Stochastic dynamical analysis for the complex infectious disease model driven by multisource noises

PLOS ONE RESEARCH ARTICLE Stochastic dynamical analysis for the complex infectious disease model driven by multisource noises Liqiong Jian1, Xinyu Bai ID2*, Shaojuan Ma ID2 1 The Blood Center of Ningxia Hui Autonomous Region, Yinchuan, China, 2 School of Mathematics and Information Science, North Minzu University, Yinchuan, China * a1111111111 a1111111111 a1111111111 a1111111111 a1111111111 OPEN ACCESS Citation: Jian L, Bai X, Ma S (2024) Stochastic dynamical analysis for the complex infectious disease model driven by multisource noises. PLoS ONE 19(1): e0296183. https://doi.org/10.1371/ journal.pone.0296183 Editor: Minyu Feng, Southwest University, CHINA Abstract This paper mainly studies the dynamical behavior of the infectious disease model affected by white noise and Lévy noise. First, a stochastic model of infectious disease with secondary vaccination affected by noises is established. Besides, the existence and uniqueness of the global positive solution for the stochastic model are proved based on stochastic differential equations and Lyapunov function, then the asymptotic behavior of the disease-free equilibrium point is studied. Moreover, the sufficient conditions for the extinction of the disease are obtained and the analysis showed that different noise intensity could affect the extinction of infectious disease on different degree. Finally, the theoretical results are verified by numerical simulation and some suggestions have been put forward on how to prevent the spread of diseases are presented. Received: September 27, 2023 Accepted: December 7, 2023 Published: January 4, 2024 Copyright: © 2024 Jian et al. This is an open access article distributed under the terms of the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Data Availability Statement: All relevant data are within the paper. Funding: This work was supported by the grants from the National Natural Science Foundation of China (No. 12362005), Ningxia higher education first-class discipline construction funding project (NXYLXK2017B09), Major Special project of North Minzu University (No. ZDZX201902). The funders had role in supervision, writing, decision to publish and review of the manuscript. Competing interests: The authors have declared that no competing interests exist. 1 Introduction Infectious diseases can be transmitted between people, animals and goods which always threaten human survival and take great challenges to whole world [1–3]. In recent decades, a large number of mathematical models for infectious diseases have been built and studied widely to realize the infectious disease. The basic mathematical model representing the dynamical behavior of the three main populations which include the susceptible(S(t)), the infected(I(t)) and the recovered(R(t)), was firstly proposed in 1927 by Kermack and McKendricks [4] called SIR model. Two metapopulation SIR models from individual and population perspectives were proposed in reference [5], which studied the significant influence of contactdependent infection and migration on epidemic propagation. Khyar et al. [6] considered the multi strain SEIR epidemic model with general incidence rate and gave the equilibrium point stability theorem of different strains. Reference [7] analyzed modified SLIR model with nonlinear incidence and equilibriums of the proposed model are both globally asymptotically stable. Gumel et al. [8] proposed extended models of COVID-19 in which the stability of the equilibrium point and parameter estimation was studied. As we all know, the accurate modeling can more effectively explore the mechanism of the infectious disease. Therefore, the main factors reflected the practical infectious diseases must be considered, such as delay-time, vaccination and random disturbance. Vaccination [9–12] has PLOS ONE | https://doi.org/10.1371/journal.pone.0296183 January 4, 2024 1 / 32 PLOS ONE always been one of the effective measures to control infectious diseases. Xing et al. [13] studied a recurrent nonautonomous SVIR epidemic model with vaccination, who proved the existence and uniqueness of globally attractive near periodic solutions for the model. A deterministic SVIRS epidemic model with Holling type II incidence rate and vaccination was investigated in reference [14], which explicitly discussed the local stability of the disease equilibrium and the existence of Hopf bifurcation. In order to enhance the immune effect and increase the probability of antibody production, most vaccines adopt a vaccination program of two or more doses. Gabrick et al. [15] propose a SEIR model with two doses of vaccine administration, and analyze that administering two doses of vaccine can significantly reduce the number of infections. Omar et al. [16] generated fractional order model based on the secondary vaccination and analyzed various vaccination strategies. Reference [17] established the SIRS model by introducing vaccination passes and made predictions based on real-world parameter values. In real life, infectious diseases are inevitably influenced by various random factors during the transmission process, so considering the influence of random factors in infectious disease models will be more practical. In addition, many studies shown that environmental fluctuations also have a huge impact on the development of epidemics with vaccination. Therefore, stochastic differential equation model [18–20] became a more appropriate method for modeling epidemic diseases. A stochastic cholera model with saturation recovery rate is discussed in the reference [21], then the optimal control is added and studied to provide a theoretical basis for the prevention and control of cholera. Zhang et al. [22] established a stochastic SVIR model with general incidence rate, who obtained the sufficient conditions of the extinction and persistence for the model affected by white noise. Reference [23] built a SIVS epidemic model with white noise and gave sufficient conditions of the existence for the periodic solutions. It can be found in reference [24] that the random threshold of the outcome for the stochastic SIS model with vaccination can be determined in case the white noises are small. The SVIR model with white noise was proposed in reference [25], which showed that that environmental white noise is helpful for controlling the disease. Reference [26] further shown that the disease gradually disappeared due to the influence of environmental white noise in the stochastic COVID19 model. Wang et al. [27] studied the influence of vaccination rates, vaccine effectiveness and immune loss rates on infectious disease in a stochastic mathematical model with vaccination. We can conclude from the above references that appropriate white noise intensity can accelerate the extinction of diseases under some conditions. However, some sudden environmen (...truncated)