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ISRN Applied Mathematics
Volume 2012 (2012), Article ID 864795, 25 pages
http://dx.doi.org/10.5402/2012/864795
Research Article

HIV/AIDS Dynamics with Three Control Strategies: The Role of Incidence Function

1Mathematics Department, University of Dar es Salaam, Dar es Salaam, Tanzania
2Department of Applied Mathematics, National University of Science and Technology, Bulawayo, Zimbabwe

Received 3 March 2012; Accepted 29 April 2012

Academic Editors: H. Akçay, C. Lu, and G. Psihoyios

Copyright © 2012 Emmanuelina L. Kateme et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

Abstract

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 [14] 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 𝜆st=𝛽𝐼1+𝑘1𝐼2+𝑘2𝑃+𝑘3𝐴𝑁,(2.1) 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 𝜆sat=𝛽𝐼1+𝑘1𝐼2+𝑘2𝑃+𝑘3𝐴𝐼1+𝜔1+𝐼2+𝑃+𝐴,(2.2) 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 𝑑𝑆𝑑𝑡=Λ𝜆𝑆(𝜇+𝜏)𝑆,(2.3) 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, 𝑑𝐼1𝑑𝑡=𝜆𝑆𝜌𝐼1𝜃𝐼1𝜈𝐼1(𝜇+𝜏)𝐼1.(2.4) 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 𝑑𝐼2𝑑𝑡=𝜌𝐼1𝜎𝐼2𝜂𝐼2(𝜇+𝜏)𝐼2.(2.5) 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 𝑑𝑃𝑑𝑡=𝜃𝐼1+𝜎𝐼2(𝛼+𝜇)𝑃.(2.6) 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, 𝑑𝐴𝑑𝑡=𝜈𝐼1+𝜂𝐼2+𝛼𝑃(𝛿+𝜇)𝐴.(2.7)

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: 𝑑𝑆𝑑𝑡=Λ𝜆𝑆(𝜇+𝜏)𝑆,𝑑𝐼1𝑑𝑡=𝜆𝑆𝜌𝐼1𝜃𝐼1𝜈𝐼1(𝜇+𝜏)𝐼1,𝑑𝐼2𝑑𝑡=𝜌𝐼1𝜎𝐼2𝜂𝐼2(𝜇+𝜏)𝐼2,𝑑𝑃𝑑𝑡=𝜃𝐼1+𝜎𝐼2(𝛼+𝜇)𝑃,𝑑𝐴𝑑𝑡=𝜈𝐼1+𝜂𝐼2+𝛼𝑃(𝛿+𝜇)𝐴.(3.1) A schematic flow diagram of the model structure is depicted in Figure 1.

