# Mathematical Analysis of an SIVRWS Model for Pertussis with Waning and Naturally Boosted Immunity

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## Abstract

**:**

## 1. Introduction

## 2. Model Formulation

**Proposition**

**1.**

## 3. Equilibrium and Stability Analyses

#### 3.1. Infection-Free Equilibrium and the Control Reproduction Number

**Proposition**

**2.**

#### 3.2. Endemic Equilibriums and the Bifurcation Direction

**Proposition**

**3.**

**Proposition**

**4.**

## 4. Uniform Persistence

- $\left({C}_{1}\right)$
- For $\frac{d{X}_{1}}{dt}=F({X}_{1},0)$, ${X}_{1}^{0}$ is globally asymptotically stable.
- $\left({C}_{2}\right)$
- $G({X}_{1},{X}_{2})=A{X}_{2}-\widehat{G}({X}_{1},{X}_{2})$, where $\widehat{G}({X}_{1},{X}_{2})\u2a7e0$ for $({X}_{1},{X}_{2})\in \Omega $,

- where the Jacobian matrix $A={D}_{{X}_{2}}G({X}_{1}^{0},0)$ has all non-negative off-diagonal elements, and $\Omega $ is the region where the model makes biological sense. Then, we present the following proposition, whose proof is deferred to Appendix C.

**Proposition**

**5.**

**Proposition**

**6.**

**Proposition**

**7.**

## 5. Controllability of the Infection

**Proposition**

**8.**

## 6. Impact of Waning and Natural Immune Boosting on Disease Outcomes

## 7. Summary, Conclusions and Future Work

- The higher the natural boosting immunity is, the lower the endemic prevalence of infection is. In other words, ignoring the natural boosting of immunity overestimates the endemic prevalence of infection;
- The faster the progression of secondary susceptible individuals is, the higher the endemic prevalence of infection is;
- The shorter the period of immunity acquired by either vaccination or experiencing natural infection, the higher the reproduction number and the endemic prevalence of infection, and therefore, the higher the effort needed to eliminate the infection is.

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Positivity and Boundedness—Proof of Proposition 1

**Proof.**

## Appendix B. Direction of Bifurcation by Using Implicit Function Theorem

## Appendix C. Proof of Proposition 5

**Proof.**

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**Figure 2.**A bifurcation diagram showing the endemic prevalence of infection as a function of the control reproduction number, with parameter values as shown in Table 3.

**Figure 3.**Time-dependent solutions, with various initial conditions, for the proportion of infected individuals $I\left(t\right)$. Each colored curve represents a solution. Simulations have been done with parameter values as shown in Table 3, except $\beta $ that has been chosen to keep the reproduction number ${\Re}_{v}=0.8<1$.

**Figure 4.**Time-dependent solutions, with various initial conditions, for the proportion of non-infected sub-populations. Each colored curve represents a solution. The subfigures (

**A**–

**D**) show solutions of the proportion of primary susceptible ${S}_{1}\left(t\right)$, vaccinated $V\left(t\right)$, waned $W\left(t\right)$, and secondary susceptible ${S}_{2}\left(t\right)$ individuals, respectively. Simulations have been done with parameter values as shown in Table 3, except $\beta $ that has been chosen to keep the reproduction number ${\Re}_{v}=0.8<1$. The dotted horizontal line in each subfigure represents the infimum.

**Figure 5.**Time-dependent solutions, with various initial conditions, for the proportion of non-infected sub-populations. Each colored curve represents a solution. The subfigures (

**A**–

**D**) show solutions of the proportion of primary susceptible ${S}_{1}\left(t\right)$, vaccinated $V\left(t\right)$, waned $W\left(t\right)$, and secondary susceptible ${S}_{2}\left(t\right)$ individuals, respectively. Simulations have been done with parameter values as shown in Table 3, where the reproduction number ${\Re}_{v}=13>1$. The dotted horizontal line in each subfigure represents the infimum.

**Figure 6.**Time-dependent solutions, with various initial conditions, for the proportion of infected individuals $I\left(t\right)$, with parameter values as shown in Table 3, where the reproduction number ${\Re}_{v}=13>1$. Each colored curve represents a solution. Part (

**A**) shows the results, with Y-axis being in a linear scale, while part (

**B**) shows it in a logarithmic scale.

**Figure 7.**The critical contact rate ${\beta}_{v}$ as a function of the vaccination coverage level p for various values of ${P}_{V},{P}_{W}$ and r, while keeping other model parameters’ value as shown in Table 3. The curve divides the ($p,\beta $) plane into the two regions of disappearance (below the curve) and persistence (above the curve) of infection. The subfigures (

**A**,

**B**) show that the disappearance region extends with the increase in the proportions ${P}_{V}$ and ${P}_{W}$, respectively, while the subfigure (

**C**) shows that it increases with the decrease in the relative susceptibility r of secondary (with respect to primary) susceptible individuals.

