Predicting the effectiveness of drug interventions with ‘HIV counseling & testing’ (HCT) on the spread of HIV/AIDS: a theoretical study
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Abstract
In this paper, a dynamic compartmental model is constructed for the transmission of HIV/AIDS receiving drug treatment and knowledge from effective awareness programs through media. Using stability theory of differential equations the model is analyzed qualitatively. The equilibrium points in the local and global stability proof are found to be stable under certain conditions. Further, we use Pontryagin’s Minimum Principle in the timedependent constant control case to derive necessary conditions for the optimal control of the disease. Sensitivity analysis is also performed to check the robustness of the model with respect to small changes in parametric values of the system. In order to predict the long term dynamics of the disease, projections are made. These studies reveal that HIV incidence could be substantially reduced by improving the testandtreat strategy and implication of media awareness.
Keywords
Epidemic model HIV Drugdosing Awareness programs Basic reproductive number Local & global stability Numerical simulation Sensitivity analysis1 Introduction
According to the report from the US Centers for Disease Control and Prevention, acquired immune deficiency syndrome (AIDS) was first identified as a distinct infectious disease in the year 1981. Since then, more than 70 million people have been infected and about 35 million people have already died of this disease. World Health Organization (WHO) reported that globally, there were 36.9 million people who were living with HIV at the end of 2017 and 940,000 people died of HIVrelated illnesses worldwide in the same year. Can you imagine what would happen if this number of deaths still happen for next few years?
Nevertheless, there is no room for complacency. Countries need to live up their commitment, adopted by the United Nations General Assembly in September 2015, to end the AIDS epidemic as a public health threat by 2030. The immediate challenge is to reach the FastTrack targets for 2020, and a rational step should be immediately taken to make people aware about the disease, and its preventive measures, through the media, as HIVrelated deaths are still unacceptably high.
Recent studies suggested that education and media have a great visitation in preventing the spread of HIV/AIDS among married couples in Bangladesh [1, 2]. Broadcast media, being the primary source of information, can not only increases the governmental health care involvement to control the spread of HIV but also makes people acquainted with the disease. Del Valle et al. (2004) studied the effects of education, temporary vaccination, and treatment on HIV transmission in a homosexually active population [3]. It was suggested by them that along with proper vaccination and treatment, awareness programs could be one of the effective solutions in reduction of such diseases. Gumel et al. (2004) worked on determining the optimal vaccine coverage and efficacy levels needed for communitywide eradication of HIV [4]. Baryarama et al. (2005) worked on a mathematical model and explained that there is a tendency for the epidemic to stabilize at higher numbers of infectives and AIDS cases than the minimum numbers attained during the first decline of the epidemic [5]. HoveMusekwa et al. (2009) worked to determine the effects of carriers and randomly screened carriers, who are aware of their status, on the transmission of HIV [6]. Tripathi et al. (2007) also proposed a model presented without interventions, which considers infection leading to asymptomatic HIV infectives who are later screened and finally develop AIDS [7]. A similar approach was undertaken earlier by Hyman et al. (2003) with differential infectivity and staged progression models [8]. In our recent works, we have shown that the incidences can be controlled if people take obligatory precautions to make themselves protected [9, 10]. They can minimize the risk if they become wellinformed and aware about the prevalence. Other research articles are also studied to improve this modeling process [11, 12, 13, 14].
These studies have motivated us to formulate a mathematical model that incorporates both of these events: testandtreat program and media awareness. In the modeling process, we assume that the growth rate of the cumulative density of awareness programs driven by the media is proportional to the number of untreated infectives present in the population. The whole population is divided into five separate classes; highrisk unaware susceptible class, class of nondiagnosed HIV infected individuals, diagnosed class of HIVpositive individuals who have not yet progressed to AIDS, class of those individuals with clinical AIDS, and aware susceptible class. We also discuss the equilibrium points and their stability. Finally, we solve the model numerically and then discuss the results from the biological aspect.
2 The model
Here, the constant recruitment rate in the susceptible population is Π either by birth or immigration. There is a constant natural death rate d. β is the product of the effective contact rate between susceptible and infected individuals to result in HIV infection and the transmission probability of HIV per contact. \(\lambda_{1}\), \(\lambda_{2}\) are the modification factors accounting for varying levels of the activity and infectiousness of the diagnosed HIVpositive individuals and the AIDS patients, respectively. The dissemination rate of awareness among susceptible, at which they form the aware class, is represented by c. The transfer rate from aware susceptible to unaware susceptible class is denoted as ω. δ is the diagnosis rate and \(d_{I}\) is the natural death rate of infected individuals. ρ is the proportion of diagnosed individuals who have not yet developed to AIDS (\(0\leq \rho \leq 1\)). Here, \(\alpha_{I}\) and \(\alpha_{D}\) are additional death rates for the diagnosed HIVpositive individuals and for those who are AIDS infected, respectively. The parameter μ is the proportionality constant which governs the implementation of awareness programs and \(\mu_{0}\) denotes the depletion rate of these programs due to ineffectiveness, public interests, social problems, etc.
Throughout this paper, to shorten the notation, we use the following notations: \(\mu_{1}=\delta + d + d_{I}\), \(\mu_{2}=\xi ' + d +\alpha_{I}\), \(\mu_{3}=d+\alpha_{A}\), \(\mu_{4}=d+\omega \).
2.1 Model properties:
Theorem 2.1
The solutions of system (1) with initial conditions satisfy\(S(t)>0\), \(I(t)>0\), \(I_{D}(t)>0\), \(I_{DA}(t)>0\), \(S_{+}(t)>0\), \(M(t)>0\)for all\(t>0\). The region\(\mathcal{D}\in \mathbb{R}^{6}_{+}\)is positively invariant and attracting with respect to system (1).
Proof
3 Analysis of equilibria
In this section, we focus on the existence and stability of equilibria for system (1). The equilibrium points are obtained by equating the righthand side of each equation in (1) to zero and it is found that system (1) has two nonnegative equilibria.
3.1 Diseasefree equilibrium (DFE) and basic reproduction number \(\mathcal{R}_{0}\)

