# Effect of Transmission and Vaccination on Time to Dominance of Emerging Viral Strains: A Simulation-Based Study

## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

`R`statistical software [30].

#### 2.1. Stochastic Multi-Strain SIR Model with Cross-Immunity

#### 2.1.1. Modified Multi-Strain SIR Model

#### 2.1.2. Continuous-Time Markov Chain Multi-Strain SIR Model

#### 2.2. Time to Dominance (TTD)

#### 2.3. Simulations and Factorial Experiment

`emmeans`package in

`R`[41]. A significance level of 0.05 was used in testing hypotheses and constructing confidence intervals.

## 3. Results

#### 3.1. Interaction Plots

#### 3.2. Vaccination Rate $\nu $

#### 3.3. Initial Vaccination Coverage $v\left(0\right)$

#### 3.4. Relative Transmissibility of Emergent Strain (ESTR)

## 4. Discussion

## Funding

## Institutional Review Board Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Abbreviations

SIR Model | Susceptible-Infected-Recovered Model |

TTD | Time to Dominance |

ESTR | Emergent Strain Transmission Ratio |

## Appendix A

Coefficient | Definition |
---|---|

s | Proportion of susceptible individuals |

v | Proportion of vaccinated individuals |

${i}_{1}$ | Proportion of individuals infected by the existing strain |

${i}_{2}$ | Proportion of individuals infected by the emergent strain |

${r}_{1}$ | Proportion of individuals that recovered from the existing strain |

${r}_{2}$ | Proportion of individuals that recovered from the emergent strain |

$\beta $ | Transmission coefficient of the existing strain |

${\beta}^{\prime}$ | Transmission coefficient of the emergent strain |

$\gamma $ | Recovery coefficient of the existing strain |

${\gamma}^{\prime}$ | Recovery coefficient of the existing strain |

$\nu $ | Vaccination coefficient |

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**Figure 1.**The schematic diagram of the multi-strain SIR model for emerging viral diseases with cross-immunity. The diagram includes the compartments and transition rates between compartments. Definitions of the transition coefficients are available in Table A1.

**Figure 2.**A representative simulation run for the factor combination ($\beta $, ${\beta}^{\prime}/\beta $, $\nu $, $v\left(0\right)$) = (1.4, 1.5, 0.5, 0.25). The proportion of emergent strain infections follows a logistic growth curve.

**Figure 3.**A sample run for the factor combination ($\beta $, ${\beta}^{\prime}/\beta $, $\nu $, $v\left(0\right)$) = (1.4,1.25,0,0). The proportion of emergent strain infections among the infected did not go above 0.5, despite observing a steady increase in infections.

**Figure 4.**A sample run for the factor combination ($\beta $, ${\beta}^{\prime}/\beta $, $\nu $, $v\left(0\right)$) = (2.1, 2, 0.5, 0.5). Both proportions dropped to zero at time $t=12$, due to complete recovery after complete infection of all infected individuals in the population.

**Figure 5.**The interaction plots of the vaccination rate and emergent strain transmission ratio (ESTR) for different combinations of the initial vaccination coverage (initialvaxx), and transmission coefficient of the existing strain (beta).

Factor | Simulation Parameter | Values |
---|---|---|

Transmission coefficient of existing strain | $\beta $ | 1.4, 1.75, 2.1 |

Emergent strain transmission ratio (ESTR) | ${\beta}^{\prime}/\beta $ | 1.25, 1.5, 1.75, 2 |

Vaccination coefficient | $\nu $ | 0, 0.25, 0.5 |

Initial vaccination coverage | $v\left(0\right)$ | 0, 0.25, 0.5, 0.75 |

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**MDPI and ACS Style**

Fudolig, M. Effect of Transmission and Vaccination on Time to Dominance of Emerging Viral Strains: A Simulation-Based Study. *Microorganisms* **2023**, *11*, 860.
https://doi.org/10.3390/microorganisms11040860

**AMA Style**

Fudolig M. Effect of Transmission and Vaccination on Time to Dominance of Emerging Viral Strains: A Simulation-Based Study. *Microorganisms*. 2023; 11(4):860.
https://doi.org/10.3390/microorganisms11040860

**Chicago/Turabian Style**

Fudolig, Miguel. 2023. "Effect of Transmission and Vaccination on Time to Dominance of Emerging Viral Strains: A Simulation-Based Study" *Microorganisms* 11, no. 4: 860.
https://doi.org/10.3390/microorganisms11040860