Effect of awareness programs in controlling the disease HIV/AIDS: an optimal control theoretic approach
- Priti Kumar Roy^{1}Email author,
- Shubhankar Saha^{1} and
- Fahad Al Basir^{1}
https://doi.org/10.1186/s13662-015-0549-9
© Roy et al. 2015
Received: 2 March 2015
Accepted: 18 June 2015
Published: 15 July 2015
Abstract
Research reveals that most HIV infections come from undiagnosed people and untreated people, but principally from unaware people. Human awareness results in the reduction of susceptibility to infection, naturally, in the epidemiological study this factor should be included. Researchers have produced a wealth of information about the disease, including a number of critical tools and interventions to diagnose, prevent and treat HIV. Broadcast media have tremendous reach and influence, particularly with young people, so in this study, we incorporate the influence of media and investigate the effect of awareness program in disease outbreaks. We divided the whole susceptible and infected populations into two sub-classes: ‘unaware class’ and ‘aware class’. A detailed mathematical analysis has been carried out and numerical simulations have been performed to show the role of some of the important model parameters.
Keywords
1 Introduction
It has been nearly thirty years since the first cases of human immunodeficiency virus (HIV) garnered the world’s attention. Without treatment and awareness, the virus slowly debilitates a person’s immune system until they fall prey to illness. The epidemic has claimed the lives of nearly 35.0 million worldwide [1] and has affected many more. In India, approximately 3.5 million [3.3 to 4.1 million] people are acquiring HIV every year and more than 1.93 to 3.04 million people [2] are living with HIV. Unless we take bold actions, however, we anticipate a new era of rising infections and even greater challenges in serving people living with HIV and higher health care costs.
HIV/AIDS epidemic is a serious, growing public health problem worldwide. The cause is known and the principal routes of transmission understood, but resources for treating HIV infected patients and for combating the spread of the virus are limited. Though the best way of disease control is mass vaccination, but for the case of HIV, the usage of vaccination is very costly and conferred immunities are temporary. We have observed that basic mathematical models [3–8] mainly deal with the interaction between susceptible and infective individuals. In view of controlling HIV/AIDS, there are many antiretroviral therapies (ART) available nowadays [9–14] which help the immune system in preventing the infection even though it is not possible to cure it. There are many other factors such as unsafe sex and low condom usage, injecting drug use, widespread stigma, education, migration and mobility etc. which also affect the wideness of this infectious disease. Awareness programs introduce people with the disease and help them to take precautions to reduce the chances of being infected. Moreover, it is important to mention that awaking people through media in the population results in less interaction between susceptible and infected individuals, which lowers the disease transmission rate into susceptible individuals. So, awareness factor can be considered as a strong tool against the expansion of HIV/AIDS. Saying particulary, awareness factor has a great visitation not only on the behavioral changes in individuals, but it also helps governmental health care interventions to control the spread of the disease HIV/AIDS.
Many mathematical models have been developed to monitor HIV/AIDS and explore the impact of intervention strategies that are being implemented. Misra et al. [15, 16] studied the effect of awareness programs through media on disease dynamics in a variable population with immigration. Van Segbroeck et al. [17] analytically studied the disease dynamics of a well-mixed population with rescaled infectiousness where the contact network reshaping occurs much faster than disease spreading. Recently, Samanta and Chattopadhyay [18] have studied a slow-fast dynamics with the effect of awareness program in disease outbreak.
In this present study, we have analyzed a simple SI network epidemic model to study the impact of awareness programs conducted through media campaigning on the spread of HIV/AIDS in a variable population with immigration. However, these results will fall into the non-network epidemic models category. We assume that the growth rate of the cumulative density of awareness programs driven by media is proportional to the number of the infective present in the population. We also assume that the awareness programs against the disease will alert the susceptible to isolate themselves from infective individuals and to form a separate class. We have formulated the mathematical model and also discuss the equilibrium points and their stability. Lastly, we use an optimal control theory paradigm to our mathematical model, in which the contact processes between unaware susceptible and infected classes with aware susceptible and infected classes are controlled. We also solve the model numerically and then discuss the analytical and numerical results according to biological aspect.