(A1) \(\mathcal{F}_{i}(x) \geq 0\), \(\mathcal{V}_{i}^{+}(x) \geq 0\), \(\mathcal{V}_{i}^{}(x) \geq 0\) for any \(x \geq 0\);

(A2) if \(x_{i}=0\), then \(\mathcal{V}_{i}^{}=0\);

(A3) \(\mathcal{F}_{i}=0\) for \(i>3\);

(A4) if \(x \in \mathcal{X}_{S}\), then \(\mathcal{F}_{i}(x)=0\) and \(\mathcal{V}_{i}^{+}(x)=0\) for \(i=1, 2, 3\);

(A5) if \(x_{0}\) is DFE, then eigenvalues of the Jacobi matrix \(Df(x_{0})\) restricted to the subspace \(\mathcal{F}=0\) have all eigenvalues with negative real parts.
Corollary 3.1
If\(\frac{\beta \varPi }{d\mu_{1}} (1+\lambda_{1}\frac{ \rho \delta }{\mu_{2}}+\lambda_{2}\frac{\delta [\rho \xi '+(1\rho )\mu_{2}]}{\mu _{2}\mu_{3}} ) <1\), then the diseasefree equilibrium\(E_{0}\)is locally asymptotically stable.
Notice that, the inequality \(\mathcal{R}_{0}<1\) is equivalent to \(S^{*}>S_{0}\) which means that the positive equilibrium EE does not exist. On the other hand, if EE exists, then DFE is unstable.
3.2 Endemic equilibrium (EE) and its stability
Remark
Observe from the expression of \(I^{*}\) that when \(\mathcal{R}_{0}>1\), the endemic equilibrium exists. We thus have the existence condition of the endemic equilibria. It is also important to note that \(\frac{\partial I^{*}}{\partial c}\) and \(\frac{\partial I ^{*}}{\partial \delta }\) are negative. This means, as long as infected individuals become aware and go under treatment, the equilibrium level of unaware and untreated infectives starts to decrease. This shows that the dissemination rate and the treatment rate really have significant input in HIV control.
In order to attain full characterization of the endemic equilibrium \(E^{*}\), we study the asymptotic stability behavior using Lyapunov’s stability theory. If this function has only a single minimum, i.e., an equilibrium point, and it is strictly decreasing along all nonequilibrium solutions, then all solutions tend to the equilibrium point where the scalar function (Lyapunov function) is minimum. The results obtained by performing local and global stability analysis of the obtained equilibria are stated in the following theorems.
Theorem 3.2
In the next theorem, we show that the endemic equilibrium point \(E^{*}(S^{*},I^{*},I_{D}^{*}, I_{DA}^{*}, S_{+}^{*},M^{*})\) is globally asymptotically stable.
Theorem 3.3
For the proof of Theorems 3.2 and 3.3 see Appendices 1 and 2, respectively.
4 Optimal control strategy
4.1 The optimality system
We begin this section by noting that the existence of an optimal control pair that can be obtained using a result from Fleming et al. (2012) [17]. It is rather straightforward to show that the righthand sides of system (1) are bounded by a linear function of the state and control variables, and the integrand of the objective function (7) is concave on \(\mathcal{U}\) and is bounded below. These bounds give one the compactness needed to establish the existence of optimal controls using standard arguments (see [17]).
Theorem 4.1
If the objective cost function\(J(u_{1},u_{2})\)over\(\mathcal{U}\)attains its minimum for the optimal control\(u^{*}=(u_{1}^{*},u_{2} ^{*})\)corresponding to the endemic equilibrium\((S^{*},I^{*}, I^{*} _{D}, I^{*}_{DA}, S^{*}_{+},M^{*})\), then there exist adjoint functions\(\xi_{1}\), \(\xi_{2}\), \(\xi_{3}\), \(\xi_{4}\), \(\xi_{5}\), \(\xi_{6}\)satisfying equations (10) along with the transversality condition\(\xi_{i}(t_{f})=0\) (\(i=1,2,\ldots,6\)).
5 Numerical simulation
List of parameters used for system (1)
Parameter  Definition  Value/year  References 

