# Stochastic SIS Modelling: Coinfection of Two Pathogens in Two-Host Communities

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## Abstract

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## 1. Introduction

## 2. Materials and Methods

#### 2.1. Continuous Time Markov Chain Model

- The rate at which the two infectious classes, ${I}_{1}(t)$ and ${I}_{2}(t)$, can be recovered are ${\Phi}_{1}$ and ${\Phi}_{2}$ respectively,${I}_{1}(t)\stackrel{{\Phi}_{1}}{\to}\varnothing $ and ${I}_{2}(t)\stackrel{{\Phi}_{2}}{\to}\varnothing $.
- The two susceptible classes, ${S}_{1}(t)$ and ${S}_{2}(t)$, can move to infectious classes, ${I}_{1}(t)$ and ${I}_{2}(t)$, at the disease transmission rates ${\beta}_{11},{\beta}_{12},{\beta}_{21}$, and ${\beta}_{22}$, such that${S}_{1}(t)+{I}_{1}(t)\stackrel{{\beta}_{11}}{\to}2{I}_{1}(t)$,${S}_{1}(t)+{I}_{2}(t)\stackrel{{\beta}_{12}}{\to}2{I}_{2}(t)$,${S}_{2}(t)+{I}_{1}(t)\stackrel{{\beta}_{21}}{\to}2{I}_{1}(t)$,${S}_{2}(t)+{I}_{2}(t)\stackrel{{\beta}_{11}}{\to}2{I}_{2}(t)$.

#### 2.2. Basic Reproduction Number

#### 2.3. Multi-Type Branching Process

**Theorem**

**1.**

## 3. Results

#### Numerical Examples

## 4. Discussion

## 5. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

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**Figure 1.**One stochastic realisation approximates the deterministic models for ${\beta}_{11}=0.02$, ${\beta}_{12}=0.01$, ${\beta}_{21}=0.02$, ${\beta}_{22}=0.02$, given (

**a**) the sample paths of Equations (6) with ${\Phi}_{1}={\Phi}_{2}=0.2$, ${S}_{1}(0)=100$, ${I}_{1}(0)=2$, ${S}_{2}(0)=90$, and ${I}_{2}(0)=3$; (

**b**) the sample paths of Equations (8) with the initial conditions ${S}_{1}(0)=100$, ${I}_{1}(0)=2$, ${S}_{2}(0)=90$, and ${I}_{2}(0)=3$; (

**c**) the sample paths of Equations (6) for ${\Phi}_{1}={\Phi}_{2}=0.2$, ${S}_{1}(0)=50$, ${I}_{1}(0)=2$, ${S}_{2}(0)=45$, and ${I}_{2}(0)=3$; and (

**d**) the sample paths of Equations (8) with the initial conditions ${S}_{1}(0)=50$, ${I}_{1}(0)=2$, ${S}_{2}(0)=45$, and ${I}_{2}(0)=3$.

**Figure 2.**Solutions of the compartments, (

**a**) ${S}_{1}(t)$, (

**b**) ${S}_{2}(t)$, (

**c**) ${I}_{1}(t)$, (

**d**) ${I}_{2}(t)$, of the epidemic model (6) at $t=20$ based on 1000 stochastic realisations. The parameters used for determining the distributions are ${\beta}_{11}=0.02$, ${\beta}_{12}=0.01$, ${\beta}_{21}=0.02$, ${\beta}_{22}=0.02$, ${\Phi}_{1}={\Phi}_{2}=0.2$, ${S}_{1}(0)=100$, ${I}_{1}(0)=2$, ${S}_{2}(0)=90$, and ${I}_{2}(0)=3$.

**Table 1.**Assumptions of the ${S}_{1}{I}_{1}{S}_{2}{I}_{2}$ continuous time Markov chain (CTMC) model.

Event | Transition Between t and $\mathit{t}+\mathit{\delta}\mathit{t}$ | Probability |
---|---|---|

$(i)$ Mortality of ${I}_{1}(t)$ | $({n}_{1},{n}_{2}+1,{n}_{3},{n}_{4})$→$({n}_{1},{n}_{2},{n}_{3},{n}_{4})$ | ${\Phi}_{1}({n}_{2}+1)\delta t+0{(\delta t)}^{2}$ |

