The type of incidence function used in epidemiological models is generally a matter of choice and convenience. Basic deterministic HIV/AIDS models with standard and saturated incidences are developed and extended to include three control measures, namely, public health educational campaigns, condom use, and treatment. The potential impact of these incidence functions on long-term projection of the disease dynamics is theoretically assessed, both qualitatively and quantitatively. Conditions for the stability of the model steady states are provided. The model with saturated incidence yields a higher number of secondary infections compared to the same outcome in the standard incidence formulation.
1. Introduction
HIV/AIDS prevention and control is a public health priority in light of the global pandemic and the elevated death toll (27 million since the disease was identified in 1981). To assess the impact of incidence functions in the estimation of the long-term dynamics of the disease, mathematical models of infectious diseases are useful tools for comparing control strategies and identifying key disease drivers as well as important areas of uncertainty that may be prioritized for urgent research. Large amount of work done on modeling the spread of HIV has been largely restricted to ordinary differential equations, though studies which have incorporated the combination of condom use, public health education campaigns, and treatment of infected individuals for eradication of the epidemic in the saturation incidence are uncommon (see [1–4] and the references therein).
Standard incidence models with constant total population are essentially mass action models [5], but reality is somewhere in between these two popular formulations. They appear to be the most widely used in the mathematical epidemiology repertoire, and some studies have suggested that the standard incidence formulation is more realistic for human diseases. To model the inhibition effect resulting from behavioral change or the crowding effect of infected individuals, neither mass action nor standard incidence functions is adequate. This is better captured using a saturation incidence formulation. The role of saturation and standard incidence functions on the dynamics of HIV is assessed with and without control measures.
Although the choice of one formulation over the other really depends on the disease being modeled and more often on the need for analytical tractability, it is imperative to assess the impact of this choice on initial disease threshold. We are not aware of any study that has investigated this impact, and it is our hope that this study will shed some light on whether using one formulation over the other over or underestimate the burden of the disease.
The rest of the paper is organized as follows: the model framework is presented in Section 2, followed by the basic model and its analysis in Section 3. The model with control measures is formulated and analyzed it Section 4, with graphical representations provided for illustration. Section 5 concludes the paper.
2. Model Framework
The proposed deterministic HIV/AIDS model subdivides the total population at time t, denoted by N(t), into susceptibles or HIV negatives, S(t), HIV positives or infectives who do not know their status, I1(t), HIV positives or infectives aware of their status, I2(t), pre-AIDS population, P(t), and full blown AIDS individuals, A(t). The total sexually active population size at time t is thus given by N(t)=S(t)+I1(t)+I2+P(t)+A(t). Some models have assumed that only a fraction of individuals in the AIDS class are sexually active but fail to give a clear estimate of this fraction [4, 6]. However, the long-term dynamics of the disease is not affected by this small withdrawal, and, for this reason, such categorization is not considered herein.
Suppose that β is the effective contact rate (i.e., the average number of contacts sufficient to transmit infection) per individual per unit time. Then βI1/N is the average number of contacts a susceptible individual makes with infective individuals per unit time. Therefore, the number of new infections coming from susceptibles is λS where λ=βI1/N represents the force of infection (this rate at which susceptible individuals contract the disease is measured by reference to the serological status of an individual through time with respect to antibodies specific to the infectious agent in question). When the effective contact rate β is constant, the force of infection is referred to as a standard incidence (frequency-dependent incidence), and I/N represents the fraction of the infected population. When the total population size N is quite large, since the number of contacts made by an infective per unit time should be limited or should grow less rapidly as the total population size N increases, the constant contact rate, β, may be more realistic [7].
However, the contact rate is not always linear. As the number of infectives rises, the number of susceptibles usually declines making it more difficult for contacts to be made. This means that there is a saturation effect in the contact rate as the number of infectives increases. The dynamics of an epidemic to a large extent is determined by how new infections are generated and the population mixing pattern. Incidence functions determine the rise and fall of epidemics. The incidence rate of a disease is the rate at which new cases of infection arise in a population and play an important role in the study of mathematical epidemiology [7].
The population mixes homogeneously, which means that susceptible individuals are equally likely to be infected by an infectious individual. Homogeneity is equivalent to assuming that the conditional mean distribution of parameters of interest is invariant across population subgroups. Nonlinearities can be approximated by a variety of forms. Xiao and Ruan [8] considered a saturation incidence rate of the form kIS/(1+αI2) where kI measures the infection force of the disease and 1/(1+αI2) describes the psychological or inhibitory effect from the behavioral change of the susceptible individuals when the number of infectives is very large (k,α>0). Liu et al. [9] focussed on the incidence rate λIpSq and noted that the global results obtained may be less generally applicable than the local results where p and q are positive parameters. Zhang and Ma [7] considered a saturation contact rate C(N)=bN/(1+bN+1+2bN) where b is a nonnegative parameter. In what follows, we will restrict our attention to a standard incidence function of the form βI/N and a saturation incidence function of the form βI/(1+αI) to investigate how the dynamics, in terms of the reproduction numbers changes with change in the incidence function. The force of infection for the standard incidence model is given by
(2.1)λst=β(I1+k1I2+k2P+k3A)N,
where β is the effective contact rate. The modification parameters k1<1,k3>1 and k2<k3 account for the relative infectivity of individuals in the I2,P, and A classes when compared to individuals in the I1 class. This implies that individuals in the A class are more infectious than those in the I1 class due to their higher viral load. Likewise, individuals in the I2 class tend to infect less compared to those in the I1 class because of their status awareness (individuals in the I2 class may choose to use preventive measures and change their behavior and thus may contribute little in spreading the infection). Individuals in the P class are less infectious than those in the A class because of their lower viral load but more infectious than those in the I1 and I2 classes. The modification parameters thus satisfy the following relation, k1<k2<k3.
