# Dynamical Behaviour of a Modified Tuberculosis Model with Impact of Public Health Education and Hospital Treatment

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## Abstract

**:**

## 1. Introduction

## 2. Materials and Methods

#### 2.1. Clinical Assumptions of the TB Model with Public Health Education and Hospital Treatment

- There is a constant recruitment rate to the susceptible population, and natural cause of death affects individuals in all compartments, with an extra TB-induced death rate in the infected class.
- We assume that, at any moment, the educated susceptible group may act as ignorantly and enter the class of susceptible at a constant rate $\varphi ,$ i.e., loss of temporary protection [16].
- We further assumed that some uncured TB patients wanted to be discharged and continue treatment at home at a rate ${\eta}_{2},$ [20].
- Both infected individuals at home and in the hospital experience the infections effect at the rate $\lambda $.
- The infected compartment is divided into two groups, namely, infected individuals who received treatment at home ${I}_{\iota 1},$ and infected individuals receiving treatment in hospital ${I}_{\iota 2}$.
- The recovered individual may be again infected by an infectious individual [25].

#### 2.2. Basic Properties of the TB Model with Public Health Education and Hospital Treatment

#### Positivity of Solutions

**Theorem 1.**

**Proof**of Theorem 1

#### 2.3. Invariant Region

**Theorem 2.**

**Proof**of Theorem 2

## 3. Existence of Equilibrium

#### 3.1. Tuberculosis Free Equilibrium (TFE)

#### 3.2. Calculation of Effective Reproduction Number $\left({R}_{ph}\right)$ For the System Model (23)

#### 3.3. Local Stability of TB Free Equilibrium

**Theorem 3.**

**Proof**

**of Theorem 3**

#### 3.4. Global Stability of the TB Free Equilibrium

**Theorem 4.**

**Proof**of Theorem 4

#### 3.5. Endemic Equilibrium State $\left({T}^{*}\right)$

**Theorem 5.**

**Theorem 6.**

- 1.
- One or more endemic equilibria when ${R}_{ph}<1.$
- 2.
- A unique endemic equilibrium when ${R}_{ph}>1.$
- 3.
- No endemic equilibrium otherwise.

## 4. Local Stability of Endemic Equilibrium

**Theorem 7.**

## 5. Sensitivity Analysis

## 6. Numerical Results and Discussions

## 7. Conclusions

- Considering a stochastic model approach. This will result in more realistic TB model dynamics.
- Since the spread of tuberculosis affects all age groups, it is crucial to consider the dynamics of the TB model by incorporating an age-structured model.
- Real data will also be considered because collecting data for TB patients is difficult in epidemiological models; as a result, we use data collected or estimated from literature sources. Once we have real-world data for TB patients, we can compare it to theoretical outcomes.
- Analyzing the dynamics of the TB model using a fractional order differential equation (FODE). It will be extremely interesting to use a FODE to examine the dynamics of TB model.

## Author Contributions

## Funding

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A

**Proof**of Theorem 5

## Appendix B

**Proof**of Theorem 7

## References

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**Figure 2.**The bifurcation diagram of force of infection against ${R}_{ph}$ which illustrates a backward bifurcation for the system model (23).

**Figure 3.**The relationship between $\psi $ (

**a**), ${\eta}_{1}$ (

**b**), ${\kappa}_{1}$ (

**c**), and ${\kappa}_{2}$ (

**d**) and the effective reproduction number ${R}_{ph}$.

**Figure 4.**Simulations of system (23) showing the behaviour of susceptible and educated susceptible. Parameters used are $\Lambda =\mathrm{450,862},\phantom{\rule{0.166667em}{0ex}}\beta =0.86,\nu =0.25,\varphi =0.2,\sigma =0.65,z=0.11,\mu =0.02041,$$\psi =0.5,\phantom{\rule{0.166667em}{0ex}}{\eta}_{1}=0.02,\phantom{\rule{0.166667em}{0ex}}{\eta}_{2}=0.02,\phantom{\rule{0.166667em}{0ex}}{\gamma}_{1}=0.5,\phantom{\rule{0.166667em}{0ex}}{\gamma}_{2}=0.6,\phantom{\rule{0.166667em}{0ex}}{\kappa}_{1}=0.03,\phantom{\rule{0.166667em}{0ex}}{\kappa}_{2}=0.3,\phantom{\rule{0.166667em}{0ex}}{\delta}_{1}=0.2,\phantom{\rule{0.166667em}{0ex}}{\delta}_{2}=0.12.$

**Figure 5.**Impact of awareness rate $\left(\psi \right)$ on ${S}_{\iota},{E}_{\iota},{I}_{\iota 1},\mathrm{and}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{0.166667em}{0ex}}{I}_{\iota 2},$ varying $\psi .$ Other parameters are given as $\Lambda =\mathrm{450,862},\phantom{\rule{0.166667em}{0ex}}\beta =0.86,\nu =0.25,\varphi =0.2,\sigma =0.65,z=0.11,\mu =0.02041,\phantom{\rule{0.166667em}{0ex}}\psi =0.5,\phantom{\rule{0.166667em}{0ex}}{\eta}_{1}=0.02,$${\eta}_{2}=0.02,\phantom{\rule{0.166667em}{0ex}}{\gamma}_{1}=0.5,\phantom{\rule{0.166667em}{0ex}}{\gamma}_{2}=0.6,\phantom{\rule{0.166667em}{0ex}}{\kappa}_{1}=0.03,\phantom{\rule{0.166667em}{0ex}}{\kappa}_{2}=0.3,\phantom{\rule{0.166667em}{0ex}}{\delta}_{1}=0.2,\phantom{\rule{0.166667em}{0ex}}{\delta}_{2}=0.12.$

