J. Nig. Soc. Phys. Sci. 5 (2023) 1389 Journal of the Nigerian Society of Physical Sciences Understanding the Transmission Dynamics and Control of HIV Infection: A Mathematical Model Approach Saheed Ajaoa,∗, Isaac Olopadeb, Titilayo Akinwumia, Sunday Adewalec, Adelani Adesanyaa aDepartment of Mathematics and Computer Science, Elizade University, Ilara−mokin, Ondo State. Nigeria bDepartment of Mathematics and Statistics, Federal University Wukari, Wukari, Taraba State. Nigeria. cDepartment of Pure and Applied Mathematics, Ladoke Akintola University of Technology, Ogbomoso, Oyo State. Nigeria. Abstract New challenges like the outbreak of new diseases, government policies, war and insurgency etc. present distortion, delay and denial of persons’ access to ART, thereby fuelling the spread and increasing the burden of HIV/AIDS. A mathematical model is presented to study the transmission dynamics and control of HIV infection. The qualitative and quantitative analyses of the model are carried out. It is shown that the disease-free equilibrium of the model is globally asymptotically stable whenever the basic reproduction number is less than unity. It is also shown that a unique endemic equilibrium exists whenever the basic reproduction number exceeds unity and that the model exhibits a forward bifurcation. Furthermore, the Lyapunov function is used to show that the endemic equilibrium is globally asymptotically stable for a special case of the model whenever the associated basic reproduction number is greater than unity. The model is calibrated to the data on HIV/AIDS prevalence in Nigeria from 1990 to 2019 and it represents reality. The numerical simulations on the global stability of disease-free equilibrium and endemic equilibrium justify the analytic results. The fraction of the detected individuals who are receiving treatment and stay in the treatment class plays a significant role as it influences the population of the latently-infected individuals and AIDS class as the treatment prevents the individuals from progressing into the AIDS class. DOI:10.46481/jnsps.2023.1389 Keywords: HIV/AIDS, Basic reproduction number, Global stability, Bifurcation, Lyapunov function. Article History : Received: 09 February 2023 Received in revised form: 22 March 2023 Accepted for publication: 26 March 2023 Published: 19 April 2023 c© 2023 The Author(s). Published by the Nigerian Society of Physical Sciences under the terms of the Creative Commons Attribution 4.0 International license (https://creativecommons.org/licenses/by/4.0). Further distribution of this work must maintain attribution to the author(s) and the published article’s title, journal citation, and DOI. Communicated by: P. Thakur 1. Introduction Efficient and effective testing is a gateway to HIV treatment and it is an important element of efforts to stop the AIDS epi- demic [1]. A positive diagnosis allows an HIV-infected per- son to receive antiretroviral therapy (ART) [2]. The ART sup- ∗Corresponding author tel. no: +2348060548485 Email address: saheed.ajao@elizadeuniversity.edu.ng (Saheed Ajao ) presses the replication of the virus and this prevents transmis- sion to one’s sexual partner. Thus, early access to antiretroviral therapy (ART) and support for continued treatment is important not only to improve the health status of HIV-infected individu- als but also to prevent the transmission of HIV [3]. In 2021, 28.7 million people were receiving ART globally, and the global ART coverage was 75%. At the end of 2021, only 52% of children aged 0 to 14 years had received ART and World Health Organization recommends that more efforts should be put in place to scale up treatment, most especially for 1 Ajao et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1389 2 children and adolescents [3]. In the WHO African region, where two-thirds of the disease burden exist, access to treatment is an issue for people living with HIV due to many factors. In South Africa, discrimination and stigmatization still remain major obstacles to HIV response efforts which also affect children. According to a UN report, African children are