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Extinction and persistence of a stochastic SICA epidemic model with standard incidence rate for HIV transmission
Advances in Difference Equations volume 2021, Article number: 260 (2021)
Abstract
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.
Introduction
To the best of our knowledge, the human immunodeficiency virus (HIV) is a retrovirus that causes HIV infection and, over time, acquired immunodeficiency syndrome (AIDS). There is no cure or vaccine to AIDS. However, antiretroviral (ART) treatment improves health, prolongs life, and substantially reduces the risk of HIV transmission. As more people receive antiretroviral therapy, the number of new HIV infections worldwide is approximately 2.3 million, a 33 per cent decline in new infections compared to 2001. At the same time, the number of AIDS deaths is also declining, with approximately 1.6 million AIDS deaths in 2012, down from 2.3 million in 2005 [1]. This goes to show that access to antiretroviral therapy has a huge impact on HIV prevention. Parvaiz et al. consider a fractionalorder HIV epidemic model with the inclusion of prostitution in the population and its consequences on the disease transmission [2]. In [3], the authors consider a nonlinear fractional order epidemic model for HIV transmission and analyze by including an extra compartment, namely the exposed class, to the basic SIR epidemic model. They show through numerical simulations that the control measures effectively increase the quality of life and age limit of the HIV patients. The authors in [4] consider the following model:
where the parameters are:

\(N(t)\): The total population at time t;

\(S(t)\): Susceptible individuals at time t;

\(I(t)\): HIVinfected individuals with no clinical symptoms of AIDS at time t;

\(C(t)\): HIVinfected individuals under ART treatment with a viral load remaining low at time t;

\(A(t)\): HIVinfected individuals with AIDS clinical symptoms at time t;

λ: Recruitment rate;

μ: Natural death rate;

β: HIV transmission rate;

γ: HIV treatment rate for I individuals;

ρ: Default treatment rate for I individuals;

α: AIDS treatment rate;

ω: Default treatment rate for C individuals;