tab1
Table 1: State variables of the HIV/AIDS basic model.
tab2
Table 2: Parameters of the HIV/AIDS basic model.
864795.fig.001
Figure 1: 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 forall𝑡0. The model is well defined in Γ={(𝑆,𝐼1,𝐼2,𝑃,𝐴)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 𝐸0=Λ𝜇+𝜏,0,0,0,0.(3.2) 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], 𝑅st0=𝛽𝐸1+𝛽𝑘1𝜌𝐸1𝐵+𝛽𝑘2(𝜎𝜌+𝜃𝐵)𝐸1+𝐵𝐶𝛽𝑘3[]𝛼(𝜎𝜌+𝜃𝐵)+𝐶(𝜂𝜌+𝜈𝐵)𝐸1,𝐵𝐶𝐷(3.3) where 𝐸1=𝜌+𝜃+𝜈+𝜇+𝜏,𝐵=𝜎+𝜂+𝜇+𝜏,𝐶=𝜇+𝛼,𝐷=𝛿+𝜇.(3.4) 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 𝑅st0<1 and unstable if 𝑅st0>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 𝑅𝐼10=𝛽𝜌+𝜃+𝜈+𝜇+𝜏,𝑅𝐼20=𝛽𝑘1,𝑅𝜎+𝜂+𝜇+𝜏𝑃0=𝛽𝑘2𝜇+𝛼,𝑅𝐴0=𝛽𝑘3,𝛿+𝜇(3.5) respectively, where 𝑅𝐼10 is the contribution to the reproduction number by unaware infectives 𝐼1, 𝑅𝐼20 is the contribution to the reproduction number by aware infectives 𝐼2, 𝑅𝑃0 is the contribution to the reproduction number by pre-AIDS individuals 𝑃, and 𝑅𝐴0 is the contribution to the reproduction number by AIDS individuals 𝐴. The terms in (3.3) can be interpreted as follows.(i)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. (ii)𝜌/(𝜌+𝜃+𝜈+𝜇+𝜏) is the proportion of individuals who become aware of their infection by progression from compartment 𝐼1 to 𝐼2.(iii)𝜃/(𝜌+𝜃+𝜈+𝜇+𝜏) is the proportion of individuals who develop symptoms and progress from compartment 𝐼1 to 𝑃.(iv)𝜈/(𝜌+𝜃+𝜈+𝜇+𝜏) is the proportion of individuals who develop full blown AIDS from compartment 𝐼1.(v)𝜎/(𝜎+𝜂+𝜇+𝜏) is the proportion of individuals who develop symptoms from compartment 𝐼2.(vi)𝜂/(𝜎+𝜂+𝜇+𝜏) is the proportion of individuals who develop full blown AIDS from compartment 𝐼2.(vii)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: Λ𝝀st𝑆(𝜇+𝜏)𝑆𝝀=0,st𝑆(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,(3.1) where 𝜆st=𝛽𝐼1+𝑘1𝐼2+𝑘2𝑃+𝑘3𝐴𝑁.(3.7) From (3.1), the steady-state expressions for 𝐼2,𝑃, and 𝐴 in terms of 𝐼1 are given by 𝐼2=𝜔1𝐼1,𝑃=𝜔2𝐼1,𝐴=𝜔3𝐼1,(3.8) where 𝜔1=𝜌𝜎+𝜂+𝜇+𝜏,𝜔2=𝜃+𝜎𝜔1𝜇+𝛼,𝜔3=𝜈+𝜂𝜔1+𝛼𝜔2.𝛿+𝜇(3.9) From (3.7), 𝜆st=𝜙𝐼1𝑁,where𝜙=𝛽+𝛽𝑘1𝜔1+𝛽𝑘2𝜔2+𝛽𝑘3𝜔3.(3.10) Substituting for 𝜆st in the second equation of (3.1), we have either 𝐼1=0 or 𝜙𝑆(𝜌+𝜃+𝜈+𝜇+𝜏)𝑁=0.(3.11) The total population 𝑁 at steady state can be written as 𝑁=𝑆+𝜔4𝐼1,with𝜔4=1+𝜔1+𝜔2+𝜔3.(3.12) Substituting for 𝑁 in (3.11), we obtain 𝐼1=𝜗1𝜔4𝑆,(3.13) where 𝜗=𝜙/(𝜌+𝜃+𝜈+𝜇+𝜏). After some little rearrangement, we obtain 𝜗=𝑅st0.(3.14) Substituting (3.10) and (3.13) in the first equation of (3.1) gives 𝑆=Λ𝑅st0𝜔4𝜙𝑅st01+𝑅st0𝜔4.(𝜇+𝜏)(3.15) 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 𝑆=Λ𝑅st0𝜔4𝜙𝑅st01+𝑅st0𝜔4,𝐼(𝜇+𝜏)1=𝑅st01𝜔4Λ𝑅st0𝜔4𝜙𝑅st01+𝑅st0𝜔4,𝐼(𝜇+𝜏)2=𝜔1𝐼1,𝑃=𝜔2𝐼1,𝐴=𝜔3𝐼1.(3.16) We thus have the following result on the existence of the endemic equilibrium point.

