Global Stability of Switched HIV/AIDS Models with Drug Treatment Involving Caputo-Fractional Derivatives

In this paper, we formulate and investigate new switched HIV/AIDS models with drug treatment involving Caputo-fractional derivatives. Initially, due to the fractional derivative order related to the memory and hereditary effects and supposing that the model coefficients are time-varying parameters, we develop a Caputo-fractional order HIV/AIDS models with switching parameters and study their dynamics utilizing Lyapunov–Razumikhin technique. Furthermore, the results show that the fractional derivative 
 
 α
 
 (
 
 0
 <
 α
 <
 1
 
 ) and the switching parameters are related to the critical threshold value (
 
 
 R
 ^
 
 
 or 
 
 
 R
 ¯
 
 
 ) which ensures disease eradication under the condition of 
 
 
 R
 ^
 
 <
 1
 
 or 
 
 
 R
 ¯
 
 <
 1
 
 . Then, a treatment compartment is introduced into the above model from the asymptomatic infected individuals until the full blown AIDS individuals. Novel sufficient conditions on the threshold value are derived to verify that the disease is eventually cleared as the critical threshold parameter is below unity. Finally, some simulations are employed to support the main results and one future research direction is presented.


Introduction
HIV/AIDS has been one of the most deadly diseases around the world since the first patients were documented in 1981. It has seriously affected human health and even life. According to clinical trials, the virus will go through life once infected with HIV. In the absence of drug treatment, the average life span of patients from HIV infection to AIDS is within 5-10 years [1]. With effective drug therapies, treated individuals can prolong their survival and improve their life quality but do not cure for HIV or AIDS. us, it is necessary to propose effective methods for prevention and control of AIDS.
In recent years, many epidemic issues have been studied by modelling mathematical models to reveal transmission of the disease or predict tendencies of the disease. In particular, Anderson et al. [2] investigated the dynamics of the initial HIV models using ordinary differential equations. e models have been improved by adding into various factors on the disease [3][4][5]. Okosun et al. [6] developed HIV/AIDS with treatment and screening of unaware infectives and investigated optimal control of the treatment. Pitchaimani and Monica [7] incorporated three time delays into a HIV-1 infection model and gave the existence of Hopf bifurcation of the model. Silva and Torres [8] introduced the fractional order into an HIV/AIDS model and surveyed local and uniform stability of its disease-free equilibrium. Huo et al. [9] added anti-HIV preventive vaccines into a fractional order HIV model and showed that the model has rich phenomenon of backward bifurcation with different dosages of vaccines and fractional derivative orders. In their models, they show that fractional order models are related to memory, history, or nonlocal property, which seems to better display complex behavior of real-world phenomena. Nowadays, fractional order models have been wide applied in many fields such as applied mathematics, engineering, economics, biology, and medicine [10][11][12]. Modelling the dynamical behavior of the epidemic diseases by fractional derivative models has more effective than interorder modelling [13][14][15]. For instance, Ullah et al. [16] studied the dynamics of fractional order derivative tuberculosis model and proved the existence and uniqueness of equilibria. Kheiri and Jafari [17] proposed a multipatch fractional order derivative epidemic model and derived relationship between the value of objective functional and the fractional derivative order. Khan and Atangana [18] formulated novel corona virus models with fractional derivative and studied the relationship between the fractional order parameters and the infected compartments. Naik et al. [19] modelled a fractional order HIV-1 model and verified the existence of equilibria and their asymptotical stability.
Seasonal changes in a fluctuating environment often have an effect on the transmission of an infectious disease, which naturally causes that the biological and environment parameters are varying in time. For instance, the spread of dengue disease (transmitted by Aedes aegypti) is fast in summer, and the spread is low in the winter. It has been shown that the transmission speed of childhood infections is the highest at the start of the school year and then declines gradually [20,21]. e research works on the infection disease models affected due to seasonality (such as changes in host immunity or host behavior) have been carried out [22][23][24]. In most of the epidemic models, the coefficients are traditionally assumed to be constant or smoothly varying functions. However, in some cases (such as vacation for school children), the models parameters are assumed to change abruptly in time [25][26][27][28]. is type of parameter is called switching parameter, and the model is called the switched system. Since this type of epidemic model involves multiple subsystems and subject investigated is relatively complex, research work is relatively few [29][30][31]. us, it is extremely significant to incorporate switching into epidemic models and study their dynamics. is paper mainly develops and investigates switched HIV/AIDS models with drug treatment involving Caputofractional derivatives. Specifically, the models' parameters under the influence of seasonality are supposed to be varying in time and change their function forms, and the derivative of the model is assumed to be fractional order with its memory property, which leads to a novel HIV/AIDS model. New global stability conditions of the disease-free equilibrium are derived to examine the dynamics of the disease based on Lyapunov-Razumikhin technique. Moreover, treatment strategies are proposed and evaluated to show that the drug therapy has important influence on the stability of the model. is paper is organized as follows. e classic HIV/AIDS model is extended as a fractional switched HIV/AIDS model, and threshold conditions are obtained to illustrate that the disease-free equilibrium is globally asymptotically stable in Section 2. In Section 3, treatment strategies are applied into the above model, and sufficient conditions on the disease eradication are derived. Section 4 gives numerical examples to verify the proposed results. Conclusions are presented in Section 5.