d: AIDS induced death rate.
They found that when \(R_{0}<1\), the diseasefree equilibrium of system (1.1) is asymptotically stable; when \(R_{0}>1\), the diseasefree equilibrium is unstable and there is a globally asymptotically stable endemic equilibrium. Here, \(R_{0}= \frac{\beta (\alpha +\mu +d)(\mu +\omega )}{\mu [(\mu +\omega )(\rho +\alpha +\mu +d)+\gamma (\alpha +\mu +d)+\rho d]+\rho \omega d}\) is the basic reproduction number.
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 [8–12]. 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 [13–16].
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 \(\beta \rightarrow \beta +\sigma \,dB(t)\), where \(B(t)\) is a standard Brownian motion with intensity \(\sigma ^{2}>0\). 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 \((\Omega , \mathcal{F}, \{\mathcal{F}\}_{t\geq 0}, \mathbb{P})\) be a complete probability space with filtration \(\{\mathcal{F}\}_{t\geq 0}\) satisfying the usual conditions (that is to say, it is increasing and right continuous while \(\mathcal{F}_{0}\) contains all \(\mathbb{P}\)null sets). On the other hand, we define \(\mathbb{R}_{+}^{d}=\{x\in \mathbb{R}^{d}x_{i}>0\mbox{ for all }1\leq i\leq d\}\).
Generally speaking, consider the ddimensional stochastic differential equation
where \(f(t,x(t))\) is a function in \(\mathbb{R}^{d}\) defined in \([t_{0},\infty ]\times \mathbb{R}^{d}\), and \(g(x(t), t)\) is a \(d\times {m}\) matrix, f, g are locally Lipschitz functions in x. \(B_{t}\) denotes an mdimensional standard Brownian motion defined on the complete probability space \((\Omega ,\mathcal{F},\{\mathcal{F}\}_{t\geq 0},\mathbb{P})\). Denote by \(C^{2,1}(\mathbb{R}^{d}\times [t_{0},\infty ];\mathbb{R}_{+})\) the family of all nonnegative functions \(V (x(t), t)\) defined on \(\mathbb{R}^{d}\times [t_{0},\infty ]\) such that they are continuously twice differentiable in x and once in t. We define the differential operator L of equation (1.3) by [17]
If L acts on a function \(V\in {C^{2,1}(\mathbb{R}^{d}\times [t_{0},\infty ],\mathbb{R}_{+})}\), then
where \(V_{t}(x,t)=\frac{\partial {V}}{\partial {t}}\), \(V_{x}(x,t)=( \frac{\partial {V}}{\partial {x_{i}}},\ldots, \frac{\partial {V}}{\partial {x_{d}}})\), \(V_{xx}(x,t)=( \frac{\partial ^{2}{V}}{\partial {x_{i}}\partial {x_{j}}})_{d\times {d}}\).
From Itô’s formula, if \(x(t)\in \mathbb{R}^{d}\), then
Existence and uniqueness of positive solution
Theorem 2.1
There is a unique solution \((S (t ),I (t ),C (t ),A (t ) )\) of system (1.2) on \(t\geq 0\) for any initial value \((S(0),I(0),C(0),A(0))\in \mathbb{R}_{+}^{4}\), and the solution will remain in \(\mathbb{R}_{+}^{4}\) with probability one, namely \((S (t ),I (t ),C (t ),A (t ) ) \in \mathbb{R}_{+}^{4}\) for all \(t\geq 0\) almost surely. Moreover,
where \(N(t)=S (t )+I (t )+C (t )+A (t )\).
Proof
We can easily know that the coefficients of system (1.2) are locally Lipschitz continuous, then for any given initial value \((S(0),I(0),C(0),A(0))\in \mathbb{R}_{+}^{4} \), there is a unique local solution \((S(t),I(t),C(t),A(t))\) on \(t\in [0,\tau _{e})\), where \(\tau _{e}\) is the explosion time (see [17]). To show that this solution is global, we only need to prove that \(\tau _{e}=\infty \) almost surely. Let \(k_{0}\geq 0\) be sufficiently large so that \((S(0),I(0),C(0),A(0))\) all lie within the interval \([\frac{1}{k_{0}},k_{0}]\). For each integer \(k\geq {k_{0}}\), define the following stopping time:
where throughout this paper, we set \(\inf \emptyset =\infty \) (as usual ∅ denotes the empty set). According to the definition of the stopping time, \(\tau _{k}\) is increasing as \(k\rightarrow \infty \). Set \(\tau _{\infty }{=}\lim_{k\rightarrow \infty }\tau _{k}\), whence \(\tau _{\infty }\leq \tau _{e}\) almost surely. Namely, we need to show that \(\tau _{\infty }{=}\infty \) almost surely. We assumed that there exists a pair of constants \(T>0\) and \(\epsilon \in (0,1 )\) such that
As a result, there is an integer \(k_{1}\geq {k_{0}}\) such that
Now define a \(C^{2}\)function \(V:\mathbb{R}_{+}^{4}\rightarrow \mathbb{R}_{+}\) by
Applying Itô’s formula, we obtain
where
Thus
Integrating both sides of (2.3) from 0 to \(T\wedge {\tau _{k}}\) and taking expectations, we can obtain
Set \(\Omega _{k}=\{\tau _{k}\leq {t}\}\) for \(k\geq {k_{1}}\) by (2.2), \(P (\Omega _{k} )\geq \epsilon \). Notice that, for every \(\omega \in \Omega _{k}\), there is at least one of \((S (\tau _{k},\omega ),I (\tau _{k},\omega ),C (\tau _{k},\omega ),A (\tau _{k},\omega ) )\) that equals to k or \(\frac{1}{k}\). Consequently,
