A stochastic mathematical model of two different infectious epidemic under vertical transmission

Xunyang Wang; Canyun Huang; Yixin Hao; Qihong Shi; Xunyang Wang; Canyun Huang; Yixin Hao; Qihong Shi

doi:10.3934/mbe.2022101

Mathematical Biosciences and Engineering

2022, Volume 19, Issue 3: 2179-2192. doi: 10.3934/mbe.2022101

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A stochastic mathematical model of two different infectious epidemic under vertical transmission

1.
Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou 730050, China
2.
State Grid Gansu Electric Power Research Institute, Lanzhou 730070, China

Received: 11 October 2021 Revised: 02 December 2021 Accepted: 10 December 2021 Published: 29 December 2021

In this study, considering the effect of environment perturbation which is usually embodied by the alteration of contact infection rate, we formulate a stochastic epidemic mathematical model in which two different kinds of infectious diseases that spread simultaneously through both horizontal and vertical transmission are described. To indicate our model is well-posed and of biological significance, we prove the existence and uniqueness of positive solution at the beginning. By constructing suitable $ Lyapunov $ functions (which can be used to prove the stability of a certain fixed point in a dynamical system or autonomous differential equation) and applying $ It\hat{o} $'s formula as well as $ Chebyshev $'s inequality, we also establish the sufficient conditions for stochastic ultimate boundedness. Furthermore, when some main parameters and all the stochastically perturbed intensities satisfy a certain relationship, we finally prove the stochastic permanence. Our results show that the perturbed intensities should be no greater than a certain positive number which is up-bounded by some parameters in the system, otherwise, the system will be surely extinct. The reliability of theoretical results are further illustrated by numerical simulations. Finally, in the discussion section, we put forward two important and interesting questions left for further investigation.
- vertical transmission,
- $ It\hat{o} $'s formula,
- stochastic ultimate boundedness,
- stochastic permanence
Citation: Xunyang Wang, Canyun Huang, Yixin Hao, Qihong Shi. A stochastic mathematical model of two different infectious epidemic under vertical transmission[J]. Mathematical Biosciences and Engineering, 2022, 19(3): 2179-2192. doi: 10.3934/mbe.2022101

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Abstract

In this study, considering the effect of environment perturbation which is usually embodied by the alteration of contact infection rate, we formulate a stochastic epidemic mathematical model in which two different kinds of infectious diseases that spread simultaneously through both horizontal and vertical transmission are described. To indicate our model is well-posed and of biological significance, we prove the existence and uniqueness of positive solution at the beginning. By constructing suitable $ Lyapunov $ functions (which can be used to prove the stability of a certain fixed point in a dynamical system or autonomous differential equation) and applying $ It\hat{o} $'s formula as well as $ Chebyshev $'s inequality, we also establish the sufficient conditions for stochastic ultimate boundedness. Furthermore, when some main parameters and all the stochastically perturbed intensities satisfy a certain relationship, we finally prove the stochastic permanence. Our results show that the perturbed intensities should be no greater than a certain positive number which is up-bounded by some parameters in the system, otherwise, the system will be surely extinct. The reliability of theoretical results are further illustrated by numerical simulations. Finally, in the discussion section, we put forward two important and interesting questions left for further investigation.

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