Stability analysis of HIV / AIDS dynamics: Modelling the tested and untested populations
AJOY DUTTA1,2and PRAVEEN KUMAR GUPTA1 ,∗
1Department of Mathematics, National Institute of Technology Silchar, Silchar 788 010, India
2Department of Mathematics, Pandit Deendayal Upadhyaya Adarsha Mahavidyalaya, Biswanath 784 184, India
∗Corresponding author. E-mail: pkguptaitbhu@gmail.com, praveen@math.nits.ac.in MS received 14 July 2020; revised 17 August 2021; accepted 28 September 2021
Abstract. In this manuscript, the dynamic behaviour of a nonlinear HIV/AIDS dynamics model using HIV infected population is proposed. Here, we divide the HIV infected population into two subclasses: the tested and untested HIV infected populations. The novel part of the model is that when susceptible population interacts with the untested HIV infected population, the susceptible population is shifted to untested HIV infected population.
Otherwise, it is transferred to tested HIV infected population, while many researchers have taken this infection as tested HIV infected population. For infection-free equilibrium point, we determine the basic reproduction number and explore the existence and local stability of equilibrium point. For two endemic equilibrium states, we investigate the positivity and stability of equilibrium points and determine the conditions where it exists. The Routh–Hurwitz criterion and Bellman and Cooke’ theorem are used to establish the stability of the non-infected and two endemic equilibrium states. Numerical simulations for all equilibria are also carried out to examine the behaviour of the system in different dynamical regimes.
Keywords. HIV/AIDS dynamics model; tested and untested HIV infected populations; stability analysis;
Bellman–Cooke’s theorem.
PACS Nos 02.30.Hq; 02.60.Cb; 02.70.c; 05.45.a
1. Introduction
The last three decades’ research shows that majority of the HIV infections come from undiagnosed and untreated populace, but predominantly from unaware people [1–4]. The human awareness is the main conse- quence of decrease in sensitivity of infection. Resear chers have created a good collection of information about the disease, as well as a numeral of critical tools and intervention to identify, prevent and treat the dis- ease. Generally, HIV infection transfers from an infected person to the victim through blood transfusion and sex- ual intercourse. It has been estimated that approximately 80% of HIV infection resulted through abnormal sexual intercourse [5]. The history of mathematical modelling shows that modelling is an important tool to analyse the spread and control of HIV disease. The objective of any modelling section is to provide as much information as possible from the available data and to suggest a precise picture of both the knowledge and ambiguity about the real problem [6–8]. The study of HIV transmission and
the dynamics of the disease have been of great interest to both applied mathematicians and biologists.
The literature survey shows that many mathematical models have been developed and analysed to study the effects of information and awareness on the spread and control of HIV/AIDS disease [3,9,10]. Safielet al[11]
examined the effect of screening and treatment on the transmission of HIV/AIDS infection in a population, and observed that the screening of untested HIV infected population and treatment of screened HIV infected pop- ulation reduces the transmission of the disease. Royet al [3] presented plans which suggest the impact of human awareness programs in the reduction of susceptibility to infection. Misraet al[1] studied the effect of awareness programs via newspaper and social media on epidemic disease in a variable size population with immigration through mathematical model. Recently, Kauret al[12]
studied the consequence of awareness and counselling on the transmission of HIV and the spread of HIV infection in the endemic region using a nonlinear math- ematical model. They revealed that with some suitable 0123456789().: V,-vol
Figure 1. Flow diagram of the deterministic model of HIV/AIDS.
choice of parameters corresponding to awareness, screening and counselling, the disease can be controlled within a reasonable duration of time.
In this article, we study the power of media and aware- ness programs organised by government/public sectors, and investigate the effect of awareness program in the proposed model for disease outbreaks. We separated the entire infected HIV population into two sub-classes: ‘the untested class’ and ‘tested class’. This article is organ- ised as follows: A detailed description of the proposed model and basic properties are discussed in §2 and §3 has given the analysis of the model. In §4, numerical simulations have been performed using Mathematica software to demonstrate the role of important param- eters in the model.