where \(a\wedge {b}\) denotes the minimum of a and b. In view of (2.4) and (2.5), we have
where \(1_{\Omega _{k}}\) is the indicator function of \(\Omega _{k}\). Let \(k\rightarrow \infty \) lead to the contradiction
Therefore, we must have \(\tau _{\infty }=\infty \) almost surely.
In view of system (1.2), we have
Solving this equation, we obtain that
which implies that
On the other hand, we have
Then we can obtain
which implies that
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 \(\langle X(t)\rangle =\frac{1}{t}\int _{0}^{t}X(s)\,ds\).
Theorem 3.1
If \(R_{1}<1\) or \(\sigma ^{2}\leq \beta \) and \(R_{2}<1\) hold, then the disease \(I(t)\) will die out exponentially with probability one, that is,
where
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
In the light of (3.2) and (3.3), if \(R_{1}<1\), then
which implies that
On the other hand, we consider the function \(f(x)=\beta x \frac{\sigma ^{2}x^{2}}{2}\), where \(x\in (0,1]\). One can obtain that if \(\frac{\sigma }{\sqrt{2}}\leq \frac{\beta \sqrt{2}}{2\sigma }\), that is, \(\sigma ^{2}\leq \beta \), \(f(x)\) has the max value \(f(1)=\beta \frac{\sigma ^{2}}{2}\). Let \(x= \frac{ S(t)I(t)}{N(t)(I(t)+C(t)+A(t))}\), we have
Therefore,
Hence, if \(R_{2}<1\), we obtain
which implies that
Based on the above analysis, in view of (2.1), if \(R_{1}<1\) or \(\sigma ^{2}\leq \beta \) and \(R_{2}<1\), we have
The proof is completed. □
Theorem 3.2
For any initial value \((S(0),I(0),C(0),A(0))\in \mathbb{R}_{+}^{4}\), if \(R_{3}>1\), the disease is persistence in mean. Furthermore,
where
Proof
Integrating system (1.2) from 0 to t, we can obtain
Then
where
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 \(N(t)>\frac{\lambda }{\mu +d}\), we have
Hence
In the light of (3.6) and (3.7), we have
Furthermore,
where
If \(R_{3}>1\), then
□
Numerical simulations
In this section, we use Milstein’s method [18] to simulate stochastic model (1.2) with a numerical scheme for stochastic model (1.2) given by
where \(\xi _{k}\), \(k = 1,2,\ldots ,n\), are independent Gaussian random variables \(N(0,1)\).
Firstly, we choose \(\sigma =0.8\), \(\gamma =0.07\), \(\omega =0.9\), 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. 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 \(\sigma =0.645\), \(\gamma =0.05\), \(\omega =0.8\), 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 \(\gamma =0.07\), \(\omega =0.9\), and \(\sigma =0.8,1,1.2,1.4\), 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 \(\sigma =0.8\) and choose \(\beta =0.2, 0.4, 0.6, 0.8, 1, 1.2\) 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 \(\sigma =0.2\), \(\gamma =0.01\), \(\omega =0.1\) and other parameter values given by Table 1. In this case, we have
then the disease \(I(t)\) will be persistence in mean, namely, the disease will prevail (see Theorem 3.2 and Fig. 5(b)).
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 \(R_{1}=\frac{\beta ^{2}}{2\sigma ^{2}\mu }<1\) or \(\sigma ^{2}<\beta \) and \(R_{2}=\frac{\beta }{(\mu +\frac{\sigma ^{2}}{2})}<1\), disease will die out (see Theorem 3.1 and Fig. 1 and Fig. 2). Furthermore, \(\lim_{t\rightarrow +\infty }S(t)=\frac{\lambda }{\mu }\) (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).
Finally, if \(R_{3}=\frac{\beta }{\mu +\gamma +\rho +\frac{\sigma ^{2}}{2}}>1\), the disease will be persistence in mean, namely, the disease will prevail (see Theorem 3.2 and Fig. 5).
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 [23–26] 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 [27–29]). We leave these investigations for future work.
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The authors would like to thank the editor and the anonymous reviewers for their valuable comments and constructive suggestions.
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This research was supported by Program for Tianshan Innovative Research Team of Xinjiang Uygur Autonomous Region, China (2020D14020).
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Wang, X., Wang, C. & Wang, K. Extinction and persistence of a stochastic SICA epidemic model with standard incidence rate for HIV transmission. Adv Differ Equ 2021, 260 (2021). https://doi.org/10.1186/s1366202103392y
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Keywords
 SICA epidemic model
 Extinction
 Persistence in mean
 Stochastic perturbation