www.biomathforum.org/biomath/index.php/biomath ORIGINAL ARTICLE Analysis of a virus-resistant HIV-1 model with behavior change in non-progressors Rabiu Musa, Robert Willie, Nabendra Parumasur School of Mathematics, Statistics and Computer Science University of KwaZulu-Natal, Durban, South Africa. rabiumusa003@gmail.com, willier@ukzn.ac.za, parumasurn1@ukzn.ac.za Received: 26 February 2020, accepted: 14 June 2020, published: 8 August 2020 Abstract—We develop a virus-resistant HIV-1 mathematical model with behavior change in HIV- 1 resistant non-progressors which was analyzed for both partial and total abstinence cases. The model has both disease-free and endemic equilib- rium points that are locally asymptotically stable depending on the value of the associated threshold quantities RT and R ′ T . In both cases, a non- linear Goh–Volterra Lyapunov function was used to prove that the endemic equilibrium point is globally asymptotically stable for special case while the method of Castillo-Chavez was used to prove the global asymptotic stability of the disease-free equilibrium point. In both the analytic and numer- ical results, this study shows that in the context of resistance to HIV/AIDS, total abstinence can also play an important role in protection against this notorious infectious disease. Keywords-Resistance; Behavior change; Partial & Total Abstinence; Goh–Volterra Lyapunov function. AMS Subject Classification: 92Bxx, 92B05. I. INTRODUCTION As it was reported in the 1980s, the human immunodeficiency virus (HIV), and the later stage of infection through cell depletion known as AIDS has continue to play a leading role in the series of the greatest ever infectious disease. United Nations Program on HIV/AIDS (UNAIDS) and the World Health Organization (WHO) have already provided the estimates of the number of cases since the 1980s. More than 30 million people are currently HIV positive. According to the current trends, at least 7300 people are infected with HIV and a minimum of 5000 die from AIDS-related causes including at-least 690 children on a daily basis (UNAIDS, 2009). This means that for every five HIV positive individuals, at least four of them including adults and children die from the infection daily [10], [32]. The two main types of HIV are HIV-1 and HIV- 2. The most dangerous that has spread worldwide is HIV-1 while the latter is less pathogenic and less spread since it’s confined to West African countries. The test carried out on one can not Copyright: c©2020 Musa et al. This article is distributed under the terms of the Creative Commons Attribution License (CC BY 4.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original author and source are credited. Citation: Rabiu Musa, Robert Willie, Nabendra Parumasur, Analysis of a virus-resistant HIV-1 model with behavior change in non-progressors, Biomath 9 (2020), 2006143, http://dx.doi.org/10.11145/j.biomath.2020.06.143 Page 1 of 17 http://www.biomathforum.org/biomath/index.php/biomath https://creativecommons.org/licenses/by/4.0/ https://creativecommons.org/licenses/by/4.0/ http://dx.doi.org/10.11145/j.biomath.2020.06.143 R. Musa, R. Willie, N. Parumasur, Analysis of a virus-resistant HIV-1 model with behavior change in ... sufficiently detect the other due to large genetic differences between them. Immediately after HIV infection, the lympho- cytes, or white blood cells, known as CD4+ T cells are the major target. Therefore, the anti- HIV antibody and cytotoxic T cell production by the immune system is consequently initiated. An HIV positive individual is not classified as having AIDS until CD4+T cell count which is approxi- mately around 1000mm−3 depletes to 150mm−3 or thereabout. Since CD4+T plays a very important role in the body immune mechanism, deterioration and depletion result in acquired immunodeficiency syndrome called AIDS. The average number of times it takes HIV to develop to AIDS is depen- dent on the strength of the immune system of the victim [23]. It is therefore pertinent to study methods of HIV prevention. Different control strategies such as behavior change due to HIV awareness cam- paign, reduction in sexual partners, anti retro- viral treatment ART etc. have collectively played important roles in combating the menace. They are still very much relevant due to unavailability of vaccine. The use of condom has also played an important role and can possibly prevent HIV transmission almost perfectly. Other intervention methods that can concurrently prevent both sexes are still very much needed. Recently, an experi- mental product containing a drug that can prevent rectal and vaginal transmission of HIV and other sexually transmitted diseases was detected but unfortunately did not see the light of the day due to the fact that the gel is ineffective with high HIV infection risk [24]. Other efforts such as the HIV vaccine and diaphragm technique fail to manifest to any meaningful impact [6], [21]. From biological point of view, HIV resistance is known as the genetic mutation in the DNA that delays AIDS progression or aids production of permanent immunity (i.e. no progression) to AIDS. This kind of mutation which is known as CCR5-delta 32 plays an important role in the development of the two kinds of HIV resistance known. This CCR5-delta 32 breaks and distorts the HIV’s ability to deplete and destroy the im- munity of the CD4+T cells. The mutation makes the CCR5 co-receptor on the outside of cells to develop at a smaller rate than usual and no longer sit outside of the cell. This co-receptor is