2. Deterministic model of HIV/AIDS
2.1 System description
In this article, we have considered the infection of HIV/AIDS, spread in a host population of four classes:
S(t) is the size of the susceptible population at time t; I1(t) is the size of the untested HIV infected pop- ulation at time t; I2(t) is the size of the tested HIV infected population at timetandA(t)is the size of the AIDS population at timet. In the anticipated model, we have proposed that when untested HIV infected pop- ulation interacts with the susceptible population, then susceptible population have been shifted to untested HIV infected population, while many researchers [1,5, 10,13,14] have taken this population as the tested HIV infected population.
The schematic diagram (figure1) leads to the follow- ing system of ordinary differential equations:
dS dt =r −
δ1γ1I1
N +δ2γ2I2
N
S−dS (1)
dI1
dt = δ1γ1I1S
N +σI1−(k1+d+τ)I1 (2) dI2
dt = δ2γ2I2S
N +τI1−(k2+d)I2 (3) dA
dt =k1I1+k2I2−(k3+d)A (4) with
S(0)=S0, I1(0)= I10, I2(0)= I20, A(0)= A0, (5) where N(t) is the total population at time t, r is the rate of recruitment of susceptible population into the total population,k1 is the conversion rate of untested HIV infected population to AIDS population,k2 is the conversion rate of tested HIV infected population to AIDS population,k3is the AIDS-induced mortality rate, d is the natural death rate, δ1 is the contact rate of untested HIV infected population,δ2is the contact rate of the tested HIV infected population, γ1 and γ2 are the numbers of various parameters (e.g., sexual part- ners, syringe, blood etc.) of the untested and tested HIV infected individual, respectively. Since untested HIV infected population can easily be in contact with these factors while tested HIV infected population is more conscious,γ1 > γ2. Here,σ is the rate of recruit- ment of infective immigrants into the untested infected population, and also some untested HIV infected popu- lation will stay in the same class due to the awareness programs (running by NGOs, Government and many more agencies for controlling the HIV/AIDS), is the conversion rate of untested to tested HIV infected pop- ulation. If the size of the total population is taken as N = S+ I1 +I2 + A, the proposed model (1)–(5) is reduced as follows:
dN
dt =r+σI1−k3A−d N (6)
dI1
dt = δ1γ1I1(N −I1−I2− A)
+σI1−(k1N+d+τ)I1 (7) dI2
dt = δ2γ2I2(N −I1−I2−A)
+τI1−(k2+Nd)I2 (8) dA
dt =k1I1+k2I2−(k3+d)A (9) with
N(0)= N0, I1(0)= I10, I2(0)= I20, A(0)= A0. (10) 2.2 Basic properties
2.2.1 Non-negative solutions. When we study the dynamics of the population model, it is extremely impor- tant to make sure that the dynamics prediction about the model is closer to the real world. In this sequence, the first objective is that the solutions should always be pos- itive. Therefore, we establish the following lemmas with the help of [2]:
Lemma 2.1. The exact solutionsN(t),I1(t),I2(t),A(t) of model(6)–(10)with the given initial valuesN(0) >
0,I1(0)≥0,I2(0)≥0,A(0)≥0are non-negative for allt>0(see[1,15,16]).
Proof. From eq. (1), the expression is defined as d
dt
S(t)exp
t
0
δ1γ1I1(μ)
N(μ) +δ2γ2I2(μ) N(μ) +d
dμ
=r exp
t
0
δ1γ1I1(μ)
N(μ) +δ2γ2I2(μ) N(μ) +d
dμ
(11) or
S(t)exp t
0
δ1γ1I1(μ)
N(μ) +δ2γ2I2(μ) N(μ) +d
dμ
−S(0)= t
0
r exp
x 0
δ1γ1I1(μ) N(μ) +δ2γ2I2(μ)
N(μ) +d
dμ
dx. Hence
S(t)=exp
− t
0
δ1γ1I1(μ)
N(μ) +δ2γ2I2(μ) N(μ) +d
dμ
× t
0
r exp
x 0
δ1γ1I1(μ)
N(μ) +δ2γ2I2(μ) N(μ) +d
dμ
dx+S(0)exp
− t
0
δ1γ1I1(μ) N(μ) +δ2γ2I2(μ)
N(μ) +d
dμ
>0. (12)
Subsequently, we can also illustrate I1(t) ≥ 0, I2(t) ≥ 0 and A(t) ≥ 0. Therefore, the solutions S(t),I1(t),I2(t),A(t)of model (1)–(5) are always pos- itive for all t > 0. Hence, N(t)is also positive for all
t >0.
