# Optimal Control Strategy for TB-HIV/AIDS Co-Infection Model in the Presence of Behaviour Modification

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Mathematical Model with Behavior Change and Treatment

${A}_{1}=b+m+\mu $ | ${A}_{2}=\mu +{d}_{T}$ | ${A}_{3}=\epsilon +\mu $ |

${A}_{4}={\tau}_{1}+{b}_{1}+\mu $ | ${A}_{5}=(1-r)p+{\epsilon}_{1}+\mu +{d}_{T}$ | ${A}_{6}=\omega +\mu $ |

${A}_{7}=\mu +{d}_{A}$ | ${A}_{8}=\xi +\mu $ | ${A}_{9}={\psi}_{1}+\mu +{d}_{TA}$ |

#### 2.1. Positivity and Boundedness of Solutions

**Lemma**

**1.**

#### 2.2. Analysis of the Sub-Models

#### 2.2.1. TB-Only Model

#### 2.2.2. Local Stability of Disease Free Equilibrium

**Definition**

**1.**

**Lemma**

**2.**

#### 2.2.3. HIV-Only Model

#### 2.2.4. Local Stability of Disease Free Equilibrium

#### 2.3. Analysis of the Full Model

## 3. Formulation of the Control

- Preventive Education
- (a)
- Different programs have been designed so far to enlighten the population about the risk factors of TB disease and about possible preventive mechanisms. Such preventive mechanisms are self-protective actions, like using masks and gloves while contacting TB infected individuals and consistently ventilating rooms and vehicles that are used commonly by other individuals who are especially more likely to be infectious of a TB disease.By applying such self-initiated protective measures an individual can reduce the risk of contracting the disease. Let the current level of preventive education campaigns about TB disease by various agents have convinced up to $100\times ({\alpha}_{T0}\times e)\%$ (for some $1>{\alpha}_{T0}>0)$ of the population to effectively participate in the self protective schemes available to them. If more choices of self-protective measure are offered to the population and if the awareness campaigns are intensified, more individuals may decide to choose and use at least one of these preventive measures. This helps individuals to reduce their risk of being infected by TB. This could be considered as an effort made by individuals and health care campaigners to help susceptible individuals from getting infected easily by TB.On the other hand, the same educational information can help individuals who are infected by TB but are not taking part in any of the self-protective actions about the disease so that they can change their risky behavior. These individuals may need to visit appropriate health centers regularly and take the prescribed medicine properly until the end of the specified time given from doctors. Moreover, they need to take any preventive actions against HIV so that they will not be co-infected by HIV.Assume that the control function ${u}_{1}\left(t\right)$ measures the rate at which additional susceptible individuals are convinced to take part in behaviour modification about TB disease. Then, its application in the dynamics is modelled by simply replacing the term ${\alpha}_{T}$ in the model system (3) by $({\alpha}_{T0}+{u}_{1}\left(t\right))$. We assume that the larger the proportion of the educated class, the lower will be the proportion of individuals in the susceptible population. Because of practicality and economic limitations on the maximum rate of convincing individuals for behaviour modification, we assume that ${\alpha}_{Tmax}>0$ to be the maximum rate such that $0\le {\alpha}_{T0}+{u}_{1}\left(t\right)\le {\alpha}_{Tmax}\le 1$.
- (b)
- The expansion and improvement of HIV and AIDS education around the world is critical to preventing the spread of HIV [37]. Those convinced to apply any of the preventive mechanisms against HIV infection, will enjoy a reduced risk of infection by HIV. Therefore, they will be better off as compared to individuals with risky behaviour.Moreover, it is also important to educate (or enlighten) people who are already infected by HIV so that they take maximum possible protective action against TB as their immune system is most likely be compromised due to the HIV infection.Assume that the control function ${u}_{2}\left(t\right)$ measures the rate at which additional susceptible individuals are convinced to take part in behaviour modification about HIV/AIDS disease. Then its application in the dynamics is modelled by simply replacing the term ${\alpha}_{H}$ in (3) by $({\alpha}_{H0}+{u}_{2}\left(t\right))$. Because of practicality and economic limitations on the maximum rate of convincing individuals for behaviour modification, we also assume that ${\alpha}_{Hmax}>0$ to be the maximum rate as indicated in [37] and $0\le {\alpha}_{H0}+{u}_{2}\left(t\right)\le {\alpha}_{Hmax}\le 1$.