Π  Constant recruitment rate in the susceptible population  15  [6] 
β  Contact rate between susceptible and infected individuals  0.002–0.2  [6] 
\(\lambda _{1}\)  Modification factor  0.01  Estimated 
\(\lambda _{2}\)  Modification factor  0.01–0.8  Estimated 
μ  The proportionality constant which governs the implementation of awareness programs  0.025  [9] 
d  Natural death rate of susceptible  0.01–0.025  
\(d_{I}\)  Natural death rate of infected individuals  0.005–0.01  [9] 
c  Transfer rate of susceptible from unaware to aware class  0.00002–0.02  [9] 
ω  Transfer rate from aware susceptible to unaware susceptible class  0.4  
ρ  The proportion of diagnosed individuals who have not yet developed to AIDS  0.75  Estimated 
δ  Diagnosis rate  0.304  [9] 
\(\alpha _{I}\)  Additional death rates for the diagnosed HIVpositive individuals  0.0172  [6] 
\(\alpha _{A}\)  Additional death rates for those who are AIDS infected  0.0138  [6] 
\(\mu _{0}\)  Depletion rate of ineffective media programs  0.06  [9] 
ξ  Rate of progression from HIV diagnosis to the AIDS class  0.4  Estimated 
η  Effectiveness of the drug input  0.6  Estimated 
\(u_{1}\), \(u_{2}\)  Control variables  0.0–1.0  Estimated 
Initially, to confirm the feasibility of our analysis regarding existence and its stability conditions for system (1), we have carried out some numerical simulations by selecting the following parametric set as described in Table 1. For the set of parameter values given in Table 1, it may be checked that the condition of existence of an endemic equilibrium \(E^{*}\) and the stability conditions are satisfied. The eigenvalues of the Jacobian matrix corresponding to the equilibrium \(E^{*}\) of the model system (1) are obtained as −0.8689, −0.7432, \(0.3938\pm 0.1232i\), and \(0.07094 \pm 0.0383i\). We note that all eigenvalues of the Jacobian matrix are either negative or have negative real parts. Hence, the endemic equilibrium \(E^{*}\) is locally asymptotically stable for the above set of parameter values.
6 Conclusion
In this paper, we have established an epidemic model to investigate the likely impact of awareness campaigning driven by media along with screening and treatment on the dynamics of HIV/AIDS. This model looks at the recently launched HIV counseling and testing (HCT) campaign followed by awareness campaigning, to model its feasible impact on the dynamics of the disease. HIV awareness and prevalence of the disease are inversely correlated with each other and depend upon human behavior. The model analysis shows how the inclusion of awareness modifies the contact structure and thereby affects the disease states. The model exhibits two equilibria, namely diseasefree equilibrium and endemic equilibrium. We have studied the existence and stability of the diseasefree and endemic equilibria. We obtain the basic reproduction number \((\mathcal{R}_{0})\) which determines the persistence of the disease. For \(\mathcal{R}_{0}\) below the unity, disease cannot persist in the system, whereas for \(\mathcal{R}_{0}\) above the unity, disease coexists in the system. We have also observed that the basic reproduction number \(\mathcal{R} _{0}\) contains terms like δ, \(\xi '\) etc., but does not contain any awareness related terms. However, HCT itself has very little impact on reducing the prevalence of HIV unless the efficacy of the campaigns exceeds an evaluated threshold. We prove the local and global asymptotic stability of the diseasefree equilibrium. Also, for \(\mathcal{R}_{0}>1\), the global stability of the endemic equilibrium is also derived by constructing a Lyapunov function.
Developing an optimal strategy that minimizes the total number of infected individuals and the costs associated, drugdosing and the implementation of awareness campaigns, we have extended our proposed model, where we consider that drugdosing and implementation of awareness campaigns are not constants but vary with time. The obtained optimality system (8) subject to (9) measures a cost effective way of controlling the disease by expanding the number of effective awareness campaigning. Also, drugdosing through antiviral therapy is equally important, which decelerates the AIDS progression significantly due to reduction in viral load. However, we should keep in mind that we should always run awareness campaigning in a cabalistic way so that people always keep receiving the latest information about the disease. Since it is observed that if the awareness of the local prevalence of a disease is not addressed by the media or local health authorities, it is more likely to be raised by the acts of informal information spread. If the information about infectious disease is disseminated in the population, people regulate their behavior according to their awareness level.
Notes
Acknowledgements
The authors would like to give special thanks to Prof. Urszula Foryś, Faculty of Mathematics, Informatics and Mechanics, Institute of Applied Mathematics and Mechanics, Warsaw University for her constructive and insightful comments, which helped us to improve the quality of this work.
Authors’ contributions
All authors contributed equally and significantly in this manuscript, and they read and approved the final manuscript.
Funding
INSPIRE Program (IF131081), Department of Science and Technology, Government of India is gratefully acknowledged for its financial support.
Competing interests
The authors declare that they have no competing interests.
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