Lemma 3.2. If 𝑅st01, system (3.1) has a unique disease-free equilibrium 𝐸0. If 𝑅st0>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 𝑑𝑥1𝑑𝑡=𝑓1𝑥=Λ𝛽2+𝑘1𝑥3+𝑘2𝑥4+𝑘3𝑥5𝑥1+𝑥2+𝑥3+𝑥4+𝑥5𝑥1(𝜇+𝜏)𝑥1𝑑𝑥2𝑑𝑡=𝑓2𝑥=𝛽2+𝑘1𝑥3+𝑘2𝑥4+𝑘3𝑥5𝑥1+𝑥2+𝑥3+𝑥4+𝑥5𝑥1(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.(3.17) The Jacobian matrix of system (3.17) at the disease-free equilibrium is given by 𝐉𝐄𝟎=(𝜇+𝜏)𝛽𝛽𝑘1𝛽𝑘2𝛽𝑘30𝛽𝐸1𝛽𝑘1𝛽𝑘2𝛽𝑘30𝜌𝐵000𝜃𝜎𝐶00𝑣𝜂𝛼𝐷.(3.18) Suppose that 𝛽=𝛽 is chosen as a bifurcation parameter. Consider the case when 𝑅st0=1, then solving for 𝛽 from 𝑅st0=1 gives 𝛽=𝛽=𝐸1𝐵𝐶𝐷𝐵𝐶𝐷+𝑘1𝜌𝐶𝐷+𝑘2.𝐷(𝜎𝜌+𝜃𝐵)+𝑘3(𝛼(𝜎𝜌+𝜃𝐵)+𝐶(𝜂𝜌+𝑣𝐵))(3.19) 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: (𝜇+𝜏)𝑣1𝛽𝑣2𝛽𝑘1𝑣3𝛽𝑘2𝑣4𝛽𝑘3𝑣5=0,𝛽𝐸1𝑣2+𝛽𝑘1𝑣3𝛽𝑘2𝑣4+𝛽𝑘3𝑣5=0,𝜌𝑣2𝐵𝑣3=0,𝜃𝑣2+𝜎𝑣3𝐶𝑣4=0,𝜈𝑣2+𝜂𝑣3+𝛼𝑣4𝐷𝑣5=0.(3.20) The Jacobian matrix (3.18) has a right eigenvector 𝑉=[𝑣1,𝑣2,𝑣3,𝑣4,𝑣5]𝑇 associated with the zero eigenvalue given by 𝑣1=𝑧1𝑣3,𝑣2=𝐵𝑣3𝜌,𝑣3=𝑣3>0,𝑣4=𝑧1𝑣3,𝑣5=𝑧2𝑣3,(3.21) where 𝑧1=𝜃𝐵+𝜎𝜌𝐶𝜌,𝑧2=𝜈𝐵+𝜂𝜌+𝛼𝜌𝑧1𝐷𝜌.(3.22)𝐵,𝐶, and 𝐷 are defined in (3.4). We then transpose the Jacobian matrix (3.18) to calculate the left eigenvector 𝐉𝐄𝟎𝐓=(𝜇+𝜏)0000𝛽𝛽𝐸1𝜌𝜃𝜈𝛽𝑘1𝛽𝑘1𝐵𝜎𝜂𝛽𝑘2𝛽𝑘20𝐶𝛼𝛽𝑘3𝛽𝑘300𝐷,(3.23) which leads to the following system of equations: (𝜇+𝜏)𝜂1=0,𝛽𝜂1+𝛽𝐸1𝜂2+𝜌𝜂3+𝜃𝜂4+𝜈𝜂5=0,𝛽𝑘1𝜂1+𝛽𝑘1𝜂2𝐵𝜂3+𝜎𝜂4+𝜂𝜂5=0,𝛽𝑘2𝜂1+𝛽𝑘2𝜂2𝐶𝜂4+𝛼𝜂5=0,𝛽𝑘3+𝛽𝑘3𝐷=0.(3.24) Thus, solving system (3.24) yields the following left eigenvector 𝜂=[𝜂1,𝜂2,𝜂3,𝜂4,𝜂5]𝑇 associated with the zero eigenvalue, where 𝜂1=0,𝜂2=𝑧5𝐷𝜂3𝑧4𝛽𝑘3,𝜂3=𝜂3>0,𝜂4=𝑧6𝜂3,𝜂5=𝑧5𝜂3𝑧4,𝑧4=𝜎𝛽𝐸1𝐷+𝜈𝛽𝑘3𝜃𝛽𝑘1𝐷+𝜂𝛽𝑘3𝜎𝛽𝑘3,𝑧5=𝜎𝜌+𝜃𝐵𝜎,𝑧6=𝐵𝛽𝑘3+𝛽𝑘1𝐷+𝜂𝛽𝑘3𝑧5𝑧4𝜎𝛽𝑘3.(3.25)

