Stationary distribution and extinction of a stochastic model of syphilis transmission in an MSM population with telegrap
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Stationary distribution and extinction of a stochastic model of syphilis transmission in an MSM population with telegraph noises Yaxin Zhou1 · Wenjie Zuo1
· Daqing Jiang1,2 · Mingyu Song1
Received: 16 September 2020 / Revised: 23 October 2020 / Accepted: 24 October 2020 © Korean Society for Informatics and Computational Applied Mathematics 2020
Abstract This paper is concerned with the dynamical behaviors of a model of syphilis transmission disturbed by both white noises and telegraph noises. Multiple infections and treatment stages are considered, which include and extend the existing ones. The existence and ergodicity of the stationary distribution are obtained by constructing a suitable Lyapunov function, which determines a critical value R0∗ corresponding to the control reproduction number Rc of the corresponding determined system. In addition, a sufficient criteria for extinction of the diseases is derived. Finally, the numerical simulations illustrate our theoretical results, which show that, the stronger white noises can result in the extinction of the diseases and telegraph noises can strength the stability of the system. Keywords Syphilis transmission · Telegraph noises · White noises · Stationary distribution · Ergodicity Mathematics Subject Classification 34K25 · 34F05 · 37A30 · 37H30
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Wenjie Zuo [email protected]
1
College of Science, China University of Petroleum (East China), Qingdao 266580, People’s Republic of China
2
Nonlinear Analysis and Applied Mathematics (NAAM)-Research Group, King Abdulaziz University, Jeddah, Saudi Arabia
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Y. Zhou
1 Introduction Nonlinear dynamical problems have been and will long be the one of the most important hot topics in mathematical biology, engineering and physics. For most real-life nonlinear problems, it is always impossible to find the exact analytic solutions. Bayat and Pakar [1–3] used the variational approach method to find their approximate solutions. In recent years, many analytical and numerical methods [4–9] have been developed for disclosing nonlinear phenomena. All kinds of sexually transmitted diseases are caused by bacteria: for example, syphilis is caused by the bacterium Treponema pallidum (T. pallidum) subspecies [10]. Since T. pallidum can be latent in the body, people may be attacked when the immunity declines and it is first obtained from inoculated rabbits, which has been recognized as the greatest obstacle in syphilis research [10]. In human, an untreated syphilis infection undergoes multiple stages, including exposed stage, primary stage, secondary stage, latent stage and tertiary stage [10,11]. Primary and secondary syphilis has long been the one of the public health issues in the United States [12]. Syphilis in men who have sex with men (MSM) has been increasing [13,14], though there had been a decrease in syphilis infection worldwide [15]. Although effective treatment has been available since the introduction of penicillin in the mid-twentieth century, syphilis remains an important global health problem [16,17]. Many of the
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