Dynamics of an HIV/AIDS transmission model with protection awareness and fluctuations

Infectious diseases caused one-quarter of the global deaths ref1 . Various factors such as media campaigns, population migration and temperature changes influenced the spreading of infectious diseases. Since June 6 of 1981, the first global case of HIV (Human Immunodeficiency Virus) infection was announced, human beings have been fighting against HIV for over four decades. As of the year 2022, about 37.7 million individuals have been infected with HIV ref2 . As we have known today, the transmission of HIV took place through blood, semen, cervical (or, vaginal secretions) and breast milk as well. Especially, an infected individual was unaware of the protection and was lack of active treatment, HIV often broke down the immune system of the infected individual and eventually turned into the acquired immune deficiency syndrome (AIDS). Although there was no drug or vaccine for HIV, the antiretroviral therapy (ART) could prolong the life expectancy of an infected individual and make it approach that of the uninfected individuals. Meanwhile, the infected individuals with ART treatment do not retransmit HIV to their sexual partners ref2 . From 2000 to 2018, the number of new HIV infected individuals fell by 37%, HIV-related deaths fell by 45%, and ART treatment saved 13.6 million individuals. At the end of 2018, about 23.3 million individuals had received ART treatment ref2 . Thus, UNAIDS puts forward the 90%-90%-90% plan(90% of AIDS infected people know they are infected, 90% of confirmed AIDS patients to be treated, and 90% of HIV in treated patients’ body is suppressed), and set the great goal of eliminating the AIDS epidemic by 2030 in UN General Assembly Resolution 70/266.

Mathematical modelling of HIV/AIDS and its kinetic behavior analysis can well predict the development trend of HIV/AIDS. Many scholars have already studied the HIV/AIDS model and its kinetic behavior. For example, Silva and Torres ref3 obtained the results on the global stability of the HIV/AIDS model by considering bilinear incidence rates. Ghosh et al. ref4 studied the effect of medias and self-imposed psychological fears on disease dynamics by separating the susceptible into the unaware and the aware individuals. Later, Zhao et al. ref6 modified the model established by Fatmawati et al. ref5 , and considered piecewise fractional differential equations and investigated the effect of protection awareness on HIV transmission. However, the transmissions of infectious disease were inevitably affected by the environmental noises in the real circumstances. In other words, the numbers of the individuals in each compartment were usually fluctuated due to the emergence of infectious disease and control as well by local governments. Therefore, the epidemic models with fluctuations in practice were necessary to investigate their long-term dynamics. For instance, Mao et al. ref7 found that the small fluctuations to the deterministic models effectively suppress the rapid increment of the population. Other recent epidemic models in ref8 ; ref9 ; ref10 ; ref11 ; ref12 ; ref13 ; ref13.2 ; ref13.3 also governed the fluctuations to describe the diversities of their models. More precisely, ref13.2 ; ref13.3 found that small fluctuations produced the long-term persistence, and large fluctuations led to the extinction of infectious diseases in stochastic HIV/AIDS models. Liu ref13.4 discovered that the higher order fluctuations made HIV/AIDS eradicative under sufficient conditions. Meanwhile, Wang ref13.5 also figured out the extinction and persistence in the mean depended on the fluctuations of main parameters.

We formulate a stochastic HIV/AIDS model with protection awareness by considering the environmental noises into the model of Fatmawati et al. ref5 . We next provide the expression of the basic reproduction number for the deterministic model. In Section 3, we prove theoretically that there exists a unique positive solution to the stochastic model, and the existence of a unique ergodic stationary distribution is investigated. Further, we give a new threshold $R_{0}^{e}$ for the extinction of HIV/AIDS, and the corresponding numerical simulations are demonstrated to verify the theoretical results. Then we conclude that the detailed publicity has great impact on the control of AIDS compared with the extensive publicity. Moreover, the continuous antiretroviral therapy (ART) is helpful to control the number of the individuals with HIV/AIDS. Meanwhile, we figure out the main improvements compared with other contributions, and give some suggestions to control the long-term dynamics of HIV/AIDS.

The number that an infected contacts with the susceptible per unit of time is called the contact number, which is usually related to the total population $N$ and denoted as $U(N)$. Let the probability of infection per exposed individual be $\beta_{0}$, and $\beta_{0}U(N)$ is called the effective exposure number, which represents the infectivity of an infected individual per unit of time. The total population is usually separated by the susceptible individuals, the immune individuals, and the exposed individuals. So, the proportion of the susceptible individuals in the whole population is ${\frac{S(t)}{N(t)}}$, which is not being infected by the infected individuals. Then, the number of the susceptible being infected effectively at time $t$ is

which is called the incidence rate. We assume in this paper that the contact rate between the susceptible and the infected is proportional to the total population, i.e., $U(N)=kN(t)$. Let $\beta=\beta_{0}k$, then the incidence rate is rewritten as $\beta S(t)I(t)$, which is called the bilinear incidence rate. Many researchers have applied the bilinear incidence rates into their HIV/AIDS models ref14 ; ref15 ; ref16 ; ref17 for further discussions.