**Figure 6.**Impact at which educated susceptible lose awareness $\left(\varphi \right)$ on ${I}_{\iota 1},\phantom{\rule{0.166667em}{0ex}}\mathrm{and}\phantom{\rule{0.166667em}{0ex}}{I}_{\iota 2},$ varying $\varphi .$ Other parameters are given as $\Lambda =\mathrm{450,862},\phantom{\rule{0.166667em}{0ex}}\beta =0.86,\nu =0.25,\varphi =0.2,\sigma =0.65,z=0.11,\mu =0.02041,$$\psi =0.5,\phantom{\rule{0.166667em}{0ex}}{\eta}_{1}=0.02,\phantom{\rule{0.166667em}{0ex}}{\eta}_{2}=0.02,\phantom{\rule{0.166667em}{0ex}}{\gamma}_{1}=0.5,\phantom{\rule{0.166667em}{0ex}}{\gamma}_{2}=0.6,\phantom{\rule{0.166667em}{0ex}}{\kappa}_{1}=0.03,\phantom{\rule{0.166667em}{0ex}}{\kappa}_{2}=0.3,\phantom{\rule{0.166667em}{0ex}}{\delta}_{1}=0.2,\phantom{\rule{0.166667em}{0ex}}{\delta}_{2}=0.12.$

**Figure 7.**Impact of reduction of infection rate as a result of ${I}_{2}$$\left(z\right)$ on ${I}_{\iota 1},\phantom{\rule{0.166667em}{0ex}}\mathrm{and}\phantom{\rule{0.166667em}{0ex}}{I}_{\iota 2},$ varying $z.$ Other parameters are given as $\Lambda =\mathrm{450,862},\phantom{\rule{0.166667em}{0ex}}\beta =0.86,\nu =0.25,\varphi =0.2,\sigma =0.65,z=0.11,\mu =0.02041,$$\psi =0.5,\phantom{\rule{0.166667em}{0ex}}{\eta}_{1}=0.02,\phantom{\rule{0.166667em}{0ex}}{\eta}_{2}=0.02,\phantom{\rule{0.166667em}{0ex}}{\gamma}_{1}=0.5,\phantom{\rule{0.166667em}{0ex}}{\gamma}_{2}=0.6,\phantom{\rule{0.166667em}{0ex}}{\kappa}_{1}=0.03,\phantom{\rule{0.166667em}{0ex}}{\kappa}_{2}=0.3,\phantom{\rule{0.166667em}{0ex}}{\delta}_{1}=0.2,\phantom{\rule{0.166667em}{0ex}}{\delta}_{2}=0.12.$

**Figure 8.**Impact of progression rate $\left({\eta}_{1}\right)$ from ${I}_{\iota 1}$ to ${I}_{\iota 2}$ on ${I}_{\iota 1},\phantom{\rule{0.166667em}{0ex}}\mathrm{and}\phantom{\rule{0.166667em}{0ex}}{I}_{\iota 2},$ varying ${\eta}_{1}.$ Other parameters are given as $\Lambda =\mathrm{450,862},\phantom{\rule{0.166667em}{0ex}}\beta =0.86,\nu =0.25,\varphi =0.2,\sigma =0.65,z=0.11,\mu =0.02041,$$\psi =0.5,$${\eta}_{1}=0.02,\phantom{\rule{0.166667em}{0ex}}{\eta}_{2}=0.02,\phantom{\rule{0.166667em}{0ex}}{\gamma}_{1}=0.5,$${\gamma}_{2}=0.6,\phantom{\rule{0.166667em}{0ex}}{\kappa}_{1}=0.03,\phantom{\rule{0.166667em}{0ex}}{\kappa}_{2}=0.3,\phantom{\rule{0.166667em}{0ex}}{\delta}_{1}=0.2,\phantom{\rule{0.166667em}{0ex}}{\delta}_{2}=0.12.$

**Table 1.**Interpretation of the Variables and Parameters of the TB Model with Public Education and hospital Treatment (23).

State Variables | Explanation |
---|---|

${S}_{\iota}$ | The number of individuals who are susceptible |

${P}_{\iota}$ | The number of individuals who are educated susceptible |

${E}_{\iota}$ | The number of individuals who are exposed to TB |

${I}_{\iota 1}$ | The number of infected individuals at home |

${I}_{\iota 2}$ | The number of infected individuals at hospital |

${R}_{\iota}$ | The number of individuals who have recovered |

${N}_{\iota}$ | Human population size |

Parameters | |

$\Lambda $ | Inflow of recruitment into susceptible class |

$\mu $ | Per capita natural death rate of humans |

$\psi $ | Information dissemination (awareness rate) |

$\varphi $ | Rate at which the educated susceptible become susceptible |

$\nu $ | Reduction of infection rate as a result of awareness |

$\beta $ | Transmission rate |

z | Reduction of infection rate as a result of individuals in the hospital |

${\kappa}_{1}$ | Progression rate from ${E}_{\iota}$ to ${I}_{\iota 1}$ |

${\kappa}_{2}$ | Progression rate from ${E}_{\iota}$ to ${I}_{\iota 2}$ |

${\gamma}_{1}$ | recovery rate for ${I}_{\iota 1}$ |

${\gamma}_{2}$ | recovery rate for ${I}_{\iota 2}$ |

$\sigma $ | Mod. parameter for re-infection among the recovered individuals |

${\eta}_{1}$ | Progression rate from ${I}_{\iota 1}$ to ${I}_{\iota 2}$ |

${\eta}_{2}$ | Progression rate from ${I}_{\iota 2}$ to ${I}_{\iota 1}$ |

${\delta}_{1}$ | TB induced mortality rate for ${I}_{\iota 1}$ |

${\delta}_{2}$ | TB induced mortality rate for ${I}_{\iota 2}$ |

Parameters | Description | Sensitivity Index |
---|---|---|

$\mu $ | Per capita natural death rate of humans | −0.6970 |

$\psi $ | Information dissemination (awareness rate) | −0.2554 |

$\varphi $ | The rate at which susceptible individuals lose awareness | +0.2454 |

$\nu $ | Reduction in risk of infection due to awareness | +0.1103 |

$\beta $ | Transmission rate for contact with ${I}_{1}$ | +1.0000 |

z | Reduction of infection rate as a result of ${I}_{2}$ | +0.4476 |

${\kappa}_{1}$ | Progression rate from ${E}_{\iota}$ to ${I}_{1}$ | +0.4228 |

${\kappa}_{2}$ | Progression rate from ${E}_{\iota}$ to ${I}_{2}$ | −0.2147 |

${\gamma}_{1}$ | Treatment rate for ${I}_{\iota 1}$ | −0.2573 |

${\gamma}_{2}$ | Treatment rate for ${I}_{\iota 2}$ | +0.5570 |

${\eta}_{1}$ | Progression rate from ${I}_{\iota 1}$ to ${I}_{\iota 2}$ | −0.6832 |

${\eta}_{2}$ | Progression rate from ${I}_{\iota 2}$ to ${I}_{\iota 1}$ | +0.1845 |

${\delta}_{1}$ | TB induced mortality rate for ${I}_{\iota 1}$ | −0.2058 |

${\delta}_{2}$ | TB induced mortality rate for ${I}_{\iota 2}$ | −0.3714 |

**Table 3.**The Parameters and Baseline Values of the Model with Public Health Education and Hospital Treatment (23).

Parameters | Baseline Values | Ranges | References |
---|---|---|---|

$\Lambda $ | 3,768,410 year${}^{-1}$ | [3,000,000, 4,000,000] | [42] |

$\mu $ | 0.02041 year${}^{-1}$ | [0.0143, 0.03] | [23] |

$\psi $ | Variable | [0–1] | Assumed |

$\varphi $ | Variable | [0–1] | Assumed |

$\nu $ | Variable | [0–1] | Assumed |

$\beta $ | Variable | Varied | Assumed |

$\sigma $ | 0.25 | [0–1] | [25,43] |

z | 0.11 | [0–0.9] | [16] |

${\kappa}_{1}$ | Variable | [0–1] | Assumed |

${\kappa}_{2}$ | Variable | [0–1] | Assumed |

${\gamma}_{1}$ | 0.09 year${}^{-1}$ | [0–1] | [20] |

${\gamma}_{2}$ | 0.72 year${}^{-1}$ | [0–1] | [20] |

${\eta}_{1}$ | Variable | [0–1] | assumed |

${\eta}_{2}$ | Variable | [0–1] | assumed |

${\delta}_{1}$ | 0.2 year${}^{-1}$ | - | [44] |

${\delta}_{2}$ | 0.02 year${}^{-1}$ | - | [45] |

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**MDPI and ACS Style**

Sulayman, F.; Abdullah, F.A.
Dynamical Behaviour of a Modified Tuberculosis Model with Impact of Public Health Education and Hospital Treatment. *Axioms* **2022**, *11*, 723.
https://doi.org/10.3390/axioms11120723

**AMA Style**

Sulayman F, Abdullah FA.
Dynamical Behaviour of a Modified Tuberculosis Model with Impact of Public Health Education and Hospital Treatment. *Axioms*. 2022; 11(12):723.
https://doi.org/10.3390/axioms11120723

**Chicago/Turabian Style**

Sulayman, Fatima, and Farah Aini Abdullah.
2022. "Dynamical Behaviour of a Modified Tuberculosis Model with Impact of Public Health Education and Hospital Treatment" *Axioms* 11, no. 12: 723.
https://doi.org/10.3390/axioms11120723