neglected when it comes to HIV treat- ment [4]. UNAIDS reported that despite the global scientific advances in providing better treatment for children and adults, children with HIV in Indonesia have difficulties accessing an- tiretroviral therapy. The deeply rooted societal and gender in- equalities create barriers to women, children, and adolescents accessing quality prevention and care services and thereby mak- ing the situation worse [5]. The report of [6] explained how government policy creates a barrier to HIV treatment. Several factors have been identified as barriers to HIV treatment (see [7-10]). The modelling of the transmission of infectious diseases is now influencing the theory and practice of disease management and control. Mathematical modelling now plays a significant role in policy decision-making regarding the epidemiology of diseases in many countries [12]. Several models have been de- veloped to study the dynamics of HIV transmission (see [13- 16]). Apentang et al [17] studied the impact of the implemen- tation of HIV prevention policies therapy and control strategies among HIV/AIDS new cases in Malaysia. The study revealed that the use of condoms and uncontaminated needle-syringes are important intervention control strategies. Yang et al. [18] studied the global dynamics of an HIV model, which incorpo- rates senior male clients and their results showed that diagno- sis, treatment and education have a positive impact on control- ling HIV transmission, while senior male clients increase the number of new cases of HIV and prolong the time of the out- break. Dubey et al. [19] modelled the role of acquired immune response and antiretroviral therapy in the dynamics of HIV in- fection. Ghosh et al. [20] worked on an HIV/AIDS model of an SI-type with the inclusion of media and self-imposed psycho- logical fear and their results revealed that awareness is more effective in eradicating HIV infection. The existing models in the literature failed to consider the partitioning of detected individuals who are receiving treatment and those who do not access treatment. Hence, we proposed a mathematical model to study the transmission dynamics of HIV in Nigeria. We assume that a fraction of individuals that are detected moves to the treatment class while the remaining fraction moves to the AIDS class. The paper is structured as follows: section 2 contains the method used in the study, Section 3 has the numerical simula- tions and discussion of results, while the conclusion follows in section 4. 2. Method 2.1. Model Formulation Here, we give the description of how the HIV model is de- signed and formulated. The overall population of humans at the time (t) is denoted by N(t) and is partitioned into six dis- tinct classes namely: the susceptible population S (t), the HIV- latently infected L, the HIV-infected undetected class HU , the HIV-infected detected class HD, the treatment class HW , and the AIDS class A. Thus N(t) = S (t) + L(t) + HU (t) + HD(t) + HW (t) + A(t) (1) The susceptible individuals are assumed to be recruited into the population at the rate π and get infected after effective contact with HIV-infected people in the latent, undetected, detected, treatment and AIDS classes at the rate λ, which is given by λ = β(L + η1 HU + η2 HD + η3 HW + η4 A) N (2) where β is the contact rate, η1,η2,η4 ≥ 1 and η3 ≤ 1 are the modification parameters which compare the level of transmissi- bility of the disease in HU, HD, A, and HW classes with respect to people in L class. It is assumed that a fraction � of newly infected individuals progresses to the latently-infected class and the remaining frac- tion with compromised immunity moves to the HIV-undetected class. A fraction ω of people in the latent class who are de- tected progresses to the detected class while the other fraction 1 − ω proceeds to the undetected class. The population of the HIV-detected class increases as a result of the detection of the undetected individuals at the rate γ and diminishes as a result of fraction α of detected individuals who are receiving treatment that progresses to treatment class