similar to a door that allows HIV passage into the cell where within a second locks “the door” which consequently prevents HIV entrance into the CD4+T cells [14]. This genetic mutation has been reported to be inborn. There are very few paper on resistant mathematical model, some of them are [25], [13] and [15] but the resistance was modeled on influenza and SARS which is quite different from HIV/AIDS. This still remains a biological research question needed to be answered. Research has shown that some people develop resistance to the killer HIV-1 virus [22], [28]. In fact, a report in [18] shows that though this resis- tance is rare but actually exists. Virus resistance can be understood in two scenarios. First, there are cases of individuals that are exposed to HIV but after a long period of times, diagnosis shows that they are uninfected. This case of exposed un- infected have been detected from among infants of infected mothers, health workers during treatment of infected individuals, commercial sex workers, individuals having unprotected sex with seropos- itive partners etc. The second category is HIV infected individuals with low or no progression to AIDS as expected under normal circumstances. They live with the virus for many years with an absolutely low level of HIV-1 RNA or no loss of CD4+ cells that has been identified among various individuals such as children and homosexual men and women mutation [18], [8]. In 2014, the report in [12] confirms that some people show partial or absolutely complete in- born resistance to the HIV virus . The major or main contributor to this strange development is a mutation of the gene encoding CCR5 which acts as a co-receptor for HIV. CCR5 may even be defective in some individuals which will enhance protection against disease. These individuals live a normal life since the HIV-1 virus cannot bind itself to it and its perhaps here that the key to Biomath 9 (2020), 2006143, http://dx.doi.org/10.11145/j.biomath.2020.06.143 Page 2 of 17 http://dx.doi.org/10.11145/j.biomath.2020.06.143 R. Musa, R. Willie, N. Parumasur, Analysis of a virus-resistant HIV-1 model with behavior change in ... overcome the disease lies hidden. Estimation later shows that the proportion of individuals under this category is less than 1%. Similar occurrences make leading Oxford University researcher Sarah Rowland-Jones to believe continual exposure is a requirement for maintaining immunity after which 15 proteins were identified to be unique to those virus-free sex workers [3], [2]. A genetic mutation that blocks HIV which may hold the key to future treatment was also studied in 2016 by [9]. In 2010, [4] identified factors such as APOBEC3G, Toll-like receptors, acute-phase amyloid A protein, interleukin-22, APOBEC3G and natural killer cells as the main reason why some people do not even seroconvert let alone progressing to AIDS despite multiple HIV expo- sure. More interesting reasons behind this strange occurrence has been examined by the university of Minnesota in 2014 [33] and by [17] in 2013. Another interesting factor that influence the spread of HIV/AIDS is change in sexual behavior towards sex. This is caused by the infectiousness nature, high death rate and stigmatization encountered by victims of HIV/AIDS. This has subsequently affect the transmission of the disease in recent years. Behavior change intervention will help individ- uals change their drug-using behaviors and sexual behavior that put them at a high risk of contracting HIV. It also creates skills and knowledge that can influence their motivation and ability to kick start behavior change. Couples, peer groups, individu- als, communities or institutions can be targeted on a multiple level. This behavior change can also be motivated through skills-building, motivational or educational approach. Interventions can target different kind of behaviors such as condom usage, number of sexual partners, correct use of best prevention approach etc. Though many researchers have developed different models to examine the dynamics of the virus, HIV-1 mathematical model where infected individuals gain resistance to ac- quisition of HIV and resistance to deterioration of HIV incorporating behavior change in form of partial and total abstinence is still a biological question needed to be answered. Researchers like [31], [19] have done commend- able work in tackling the menace of the deadly virus, in this research, we present a new virus- resistant HIV-1 model with behavior change. This behavior change to avoid infection happens as a result of the wide spread of the agony and death caused by HIV/AIDS. This change happens either partially or totally. Those who show partial abstinence are those that only reduced their sexual partners but still involve in HIV-risk activities or live in endemic environment while those who totally abstain are those who maintain only one sexual partner and do away from all HIV-risk activities or exposed and endemic environment. Mathematical modeling has become an effec- tive tool in studying infectious disease by many researchers. It shall be used again here to study the dynamics of resistance in HIV-1 transmission and how it produce significant reduction rate in the community. We hope it helps policy-makers and public health workers in the epidemic control. Several researchers like [20], [19], [1] and references therein have published commendable research output about transmission dynamics of HIV/AIDS. They have also studied control and prevention strategies of this notorious epidemic. In order to further extend, compliment and contribute to the work of the aforementioned researchers, a new comprehensive model has been designed. The model extends the work of the aforementioned researchers by, for instance, 1) Considering the influence of virus-resistance i.e. resistance to acquisition and resistance to deterioration. 