### Abstract

Stochastic epidemic models, generally more realistic than deterministic counterparts, have often been seen too complex for rigorous mathematical analysis because of level of details it requires to comprehensively capture the dynamics of diseases. This problem further becomes intense when complexity of diseases increases as in the case of vector-borne diseases (VBD). The VBDs are human illnesses caused by pathogens transmitted among humans by intermediate species, which are primarily arthropods. In this study, a stochastic VBD model is developed and novel mathematical methods are described and evaluated to systematically analyze the model and understand its complex dynamics. The VBD model incorporates some relevant features of the VBD transmission process including demographical, ecological and social mechanisms, and different host and vector dynamic scales. The analysis is based on dimensional reductions and model simplifications via scaling limit theorems. The results suggest that the dynamics of the stochastic VBD depends on a threshold quantity R_{0}, the initial size of infectives, and the type of scaling in terms of host population size. The quantity R_{0} for deterministic counterpart of the model is interpreted as a threshold condition for infection persistence as is mentioned in the literature for many infectious disease models. Different scalings yield different approximations of the model, and in particular, if vectors have much faster dynamics, the effect of the vector dynamics on the host population averages out, which largely reduces the dimension of the model. Specific scenarios are also studied using simulations for some fixed sets of parameters to draw conclusions on dynamics.

Original language | English (US) |
---|---|

Pages (from-to) | 42-65 |

Number of pages | 24 |

Journal | Mathematical Biosciences |

Volume | 309 |

DOIs | |

State | Published - Mar 1 2019 |

### Fingerprint

### Keywords

- Fast and slow dynamics
- Functional central limit theorem
- Functional law of large numbers
- Multiscale analysis
- Quasi-stationary distributions
- SIS compartment model
- Time to extinction
- Vector-borne disease model

### ASJC Scopus subject areas

- Statistics and Probability
- Modeling and Simulation
- Biochemistry, Genetics and Molecular Biology(all)
- Immunology and Microbiology(all)
- Agricultural and Biological Sciences(all)
- Applied Mathematics

### Cite this

*Mathematical Biosciences*,

*309*, 42-65. https://doi.org/10.1016/j.mbs.2019.01.003

**Approximation methods for analyzing multiscale stochastic vector-borne epidemic models.** / Liu, Xin; Mubayi, Anuj; Reinhold, Dominik; Zhu, Liu.

Research output: Contribution to journal › Article

*Mathematical Biosciences*, vol. 309, pp. 42-65. https://doi.org/10.1016/j.mbs.2019.01.003

}

TY - JOUR

T1 - Approximation methods for analyzing multiscale stochastic vector-borne epidemic models

AU - Liu, Xin

AU - Mubayi, Anuj

AU - Reinhold, Dominik

AU - Zhu, Liu

PY - 2019/3/1

Y1 - 2019/3/1

N2 - Stochastic epidemic models, generally more realistic than deterministic counterparts, have often been seen too complex for rigorous mathematical analysis because of level of details it requires to comprehensively capture the dynamics of diseases. This problem further becomes intense when complexity of diseases increases as in the case of vector-borne diseases (VBD). The VBDs are human illnesses caused by pathogens transmitted among humans by intermediate species, which are primarily arthropods. In this study, a stochastic VBD model is developed and novel mathematical methods are described and evaluated to systematically analyze the model and understand its complex dynamics. The VBD model incorporates some relevant features of the VBD transmission process including demographical, ecological and social mechanisms, and different host and vector dynamic scales. The analysis is based on dimensional reductions and model simplifications via scaling limit theorems. The results suggest that the dynamics of the stochastic VBD depends on a threshold quantity R0, the initial size of infectives, and the type of scaling in terms of host population size. The quantity R0 for deterministic counterpart of the model is interpreted as a threshold condition for infection persistence as is mentioned in the literature for many infectious disease models. Different scalings yield different approximations of the model, and in particular, if vectors have much faster dynamics, the effect of the vector dynamics on the host population averages out, which largely reduces the dimension of the model. Specific scenarios are also studied using simulations for some fixed sets of parameters to draw conclusions on dynamics.

AB - Stochastic epidemic models, generally more realistic than deterministic counterparts, have often been seen too complex for rigorous mathematical analysis because of level of details it requires to comprehensively capture the dynamics of diseases. This problem further becomes intense when complexity of diseases increases as in the case of vector-borne diseases (VBD). The VBDs are human illnesses caused by pathogens transmitted among humans by intermediate species, which are primarily arthropods. In this study, a stochastic VBD model is developed and novel mathematical methods are described and evaluated to systematically analyze the model and understand its complex dynamics. The VBD model incorporates some relevant features of the VBD transmission process including demographical, ecological and social mechanisms, and different host and vector dynamic scales. The analysis is based on dimensional reductions and model simplifications via scaling limit theorems. The results suggest that the dynamics of the stochastic VBD depends on a threshold quantity R0, the initial size of infectives, and the type of scaling in terms of host population size. The quantity R0 for deterministic counterpart of the model is interpreted as a threshold condition for infection persistence as is mentioned in the literature for many infectious disease models. Different scalings yield different approximations of the model, and in particular, if vectors have much faster dynamics, the effect of the vector dynamics on the host population averages out, which largely reduces the dimension of the model. Specific scenarios are also studied using simulations for some fixed sets of parameters to draw conclusions on dynamics.

KW - Fast and slow dynamics

KW - Functional central limit theorem

KW - Functional law of large numbers

KW - Multiscale analysis

KW - Quasi-stationary distributions

KW - SIS compartment model

KW - Time to extinction

KW - Vector-borne disease model

UR - http://www.scopus.com/inward/record.url?scp=85060165309&partnerID=8YFLogxK

UR - http://www.scopus.com/inward/citedby.url?scp=85060165309&partnerID=8YFLogxK

U2 - 10.1016/j.mbs.2019.01.003

DO - 10.1016/j.mbs.2019.01.003

M3 - Article

VL - 309

SP - 42

EP - 65

JO - Mathematical Biosciences

JF - Mathematical Biosciences

SN - 0025-5564

ER -