- Treatment of infected individuals
- (a)
- TB treatment for individuals who are infected by TB bacteria.TB infected individuals can be treated with appropriate medicine and become non-infectious within an average treatment period of 6 months [11]. Such treatments not only help the infected individuals to recover from the disease, but also make them non infectious and thereby reducing the force of infection in the dynamics of the disease. Therefore, investing on treatments also have a positive impact on the reduction of the burden of the disease in the society in general.Assume that the control function ${u}_{3}\left(t\right)$ measures the rate at which additional infectious individuals are recruited to receive TB treatment at any time t. If the current rate of TB treatment is ${\tau}_{0}$ proportion from the total infected people, this control measure will be introduced in the dynamics as $({\tau}_{0}+{u}_{3}\left(t\right))$ by replacing the parameter $\tau $. In addition, we assumed that only an r fraction of people from the ${I}_{TH}$ classes are recruited to receive TB treatment while others receive both types simultaneously. Due to economic and logistic reasons, there could be limitations on the maximum rate to be achieved. Thus, we assume that the constant ${\tau}_{max}\le 1$ represents the maximum rate of recruitment for treatment of infected individuals with $0\le {\tau}_{0}+{u}_{3}\left(t\right)\le {\tau}_{max}\le 1$.
- (b)
- Treating HIV infected individual using ARV.Similar to the case of TB, treating HIV infected individuals with ARV will reduce their level of infectiousness by suppressing their viral load while helping them to regain their immunity thereby get a better quality of life. This treatment may also reduce the rate of co-infection by TB.We assume that the rate of recruiting individuals to receive ART is the same for both ${I}_{H}$ and ${A}_{H}$ classes, and we take $\psi =\sigma $. Moreover, it is assumed in this work that the rate of receiving treatment for both HIV and TB simultaneously is taken to be the maximum possible (which is p) and we require no additional effort in this regard.Let the control function ${u}_{4}\left(t\right)$ measures the rate at which additional infected individuals with HIV virus are recruited to receive ARV at any time t. If the current rate of recruitment is ${\sigma}_{0}$ proportion from among all HIV infected individuals, then this control measure can be introduced in the dynamics as ${\sigma}_{0}+{u}_{4}\left(t\right)$ in place of the parameter $\sigma $ and $\psi $ (which are assumed to be equal). Similar to the TB case we assume that there is a limitation on the maximum rate of treating people with ARV. Thus, we may represent the maximum recruitment rate to be ${\sigma}_{max}\le 1$ and therefore, we have $0\le {\sigma}_{0}+{u}_{4}\left(t\right)\le {\sigma}_{max}\le 1$.