**Figure 8.**The endemic prevalence of infection as a function of the control reproduction number with various values of the natural boosting immunity parameter g, where all other parameters have been kept fixed and with values as shown in Table 3.

**Figure 9.**The endemic prevalence of infection ${I}^{*}$ as a function of the control reproduction number ${\Re}_{v}$ for several values of the rate $\sigma $ while the remaining parameters’ values have been kept as in Table 3.

**Figure 10.**The endemic prevalence of infection ${I}^{*}$ as a function of the control reproduction number ${\Re}_{v}$ for various values of the rate $\alpha $, while the other parameters’ value added are kept as shown in Table 3.

Symbol | Description |
---|---|

${S}_{1}\left(t\right)$ | Time-dependent proportion of individuals who have never experienced the infection. |

$V\left(t\right)$ | Time-dependent proportion of individuals who got vaccinated immediately after birth. |

$I\left(t\right)$ | Time-dependent proportion of individuals who are infected and capable of transmitting the infection. |

$R\left(t\right)$ | Time-dependent proportion of individuals who recently recovered from the infection or whose immunity has been naturally boosted due to re-exposure to the infection. |

$W\left(t\right)$ | Time-dependent proportion of individuals, whose immunity acquired by vaccination or due to recovery after infection, declined but are still protected from acquiring the infection. |

${S}_{2}\left(t\right)$ | Time-dependent proportion of individuals who partially lost their acquired immunity. |

Parameter | Description | Dimension |
---|---|---|

$\mu $ | Birth and natural death rate. | Time${}^{-1}$ |

$\beta $ | Successful contact rate between primary susceptible and infected individuals. | Time${}^{-1}$ |

$\gamma $ | Rate of recovery from the infection. | Time${}^{-1}$ |

$\sigma $ | Recovered individuals’ rate of waning immunity | Time${}^{-1}$ |

$\alpha $ | The progression rate of waned individuals to become secondary susceptible | Time${}^{-1}$ |

p | Proportion of newborns who get vaccinated immediately after birth. | Dimensionless |

b | Relative loss in vaccine-acquired immunity with respect to naturally acquired immunity. | Dimensionless |

r | Relative susceptibility of secondary susceptible with respect to primary susceptible individuals. | Dimensionless |

g | A rescaling parameter that accounts for naturally boosting the immunity due to re-exposure to the infection. | Dimensionless |

Parameter | Value | Range | Unit | Reference |
---|---|---|---|---|

$\mu $ | $1/73$ | - | per year | [26] |

${\Re}_{v}$ | 13 | [12–15] | Dimensionless | [27] |

$\beta $ | - | - | per year | Assumed |

$\gamma $ | 25 | [8.67–52] | per year | [12,28,29] |

$\sigma $ | 0.05 | [0.04–0.25] | per year | [12,15,28,29] |

$\alpha $ | 0.16 | - | per year | Assumed |

p | 0.83 | [0.8–0.9] | Dimensionless | [30] |

$b\sigma $ | 0.1 | [0.083–0.25] | Dimensionless | [12,15,28,29] |

b | 2 | - | Dimensionless | [12] |

r | $0.8$ | [0, 1] | Dimensionless | [12] |

g | $0.3$ | - | Dimensionless | Assumed |

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**MDPI and ACS Style**

Safan, M.; Barley, K.; Elhaddad, M.M.; Darwish, M.A.; Saker, S.H.
Mathematical Analysis of an SIVRWS Model for Pertussis with Waning and Naturally Boosted Immunity. *Symmetry* **2022**, *14*, 2288.
https://doi.org/10.3390/sym14112288

**AMA Style**

Safan M, Barley K, Elhaddad MM, Darwish MA, Saker SH.
Mathematical Analysis of an SIVRWS Model for Pertussis with Waning and Naturally Boosted Immunity. *Symmetry*. 2022; 14(11):2288.
https://doi.org/10.3390/sym14112288

**Chicago/Turabian Style**

Safan, Muntaser, Kamal Barley, Mohamed M. Elhaddad, Mohamed A. Darwish, and Samir H. Saker.
2022. "Mathematical Analysis of an SIVRWS Model for Pertussis with Waning and Naturally Boosted Immunity" *Symmetry* 14, no. 11: 2288.
https://doi.org/10.3390/sym14112288