2) Incorporating the change of behavior class whose rate of progression is either through partial abstinence or total abstinence. 3) Including a compartment (I1) for slow pro- gressors. These are the category of people with partial resistance to the virus. 4) Including a compartment (I2) for non pro- gressors. These are the category of people with complete resistance to the virus and do not move to AIDS compartment (A). 5) Including a compartment (I3) for fast pro- Biomath 9 (2020), 2006143, http://dx.doi.org/10.11145/j.biomath.2020.06.143 Page 3 of 17 http://dx.doi.org/10.11145/j.biomath.2020.06.143 R. Musa, R. Willie, N. Parumasur, Analysis of a virus-resistant HIV-1 model with behavior change in ... gressors. These are the category of people with no resistance to the virus. All these instances have not been considered be- fore. The paper is organized as follows. Section 2 entails model formulation and assumptions while section 3 contains basic properties of the model. This is followed by the analysis of the sub-model (model with total abstinence) and that of the full model (model with partial abstinence) in section 4. Section 5 presents the numerical simulation and discussion of results while the last section contains the conclusion, acknowledgment and disclosure statement. II. MODEL FORMULATION AND MODEL ASSUMPTIONS We formulate an HIV-1 resistant and behavior change model by splitting the total human popula- tion at time t, denoted by N(t), into six mutually- exclusive compartments of susceptible individuals S(t), slow progressor HIV-1 infected class I1(t), non progressor HIV-1 infected class I2(t), fast progressor HIV-1 infected class I3(t), behavior change class I4 and AIDS class A such that N(t) = S(t) +I1(t) +I2(t) +I3(t) +I4(t) +A(t). It is worth noting that the AIDS class consists of weak and unhealthy infected individuals that are assumed to be sexually inactive. Sexually active individuals are recruited into the susceptible population at a constant rate B. The susceptible individuals acquire the virus through effective contact with an HIV-1 positive and in- fectious individuals at the rate λ given by λ = β(I3 + σ1I1 + σ2I2 + σ3I4) N , (1) where β in (1) denotes the effective contact rate that is capable of leading to infection, 0 ≤ σ1 ≤ 1 denotes the modification parameter that account for the assumed reduction in the transmission of virus by the slow progressor HIV-1 infected class I1 in comparison to the fast progressor HIV-1 infected individuals in I3, 0 ≤ σ2,σ3 ≤ 1 are the modification parameters accounting for the assumed reduction of infectiousness by I2 and I4 classes in comparison to the slow-progressor and fast progressor classes I1 and I3 respectively. So that σ3 < σ2 < σ1 < 1, σ3 ≥ 0. (2) The acquisition of infection by the slow progres- sor HIV-1 infected individuals I1 occur at the rate α1λ, that of I2 occur at the rate α2λ and that of I3 at the rate α3λ. Natural death occur constantly to anybody at the rate µ and rate of progression from I1 to AIDS class A at the rate ρ1. Therefore, the rate of change of the total population of the susceptible and and slow progressor classes is respectively given by Ṡ(t) =B − (α1 + α2 + α3)λS −µS, İ1(t) =α1λS −ρ1I1 −µI1, where · represents derivative with respect to time. The non-progressor HIV-1 infected class is gener- ated by the break-through of infection of suscep- tible class at the rate α2λ, total abstinence due to behavior change at the rate γ1, partial abstinence from I4 due to behavior change at the rate γ2 and natural death at the rate µ so that we have İ2(t) = α2λS −γ1I2 + γ2I4 −µI2. Similarly, we compose the fast progressor class by the break-through of infection of the susceptible class at the rate α3λ, AIDS acquisition at the rate ρ2 so that the class is given by İ3(t) = α3λS −ρ2I3 −µI3. The behavior change class is formulated through the total abstinence of non progressors at the rate γ1 and partial abstinence at the rate γ2 given by İ4(t) = γ1I2 −γ2I4 −µI4. While incorporating the behavior change in the model, we deliberately focused on the behavior change of the non-progressors HIV-1 infected in- dividuals even though, it is imperative that all individuals can change their behavior at any given time. This is because this class of individuals are the most dangerous class just that they won’t show Biomath 9 (2020), 2006143, http://dx.doi.org/10.11145/j.biomath.2020.06.143 Page 4 of 17 http://dx.doi.org/10.11145/j.biomath.2020.06.143 R. Musa, R. Willie, N. Parumasur, Analysis of a virus-resistant HIV-1 model with behavior change in ... Fig. 1. Flow chart of the model. any sign of AIDS. And finally, the AIDS class is given by Ȧ(t) = ρ1I1 + ρ2I3 − (µ + τ)A, where τ is the AIDS-induced death rate. Since progression are not the same, we have α3 > α1 > α2, α1 + α2 + α3 = 1 (3) where 0 < α1, α2, α3 < 1. The resultant math- ematical model for the transmission dynamics of HIV-1 incorporating virus resistance and behavior change through partial and total abstinence using a set of non-linear autonomous set of differential equations is given by: dS dt =B − (α1 + α2 + α3)λS −µS, (4) dI1 dt =α1λS −K1I1, (5) dI2 dt =α2λS + γ2I4 −K2I2, (6) dI3 dt =α3λS −K3I3, (7) dI4 dt =γ1I2 −K4I4, (8) dA dt =ρ1I1 + ρ2I3 −K5A, (9) where K1 = ρ1 + µ,K2 = γ1 + µ,K3 = ρ2 + µ, K4 = γ2 + µ,K5 = µ + τ, with initial condition S(0) > 0, I1(0) > 0, I2(0) > 0, I3(0) > 0, I4(0) > 0, A(0) > 0. (10) The flow chart of this model is given in Figure 1. III. BASIC PROPERTIES OF THE MODEL Since the model is a dynamical system, it it is therefore imperative to ensure that it is biologically meaningful through the establishment of its posi- tivity solution and boundedness at all time t ≥ 0. A. Positivity and boundedness of the Model. Lemma III.1. The closed set Γ= { (S,I1,I2,I3,I4,A)∈R6+|S+I1+...