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 𝜓1=𝑧5𝐷𝜂3(𝜇+𝜏)𝜎𝛽𝑘3𝑣23Λ𝑘3𝜎𝛽𝐷+𝜎𝜈𝛽𝑘3𝜎𝐸1𝐷+𝜃𝛽𝑘1𝐷+𝜂𝛽𝑘3𝑔1+𝑔2+𝑔3+𝑔4,(3.26) where 𝑔1=1𝜌2𝛽𝜌+1+𝑘1+1+𝑘2𝑧1+1+𝑘3𝑧2,𝑔2=1+𝑘1𝛽𝜌+2𝑘1+𝑘1+𝑘2𝑧1+𝑘1+𝑘3𝑧2,𝑔3=𝑧11+𝛽2𝛽𝜌+𝑘1+𝑘2+𝑘2𝑧1+𝑘2+𝑘3𝑧2,𝑔4=𝑧21+𝑘3𝛽𝜌+𝑘1+𝑘3+𝑘2+𝑘3𝑧1+2𝑘2𝑧2,(3.27) and 𝑧1,𝑧2,𝑧5 are defined in (3.22) and (3.25). It can be shown that 𝜓1<0 if the following inequality holds: 𝜎𝛽𝐷+𝜎𝜈𝛽𝑘3<𝜎𝐸1𝐷+𝜃𝛽𝑘1𝐷+𝜂𝛽𝑘3.(3.28) 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 𝑅st0>1 and unstable otherwise.

Theorem 3.4. The disease-free equilibrium of the basic model with standard incidence rate is globally asymptotically stable if 𝑅st01.

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 𝑅0sat=𝛾6𝑅𝐼10+𝛾7𝑅𝐼20+𝛾8𝑅𝑃0+𝛾9𝑅𝐴0,(3.29) where 𝑅𝐼10=𝛽𝜌+𝜃+𝜈+𝜇+𝜏,𝑅𝐼20=𝛽𝑘1𝜎+𝜂+𝜇+𝜏,𝑅𝑃0=𝛽𝑘2,𝑅𝜇+𝛼𝐴0=𝛽𝑘3𝛿+𝜇,𝛾6=Λ𝜇+𝜏,𝛾7=𝜌𝛾6𝐸1,𝛾8=(𝜎𝜌+𝜃𝐵)𝛾6𝐸1𝐵,𝛾9=(𝛼(𝜎𝜌+𝜃𝐵)+𝐶(𝜂𝜌+𝜈𝐵))𝛾6𝐸1,𝐵𝐶(3.30) with the expressions of 𝐸1,𝐵,𝐶,𝐷 given by (3.4). Equation (3.29) can be rewritten as 𝑅0sat=Λ𝑅𝜇+𝜏st0.(3.31) The reproduction number for the saturated incidence model is greater than that for the standard incidence model, that is, 𝑅0sat>𝑅st0 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 𝑅0sat1. 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 𝑅0sat1.

