Extinction and persistence of a stochastic SICA epidemic model with standard incidence rate for HIV transmission

In this paper, a stochastic SICA epidemic model with standard incidence rate for HIV transmission is proposed. The sufficient conditions of the extinction and persistence in mean for the disease are established. Numerical simulations show that random perturbations can suppress disease outbreaks and the risk of HIV transmission can be reduced by reducing the transmission coefficient of HIV while increasing the strength of the stochastic perturbation.

The above studies did not consider the effect of white noise in the environment on the model; in fact, infectious diseases are inevitably affected by random white noise in the environment. May [5] finds that because of the fluctuation of the environment, the parameters of the deterministic system, such as the death rate and the transmission coefficient, and other parameters of the deterministic system show a certain degree of random fluctuation. Therefore, the transmission coefficient may be affected by many environmental factors, such as temperature, wind, rain, and snow. In [6], the authors extend the classical SIS epidemic model from a deterministic framework to a stochastic one and formulate it as a stochastic differential equation (SDE) for the number of infectious individuals I(t). They discuss perturbation by stochastic noise. In the case of persistence they show the existence of a stationary distribution and derive expressions for its mean and variance. In [7], the authors present the threshold of a stochastic SIQS epidemic model which determines the extinction and persistence of the disease and find that noise can suppress the disease outbreak. Therefore, when establishing the corresponding mathematical model, we must consider the impact of white noise on the disease. Many scholars have introduced white noise into the infectious disease model [8][9][10][11][12]. In addition, there are a number of other types of stochastic models that have been developed to further explain that stochastic factors are integral to the modeling of infectious diseases, see [13][14][15][16].
In this paper, our aim is to introduce random white noise in the environment into the deterministic model and to study the effect of random disturbance on the number of HIV infected people and the conditions between the random disturbance and the parameters of the model, and if the number of HIV infected people can be controlled. Motivated by [6], we consider here random white noise in the environment, which is assumed to demonstrate itself as fluctuations in the parameter β, so that β → β + σ dB(t), where B(t) is a standard Brownian motion with intensity σ 2 > 0. Hence, we can derive the following stochastic model: This paper is organized as follows. In Sect. 2, we prove that there is a unique global positive solution for system (1.2). In Sect. 3, we show that the disease goes to extinction exponentially under certain conditions and the persistence of the disease, that is to say, the disease will prevail. In Sect. 4, we carry out the numerical simulations to demonstrate the analytical results. In Sect. 5, we give some conclusions.
Throughout this paper, we let ( , F, {F } t≥0 , P) be a complete probability space with filtration {F } t≥0 satisfying the usual conditions (that is to say, it is increasing and right continuous while F 0 contains all P-null sets). On the other hand, we define R d Generally speaking, consider the d-dimensional stochastic differential equation  (1.2) on t ≥ 0 for any initial value (S(0), I(0), C(0), A(0)) ∈ R 4 + , and the solution will remain in R 4 + with probability one, namely (S(t),

Theorem 2.1 There is a unique solution (S(t), I(t), C(t), A(t)) of system
Proof We can easily know that the coefficients of system (1.2) are locally Lipschitz continuous, then for any given initial value (S(0), where τ e is the explosion time (see [17]). To show that this solution is global, we only need to prove that τ e = ∞ almost surely. Let k 0 ≥ 0 be sufficiently large so that (S(0), I(0), C(0), A(0)) all lie within the interval [ 1 k 0 , k 0 ]. For each integer k ≥ k 0 , define the following stopping time: where throughout this paper, we set inf ∅ = ∞ (as usual ∅ denotes the empty set). According to the definition of the stopping time, τ k is increasing as k → ∞. Set τ ∞ = lim k→∞ τ k , whence τ ∞ ≤ τ e almost surely. Namely, we need to show that τ ∞ =∞ almost surely. We assumed that there exists a pair of constants T > 0 and ∈ (0, 1) such that As a result, there is an integer k 1 ≥ k 0 such that Applying Itô's formula, we obtain where Thus Integrating both sides of (2.3) from 0 to T ∧ τ k and taking expectations, we can obtain where a ∧ b denotes the minimum of a and b. In view of (2.4) and (2.5), we have Therefore, we must have τ ∞ = ∞ almost surely. In view of system (1.2), we have Solving this equation, we obtain that On the other hand, we have Then we can obtain The proof of Theorem 2.1 is complete.

Extinction and persistence in mean
In this section, we discuss under what conditions the disease will be extinct and the persistence of the disease, namely, under what condition the disease will prevail. For convenience, firstly, we define X(t) = 1 t t 0 X(s) ds.
Theorem 3.1 If R 1 < 1 or σ 2 ≤ β and R 2 < 1 hold, then the disease I(t) will die out exponentially with probability one, that is, .