+I4+A≤ B µ } is attracting and positively invariant with respect to the model equation (4)-(9). Proof: From (4), we define an integrating factor as ξ(t) = exp {∫ t o [µ + (α1 + α2 + α3)λ(η)]dη } , Biomath 9 (2020), 2006143, http://dx.doi.org/10.11145/j.biomath.2020.06.143 Page 5 of 17 http://dx.doi.org/10.11145/j.biomath.2020.06.143 R. Musa, R. Willie, N. Parumasur, Analysis of a virus-resistant HIV-1 model with behavior change in ... where λ(η) = λ(I1,I2,I3,I4). So that the solution of (4) is given by S(t)ξ(t) = B ∫ t o ξ(t)dt, which can be re-written as S(t) exp {∫ t o [µ+(α1 +α2 +α3)λ(η)]dη } = S(0) +B ∫ t o [ exp {∫ s o [µ+(α1 +α2 +α3)λ(η)]dη }] ds, which implies S(t) exp { µt+ ∫ t o (α1 +α2 +α3)λ(η)dη } = S(0) +B ∫ t o [ exp { µs+ ∫ s o (α1 +α2 +α3)λ(η)dη }] ds so that S(t) =B ∫ t o [ exp { µs+ ∫ s o (α1 +α2 +α3)λ(η)dη }] ds × exp { −µt− ∫ t o (α1 +α2 +α3)λ(η)dη } + S(0) exp { −µt− ∫ t o (α1 +α2 +α3)λ(η)dη } , where S(0) is an initial condition for S(t) and hence it is a constant. This expression guarantees the positivity of the state variable S(t) under the condition that S(0) > 0 which consequently ensures the positivity of I1(t), I2(t), I3(t), I4(t) and A(t) provided that (10) is satisfied for all time t ≥ 0. Furthermore, addition of (4)-(9) gives dN(t) dt = B −µN(t) − τAw� (11) dN(t) dt ≤ B −µN(t), whose solution is N(t) ≤ B µ + [ N(0) − B µ ] exp (−µt), (12) lim t→∞ N(t) ≤ B µ + lim t→∞ [ N(0)− B µ ] exp (−µt) = B µ . This shows the boundedness of the solution above by B µ in the domain defined by the provision of Lemma III.1. Therefore, the model is epidemically well-posed and mathematically meaningful since all the state variables are non-negative for all t ≥ 0. Hence, it is sufficient to study and analyze the model in Γ [26], [27]. This completes the proof. IV. ANALYSIS OF THE MODEL A. Analysis of the Model with Total Abstinence of Non-progressors Here, we analyze the model for non-progressors that change their behavior through total abstinence from all means of contracting HIV-1 and from all HIV-1 endemic environments i.e. γ2 = 0,σ3 = 0 so that equation (4)-(9) becomes dS dt =B − (α1 + α2 + α3)λ1S −µS, (13) dI1 dt =α1λ1S −K1I1, (14) dI2 dt =α2λ1S −K2I2, (15) dI3 dt =α3λ1S −K3I3, (16) dI4 dt =γ1I2 −µI4, (17) dA dt =ρ1I1 + ρ2I3 −K5A, (18) where λ1 = β(I3 + σ1I1 + σ2I2) N . (19) All model parameters are positive. B. Local Stability of Disease-Free equilibrium (DFE) The disease-free equilibrium of (13)-(18) is given by ψ∗1 = (S ∗,I∗1,I ∗ 2,I ∗ 3,I ∗ 4,A ∗) = ( B µ , 0, 0, 0, 0, 0 ) . (20) This shows that N∗ = S∗ = B µ and S∗ N∗ = 1 Biomath 9 (2020), 2006143, http://dx.doi.org/10.11145/j.biomath.2020.06.143 Page 6 of 17 http://dx.doi.org/10.11145/j.biomath.2020.06.143 R. Musa, R. Willie, N. Parumasur, Analysis of a virus-resistant HIV-1 model with behavior change in ... at disease-free equilibrium point ψ∗1. By employing the next generation method [7], [34], F1 (the new infection terms) and V1 (transfer terms) are expressed as F1 =   α1σ1β βσ2α1 βα1 0 0 α2σ1β βσ2α2 βα2 0 0 α3σ1β βσ2α3 βα3 0 0 0 0 0 0 0 0 0 0 0 0   , V1 =   K1 0 0 0 0 0 K2 0 0 0 0 0 K3 0 0 0 −γ1 0 µ 0 −ρ1 0 −ρ2 0 K5   . Taking ρ as the spectral radius (magnitude of the dominate eigenvalue) of the next generation matrix F1V−11 , the reproduction number is given by R ′ T = β(α1K2K3σ1 + α2K1K3σ2 + α3K1K2) K1K2K3 . (21) The quantity R ′ T represents the measure of average number of new virus infection of HIV-1 developed by a single HIV-1 infected individual in a popu- lation where there are people who practice total abstinence and are completely susceptible. Hence, we present the following Lemma. Lemma IV.1. The DFE of the reduced model (13)- (18) with total abstinence is locally asymptotically stable (LAS) if R ′ T < 1, and unstable if R ′ T > 1. The proof is standard and can be established using theorem 2 of [34]. C. Existence of Endemic Equilibrium The reduced model with total abstinence has a unique positive endemic equilibrium point (EEP). This is the point where at least one of the virus infected compartments is non-zero. Let ψ∗∗1 = (S ∗∗,I∗∗1 ,I ∗∗ 2 ,I ∗∗ 3 ,I ∗∗ 4 ,A ∗∗) (22) be the endemic equilibrium point. We further de- fine the force of infection as λ∗∗1 = β(I∗∗3 + σ1I ∗∗ 1 + σ2I ∗∗ 2 ) N∗∗ . (23) Solving equation (13)-(18) in terms of the force of infection λ∗∗1 at steady-state gives: S∗∗= B µ + (α1 + α2 + α3)λ ∗∗ 1 , I∗∗1 = α1Bλ ∗∗ 1 K1[µ + (α1 + α2 + α3)λ ∗∗ 1 ] , I∗∗3 = Bλ∗∗1 K3[µ + (α1 + α2 + α3)λ ∗∗ 1 ] , I∗∗4 = γ1Bα2λ ∗∗ 1 K2µ[µ + (α1 + α2 + α3)λ ∗∗ 1 ] , (24) A∗∗= Bλ∗∗1 (ρ1α1K3 + K1ρ2) K1K3K5[µ + (α1 + α2 + α3)λ ∗∗ 1 ] , I∗∗2 = α2Bλ ∗∗ 1 K2[µ + (α1 + α2 + α3)λ ∗∗ 1 ] , N∗∗= BK1K3K5f1−τBλ∗∗1 (ρ1α1K3 +K1ρ2) µK1K3K5f1 , where f1 = µ + (α1 + α2 + α3)λ∗∗1 . Substituting all the equations in (24) into (23), it can be shown that the non-zero equilibria of the model satisfy the following linear equation in terms of λ∗∗1 : aoλ ∗∗ 1 + a1 = 0, (25) where ao = α1µK2K3(µ + τ + ρ1) + K1K2[µα3(ρ2 + µ + τ) + K3K5α2], (26) a1 = µK1K2K3K5(1 −R ′ T ). (27) Clearly, ao > 0, a1 ≥ 0 if and only if R ′ T ≤ 1 so that λ∗∗1 = − a1 ao ≤ 0. This shows that no ex- istence of positive endemic equilibrium whenever R ′ T ≤ 1. Hence, the endemic equilibrium point ψ∗∗1 exists and unique whenever R ′ T > 1. We claim the following result. Lemma IV.2. The endemic equilibrium point (EEP) of the reduced model (13)-(18) with total abstinence is locally asymptotically stable (LAS) if R ′ T > 1. D. Global Stability of DFE To establish the global stability of DFE points, we adopt the approach of [5] to re express (13)- (18) in the following vector form Ẋ = L(X,Y ), (28) Biomath 9 (2020), 2006143, http://dx.doi.org/10.11145/j.biomath.2020.06.143 Page 7 of 17 http://dx.doi.org/10.11145/j.biomath.2020.06.143 R. Musa, R. Willie, N. Parumasur, Analysis of a virus-resistant HIV-1 model with behavior change in ... Ẏ = M(X,Y ),M(X, 0) = 0, (29) where the vector X = (S) denotes the HIV-1 uninfected compartment of the system and Y = (I1,I2,I3,I4,A) ∈ R5+ represents the HIV- 1 infected compartments. Using the DFE point to establish the stability analysis, the following two conditions must be satisfied: N1 : For Ẋ(t) = L(Xo, 0), Xo is globally asymptotically stable. N2 : M(X,Y ) = JY −M̂(X,Y ), M̂(X,Y ) ≥ 0 for X,Y ∈ Ωm where J = ∂M∂Y (X o, 0). For this analysis, the expressions for J1,Y1,M̂1 and M1 are for the reduced model with the same definition as above while expressions for J,Y,M̂ and M are for the full model with the same definition. From our model equation, we obtain the Jacobian matrix of only the infected compartment at DFE as follows: J1 =  α1σ1βS ∗ N∗ −K1 βσ2α1S ∗ N∗ βα1S ∗ N∗ 0 0 α2σ1βS ∗ N∗ α2σ2βS ∗ N∗ −K2 βα2S ∗ N∗ 0 0 α3σ1βS ∗ N∗ α3σ2βS ∗ N∗ βα3S ∗ N∗ −K3 0 0 0 γ1 0 −µ 0 ρ1 0 ρ2 0 −K5   J1Y1 =J1   I1 I2 I3 I4 A  =   βα1(σ1I1+σ2I2+I3)S ∗ N∗ −K1I1 βα2(σ1I1+σ2I2+I3)S ∗ N∗ −K2I2 βα3(σ1I1+σ2I2+I3)S ∗ N∗ −K3I3 γ1I2 −µI4 ρ1I1 + ρ2I3 −K5A   M̂1(X,Y ) = J1Y1 −M1(X,Y ) ≥ 0 where M1(X,Y ) =   βα1(σ1I1+σ2I2+I3)S N −K1I1 βα2(σ1I1+σ2I2+I3)S N −K2I2 βα3(σ1I1+σ2I2+I3)S N −K3I3 γ1I2 −µI4 ρ1I1 + ρ2I3 −K5A   and M̂1(X,Y ) =   βα1(σ1I1 +σ2I2 +I3) ( 1− S N ) βα2(σ1I1 +σ2I2 +I3) ( 1− S N ) βα3(σ1I1 +σ2I2 +I3) ( 1− S N ) 0 0   , Since S ≤ N, this shows that M̂1(X,Y ) ≥ 0. It can be seen that limt→∞X(t) = Xo and J is an M-matrix, thus Xo is globally asymptotically stable, hence, N1 is satisfied. Also, M̂1(X,Y ) ≥ 0 for (X,Y ) ∈ Ωm. Hence, N2 is satisfied and Eo is globally asymptotically stable whenever R ′ T < 1. E. Global Stability of Endemic Equilibrium Point Following the provision of Lemma IV.2, we establish the following theorem. Theorem IV.3. The endemic equilibrium point of the reduced model (13)-(18) is globally asymptot- ically stable (GAS) whenever R ′ T > 1. Proof: Using the idea of [1], we construct the Lyapunov function: B = 6∑ k=1 AkBk, Ak > 0, (30) where Ak is a constant and Bk is given by Bk = ∫ f f∗∗k ( 1 − f∗∗k x ) dx, (31) for f∗∗k ∈ W = {S,I1,I2,I3,I4,A} , where k = 1, 2, 3, 4, 5, 6. This vividly shows that Bk is positive definite, continuous and differen- tiable in Γ. Hence, Bk ∈ C ′ [Γ,R+]. Differentiat- ing B partially with respect to each fk we have ∂B ∂fk = Ak ( 1 − f∗∗k fk ) , (32) so that ∂B ∂fk = 0 =⇒ Ak ( 1 − f∗∗k fk ) = 0. Biomath 9 (2020), 2006143, http://dx.doi.org/10.11145/j.biomath.2020.06.143 Page 8 of 17 http://dx.doi.org/10.11145/j.biomath.2020.06.143 R. Musa, R. Willie, N. Parumasur, Analysis of a virus-resistant HIV-1 model with behavior change in ... Differentiating (32) again partially with respect to each fk gives ∂2B ∂f2k = Akf ∗∗ k f2k , k = 1, ..., 6. (33) From (32), if fk = f∗∗k , then S = S ∗∗,I1 = I∗∗1 ,I2 = I ∗∗ 2 ,I3 = I ∗∗ 3 ,I4 = I ∗∗ 4 ,A = A ∗∗. This clearly shows that the endemic equilibrium point is the only stationary point of B. Since (30) is always positive, it means that the endemic equilibrium is a global minimum point of the function B for all fk ∈ Γ ⊆ R6+. Next is to establish that the function B is a Lyapunov function which can be done by proving that B is negative definite. The time derivative of B is given by dB dt = 6∑ k=1 Ak ( 1 − f∗∗k fk ) ḟk, (34) which is negative definite for all time t > 0. It is worth noting here that for all f∗∗k ∈ Γ, ḟk ≤ Ṅ which makes equation (34) to be dB dt ≤ 6∑ k=1 Ak ( 1 − f∗∗k fk ) Ṅ. (35) From equation (12), we obtain the derivative dN dt = µ ( B µ −N(0) ) exp(−µt). (36) Substituting (36) in (35), we have dB dt ≤ 6∑ k=1 Ak ( 1− f∗∗k fk ) µ ( B µ −N(0) ) exp(−µt). (37) When t → ∞, dB dt ≤ 0 which means that the total initial population N(0) is within the basin Γ i.e. N(0) ≤ B µ . Also when the initial population is outside the basin of attraction i.e. N(0) ≥ B µ as t → ∞, then dB dt ≤ 0 and hence, the right- hand side of (37) is negative definite. This proves that irrespective of the size of the initial population N(0), the left hand side is always less or equal to zero as t > 0. This consequently clarifies that the constructed function B is a Lyapunov type and can be used to establish the global stability of the system. Moreover, dB dt = 0 if and only if S = S∗∗,I1 = I ∗∗ 1 ,I1 = I ∗∗ 1 ,I2 = I ∗∗ 2 , I3 = I ∗∗ 3 ,I4 = I ∗∗ 4 ,A = A ∗∗, and the largest positive invariant subset of Γ that satisfies dB dt = 0 is the singleton ψ∗∗1 . Hence, ψ ∗∗ 1 is a unique endemic equilibrium point of the system (13)-(18) which is GAS in Γ. F. Analysis of the Full Model G. Local Stability of DFE In this section, we shall analyze the full model just as we did for the sub-model in the previous section. It is worth noting that the full