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, (1𝑏), 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 (1𝑘)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 𝜆0st=𝛽𝐼1+𝑘1𝐼2+𝑘2𝑃+𝑘4𝑇+𝑘3𝐴𝑁,𝜆0sat=𝛽𝐼1+𝑘1𝐼2+𝑘2𝑃+𝑘4𝑇+𝑘3𝐴𝐼1+𝜔1+𝐼2+𝑃+𝑇+𝐴(4.1) 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: 𝑑𝑆𝑑𝑡=(1𝑏)Λ𝜆0𝑆(1𝑝)(𝑟+𝜇+𝜏)𝑆,𝑑𝐸𝑑𝑡=𝑏Λ+𝑟𝑆𝜆0𝐸(1𝑘)(1𝑝)(𝜇+𝜏)𝐸,𝑑𝐼1𝑑𝑡=𝜆0𝑆(1𝑝)+𝜆0𝐸(1𝑘)(1𝑝)(𝜌+𝜈+𝜃+𝜇+𝜏)𝐼1,𝑑𝐼2𝑑𝑡=𝜌𝐼1𝜎+𝜌1𝐼+𝜂+𝜇+𝜏2,𝑑𝑃𝑑𝑡=𝜃𝐼1+𝜎𝐼2𝜌2+𝜇+𝛼𝑃,𝑑𝑇𝑑𝑡=𝜌1𝐼2+𝜌2𝛼𝑃1+𝜇+𝜏𝑇,𝑑𝐴𝑑𝑡=𝜈𝐼1+𝜂𝐼2+𝛼𝑃+𝛼1𝑇(𝛿+𝜇)𝐴,(4.2) with initial conditions 𝑆(0)=𝑆0,𝐸(0)=𝐸0,𝐼1(0)=𝐼01,𝐼2(0)=𝐼02,𝑃(0)=𝑃0,𝑇(0)=𝑇0,𝐴(0)=𝐴0.

tab3
Table 3: Additional state variables of the model with control measures.
tab4
Table 4: Additional parameters of the model with control measures.
864795.fig.002
Figure 2: HIV/AIDS model structure with control measures.
4.1. Model with Standard Incidence

The disease-free equilibrium 𝐸02 of (4.2) is 𝐸02=(1𝑏)Λ,𝑟+𝜇+𝜏𝑏Λ(𝜇+𝜏)+Λ𝑟(𝜇+𝜏)(𝑟+𝜇+𝜏),0,0,0,0,0.(4.3)

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 𝑅st𝑒 is given by 𝑅st𝑒=𝛾1𝑅𝐼1𝑒+𝛾2𝑅𝐼2𝑒+𝛾3𝑅𝑃𝑒+𝛾4𝑅𝑇𝑒+𝛾5𝑅𝐴𝑒,(4.4) where 𝑅𝐼1𝑒=𝛽𝜌+𝜃+𝜈+𝜇+𝜏,𝑅𝐼2𝑒=𝛽𝑘1𝜎+𝜌1+𝜂+𝜇+𝜏,𝑅𝑃𝑒=𝛽𝑘2𝜌2,𝑅+𝜇+𝛼𝑇𝑒=𝛽𝑘4𝛼1+𝜇+𝜏,𝑅𝐴𝑒=𝛽𝑘3,𝛾𝛿+𝜇1=(1𝑝)(1𝑏)(𝜇+𝜏)+(𝑏(𝜇+𝜏)+𝑟)(1𝑘)𝑟+𝜇+𝜏,𝛾2=𝛾1𝜌𝐸1,𝛾4=𝛾1𝜌1𝜌𝐶1+𝜌2𝜃𝐵1+𝜌2𝜎𝜌𝐸1𝐵1𝐶1,𝛾3=𝛾1𝜃𝐵1+𝜎𝜌𝐸1𝐵1,𝛾5=𝛾1𝜈𝑙𝐶1𝐵1+𝜂𝜌𝑙𝐶1+𝛼𝜃𝑙𝐵1+𝛼𝑙𝜎𝜌+𝛼1𝜌1𝜌𝐶1+𝛼1𝜌2𝜃𝐵1+𝛼1𝜌2𝜎𝜌𝐸1𝐵1𝐶1𝑙,𝐵1=𝐵+𝜌1,𝐶1=𝐶+𝜌2,𝑙=𝛼1+𝜇+𝜏.(4.5)𝐵,𝐶, and 𝐷 are defined in (3.4) above. 𝑅𝐼1𝑒 is the contribution to the effective reproduction number by unaware infectives 𝐼1,𝑅𝐼2𝑒 is the contribution to the effective reproduction number by aware infectives 𝐼2,𝑅𝑃𝑒 is the contribution to the effective reproduction number by pre-AIDS individuals 𝑃,𝑅𝑇𝑒 is the contribution to the effective reproduction number by individuals who are under treatment 𝑇, and 𝑅𝐴𝑒 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,𝑅st𝑒 reduces to 𝑅st0 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 𝑅st𝑒<1 and unstable otherwise.