2 Formulation of mathematical model with basic assumptions
Furthermore, the total susceptible and infected, each population is divided into two sub-classes such that \(S(t)=S_{-}(t)+S_{+}(t)\) and \(I(t)=I_{-}(t)+I_{+}(t)\) due to the awareness programs driven by the media \(M(t)\), where the ‘−’ sign denotes the unaware class and ‘+’ sign represents the aware class. As the awareness disperses, people respond to it and change their behavior to alter their susceptibility. It is also assumed that due to awareness programs, aware susceptible and infected individuals avoid being in contact with the infective, form separate ‘aware classes’ (\(S_{+}(t)\) and \(I_{+}(t)\)) and do not get involved in sexual relations or in any other means that causes AIDS. But due to lack of their memory, a portion of aware people ignore the indemnities and become unaware again. We assume that this transfer rate from aware class to unaware class is \(\lambda_{i} \) (\(i=1,2\)).
3 Equilibrium points and stability analysis
System (3) has two non-negative equilibria.
3.1 Disease-free equilibrium point
The disease-free equilibrium point \(E^{0}\) always exists, and it is locally asymptotically stable when \(R_{0}<1\) and unstable when \(R_{0}>1\) [20]. We use the threshold \(R_{0}\) to answer the question of whether the infection can be established. When \(R_{0}>1\), HIV infection can take hold. Otherwise the infection will be eliminated.
3.2 Endemic equilibrium point
Thus, we can conclude the following theorem.
Theorem 3.1
The disease-free equilibrium (DFE) \(E^{0}\) exists and is locally asymptotically stable when \(R_{0}=\frac{\Pi\beta}{d (d+\delta_{I})}<1\). Whenever \(R_{0}=\frac{\Pi\beta}{d (d+\delta_{I})}>1\), \(E^{0}\) becomes unstable and the endemic equilibrium point \(E^{*}\) exists.
4 Optimal control strategy
The parameters \(P>0\) and \(Q>0\) are dimensionless weight functions on the benefit of the cost. These are the costs of per media campaigning for unaware susceptible and unaware infected populations, respectively, per unit time.
By using the Pontryagin minimum principle and for the existence condition of the optimal control theory [24], we obtain the following theorem.
Theorem 4.1
Proof
Using (24), we can get the adjoint system (15) corresponding to systems (12) and (13). The optimal system consists of the state system (12), satisfying the initial condition \(S_{-}(0)=S_{0-}\), \(S_{+}(0)=S_{0+}\), \(I_{-}(0)=I_{0-}\), \(I_{+}(0)=I_{0+}\), \(M(0)=M_{0}\), and the adjoint system (15), satisfying transversality condition \(\xi_{i}(t_{f})=0\) for \(i=1,2,3,4,5\). □
4.1 Uniqueness of the optimal control
To prove the uniqueness of the optimality system, we use a simple lemma for the small time interval.
Lemma 1
[25]
The function \(u^{*}(s)=\min(\max (s,a),b)\) is Lipschitz continuous in s, where \(a< b\) are some fixed positive constants.
Theorem 4.2
The solution of the optimal system is unique for sufficiently small \([t_{0},t_{f}]\).
Proof
Suppose that \((S_{-}, S_{+}, I_{-}, I_{+}, M, \xi_{1}, \xi_{2}, \xi_{3},\xi_{4}, \xi_{5})\) and \((\bar{S}_{-},\bar{S}_{+}, \bar{I}_{-},\bar{I}_{+},\bar{M},\bar{\xi}_{1},\bar{\xi}_{2},\bar{\xi}_{3}, \bar{\xi}_{4},\bar{\xi}_{5})\) are two solutions of systems (12) and (15) such that for \(\theta>0\), \(S_{-}=e^{\theta t}p_{1}\), \(S_{+}=e^{\theta t}p_{2}\), \(I_{-}=e^{\theta t}p_{3}\), \(I_{+}=e^{\theta t}p_{4}\), \(M=e^{\theta t}p_{5}\), \(\xi_{1}=e^{-\theta t}q_{1}\), \(\xi_{2}=e^{-\theta t}q_{2}\), \(\xi_{3}=e^{-\theta t}q_{3}\), \(\xi_{4}=e^{-\theta t}q_{4}\), \(\xi_{5}=e^{-\theta t}q_{5}\) and \(\bar{S}_{-}=e^{\theta t}\bar{p}_{1}\), \(\bar{S}_{+}=e^{\theta t}\bar{p}_{2}\), \(\bar{I}_{-}=e^{\theta t}\bar{p}_{3}\), \(\bar{I}_{+}=e^{\theta t}\bar{p}_{4}\), \(\bar{M}=e^{\theta t}\bar{p}_{5}\), \(\bar{\xi}_{1}=e^{-\theta t}\bar{q}_{1}\), \(\bar{\xi}_{2}=e^{-\theta t}\bar{q}_{2}\), \(\bar{\xi}_{3}=e^{-\theta t}\bar{q}_{3}\), \(\bar{\xi}_{4}=e^{-\theta t}\bar{q}_{4}\), \(\bar{\xi}_{5}=e^{-\theta t}\bar{q}_{5}\).
Thus, the system has a unique optimal solution for a small time interval. If the state equation has the initial condition and the adjoint equation has the final time condition, then the optimal controls \(u^{*}_{1}\) and \(u^{*}_{1}\) give a unique and optimal control strategy for the density of awareness programs driven by the media in the region under consideration. □
5 Numerical simulation
List of parameters used for system ( 2 )
Parameter | Definition | Reference | Assigned value (day \(^{\boldsymbol {-1}}\) ) |
---|---|---|---|
Π | Constant recruitment rate | 12 | |
β | Disease transmission rate | [18] | 0.0025 |
\(\alpha_{1}\) | Contact rate between unaware susceptible with media | [16] | 0.0002 |
\(\lambda_{1}\) | Transfer rate of people from aware individuals to unaware susceptible class | 0.0052 | |
d | Natural death rate | 0.005 | |
\(\delta_{I}\) | Additional death rate due to infection | [6] | 0.007 |
\(\alpha_{2}\) | Contact rate between unaware infected with media | - | 0.1 |
\(\lambda_{2}\) | Transfer rate of people from aware individuals to unaware infected class | - | 0.0015 |
η | Rate of implementation of awareness programs | [15] | 0.005 |
\(\eta_{0}\) | Depletion rate of awareness program due to ineffectiveness | [15] | 0.06 |
6 Discussion and conclusion
In this research article, we deal with a non-linear SI mathematical model reflecting the effect of awareness programs on a certain population with constant recruitment rate. We have studied the impact of awareness as a novel intervention for the control of epidemiological diseases. In the modeling process, it is assumed that media campaigns create awareness regarding personal protection as well as control AIDS. As a result, behavioral changes (transfer from unaware to aware) occur within the human population, which results in the formation of a new class, i.e., aware class. Individuals of this class not only protect themselves from the infection, but being aware they also take part in reducing AIDS by taking precautions. Our analytical study shows that the basic reproduction number \(R_{0}\), which determines the existence of the disease, does not contain any awareness related terms. As a result, the persistence of the disease does not depend on awareness programs. However, the awareness programs reduce the infection rate, shorten the rate of disease transmission and cut down the size of the disease. Numerical simulations, which are very realistic, add an extra dimension to our analytic conclusions. On the basis of the existence conditions and analytical results, we showed that the system has a unique optimal control pair \((u^{*}_{1},u_{2}^{*})\) for which the cost function will be minimum and outcomes will be time worthy. This implies that the presence of awareness in the population makes the disease expedition difficult and shorter. But in practical sense, disease remains endemic, because factors like low education, ignorance in taking precautions, social problems, immigration etc. play negative roles in the system. We discuss a simple model that captures some important features, and we believe these findings may help in controlling AIDS through awareness.
Declarations
Acknowledgements
The authors would like to thank the referees for their constructive and insightful comments, which helped us to improve the quality of this work. This work is supported by the Department of Science and Technology, Government of India.
