Global Analysis of a Novel Nonlinear Stochastic SIVS Epidemic System with Vaccination Control

This paper proposes a stochastic SIVS epidemic system with nonlinear saturated infection rate under vaccination and investigates the dynamics predicted by the model. By using Itô’s formula and Lyapunov methods, we first study the existence and uniqueness of global positive solution. Then we investigate the stochastic dynamics of the system and obtain the thresholds which govern the extinction and the spread of the epidemic disease. Results show that large stochastic noises can lead to the extinction of epidemic diseases; that is, stochastic disturbances can suppress the outbreak of epidemic diseases. Finally, we carry out a series of numerical simulations to demonstrate the performance of our theoretical findings.


Introduction
Infectious diseases pose a serious threat to public health around the world.Therefore, the study of epidemic diseases has been part of the burning issues of scientists.Mathematically, after the pioneering work of Kermack and McKendrick on the epidemic regularity of the Black Plague in London by the famous SIR model [1], mathematical models have been used extensively by scientists to study the spread and control of diseases [2][3][4][5][6][7].There are many ways to suppress the spread of disease, for instance, cutting off transmission routes, paying attention to food hygiene, and vaccination.Vaccination is an effective method of preventing infectious diseases and many scientists have explored the effect of vaccination on diseases [8][9][10][11][12][13][14][15][16][17].For instance, Li and Ma [8]  ( The incidence rate in model ( 1) is bilinear, which means that the number of people infected by a patient within a unit time is proportional to the total number of susceptible individuals in the environment.However, the number of susceptible individuals to contact with a patient within a unit time is limited, and many scholars have noted that the saturated incidence is more accurate to describe the spread of epidemic [18][19][20][21].Besides, the WHO reports that licensed vaccines are currently available to prevent or contribute to the prevention and control of twenty-five preventable infections.But the effect of vaccination is not absolute.In other words, some people who are vaccinated still have the risk of being infected [22,23].Moreover, epidemic diseases may be subject to uncertain environmental disturbances, such as fluctuations of birth rate, death rate, and infection rate.These phenomena can be characterized by stochastic processes.Numerous scholars have introduced stochastic interferences into differential systems, and the stochastic dynamics of such systems were investigated [24][25][26][27][28][29][30][31][32][33][34][35][36].
Motivated by the above works, in this paper, we suppose the following: Then model (1) becomes where (), (), and (), respectively, stand for the density of susceptible and infective and vaccinated individuals at time ; Λ is a constant input of new numbers into the population;  means a fraction of vaccinated newborn;  represents the natural death rate of (), (), and ();  is the proportional coefficient of vaccinated susceptible;  is the recovery rate of ();  stands for the rate of losing their immunity for vaccinated individuals; and  is the disease-caused death rate of infectious individuals ().Throughout this paper, let (Ω, F, {F} ≥0 , P) be a complete probability space with a filtration {F  } ≥0 satisfying the usual conditions (i.e., it is increasing and right continuous while F 0 contains all P-null sets).Function   () ( = 1, 2) is a Brownian motion defined on the complete probability space Ω, and the intensity of   () is   ( = 1, 2).For an integrable function () on [0, +∞), we define ⟨()⟩ = (1/) ∫  0 ().

The Existence and Uniqueness of Global Positive Solutions
Due to physical meaning, variables (), (), and () in model (2) should remain nonnegative for  ≥ 0. We next prove that this is actually the case and, furthermore, the positive solution is unique.
When  → ∞, we have This is a contradiction.So  ∞ = ∞ and we complete the proof.

Extinction
Theorem 3. Let ((), (), ()) be the solution of model (1) with initial value ((0), (0), (0)) ∈  3  + .If one of the following conditions holds: Proof.Applying Itô's formula to ln (), we have Mathematical Problems in Engineering Integrating from 0 to  and dividing both sides of ( 18) by , one has 2 where are real-valued continuous martingales.Thus By the strong law of large numbers [37], we have lim Taking the limit superior on both sides of (32), we have Applying (36) and lim →+∞ () = 0, there exist two arbitrarily small constants  1 ,  2 > 0 such that when  > , we have and so we have
Taking the limit superior on both sides of (58), we have lim sup This completes the proof.

Conclusions and Numerical Simulations
This paper investigates a stochastic epidemic system with nonlinear incidence rate under vaccination.We first demonstrate the existence and uniqueness of global positive solution of model ( 2).Then we study the persistence in mean and extinction of the stochastic SIV system.Our result shows that large noises can lead to the extinction of epidemic diseases.In addition, notice that the basic reproduction number for epidemic disease to die out is negatively related to the intensity of interference, which indicates that stochastic impact can suppress the epidemic diseases.Compared with the existing work in [8,10], the model constructed in this paper also considered the efficiency of vaccination.Authors in [8,10] considered the case that the average contact rate of an infective individual explored to a vaccinated individual per unit time at time  is  2 = 0, and we study the more general case 0 ≤  2 ≤  1 , which is more consistent with the actual situation.In addition, since the number of susceptible and vaccinated individuals to contact with a patient within a unit time is limited, in this paper, we employ a saturated incidence to describe the spread of an epidemic; that is,   /(1 +   ),  = 1, 2.