#### Existence and Characterization of Optimal Control Solution

**Theorem**

**1**(Existence of optimal control solution).

**Proof.**

- A.
- B.
- The state system can be written as a linear function of the control variables with coefficients dependent on time and the state variables.
- C.
- The integrand L in (19) from objective functional with $L(\mathbf{x},\mathbf{u},t)={C}_{1}{I}_{T}\left(t\right)+{C}_{2}{I}_{H}\left(t\right)+{C}_{3}{I}_{TH}\left(t\right)+{C}_{4}{A}_{H}\left(t\right)+{C}_{5}{A}_{T}\left(t\right)+{D}_{1}{u}_{3}\left(t\right)\left(\right)open="("\; close=")">{I}_{T}\left(t\right)+{I}_{TH}\left(t\right)+\frac{{B}_{1}}{2}{u}_{1}^{2}\left(t\right)+\frac{{B}_{2}}{2}{u}_{2}^{2}\left(t\right)+\frac{{B}_{3}}{2}{u}_{3}^{2}\left(t\right)+\frac{{B}_{4}}{2}{u}_{4}^{2}\left(t\right)$ is convex on $\mathcal{U}$, and additionally it satisfies $L(\mathbf{x},\mathbf{u},t)\ge {\delta}_{1}\mid ({u}_{1},{u}_{2},{u}_{3},{u}_{4}){\mid}^{\beta}-{\delta}_{2}$ where ${\delta}_{1}>0$ and $\beta >1$.In order to establish condition A, we refer to Picard-Lindelöf’s theorem from [38,39]. If the solutions to the state equations are bounded and if the state equations are continuous and Lipschitz in the state variables, then there is a unique solution corresponding to every admissible control $\mathcal{U}$.It is indicated that the total population is bounded below by a positive nonzero number ${N}_{0}$ and bounded above by $\frac{\Pi}{\mu}$ as well as each of the state variables are bounded. With the bounds established above, it follows that the state system is continuous and bounded. It is equally direct to show the boundedness of the partial derivatives with respect to the state variables in the state system, which establishes that the system is Lipschitz with respect to the state variables (see [40]). This completes the proof that condition A holds.Condition B is verified by observing the linear dependence of the state equations on controls ${u}_{1},{u}_{2},{u}_{3}$ and ${u}_{4}$. Finally, to verify condition C by definition from [41,42] any constant, linear and quadratic functions are convex. Therefore, $L(\mathbf{x},\mathbf{u},t)$ is convex on $\mathcal{U}$. To prove the bound on the L we note that by the definition of $\mathcal{U}$, we have$$\begin{array}{c}{B}_{4}{u}_{4}^{2}\le {B}_{4}\phantom{\rule{0.222222em}{0ex}}since\phantom{\rule{0.222222em}{0ex}}{u}_{4}\in [0,1]\hfill \\ \frac{{B}_{4}}{2}{u}_{4}^{2}\le \frac{{B}_{4}}{2},\phantom{\rule{2.em}{0ex}}\frac{{B}_{4}}{2}{u}_{4}^{2}-\frac{{B}_{4}}{2}\le 0\hfill \\ L(\mathbf{x},\mathbf{u},t)={C}_{1}{I}_{T}\left(t\right)+{C}_{2}{I}_{H}\left(t\right)+{C}_{3}{I}_{TH}\left(t\right)+{C}_{4}{A}_{H}\left(t\right)+{C}_{5}{A}_{T}\left(t\right)\hfill \\ +{D}_{1}{u}_{3}\left(t\right)\left(\right)open="("\; close=")">{I}_{T}\left(t\right)+{I}_{TH}\left(t\right)+{D}_{2}{u}_{4}\left(t\right)\left(\right)open="("\; close=")">{L}_{TH}\left(t\right)+{I}_{H}\left(t\right)+{R}_{H}\left(t\right)+{A}_{H}\left(t\right)\hfill \end{array}+\frac{{B}_{1}}{2}{u}_{1}^{2}\left(t\right)+\frac{{B}_{2}}{2}{u}_{2}^{2}\left(t\right)+\frac{{B}_{3}}{2}{u}_{3}^{2}\left(t\right)+\frac{{B}_{4}}{2}{u}_{4}^{2}\left(t\right)\hfill \\ \ge \frac{{B}_{1}}{2}{u}_{1}^{2}\left(t\right)+\frac{{B}_{2}}{2}{u}_{2}^{2}\left(t\right)+\frac{{B}_{3}}{2}{u}_{3}^{2}\left(t\right)+\frac{{B}_{4}}{2}{u}_{4}^{2}\left(t\right)-\frac{{B}_{4}}{2}\hfill \\ \Rightarrow L(\mathbf{x},\mathbf{u},t)\ge min(\frac{{B}_{1}}{2},\frac{{B}_{2}}{2},\frac{{B}_{3}}{2},\frac{{B}_{4}}{2})({u}_{1}^{2}+{u}_{2}^{2}+{u}_{3}^{2}+{u}_{4}^{2})-\frac{{B}_{4}}{2}\hfill \\ \Rightarrow L(\mathbf{x},\mathbf{u},t)\ge min(\frac{{B}_{1}}{2},\frac{{B}_{2}}{2},\frac{{B}_{3}}{2},\frac{{B}_{4}}{2}){\left|({u}_{1},{u}_{2},{u}_{3},{u}_{4})\right|}^{2}-\frac{{B}_{4}}{2}\hfill \\ \mathrm{Therefore},L(\mathbf{x},\mathbf{u},t)\ge {\delta}_{1}|({u}_{1},{u}_{2},{u}_{3},{u}_{4}){|}^{\beta}-{\delta}_{2};\mathrm{where}\phantom{\rule{0.277778em}{0ex}}{\delta}_{1}=min((\frac{{B}_{1}}{2},\frac{{B}_{2}}{2},\frac{{B}_{3}}{2},\frac{{B}_{4}}{2}),{\delta}_{2}=\frac{{B}_{4}}{2}\phantom{\rule{3.33333pt}{0ex}}\phantom{\rule{3.33333pt}{0ex}}\beta =2.\hfill $$