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
Authors’ Affiliations
References
- www.who.int/gho/hiv/en
- http://www.worldbank.org/en/news/feature/2012/07/10/hiv-aids-india
- Del Valle, S, Evangelista, AM, Velasco, MC, Kribs-Zaleta, CM, Schmitz, SH: Effects of education, vaccination and treatment on HIV transmission in homosexuals with genetic heterogeneity. Math. Biosci. 187, 111-133 (2004) MathSciNetView ArticleMATHGoogle Scholar
- Gumel, AB, Moghadas, SM, Mickens, RE: Effect of a preventive vaccine on the dynamics of HIV transmission. Commun. Nonlinear Sci. Numer. Simul. 9, 649-659 (2004) MathSciNetView ArticleMATHGoogle Scholar
- Baryarama, F, Luboobi, LS, Mugisha, JYT: Periodicity of the HIV/AIDS epidemic in a mathematical model that incorporates complacency. Am. J. Infect. Dis. 1(1), 55-60 (2005) View ArticleGoogle Scholar
- Hove-Musekwaa, SD, Nyabadza, F: The dynamics of an HIV/AIDS model with screened disease carriers. Comput. Math. Methods Med. 10(4), 287-305 (2009) MathSciNetView ArticleGoogle Scholar
- Roy, PK, Chatterjee, AN, Greenhalgh, D, Khan, QJA: Long term dynamics in a mathematical model of HIV-1 infection with delay in different variants of the basic drug therapy model. Nonlinear Anal., Real World Appl. 14, 1621-1633 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Roy, PK, Chatterjee, AN: Effect of HAART on CTL mediated immune cells: an optimal control theoretic approach. In: Electrical Engineering and Applied Computing, vol. 90, pp. 595-607. Springer, Berlin (2011) View ArticleGoogle Scholar
- Roy, PK, Chatterjee, AN: Anti-viral drug treatment along with immune activator IL-2: a control-based mathematical approach for HIV infection. Int. J. Control 85(2), 220-237 (2012) MathSciNetView ArticleMATHGoogle Scholar
- Roy, PK, Chowdhury, S, Chatterjee, AN, Chottopadhyay, J, Norman, R: A mathematical model on CTL mediated control of HIV infection in a long term drug therapy. J. Biol. Syst. 21(3), 1350019 (2013) View ArticleGoogle Scholar
- Wodarz, D, Nowak, MA: Specific therapy regimes could lead to long-term immunological control to HIV. Proc. Natl. Acad. Sci. USA 96(25), 14464-14469 (1999) View ArticleGoogle Scholar
- Wodarz, D, May, RM, Nowak, MA: The role of antigen-independent persistence of memory cytotoxic T lymphocytes. Int. Immunol. 12, 467-477 (2000) View ArticleGoogle Scholar
- Kim, WH, Chung, HB, Chung, CC: Optimal switching in structured treatment interruption for HIV therapy. Asian J. Control 8(3), 290-296 (2006) MathSciNetView ArticleGoogle Scholar
- Liu, R, Wu, J, Zhu, H: Media/psychological impact on multiple outbreaks of emerging infectious diseases. Comput. Math. Methods Med. 8(3), 153-164 (2007) MathSciNetView ArticleMATHGoogle Scholar
- Misra, AK, Sharma, A, Singh, V: Effect of awareness programs in controlling the prevalence of an epidemic with time delay. J. Biol. Syst. 19, 389-402 (2011) MathSciNetView ArticleMATHGoogle Scholar
- Samanta, S, Rana, S, Sharma, A, Misra, AK, Chaattopadhyay, J: Effect of awareness programs by media on the epidemic outbreaks: a mathematical model. Appl. Math. Comput. 219, 6965-6977 (2013) MathSciNetView ArticleMATHGoogle Scholar
- Van Segbroeck, S, Santos, FC, Pacheco, JM: Adaptive contact networks change effective disease infectiousness and dynamics. PLoS Comput. Biol. 6(8), e10008 (2010) Google Scholar
- Samanta, S, Chattopadhaya, J: Effect of awareness program in disease outbreak - a slow-fast dynamics. Appl. Math. Comput. 237, 98-109 (2014) MathSciNetView ArticleGoogle Scholar
- Heffernan, JM, Smith, RJ, Wahl, LM: Perspectives on the basic reproductive ratio. J. R. Soc. Interface 2(4), 281-293 (2005) View ArticleGoogle Scholar
- Van den Driessche, P, Watmough, J: Reproduction numbers and sub threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci. 180, 29-48 (2000) View ArticleGoogle Scholar
- Pontryagin, LS, Boltyanskii, VG, Gamkrelidze, RV, Mishchenko, EF: Mathematical Theory of Optimal Processes, vol. 4. Gordon & Breach, New York (1986) MATHGoogle Scholar
- Kamien, M, Schwartz, NL: Dynamic Optimisation, 2nd edn. North-Holland, New York (1991) MATHGoogle Scholar
- Swan, GM: Application of Optimal Control Theory in Biomedicine. Dekker, New York (1984) Google Scholar
- Fleming, W, Rishel, R: Deterministic and Stochastic Optimal Controls. Springer, New York (1975) View ArticleGoogle Scholar
- Joshi, HR: Optimal control of an HIV immunology model. Optim. Control Appl. Methods 23, 199-213 (2002) View ArticleGoogle Scholar