Model Development
Noting that memory and hereditary properties of fractional order derivatives can make complex behavioral of epidemic models, we will extend the integer order HIV/AIDS models by introducing a fractional order HIV/AIDS model with switching parameters. Assume that the total individuals N(t) consist of the susceptible individuals S(t), infected individuals I 1 (t) with asymptomatic, infected individuals I 2 (t) with symptomatic, and individuals diagnosed with AIDS A(t), that is, N(t) � S(t) + I 1 (t) + I 2 (t). On the contrary, the variety of seasons can made biological parameters change abruptly in time. It is assumed the model coefficients are as switching parameters. Assume these parameters are controlled by a piecewise continuous (from the left) switching signal . ., and I is the set of all switching rules. Assume that p σ is a switched transmission rate between susceptible individuals and individuals with asymptomatic, q σ is a switched transmission rate between the susceptible individuals and individuals with symptomatic, η σ is a switched transmission rate between the susceptible individuals and individuals with individuals diagnosed with AIDS, e σ r σ is a switched transmission rate from individuals with asymptomatic moving to individuals with symptomatic, (1 − e σ ) r σ is a switched transmission rate from individuals with asymptomatic moving to individuals diagnosed with AIDS, φ σ r σ is a switched transmission rate from individuals with symptomatic moving to individuals diagnosed with AIDS, and a σ , b σ , and c σ are mortality from diseases of individuals with asymptomatic, individuals with symptomatic, and individuals diagnosed with AIDS, respectively. λ denotes a recruitment rate of susceptible individuals. μ represents natural mortality rate of four individuals classes. us, the modified models is presented by giving the main results, we introduce the following definitions and lemmas of Riemann-Liouville and Caputo fractional derivative.
Definition 3 (see [34]). e Caputo derivative with fractional order 0 < α < 1 of a function f: Definition 4 (see [34]). e constant x * is an equilibrium point Lemma 1 (see [34]). Let f, g: Lemma 2 (see [34]). Let x(t) and y(t) be continuous realvalued functions and nonnegative in t ∈ [π 1 , π 2 ]. If h is a nonnegative constant and x(t) satisfies the integral inequality then Now, we give the following lemma to show the bounded of the solution for system (1): Proof. From the equations of system (1), it follows that Based on the Laplace transform to (8), it follows that erefore, we will derive the threshold value of system (1) by calculating the spectral radius of a next generation integral operator and investigate global stability of Q 0 in the feasible region on the basis of the Lyapunov-Razumikhin method.
is a solution of system (1). Suppose that α ∈ (0, 1) and then the disease in system (1) dies out theoretically. In other words, the disease-free equilibrium Q 0 is globally asymptotically stable.
Proof. Construct the following set of Lyapunov functions: where By Lemma 1 and taking the derivative of along system (1), we have Noting that B σ > 0 and E σ > 0, it follows that Discrete Dynamics in Nature and Society 3 (13) can be written as On the contrary, taking θ � min σ∈ 1,2,...,m Combining equations (14) and (15), it follows that Assume that L(t) � I 1 (t) + I 2 (t) + A(t) and H � max σ∈ 1,2,...,m { } θ/TG σ . Taking the fractional integral c t 0 D − α t on both sides of (16), for t ∈ (t k− 1 , t k ], we have By Lemma 2, it leads to In addition, it follows that, for t ∈ (0, t 1 ], Since R < 1, it can be deducted that Note that I 1 ,I 2 , A ≥ 0, and I 1 , I 2 , and A converge to zero exponentially, and hence, S approaches S. In other words, the disease in system (1) dies out theoretically.
Next, biological and environmental parameters are periodic variation due to the seasonal changes, and we will assume that a switching rule σ for system (1) is periodic and satisfies the following conditions [35]: ω denotes one switching period, and t k − t k− 1 � ω k with ω k+m � ω k and ω � ω 1 + · · · + ω m . Take I periodic be the set of periodic switching rule and I periodic ⊂ I. us, the following results are presented.
Proof. Assume that the switching rule σ is periodic. By the proof of eorem 1, for t � t 0 + ω, it follows that 4 Discrete Dynamics in Nature and Society According to the condition of R < 1, we can get ζ < 1. For some integer h � 1, 2, . . ., e sequence L(t 0 + hω) ⟶ 0, h ⟶ ∞. In general, for t ∈ (t k− 1 , t k ] and t 0 + hω < t k ≤ t 0 + (h + 1)ω, it follows that By the facts that I 1 , I 2 , A ≥ 0, we can deduct that I 1 , I 2 , and A converge to zero, and hence, S approaches S, which means that the disease in system (1) dies out theoretically.