The threshold quantity 𝑅st𝑒 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 𝜆0st as follows: 𝑆=(1𝑏)Λ𝜆0st(1𝑝)+(𝑟+𝜇+𝜏),𝐼1=𝑗𝜆0st(1𝑝)𝐸1,𝐸=𝑏Λ𝜆0st+𝑑+𝑒𝑟𝑎𝜆0st𝜆𝑑+𝑒0st(1𝑝)+𝑐,𝐼2=𝑗𝜌𝜆0st(1𝑝)𝐵1𝐸1,𝑃=𝑗𝑓2𝜆0st(1𝑝)𝐸1,𝑇=𝑗𝑓3𝜆0st(1𝑝)𝐸1,𝐴=𝑗𝑓4𝜆0st(1𝑝)𝐷𝐸1,(4.6) where 𝜆0st=𝛽𝐼1+𝑘1𝐼2+𝑘2𝑃+𝑘4𝑇+𝑘3𝐴𝑁,𝑎𝑗=𝜆st(+𝑓1𝑝)+𝑐1𝜆0st+𝑑+𝑒𝑑𝑟Λ𝜆0st𝜆(1𝑝)+𝑐0st,𝑓𝑑+𝑒𝑎=(1𝑏)Λ,𝑐=𝑟+𝜇+𝜏,𝑑=(1𝑘)(1𝑝),𝑒=𝜇+𝜏,1=(1𝑘)𝑏Λ,𝑓2=𝜃𝐶1+𝜎𝜌𝐶1𝐵1,𝑓3=𝜌𝜌1𝑙𝐵1+𝜌2𝑙𝑓2,𝑓4=𝜈+𝜂𝜌𝐵1+𝛼𝑓2+𝛼1𝑓3,𝑙=𝛼1+𝜇+𝜏.(4.7) Substituting (4.6) into the expression for 𝜆0st at steady state and after a little rearrangement, we obtain 𝑔𝜆0st=𝑏11𝜆02st+𝑏21𝜆0+𝑏31=0,(4.8) where 𝑏11=𝑓5𝑎𝑏1𝑑+𝑓1𝑓5𝑏21,𝑏21=𝑎𝐸1𝑑+𝑏1𝑏Λ𝐸1+𝑓5𝑏1𝑎𝑒+𝑓5𝑓1𝑏1𝑐+𝑓5𝑏1𝑑𝑟Λ𝑎𝑏1𝑑𝑓6+𝑓6𝑓1𝑏21,𝑏31=𝑏1𝑑𝑟Λ𝑓6+𝑟𝑎𝐸1+𝑎𝐸1𝑐+𝑏Λ𝐸1𝑐𝑅1st𝑒𝑏,𝑓5𝜌=1+𝐵1+𝑓2+𝑓3+𝑓4𝐷,𝑓6=𝛽+𝛽𝑘1𝜌𝐵1+𝛽𝑘2𝑓2+𝛽𝑘4𝑓3+𝛽𝑘3𝑓4𝐷.(4.9) Both 𝑏11>0 and 𝑏21>0 if 𝑎𝑏1𝑑𝑓6+𝑓6𝑓1𝑏21<𝑎𝐸1𝑑+𝑏1𝑏Λ𝐸1+𝑓5𝑏1𝑎𝑒+𝑓5𝑓1𝑏1𝑐+𝑓5𝑏1𝑑𝑟Λ, while 𝑏31>0 if 𝑅st𝑒<1 and 𝑟𝑎𝐸1+𝑎𝐸1𝑐+𝑏Λ𝐸1𝑐(1(𝑅st𝑒/𝑏))>𝑏1𝑑𝑟Λ𝑓6. By the Routh-Hurwiz criterion, the quadratic (4.8) has no positive root; hence, no endemic equilibrium exists when 𝑅st𝑒<1. When 𝑅st𝑒=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 𝑅st𝑒=1. The case when 𝑅st𝑒>1 makes 𝑏31<0 if 𝑏1𝑑𝑟Λ𝑓6+𝑏Λ𝐸1𝑐(1(𝑅st𝑒/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 𝑅st𝑒1.