$(ii)$ Mortality of ${I}_{2}(t)$ | $({n}_{1},{n}_{2},{n}_{3},{n}_{4}+1)$→$({n}_{1},{n}_{2},{n}_{3},{n}_{4})$ | ${\Phi}_{2}({n}_{4}+1)\delta t+0{(\delta t)}^{2}$ |

$(iii)$${I}_{1}(t)$ infects ${S}_{1}(t)$ | $({n}_{1}+1,{n}_{2}-1,{n}_{3},{n}_{4})$→$({n}_{1},{n}_{2},{n}_{3},{n}_{4})$ | ${\beta}_{11}({n}_{1}+1)({n}_{2}-1)\delta t+0{(\delta t)}^{2}$ |

$(iv)$${I}_{2}(t)$ infects ${S}_{1}(t)$ | $({n}_{1}+1,{n}_{2},{n}_{3},{n}_{4}-1)$→$({n}_{1},{n}_{2},{n}_{3},{n}_{4})$ | ${\beta}_{12}({n}_{1}+1)({n}_{4}-1)\delta t+0{(\delta t)}^{2}$ |

$(v)$${I}_{1}(t)$ infects ${S}_{2}(t)$ | $({n}_{1},{n}_{2}-1,{n}_{3}+1,{n}_{4})$→$({n}_{1},{n}_{2},{n}_{3},{n}_{4})$ | ${\beta}_{21}({n}_{2}-1)({n}_{3}+1)\delta t+0{(\delta t)}^{2}$ |

$(vi)$${I}_{2}(t)$ infects ${S}_{2}(t)$ | $({n}_{1},{n}_{2},{n}_{3}+1,{n}_{4}-1)$→$({n}_{1},{n}_{2},{n}_{3},{n}_{4})$ | ${\beta}_{22}({n}_{3}+1)({n}_{4}-1)\delta t+0{(\delta t)}^{2}$ |

**Table 2.**The probabilities of extinction P${}_{01}$ and P${}_{02}$ of two infectious classes ${I}_{1}(t)$ and ${I}_{2}(t)$, while Approx 1 and 2 are their respective numerical estimations using CTMC for ${\beta}_{11}={\beta}_{12}={\beta}_{21}={\beta}_{22}=0.2$, ${\Phi}_{1}$ = ${\Phi}_{2}$ = 0.3 based on ${10}^{6}$ sample paths.

${\mathit{I}}_{1}(0)$ | ${\mathit{I}}_{2}(0)$ | P${}_{01}$ | Approx 1 | P${}_{02}$ | Approx 2 |
---|---|---|---|---|---|

1 | 1 | $0.7500$ | $0.4997$ | $0.7500$ | $0.5003$ |

2 | 1 | $0.5625$ | $0.3333$ | $0.7500$ | $0.6667$ |

1 | 2 | $0.7500$ | $0.6664$ | $0.5625$ | $0.3336$ |

2 | 2 | $0.5625$ | $0.5001$ | $0.5625$ | $0.4999$ |

3 | 2 | $0.4219$ | $0.3995$ | $0.5625$ | $0.6004$ |

2 | 3 | $0.5625$ | $0.6000$ | $0.4219$ | $0.4000$ |

3 | 3 | $0.5000$ | $0.5000$ | $0.4219$ | $0.4219$ |

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**MDPI and ACS Style**

Abdullahi, A.; Shohaimi, S.; Kilicman, A.; Hafiz Ibrahim, M.; Salari, N.
Stochastic SIS Modelling: Coinfection of Two Pathogens in Two-Host Communities. *Entropy* **2020**, *22*, 54.
https://doi.org/10.3390/e22010054

**AMA Style**

Abdullahi A, Shohaimi S, Kilicman A, Hafiz Ibrahim M, Salari N.
Stochastic SIS Modelling: Coinfection of Two Pathogens in Two-Host Communities. *Entropy*. 2020; 22(1):54.
https://doi.org/10.3390/e22010054

**Chicago/Turabian Style**

Abdullahi, Auwal, Shamarina Shohaimi, Adem Kilicman, Mohd Hafiz Ibrahim, and Nader Salari.
2020. "Stochastic SIS Modelling: Coinfection of Two Pathogens in Two-Host Communities" *Entropy* 22, no. 1: 54.
https://doi.org/10.3390/e22010054