Modelling and analysis of the HIV/AIDS epidemic with fast and slow asymptomatic infections in China from 2008 to 2021

1.   Introduction

Human Immunodeficiency Virus (HIV) is a virus that attacks the human immune system. When the virus severely compromises the body's immune system ^[1], it results in acquired immunodeficiency syndrome (AIDS), which is an advanced stage of HIV infection. AIDS caused by HIV infection has been widely spread in different countries and regions since the first HIV case was discovered in the United States in 1981 ^[2]. In 2021, 38.4 million people were living with HIV globally, and as many as 650,000 people have died from HIV-related causes^[3]. In China, Yunnan Province was the first region to experience an AIDS epidemic, which has continuously spread through most of China since then ^[4,5,6]. In the past few years, the number of people living with HIV in China has rapidly increased. By the end of 2020, 1,053,000 people have been diagnosed with HIV, with 351,000 cumulative deaths reported in China. Moreover, an increasing number of HIV-infected patients are beginning to develop clinical AIDS ^[7]. More details on the evolution of the HIV/AIDS epidemic in China can be found in ^{[8,9,10,11,12]}.

In recent years, many scholars have adopted the idea of compartment modelling, which divides the total population into corresponding categories, to study the transmission and development of AIDS within the population and to reflect the dynamics of the HIV epidemic. These models can provide short- or long-term predictions of HIV/AIDS incidences and have become an important tool in analyzing the transmission and control of HIV/AIDS. Hyman et al. ^[13] developed a staged progression model which subdivided the infected population into $n$ subgroups. Each infected subgroup has a different impact on disease transmission. Mccluskey ^[14] studied an HIV/AIDS model with multistage infection. Naresh et al. ^[15] proposed a nonlinear mathematical model to analyze the effect of contact tracing to reduce the transmission of HIV/AIDS in a homogeneous population. In the model, the population was divided into four subcategories: susceptible individuals, HIV-positive infected individuals who did not know they were infected, HIV-positive infected individuals who knew they were infected, and people living with AIDS. Hattaf and Dutta ^[16] developed an epidemic model with three classes and considered the cure of infected individuals during the latent phase. Wattanasirikosone et al. ^[17] considered two groups of infected individuals-HIV and AIDS-and introduced a mathematical model with treated and monitored patients. Xue and Sun et al. ^[18] developed a six-dimensional compartmental model to study the dynamics of HIV transmission; the population was divided into susceptible, latent, consciously infected, unconsciously infected, treated, and AIDS patients. They proved the existence and stability of disease-free and endemic equilibria. Cheneke et al. ^[19] constructed a susceptible-window period-HIV-AIDS (SWIA) model of HIV/AIDS transmission incorporating stage progression. Suppose that a population $N$ is divided into susceptible ($S$), window stage ($W$), HIV-stage ($I$), and AIDS-stage ($A$) individuals. Their model is shown below:

where $\lambda$ is the recruitment rate of the susceptible population, $\beta_1$, $\beta_2$ and $\beta_3$ are the transmission rates of $W$, $I$ and $A$, respectively, the parameter $\rho$ is the transfer rate of individuals from the window phase to the HIV phase, and $\eta$ is the transfer rate of individuals from the HIV phase to transfer to the AIDS phase. Here, $\mu$ and $\delta$ are the natural mortality rate and the AIDS mortality rate, respectively. Through the aforementioned analysis, these models often ignore the impact of asymptomatic infection. Thus, asymptomatic infections should be considered in HIV/AIDS transmission.

In practice, HIV/AIDS patients can develop multiple opportunistic infections due to immunodeficiency, and diseases such as tuberculosis and hepatitis C are secondary infections with both high morbidity and mortality ^[20,21,22]. The co-infection may increase the risk of AIDS and seriously affects HIV/AIDS patients ^[23,24,25]. Compared to HIV/AIDS patients without other diseases, the patients co-infected with HIV/AIDS have a shorter asymptomatic period. Thus, it should distinguish between HIV/AIDS patients with or without co-infection by either describing the fast or slow progression, respectively. Now that co-infections have become a serious public health challenge, our findings can provide effective recommendations to public health authorities and have important implications in terms of improving the quality of patient survival. In our work, a five-dimensional mathematical model is developed to describe a more complex system of HIV transmission dynamics. Two different asymptomatic infections are emphasized, generally agreeing with the actual epidemic. This paper differs from other models of HIV/AIDS infections in that it takes the fast and slow asymptomatic phases that characterize the course of HIV infection due to co-infection into account for the model.

The remaining sections are organized as follows. Section 2 collects data on the number of HIV/AIDS patients and death cases reported in China from January 2008 to December 2021. In Section 3, we establish a five-dimensional system focusing on asymptomatic compartments, considering a model of the HIV epidemic with two asymptomatic infections. Moreover, the global asymptotic stability of disease-free and endemic equilibria is analyzed. Based on the data set, a parameter estimation is performed in Section 4 using a genetic algorithm. The parameter estimation results are used to analyze the model in Section 5. A concluding remark is provided in Section 6.

2.   HIV/AIDS data

HIV is mainly transmitted through blood, from mother-to-child, and by sexual contact. It will attack the immune system after entering the human body, resulting in different degrees of immune dysfunction. Generally, HIV has an average incubation period of 10 years within the human body. The progression for HIV-infected patients to AIDS usually requires a natural process. It is clinically divided into three stages. (ⅰ) The first stage of the acute infection period, usually classified as the primary HIV infection, includes body symptoms such as fevers, sore throats, and rashes, of which usually return to normal after two to three weeks. (ⅱ) The second stage is classified as the asymptomatic infection period, where individuals infected with HIV do not have any symptoms; infection is only detected by positive HIV antibodies within blood tests. Patients do not display any symptoms and live and work for many years as normal. (ⅲ) The third stage is classified as the AIDS period, in which the immune function is completely broken down and the patient develops various severe syndromes until their inevitable death. Thus, mathematical models are used to study the developmental process of HIV and HIV co-infection with other diseases, reveal the infection pattern, and predict the changing trend.

With the expansion of HIV testing, the number of AIDS patients has been increasing. The number of AIDS patients in 2021 was 4.9 times that of 2008. The number of living HIV and AIDS patients has officially exceeded 1 million. For HIV-infected and AIDS patients, male cases were higher than female ones. Curbing the AIDS epidemic is crucial for China. The onset and death data from the National Health Commission of the People's Republic of China was collected from January 1, 2008, to December 31, 2021. According to the Chinese Center for Disease Control and Prevention, by March 31, 2018, the number of living HIV/AIDS patients reached 789,617, and the number of death cases reached 245,498. By calculation, the cumulative number of surviving HIV/AIDS cases was approximately 361,270, and the number of deaths was approximately 245,000 by the end of 2007. Figure 1 shows the cumulative number of HIV/AIDS survivors, deaths from the disease, and the case fatality rate.

3.   An HIV/AIDS model and dynamic properties

In our work, infected individuals are divided into acutely, slowly, and rapidly asymptomatic infected individuals. Suppose that all AIDS patients will self-isolate and not spread the infection. Patients with HIV/AIDS can develop multiple opportunistic infections due to immunodeficiency. When HIV-infected patients are co-infected with other diseases, the disease progresses rapidly, and the asymptomatic period becomes shorter. Unless HIV-infected patients are promptly screened and treated for other diseases, the progression to AIDS will be accelerated. In contrast to other models of HIV/AIDS epidemics, this paper considers the course of infection in HIV co-infection with other diseases. We will distinguish between co-infected and unco-infected AIDS patients by describing the rapid and slow progression of HIV/AIDS. An HIV/AIDS epidemiological model containing two different asymptomatic infections is developed to analyze the transmission of HIV between individuals. The population is divided into five compartments: susceptible ($S$), acute infection ($W$), fast asymptomatic infection ($I_s$), slow asymptomatic infection ($I_l$), and AIDS ($A$). The total number of individuals at time $t$ is $N(t)$, that is, $N(t) = S(t) + W(t) + I_s(t) + I_l(t) + A(t)$. The model is established as follows:

where all parameters of model (1) are defined in Table 1. Figure 2 shows a schematic diagram of the HIV/AIDS model for the two asymptomatic infections.

For convenience, let $p_1 = p\varepsilon$, $p_2 = (1-p)\varepsilon$, $m = -(\omega_1+\sigma_1+\mu)$, $n = -(\omega_2+\sigma_2+\mu)$. Then, model (1) is rewriten as follows:

In model (2), the disease-free equilibrium point is $E_0 = (\frac{b}{\mu}, 0, 0, 0, 0)$. Based on the model, it is easy to obtain the following result.

Lemma 1. Let $S(0)\geq0$, $W(0)\geq0$, $I_s(0)\geq0$, $I_l(0)\geq0$, $A(0)\geq0$ be the initial values of model (2). Then, the feasible region

is a positive invariant set, where $N(t) = S(t)+W(t)+I_s(t)+I_l(t)+A(t)$.

Lemma 1 shows that $\lim\limits_{t\rightarrow \infty}\sup N(t)\leq \frac{b}{\mu}$. It means that $S(t), W(t), I_s(t)$, $I_l(t)$ and $A(t)$ are ultimately bounded. The basic reproduction number $R_0$ is the threshold parameter of the epidemic models. Here, we calculate it using the next-generation matrix method ^[26]. Let $Y = (W(t), I_s(t), I_l(t), A(t))^T$; model (2) can be written as follows:

where

The Jacobian matrix of $\mathscr{F}(x)$ and $\mathscr{V}(x)$ can be obtained as follows:

Thus, the basic reproduction number $R_0$ of model (2) is as follows:

Since $mn-\omega_1\omega_2 = (\omega_1+\sigma_1+\mu)(\omega_2+\sigma_2+\mu)-\omega_1\omega_2 > 0$, $p_2\omega_2-p_1n = (1-\varepsilon)p\omega_2+p\varepsilon(\omega_2+\sigma_2+\mu) > 0$, $p_1\omega_1-p_2m = p\varepsilon\omega_1+(1-p)\varepsilon(\omega_1+\sigma_1+\mu) > 0$, we have $R_0 > 0$.

If $R_0 > 1$, model (2) has a positive equilibrium $E^* = (S^*, W^*, I_s^*, I_l^*, A^*)$ given by the following:

Theorem 1. If $R_0 < 1$, the disease-free equilibrium $E_0 = (\frac{b}{\mu}, 0, 0, 0, 0)$ of model (2) is locally asymptotically stable.

Proof. If $R_0 < 1$, we have $\frac{\beta_1b}{\mu(\varepsilon+\mu)} < 1$. Denote $f = \beta_1\frac{b}{\mu}-(\varepsilon+\mu) < 0$, $l = -(d+\mu) < 0$. The Jacobian matrix of model (2) at $E_0$ can be obtained as follows:

Let

The characteristic equation is formed by the following:

where $a_0 = 1$, $a_1 = -(m+n+f) > 0$, $a_2 = mf+mn+nf-\omega_1\omega_2-p_1\beta_2\frac{b}{\mu}-p_2\beta_3\frac{b}{\mu}$, $a_3 = (\omega_1\omega_2-mn)f+np_1\beta_2\frac{b}{\mu}-p_2\omega_2\beta_2\frac{b}{\mu}+mp_2\beta_3\frac{b}{\mu}-p_1\omega_1\beta_3\frac{b}{\mu}.$

When $R_0 < 1$, we can calculate the following:

There is $\tau \in N^*$ such that $(-p_1\beta_2\frac{b}{\mu}-p_2\beta_3\frac{b}{\mu})\tau = p_1\beta_2\frac{b}{\mu}n+p_2\beta_3\frac{b}{\mu}m$. Hence, we obtain the following:

If $-1 < f < 0$, then $1+f > 0$. Clearly, $a_2 > 0$. If

we have $a_2 > 0$. Since $R_0 < 1$, we obtain the following:

Then, $a_3 = (\omega_1\omega_2-mn)f+np_1\beta_2\frac{b}{\mu}-p_2\omega_2\beta_2\frac{b}{\mu}+mp_2\beta_3\frac{b}{\mu}-p_1\omega_1\beta_3\frac{b}{\mu} > 0$. It follows that

If $f\leq min\{\omega_1, \omega_2\}$, $n\leq min\{p_1, \beta_2, \frac{b}{\mu}\}$, and $m\leq min\{p_2, \beta_3, \frac{b}{\mu}\}$, then we have $a_1a_2-a_0a_3 > 0$. According to the Hurwitz criterion, the disease-free equilibrium $E_0$ of model (2) is locally asymptotically stable. □

Theorem 2. If $R_0 < 1$, the disease-free equilibrium $E_0 = (\frac{b}{\mu}, 0, 0, 0, 0)$ of model (2) is globally asymptotically stable.

Proof. Let $H$ and $K$ be two positive constants. We construct a Lyapunov function $V = W+HI_s+KI_l$. The derivative of $V$ is calculated as follows:

Substituting the model equations into Eq (4), we obtain the following:

With $S_0 = \frac{b}{\mu}$, we have the following:

To obtain the values of $H$ and $K$, we can use the following:

From Eq (5), we can calculate the following:

Hence, we obtain the following:

When $R_0 < 1$, we have the following:

It follows that

According to LaSalle's invariance principle, the disease-free equilibrium $E_0$ of model (2) is globally asymptotically stable. □

Theorem 3. If $R_0 > 1$, then system (2) is a uniform persistence with respect to $(X_0, \partial X_0)$, i.e., there exists $\xi > 0$ such that any positive solution $(S(t), W(t), I_s(t), I_l(t), A(t))$ of system (2) in $X_0$ satisfies the following:

Proof. We begin by defining the following symbols:

It is obvious that $X$ and $X_0$ are positivity invariants and system (2) is a point dissipative. In the following, we will show that system (2) is uniformly persistent with respect to $(X_0, \partial X_0)$. For this purpose, we denote the following:

Now, we prove that $M_\partial = \{(S(0), 0, 0, 0, 0 | S(0)\geq 0\}$. For convenience, let $M = \{(S(0), 0, 0, 0, 0| S(0)\geq 0 \}$. It is clear that $M \subset M_\partial$ and it is only necessary to show that $M_\partial \subset M$. If $(S(0), W(0), I_s(0), I_l(0), A(0)) \in M_\partial$, then $W(0) = I_s(0) = I_l(0) = A(0) = 0$. By contradiction, assume at least one of $W(0)$, $I_s(0)$, $I_l(0)$ or $A(0)$ is greater than zero (e.g., $W(0) > 0$), then we can obtain $I_s(0)$, $I_l(0)$ and $A(0)$, which are all greater than zero in a certain interval, such as $[0, T_1]$. In fact, for $t \in [0, T_1]$, from the inequality $\frac{dW(t)}{dt} |_{(2)} \geq -(\varepsilon+\mu)W(t)$, we have the following:

Similarly, we have $I_s(t)\geq I_s(0)e^{mT_1} > 0$, $I_l(t)\geq I_l(0)e^{nT_1} > 0$, and $A(t)\geq A(0)e^{-(d+\mu)T_1} > 0$.

Consequently, $W(0) > 0$ implies that $I_s(t) > 0$, $I_l(t) > 0$ and $A(t) > 0$ for $t \in [0, T_1]$. From the definition of $M_\partial$, we know that any point in $\partial X_0$ with $W(t) > 0$ can not belong to $M_\partial$. A similar idea and procedure shows that any point in $\partial X_0$ other than $(S(0), 0, 0, 0, 0)$ can not belong to $M_\partial$. Hence, it is obvious that $M_\partial = \{(S(0), 0, 0, 0, 0 | S(0)\geq 0\}$.

The disease-free equilibrium point $E_0$ is the only equilibrium point in $M_\partial$. Next, we prove that $W^s(E_0)\cap X_0 = \emptyset$ when $R_0 > 1$, where $W^s(E_0)$ is a stable manifold of $E_0$. If it is not true, then there exists a solution $(S, W, I_s, I_l, A)$ in $X_0$ such that

Thus, for any constant $\epsilon > 0$, there exists a $T^* > 0$ such that for any $t\geq T^*$, there is $S(t)\geq S(0)-\epsilon$. Define the following functions $U(t) = W(t)$. When $t\geq T^*$, we have the following:

Since $R_0 > 1$, we have $\beta_1S(0)-(\varepsilon+\mu) > \frac{\beta_2b(p_2\omega_2-p_1n)+\beta_3b(p_1\omega_1-p_2m)}{\mu(mn-\omega_1\omega_2)}$, and it is possible to pick $\epsilon$ small enough so that $\beta_1(S(0)-\epsilon)-(\varepsilon+\mu) > 0$, $\beta_2(S(0)-\epsilon) > 0$ and $\beta_3(S(0)-\epsilon) > 0$. Therefore, $U(t)$ is an increasing function on $t\geq T^*$. It follows that $U(t)$ does not tend to be zero when $t\rightarrow \infty$, which is a contradiction. Thus, we have $W^s(E_0)\cap X_0 = \emptyset$. Since $E_0$ is globally asymptotically stable in $M_\partial$, $E_0$ is an isolated invariant set and is acyclic. According to ^[27], system (2) is uniformly persistent with respect to $(X_0, \partial X_0)$. □

Theorem 4. If $R_0 > 1$ and $\frac{\beta_2\omega_2}{p_1}\leq\frac{\beta_3\omega_1}{p_2}$, the endemic equilibrium $E^* = (S^*, W^*, I_s^*, I_l^*, A^*)$ of model (2) is globally asymptotically stable.

Proof. By Theorem 3, we know that variables $S$, $W$, $I_s$, $I_l$ and $A$ are positive when $R_0 > 1$. We let $B$ and $C > 0$, and construct the following Lyapunov function:

where $B = \frac{\beta_2S^*I_s^*}{p_1W^*}$, $C = \frac{\beta_3S^*I_l^*}{p_2W^*}$.

Let $V(S) = S-S^*-S^*\ln S$, $V(W) = W-W^*-W^*\ln W$, $V(I_s) = I_s-I_s^*-I_s^*\ln I_s$, $V(I_l) = I_l-I_l^*-I_l^*\ln I_l$. The four functions $V(S)$, $V(W)$, $V(I_s)$, and $V(I_l)$ are derived alongside the solution of system (2):

Hence,

Using the inequality $1-x+\ln x \leq 0$ for $x > 0$ with equality holding if and only if $x = 1$ yield,

From the basic inequality, $2-\frac{S}{S^*}-\frac{S^*}{S}\leq0$. Based on the inequality $1-x+\ln x \leq0$ for $x > 0$, we obtain $1-\frac{S^*}{S}+\ln\frac{S^*}{S}\leq0$ and $\frac{I_l}{I_l^*}-\ln\frac{I_l}{I_l^*}-1\geq0$. If $\frac{\beta_2\omega_2}{p_1}\leq\frac{\beta_3\omega_1}{p_2}$, then $\frac{\beta_2\omega_2}{p_1}-\frac{\beta_3\omega_1}{p_2}\leq0$. Thus, $\dot{V}\leq0$, and $\dot{V} = 0$ if and only if $S = S^*$, $W = W^*$, $I_s = I_s^*$ and $I_l = I_l^*$. The endemic equilibrium point $E^*$ is globally asymptotically stable for $R_0 > 1$ by LaSalle's invariance principle. □

4.   Parameter estimation

For model (1), there is a total of fifteen unknown parameters in Table 1. Some of them are determined from population data, and the remaining parameters are estimated from AIDS patient data. The birth and natural mortality rates from 2008 to 2021 were obtained from the National Bureau of Statistics of China ^[28], see Table 2. The mean values of birth and natural mortality rates are 0.01187 and 0.0071, respectively. The whole population of China in 2007 was 1,321,290,000 ^[28]. Based on the mean of birth and death rates, the recruitment rate can be estimated by the following:

and the natural death rate is $\mu = \frac{0.0071}{12} = 0.00059$. The rate of AIDS death is $d = 0.35$ ^[29].

Other unknown parameters include $\beta_1$, $\beta_2$, $\beta_3$, $p$, $\varepsilon$, $\sigma_1$, $\sigma_2$, $\omega_1$, $\omega_2$, and three initial values $W_0$, $I_{s0}$, and $I_{l0}$. A genetic algorithm ^[30] is an optimization algorithm for adaptive searching. Compared with other algorithms, the genetic algorithm has better estimation results. It has been applied to neural networks ^[31], combinatorial optimization ^[32], artificial intelligence ^[33], genetic programming ^[34], data mining, and other fields. For the given parameter ranges (see Table 3), all parameters are randomly generated with a certain number of initial populations. The fitness index is an important indicator to evaluate whether the parameters conform to the HIV/AIDS model. Generally, the square sum of the difference between the real HIV/AIDS data and the simulated data of the model is used as a fitness index, defined as follows:

where $A_l(t)$ denotes the real cumulative HIV/AIDS infection data and $\bar{A_l}(t)$ denotes the cumulative infection data in the model (1). In genetic evolution, it is simulated for the parameter set using three genetic operators. Each generation's set of parameters is recorded and compared with the updated set of parameters in the next generation. The updated set of parameters is always recorded until the evolution is completed with the final global optimal set. The accuracy and validity of the method are verified by fitting it to the real cumulative case data of HIV/AIDS. The results of the parameter estimation are shown in Table 3.

5.   Model analysis

5.1.   Validation of the model

The initial values and parameter estimators are brought into model (1), according to Table 3. First, we validate the fitting model with the observed HIV data from January 2008 to December 2021. Figure 3 shows the curves of the observed HIV cases and the model. The green dots indicate the observed real data and the gray curve is the fitting curve of model (1). As shown by the real data, the cumulative number of HIV cases has been increasing since 2008, and the model better reflects this trend.

Figure 4 shows the residual analysis of the fitting results of model (1). When the confidence interval of the residuals contains a zero, it reveals that the model can fit the original data well. Otherwise, it can be considered as an outlier. As can be seen from Figure 4, the residual confidence interval almost contains zero points. It reflects that model (1) fits well and further indicates the validity of the genetic algorithm.

The stability of the model is shown in Figures 5 and 6. Figure 5 shows the global asymptotic stability of the disease-free equilibrium point. When $R_0 < 1$, the disease-free equilibrium first decreases and then stabilizes. In Figure 6, the epidemic equilibrium point tends to stabilize when $R_0 > 1$. The values of $W$, $I_s$, $I_l$ and $A$ fluctuate and then stabilize.

5.2.   Sensitivity analysis

The basic reproduction number $R_0$ is an important threshold value for the spread of HIV and is closely related to the parameters $\beta_1$, $\beta_2$, $\beta_3$, $p$, $\varepsilon$, $\sigma_1$, $\sigma_2$, $\omega_1$ and $\omega_2$. Considering all other parameters, Figure 7 shows that $R_0$ gradually increases when $\beta_1$, $\beta_2$, $\beta_3$, $p$ or $\omega_2$ is larger, but gradually decreases when $\varepsilon$, $\sigma_1$, $\sigma_2$ or $\omega_1$ increases. This means that reducing the rate of effective contacts with HIV-infected people and raising the rate of treatment for co-infections could curb the AIDS epidemic in China.