Remark 2.
Even though threshold value R σ of some subsystems in system (1) is greater than one, Q 0 is globally asymptotically stable as long as threshold value R of system (1) is less than one.

Treatment Strategies
In this section, we incorporate a treatment compartment U(t) into the fractional order HIV/AIDS model with switching parameters. Assume that infected individuals I 1 (t) with asymptomatic receive drug treatment and the proportion of effective treatment is τ 1 , infected individuals I 2 (t) with symptomatic receive drug treatment and the proportion of effective treatment is τ 2 , and individuals diagnosed with AIDS A(t) receive drug treatment and the proportion of effective treatment is τ 3 . erefore, a new fractional order HIV/AIDS model with switching parameters and treatment compartment can be written as follows: According to Definition 4, the disease-free equilibrium of system (22) is E 0 � (S, 0, 0, 0, 0), in which S � λ/μ. For any initial value (S(t 0 ), I 1 (t 0 ), I 2 (t 0 ), A(t 0 ), U(t 0 )) of system (22), the region Ω � (S(t), I 1 (t), I 2 (t), A(t), U(t)): S(t) ≥ 0, I 1 (t) ≥ 0, I 2 (t) ≥ 0, A(t) ≥ 0, S(t) + I 1 (t) + I 2 (t) + A(t) + U(t) ≤ λ/μ} is a positive invariant set. In the following, we analysis the global asymptotical stability of E 0 for system (22).
On the contrary, take the treatment parameter τ 1 � τ 2 � τ 3 � 0.1, and U(0) � 1.0. e other parameters are the same as Figure 1. It can be obtained that R * � 0.1259 < 1 by eorem 4. Figure 3 shows that the disease dies out when R * < 1, which consists of the result given in by eorem 4. Moreover, Figure 4 describes the individuals' behaviors with a different fractional order α. Compared with the higher values of the fractional order, it can be found that the treatment is more effective when taking small value.

Conclusions
Research on HIV/AIDS epidemic models involving fractional order derivative 0 < α < 1, switching parameters, and treatment compartment are becoming one of the key areas in mathematical theory of epidemiology. As we all known, the research works on the switched HIV/AIDS epidemic models are very few [36][37][38]. is paper has studied the global asymptotical stability of new fractional order HIV/AIDS models with switching parameters and treatment compartment. By constructing Lyapunov-Razumikhin technique, new threshold values are obtained to ensure eradication of the disease. More specifically, when R < 1 or R < 1, the disease-free equilibrium Q 0 of system (1) is globally asymptotically stable; in other words, the disease in system (1) dies out theoretically. Moreover, it has been shown that fractional order and switching parameters are related to threshold values, which have a significant effect on extinction of disease. On the contrary, when incorporating treatment compartments into the fractional order HIV/ AIDS model with switching parameters, a new HIV/AIDS model is developed.
e obtained results show that the disease can be cleared theoretically when the threshold value is less than one. In addition, numerical results show when the fractional order increases, the dynamics of the asymptomatic infected individuals and the symptomatic infected individuals increase while the susceptible individuals decrease. One future work is to investigate the endemic case for fractional order switched HIV/AIDS models.

Data Availability
e data used to support the findings of this study are included within the article.

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