In the model with saturated incidence rate, λsat (the force of infection at time t) is given by
(2.2)λsat=β(I1+k1I2+k2P+k3A)1+ω(I1+I2+P+A),
where 1/(1+ω(I1+I2+P+A)) measures the inhibition effect from behavioral change of susceptible individuals when their number increases or from the crowding effect of the infective individuals [8]. The nonnegative constant ω is the parameter that measures the extent of psychological or inhibitory effect (detrimental effect if 0<ω<1, beneficial or positive effect if ω>1). The nonmonotonic functions in (2.1) and (2.2) capture the psychological effects of increasing infectives in the population [10]. For a very large number of infective individuals, the force of infection may decrease as this number increases due to the fact that in the presence of large number of infectives, the population may tend to reduce the number of contacts per unit time [8].
The sexually mature susceptible individuals are recruited into the population at a constant rate Λ. This subpopulation is reduced by infection, following effective contact with infected individuals at the rate λst for the model with standard incidence rate. It is reduced further by natural death at a rate μ and emigration at a rate τ. Thus, the rate of change of susceptible individuals with time is given by
(2.3)dSdt=Λ-λS-(μ+τ)S,
with λ given by (2.1) and (2.2) for the standard incidence and the saturated incidence models, respectively.
Once an individual is infected, he/she becomes infectious and remain infectious (since treatment only suppresses the viral load). The population of HIV-positive individuals or infectives who do not know their status is increased by infection of susceptible individuals at the rate λ. The former is decreased by screening which leads to status awareness at the rate ρ, development of clinical symptoms, progression to full blown AIDS, and natural death at the rates θ,ν, and μ, respectively. This population is further decreased by emigration at a rate τ. Hence,
(2.4)dI1dt=λS-ρI1-θI1-νI1-(μ+τ)I1.
The population of HIV positives or infectives aware of their HIV status is generated by screening of unaware infectives at the rate ρ and decreased by development of clinical symptoms at the rate σ, development of full blown AIDS at the rate η, natural death at the rate μ, and emigration at the rate τ, so that
(2.5)dI2dt=ρI1-σI2-ηI2-(μ+τ)I2.
The pre-AIDS population is generated following development of clinical symptoms at the rate θ (for unaware infectives) and σ (for aware infectives). The model assumes that there is no emigration of pre-AIDS individuals to other countries because they are symptomatic and physically weak compared to I1 and I2 individuals. This subpopulation is diminished by progression to full blown AIDS at the rate α and natural death at the rate μ, so that
(2.6)dPdt=θI1+σI2-(α+μ)P.
Finally, the population of individuals with AIDS is increased by progression to full blown AIDS (at the rate ν for unaware infectives, η for aware infectives, α for pre-AIDS individuals). It is decreased by natural death at the rate μ and by disease-induced mortality at the rate δ. Thus,
(2.7)dAdt=νI1+ηI2+αP-(δ+μ)A.
3. The Basic Model
Putting the above formulations and assumptions together give the following system of differential equations for the transmission dynamics of HIV/AIDS. The associated model variables and parameters are described in Tables 1 and 2, respectively:
(3.1)dSdt=Λ-λS-(μ+τ)S,dI1dt=λS-ρI1-θI1-νI1-(μ+τ)I1,dI2dt=ρI1-σI2-ηI2-(μ+τ)I2,dPdt=θI1+σI2-(α+μ)P,dAdt=νI1+ηI2+αP-(δ+μ)A.
A schematic flow diagram of the model structure is depicted in Figure 1.
State variables of the HIV/AIDS basic model.
Symbol
Description
S(t)
Susceptibles or HIV negatives at time t
I1(t)
HIV-infected individuals unaware of their status
I2(t)
HIV-infected individuals aware of their status
P(t)
Pre-AIDS individuals at time t
A(t)
AIDS individuals at time t
Parameters of the HIV/AIDS basic model.
Symbol
Description
Λ
Recruitment rate of susceptibles into the sexually active population
β
Probability of transmission
μ
Natural mortality rate
δ
Disease-induced mortality rate
σ
Rate at which aware infectives develop symptoms
θ
Rate at which unaware infectives develop symptoms
ρ
Rate of status awareness due to screening method
α
Progression rate of pre-AIDS individuals to full blown AIDS
ν
Rate at which unaware infectives develop full blown AIDS
η
Rate at which aware infectives develop full blown AIDS
τ
Emigration rate of pre-AIDS and AIDS patients
ω
Measure of the psychological or inhibitory effect
The basic model compartments and flow.
Since system (3.1) monitors a hypothetical human population, it is assumed that all the state variables and parameters are nonnegative forallt≥0. The model is well defined in Γ={(S,I1,I2,P,A)∈ℝ+5:N(t)≤Λ/μ}, which is positively invariant and attracting [11].