Theorem 4.3. If 𝑅st𝑒1, the disease-free equilibrium is globally asymptotically stable and unstable if 𝑅st𝑒>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 𝑅𝑒sat=Λ𝑅st𝑒.𝜇+𝜏(4.10) 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>𝑅st𝑒 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 𝜆0sat as follows: 𝑆=(1𝑏)Λ𝜆0sat(1𝑝)+(𝑟+𝜇+𝜏),𝐼1=𝑗2𝜆0sat(1𝑝)𝐸1,𝐸=𝑏Λ𝜆0sat+𝑑+𝑒𝑟𝑎𝜆0sat𝜆𝑑+𝑒0sat(1𝑝)+𝑐,𝐼2=𝑗2𝜌𝜆0sat(1𝑝)𝐵1𝐸1,𝑃=𝑗2𝑓2𝜆0sat(1𝑝)𝐸1,𝑇=𝑗2𝑓3𝜆0sat(1𝑝)𝐸1,𝐴=𝑗2𝑓4𝜆0sat(1𝑝)𝐹𝐸1,(4.2) where 𝑗2=𝑎𝜆0sat+𝑓(1𝑝)+𝑐1𝜆0sat+𝑑+𝑒𝑑𝑟Λ𝜆0sat𝜆(1𝑝)+𝑐0sat,𝜆𝑑+𝑒0sat=𝛽𝐼1+𝑘1𝐼2+𝑘2𝑃+𝑘4𝑇+𝑘3𝐴𝐼1+𝜔1+𝐼2+𝑃+𝑇+𝐴,(4.12) 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 𝜆0sat (at steady state) and after some rearrangements, we obtain 𝑧1𝜆0sat=𝑎1𝜆02sat+𝑎2𝜆0sat+𝑎3=0,(4.13) where 𝑎1=𝐸1𝑑(1𝑝)+𝜔𝑓5𝑎𝑑(1𝑝)+𝜔𝑓5𝑓1(1𝑝)2,𝑎2=𝐸1𝑐(1𝑝)+𝐸1𝑒(1𝑝)+𝜔𝑓5𝑎(1𝑝)+𝜔𝑓5𝑓1𝑐(1𝑝)+𝜔𝑓5𝑓𝑑𝑟Λ(1𝑝)6𝑎𝑑(1𝑝)+𝑓6𝑓1(1𝑝)2,𝑎3=𝐸1𝑒𝑐1𝑅𝑒sat𝑓6𝑑𝑟Λ(1𝑝).(4.14) It is evident that 𝑎1>0,𝑎2>0 if 𝑓6𝑎𝑑(1𝑝)+𝑓6𝑓1(1𝑝)2<𝐸1𝑐(1𝑝)+𝐸1𝑒(1𝑝)+𝜔𝑓5𝑎(1𝑝)+𝜔𝑓5𝑓1𝑐(1𝑝)+𝜔𝑓5𝑑𝑟Λ(1𝑝), while 𝑎3>0 if 𝑅𝑒sat<1 and 𝑓6𝑑𝑟Λ(1𝑝)<𝐸1𝑒𝑐(1𝑅𝑒sat). 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 𝑎1𝜆02sat+𝑎2𝜆0sat𝑎3=0, so that 𝜆0sat=(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;𝐼01=50,000;𝐼02=30,000;𝑃0=35,000;𝑇0=20,000;𝐴0=25,000. Some of the parameter values are assumed for the purpose of illustration.

tab5
Table 5: Models parameter values.

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.

fig3
Figure 3: 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 𝑅st𝑒=1.0159 for the standard incidence and 𝑅sat𝑒=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.

fig4
Figure 4: 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.

fig5
Figure 5: 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.

fig6
Figure 6: 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.

fig7
Figure 7: 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 𝑅st01 and 𝑅0sat1 for standard incidence and saturated incidence, respectively. If both 𝑅st0 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>𝑅st0. 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 𝑅st𝑒,𝑅𝑒sat1 and when 𝑅st𝑒,𝑅𝑒sat>1, respectively. Note that 𝑅𝑒sat>𝑅st𝑒.

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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