model has the same DFE as the sub-model given by equation (20) which exists in the same region Γ. We employ the same next generation matrix to establish the reproduction number as follows: F =   α1σ1β βσ2α1 βα1 βα1σ3 0 α2σ1β βσ2α2 βα2 βα2σ3 0 α3σ1β βσ2α3 βα3 βα3σ3 0 0 0 0 0 0 0 0 0 0 0   , V =   K1 0 0 0 0 0 K2 0 −γ2 0 0 0 K3 0 0 0 −γ1 0 K4 0 −ρ1 0 −ρ2 0 K5   . Taking ρ as the spectral radius (magnitude of the dominate eigenvalue) of the next generation matrix FV−1, the reproduction number is given by RT = [ P + Q K1K3(K2K4−γ1γ2) ] , (38) where P = (K2K4−γ1γ2)(α1K3σ1 +α3K1), Q = α2K1K3(γ1σ3 +K4σ2). Lemma IV.4. The disease-free equilibrium point (DFE) of the full model (4)-(9) with partial ab- stinence is locally asymptotically stable (LAS) if RT < 1 and unstable otherwise. Biomath 9 (2020), 2006143, http://dx.doi.org/10.11145/j.biomath.2020.06.143 Page 9 of 17 http://dx.doi.org/10.11145/j.biomath.2020.06.143 R. Musa, R. Willie, N. Parumasur, Analysis of a virus-resistant HIV-1 model with behavior change in ... H. Existence of Endemic Equilibrium The full model with partial abstinence has a unique positive endemic equilibrium point (EEP). This is the point where at least one of the virus infected compartments is non-zero. Let ψ∗∗ = (S∗∗,I∗∗1 ,I ∗∗ 2 ,I ∗∗ 3 ,I ∗∗ 4 ,A ∗∗) (39) be the endemic equilibrium point. We further de- fine the force of infection as λ∗∗ = β(I∗∗3 + σ1I ∗∗ 1 + σ2I ∗∗ 2 + σ3I ∗∗ 4 ) N∗∗ . (40) Solving equation (4)-(9) in terms of the force of infection λ∗∗ at steady-state we obtain: S∗∗ = B µ + (α1 + α2 + α3)λ∗∗ , I∗∗1 = α1Bλ ∗∗ K1[µ + (α1 + α2 + α3)λ∗∗] , I∗∗3 = Bλ∗∗ K3[µ + (α1 + α2 + α3)λ∗∗] , (41) A∗∗ = Bλ∗∗(ρ1α1K3 + K1ρ2) K1K3K5[µ + (α1 + α2 + α3)λ∗∗] , I∗∗2 = α2Bλ ∗∗K4 f1(K2K4 −γ1γ2) , N∗∗= BK1K3K5f1−τBλ∗∗(ρ1α1K3 +K1ρ2) µK1K3K5f1 , I∗∗4 = γ1Bα2λ ∗∗K4 K4[µ+(α1 +α2 +α3)λ∗∗](K2K4−γ1γ2) , where f1 = µ + (α1 + α2 + α3)λ∗∗. Substituting all the equations in (41) into (40), it can be shown that the non-zero equilibria of the model satisfy the following linear equation in terms of λ∗∗: a2λ ∗∗ + a3 = 0, (42) where a2 =α3µK1(ρ2 +µ+τ)+α1K3µ(µ+ρ1 +τ) + K1K3K5α2 > 0 (43) a3 =µK1K3K5(1 −RT ). (44) Clearly, a2 > 0, a3 ≥ 0 if and only if RT ≤ 1 so that λ∗∗ = −a3 a2 ≤ 0 which shows no existence of positive endemic equilibrium whenever RT ≤ 1. Hence, the endemic equilibrium point ψ∗∗ exists and unique whenever RT > 1. We claim the following result. Lemma IV.5. The endemic equilibrium point (EEP) of the full model (4)-(9) with partial ab- stinence is locally asymptotically stable (LAS) if RT > 1. I. Global Stability of DFE of the full model We will establish the proof using the same approach as in Section IV.C as follows: M̂(X,Y ) = JY −M(X,Y ) =   βα1(σ1I1 +σ2I2 +I3 +σ3I4)S ∗ N∗ −K1I1 βα2(σ1I1+σ2I2+I3+σ3I4)S ∗ N∗ −K2I2 +γ2I4 β(σ1I1 +σ2I2 +I3 +σ3I4)S ∗ N∗ −K3I3 γ1I2 −K4I4 ρ1I1 + ρ2I3 −K5A   −   βα1(σ1I1 +σ2I2 +I3 +σ3I4)S ∗ N∗ −K1I1 βα2(σ1I1+σ2I2+I3 +σ3I4)S ∗ N∗ −K2I2 +γ2I4 β(σ1I1 +σ2I2 +I3 +σ3I4)S ∗ N∗ −K3I3 γ1I2 −K4I4 ρ1I1 + ρ2I3 −K5A   =   βα1(σ1I1 +σ2I2 +I3 +σ3I4) ( 1− S N ) βα2(σ1I1 +σ2I2 +I3 +σ3I4) ( 1− S N ) βα3(σ1I1 +σ2I2 +I3 +σ3I4) ( 1− S N ) 0 0   ≥ 0, where S ∗ N∗ ≤ 1 at DFE and since S ≤ N, this shows that M̂(X,Y ) ≥ 0. It can be seen that limt→∞X(t) = X o and J is an M-matrix, thus Xo is globally asymptotically stable, hence, N1 is satisfied. Also, M̂(X,Y ) ≥ 0 for (X,Y ) ∈ Ωm. Hence, N2 is satisfied and Eo is globally asymp- totically stable whenever RT < 1. Biomath 9 (2020), 2006143, http://dx.doi.org/10.11145/j.biomath.2020.06.143 Page 10 of 17 http://dx.doi.org/10.11145/j.biomath.2020.06.143 R. Musa, R. Willie, N. Parumasur, Analysis of a virus-resistant HIV-1 model with behavior change in ... J. Global Stability of The Endemic Equilibrium We consider the special case where the virus- induced death rate τ is negligible. This is very much realistic since HIV-1 positive individuals under treatment can live many years hail and healthy without dying of the virus. Substituting τ = 0 into (11), as t →∞ gives N → B µ . Putting this in equation (1), we have λ = β1(I3 + σ1I1 + σ2I2 + σ3I4) (45) where β1 = βµ B . Theorem IV.6. The endemic equilibrium point of the full model (4)-(9) is globally asymptotically stable (GAS) whenever RT > 1. Proof: Since Lemma IV.5 has already been established, we construct the following non-linear Lyapunov function for the system (4)-(9) as fol- lows: L = β1 f2µ ∫ S S∗ ( 1− S∗ x ) dx+ 1 α1α2 ∫ I1 I∗1 ( 1− I∗1 x ) dx + γ2 α2 ∫ I2 I∗2 ( 1− I∗2 x ) dx+ α1 β1α3 ∫ I3 I∗3 ( 1− I∗3 x ) dx + β21 K2K3γ1 ∫ I4 I∗4 ( 1− I∗4 x ) dx+ 1 ρ1ρ2 ∫ A A∗ ( 1− A∗ x ) dx. The derivative of L along the solution of the system (4)-(9) is given by L̇ = β1 f2µ ( 1 − S∗ S ) Ṡ + 1 α1α2 ( 1 − I∗1 I1 ) İ1 + γ2 α2 ( 1 − I∗2 I2 ) İ2 + α1 β1α3 ( 1 − I∗3 I3 ) İ3 + β21 K2K3γ1 ( 1 − I∗4 I4 ) İ4 + 1 ρ1ρ2 ( 1 − A∗ A ) Ȧ. Using (4)-(9), we have L̇= β1 f2µ [ B−(f2λ+µ)S− S∗∗ S {B−(f2λ+µ)S} ] + 1 α1α2 [ α1λS−K1I1− I∗∗1 I1 {α1λS−K1I1} ] + γ2 α2 [ α2λS+γ2I4−K2I2− I∗∗2 I2 {α2λS+γ2I4−K2I2} ] + α1 β1α3 [ α3λS −K3I3 − I∗∗3 I3 {α3λS −K3I3} ] + β21 K2K3γ1 [ γ1I2−K4I4− I∗∗4 I4 {γ1I2−K4I4} ] + 1 ρ1ρ2 [ ρ1I1 + ρ2I3 −K5A − A∗∗ A {ρ1I1 + ρ2I3 −K5A} ] , (46) where f2 = α1 +α2 +α3. At endemic equilibrium point of (4)-(9), we have the following expres- sions. B = µS∗∗+f2(I ∗∗ 3 +σ1I ∗∗ 1 +σ2I ∗∗ 2 +σ3I ∗∗ 4 )S ∗∗, K1 = α1β1(I ∗∗ 3 + σ1I ∗∗ 1 + σ2I ∗∗ 2 + σ3I ∗∗ 4 )S ∗∗ I∗∗1 , K2 = γ2I ∗∗ 4 +α2β1(I ∗∗ 3 +σ1I ∗∗ 1 +σ2I ∗∗ 2 +σ3I ∗∗ 4 )S ∗∗ I∗∗2 , K3 = α3β1(I ∗∗ 3 + σ1I ∗∗ 1 + σ2I ∗∗ 2 + σ3I ∗∗ 4 )S ∗∗ I∗∗3 , K4 = γ1I ∗∗ 2 I∗∗4 , K5 = ρ1I ∗∗ 1 + ρ2I ∗∗ 3 A∗∗ (47) Substituting expressions in (47) into (46), after some simplifications and factorization, we have L̇ = β1S ∗∗ f2 ( 2 − S S∗∗ − S∗∗ S ) + β21 µ (I∗∗3 +σ1I ∗∗ 1 +σ2I ∗∗ 2 +σ3I ∗∗ 4 )S ∗∗ ( 2− S S∗∗ − S∗∗ S ) + β1 α2 [ (I∗∗3 +σ1I ∗∗ 1 + σ2I ∗∗ 2 +σ3I ∗∗ 4 )S ∗∗ ( 2− I1 I∗∗1 − I∗∗1 I1 ) + (I3+σ1I1+σ2I2+σ3I4)S ( 2− I1 I∗∗1 − I∗∗1 I1 )] + ( 1 ρ2 + 1 ρ1 )( 2 − A A∗∗ − A∗∗ A ) Biomath 9 (2020), 2006143, http://dx.doi.org/10.11145/j.biomath.2020.06.143 Page 11 of 17 http://dx.doi.org/10.11145/j.biomath.2020.06.143 R. Musa, R. Willie, N. Parumasur, Analysis of a virus-resistant HIV-1 model with behavior change in ... +γ2 [ β1(I3+σ1I1+σ2I2+σ3I4)S ( 2− I2 I∗∗2 − I∗∗2 I2 ) + β1(I ∗∗ 3 +σ1I ∗∗ 1 +σ2I ∗∗ 2 +σ3I ∗∗ 4 )S ∗∗ ( 2− I2 I∗∗2 − I∗∗2 I2 )] + ( γ22 α2 + α1β 2 1 K2K3 ) I22I ∗∗ 3 I∗∗4 ( 3− I∗∗2 I2 − I2I ∗∗ 3 I∗∗2 I3 − I∗∗2 I3I4 I2I ∗∗ 3 I ∗∗ 4 ) . Consequently, since the arithmetic mean exceeds the geometric mean, then we have 2 − S S∗∗ − S∗∗ S ≤ 0, 2 − A A∗∗ − A∗∗ A ≤ 0, 2 − I1 I∗∗1 − I∗∗1 I1 ≤ 0, 2 − I2 I∗∗2 − I∗∗2 I2 ≤ 0, 3 − I∗∗2 I2 − I2I ∗∗ 3 I∗∗2 I3 − I∗∗2 I3I4 I2I ∗∗ 3 I ∗∗ 4 ≤ 0. Since S ≥ 0,I1 ≥ 0,I2 ≥ 0,I3 ≥ 0,I4 ≥ 0,A ≥ 0 and Lemma IV.5 is satisfied, it follows that L̇ ≤ 0 since all other model parameters are non-negative for RT > 1. Furthermore, L̇ = 0 if and only if S = S∗∗,I1 = I∗∗1 ,I2 = I ∗∗ 2 ,I3 = I∗∗3 ,I4 = I ∗∗ 4 ,A = A ∗∗. Thus, L is a Lyapunov function of the subsystem (4)-(9) on Γ. It therefore follows by LaSalle’s Invariance Principle [16] that the subsystem (4)-(9) has a globally asymptoti- cally stable endemic equilibrium point ψ∗∗. The result presented here shows that for a special case (τ = 0), the virus will consistently persist in the community whenever the associated reproduction number RT > 1. V. NUMERICAL SIMULATION AND DISCUSSION OF RESULTS In this section, we shall carry out the numerical simulation of the model to corroborate the analytic results. We shall solve the model equation (4)- (9) numerically and present the results graphically using Maple 18 and Python mathematical soft- ware. A 3D surface plot shall also be presented to examine the relationship between the reproduction number, the partial abstinence rate γ2 and σ3 which is the modification parameter which account for the assumed reduction of infectiousness by the behavior change class I4. Table 1: Hypothetical Value of Parameters Parameter Value (per year) Source B 5600 Estimated α1 0.25 Estimated α2 0.10 Estimated β 0.015 Estimated µ 0.016 Estimated α3 0.65 Estimated ρ1 0.12 Estimated γ1 1.00 [30], [20] γ2 0.95 Estimated ρ2 0.75 Estimated τ 0.0909 [11] σ1 0.85 Estimated σ2 0.55 Estimated σ3 0.008 [29] Table 2: Initial Conditions S(0) I1(0) I2(0) I3(0) I4(0) A(0) 450 10 8 5 10 15 400 40 25 20 10 5 300 70 50 40 30 10 200 95 65 53 47 40 100 120 88 67 65 60 To start with, we will show numerically that the disease-free equilibrium ψ∗ is locally asymptot- ically stable. The parameter values presented in Table 1 and the initial conditions shown in Table 2 shall be used. Considering the case when the reproduction number is less than unity i.e. RT = 0.025 < 1, the graphical solution of model equation (4)-(9) is given in fig.2 - fig.7. It can be seen that only the susceptible population S = 500 survive while the infected population in the slow progression class I1, non progression class I2, fast progression class I3, behavior change class I4 and AIDS class A goes into extinction. This confirms that the DFE of (4)-(9) as presented in Lemma (IV.4) is locally asymptotically stable whenever RT < 1. Biomath 9 (2020), 2006143, http://dx.doi.org/10.11145/j.biomath.2020.06.143 Page 12 of 17 http://dx.doi.org/10.11145/j.biomath.2020.06.143 R. Musa, R. Willie, N. Parumasur, Analysis of a virus-resistant HIV-1 model with behavior change in ... Fig. 2 and Fig. 3 Showing The Behavior of Both Susceptible and Slow Progression Populations when RT is Less Than Unity. Fig. 4 and Fig. 5 Showing The Behavior of Both Non Progression and Fast Progression Populations when RT is less than Unity. Fig. 6 and Fig. 7 Showing The Behavior of Both Fast Progression and AIDS Populations when RT is Less Than Unity. Biomath 9 (2020), 2006143, http://dx.doi.org/10.11145/j.biomath.2020.06.143 Page 13 of 17 http://dx.doi.org/10.11145/j.biomath.2020.06.143 R. Musa, R. Willie, N. Parumasur, Analysis of a virus-resistant HIV-1 model with behavior change in ... Fig. 8 and Fig. 9 Showing The Behavior of Both Susceptible and Slow Progression Populations when RT is Greater Than Unity. Fig. 10 and Fig. 11 Showing The Behavior of Both Non Progression and Fast Progression Populations when RT is Greater Than Unity. Fig. 12 and Fig. 13 Showing The Behavior of Both Fast Progression and AIDS Populations when RT is Greater Than Unity. Biomath 9 (2020), 2006143, http://dx.doi.org/10.11145/j.biomath.2020.06.143 Page 14 of 17 http://dx.doi.org/10.11145/j.biomath.2020.06.143 R. Musa, R. Willie, N. Parumasur, Analysis of a virus-resistant HIV-1 model with behavior change in ... Considering the case when the reproduction number is greater than unity i.e. RT = 5.903 > 1, the graphical solution of model equation (4)-(9) is given in fig.8 - fig.13. It can be observed that when the reproduction number RT > 1, all the populations survive that’s [S∗∗,I∗∗1 ,I ∗∗ 2 ,I ∗∗ 3 ,I ∗∗ 4 ,A ∗∗] = [0.0155, 0.6575, 0.9452, 0.8105, 0.5421, 0.7813], which clearly indicates that the population con- verges or tends to the endemic equilibrium points ψ∗∗ whenever RT > 1. It can also be seen that the susceptible population reduces drastically because of the reproduction number being greater than unity