3.1. The Basic Model with Standard Incidence
The basic model with standard incidence has a disease-free equilibrium given by
(3.2)E0=(Λμ+τ,0,0,0,0).
Its reproduction number defined as the expected number of secondary infections generated by a single infective introduced into a naive/susceptible population in its entire period of infectiousness plays a vital role in the control and eradication of the disease. The threshold condition for the persistence or eradication of a disease determines whether an infection can invade and persist in a population [12]. Using the next generation operator [13],
(3.3)R0st=βE1+βk1ρE1B+βk2(σρ+θB)E1BC+βk3[α(σρ+θB)+C(ηρ+νB)]E1BCD,
where
(3.4)E1=ρ+θ+ν+μ+τ,B=σ+η+μ+τ,C=μ+α,D=δ+μ.
Thus, from Theorem 2 in van den Driessche and Watmough [13], the following result holds.
Lemma 3.1.
The disease-free equilibrium E0 of the basic model with standard incidence is locally asymptotically stable if R0st<1 and unstable if R0st>1.
Lemma 3.1 implies that the disease can be eliminated from the population if the initial size of the subpopulations are in the basin of attraction of the disease-free equilibrium E0. The contributions of I1,I2,P, and A are
(3.5)R0I1=βρ+θ+ν+μ+τ,R0I2=βk1σ+η+μ+τ,R0P=βk2μ+α,R0A=βk3δ+μ,
respectively, where R0I1 is the contribution to the reproduction number by unaware infectives I1, R0I2 is the contribution to the reproduction number by aware infectives I2, R0P is the contribution to the reproduction number by pre-AIDS individuals P, and R0A is the contribution to the reproduction number by AIDS individuals A. The terms in (3.3) can be interpreted as follows.
1/(ρ+θ+ν+μ+τ),1/(σ+η+μ+τ),1/(μ+α), and 1/(δ+μ) are the average times an individual spends in either of the following disease classes I1,I2,P, and A, respectively.
ρ/(ρ+θ+ν+μ+τ) is the proportion of individuals who become aware of their infection by progression from compartment I1 to I2.
θ/(ρ+θ+ν+μ+τ) is the proportion of individuals who develop symptoms and progress from compartment I1 to P.
ν/(ρ+θ+ν+μ+τ) is the proportion of individuals who develop full blown AIDS from compartment I1.
σ/(σ+η+μ+τ) is the proportion of individuals who develop symptoms from compartment I2.
η/(σ+η+μ+τ) is the proportion of individuals who develop full blown AIDS from compartment I2.
α/(μ+α) is the proportion of individuals who develop full blown AIDS from compartment P.
The endemic equilibrium is obtained by solving the following system:
(3.1)Λ-λst*S*-(μ+τ)S*=0,λst*S*-(ρ+θ+ν+μ+τ)I1*=0,ρI1*-(σ+η+μ+τ)I2*=0.θI1*+σI2*-(μ+α)P*=0,νI1*+ηI2*+αP*-(δ+μ)A*=0,
where
(3.7)λst*=β(I1*+k1I2*+k2P*+k3A*)N*.
From (3.1), the steady-state expressions for I2*,P*, and A* in terms of I1* are given by
(3.8)I2*=ω1I1*,P*=ω2I1*,A*=ω3I1*,
where
(3.9)ω1=ρσ+η+μ+τ,ω2=θ+σω1μ+α,ω3=ν+ηω1+αω2δ+μ.
From (3.7),
(3.10)λst*=ϕI1*N*,whereϕ=β+βk1ω1+βk2ω2+βk3ω3.
Substituting for λst* in the second equation of (3.1), we have either I1*=0 or
(3.11)ϕS*-(ρ+θ+ν+μ+τ)N*=0.
The total population N at steady state can be written as
(3.12)N*=S*+ω4I1*,withω4=1+ω1+ω2+ω3.
Substituting for N* in (3.11), we obtain
(3.13)I1*=(ϑ-1ω4)S*,
where ϑ=ϕ/(ρ+θ+ν+μ+τ). After some little rearrangement, we obtain
(3.14)ϑ=R0st.
Substituting (3.10) and (3.13) in the first equation of (3.1) gives
(3.15)S*=ΛR0stω4ϕ(R0st-1)+R0stω4(μ+τ).
The first case I1*=0 results in the disease-free equilibrium point. The second case gives the endemic equilibrium point E*=(S*,I1*,I2*,P*,A*), where
(3.16)S*=ΛR0stω4ϕ(R0st-1)+R0stω4(μ+τ),I1*=(R0st-1ω4)(ΛR0stω4ϕ(R0st-1)+R0stω4(μ+τ)),I2*=ω1I1*,P*=ω2I1*,A*=ω3I1*.
We thus have the following result on the existence of the endemic equilibrium point.
Lemma 3.2.
If R0st≤1, system (3.1) has a unique disease-free equilibrium E0. If R0st>1, there exists a unique endemic equilibrium point E* whose coordinates are given by (3.16).