2.2.2 Invariant regions.
Lemma 2.2. The feasible regionis defined as
= (N(t),I1(t),I2(t),A(t))∈ R+4,0< N(t)≤ r d
,
(13) where R4+denotes the non-negative cone and its lower dimensional faces are positive invariant whenσI1 ≤ k3A, for model(6)–(10) with initial conditions in R4+ (see[1,15,16]).
Proof. From eq. (6) of model (6)–(10), we write dN
dt =r+σI1−k3A−dN or
dN
dt +dN ≤r if
σI1≤k3A. It follows that N(t)≤ r
d −N0e−dt,
whereN0represents the initial values of the total popula- tion. Thus, limt→∞Sup(N(t))≤r
d. It implies that the region = (N(t),I1(t),I2(t),A(t))∈ R4+,0< N(t)≤ r
d is a positively invariant set for model (6)–(10), when
σI1≤k3A.
3. Analysis of the model
3.1 The non-infected steady state and the basic reproduction number
It is easy to determine that there exists the following non-infected steady state for model (6)–(10):
E0=(N0∗,I10∗,I20∗,A∗0)=r
d,0,0,0
. (14)
Now, we shall illustrate the basic reproduction num- ber for model (6)–(10) by using next generation matrix method [17]. Let X = (I10∗,I20∗,A∗0,N0∗). Model (6)–(10) can be rewritten as
dX
dt =M(X)−N(X), (15)
whereM(X)is the rate of appearance of new infections, N(X)is the rate of transfer of individuals and
M(X)=
⎡
⎢⎢
⎢⎢
⎢⎣
δ1γ1I1(N−I1−I2−A)
N +σI1
δ2γ2I2(N −I1−I2− A) N0
0
⎤
⎥⎥
⎥⎥
⎥⎦
N(X)=
⎡
⎢⎣
(k1+d+τ)I1
(k2+d)I2−τI1
(k3+d)I2−k1I1−k2I2
−r −σI1+k3+d N
⎤
⎥⎦. (16)
The Jacobian matrices ofM(X)andN(X)at the non- infected equilibrium point (E0 ) are
D M(E0)=
⎡
⎢⎣
γ1δ1+σ 0 0 0 0 γ2δ2 0 0
0 0 0 0
0 0 0 0
⎤
⎥⎦,
D N(E0)=
⎡
⎢⎣
k1+d+τ 0 0 0
−τ k2+d 0 0
−k1 −k2 k3+d 0
−σ 0 k3 d
⎤
⎥⎦.
(17) Thus, the basic reproduction number, denoted by0
[16,17], for model (6)–(10) is calculated as 0 =max
δ1γ1+σ
k1+d+τ, δ2γ2
k2+d
=max(1,2) (18) where
1 = δ1γ1+σ k1+d+τ and
2 = δ2γ2
k2+d.
3.2 Stability analysis of the non-infected steady state Theorem 3.1. The non-infected steady state E0 is locally asymptotically stable in if 0 < 1. Other- wise, it is unstable.
Proof. For non-infected steady state E0, the Jacobian matrix of model (6)–(10) is
J(E0)=
⎡
⎢⎣
−d σ 0 k3
0 δ1γ1+σ −(k1+d+τ) 0 0
0 τ δ2γ2−(k2+d) 0
0 k1 k2 −(k3+d)
⎤
⎥⎦. (19)
The characteristic equation of the Jacobian matrix then becomes
(λ+d)(λ+k3+d)(λ2+a1λ+a2)=0 (20) where
a1 =(k1+d+τ)−(δ1γ1+σ)+k2+d −δ2γ2
=(k1+d+τ)(1− 1)+(k2+d)(1− 2), a2 = [(k1+d +τ)−(δ1γ1+σ)](k2+d −δ2γ2)
=(k1+d+τ)(1− 1)(k2+d)(1− 2).
From eq. (20), two values ofλare−dand−(k3+d). The other two values ofλare illustrated by the quadratic equation
(λ2+a1λ+a2)=0. (21)
Now, from eqs (18) and (20), 0 < 1 and therefore 1 < 1 and2 < 1. Hence,a1 > 0,a2 > 0. There- fore, the remaining two values ofλare also positive by Routh–Hurwitz criterion. As a result, when 0 < 1, the non-infected steady state E0 is locally asymptoti- cally stable. Biologically, it means that early precaution and awareness of HIV/AIDS is a very useful control
strategy.
3.3 Stability analysis of the endemic equilibrium states Apart from a non-infected steady stateE0, by successful computation, two more endemic equilibrium points in model (6)–(10) exist, which are as follows:
E1=(N1∗,I11∗,I21∗,A∗1) for 0 >1 and
E2=(N2∗,I12∗,I22∗,A∗2) for 0>1.
In a very rare situation, if there is no translation rate of tested HIV infected population to AIDS population (i.e.,k2 =0) then the first endemic point E1 will exist,
and for this special case we analyse the behaviour of model (6)–(10), where
N1∗= r
d, I11∗ =0, I12∗ = r(δ2γ2−d) dδ2γ2
and
A∗1 =0. (22)
From eqs (18) and (22), we clearly observe that the populations of equilibrium point E1, are either zero or positive ifδ2γ2−d >0, i.e.,0>1.
Theorem 3.2. The endemic equilibrium pointE1of sys- tem (6)–(10) is locally asymptotically stable if δ2γ2 − d >0andk1+d+τ > σ +d(δ1γ1/δ2γ2).
Proof. For infected steady stateE1, the Jacobian matrix of model (6)–(10) is
J(E1)=
⎡
⎢⎢
⎢⎢
⎢⎢
⎣
−d σ 0 −k3
0 σ −(k1+d+τ)+d δ1γ1
δ2γ2
0 0
(δ2γ2−d)2
δ2γ2 τ −(δ2γ2−d) −(δ2γ2−d)−(δ2γ2−d)
0 k1 0 −(k3+d)
⎤
⎥⎥
⎥⎥
⎥⎥
⎦
. (23)
The characteristic equation of the Jacobian matrix (23) is
(λ+d)×(λ+(k3+d))×(λ+(δ2γ2−d))
×
λ+k1+d+τ −σ−d δ1γ1
δ2γ2
=0. (24) Thus, by direct calculation, we can easily examine that all four eigenvalues (λ)−d,(k3+d),−(δ2γ2−d)and
−(k1+d+τ)+σ+d(δ1γ1/δ2γ2)are negative if(δ2γ2− d) > 0 and k1 +d +τ > σ +d(δ1γ1/δ2γ2), i.e., 0 > 1. As a result, the infected steady state E1 is
locally asymptotically stable.
Theorem 3.3 [13]. LetH(λ)=h(λ,eλ), whereh(λ,eλ) is a polynomial with a principal term. The function H(i q)is now separated into real and imaginary parts, that is, we set H(i q)= F(q)+i G(q). If all the zeros of the functionH(λ)lie to the left side of the imaginary axis, then the zeros of the functionsF(q)andG(q)are real, alternating, and
F(q)G(q)−F(q)G(q) >0, (25) for eachq. In addition, in order that all the zeros of the function lie to the left of the imaginary axis, it is sufficient that one of the following conditions be satisfied:
(i) All the zeros of the functions F(q) and G(q) are real and alternative and inequality (25) is satisfied for at least one value ofq.
(ii) All the zeros of the function F(q)are real and, for each zero, inequality (25) is satisfied, i.e., F(q)G(q) <0.
(iii) All the zeros of the functionG(q)are real and for each of these zeros, inequality(25)is satisfied, i.e., F(q)G(q) >0.
In another situation, if the translation rate of the tested HIV infected population to AIDS population is contin- uous, then one more endemic point E2 will exist, and for this case we also analyse the behaviour of model (6)–(10), where
N2∗= rδ1γ1
D [(k1(d+k2)+(d+τ)(d+k3) +k2(d+τ +k3))δ1γ1+(σ −d−τ−k1) +(d+k1+k3)δ2γ2)]
I12∗ = 1
D[(r(d+k3)(d +τ +k1)(1−1)
×((d+k2)δ1γ1+(d+τ +k1)(1−1)δ2γ2))]
I22∗ = δ1γ1τr
D [(d+k3)(d+τ +k1)(1−1)]
A∗2 = 1
D[(r(d+τ+k1)(1−1)(τk2δ1γ1
+k1((d+k2)δ1γ1+(σ−d−τ −k1)δ2γ2))]
D=(d−σ +τ +k1)[(d+k2)(d+k3)γ12δ12
+(σ −d−τ−k1)(dσ +(σ −k1)k3)γ2δ2
+γ1δ1(d(dσ +(σ −k1)k3) +k2(dσ +(σ −τ −k1)k3)
+(σ −d−k1)(d +k3)γ2δ2)]. (26) From eqs (18) and (26), we clearly observe that the populations of equilibrium pointE2are positive ifσ− d−τ −k1 >0 and D>0.
