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      An Optimal Treatment Control of TB-HIV Coinfection

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      Journal: International Journal of Mathematics and Mathematical Sciences
      Publisher: Hindawi Limited

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          Abstract

          An optimal control on the treatment of the transmission of tuberculosis-HIV coinfection model is proposed in this paper. We use two treatments, that is, anti-TB and antiretroviral, to control the spread of TB and HIV infections, respectively. We first present an uncontrolled TB-HIV coinfection model. The model exhibits four equilibria, namely, the disease-free, the HIV-free, the TB-free, and the coinfection equilibria. We further obtain two basic reproduction ratios corresponding to TB and HIV infections. These ratios determine the existence and stability of the equilibria of the model. The optimal control theory is then derived analytically by applying the Pontryagin Maximum Principle. The optimality system is performed numerically to illustrate the effectiveness of the treatments.

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          Modeling TB and HIV co-infections.

          Zhilan Feng, Lih-Ing W Roeger, Carlos Castillo-Chavez (2009)
          Tuberculosis (TB) is the leading cause of death among individuals infected with the human immunodeficiency virus (HIV). The study of the joint dynamics of HIV and TB present formidable mathematical challenges due to the fact that the models of transmission are quite distinct. Furthermore, although there is overlap in the populations at risk of HIV and TB infections, the magnitude of the proportion of individuals at risk for both diseases is not known. Here, we consider a highly simplified deterministic model that incorporates the joint dynamics of TB and HIV, a model that is quite hard to analyze. We compute independent reproductive numbers for TB (R1) and HIV (R2) and the overall reproductive number for the system, R=max{R1, R2}. The focus is naturally (given the highly simplified nature of the framework) on the qualitative analysis of this model. We find that if R 1, does not necessarily guarantee the stability of the HIV-only equilibrium EH, and it is possible that TB can coexist with HIV when R2 > 1. In other words, in the case when R1 1 (or when R1 > 1 and R2 > 1), we are able to find a stable HIV/TB coexistence equilibrium. Moreover, we show that the prevalence level of TB increases with R2 > 1 under certain conditions. Through simulations, we find that i) the increased progression rate from latent to active TB in co-infected individuals may play a significant role in the rising prevalence of TB; and ii) the increased progression rates from HIV to AIDS have not only increased the prevalence level of HIV while decreasing TB prevalence, but also generated damped oscillations in the system.
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            An optimal control strategy to reduce the spread of malaria resistance

            Fatmawati, Hengki Tasman (2015)
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              Mathematical Analysis of the Effects of HIV-Malaria Co-infection on Workplace Productivity

              Baba Seidu, Oluwole D. Makinde, Ibrahim Y. Seini (2015)
              In this paper, a nonlinear dynamical system is proposed and qualitatively analyzed to study the dynamics and effects of HIV-malaria co-infection in the workplace. Basic reproduction numbers of sub-models are derived and are shown to have LAS disease-free equilibria when their respective basic reproduction numbers are less than unity. Conditions for existence of endemic equilibria of sub-models are also derived. Unlike the HIV-only model, the malaria-only model is shown to exhibit a backward bifurcation under certain conditions. Conditions for optimal control of the co-infection are derived using the Pontryagin's maximum principle. Numerical experimentation on the resulting optimality system is performed. Using the incremental cost-effectiveness ratio, it is observed that combining preventative measures for both diseases is the best strategy for optimal control of HIV-malaria co-infection at the workplace.
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                Author and article information

                Journal
                Title: International Journal of Mathematics and Mathematical Sciences
                Abbreviated Title: International Journal of Mathematics and Mathematical Sciences
                Publisher: Hindawi Limited
                ISSN (Print): 0161-1712
                ISSN (Electronic): 1687-0425
                Publication date Created: 2016
                Publication date (Print): 2016
                Volume: 2016
                Issue:
                Pages: 1-11
                Article
                DOI: 10.1155/2016/8261208
                SO-VID: 5cb959c3-d23f-4a10-ad45-0775bb575d71
                Copyright © © 2016
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                http://creativecommons.org/licenses/by/4.0/

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