3.1.1. Local Stability of the Endemic Equilibrium
Here, the center manifold approach [14] as described by Theorem 4.1 in Castillo-Chavez and Song [15] will be used. To apply the said theorem, it is intuitive to rewrite system (3.1) after a change of variables: S=x1,I1=x2,I2=x3,P=x4,A=x5. In vector form, system (3.1) takes the form dX/dt=f(X), where X=[x1,x2,x3,x4,x5]T and [·]T denotes the matrix transpose. We can thus write the system as
(3.17)dx1dt=f1=Λ-β(x2+k1x3+k2x4+k3x5x1+x2+x3+x4+x5)x1-(μ+τ)x1dx2dt=f2=β(x2+k1x3+k2x4+k3x5x1+x2+x3+x4+x5)x1-(ρ+θ+ν+μ+τ)x2dx3dt=f3=ρx2-(σ+η+μ+τ)x3,dx4dt=f4=θx2+σx3-(μ+α)x4,dx5dt=f5=νx2+ηx3+αx4-(δ+μ)x5.
The Jacobian matrix of system (3.17) at the disease-free equilibrium is given by
(3.18)JE0=[-(μ+τ)-β-βk1-βk2-βk30β-E1βk1βk2βk30ρ-B000θσ-C00vηα-D].
Suppose that β=β* is chosen as a bifurcation parameter. Consider the case when R0st=1, then solving for β from R0st=1 gives
(3.19)β=β*=E1BCDBCD+k1ρCD+k2D(σρ+θB)+k3(α(σρ+θB)+C(ηρ+vB)).
We calculate the right and left eigenvectors associated with JE0. Note that 0 is a simple eigenvalue of JE0. From (3.18), we obtain the following equations:
(3.20)-(μ+τ)v1-βv2-βk1v3-βk2v4-βk3v5=0,(β-E1)v2+βk1v3βk2v4+βk3v-5=0,ρv2-Bv3=0,θv2+σv3-Cv4=0,νv2+ηv3+αv4-Dv5=0.
The Jacobian matrix (3.18) has a right eigenvector V=[v1,v2,v3,v4,v5]T associated with the zero eigenvalue given by
(3.21)v1=-z1v3,v2=Bv3ρ,v3=v3>0,v4=z1v3,v5=z2v3,
where
(3.22)z1=θB+σρCρ,z2=νB+ηρ+αρz1Dρ.B,C, and D are defined in (3.4). We then transpose the Jacobian matrix (3.18) to calculate the left eigenvector
(3.23)(JE0)T=[-(μ+τ)0000-ββ-E1ρθν-βk1βk1-Bση-βk2βk20-Cα-βk3βk300-D],
which leads to the following system of equations:
(3.24)-(μ+τ)η1=0,-βη1+(β-E1)η2+ρη3+θη4+νη5=0,-βk1η1+βk1η2-Bη3+ση4+ηη5=0,-βk2η1+βk2η2-Cη4+αη5=0,-βk3+βk3-D=0.
Thus, solving system (3.24) yields the following left eigenvector η=[η1,η2,η3,η4,η5]T associated with the zero eigenvalue, where
(3.25)η1=0,η2=-z5Dη3z4βk3,η3=η3>0,η4=z6η3,η5=-z5η3z4,z4=σ((β-E1)D+νβk3)-θ(βk1D+ηβk3)σβk3,z5=σρ+θBσ,z6=Bβk3+(βk1D+ηβk3)z5z4σβk3.
The local bifurcation analysis near β=β* is then determined by the signs of two associated constants, denoted herein by ψ1 and ψ2. In general, using β as the bifurcation parameter ensures that ψ2>0 [13], and, for this reason, the expression for ψ2 is not derived. ψ1 is given by
(3.26)ψ1=z5Dη3(μ+τ)σβk3v32Λk3(σβD+σνβk3-(σE1D+θ(βk1D+ηβk3)))(g1+g2+g3+g4),
where
(3.27)g1=1ρ[2βρ+1+k1+(1+k2)z1+(1+k3)z2],g2=(1+k1)βρ+2k1+(k1+k2)z1+(k1+k3)z2,g3=z1[(1+β2)βρ+k1+k2+k2z1+(k2+k3)z2],g4=z2[(1+k3)βρ+k1+k3+(k2+k3)z1+2k2z2],
and z1,z2,z5 are defined in (3.22) and (3.25). It can be shown that ψ1<0 if the following inequality holds:
(3.28)σβD+σνβk3<σE1D+θ(βk1D+ηβk3).
Since ψ1<0, this precludes the phenomenon of backward bifurcation, and, consequently, local and global stability follows, in which case elimination of the disease may be guaranteed via the available control measures. From the above mentioned, the following results are established.
Theorem 3.3.
The endemic equilibrium E* is locally asymptotically stable if R0st>1 and unstable otherwise.
Theorem 3.4.
The disease-free equilibrium of the basic model with standard incidence rate is globally asymptotically stable if R0st≤1.
3.2. The Basic Model with Saturated Incidence
The basic model with saturated incidence has the same disease-free equilibrium as the standard incidence model given by (3.2). Using the next generation operator method as described by van den Driessche and Watmougth [13] and the notation therein, the average number of new infections generated by a single infected individual in a completely susceptible population [16] for model system (3.1) is
(3.29)R0sat=γ6R0I1+γ7R0I2+γ8R0P+γ9R0A,
where
(3.30)R0I1=βρ+θ+ν+μ+τ,R0I2=βk1σ+η+μ+τ,R0P=βk2μ+α,R0A=βk3δ+μ,γ6=Λμ+τ,γ7=ργ6E1,γ8=(σρ+θB)γ6E1B,γ9=(α(σρ+θB)+C(ηρ+νB))γ6E1BC,
with the expressions of E1,B,C,D given by (3.4). Equation (3.29) can be rewritten as
(3.31)R0sat=Λμ+τR0st.
The reproduction number for the saturated incidence model is greater than that for the standard incidence model, that is, R0sat>R0st since Λ>(μ+τ); hence, the saturated incidence may overestimate the number of secondary infection compared to the same outcome when the standard incidence formulation is used. From Theorem 2 in van den Driessche and Watmough [13] and the fact that the disease-free equilibrium is the same as in (3.2), the following result holds.
Lemma 3.5.
The disease-free equilibrium (3.2) of the basic model with saturated incidence rate is both locally and globally asymptotically stable if R0sat<1 and unstable otherwise.
It can be shown that the saturated incidence model has no endemic equilibrium when R0sat≤1. Thus, we claim the following result.
Theorem 3.6.
The disease-free equilibrium of the basic model with saturated incidence is globally asymptotically stable if R0sat≤1.
The consequence of Theorem 3.6 vis-a-vis backward bifurcation is that the saturated incidence model does not exhibit backward bifurcation; consequently, the endemic equilibrium is unique and is globally asymptotically stable whenever R0sat>1. This concludes the analysis of the basic models. The next section extends these models by incorporating some control measures.
4. HIV/AIDS Model with Control Measures
Three control measures, namely, public health educational campaigns, condom use, and treatment of infected individuals, are incorporated into the basic model (3.1). In the following, we describe the additional variables and parameters added to the extended model. Educated and individuals under treatment are denoted by the variables E(t) and T(t), respectively. Susceptible individuals are recruited into the sexually active population at the rate, Λ, a proportion b of which is assumed to be educated but susceptibles and move to the educated class, E. The complementary proportion, (1-b), is susceptible and moves to the susceptible class, S, who is educated at a constant rate, r, and moves into the E class. Susceptible individuals acquire infection upon effective contact with an infected individual at the rate λ0 and move to the class I1 of infected individuals who are not aware of their infection. Educated individuals are infected at the rate (1-k)λ0 and move to the class I1, where k measures the overall effectiveness of the public health educational campaign, 0<k<1. The two extreme values are excluded because k=0 implies that education is useless, while k=1 implies that education is completely effective.
A proportion p of individuals use condoms. Since individuals in the I2-class are aware of their infection, they seek treatment at the rate ρ1. Infected individuals who have developed symptoms of the disease seek treatment at the rate, ρ2. Individuals in the T class interact with the rest of the community, but they are less infectious than those in the I1 class, because the use of treatment significantly reduces the viral load [5]. k4 is the relative infectivity of individuals in the I1 class and 0<k4<1 (0 means the drug is useless, and 1 means the drug is completely effective in stopping HIV transmission). They develop full blown AIDS at the rate α1 with 0<α1<1 and move to the AIDS class A. Further, there is a constant emigration of educated susceptibles and those on treatment at the rate τ. From the aforesaid, the forces of infection are now defined as
(4.1)λst0=β(I1+k1I2+k2P+k4T+k3A)N,λsat0=β(I1+k1I2+k2P+k4T+k3A)1+ω(I1+I2+P+T+A)
for the standard incidence and saturated incidence models, respectively, with k4<k1<k2<k3. The model flow chart is shown in Figure 2. The additional state variables and parameters are described, respectively, in Tables 3 and 4. The above and previous assumptions lead to the following system of nonlinear ordinary differential equations describing the disease dynamics when control measures are implemented:
(4.2)dSdt=(1-b)Λ-λ0S(1-p)-(r+μ+τ)S,dEdt=bΛ+rS-λ0E(1-k)(1-p)-(μ+τ)E,dI1dt=λ0S(1-p)+λ0E(1-k)(1-p)-(ρ+ν+θ+μ+τ)I1,dI2dt=ρI1-(σ+ρ1+η+μ+τ)I2,dPdt=θI1+σI2-(ρ2+μ+α)P,dTdt=ρ1I2+ρ2P-(α1+μ+τ)T,dAdt=νI1+ηI2+αP+α1T-(δ+μ)A,
with initial conditions S(0)=S0,E(0)=E0,I1(0)=I10,I2(0)=I20,P(0)=P0,T(0)=T0,A(0)=A0.
Additional state variables of the model with control measures.
Symbol
Description
E(t)
Educated individuals at time t
T(t)
Treated individuals at time t
Additional parameters of the model with control measures.
Symbol
Description
b
Proportion of educated individuals
k
Education effectiveness
p
Proportion of individuals who use condoms
r
Rate at which susceptible individuals are educated
ρ1
Rate at which aware infectives seek treatment
ρ2
Rate at which pre-AIDS/asymptomatic individuals seek treatment
α1
Rate at which treated individuals develop full blown AIDS
τ
Emigration rate of educated and treated patients
HIV/AIDS model structure with control measures.
4.1. Model with Standard Incidence
The disease-free equilibrium E02 of (4.2) is
(4.3)E02={(1-b)Λr+μ+τ,bΛ(μ+τ)+Λr(μ+τ)(r+μ+τ),0,0,0,0,0}.