**Optimality Conditions**:

- (i)
- for the control ${u}_{1}$ we must have,$$\frac{\partial H}{\partial {u}_{1}}={B}_{1}{u}_{1}-{h}_{1}{e}_{T}S+{h}_{2}{e}_{T}S=0\Rightarrow {\overline{u}}_{1}=\frac{1}{{B}_{1}}\left(\right)open="("\; close=")">{h}_{1}-{h}_{2}$$
- (ii)
- for the control ${u}_{2}$ we must have$$\frac{\partial H}{\partial {u}_{2}}={B}_{2}{u}_{2}-{h}_{1}{e}_{H}S+{h}_{6}{e}_{H}S=0\Rightarrow {\overline{u}}_{2}=\frac{1}{{B}_{2}}\left(\right)open="("\; close=")">{h}_{1}-{h}_{6}$$
- (iii)
- for the control ${u}_{3}$ we must have$$\begin{array}{ccc}\hfill \frac{\partial H}{\partial {u}_{3}}& =& {D}_{1}{I}_{T}+{B}_{3}{u}_{3}-{h}_{4}{I}_{T}+{h}_{5}{R}_{T}+{h}_{10}{\epsilon}_{1}{I}_{TH}-{h}_{13}{\epsilon}_{1}{I}_{TH}=0\hfill \\ & \Rightarrow & {\overline{u}}_{3}=\frac{1}{{B}_{3}}\left(\right)open="["\; close="]">({h}_{4}-{D}_{1}){I}_{T}-{h}_{5}{R}_{T}+({h}_{13}-{h}_{10}){\epsilon}_{1}{I}_{TH}.\hfill \end{array}$$
- (iv)
- Similarly, for the control ${u}_{4}$ we must have$$\begin{array}{ccc}\hfill \frac{\partial H}{\partial {u}_{4}}& =& {D}_{2}{I}_{H}+{B}_{4}{u}_{4}-{h}_{7}{I}_{H}-{h}_{8}{L}_{TH}-{h}_{10}{R}_{H}-{h}_{11}{A}_{H}+{h}_{12}({L}_{TH}+{I}_{H}+{R}_{H}+{A}_{H})=0\hfill \\ & \Rightarrow & {\overline{u}}_{4}=\frac{1}{{B}_{4}}\left(\right)open="["\; close="]">({h}_{7}-{h}_{12}-{D}_{2}){I}_{H}+({h}_{8}-{h}_{12}){L}_{TH}+({h}_{10}-{h}_{12}){R}_{H}+({h}_{11}-{h}_{12}){A}_{H}.\hfill \end{array}$$