According to Theorem 3.3, for model (6)–(10), the characteristic equation of the Jacobian matrix at endemic point E2 declares that the eigenvalues are real and negative. A detailed discussion is given in Appendix. It is specified by the inequality
F(0)G(0)−F(0)G(0) >0, (27)
Table 1. Variation of conversion rate from untested to tested HIV infected population.
r k1 k2 k3 σ τ γ1 δ1 γ2 δ2 d J Remark
10 0.4 0.5 1 1 0.1 0.2 5 0.06 4 0.02 0.1777524 Stable
10 0.4 0.5 1 1 0.2 0.2 5 0.06 4 0.02 0.0803119 Stable
10 0.4 0.5 1 1 0.3 0.2 5 0.06 4 0.02 0.0343881 Stable
10 0.4 0.5 1 1 0.4 0.2 5 0.06 4 0.02 0.0133213 Stable
10 0.4 0.5 1 1 0.5 0.2 5 0.06 4 0.02 0.0041797 Stable
10 0.4 0.5 1 1 0.6 0.2 5 0.06 4 0.02 0.0006140 Stable
10 0.4 0.5 1 1 0.7 0.2 5 0.06 4 0.02 −0.0004830 Unstable
10 0.4 0.5 1 1 0.8 0.2 5 0.06 4 0.02 −0.0005999 Unstable
which is the stability condition for the endemic equilib- rium state E2. From eqs (A.2)–(A.5) of Appendix, the inequality (27) becomes,
J =F(0)G(0) >0.
According to Bellman and Cooke [18], the endemic equilibrium state E2 will be stable, when J is greater than zero.
From table1, we observe that if we increase the con- version rate of the untested infected to the tested infected population, i.e., τ (0 < τ < 1), then Bellman coeffi- cient (J) decreases from positive to negative with one threshold point. From Theorem 3.3, equilibrium point is unstable ifJ is negative, otherwise it is stable. There- fore, if we assume 0 < τ < 0.6, the solution will be stable. Thus, we conclude that endemic equilibrium state E2 is stable when we control the conversion rate from untested to tested HIV infected population, i.e., with the help of awareness programmes, we can control the disease.
4. Numerical simulations
In this section, the numerical results for three different sets of parameters of system (6)–(10) are presented for supporting the analytic results obtained in §3. We have considered the parameter values r = 10, k1 = 0.4, k2 = 0.5, k3 = 1, σ = 0.5, τ = 0.6, γ1 = 0.1, δ1 = 5, γ2 = 0.05,δ2 = 2, d = 0.02 and the initial values N(0) = 5000, I1(0) = 2000, I2(0) = 1000, A(0) = 500 from [5]. Consequently, we determine the reproduction number 0 = 0.9803 from eq. (18), which is less than one. The numerical solutions of model (6)–(10) are plotted in figure2. It is straightforwardly demonstrated that for the given parameter values and initial conditions, the solution of model (6)–(10) asymp- totically approaches the non-infected equilibrium point E0(500,0,0,0).
Figure2a shows that the total population decreases within a very short time and tends to positive equilib- rium pointN0∗. Figure2b confirms that the untested HIV infected population decreases drastically with increase in time and it tends to the equilibrium pointI10∗. Figure2c shows that for the first few days the tested HIV infected population increases swiftly and later on decreased rad- ically to the equilibrium point I20∗. Similarly, figure2d clearly demonstrates that in the first two days, the popu- lation of AIDS increases, thereafter it decreases rapidly and tends to the equilibrium point A∗0.
Now, we have chosen a different set of parameter val- ues, i.e., r = 10,k1 = 0.375,k2 = 0,k3 = 1, σ = 0.5, τ = 0.6, γ1 = 0.1, δ1 = 5, γ2 = 0.0101, δ2 = 2,d = 0.02. We calculated the reproduction number 0 = 1.0050 from eq. (18), which is greater than one.
In this special case, we studied the dynamical behaviour of semi-endemic equilibrium state E1(500,0,4.95,0) fork2 = 0, where the tested HIV infected populations are not transferring directly in AIDS population. The numerical solutions of model (6)–(10) are exhibited in figure3. This figure can be used as an evidence for the specific parameter values and initial conditions, which are taken as above. In this case, the solution of the model approaches asymptotically at the first endemic equilib- rium pointE1, which is also stable.
Figure3a shows that total population decreases expo- nentially with increase in time and tends to the equilib- rium pointN1∗. Figure3b confirms that the untested HIV infected population decreases drastically within a very short time period, after which it tends to the equilibrium point I10∗. Figure3c demonstrates that the first week is very crucial for the population because in this period the tested HIV infected population increases, later it decreases exponentially and tends to the equilibrium pointI20∗. Figure3d shows that for the first few days, the AIDS population increases, later on it decreases drasti- cally and tends to the equilibrium point A∗0.
Now, for one more different set of parameter val- ues, i.e. r = 10,k1 = 0.375,k2 = 0.5,k3 = 1, σ = 0.5, τ = 0.6, γ1 = 0.1, δ1 = 5, γ2 =
Figure 2. Dynamical behaviour of model (6)–(10) for 0 = 0.9803 < 1. In this case, the HIV/AIDS will not persist.
Figure 3. Dynamical behaviour of model (6)–(10) for 0 = 1.0050 > 1, whenk2 = 0. In this case, the HIV infection continues but AIDS will not persist.
Figure 4. Dynamical behaviour of model (6)–(10) for 0 =1.0050, whenk2 = 0.5. In this case, the HIV/AIDS persists.
0.01, δ2 = 2,d = 0.02, we have calculated the reproduction number = 1.0050 from eq. (18), which is larger than one. In this case, we studied the dynamical behaviour of the endemic equilibrium stateE2(466.32,1.47,1.77,1.41), where no population died. The numerical results are plotted in figure4.
Figure4a clearly shows that within a few days the total compartmental population decreases drastically with increase in time. Thereafter, it tends to the equilibrium pointN0∗. Similarly, figure4b confirms that initially the untested HIV infected population decreases drastically, and after that it tends to the equilibrium pointI10∗. Fig- ure4c shows that for the first two days, the tested HIV infected population increases, later on it decreases rad- ically and tends to the equilibrium pointI20∗. Figure4d reveals that for the first two days, the AIDS population increases, later on it decreases very fast and tends to the equilibrium pointA∗0.
Biologically, we conclude from figure 2 that if we increase the number of awareness programs in the soci- ety, then tested and untested HIV infected population, and AIDS population decreases with increase in time and disease is stable. Figure 3 reflects that for some special condition (k2 = 0 ) the untested HIV infected population and AIDS population will not persist but the tested HIV infected populations will exist. So, in this situation HIV infected population can live with HIV for a long time. Similarly, figure4shows that all the tested and untested HIV infected and AIDS population will persist with increase in time but the susceptible pop- ulation is always more than the sum of the remaining population whenk2 = 0.5. Therefore, we can say the age of the infected population has increased.
5. Conclusion and discussion
In this study, a novel nonlinear mathematical model is described to study the effect of untested and tested HIV infected populations with variable size population during the spread of HIV/AIDS, where susceptible pop- ulation is divided into both untested and tested HIV infected populations. During the analysis, we obtained one non-infected equilibrium and two endemic equilib- rium points. The stability analysis segment of the paper shows that non-infected equilibrium point is locally asymptotically stable for0 <1, and the endemic equi- librium points will persist when0 >1. We have used Routh–Hurwitz criterion to establish the local stability of both non-infected equilibrium and endemic equi- librium E0(500,0,0,0), E1(500,0,4.95,0), respec- tively. And for another endemic equilibrium point E2(466.32,1.47,1.77,1.41), we have analysed the sta- bility by Bellman and Cooke’s theorem. To support the
analytical results, we have carried out numerical solu- tions for sets of parameter values which are depicted by figures 2–4, for non-infected equilibrium point E0, semi-endemic equilibrium pointE1and purely endemic equilibrium point E2, respectively. Figures 3 and 4 demonstrate that the semi- and purely endemic points E1andE2are globally stable when0 >1.
The analytical and numerical results of the model confirm that the rate of recruitment of HIV infective immigrants into the unaware infected population and the number of AIDS population decrease, if we increase the awareness programs and advertisement through any medium in the society, which confirm the physical jus- tification of the reality. Finally, we conclude that the successful investigations of unaware and aware HIV infected populations have a major impact on falling rates of HIV/AIDS. This has happened because of the awareness of the HIV-infected population to take neces- sary precautionary measures to prevent the spread of the
disease. Awareness programs that pursue these strate- gies can successfully shrink the spread of HIV/AIDS in a population.
Acknowledgements
The first author Ajoy Dutta is extending his gratitude to the Ministry of Human Resource and Development, Govt. of India for providing him the Ph.D. fellowship under the supervision of Dr. Praveen Kumar Gupta. The authors are also very thankful to the editor and anoma- lous reviewers for his/her valuable suggestions.
Appendix
The characteristic equation of the Jacobian matrix at the endemic stateE2for model (6)–(10) is
H(λ)=
λ4+λ3
k1+k2+k3+ N2(4d−σ +τ)+(2I12−N2)γ1δ1+(2I22−N2)γ2δ2
N2
+λ2 1
N22(3d N22(2d−σ +τ)+3d N22k3−N22σk3+N22τk3+6d I12N2γ1δ1−3d N22γ1δ1
−AI12σγ1δ1−I122σγ1δ1−I12I22σγ1δ1+I12N2τγ1δ1+2I12N2k3γ1δ1−N22k3γ1δ1
+γ2((2I22−N2)N2(3d−σ+τ +k3)+(3I12I22−2(I12+I22)N2+N22)γ1δ1)δ2
+N2k2(N2(3d−σ +τ)+N2k3+(2I22−N2)γ1δ1+I22γ2δ2)+N2k1(3d N2+N2k2+N2k3
+I12γ1δ1+(2I22−N2)γ2δ2))
+λ 1
N23(d2N23(4d−3σ −3τ)+3d2N23k3−2d N23σk3
+2d N23τk3+6d2N22I12γ1δ1−3d2N23γ1δ1−2AI12N2σγ1δ1−2d I12N1σγ1δ1−2d I12I22σγ1δ1
+2d I1N12τγ1δ1+4d I1N12k3γ1δ1−2d N13k3γ1δ1−AI1N1σk3γ1δ1−I122 N2σk3γ1δ1
−I22I12N2σk3γ1δ1+I12N22τk3γ1δ1+γ2((2I22−N2)N22(d(3d −2σ +2τ)+(2d −σ +τ)k3)) +(2d N2(3I22I12−2(I22+I12)N2+N22)+I12(A+I22+I12)(−I12+N2)σ +N2(3I22I12
−2(I22+I12)N2+N22)k3)γ1δ1)δ2+N2k1(d N2+(3d N2+2I12γ1δ1)+γ2(2d(2I22−N2)N2
+I12(I22−N2)γ1δ1)δ2+N2k2(2d N2+N2k3+I12γ1δ1+I22γ2δ2) +k3(2d N22+I12(A+I22+I12)γ1δ1+(2I22−N2)N2γ2δ2))
+N2k2(k3(N22(2d−σ+τ)+γ1δ1(2I22−N2)N2+I22(A+I22+I12)γ2δ2) +γ1δ1(2d(2I12−N2)N2−I12(A+I22+I12)σ+I12N2τ +I22(I12−N2)γ2δ2) +N2(d N2(3d−2σ +2τ)+I22(2d−σ +τ)γ2δ2)
+ 1
N23
k1(N2(d +k2)(d N22(d+k3) +I12(d N2+(A+I22+I12)k3)γ1δ1)+γ2(d(2I22−N2)N22(d +k3)
+I22N2k2(d N2+(A+I22+I12)k3)+I12(I12−N2)(d N2+(A+I22+I12)k3)γ1δ1)δ2) +(d+k3)(d N22(d−σ+τ)(d N2+(2I12−N2)γ2δ2)+γ1δ1(−d N2(d N2(−2I12+N2)
+I12(A+I22+I12)σ−I12N2τ)+(d N2(3I22I12−2(I22+I12)N2+N22)+I12(A+I22+I12)
(−I12+N2)σ)γ2δ2))+k2(k3(d N23(d−σ +τ)+N2(d(2I12−N2)N2
−I12(A+I22+I12)(σ−τ))γ1δ1+I22(A+I22+I12)γ2(N2(d −σ +τ) +(I12−N2)γ1δ1)δ2)+d N2(N2(d−σ+τ)(d N2+I22γ2δ2)γ1δ1
×(d(2I12−N2)N2−I12(A+I22+I12)σ +I12N2τ+I22(I12−N2)γ2δ2)))
. (A.1)
As a result of Theorem 3.3, first we substituteλ=i q into eq. (A.1), and we write H(i q) = F(q)+i G(q), whereF(q)andG(q)are the real and imaginary parts of H(i q). To define zeros, substituteq =0 and calculate F(0),G(0),F(0)andG(0), as follows:
F(0)= 1
N23(k1(N2(d+k2)(d N22(d+k3) +I12(d N2+(A+I22+I12)k3)γ1δ1) +γ2(d(2I22−N2)N22(d+k3) +I22N2k2(d N2+(A+I22+I12)k3) +I12(I22−N2)(d N2+(A+I22+I12)k3)
×γ1δ1)δ2)+(d +k3)(d N22(d−σ +τ)(d N2
+(2I12−N2)γ2δ2)+γ1δ1(−d N2(d N2(−2I12
+N2)+I12(A+I22+I12)σ
−I12N2τ)+(d N2(3I22I12−2(I22+I12)N2 +N22)+I12(A+I22+I12)(−I12
+N2)σ)γ2δ2))+k2(k3(d N23
×(d−σ +τ)+N2(d(2I12−N2)N2
−I12(A+I22+I12)(σ −τ))γ1δ1
+I22(A+I22+I12)γ2(N2(d−σ+τ) +(I12−N2)γ1δ1)δ2)
+d N2(N2(d−σ+τ)(d N2+I22γ2δ2) +γ1δ1(d(2I12−N2)N2
−I12(A+I22+I12)σ
+I12N2τ +I22(I12−N2)γ2δ2))) (A.2)
G(0)=0 (A.3)
F(0)=0 (A.4)
G(0)= 1
N23(d2N23(4d−3σ −3τ)
+3d2N23k3−2d N23σk3+2d N23τk3
+6d2N22I12γ1δ1−3d2N23γ1δ1
−2Ad I12N2σγ1δ1−2d I12N1σγ1δ1
−2d I12I22N1σγ1δ1+2d I1N12τγ1δ1
+4d I1N12k3γ1δ1−2d N13k3γ1δ1
−AI1N1σk3γ1δ1−I122 N2σk3γ1δ1
−I22I12N2σk3γ1δ1+I12N22τk3γ1δ1
+γ2((2I22−N2)N22(d(3d−2σ +2τ)
+(2d−σ+τ)k3))+(2d N2(3I22I12
−2(I22+I12)N2+N22) +I12(A+I22+I12)(−I12
+N2)σ +N2(3I22I12−2(I22+I12)N2
+N22)k3)γ1δ1)δ2+N2k1(d N2+(3d N2 +2I12γ1δ1)+γ2(2d(2I22−N2)N2 +I12(I22−N2)γ1δ1)δ2+N2k2(2d N2
+N2k3+I12γ1δ1+I22γ2δ2)
+k3(2d N22+I12(A+I22+I12)γ1δ1+(2I22
−N2)N2γ2δ2))+N2k2(k3(N22(2d−σ +τ) +γ1δ1(2I22−N2)N2+I22(A+I22
+I12)γ2δ2)+γ1δ1(2d(2I12
−N2)N2−I12(A+I22+I12)σ +I12N2τ +I22(I12
−N2)γ2δ2)+N2(d N2(3d−2σ +2τ) +I22(2d−σ+τ)γ2δ2))
. (A.5)
For model (6)–(10), the characteristic eq. (A.1) shows that the eigenvalues are real and negative if it follows the inequality,
F(0)G(0)−F(0)G(0) >0 (A.6) which is the stability condition. From eqs (A.2)–(A.5), inequality (A.6) becomes
J ≡ F(0)G(0) >0. (A.7)
In [18], Bellman and Cooke state that the equilibrium point will be stable when J >0.
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