Again, using the next generation matrix operator [13], the control measure (public health educational campaigns, condom use, and treatment) induced effective reproduction number of the model with standard incidence denoted by Rest is given by
(4.4)Rest=γ1ReI1+γ2ReI2+γ3ReP+γ4ReT+γ5ReA,
where
(4.5)ReI1=βρ+θ+ν+μ+τ,ReI2=βk1σ+ρ1+η+μ+τ,ReP=βk2ρ2+μ+α,ReT=βk4α1+μ+τ,ReA=βk3δ+μ,γ1=(1-p)((1-b)(μ+τ)+(b(μ+τ)+r)(1-k)r+μ+τ),γ2=γ1ρE1,γ4=γ1(ρ1ρC1+ρ2θB1+ρ2σρ)E1B1C1,γ3=γ1(θB1+σρ)E1B1,γ5=γ1(νlC1B1+ηρlC1+αθlB1+αlσρ+α1ρ1ρC1+α1ρ2θB1+α1ρ2σρ)E1B1C1l,B1=B+ρ1,C1=C+ρ2,l=α1+μ+τ.B,C, and D are defined in (3.4) above. ReI1 is the contribution to the effective reproduction number by unaware infectives I1,ReI2 is the contribution to the effective reproduction number by aware infectives I2,ReP is the contribution to the effective reproduction number by pre-AIDS individuals P,ReT is the contribution to the effective reproduction number by individuals who are under treatment T, and ReA is the contribution to the effective reproduction number by individuals with full blown AIDS. When there are no control measures, that is b=r=k=p=ρ1=ρ2=α1=0,Rest reduces to R0st whose expression appears in (3.3). Thus, from Theorem 2 in van de Driessche and Watmough [13], we claim the following result.
Lemma 4.1.
The disease-free equilibrium E02 of the model with control measures and standard incidence is locally asymptotically stable whenever Rest<1 and unstable otherwise.
The threshold quantity Rest is the reproduction number for the standard incidence model which measures the average number of new HIV infections generated by a single HIV-infected individual in a population where some individuals are educated, a proportion using condoms, and others are receiving treatment.
Existence of the endemic equilibrium
At steady state, the variables of the standard incidence model can be expressed in terms of λst0** as follows:
(4.6)S**=(1-b)Λλst0**(1-p)+(r+μ+τ),I1**=jλst0**(1-p)E1,E**=bΛλst0**d+e+ra(λst0**d+e)(λst0**(1-p)+c),I2**=jρλst0**(1-p)B1E1,P**=jf2λst0**(1-p)E1,T**=jf3λst0**(1-p)E1,A**=jf4λst0**(1-p)DE1,
where
(4.7)λst0**=β(I1*+k1I2*+k2P*+k4T*+k3A*)N*,j={aλst**(1-p)+c+f1λst0**d+e+drΛ(λst0**(1-p)+c)(λst0**d+e)},a=(1-b)Λ,c=r+μ+τ,d=(1-k)(1-p),e=μ+τ,f1=(1-k)bΛ,f2=θC1+σρC1B1,f3=ρρ1lB1+ρ2lf2,f4=ν+ηρB1+αf2+α1f3,l=α1+μ+τ.
Substituting (4.6) into the expression for λst0** at steady state and after a little rearrangement, we obtain
(4.8)g(λst0**)=b11λst0**2+b21λ0**+b31=0,
where
(4.9)b11=f5ab1d+f1f5b12,b21=aE1d+b1bΛE1+f5b1ae+f5f1b1c+f5b1drΛ-(ab1df6+f6f1b12),b31=-b1drΛf6+raE1+aE1c+bΛE1c(1-Restb),f5=1+ρB1+f2+f3+f4D,f6=β+βk1ρB1+βk2f2+βk4f3+βk3f4D.
Both b11>0 and b21>0 if ab1df6+f6f1b12<aE1d+b1bΛE1+f5b1ae+f5f1b1c+f5b1drΛ, while b31>0 if Rest<1 and raE1+aE1c+bΛE1c(1-(Rest/b))>b1drΛf6. By the Routh-Hurwiz criterion, the quadratic (4.8) has no positive root; hence, no endemic equilibrium exists when Rest<1. When Rest=1, b31>0 if b1drΛf6+ΛE1c<raE1+aE1e+bΛE1c, then by the Routh-Hurwitz criterion, the quadratic (4.8) has no positive root; thus, no endemic equilibrium exists when Rest=1. The case when Rest>1 makes b31<0 if b1drΛf6+bΛE1c(1-(Rest/b))>raE1+aE1e; in this case, the quadratic (4.8) has two roots with opposite signs (the negative root is epidemiologically meaningless). Hence, the following results are established.
Lemma 4.2.
The standard incidence model has no endemic equilibrium when Rest≤1.
Theorem 4.3.
If Rest≤1, the disease-free equilibrium is globally asymptotically stable and unstable if Rest>1.
4.2. Model with Saturated Incidence
This model has the same disease-free equilibrium E02 (see (4.3)) as that of the model with standard incidence. Using the next generation operator method [13], the effective reproduction number of the model with saturated incidence when the control measures are in place is given by
(4.10)Resat=ΛRestμ+τ.
Similar to the qualitative result obtained in the basic model analysis, the effective reproduction number for the saturated incidence is greater than that of standard incidence. That is Resat>Rest since Λ>(μ+τ). In the absence of public health educational campaigns, condom use, and treatment (that is, b=r=k=p=ρ1=ρ2=α1=0), Resat simply reduces to R0sat. Hence, from Theorem 2 in [13], the following result is established.
Lemma 4.4.