Fatmawati et al. ref5 developed a model of HIV/AIDS with the protection awareness. Precisely, they divided the population into five different groups, that is, the susceptible without protection awareness $(S_{u})$, the susceptible with protection awareness $(S_{a})$, the infected without ART treatment $(I)$, the infected with ART treatment $(C)$ and the infected who eventually developed into AIDS $(A)$. And, the total people size $N$ is expressed as

We simplified the model of Fatmawati et al. ref5 by considering bilinear incidence:

where $\Lambda$ is the recruitment rate, $\mu$ is the natural death rate, and $\delta$ is the mortality rate of AIDS. $\alpha$ is the migration rate from $S_{u}$ to $S_{a}$, $\beta$ is the HIV transmission rate, and $\varepsilon$ is the infection rate of $S_{a}$. The parameters $\eta$, $\nu$, $\gamma$, and $\rho$ represent the transmission rates from $C$ to $I$, $A$ to $I$, $I$ to $A$ and $I$ to $C$. The mutual migrating mechanisms of each group in the model could be demonstrated clearly in Figure 2.1. We suppose that all parameters of model (2.1) are positive initiated with

Now, we add five equations of model (2.1) and then we get

By comparing theorems, the positive invariant set of model (2.1) is derived

We only consider the biological properties of model (2.1) in the set $\Omega$. The basic regeneration number of model (2.1) can be obtained from the next generation matrix approach ref17.1 ; ref17.2 as follows:

with $k_{1}=\mu+\delta+\nu$, $k_{2}=\mu+\eta$.

Similarly to the proof of Theorem 2 in Fatmawati et al. ref5 , when $R_{0}<1$, we obtain that model (2.1) has a unique boundary equilibrium point

which is locally asymptotically stable in the set $\Omega$. We also provide the expression of an endemic equilibrium point of model (2.1) when $R_{0}>1$, that is

where

Substituting $S_{u}^{\ast}$, $S_{a}^{\ast}$, $C^{\ast}$, $A^{\ast}$ into the third equation of (2.1) and making the left side be zero, then we can get

where

Obviously $f(0)=0$. When $R_{0}>1$, we have $P_{1}<0$ and $P_{3}>0$. According to the Descartes sign rule, $f(I^{\ast})$ has a unique positive real root regardless of the sign of $P_{2}$. Therefore, the endemic equilibrium point $P^{\ast}$ exists.

Motivated by the models described by stochastic differential equations in ref18 ; ref19 ; ref20 ; ref21 ; ref22 , we introduce the environmental fluctuations into model (2.1) similar to that of Evansref22.1 and Tanref22.2 . We set that environmental fluctuations are multiplicative white noise types which are proportional to $S_{u},S_{a},I,C,A$. When $\Delta t\rightarrow 0$, we consider a Markov process

with the following descriptions:

and

We therefore derive a stochastic epidemic model as follows:

Let $i=1,2,3,4,5$, then $B_{i}(t)$ are independent standard Brownian motions with the initial values $B_{i}\left(0\right)=0$ and $\sigma_{i}^{2}>0$ are the intensities of white noises, and the initial values $X(0)=(S_{u}(0),S_{a}(0),I(0),C(0),A(0))^{T}$ as well.

Let $X(t)$ be a homogeneous markov process in $\mathbb{R}^{d}$, which satisfies the stochastic differential equation

The diffusion matrix is defined as

(i) The minimum eigenvalue of the diffusion matrix $A(X)$ is non-zero in its domain $G$ and one of its neighborhoods.

(ii) The average time $\tau$ for the path from $z$ to set $G$ is finite when $z\in\mathbb{R}^{d}\backslash G$, and $\sup_{z\in K}E^{z}\tau<\infty$ holds for each compact subset $K\in\mathbb{R}^{d}$.

Then, the Markov process $X(t)$ has a unique ergodic stationary distribution $\pi(\cdot)$. Let $f(X)$ be an integrable function of $\pi$, for all $X\in\mathbb{R}^{d}$, the following formula holds:

Now, we will give two useful results, Lemma 3.2 and Lemma 3.3, by using of Theorem 2.1 and Theorem 3.1 in ref27 . In fact, we write down the conclusions without consider the details of the proofs.

Before we start to study the dynamical behaviors of the stochastic epidemic model (2.3), the existence of a global positive solution is of importance. Next, we show that there exists a unique global positive solution to (2.3) for any given initial value.

Obviously, inequality (3.2) can be transformed into

where $\mathbb{I}_{\Omega_{n}\left(\omega\right)}$ is the index function of $\Omega_{n}\left(\omega\right)$. Let $n\rightarrow\infty$, we get

This is a contradiction. The proof is complete.

Sufficient conditions for the existence of stationary distribution and ergodicity of model (2.3) are given below, which also implies that HIV/AIDS is persistent in the mean. We demote the stochastic index

with

When $\sigma_{1}=\sigma_{2}=\sigma_{3}=\sigma_{4}=\sigma_{5}=0$, $R_{0}^{s}$ degenerates to $R_{0}$ in (2.2).

We obtained that

where

Since $G\left(S_{u},S_{a},I,C,A\right)$ is a continuous function, there must be a minimum value $\widetilde{G}$. Define a non-negative $C^{2}$-function

then we apply the Itô’s formula on $V_{1}$: