Dynamic for a Stochastic Multi-Group AIDS Model with Saturated Incidence Rate
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Wuhan Institute Physics and Mathematics, Chinese Academy of Sciences, 2020
http://actams.wipm.ac.cn
DYNAMIC FOR A STOCHASTIC MULTI-GROUP AIDS MODEL WITH SATURATED INCIDENCE RATE∗
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Qixing HAN (
School of Mathematics, Changchun Normal University, Changchun 130032, China School of Mathematics, Jilin University, Changchun 130012, China E-mail : [email protected]
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Daqing JIANG (
†
Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, Department of Mathematics, King Abdulaziz University, Jeddah, Saudi Arabia College of Science, China University of Petroleum (East China), Qingdao 266580, China E-mail : [email protected] Abstract In this paper, a stochastic multi-group AIDS model with saturated incidence rate is studied. We prove that the system is persistent in the mean under some parametric restrictions. We also obtain the sufficient condition for the existence of the ergodic stationary distribution of the system by constructing a suitable Lyapunov function. Our results indicate that the existence of ergodic stationary distribution does not rely on the interior equilibrium of the corresponding deterministic system, which greatly improves upon previous results. Key words
multi-group AIDS model; Lyapunov function; stationary distribution; persistence in the mean
2010 MR Subject Classification
1
92D30; 93E15
Introduction
The acquired immunodeficiency syndrome (AIDS) epidemic model is one of the most important epidemic models, and is used to describe a disease of the human immune system caused by the human immunodeficiency virus (HIV). AIDS is a widespread and potentially fatal disease. In 2007, it was calculated that 33.2 million people lived with the disease worldwide, and it was estimated 2.1 million AIDS death occurred, of those, 330,000 being children [1]. Therefore, it is necessary to understand the dynamical behavior of such diseases and to predict what may happen. Over the years, many researchers have paid attention to AIDS models, and achieved some important results [2–5]. ∗ Received June 5, 2019; revised July 28, 2020. The work was supported by NSF of China (11801041, 11871473), Foudation of Jilin Province Science and Technology Development (20190201130JC), Scientific Rsearch Foundation of Jilin Provincial Education Department (JJKH20181172KJ, JJKH20190503KJ) and Natural Science Foundation of Changchun Normal University. † Corresponding author: Daqing JIANG.
1884
ACTA MATHEMATICA SCIENTIA
Vol.40 Ser.B
The AIDS population is often divided into three parts: the susceptible population (S), the infected population (I), and the AIDS patient population (A). Here, I is subdivided into n groups I1 , I2 , · · · , In . We all know that incidence rate plays an important role in modelling epidemics. Some authors employ the bilinear incidence rate βSI [6, 7]. The classical deterministic multi-group AIDS epidemic model with bilinear incidence rate is described by the following n + 2 dimensional ODE: n X 0 dS(t) = µS − µS(t) − β S(t)I (t) dt, j j j=1 X
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