In this paper, we provide an effective method for estimating the thresholds of the stochastic models with time delays by using of the nonnegative semimartingale convergence theorem. Firstly, we establish the stochastic delay differential equation models for two diseases, and obtain two thresholds of two diseases and the sufficient conditions for the persistence and extinction of two diseases. Then, numerical simulations for co-infection of HIV/AIDS and Gonorrhea in Yunnan Province, China, are carried out. Finally, we discuss some biological implications and focus on the impact of some key model parameters. One of the most interesting findings is that the stochastic fluctuation and time delays introduced into the deterministic models can suppress the outbreak of the diseases, which can provide some useful control strategies to regulate the dynamics of the diseases, and the numerical simulations verify this phenomenon.
Citation: Zhimin Li, Tailei Zhang, Xiuqing Li. Threshold dynamics of stochastic models with time delays: A case study for Yunnan, China[J]. Electronic Research Archive, 2021, 29(1): 1661-1679. doi: 10.3934/era.2020085
In this paper, we provide an effective method for estimating the thresholds of the stochastic models with time delays by using of the nonnegative semimartingale convergence theorem. Firstly, we establish the stochastic delay differential equation models for two diseases, and obtain two thresholds of two diseases and the sufficient conditions for the persistence and extinction of two diseases. Then, numerical simulations for co-infection of HIV/AIDS and Gonorrhea in Yunnan Province, China, are carried out. Finally, we discuss some biological implications and focus on the impact of some key model parameters. One of the most interesting findings is that the stochastic fluctuation and time delays introduced into the deterministic models can suppress the outbreak of the diseases, which can provide some useful control strategies to regulate the dynamics of the diseases, and the numerical simulations verify this phenomenon.
[1] | An introduction to stochastic epidemic models. Mathematical epidemiology, Lecture Notes in Mathematics (2008) 1945: 81-130. |
[2] | A stochastic threshold for an epidemic model with Beddington-DeAngelis incidence, delayed loss of immunity and Lévy noise perturbation. Phys. A (2018) 507: 312-320. |
[3] | Stochastic epidemic models: A survey. Math. Biosci. (2010) 225: 24-35. |
[4] | The global dynamics for a stochastic SIS epidemic model with isolation. Phys. A (2018) 492: 1604-1624. |
[5] | A class of stochastic delayed SIR epidemic models with generalized nonlinear incidence rate and temporary immunity. Phys. A (2017) 481: 198-208. |
[6] | Correction for life expectation of population and mortality of infant in Yunnan. Maternal and Child Health Care of China (2005) 20: 681-685. |
[7] | Joint effects of mitosis and intracellular delay on viral dynamics: Two-parameter bifurcation analysis. J. Math. Biol. (2012) 64: 1005-1020. |
[8] | Global asymptotic stability of stochastic Lotka-Volterra systems with infinite delays. IMA J. Appl. Math. (2015) 80: 1431-1453. |
[9] | Dynamics of positive solutions to SIR and SEIR epidemic models with saturated incidence rates. Nonlinear Anal. Real World Appl. (2013) 14: 1286-1299. |
[10] | Asymptotic stability of a two-group stochastic SEIR model with infinite delays. Commun. Nonlinear Sci. Numer. Simul. (2014) 19: 3444-3453. |
[11] | Analysis of the deterministic and stochastic SIRS epidemic models with nonlinear incidence. Phys. A (2015) 428: 140-153. |
[12] | Stationary distribution of a stochastic delayed SVEIR epidemic model with vaccination and saturation incidence. Phys. A (2018) 512: 849-863. |
[13] | Environmental Brownian noise suppresses explosions in population dynamics. Stochastic Process. Appl. (2002) 97: 95-110. |
[14] | Dynamics of a novel nonlinear stochastic SIS epidemic model with double epidemic hypothesis. J. Math. Anal. Appl. (2016) 433: 227-242. |
[15] | HIV cell-to-cell transmission requires the production of infectious virus particles, and does not proceed through Env-mediated fusion pores. Journal of Virology (2012) 86: 3924-3933. |
[16] | An age-structured model of HIV infection that allows for variations in the production rate of viral particles and the death rate of productively infected cells. Math. Biosci. Eng. (2004) 1: 267-288. |
[17] | Yunnan Bureau of Satistic, The Sixth National Census Data Bulletin in Yunnan Province 2010, Available from: http://www.stats.gov.cn/tjsj/tjgb/rkpcgb/dfrkpcgb/201202/t20120228_30408.html. |
[18] | The Data-center of China Public Health, Available from: http://www.phsciencedata.cn/. |
[19] | Parasite population delay model of malaria type with stochastic perturbation and environmental criterion for limitation of disease. J. Math. Anal. Appl. (2009) 360: 624-630. |
[20] | Response of a deterministic epidemiological system to a stochastically varying environment. Proceedings of the National Academy of Sciences of the United States of America (2003) 100: 9067-9072. |
[21] | An intra-host mathematical model on interaction between HIV and malaria. Bull. Math. Biol. (2010) 72: 1892-1911. |
[22] | Entire solutions for nonlocal dispersal equations with bistable nonlinearity: Asymmetric case. Acta Math. Sin. (Engl. Ser.) (2019) 35: 1771-1794. |
[23] | Mathematical analysis and simulation of a Hepatitis B model with time delay: A case study for Xinjiang, China. Mathematical Biosciences and Engineering (2019) 17: 1757-1775. |
[24] | Application of stochastic inequalities to global analysis of a nonlinear stochastic SIRS epidemic model with saturated treatment function. Adv. Difference Equ. (2018) 2018: 50-71. |
[25] | Mathematical modeling for schistosomiasis with seasonal influence: A case study in Hubei, China. SIAM J. Appl. Dyn. Syst. (2020) 19: 1438-1471. |
[26] | Mathematical model of transmission dynamics of human immune-deficiency virus: A case study for Yunnan, China. Appl. Math. Model. (2016) 40: 4859-4875. |
[27] | Study on the threshold of a stochastic SIR epidemic model and its extensions. Commun. Nonlinear Sci. Numer. Simul. (2016) 38: 172-177. |