and the remaining fraction 1 −α that progresses to AIDS class. We assume that those that are receiving treatment move to the latent class at the rate φ. The AIDS class reduces as a result of death due to the disease at the rate δ. Each population size reduces as a result of nat- ural death which occurs in all classes. The flow chart of the model showing the interaction among the classes is depicted in figure 1. Thus, with the assumptions above, we present a deter- ministic model of HIV infection as follows: dS dt = π−λS −µS dL dt = �λS + φHW − (κ + µ)L dHU dt = (1 − �)λS + (1 −ω)κL − (γ + µ)HU (3) dHD dt = ωκL + γHU − (τ + µ)HD dHW dt = ατHD − (φ + µ)HW dA dt = (1 −α)τHD − (µ + δ)A where λ = β(L + η1 HU + η2 HD + η3 HW + η4 A) N (4) N = S + L + HU + HD + HW + A (5) For convenience, we re-write the above equation (3) as thus: dS dt = π−λS −µS 2 Ajao et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1389 3 Figure 1. Schematic diagram of the model dL dt = �λS + φHW − T1 L dHU dt = (1 − �)λS + W L − T2 HU (6) dHD dt = XL + γHU − T3 HD dHW dt = Y HD − T4 HW dA dt = ZHD − T5 A where T1 = κ + µ T2 = γ + µ T3 = τ + µ T4 =φ + µ T5 = µ + δ W =(1 −ω)κ X =ωκ Y =ατ Z = (1 −α)τ 2.2. Model analysis 2.2.1. Basic properties This section explores the basic dynamical features of the model (3). We claim the following: Lemma 2.1. The closed set D = { (S, L, HU, HD, HW, A) ∈ R 6 + : N ≤ π µ } is positively invariant with non-negative initial values in R6+. Proof. Summing up all the compartments of (3) with δ = 0 we have dN dt = π−µN It follows that dN dt ≤ π−µN Then N(t) ≤ N(0)e−µt + π µ (1 − e−µt) If N(0) ≤ π µ , then N(t) ≤ π µ . Hence, all solutions of the model having their starting values in D stay there for t > 0. This means that D is positively invariant and in this region, the model is considered to be epidemiologically meaningful and mathemat- ically well-posed. Hence, we can study the dynamics of the basic model (3) in D. 2.2.2. Stability of the disease-free equilibrium The disease-free equilibrium of the model (6) denoted by E1 is given by E1 = (S 0, L0, HU0, HD0, HW0, A0) = ( π µ , 0, 0, 0, 0, 0 ) (7) By adopting the approach of the Next Generation Matrix method as given by [25], the matrices F (new infection terms) and V (Transition terms) are as given below: F =  �β �βη1 �βη2 �βη3 �βη4 (1 − �)β (1 − �)βη1 (1 − �)βη2 (1 − �)βη3 (1 − �)βη4 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0  (8) and V =  T1 0 0 −φ 0 −W T2 0 0 0 −X −γ T3 0 0 0 0 −Y T4 0 0 0 −Z 0 T5  (9) Then R0 = ρ(FV−1), is given by R0 = β  (1 − �)[T1T3T4T5η1 − XY T5φη1 + T1T4T5γη2 +T1T5Yγη3 + T1T4Zγη4 + Y T5γφ] +�(XT2 + γW)[T4T5η2 + Y T5η3 + ZT4η4] +�T3T4T5(T2 + Wη1)  T5(T1T2T3T4 − XY T2φ− WYγφ) (10) where ρ is the spectral radius of the dominant eigenvalue of the matrix FV−1. Hence, using the Theorem 2 of [25], the following result is established: Lemma 2.2. The disease-free equilibrium of model (6) is lo- cally asymptotically stable whenever the basic reproduction number R0 < 1 and otherwise if R0 > 1 The threshold R0 represents the basic reproduction number of the disease, which is the average number of secondary infec- tions emanating from a single infection source in a population consisting of only the susceptible people [26]. The implication of lemma 2.2 is that a small introduction of infected individuals into the community/population will not produce a substantial outbreak of the disease when the basic reproduction number is less than unity and therefore the disease vanishes. In the next theorem, we show that the disease can be eradicated irrespective of the initial sizes of the sub-populations when R0 < 1 through the exploration of the global stability of the disease-free equi- librium. Theorem 2.3. The disease-free equilibrium (7) of the HIV model is globally asymptotically stable whenever R0 < 1 3 Ajao et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1389 4 Table 1. Description of the parameters used in the model (3) Parameter Description Value Source π Recruitment rate 618893 [21] β Contact rate 0.0785 Estimated µ Natural death rate 0.022 [22] α Fraction of detected individuals