**The adjoint (co-state) equations**:

## 4. Numerical Simulation and Results

## 5. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

## References

- World Health Organization (WHO). Global Tuberculosis Control. 2014. [Google Scholar]
- AVERT, Tuberculosis and HIV Co-Infection. Global Information and Education on HIV and AIDS. 2017. Available online: http://www.avert.org/tuberc.htm (accessed on 25 June 2017).
- Aweke, T.D.; Kassa, S.M. Impacts of vaccination and behavior change in the optimal intervention strategy for controlling the transmission of Tuberculosis. In CIM Series in Mathematical Sciences; Springer: Basel, Switzerland, 2015; Volume 2, pp. 32–55. [Google Scholar]
- CDC. Center for Disease Control and Prevention (CDC). Available online: http://www.cdc.gov/tb/publications/factsheets/drtb/mdrtb/htm (accessed on 10 July 2015).
- WHO. Tuberculosis. Fact Sheet No. 104. Available online: http://www.who.int/mediacentre/factsheets/fs104/en (accessed on August 2017).
- UNAIDS. Fact Sheet–Latest Statistics on the Status of the AIDS Epidemic. Available online: http://www.unaids.org/en/resources/fact-sheet (accessed on 01 August 2017).
- Getahun, H.; Gunneberg, C.; Granich, R.; Nunn, P. HIV infection-associated tuberculosis: The epidemiology and the response. Clin. Infect. Dis.
**2010**, 50 (Suppl. 3), S201–S207. [Google Scholar] [CrossRef] [PubMed] - AVERT, HIV & AIDS Information from AVERT.org. Available online: http://www.avert.org/worldwide-hiv-aids-statistics.htm#sthash.YzzqcNUT.dpuf (accessed on 01 August 2017).
- Silva, C.J.; Torres, D.F.M. A TB-HIV/AIDS co-infection model and optimal control treatment. Discret. Contin. Dyn. Syst. A
**2015**, 35, 4639–4663. [Google Scholar] [CrossRef] - Waaler, H.T.; Gese, A.; Anderson, S. The use of mathematical models in the study of the epidemiology of tuberculosis. Am. J. Public Health
**1962**, 52, 1002–1013. [Google Scholar] [CrossRef] - Marahatta, S.B. Multi-drug resistant tuberculosis burden and risk factors: An update. Kathmandu Univ. Med. J.
**2010**, 8, 116–125. [Google Scholar] [CrossRef] - Mishra, B.K.; Srivastava, J. Mathematical model on pulmonary and multidrug-resistant tuberculosis patients with vaccination. J. Egypt. Math. Soc.
**2014**, 22, 311–316. [Google Scholar] [CrossRef] - Jung, E.; Lenhart, S.; Feng, Z. Optimal control of treatment in a two-strain tuberculosis model. Discret. Contin. Dyn. Syst. B
**2002**, 2, 473–482. [Google Scholar] - Trauer, J.M.; Denholm, J.T.; McBryde, E.S. Construction of a mathematical model for tuberculosis transmission in highly endemic regions of the Asia-Pacific. J. Theor. Biol.
**2014**, 358, 74–84. [Google Scholar] [CrossRef] [PubMed] - Castillo-Schavez, C.; Feng, Z. To treat and not to treat: The case of tuberculosis. J. Math. Biol.
**1997**, 35, 629–656. [Google Scholar] [CrossRef] - Hansen, E. Application of Optimal Control Theory to Infectious Disease Modeling. Ph.D. Thesis, Queen’s University, Kingston, ON, Canada, 2011. [Google Scholar]
- Maliyani, M.; Mwamtobe, P.M.; Hove-Musekwa, S.D.; Tchuenche, J.M. Modelling the role of diagnosis, Treatment and Health education on Multi-Drug resistant tuberculosis dynamics. ISRN Biomath.
**2012**, 2012, 1–20. [Google Scholar] [CrossRef] - Yusuf, T.T.; Benyah, F. Optimal strategy for controlling the spread of HIV/AIDS disease: A case study of South Africa. J. Biol. Dyn.
**2012**, 6, 475–494. [Google Scholar] [CrossRef] [PubMed] - Bhunu, C.P.; Garira, W.; Magombedze, G. Mathematical Analysis of a Two Strain HIV/AIDS Model with Antiretroviral Treatment. Acta Biotheor.
**2009**, 57, 361–381. [Google Scholar] [CrossRef] [PubMed] - Naresh, M.; Tripathi, A.; Sharma, D. Modelling and analysis of the spread of AIDS epidemic with immigration of HIV infectives. Math. Comput. Model.
**2009**, 49, 880–892. [Google Scholar] [CrossRef] - Mukandavire, Z.; Gumel, A.B.; Winston, W.; Tchuenche, J.M. Mathematical Analysis of a Model for HIV-Malaria Co-Infection. Math. Biosci. Eng.
**2009**, 6, 333–362. [Google Scholar] [PubMed] - Naresh, R.; Tripathi, A. Modelling and Analysis of HIV-TB Co-Infection in Avariable Size Population; Tylor & Francis: London, UK, 2005; pp. 275–286. [Google Scholar]
- Shah, N.H.; Gupta, J. Modelling of HIV-TB Co-infection Transmission Dynamics. Am. J. Epidemiol. Infect. Dis.
