# A Generalized Mathematical Model of Toxoplasmosis with an Intermediate Host and the Definitive Cat Host

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## Abstract

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## 1. Introduction

## 2. Mathematical Model

- Cats can be susceptible (S), infected (I), or recovered/vaccinated (${V}_{R}$).
- The mouse population ${N}_{m}$ is divided into two classes, namely, susceptible (${S}_{m}$) and infectious (${I}_{m}$).
- Oocysts (O) denotes the amount of T. gondii oocysts.
- The mouse and cat populations are assumed to be constant.
- A susceptible mouse or cat moves to the infectious class after contact with T. gondii oocysts (at rates ${\beta}_{m}$ and $\beta $, respectively).
- A susceptible cat flows to the recovered/vaccinated class ${V}_{R}$ at a rate $\gamma $. An infectious cat flows to the vaccinated/recovered subpopulation ${V}_{R}\left(t\right)$ at a rate $\alpha $.
- T. gondii oocysts $O\left(t\right)$ are generated by infectious cats $I\left(t\right)$.
- ${\mu}_{0}$ is the degradation/removal rate of T. gondii oocysts in the environment.
- $\mu $ represents the birth and death rate for cats.
- ${\mu}_{c}$ represents the birth and death rate of mice.
- A proportion q of vertical transmittal is assumed in the cat population.
- A proportion p of vertical transmittal is assumed in the mouse population.

## 3. Stability Analysis

**Theorem 1.**

**Proof.**

## 4. Mathematical Model with Full Vertical Transmission in the Mouse Population

#### 4.1. Toxoplasmosis-Free Steady State Considering Full Vertical Transmission in Mice

**Remark 1.**

**Remark 2.**

#### 4.2. Global Stability of Toxoplasmosis-Free Equilibrium Points

**Theorem 2.**

**Proof.**

#### 4.3. Toxoplasmosis-Endemic Steady State

**Theorem 3.**

#### 4.4. Global Stability of the Toxoplasmosis-Endemic Equilibrium Point

**Theorem 4.**

**Proof.**

## 5. Mathematical Model without Full Vertical Transmission in the Mouse Population

#### 5.1. Stability Analysis of the Toxoplasmosis-Free Equilibrium Point

#### 5.2. Stability Analysis of the Toxoplasmosis Endemic Equilibrium Point

**Theorem 5.**

## 6. Numerical Simulations of Different Scenarios

#### 6.1. Toxoplasmosis-Free Scenario (${\mathcal{R}}_{0}<1$)

#### 6.2. Toxoplasmosis Endemic Scenario (${\mathcal{R}}_{0}>1$)

## 7. Conclusions

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**Chart of the mathematical model of generalized toxoplasmosis transmission (1) for mouse and cat populations taking into account generalized vertical transmission.

**Figure 2.**Trajectories of the subpopulations with no vaccination and ${\mathcal{R}}_{0}=0.98$. The initial subpopulations are near the toxoplasmosis-free point ${F}^{*}$. On the right-hand side, the initial amount of oocysts $O\left(0\right)$ is very large.

**Figure 3.**Trajectories of the subpopulations when ${\mathcal{R}}_{0}=0.98$. The initial subpopulations are distant from the toxoplasmosis-free steady state ${F}^{*}$. The left-hand side shows the case in which vaccination is implemented, while the right-hand side shows the case with no vaccination and a larger death/clearance rate ${\mu}_{0}$.

**Figure 4.**Dynamics of the different classes when ${\mathcal{R}}_{0}=1.04$ and the initial subpopulations are distant from the endemic equilibrium ${E}_{0}^{*}$. Note that the oocyst and infected cat populations never become extinct, as ${\mathcal{R}}_{0}>1$.

Parameter | Description | Value |
---|---|---|

$\mu $ | Death and birth rates (cats) | $1/260$ (1/weeks) [52] |

$\alpha $ | Shedding period | $1/2$ (1/weeks) [4] |

${\mu}_{0}$ | Clearance rate | $1/26$ (1/day) [4,36] |

k | Oocysts per day (cat) | $20\times {10}^{6}$ (1/day) [53] |

$\beta $ | Transmission rate | varying |

${\beta}_{m}$ | Transmission rate | $0.1\times {10}^{-8}$ |

$\gamma $ | Vaccination rate | varying |

Subpopulations | Description | Initial values |

S | Susceptible cats | 0.45 [36,54,55] |

I | Infected cats | 1/130 [36,54,55] |

${V}_{R}$ | Vaccinated/Recovered | $1-S\left(0\right)-V\left(0\right)$ |

O | Occysts | $5.92\times {10}^{6}$ |

${S}_{m}$ | Susceptible mice (proportion) | 0.9 |

${I}_{m}$ | Infected mice (proportion) | 0.1 |

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**MDPI and ACS Style**

Sultana, S.; González-Parra, G.; Arenas, A.J. A Generalized Mathematical Model of Toxoplasmosis with an Intermediate Host and the Definitive Cat Host. *Mathematics* **2023**, *11*, 1642.
https://doi.org/10.3390/math11071642

**AMA Style**

Sultana S, González-Parra G, Arenas AJ. A Generalized Mathematical Model of Toxoplasmosis with an Intermediate Host and the Definitive Cat Host. *Mathematics*. 2023; 11(7):1642.
https://doi.org/10.3390/math11071642

**Chicago/Turabian Style**

Sultana, Sharmin, Gilberto González-Parra, and Abraham J. Arenas. 2023. "A Generalized Mathematical Model of Toxoplasmosis with an Intermediate Host and the Definitive Cat Host" *Mathematics* 11, no. 7: 1642.
https://doi.org/10.3390/math11071642