that are treated 0.29 Assumed φ Progression rate from treatment class to latent class 0.201 Assumed γ Detection rate of undetected individuals 0.392 Estimated � Fraction of newly infected people with uncompromised immunity 0.9 Assumed κ Progression rate of people in the latent class 0.01 [15] η1,η2,η3,η4 Modification parameters 1.17,1.2,0.04,0.18 Assumed δ Death due to the disease 0.33 [23] τ Treatment rate 0.89 [24] ω Fraction of latent individuals that are detected 0.1 Assumed Proof. Using the Lyapunov function defined by F = E1 L + E2 HU + E3 HD + E4 HW + E5 A (11) By differentiating (11), we have F′ = E1 L ′ + E2 H ′ U + E3 H ′ D + E4 H ′ W + E5 A ′ (12) where E1 = (1 − �)β(Xη1 −γ) − N(T2 X + γW) E2 = �β(γ− Xη1) −γT1 N E3 = �βS (Wη1 + T2) + T1[(1 − �)η1βS + T2 N] E4 = ( φ[(1 − �)βS (Xη1 −γ) − N(T2 X + γW)] −βS η1[�(T2 X + γW) + γ(1 − �)] ) T4 E5 =  �βS [(T2 X + γW)(T4η2 + Yη3) + T4T3(Wη1 + T2)] +(1 − �)βS [T1T4(η2γ + T3η1) − Yφ(Xη1 −γ) +Yη3γ] + N[Yφ(T2 X + γW) − T1T2T3T4]  ZT4 Then by substituting (6) into (12), it becomes F′ = E1(�λS − T1 L + φHW ) + E2((1 − �)λS + W L (13) − T2 HU ) + E3(XL + γHU − T3 HD) + E4(Y HD − T4 HW ) + E5(ZHD − T5 A) Simplifying (13) further leads to F′ =  −NT5(YφT2 X + YφγW − T1T2T3T4) −�βS (T2 X + γW)[ZT4η4 + T5T4η2 + T5Yη3] −(1 − �)βS [T5T1T4η4γ + T5T3η1 −Y T5φ(Xη1 −γ) + T5Yη3γ + ZT4γT1η4] −�βS T5T4T3(Wη1 + T2)  A ZT4 F′ = T5 N[Yφ(T2 X + γW) − T1T2T3T4] ZT4 ( S N R0 − 1 ) A F′ ≤ ( T5 N[Yφ(T2 X + γW) − T1T2T3T4] ZT4 ) (R0 − 1)A Since S ≤ N in D, Therefore,F′ ≤ 0 if R0 ≤ 1 with F′ = 0 if and only if A = 0, L = 0, HU = 0, HD = 0, HW = 0. Also, the largest invariant set in (S, L, HU, HD, HW, A) ∈ D : F′ = 0 is the singleton E1. By LaSalle Invariance Principle [27], ev- ery solution having its starting values in D, approaches E1 as t → ∞, and therefore, the disease-free equilibrium is globally asymptotically stable whenever R0 < 1. The theorem implies that the disease can be eliminated irre- spective of the initial sizes of the subpopulations of the model whenever the basic reproduction number does not exceed unity. 2.2.3. Existence of endemic equilibrium The existence of endemic equilibrium is being investigated here and the condition for the persistence of the disease is being explored. Let E2∗ = (S ∗∗, L∗∗, H∗∗U , H ∗∗ D , H ∗∗ W , A ∗∗) represents the en- demic equilibrium state. Also, let the force of infection at en- demic equilibrium be represented by λ∗∗ = β(L∗∗ + η1 H∗∗U + η2 H ∗∗ D + η3 H ∗∗ W + η4 A ∗∗) N∗∗ (14) Then solving the model equations in terms of the λ (force of infection) at the steady state, we will get the following: S ∗∗ = π λ∗∗ + µ (15) L∗∗ = [�T2T3T4 + φYγ(1 − �)]λ∗∗S ∗∗ T2(T1T3T4 −φY X) −φYγW = Q1λ ∗∗S ∗∗ (16) 4 Ajao et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1389 5 H∗∗U = [(1 − �)(T1T3T4 −φY X) + �WT3T4]λ∗∗S ∗∗ T2(T1T3T4 −φY X) −φYγW = Q2λ ∗∗S ∗∗ (17) H∗∗D = T4[T1γ(1 − �) + �(XT2 + Wγ)]λ∗∗S ∗∗ T2(T1T3T4 −φY X) −φYγW = Q3λ ∗∗S ∗∗ (18) H∗∗W = Y [T1γ(1 − �) + �(XT2 + Wγ)]λ∗∗S ∗∗ T2(T1T3T4 −φY X) −φYγW = Q4λ ∗∗S ∗∗ (19) A∗∗ = ZT4[T1γ(1 − �) + �(XT2 + Wγ)]λ∗∗S ∗∗ T5[T2(T1T3T4 −φY X) −φYγW] = Q5λ ∗∗S ∗∗ (20) By the substitution of (16), (17), (18), (19), and (20) into (14), we have λ∗∗ = β(Q1 + η1 Q2 + η2 Q3 + η3 Q4 + η4 Q5)λ∗∗S ∗∗ S ∗∗ + Q2λ∗∗S ∗∗ + Q3λ∗∗S ∗∗ + Q4λ∗∗S ∗∗ + Q5λ∗∗S ∗∗ (21) λ∗∗ = β(Q1 + η1 Q2 + η2 Q3 + η3 Q4 + η4 Q5)λ∗∗S ∗∗ S ∗∗(1 + Vλ∗∗) (22) where V = Q1 + Q2 + Q3 + Q4 + Q5 Q1 = [�T2T3T4 + φYγ(1 − �)] T2(T1T3T4 −φY X) −φYγW Q2 = [(1 − �)(T1T3T4 −φY X) + �WT3T4] T2(T1T3T4 −φY X) −φYγW Q3 = T4[T1γ(1 − �) + �(XT2 + Wγ)] T2(T1T3T4 −φY X) −φYγW Q4 = Y [T1γ(1 − �) + �(XT2 + Wγ)] T2(T1T3T4 −φY X) −φYγW Q5 = ZT4[T1γ(1 − �) + �(XT2 + Wγ)] T5[T2(T1T3T4 −φY X) −φYγW] Hence, λ∗∗ = R0 − 1 V > 0 when R0 > 1 ∴ λ∗∗ has a positive unique solution when R0 > 1. Hence, the following result is obtained: Lemma 2.4. There exists a unique endemic equilibrium of the HIV model equation (6) whenever the basic reproduction num- ber R0 > 1. The above result indicates the existence of forward bifurca- tion which is verified in the next analysis. 