Threshold Dynamics of a Stochastic SIR Model with Vertical Transmission and Vaccination

A stochastic SIR model with vertical transmission and vaccination is proposed and investigated in this paper. The threshold dynamics are explored when the noise is small. The conditions for the extinction or persistence of infectious diseases are deduced. Our results show that large noise can lead to the extinction of infectious diseases which is conducive to epidemic diseases control.


Introduction
The history of mankind is filled with struggle with diseases. Infectious diseases such as smallpox, cholera, plague of leprosy, diphtheria, syphilis, typhus fever, malaria, rabies, and tuberculosis have threatened the health of human beings. People have realized the importance of quantitative studies on the spread of infectious diseases to predict and to control them. It can be known from referring to the literature [1][2][3][4] that, with the aid of the establishment of infectious disease models, people can understand the crucial laws of infectious diseases and provide reliable and enough information to predict and control infectious diseases. For example, as early as 1760, Bernoulli and Blower [5] proposed the first mathematical model in epidemiology for studying the spread and inoculation of smallpox. Further, in 1927, Kermack and McKendrick [6] proposed the concept of the so-called "compartmental model," in which all the population was classified into three compartments: susceptible compartment , infected compartment , and removed compartment . It is assumed in the model that the susceptible class can transform into the infective class through contact with infected individuals, and the infectives can recover through treatment so that they have permanent immunity. Therefore, it is now well known that many scholars have paid attention to models; as a result, it can be seen in the literature that a large number of mathematical models of ordinary differential equations, delay differential equations, and partial differential equations have been constructed to study the spread of infectious diseases (see, e.g., [7][8][9][10][11][12][13][14][15][16][17][18][19][20][21][22][23]). In the last decades, we observed that scholars published few papers in scientific journals related to mathematics considering infectious diseases with vertical transmission which are transmitted from parents to their offspring (e.g., [1,[24][25][26]). Although scholars neglect the effect of vertical transmission, it is very important to study the real situation of the transmission of infectious diseases. The current diseases affecting humanity such as AIDS [27][28][29][30][31], Chagas' disease [32][33][34], hepatitis B [35,36], and hepatitis C [37] are vertically transmitted. From this, it can be clearly seen that mathematical modeling including vertical transmission, horizontal transmission, and vaccination [38,39] is more realistic than without them. Therefore, in this study, we have focused our attention on this and an epidemic model involving vertical transmission and vaccination was proposed as follows [1,24] (see Figure 1 where ( ), ( ), and ( ) represent the members of the susceptible, the infectious, and the removed or the recovered members from infection, respectively. is the birth and death rate of ( ) and ( ), is the birth and death rate of ( ), is the contact rate, and (0 < < 1) is the vaccination proportion to the newborn from ( ) and ( ). Then, constant (0 < < 1) is the proportion of the offspring of infective parents that are susceptible individuals and + = 1. is the recovery rate of the infective individuals. Obviously, the total population size is normalized to one, and the basic reproductive number of system (1) is 0 = (1 − )/( + ). By constructing a Lyapunov function and using the LaSalle invariance principle, we can show that if 0 < 1, the infectionfree equilibrium 0 (1 − , 0, ) is globally asymptotically stable, while if 0 > 1, the infection-free equilibrium 0 is unstable and the endemic equilibrium * ( * , * , * ) is globally asymptotically stable.
In fact, the spread of diseases is inevitably disturbed by the influence of random factors; the stochastic epidemic system is more in line with the actual situation. Therefore, epidemic systems described by stochastic differential equations have been paid extensive attention in recent years (see, e.g., [40][41][42][43][44][45][46]). Various stochastic perturbation approaches have been introduced into epidemic systems and excellent results have been obtained. In this study, our main objective is to introduce four approaches. The first one is to analyze epidemic systems including the environment noise by using the method of time Markov chain (see, e.g., [47][48][49][50][51]). The second one is to consider the parameters' perturbation (see, e.g., ). The third one is to introduce Lévy jump noise into the system [73][74][75]. The fourth one is to investigate stochastic perturbation around the positive equilibria of deterministic systems (see, e.g., [41,42,[76][77][78]).
Parameter perturbation induced by white noises is an important and common form to describe the effect of stochasticity. In this paper, we adopt the perturbation with white noises, that is, → +( ), where ( ) is a standard Brownian motion with intensity 2 > 0. Then, the resultant system transforms into the following form: This paper is organized as follows. In Section 3, we will discuss the extinction of infectious diseases and explore the conditions leading to the extinction of infectious diseases. In Section 4, we will deduce the condition for a disease in order to be persistent.
Let { } ≥0 be a one-dimensional Brownian motion defined on the complete probability space (Ω, F, P) adapted to the filtration Let 2,1 (R × R + ; R) denote the family of all real-valued functions ( , ) defined on R × R + such that they are continuously twice differentiable in and once in . We set ) .
Lemma 1 (one-dimensional Itô's formula [40,79,80]). Let ( ) be an Itô process on ≥ 0 with the stochastic differential where is again an Itô process with the stochastic differential given by almost surely.
By using the methods from Lahrouz and Omari [81], we can prove the following lemma.

Extinction
In this section, we deduce the condition which will cause a disease to die out.
for convenience; then, we have the following results that we have mentioned in the following theorem.
Remark 6. Theorem 5 shows that when 2 > max{ , 2 / 2( + )}, the infectious disease of system (2) goes to extinction almost surely; namely, large white noise stochastic disturbance is conducive to control infectious diseases. When the white noise is not large and * < 1, the infectious disease of system (2) also goes to extinction almost surely; then, * is the threshold associated with the extinction of infectious diseases.

Let us denote
for convenience; then, we have the following results that we have mentioned in the following theorem.
Remark 9. Theorems 5 and 8 show that the condition for the disease to die out or persist depends on the intensity of white noise disturbances strongly. And small white noise disturbances will be beneficial for long-term prevalence of the disease; conversely, large white noise disturbances may cause the epidemic disease to die out.

Conclusion and Numerical Simulation
In this paper, a stochastic system with vertical transmission and vaccination is proposed. The threshold dynamics  depending on the stochastic perturbation are deduced by using the theory of stochastic differential equation and inequality technique. Our results show that the dynamics of the stochastic system are different with the deterministic case due to the effect of stochastic perturbation, and the persistent diseases in the deterministic system may be eliminated under the stochastic perturbation. In the following, by employing the Euler Maruyama (EM) method [40], we perform some numerical simulations to illustrate the extinction and persistence of the diseases in the stochastic system and corresponding deterministic system for comparison.

Conflicts of Interest
The authors declare that there are no conflicts of interest regarding the publication of this paper.