The disease-free equilibrium E02 of the HIV/AIDS model with control measures and saturated incidence is locally asymptotically stable for Resat<1 and unstable otherwise.
The threshold quantity Resat is the reproduction number for the saturated incidence model which measures the average number of new HIV infections generated by a single HIV-infected individual in a population where the three control measures (condom use, education and treatment) are implemented concurrently.
Existence of endemic equilibria
At the endemic steady state, the variables of the saturated incidence model can be expressed in terms of λsat0** as follows:
(4.2)S**=(1-b)Λλsat0**(1-p)+(r+μ+τ),I1**=j2λsat0**(1-p)E1,E**=bΛλsat0**d+e+ra(λsat0**d+e)(λsat0**(1-p)+c),I2**=j2ρλsat0**(1-p)B1E1,P**=j2f2λsat0**(1-p)E1,T**=j2f3λsat0**(1-p)E1,A**=j2f4λsat0**(1-p)FE1,
where
(4.12)j2={aλsat0**(1-p)+c+f1λsat0**d+e+drΛ(λsat0**(1-p)+c)(λsat0**d+e)},λsat0**=β(I1*+k1I2*+k2P*+k4T*+k3A*)1+ω(I1*+I2*+P*+T*+A*),
and a,c,d,e,f1,f2,f3,f4,f5,f6,B1,C1,l,D are as defined in (4.7). Substituting (4.2) into the expression for λsat0 (at steady state) and after some rearrangements, we obtain
(4.13)z1(λsat0**)=a1λsat0**2+a2λsat0**+a3=0,
where
(4.14)a1=E1d(1-p)+ωf5ad(1-p)+ωf5f1(1-p)2,a2=E1c(1-p)+E1e(1-p)+ωf5a(1-p)+ωf5f1c(1-p)+ωf5drΛ(1-p)-(f6ad(1-p)+f6f1(1-p)2),a3=E1ec(1-Resat)-f6drΛ(1-p).
It is evident that a1>0,a2>0 if f6ad(1-p)+f6f1(1-p)2<E1c(1-p)+E1e(1-p)+ωf5a(1-p)+ωf5f1c(1-p)+ωf5drΛ(1-p), while a3>0 if Resat<1 and f6drΛ(1-p)<E1ec(1-Resat). Thus, by the Routh-Hurwitz criterion, the quadratic in (4.13) has no positive root. The case Resat=1 implies a3<0, and, therefore, the quadratic in (4.13) now reads a1λsat0**2+a2λsat0**-a3=0, so that λsat0**=(-b±b2+4ac)/2a, where a=a1,b=a2, and c=-a3. For Resat>1,a3<0, in this case, the quadratic has two roots with opposite signs (the negative root is biologically meaningless, hence the uniqueness of the endemic equilibrium). Thus, we have established the following results.
Lemma 4.5.
The saturated incidence model has no endemic equilibrium when Resat<1.
Theorem 4.6.
If Resat<1, the disease-free equilibrium is globally asymptotically stable and unstable if Resat>1.
As a consequence of Theorem 4.6, the use of appropriate control measures can greatly minimize the outbreak burden and eliminate the disease in the community. The figures are generated using MatLab. In an uncontrolled outbreak, the basic reproduction number is significantly greater than the effective disease threshold number. In both basic and full models, we started the simulation at t0=0 signifying the start of implementation of control strategies. The model demographic/epidemiological data are tabulated in Table 5. The initial conditions used in the model simulations are S0=100,000;E0=70,000;I10=50,000;I20=30,000;P0=35,000;T0=20,000;A0=25,000. Some of the parameter values are assumed for the purpose of illustration.
Models parameter values.
Parameters
Value
Reference
Λ
5000 people yr^{−1}
Assumed
μ
0.0196 yr^{−1}
Assumed
β
3 yr^{−1}
Assumed
τ
0.55 yr^{−1}
Assumed
ρ
0.47 yr^{−1}
Assumed
θ
0.57 yr^{−1}
Assumed
ν
0.45 yr^{−1}
Assumed
σ
0.36 yr^{−1}
Assumed
η
0.21 yr^{−1}
Assumed
α
0.22 yr^{−1}
Assumed
δ
0.33 yr^{−1}
[3]
r
0.15 yr^{−1}
[3]
ρ1
0.57 yr^{−1}
Assumed
ρ2
0.32 yr^{−1}
Assumed
α1
0.18 yr^{−1}
Assumed
k1,k2,k3,k4
0.023, 0.15, 0.48, 0.0016
Assumed
p
0.53
[3]
b
0.2
[3]
k
0.6
[3]
ω
4.0
[8]
Dynamics of the basic model
We simulate the basic model (3.1) with standard incidence and saturated incidence, respectively. Its time series evolution is graphically shown in Figure 3, which depicts the dynamics of the sexually active population when the basic reproduction number is greater than unity for the standard and saturated incidence as shown in Figures 3(a) and 3(b), respectively. It is evident from these Figures 3(a) and 3(b) that the disease becomes endemic for both incidence functions.
The basic model with (a) standard incidence and (b) saturated incidence.