2.2.4. Bifurcation analysis The bifurcation is a phenomenon that describes the changes in the behaviour of a dynamical system as a result of changes in the parameter values or initial conditions of the model. This helps to determine if the disease can be cleared off when the basic reproduction number is less than unity. We will adopt the Center Manifold Theory [28] as described by [29][see Ap- pendix B] to establish the kind of bifurcation that the model exhibits. The Center Manifold Theory is used to determine the stability of equilibrium and plays a vital role in bifurcation the- ory because of the changes in behaviour of the system that take place on the center manifold. If β is chosen as the bifurcation parameter for model (6), then at R0 = 1, we have that β = β∗ = T5(T1T2T3T4 − XY T2φ− WYγφ) (1 − �)[T1T3T4T5η1 − XY T5φη1 + T1T4T5γη2 +T1T5Yγη3 + T1T4Zγη4 + Y T5γφ] +�(XT2 + γW)[T4T5η2 + Y T5η3 + ZT4η4] +�T3T4T5(T2 + Wη1)  (23) If the variables of (6) are changed as follows: S = x1, L = x2, HU = x3, HD = x4, HW = x5, A = x6 and we use the vector notation x = (x1, x2, x3, x4, x5, x6)T , then (6) can be re-written in the form d xdt = F(x) where F = ( f1, f2, f3, f4, f5, f6) T such that (6) becomes d x1 dt = π−λx1 −µx1 = f1 d x2 dt = �λx1 + φx5 − T1 x2 = f2 d x3 dt = (1 − �)λx1 + W x2 − T2 x3 = f3 (24) d x4 dt = X x2 + γx3 − T3 x4 = f4 d x5 dt = Y x4 − T4 x5 = f5 d x6 dt = Z x4 − T5 x6 = f6 (25) The Jacobian (24) at disease-free equilibrium E1 is given by J(E1) =  −µ −β∗ −β∗η1 −β ∗η2 −β ∗η3 −β ∗η4 0 �β∗ − T1 �β∗η1 �β∗η2 �β∗η3 + φ �β∗η4 0 (1 − �)β∗ + W (1 − �)β∗η1 − T2 (1 − �)β∗η2 (1 − �)β∗η3 (1 − �)β∗η4 0 X γ −T3 0 0 0 0 0 Y −T4 0 0 0 0 Z 0 −T5  (26) The matrix (26) has a simple zero eigenvalue at β = β∗ and hence Center Manifold Theory [28] as described by [29] can be used to analyse the dynamics of the system. The Jaco- bian matrix (26) has a right eigenvector denoted by W = (w1, w2, w3, w4, w5, w6)T and a left eigenvector V = (v1, v2, v3, v4, v5, v6)T corresponding to the zero eigenvalue. Then w1 = β∗ ( T3T4T5[w2 + η1w3] + (Xw2 + γw3) ×[T4T5η2 + T5Yη3 + T4Zη4] ) T3T4T5µ , w2 = w2 > 0, w3 = w3 > 0 w4 = Xw2 + γw3 T3 , w5 = Y [Xw2 + γw3] T3T4 , w6 = Z[Xw2 + γw3] T3T5 and v1 = 0, v2 = v2 > 0, v3 = v3 > 0, 5 Ajao et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1389 6 Figure 2. Fitting of HIV model (3) to the data of prevalence cases of HIV/AIDS infection in Nigeria between 1990 and 2019 [30]. 0 10 20 30 0 1 2 3 x 10 5 Time(Years) D e te c te d cl a ss 0 10 20 30 0 5 10 15 x 10 4 Time(Years) T re a tm e n t c la ss 0 10 20 30 0 5000 10000 15000 Time(Years) A ID S c la ss 0 20 40 6 7 8 9 10 x 10 7 Time(Years) S u ce sp ti b le c la ss 0 20 40 0 5 10 15 x 10 4 Time(Years) L a te n t c la ss 0 20 40 0 1000 2000 3000 4000 Time(Years) U n d e te ct e d cl a ss Figure 3. Plot of the different classes of the model when R0 < 1 (R0 = 0.4) with β = 0.0085 and other values used are in table 1 v4 = ( (T4T5η2 + Y T5η3 + T4Zη4)[�β∗v2 + (1 − �)β∗v3] +φY T5v2 ) T3T4T5 , v5 = β∗η3(�v2 + (1 − �)v3) + φv2 T4 , v6 = β∗η4(�v2 + (1 − �)v3) T5 Computation of a and b By finding the associated non-zero partial derivatives of F(x) at disease-free equilibrium, the associated bifurcation coefficients a and b as given by the Center Manifold The- 0 20 40 6 7 8 9 10 x 10 7 Time(Years) S u c e sp ti b le c la ss 0 20 40 0 2 4 6 8 x 10 5 Time(Years) L a te n t c la ss 0 10 20 30 0 1 2 3 x 10 4 Time(Years) U n d e te c te d c la ss 0 10 20 30 0 1 2 3 x 10 5 Time(Years) D e te c te d c la ss 0 10 20 30 0 5 10 15 x 10 4 Time(Years) T re a tm en t cl a ss 0 10 20 30 0 5000 10000 15000 Time(Years) A ID S cl a ss Figure 4. Plot of the different classes of the model when R0 > 1 (R0 = 3.3) using the values of the parameters in table 1 0 100 200 300 400 0 0.5 1 1.5 2 2.5 3 3.5 4 4.5 x 10 7 Time (Years) L + H U + H D + H W + A Figure 5. Plot of the total number of the infected classes (L+ HU + HD + HW + A) at different initial conditions when R0 = 0.4, with β = 0.0085 and other values of the parameters used are given in Table 1 ory [28][see Appendix B], are defined by a = n∑ k,i, j=1 vkwiw j ∂2 fk ∂xi∂x j (0, 0) b = n∑ k,i=1 vkwi ∂2 fk ∂xi∂β∗ (0, 0) Then, we obtain a = −2 β∗ µ π  (v2� + v3(1 − �) ) × (w2 + w3 + w4 + w5 + w6) × (η1w3 + η2w4 + η3w5 + η4w6 + w2)  (27) and b = (v2� + v3(1 − �))[w2 + w3η1 + w4η2 + w5η3 + w6η4] (28) 6 Ajao et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1389 7 0 100 200 300 400 1 1.5 2 2.5 3 3.5 4 x 10 7 Time (Years) L + H U + H D + H W + A Figure 6. Plot of the total number of the infected classes (L+ HU + HD + HW + A) at different initial conditions when R0 = 3.3, with � = 1, ω = 0 and other values of the parameters used are given in Table 1 0 5 10 15 20 25 30 0 1 2 3 4 5 6 x 10 5 Time (Years) L a te n t C la ss α=0.1 α=0.25 α=0.4 α=0.55 α=0.7 Figure 7. Graph of the population of the latent class against time when the fraction of the detected population receiving treatment is varied From (28), b is positive as usual and a is negative from (27). Since a < 0 (negative), then the HIV infection model exhibits a forward bifurcation. This implies that the epidemiological condition R0 < 1 is necessary and sufficient for the elimination of HIV infection. 2.2.5. Global stability of the endemic equilibrium We consider the global stability of endemic equilibrium of the HIV infection model for a situation when � = 1 and the frac- tion of the latently infected people that are detected equals zero (ω = 0) and also we let β∗ = βN . Then, with these assumptions the model‘s basic reproduction number when � = 1 and ω = 0 is given by Ror = β∗ π ( Yγκ T5η3 + Zγκ T4η4 + γκ T4T5η2 +κ T3T4T5η1 + T4T3T2T5 ) T5µ (T4T3T2T1 − Yγκφ) 0 5 10 15 20 25 30 0 2 4 6 8 10 12 x 10 4 Time (Years) A ID S C la ss α=0.1 α=0.25 α=0.4 α=0.55 α=0.7 Figure 8. Graph of the population of the AIDS class against time when the fraction of the detected population receiving treatment is varied Also, the model with � = 1 and ω = 0 possesses a unique endemic equilibrium point represented by E∗2r , which is given by E2r ∗ |�=1,ω=0 = (S ∗∗, L∗∗, H∗∗U , H ∗∗ D , H ∗∗ W , A ∗∗) and that S ∗∗ > 0, L∗∗ > 0, H∗∗U > 0, H ∗∗ D > 0, H ∗∗ W > 0 and A ∗∗ > 0 when Ror > 1 Theorem 2.5. The endemic equilibrium of the reduced model having � = 1 and ω = 0 is globally asymptotically stable when- ever Ror > 1. Proof. Using the Goh-Volterra type of Lyapunov function, we have M = S − S ∗∗ − S ∗∗ln S S ∗∗ + L − L∗∗ − L∗∗ln L L∗∗ + C ( HU − H ∗∗ U − H ∗∗ U ln HU H∗∗U ) + D ( HD − H ∗∗ D − H ∗∗ D ln HD H∗∗D ) + E ( HW − H ∗∗ W − H ∗∗ W ln HW H∗∗W ) + F ( A − A∗∗ − A∗∗ln A A∗∗ ) (29) where C = β∗S ∗∗[T3T4T5η1 + γ(T4T5η2 + Y T5η3 + ZT4η4)] + γYφT5 T2T3T4T5 (30) D = β∗S ∗∗[T4T5η2 + Y T5η3 + ZT4η4] + YφT5 T3T4T5 (31) E = β∗η3S ∗∗ + φ T4 (32) F = β∗η4S ∗∗ T5 (33) 7 Ajao et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1389 8 Taking the derivative of (29), we have Ṁ = Ṡ − S ∗∗ Ṡ S + L̇ − L∗∗ L̇ L +( β∗S ∗∗[T3T4T5η1 + γ(T4T5η2 + Y T5η3 + ZT4η4)] +γYφT5 ) T2T3T4T5 × ( ḢU − H ∗∗ U ḢU HU ) + β∗S ∗∗[T4T5η2 + Y T5η3 + ZT4η4] + YφT5 T3T4T5 × ( ḢD − HD ∗∗ ḢD HD ) + β∗η3S ∗∗ + φ T4 ( ḢW − H ∗∗ W ḢW HW ) + β∗η4S ∗∗ T5 ( Ȧ − A∗∗ Ȧ A ) Ṁ = 2β∗S ∗∗(L∗∗ + η1 H ∗∗ U + η2 H ∗∗ D + η3 H ∗∗ W + A ∗∗) + 2µS ∗∗ −µS − β∗S ∗∗ 2 S (L∗∗ + η1 H ∗∗ U + η2 H ∗∗ D + η3 H ∗∗ W + A ∗∗) − µS ∗∗ S − β∗S L∗∗ L (L + η1 HU + η2 HD + η3 HW + A) + φH ∗∗ W − HW L∗∗ L − LH∗∗U L∗∗HU ( β∗S ∗∗[η1 H∗∗U + η2 H ∗∗ D +η3 H∗∗W + A ∗∗] + φH∗∗W ) + ( β∗S ∗∗[η1 H ∗∗ U + η2 H ∗∗ D + η3 H ∗∗ W + A ∗∗] + φH∗∗W ) − HU H∗∗D H∗∗U HD ( β∗S ∗∗[η2 H ∗∗ D + η3 H ∗∗ W + A ∗∗] + φH∗∗W ) +( β∗S ∗∗[η2 H ∗∗ D + η3 H ∗∗ W + A ∗∗] + φH∗∗W ) − (β∗η3S ∗∗ + φ)HD H∗∗ 2 W H∗∗D HW β∗η3S ∗∗H∗∗W + φH∗∗W − β∗η4S ∗∗HD A∗∗ 2 F H∗∗D + β∗η4S ∗∗A∗∗ Then Ṁ =β∗S ∗∗L∗∗ ( 2 − S ∗∗ S − S S ∗∗ ) + µS ∗∗ ( 2 − S S ∗∗ − S ∗∗ S ) + β∗S ∗∗η1 H ∗∗ U ( 3 − S ∗∗ S − HU L∗∗S H∗∗U LS ∗∗ − LH∗∗U L∗∗HU ) + β∗S ∗∗η2 H ∗∗ D ( 4 − S ∗∗ S − HD L∗∗S H∗∗D LS ∗∗ − LH∗∗U L∗∗HU − HU H∗∗D H∗∗U HD ) +β∗S ∗∗η3 H ∗∗ W  5 − S ∗∗ S − HW L∗∗S H∗∗W LS ∗∗ − LH∗∗U L∗∗HU − HU H∗∗D H∗∗U HD − HD H∗∗W H∗∗D HW  +β∗S ∗∗η4 A ∗∗  5 − S ∗∗ S − AL∗∗S A∗∗LS ∗∗ − LH∗∗U L∗∗HU − HU H∗∗D H∗∗U HD − HD A∗∗ H∗∗D A  +φH∗∗W ( 4 − HW L∗∗ H∗∗W L − LH∗∗U L∗∗HU − HU H∗∗D H∗∗U HD − HD H∗∗W H∗∗D HW ) The arithmetic mean surpasses the geometric mean, then we have the following inequalities 2 − S S ∗∗ − S ∗∗ S ≤ 0, 2 − S ∗∗ S − S S ∗∗ ≤ 0, 3 − S ∗∗ S − HU L∗∗S H∗∗U LS ∗∗ − LH∗∗U L∗∗HU ≤ 0, 4 − S ∗∗ S − HD L∗∗S H∗∗D LS ∗∗ − LH∗∗U L∗∗HU − HU H∗∗D H∗∗U HD ≤ 0, 5 − S ∗∗ S − HW L∗∗S H∗∗W LS ∗∗ − LH∗∗U L∗∗HU − HU H∗∗D H∗∗U HD − HD H∗∗W H∗∗D HW ≤ 0, 5 − S ∗∗ S − AL∗∗S A∗∗LS ∗∗ − LH∗∗U L∗∗HU − HU H∗∗D H∗∗U HD − HD A∗∗ H∗∗D A ≤ 0, 4 − HW L∗∗ H∗∗W L − LH∗∗U L∗∗HU − HU H∗∗D H∗∗U HD − HD H∗∗W H∗∗D HW ≤ 0 Therefore L ≤ 0 when Ror > 1. Hence, by the LaSalle Invari- ance Principle [27], every solution of the model tends to E2r∗ as t →∞ for R0r > 1. The epidemiological implication of this is that HIV infec- tion will persist in the community irrespective of the initial sizes of the subpopulations of the model whenever R0r > 1. 3. Numerical Simulations and Discussion of Results We fit the HIV model to data on HIV/AIDS prevalence in Nigeria from 1990 to 2019 as presented in table 2 (see Ap- pendix. A) sourced from [30]. We use the likelihood func- tion to estimate the values of the contact rate (β) and the de- tection rate of the undetected class (γ). The estimated val- ues of β and γ are 0.0785 and 0.392 respectively. The to- tal population of Nigeria in 1990 stood at 95214256 based on [31] and the initial conditions used for the simulations are as follow: S (0) = 94999422, L(0) = 0, HU (0) = 0, HD(0) = 214834, HW (0) = 0 and A(0) = 0. The time interval of 0 to 29 corresponds to the time interval between 1990 to 2019 and the values of the parameters used for the simulations are given in table 1. The results of the numerical simulations of the model are presented in figures 2 - 8. In figure 2, we fit the HIV model to data in table 2 and also obtain the estimated values for the contact rate (β) and the de- tection rate (γ). The model fits well with the real data and thus the model represents reality. Figure 3 illustrates the behaviour of each compartment when the basic reproduction number is greater than unity. The susceptible class keeps decreasing while the other infected compartments L, HU, HD, HW and A are in- creasing which indicates the persistence of the HIV infection. Figure 4 shows the trajectories of the model when the basic re- production number is less than unity. The population of the sus- ceptible declines and the infected classes L, HU, HD, HW and A are reducing after some period, which means that the disease can be controlled if R0 < 1. Figures 5 and 6 illustrate the verification of the global sta- bility properties of the disease-free equilibrium and endemic 8 Ajao et al. / J. Nig. Soc. Phys. Sci. 5 (2023) 1389 9 equilibrium. The long-term dynamics of the model as depicted by figure 5 shows that the trajectories converge and the disease vanishes irrespective of the initial sizes of the subpopulations when the basic reproduction number is less than unity and fig- ure 6 shows that the disease persists when the basic reproduc- tion number is greater than unity. In figure 7, the plot shows the population of latently infected individuals when the fraction of the HIV-detected individuals that are receiving treatment is var- ied. The population of the latently infected increases as this fraction increases. This is due to the fact that HIV infection has no cure but can be managed. The graph in figure 8 shows that the population of the AIDS class reduces as the fraction of detected individuals receiving treatment increases. 4. Conclusion We propose a mathematical model to study the transmis- sion dynamics of HIV and conduct qualitative and quantitative analyses of the model. The model’s disease-free equilibrium is locally asymptotically stable whenever the basic reproduc- tion number is less than unity. Also, there exists a unique en- demic equilibrium for the model whenever the basic reproduc- tion number is greater than unity and it is shown that the model exhibits forward bifurcation which implies that the necessary condition R0 < 1 is sufficient for the elimination of the dis- ease. Using the Lyapunov function, we further showed that the disease-free equilibrium and endemic equilibrium are globally asymptotically stable whenever the basic reproduction num- ber is less