**2014**, 2, 1–7. [Google Scholar] - Bacaër, N.; Ouifki, R.; Pretorius, C.; Wood, R.; Williams, B. Modeling the joint epidemics of TB and HIV in a South African township. J. Math. Biol.
**2008**, 57, 557–593. [Google Scholar] [CrossRef] [PubMed] - Wang, X.; Yang, J.; Zhang, F. Dynamic of a TB-HIV Coinfection Epidemic Model with Latent Age. J. Appl. Math.
**2013**, 2013, 1–13. [Google Scholar] [CrossRef] - Roeger, L.I.; Feng, Z.; Castillo-Chavez, C. Modelling HIV-TB Co-infection. Math. Biosci. Eng.
**2009**, 6, 815–837. [Google Scholar] [CrossRef] [PubMed] - Agusto, F.B.; Adekunle, A.I. Optimal control of a two-strain tuberculosis-HIV/AIDS co-infection model. J. BioSyst.
**2014**, 119, 20–44. [Google Scholar] [CrossRef] [PubMed] - Sharomi, O.; Podder, C.N.; Gumel, A.B. Mathematical Analysis of the Transmission Dynamics of HIV/TB Co-Infection in the Presence of Treatment. Math. Biosci. Eng.
**2008**, 5, 145–174. [Google Scholar] [PubMed] - Kassa, S.M.; Ouhinou, A. Epidemiological Models with prevalence dependent endogenous self-protection measure. Math. Biosci.
**2011**, 229, 41–49. [Google Scholar] [CrossRef] [PubMed] - Wilson, D.P.; Law, M.G.; Grulich, A.E.; Cooper, D.A.; Kaldor, J.M. Relation between HIV viral load and infectiousness: A model-based analysis. Lancet
**2008**, 372, 314–320. [Google Scholar] [CrossRef] - Deeks, S.G.; Lewin, S.R.; Havlir, D.V. The end of AIDS: HIV infection as a chronic disease. Lancet
**2013**, 382, 1525–1533. [Google Scholar] [CrossRef] - Kwan, C.K.; Ernst, J.D. HIV and tuberculosis: A deadly human syndemic. Clin. Microbiol. Rev.
**2011**, 24, 351–376. [Google Scholar] [CrossRef] [PubMed] - Lakshmikantham, V.; Leela, S.; Martynyuk, A.A. Stability Analysis of Nonlinear Systems; Marcel Dekker Inc.: New York, NY, USA; Basel, Switzerland, 1989. [Google Scholar]
- Hethcote, H.W. The mathematics of infectious diseases. SIAM Rev.
**2000**, 42, 599–653. [Google Scholar] [CrossRef] - Ma, S.; Xia, Y. Mathematical Understanding of Infectious Disease Dynamics; World Scientific Publishing Co.: London, UK, 2009; Volume 16. [Google Scholar]
- Van-den-Driessche, P.; Watmough, J. Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission. Math. Biosci.
**2002**, 180, 29–48. [Google Scholar] [CrossRef] - Kassa, S.M.; Ouhinou, A. The impact of self-protective measures in the optimal interventions for controlling infectious diseases of human population. J. Math. Biol.
**2015**, 70, 213–236. [Google Scholar] [CrossRef] [PubMed] - Coddington, E.A.; Levinson, N. Theory of Ordinary Differential Equations; McGraw Hill Co. Inc.: New York, NY, USA, 1955. [Google Scholar]
- Grass, D.; Caulkins, J.P.; Feichtinger, G.; Tragler, G.; Behrens, D.A. Optimal Control of Nonlinear Processes, with Applications in Drugs, Corruption, and Terror; Springer-Verlag: Berlin/Heidelberg, Germany, 2008. [Google Scholar]
- Coddington, E.A. An Introduction to Ordinary Differential Equations; Prentice-Hall Inc.: Upper Saddle River, NJ, USA, 1961. [Google Scholar]
- Barbu, V.; Precupanu, T. Convexity and Optimization in Banach Spaces, 4th ed.; Springer Verlag: Dordrecht, The Netherlands, 2010. [Google Scholar]
- Pedregal, P. Introduction to Optimization; Springer-Verlag: New York, NY, USA, 2004. [Google Scholar]
- Bekele, B.T. Modeling Tuberculosis Dynamics in Children and Adults in the Presence of Vaccination. Master’s Thesis, Stellenbosch University, Stellenbosh, South Africa, 2010. [Google Scholar]

**Figure 1.**Schematic diagram for TB-HIV/AIDS compartmental model that includes behavior change and treatment.

**Figure 2.**Graphs for the trajectories for the sub-populations in (

**a**,

**b**) and the corresponding values for the prevalence in (

**c**,

**d**), when no control efforts are applied. In this case, parameter values ${\beta}_{1}=1.5$ and ${\beta}_{2}=0.36$ are used and the rest of the parameters are as in Table 2.

**Figure 3.**Graphs for the prevalence of (

**a**) HIV and (

**b**) TB and the corresponding values for the marginal cost in (

**c**), when various combinations of control efforts are applied. All the parameter values are as described in the caption of Figure 2.

**Figure 4.**Graphs for the Optimal controls (

**a**) when all the controls are applied and (

**b**) when only preventive controls are applied, and (

**c**) when only treatment controls are applied. All the parameter values are as described in the caption of Figure 2.

**Figure 5.**Graphs for the trajectories of the sub-populations when controls are applied. All the parameter values are as described in the caption of Figure 2.

Parameters | Description |
---|---|

$\Pi $ | Recruitment rate |

${\beta}_{1}$ | tuberculosis (TB) Transmission rate |

${\beta}_{2}$ | Human Immunodeficiency Virus (HIV) Transmission rate |

b | Endogenous reactivation rate for TB |

k | Reinfection rate for TB infection |

${b}_{1}$ | Endogenous reactivation rate of TB for individuals pre-infected with HIV |

${k}_{1}$ | Reinfection rate of TB infection for individuals pre-infected with HIV |

$\mu $ | Per capita natural mortality rate |

${d}_{T},{d}_{H},{d}_{TA}$ | Per capita TB, HIV, TB-HIV co-infection- induced death rates |

$\tau $ | TB treatment rate for ${I}_{T}$ individuals |

${\tau}_{1}$ | The rate at which individuals from ${L}_{TH}$ class recover from TB |

$\sigma $ | Rate of recruitment for ${I}_{H}$ individuals to receive HIV treatment |

m | Rate at which individual who are Latently infected with TB progress to ${I}_{T}$ |

g | The proportions of susceptible individuals who get infected with TB and move to ${L}_{T}$ |

$\delta ,\rho ,\theta ,{\theta}_{1}$ | Modification parameters |

$\epsilon $ | The rate of progression of individuals from ${I}_{H}$ class to ${A}_{H}$ |

${\epsilon}_{1}$ | The rate of progression of individuals from ${I}_{TH}$ class to ${A}_{T}$ |

$\psi $ | Rate of recruitment of individuals in ${A}_{H}$ class for HIV Treatment |

${\psi}_{1}$ | Rate of recruitment of individuals in ${A}_{T}$ class for HIV Treatment |

$\xi $ | Rate of failure to properly adhere to HIV treatment rules |

$\omega $ | The rate at which individuals from ${R}_{H}$ class progress to ${A}_{H}$ |

$\gamma $ | The average effectiveness of all existing self protective measures for TB |

${\gamma}_{1}$ | The average effectiveness of all existing self protective measures for HIV |

p | Rate at which individuals in ${I}_{TH}$ class to receive treatment for both HIV and TB |

${\alpha}_{T}$ | Rate of dissemination of information about TB disease in the population |

${\alpha}_{H}$ | Rate of dissemination of information about HIV/AIDS (Acquired Immunodeficiency |

Syndrome) disease in the population | |

r | Fraction of individuals from ${I}_{TH}$ class that receive treatments for TB |

