^{0}=1.7594\times {10}^{3}$) and various initial conditions in ${\mathbb{R}}_{+}^{4}$. The remaining parameters are consistent with those listed in Table 1. Each subfigure corresponds to a specific state of model (2). (

**a**) Infected, (

**b**) Bacteria.

**Figure 7.**Simulation of model (2) when all parameters values are as in Table 1 (so that ${\mathcal{R}}_{0}\left({Q}_{\rho}\right)=0.0023$, ${\mathcal{R}}_{0}^{0}=0.0045$, $\u03f5=50,000$ and $\xi =0.0495$) and various initial conditions chosen in ${\mathcal{D}}_{3}=]0;50,000]\times ]0;10]\times ]0;50]\times \left[{10}^{5},9.5\times {10}^{5}\right]$ and in ${\mathcal{D}}_{4}=]0;50,000]\times ]0;10]\times ]0;50]\times \left[{10}^{6},2\times {10}^{8}\right]$. Each subfigure corresponds to a specific state of model (2). (

**a**) Infected, (

**b**) Bacteria, (

**c**) Zoom on Bacteria population.

**Figure 8.**Simulation of model (2) when $\Lambda =30$, $\beta =0.001$ and $\u03f5=1000$ (so that $\xi =0.0010,$${\mathcal{R}}_{0}\left({Q}_{\rho}\right)=0.027$ and ${\mathcal{R}}_{0}^{0}=0.308$). The remaining parameters are consistent with those listed in Table 1. Various initial conditions chosen in ${\mathcal{D}}_{5}=]0;50,000]\times ]0;10]\times ]0;50]\times \left[{10}^{5},5\times {10}^{5}\right]$ and ${\mathcal{D}}_{6}=]0;50,000]\times ]0;10]\times ]0;50]\times \left[7\times {10}^{5},2\times {10}^{8}\right]$. Each subfigure corresponds to a specific state of model (2). (

**a**) Infected, (

**b**) Bacteria, (

**c**) Zoom on Bacteria population.

**Figure 9.**Simulation of model (2) when $\beta =0.1$, $\mu =1.04\times {10}^{-3}$, $\theta =0.999\times {10}^{8}$ and $\u03f5=50,000$ (so that $\xi =5.005\times {10}^{-8},{\mathcal{R}}_{0}^{0}=0.0104$ and ${\mathcal{R}}_{0}\left({Q}_{\rho}\right)=4.5370$). The remaining parameters are consistent with those listed in Table 1. Various initial conditions chosen in ${\mathcal{D}}_{7}=]0;50,000]\times ]0;10]\times ]0;50]\times \left[{10}^{5},2\times {10}^{8}\right]$. Each subfigure corresponds to the two following states of model (2): (

**a**) Infected. (

**b**) Bacteria.

**Figure 10.**Simulation of ${\mathcal{R}}_{0}^{0}$ for various values of $\theta \in [0;{10}^{6}]$ and $\rho \in [{10}^{6};{10}^{8}]$ when $r={10}^{-13}$ and the remaining parameters are consistent with those listed in Table 1.

**Figure 12.**Region of plane ($\rho $, $\theta $) in which ${Q}_{0}$ and ${Q}_{\rho}$ are stable when $r={10}^{-13}$ and the remaining parameters are consistent with those listed in Table 1. (

**a**) For $\u03f5=1$, (

**b**) For $\u03f5=50,000$.

**Figure 13.**Bifurcation Diagram of ${\mathcal{R}}_{0}^{0}$ when (

**a**) $\delta =23$ (

**b**) $\lambda =20$ and $\mu =0.05$ (

**c**) $\lambda =20$, $\mu =0.05$, and $\delta =33$. Each of the three curves in each subfigure is associated with a specific solution of Equation (28).

**Table 1.**Numerical values for the parameters of model (2).

Definition | Symbol | Estimated | Source |
---|---|---|---|

Recruitment rate | $\Lambda $ | 10 day${}^{-1}$ | Assumed |

Bacteria ingestion rate | $\beta $ | 0.0001 person${}^{-1}$ day${}^{-1}$ | Assumed |

Human population death rate | $\mu $ | $0.0104$ day${}^{-1}$ | [13] |

Bacteria shedding rate | $\delta $ | 70 cells/(mL day) | [14] |

Half-saturation constant | K | ${10}^{7}$ cells/person/mL/day | Assumed |

Cholera related death | d | 0.6 year${}^{-1}$ | Assumed |

Loss of immunity rate | $\gamma $ | 0.01 day${}^{-1}$ | Assumed |

Recovery rate | $\alpha $ | 0.045 day${}^{-1}$ | [15] |

Growth rate of Vibrios | r | 1 × 10${}^{-18}$ day${}^{-1}$ | Assumed |

Carrying capacity bacterial population | $\rho $ | 1 × 10${}^{8}$ cell/mL | Assumed |

Allee threshold bacterial population | $\theta $ | 1 × 10${}^{6}$ cell/mL | Assumed |

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Kolaye Guilsou, G.; Aziz-Alaoui, M.-A.; Houé Ngouna, R.; Archimede, B.; Bowong, S.
Gaining Profound Knowledge of Cholera Outbreak: The Significance of the Allee Effect on Bacterial Population Growth and Its Implications for Human-Environment Health. *Sustainability* **2023**, *15*, 10384.
https://doi.org/10.3390/su151310384

**AMA Style**

Kolaye Guilsou G, Aziz-Alaoui M-A, Houé Ngouna R, Archimede B, Bowong S.
Gaining Profound Knowledge of Cholera Outbreak: The Significance of the Allee Effect on Bacterial Population Growth and Its Implications for Human-Environment Health. *Sustainability*. 2023; 15(13):10384.
https://doi.org/10.3390/su151310384

**Chicago/Turabian Style**

Kolaye Guilsou, Gabriel, Moulay-Ahmed Aziz-Alaoui, Raymond Houé Ngouna, Bernard Archimede, and Samuel Bowong.
2023. "Gaining Profound Knowledge of Cholera Outbreak: The Significance of the Allee Effect on Bacterial Population Growth and Its Implications for Human-Environment Health" *Sustainability* 15, no. 13: 10384.
https://doi.org/10.3390/su151310384