To identify the most influential parameters associated with HIV-infected populations, we use the partial rank correlation coefficient (PRCC) to investigate the global sensitivity of the parameters of model (1). The PRCC measures the relationship between the state variables and each model parameter and helps to identify the most important parameters that affect HIV transmission. The sensitivity analysis of PRCC depends on the range of values of the parameters to a large extent. There exists an $\kappa\in(0, 0.1)$, and the range of parameter values is shown in Table 3. Thus, the PRCC value entirely depends on the range of $\kappa$. The results of the PRCC are shown in Figure 8. These parameters $\beta_1$, $\beta_2$, $\beta_3$, $p$ and $\omega_2$ are positively correlated with $R_0$, and $\varepsilon$, $\sigma_1$, $\sigma_2$ and $\omega_1$ are negatively correlated with $R_0$. Furthermore, we observe that $\beta_2$, $\sigma_2$, $\omega_1$ and $\omega_2$ are more sensitive to $R_0$ than other parameters. Therefore, decreasing $\beta_2$ and $\omega_2$, increasing $\omega_1$ and $\sigma_1$ can reduce $R_0$ and thus effectively slow down HIV transmission. Therefore, it is of great applied value to study the effect of changing these four parameters on the basic reproduction number. The results presented in Figure 8 suggest that lowering the proportion of asymptomatically infected individuals co-infected with other diseases is the more critical measure.

In addition, we investigate the effect of parameters on $R_0$ by the contour plot with parameters $\beta_2$, $\sigma_1$, $\omega_1$ and $\omega_2$ in Figure 9. The contour plots show a significant increase for $R_0$ with a higher $\beta_2$ and $\omega_2$. The burden of AIDS is usually exacerbated by co-infectious diseases. This means that if the public does not take preventive measures, the disease will persist in the population and spread through the community. In order to control $R_0$, it is necessary to lower $\beta_2$ and $\omega_2$, and raise $\sigma_1$ and $\omega_1$. Therefore, it is important to strengthen early diagnosis, symptom screening, and timely treatment of HIV/AIDS combined with other diseases.

5.3.   Predictions of the model

In the next numerical simulations, an uncertainty analysis is performed for two compartmental $I_s$ and $I_l$ based on the parameters in Table 3. Figure 10(a), (b) show the trends of $I_s$ and $I_l$ through the coefficient of variation (CV) $3%$ of these parameters. The mean values of $I_s$ and $I_l$ (solid line) and the $95%$ confidence intervals (dotted line) are shown at the bottom of Figure 10.

Then, we analyze the impact of various parameters on the AIDS epidemic. We take the fitting value of each compartment at the end of December 2021 as the initial value, and the estimated value of $b$, $d$, $\beta_1$, $\beta_2$, $\beta_3$, $p$, $\varepsilon$, $\sigma_1$, $\sigma_2$, $\omega_1$ and $\omega_2$ in Table 3 as the standard. When one parameter changes and other parameters are fixed, we use model (1) to analyze the trend of patients in the next 29 years. We investigate the effect of the transmission rate $\beta_2$ on the epidemic. In Figure 11(a), the number of infected persons in the HIV acute infection phase can be effectively controlled by reducing the transmission rate $\beta_2$. In addition, the epidemic dies out more quickly when the transmission rate of $\beta_2$ decreases to $0.8\beta_2$. Hence, public health organizations can enhance their control efforts by diminishing the transmission rate. For example, publicizing the dangers of AIDS can be used to slow the spread of the disease. Figure 11(b) shows the impact of reducing the proportion of HIV-infected patients with other co-infections through early diagnosis and screening, with little change in the number of rapidly asymptomatic patients. This suggests that early diagnosis and screening is not a key factor in controlling HIV transmission. Figure 12 suggests that growing the proportion of $\omega_1$ and cutting down the proportion of $\omega_2$ can significantly lessen the number of fast asymptomatic individuals, thereby inhibiting HIV transmission.

To assess the combined effect of $\beta_2$, $\omega_1$ and $\omega_2$ on HIV transmission, we use different combinations of these three parameters to model the number of infected individuals, as given in Figure 13. Figure 13(a), (b) show that the number of individuals in the acute infection and fast asymptomatic phases gradually decline as $\beta_2$ and $\omega_2$ diminishes, and $\omega_1$ expands. However, Figure 13(c), (d) reflect that the number of individuals in the acute infection and fast asymptomatic phases first tends to improve and then gradually diminishes. Figure 14 explores the trend of infected individuals for different parameter values. When $\beta_2 < 0.95\beta_2$, $\omega_1 > 1.05\omega_1$, and $\omega_2 < 0.95\omega_2$ there is a declining trend of HIV/AIDS. This means that the spread of HIV/AIDS can be effectively curbed as the transmission rate $\beta_2$ and the number of co-infected patients decreases, and the cure rate of other diseases increases.