Proof Let Q(t) = I(t) + C(t) + A(t).
Making use of Itô's formula, we can have Integrating on both sides of equation (3.1) from 0 to t, and then dividing by t, we can obtain where By the large number theorem for martingale (see [17]), we can get On the other hand, we consider the function f (x) = βx -σ 2 x 2 2 , where x ∈ (0, 1]. One can obtain that if σ Therefore,
In addition, then Furthermore, then Define where e is a constant. e = -min{-ln I(t)} to keep the nonnegativity of V (t). Applying Itô's formula, we obtain According to N(t) > λ μ+d , we have Hence In the light of (3.6) and (3.7), we have Furthermore, where If R 3 > 1, then (R 3 -1) > 0 almost surely.
In this case, we have = 0.806 < 1, σ 2β = 0.14 > 0, then the disease I(t) will die out (see Theorem 3.1 and Fig. 1(b)). In addition, the basic reproduction number of the corresponding deterministic model R 0 = 1.259 > 1, this means that the corresponding deterministic model (1.1) has an endemic equilibrium which is globally asymptotically stable, as shown in Fig. 1(a)-(d).
Secondly, we choose σ = 0.645, γ = 0.05, ω = 0.8, and other parameter values given by Table 1. In this case, we have then the disease I(t) will die out (see Theorem 3.1 and Fig. 2(b)). In addition, the basic reproduction number of the corresponding deterministic model R 0 = 1.271 > 1, this means that the corresponding deterministic model (1.1) also has an endemic equilibrium which is globally asymptotically stable, as shown in Fig. 2. Furthermore, we choose γ = 0.07, ω = 0.9, and σ = 0.8, 1, 1.2, 1.4, and other parameter values given by Table 1 to study the impact of σ on the dynamics for the SDE SICA model (1.2). In this case, we can obtain the values given in Table 2. Theorem 3.1 reveals the numerical results shown in Fig. 3.
Our results reveal that random perturbations in the environment can restrain the spread of the disease (see Fig. 2), and the bigger the intensity of the random perturbation, the    faster the disease dies out (see Fig. 3). However, deterministic models ignore this, so it is essential to introduce stochastic perturbations into deterministic models. Thirdly, we fix σ = 0.8 and choose β = 0.2, 0.4, 0.6, 0.8, 1, 1.2 and other parameters taken as in Table 1 to study the impact of β on the dynamics for the SDE SICA model (1.2). In this case, we can obtain the values given in Table 3, and the numerical results show that the smaller the transmission rate, the faster the disease dies out (see Fig. 4).
Finally, we choose σ = 0.2, γ = 0.01, ω = 0.1 and other parameter values given by then the disease I(t) will be persistence in mean, namely, the disease will prevail (see Theorem 3.2 and Fig. 5(b)).

Figure 4
The path of (S(t), I(t), C(t), A(t)) for stochastic model (1.2) when σ = 0.8 By numerical simulation, the results of numerical simulations show that HIV can be controlled by increasing the intensity of interference and reducing the transmission rates shown in Fig. 6 (e.g., increased HIV prevention campaigns, condom use, etc.)

Conclusion
This paper studied the extinction and persistence of a stochastic SICA epidemic model with standard incidence rate for HIV transmission. Firstly, we analyze that model (1.2) has a unique global positive solution for any initial value. Secondly, by Theorem 3.1, we can find that when R 1 = β 2 2σ 2 μ < 1 or σ 2 < β and R 2 = β (μ+ σ 2 2 ) < 1, disease will die out (see The-  Fig. 1 and Fig. 2). Furthermore, lim t→+∞ S(t) = λ μ (see Theorem 3.1), but for the corresponding deterministic model (1.2), R 0 > 1, there exists an endemic equilibrium, which means that a stochastic perturbation can suppress the outbreak of the disease (see Fig. 1 and Fig. 2), and the bigger the intensity of the random perturbation, the faster the disease dies out (see Fig. 3). However, deterministic models do not take this into account, so it is essential to include a stochastic element in deterministic models. In addition, we fix σ to study the impact of β on the dynamics for the SDE SICA model (1.2). In this case, we can find that the greater the rate of transmission, the higher the number of people infected (see Fig. 4).
Through numerical simulations, we can conclude that it is possible to reduce the transmission coefficient of HIV while increasing the strength of the stochastic perturbation to reduce the risk of HIV transmission, the simulation results are shown in Fig. 6.
On the other hand, in this paper, we only consider the effect of random perturbations on HIV transmission rate β, we can also study the effect of random perturbations on another parameter such as natural death rate, HIV treatment rate, AIDS induced death rate, and so on. In addition, in this paper, we only consider the effect of white noise. In fact, there are some random perturbations which cannot be modeled by white noises, for example, the telephone and Lévy noise, see [23][24][25][26] and the references therein. On the other hand, there have also been extensive numerical works to establish the positive property of numerical solutions for certain physical models (see [27][28][29]). We leave these investigations for future work.