Dynamics of the model with control measures
To assess the impact of the three control measures, the model is simulated with public health educational campaigns, condom use, and treatment as the only control measures for a time frame of 50 years (Figure 4). Figure 4(a) shows that there is a steady decrease in the susceptible population due to education and infection. Susceptible individuals join the educated susceptible population, and some are infected resulting into an increase of educated susceptibles before a further decrease due to infection. The populations of aware and unaware infectives decrease due to progression to further stages of infection and treatment of aware infectives. The number of individuals receiving treatment increases when there are more aware infectives and pre-AIDS individuals and decreases when these numbers are low. Figure 4(b) shows that susceptibles, educated susceptibles, unaware infectives, pre-AIDS, and AIDS individuals tend to their steady-state values and aware infectives approach zero. The system settles at an endemic steady-state, and the disease persists in the population, but the infection has been reduced since the reproduction number has been decreased to Rest=1.0159 for the standard incidence and Resat=5.302 for the saturated incidence. Thus, the control measures introduced are modest and some extra efforts are still needed to stem the tide of the epidemic by bringing the effective reproduction number to less than unity for the standard incidence, but the infection is still very high regardless of the effect of the aforementioned control measures, whence the need for investing in and implementing other control strategies.
Population dynamics, when intervention is instituted for 50 years, with (a) standard incidence and (b) saturated incidence.
Changes in unaware infectives with change in incidence function
Figure 5(a) shows that the number of unaware infectives decreases almost exponentially over the first 10 years and approaches a steady-state value. Figure 5(b) shows an increase in the measure of the psychological or inhibitory effect, ω, results in a decrease in the number of unaware infectives.
Change of unaware infectives with (a) standard incidence and saturated incidence for the basic model and (b) standard incidence and saturated incidence for the intervention model.
Prevalence of the disease for the standard incidence
The prevalence graph, Figure 6(a), increases with a greater gradient for a while and then stabilizes over time, which means that the disease becomes endemic in the population. It is observed in Figure 6(b) that the prevalence curve increases then drops asymptotically but does not approach zero due to reduced number of susceptible individuals over time as most susceptibles join the educated susceptible class and others become affected by the disease. This means that the disease becomes endemic in the population regardless of the effect of the current three interventions in place.
Prevalence of the disease for the (a) basic and (b) intervention models with standard incidence.
Prevalence of the disease for the saturated incidence
The numerical simulations for assessing the effects of increasing the saturation rate, ω, are given in Figure 7. An increase in the saturation rate results in a decrease in the prevalence of the infection. The contact rate is decreasing due to increasing the saturation rate but it does not go to zero, implying that the disease becomes endemic in the population despite the impact of the three interventions in place.
Prevalence of the disease for the (a) basic and (b) intervention models with saturated incidence.
5. Conclusion
A basic deterministic model of the transmission dynamics of HIV/AIDS was formulated to investigate the role of the incidence function in curtailing an outbreak of the disease. Three control measures including public health educational campaigns, condom use, and treatment were incorporated into the basic model to assess their potential impact on the transmission dynamics of the disease with change of incidence function. The qualitative features of the models were investigated.
The model has two equilibria, the disease-free equilibrium and the endemic equilibrium. Theoretical results show that the disease-free equilibrium is stable (locally and globally) when R0st≤1 and R0sat≤1 for standard incidence and saturated incidence, respectively. If both R0st and R0sat are greater than unity, then the disease will persist in the population. In this case, a unique endemic equilibrium exists, is locally and globally asymptotically stable for both models. The condition under which the disease persists in the population with saturated incidence is also derived. It was also shown that the basic reproduction number for the saturated incidence model is greater than that for the standard incidence model, that is, R0sat>R0st. The local stability of the endemic equilibrium point was proved for the standard incidence model using the center manifold theory. The models did not exhibit the phenomenon of backward bifurcation where a stable disease-free equilibrium coexists with a stable endemic equilibrium for a certain range of associated reproduction number less than unity for both incidence functions. Consequently, both the disease-free and endemic equilibria for these models are stable (both locally and globally) when Rest,Resat≤1 and when Rest,Resat>1, respectively. Note that Resat>Rest.
Numerical simulations were obtained using some demographic data from the literature and the remaining parameter values assumed for the purpose of illustration. The saturation impact is measured by the parameter ω. Though the basic as well as the effective reproduction numbers, R0sat and Resat, do not depend on ω explicitly, numerical simulations indicate that the number of infectives decreases as ω increases (Figure 7). The larger the value of ω is, the smaller the inhibitory effect is, and vice-versa. An increase in ω leads to a reduction in the prevalence of the disease.
Our results highlight the potential impact the incidence functions may play on the behavioral dynamics of the model and underscore the need for further work on the incidence functions used in HIV models. The number of HIV-infected individuals has slowed considerably following a rapid and aggressive information campaign and implementation of various control strategies in recent years. Prevalence has also fallen after peaking in the mid-1990s; consequently, a saturation incidence model currently seems to make more epidemiological sense than the standard incidence for modeling the current HIV/AIDS epidemic. However, caution should be paid in choosing and interpreting results as the choice of the incidence function can greatly influence the results by either over or underestimating the measure of initial disease spread. Whenever data is available, fitting the models to data, even though different incidence functions would probably lead to different parameter values can as well provide some information given experts opinion on the potential true course of the epidemic on which form of the incidence function to choose. This is a daunting task for most epidemic, especially given that, more often, data are not readily available and hard to come by.
Acknowledgment
E. L. Kateme acknowledges with thanks the support in part of the the Norad's programme for Master Studies (in Mathematical Modelling) at the University of Dar es Salaam.
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