than unity and greater than unity respectively. The proposed model fits with the data on HIV/AIDS prevalence in Nigeria from 1990 to 2019 as it represents the reality. The sim- ulation shows that the disease can be controlled when the basic reproduction number is less than unity and persists if otherwise. The simulations that illustrate the global stability of the model justify the analytic results. The effect of increasing the frac- tion of the detected individuals that are receiving treatment is examined and it increases the population of the latent class and reduces the population of the AIDS class, since the disease has no cure, the treatment is meant to improve the health of a pa- tient by reducing the viral load to an undetected level and pre- vent a patient from progressing into AIDS. 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Watmough, “Reproduction numbers and sub- threshold endemic equilibria for compartmental models of disease trans- mission”, Mathematical Biosciences 180 (2002) 29. [26] S. A. Ayuba, I. Akeyede & A. S.Olagunju, “Stability and Sensitivity Anal- ysis of Dengue-Malaria Co-infection Model in Endemic Stage” Journal of the Nigerian Society of Physical Sciences 3 (2021) 96. [27] J.P. LaSalle, The stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1976. [28] J. Carr, Applications of Centre Manifold Theory Applied Mathematical Sciences. Springer-Verlag, New York, 35 (1982). [29] C. Castillo-Chavez & B. Song, “Dynamical models of tuberculosis and their applications”, Math Biosci Eng 1 (2004) 361. [30] Our World in Data: Prevalence, new cases and deaths from HIV/AIDS, Nigeria,1990 to 2019. Available at https://ourworldindata.org /grapher/deaths-and-new-cases-of-hiv?country=~NGA. Ac- cessed on 16/12/2022 [31] Our World in Data: Population, 10000 BCE to 2021. Available at https: //ourworldindata.org/grapher/population?country=~NGA. Accessed on 16/12/2022 Appendix A: HIV/AIDS Prevalence in Nigeria from 1990-2019 as given by [30] Table 2. Data of HIV/AIDS Prevalence in Nigeria from 1990-2019 [30] Year 1990 1991 1992 1993 1994 Cases 214934 307403 417550 541812 674511 Year 1995 1996 1997 1998 1999 Cases 808728 940842 1065300 1177623 1271363 Year 2000 2001 2002 2003 2004 Case 1347177 1406166 1449357 1479819 1500481 Year 2005 2006 2007 2008 2009 Cases 1515892 1527636 1542107 1558937 1581336 Year 2010 2011 2012 2013 2014 Cases 1609292 1638694 1670713 1707410 1752498 Year 2015 2016 2017 208 2019 Cases 1797982 1841027 1882445 1922997 1963044 Appendix B: Theorem (Castillo-Chavez and Song [29]). Consider the fol- lowing general system of ordinary differential equations with a parameter φ. d x dt = f (x,φ), f : Rn × R → Rn and f ∈ C2(Rn × R) where 0 is an equilibrium point of the system(that is, f (0,φ) = 0 for all φ) and 1. A = Dx f (0, 0) is the linearization matrix of the system around the equilibrium 0 with φ evaluated at 0; 2. Zero is a simple eigenvalue of A and all other eigenvalues of A have negative real parts; 3. Matrix A has a right eigenvector w and a left eigenvector v corresponding to the zero eigenvalue. Let fk be the kth component of f and a = n∑ k,i, j=1 vkwiw j ∂2 fk ∂xi∂x j (0, 0) b = n∑ k,i=1 vkwi ∂2 fk ∂xi∂φ (0, 0) Then the local dynamics of the system around the equilibrium point 0 is totally determined by the signs of a and b. i. a > 0, b > 0. When φ < 0 with |φ| � 1, 0 is locally asymptotically stable, and there exists a positive unsta- ble equilibrium; when 0 < φ � 1, 0 is unstable and there exists a negative and locally asymptotically stable equi- librium; ii. a < 0, b < 0. When φ < 0 with |φ| � 1, 0 is unstable; when 0 < φ � 1, 0 is locally asymptotically stable, and there exists a positive unstable equilibrium; iii. a > 0, b < 0. When φ < 0 with |φ| � 1, 0 is unstable, and there exists a locally asymptotically stable negative equilibrium; when 0 < φ � 1, 0 is stable, and a positive unstable equilibrium appears; iv. a < 0, b > 0. When φ changes from negative to positive, 0 changes its stability from stable to unstable. Correspond- ingly a negative unstable equilibrium becomes positive and locally asymptotically stable. 10