while all the remaining infected populations increases with time. This confirms that the en- demic equilibrium points ψ∗∗ is locally asymptot- ically stable and thus, confirms the analytic results presented in Lemma IV.5. A. Effect of Partial and Total Abstinence in HIV/AIDS Transmission Here, we will observe the effect of partial and total abstinence in the transmission of HIV/AIDS. We simulate the reproduction number in equation (21) and (38). For the case of partial abstinence (i.e. when σ3 = γ2 6= 0), with the parameter values in Table 1, when γ2 = 0.95 and σ3 = 0.008, RT = 0.025. For the case of total abstinence (i.e. when σ3 = γ2 = 0), with the parameter values in Table 1, R ′ T = 0.020 meaning that R ′ T for the total abstinence is less than RT for the partial abstinence. Since our aim in epidemiology is to find all possible means to reduce the reproduction number of infectious disease, it means that those that changed their sexual attitude through total abstinence from HIV/AIDS and all factors that can cause its transmission are at lower or no risk of contacting HIV/AIDS. Hence, total abstinence is one of the key factors to be safe from HIV/AIDS. Figure 14 below shows a 3D surface plot to understand more about the relationship between the reproduction number and γ2 and σ3. We can easily observe that the higher the value of both σ3 and γ2, the higher the reproduction number and the lower their values the lower the re- production number. The lowest reproduction num- ber 0.018 is gotten when σ3 = γ2 = 0 i.e. (total abstinence). Hence, total abstinence is essential in the protection against HIV/AIDS transmission. VI. CONCLUSION, ACKNOWLEDGMENT AND DISCLOSURE STATEMENT In this study, a new virus resistant HIV-1 model with behavior change was proposed and systemati- cally analyzed for both partial and total abstinence from HIV/AIDS. Basic analysis of the model such as positivity solution, reproduction number, invariant region, establishment of both disease-free and endemic equilibrium points for both scenarios were carried out. The local asymptotic stability of the DFE and EE for both models whenever the associated reproduction number is less than unity and greater than unity respectively were proved. A non-linear Goh–Volterra Lyapunov function is used to prove that the endemic equilibrium point is globally asymptotically stable for the case when the virus-induced death rate τ = 0 while the method of Castillo-Chavez is used to prove the global asymptotic stability of the disease-free equi- librium point whenever the reproduction number is less than unity. In the numerical simulation, it was established that people with total abstinence are more protected against HIV/AIDS than those with partial abstinence and also established that the reproduction number is minimal under this same condition. Since those with resistance to HIV/AIDS do not proceed to the AIDS compart- ment, this also highlight the importance of HIV- resistance which plays an important role in the protection against HIV/AIDS. Acknowledgment The authors really acknowledge and appreciate the efforts of the unknown reviewers. The first author, Rabiu Musa, acknowledges funding from the NRF and DST of South Africa through grant number 48518. Disclosure Statement The research work forms part of the first authors Biomath 9 (2020), 2006143, http://dx.doi.org/10.11145/j.biomath.2020.06.143 Page 15 of 17 http://dx.doi.org/10.11145/j.biomath.2020.06.143 R. Musa, R. Willie, N. Parumasur, Analysis of a virus-resistant HIV-1 model with behavior change in ... Fig. 14 Showing The Effect of σ3 and γ2 on the Reproduction Number Using the Parameter Values on Table 1 when σ3 = γ2 = [0, 1]. PhD work and the co-authors are his supervisors. Data Availability Statement The numerical data and hypothetical value of parameters used to support the findings of this research are included within the article. They are either properly referenced, assumed or estimated in Table 1. Conflict of Interest No conflict of interest as far as this research is concerned. REFERENCES [1] Afassinou, K., Chirove, F., & Govinder, K. S. (2017). 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Biomath 9 (2020), 2006143, http://dx.doi.org/10.11145/j.biomath.2020.06.143 Page 17 of 17 https://www.nature.com/scitable/blog/viruses101/HIV_resistant_mutation/?isForcedMobile=Y https://www.nature.com/scitable/blog/viruses101/HIV_resistant_mutation/?isForcedMobile=Y https://doi.org/10.1016/j.mcm.2008.09.013 https://doi.org/10.1016/j.mcm.2008.09.013 www.unaids.org/en/KnowledgeCentre/HIV\Data/EpiUpdate/EpiUpdArcHIVe/2009/default.asp www.unaids.org/en/KnowledgeCentre/HIV\Data/EpiUpdate/EpiUpdArcHIVe/2009/default.asp www.sciencedaily.com/releases/2014/11/141120141750.htm www.sciencedaily.com/releases/2014/11/141120141750.htm http://dx.doi.org/10.11145/j.biomath.2020.06.143 Introduction Model Formulation and Model Assumptions Basic Properties of the Model Positivity and boundedness of the Model. Analysis of the Model Analysis of the Model with Total Abstinence of Non-progressors Local Stability of Disease-Free equilibrium (DFE) Existence of Endemic Equilibrium Global Stability of DFE Global Stability of Endemic Equilibrium Point Analysis of the Full Model Local Stability of DFE Existence of Endemic Equilibrium Global Stability of DFE of the full model Global Stability of The Endemic Equilibrium Numerical Simulation and Discussion of Results Effect of Partial and Total Abstinence in HIV/AIDS Transmission Conclusion, Acknowledgment and Disclosure Statement References