Based on the above numerical simulations, we know $R_0 = 1.7593 > 1$ for the given parameters in Table 3. If the previous control measures do not change, the HIV epidemic will be persistent and approach epidemic equilibrium. In addition, we have found that the effective rate of exposure to the disease and the proportion of co-infections in HIV/AIDS have a significant impact on reducing the number of infected individuals. When more HIV-infected people are either protected or screened, the number of people receiving treatment increases. Therefore, it needs to improve early diagnoses and screenings of HIV patients co-infected with other diseases. By increasing awareness of co-infection and improving the rate of treatment for co-infected patients, the incidence and transmission of HIV can significantly decrease, thereby alleviating pressure on public health organizations.

6.   A concluding remark

In this paper, we developed a five-dimensional mathematical model including susceptible infection, acute infection, fast asymptomatic infection, slow asymptomatic infection, and AIDS. A theoretical analysis and application were conducted based on this model. The basic reproduction number $R_0$ was obtained by the next-generation matrix method. The stability of the model was studied using the Lyapunov method. When $R_0 < 1$, the disease-free equilibrium is globally asymptotically stable and the disease gradually disappears. When $R_0 > 1$, the endemic equilibrium is globally asymptotically stable and the disease continues to spread.

We collected the number of reported HIV/AIDS cases in China from 2008 to 2021. The unknown parameters were estimated by a genetic algorithm, and the basic reproduction number $R_0 = 1.7593$ was finally calculated, which indicated that the model we built was consistent with the actual HIV/AIDS epidemic. Furthermore, a sensitivity analysis of the parameters was performed to analyze the effect of individual parameters on $R_0$, and global sensitivity was performed from the PRCC approach. We found that parameters associated with fast asymptomatic infection had a more significant effect on $R_0$. The studied results showed that the spread of the HIV/AIDS epidemic could be effectively controlled by decreasing the contact rate $\beta_2$ and improving early diagnosis, screening rates, and treatment rates for co-infected patients. Therefore, we further analyzed the impact of single and multiple measures on disease transmission. Multiple measures can be more effective in controlling the spread of HIV than one.

In summary, public health organizations can lower transmission rates by publicizing the dangers of AIDS, reduce the risk of co-infections by strengthening the awareness of self-protection among HIV-infected individuals, and improve treatment rates for co-infections by upgrading medical technology. Reference ^[18] suggests that HIV transmission can be mitigated by improving medical technology, increasing testing rates, and intensify education for unaware infected individuals. Our proposed program focuses more on the impact of co-infections on the course of AIDS than those proposed in the ^[18].

In the work, we focus on studying the HIV/AIDS epidemic with either fast or slow asymptomatic infections. However, there are still some unsolved and interesting problems. For example, some factors, such as age structure and gender, often play important roles in HIV/AIDS transmission ^[35,36,37]. The theoretical methods of the aforementioned results are important references for our subsequent research and practice.

Use of AI tools declaration

The authors declare they have not used Artificial Intelligence (AI) tools in the creation of this article.

Funding

This research was supported by the National Natural Science Foundation of China under Grant No. 12061070, the Natural Science Foundation of Xinjiang Uygur Autonomous Region under Grant No. 2021D01E13, the Innovation Project of Excellent Doctoral Students of Xinjiang University under Grant Nos. XJU2023BS017, XJU2022BS027, and the Research Innovation Program for Postgraduates of Xinjiang Uygur Autonomous Region under Grant Nos. XJ2023G016, XJ2023G017.

Conflict of interest

The authors declare that they have no conflict of interest.

Acknowledgments

We thank the editor and referees for their insightful comments and constructive suggestions.