Parameters | Description | Value | References |
---|---|---|---|

$\Pi $ | Recruitment rate | 1500 | [27] |

${\beta}_{1}$ | TB Transmission rate | Variable | |

${\beta}_{2}$ | HIV Transmission rate | Variable | |

b | Endogenous reactivation rates for TB | 0.003 | [3,43] |

k | Reinfection rates for TB infection | 0.02 | [3,43] |

${b}_{1}$ | Endogenous reactivation rates of TB for individuals pre-infected with HIV | 0.2 | Assumed |

${k}_{1}$ | Reinfection rates of TB infection for individuals pre-infected with HIV | 0.5 | Assumed |

$\mu $ | Per capita natural rate of mortality | $1/48$ | Assumed |

${d}_{T},{d}_{H},{d}_{TA}$ | TB, HIV, both TB & HIV death rates | 0.1, 0.2, 0.33 | [9,15] |

${\sigma}_{0}$ | Baseline HIV treatment rate | 0.16 | Assumed |

${\tau}_{0}$ | Baseline TB treatment rate | 0.16 | Assumed |

${\tau}_{1}$ | Rate at which individuals from ${L}_{TH}$ class recover from TB infection | 0.2 | [9,13,15] |

m | Rate at which TB-Latent individuals progress to ${I}_{T}$ | 0.5 | [9,13,15] |

g | Proportion of susceptibles individuals who get infected by TB and move to ${L}_{T}$ | 0.85 | [3] |

$\delta ,{\rho}_{1},\rho ,\theta ,{\theta}_{1}$ | Modification parameters | 1.03, 1.17, 1.07, 0.9, 1 | [9] |

${\eta}_{a},{\eta}_{c}$ | Modification parameters | 1.05, 0.08 | [29] |

$\nu ,{\nu}_{1}$ | Modification parameters | 0.75, 0.5 | Assumed |

$\epsilon $ | The rate of progression of ${I}_{H}$ to ${A}_{H}$ | 0.1 | [27] |

${\epsilon}_{1}$ | The rate of progression of ${I}_{TH}$ to ${A}_{T}$ | 0.2 | Assumed |

$\psi $ | Treatment rate for individuals in ${A}_{H}$ | 0.33 | [9] |

${\psi}_{1}$ | HIV treatment rate for ${A}_{T}$ individuals | 0.33 | [9] |

$\xi $ | Rate of failure to adhere to HIV treatments | 0.08 | Assumed |

$\omega $ | Rate at which individuals in ${R}_{H}$ progress to ${A}_{H}$ | 0.1 | Assumed |

r | fraction of individuals from ${I}_{TH}$ that receive treatment only for TB | 0.33 | Assumed |

p | Baseline rate at which individuals in ${I}_{TH}$ receive treatments for both TB & HIV | 0.40 | Assumed |

${\alpha}_{T0}$ | Baseline rate of dissemination of information about TB disease | 0.12 | Assumed |

${\alpha}_{H0}$ | Baseline rate of dissemination of information about HIV | 0.18 | [29] |

$\gamma $ | Effectiveness of existing self-preventive measures for TB | 0.93 | Assumed |

${\gamma}_{1}$ | Effectiveness of existing self-preventive measures for HIV | 0.93 | [37] |

${p}_{T*}$ | Prevalence producing half of the maximum behavioural change value for TB | 0.09 | [3] |

${p}_{H*}$ | Prevalence producing half of the maximum behavioural change value for HIV | 0.09 | [29] |

n | Level of reaction of the population for diseases | 2 | [29] |

© 2018 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (http://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Awoke, T.D.; Kassa, S.M.
Optimal Control Strategy for TB-HIV/AIDS Co-Infection Model in the Presence of Behaviour Modification. *Processes* **2018**, *6*, 48.
https://doi.org/10.3390/pr6050048

**AMA Style**

Awoke TD, Kassa SM.
Optimal Control Strategy for TB-HIV/AIDS Co-Infection Model in the Presence of Behaviour Modification. *Processes*. 2018; 6(5):48.
https://doi.org/10.3390/pr6050048

**Chicago/Turabian Style**

Awoke, Temesgen Debas, and Semu Mitiku Kassa.
2018. "Optimal Control Strategy for TB-HIV/AIDS Co-Infection Model in the Presence of Behaviour Modification" *Processes* 6, no. 5: 48.
https://doi.org/10.3390/pr6050048