American Institute of Mathematical Sciences

Advanced
  • Previous Article
    An iterative approach to $L^\infty$-boundedness in quasilinear Keller-Segel systems
  • PROC Home
  • This Issue
  • Next Article
    Positive solutions of a fractional boundary value problem with a fractional derivative boundary condition
2015, 2015(special): 621-634. doi: 10.3934/proc.2015.0621

Optimal control and stability analysis of an epidemic model with education campaign and treatment

Sanjukta Hota 1, , Folashade Agusto 2, , Hem Raj Joshi 3, and  Suzanne Lenhart 4,

1. 

Department of Mathematics and Computer Science, Fisk University, Nashville TN 37208, United States
2. 

Department of Ecology and Evolutionary Biology, University of Kansas, Lawrence, KS 66045
3. 

Department of Mathematics, Xavier University, Cincinnati, OH 45207-4441, United States
4. 

Department of Mathematics, University of Tennessee, Knoxville, TN 37996-1300

Received  September 2014 Revised  September 2015 Published  November 2015

In this paper we investigated a SIR epidemic model in which education campaign and treatment are both important for the disease management. Optimal control theory was used on the system of differential equations to achieve the goal of minimizing the infected population and slow down the epidemic outbreak. Stability analysis of the disease free equilibrium of the system was completed. Numerical results with education campaign levels and treatment rates as controls are illustrated.
Keywords: treatment and education., epidemic model, stability analysis, Optimal Control.
Mathematics Subject Classification: Primary: 58F15, 58F17; Secondary: 53C3.
Citation: Sanjukta Hota, Folashade Agusto, Hem Raj Joshi, Suzanne Lenhart. Optimal control and stability analysis of an epidemic model with education campaign and treatment. Conference Publications, 2015, 2015 (special) : 621-634. doi: 10.3934/proc.2015.0621
References:
[1]

R. M. Anderson and R. May, Infectious Diseases of Humans, Oxford University Press, (1991), New York.

[2]

D. N. Bhatta, U. R. Aryal and K. Khanal, Education: The Key to Curb HIV and AIDS Epidemic, Kathmandu Univ Med J, 42(2) (2013), 158-161.

[3]

H. Behncke, Optimal control of deterministic epidemics, Optimal Control Applications and Methods, 21 (2000), 269-285.

[4]

O. Diekmann, J. A. P. Heesterbeek and J. A. P. Metz, On the definition and computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol, 28 (1990) 503-522.

[5]

E. Fenichel, C. Castillo-Chavez and C. Villalobos, et al, Adaptive human behavior in epidemiological models, Proceedings of the National Academy of Sciences of the United States of America, 108(15) (2011), 6306-6311.

[6]

S. Funk, M. Salath and V. Jansen, Modelling the influence of human behaviour on the spread of infectious diseases: A review, J R Soc Interface, 7 (2010), 1247-1256.

[7]

E. Green, D. Halperin, V. Nantulya, and J. Hogle, Uganda's HIV Prevention Success: The Role of Sexual Behavior Change and the National Response, AIDS Behav., 10(4) (2006), 335-346.

[8]

Global Campaign for Education (GCE), Learning to survive: How education for all would save millions of young people from HIV/AIDS, London, GCE, 2004, http://www.campaignforeducation.org/resources.

[9]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42(4) (2000), 599-653.

[10]

H. R. Joshi, S. Lenhart, K. Albright and K. Gipson, Modeling the Effect of Information Campaigns On the HIV Epidemic In Uganda, Mathematical Biosciences and Engineering, 5(4) (2008), 757-770.

[11]

H. R. Joshi, S. Lenhart, S. Hota and F. Agusto, Optimal Control of SIR Model with Changing Behavior through an Education Campaign, Electronic Journal of Differential Equations, 50 (2015), 1-14.

[12]

W. O. Kermack and A. G. McKendrick, Contribution to the mathematical theory of epidemics, Proc. Roy. Soc., A 115 (1927), 700-721.

[13]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics II. The problem of endemicity, Proc. Roy. Soc., A 138 (1932), 55-83.

[14]

H. Laarabi, M. Rachik, O. E. Kahlaoui and E. H. Labriji, Optimal Vaccination Strategies of an SIR Epidemic Model with a Saturated Treatment, Universal Journal of Applied Mathematics, 1(3) (2013), 185-191.

[15]

S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, 1st Edition, Chapman and Hall/CRC, 2007.

[16]

D. Low-Beer, R. Stoneburner, T. Barnett and M. Whiteside, Knowledge Diffusion and Personalizing Risk: Key Indicators of Behavior Change in Uganda Compared to South Africa, XIII International AIDS Conference,

[17]

S. Okware, J. Kinsman, S. Onyango, A. Opio, and P. Kaggwa, Revisiting the ABC strategy: HIV prevention in Uganda in the era of antiretroviral therapy, Postgraduate Medical Journal, 81(960) (2005), 625-628.

[18]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience Publishers, 1962.

[19]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.

[20]

Control of communicable diseases and prevention of epidemics, Environmental health in emergencies and disasters: a practical guide, World Health Organization 2003, http://www.who.int/water_sanitation_health/hygiene/emergencies/em2002chap11.pdf

show all references

References:
[1]

R. M. Anderson and R. May, Infectious Diseases of Humans, Oxford University Press, (1991), New York.

[2]

D. N. Bhatta, U. R. Aryal and K. Khanal, Education: The Key to Curb HIV and AIDS Epidemic, Kathmandu Univ Med J, 42(2) (2013), 158-161.

[3]

H. Behncke, Optimal control of deterministic epidemics, Optimal Control Applications and Methods, 21 (2000), 269-285.

[4]

O. Diekmann, J. A. P. Heesterbeek and J. A. P. Metz, On the definition and computation of the basic reproduction ratio $R_0$ in models for infectious diseases in heterogeneous populations, J. Math. Biol, 28 (1990) 503-522.

[5]

E. Fenichel, C. Castillo-Chavez and C. Villalobos, et al, Adaptive human behavior in epidemiological models, Proceedings of the National Academy of Sciences of the United States of America, 108(15) (2011), 6306-6311.

[6]

S. Funk, M. Salath and V. Jansen, Modelling the influence of human behaviour on the spread of infectious diseases: A review, J R Soc Interface, 7 (2010), 1247-1256.

[7]

E. Green, D. Halperin, V. Nantulya, and J. Hogle, Uganda's HIV Prevention Success: The Role of Sexual Behavior Change and the National Response, AIDS Behav., 10(4) (2006), 335-346.

[8]

Global Campaign for Education (GCE), Learning to survive: How education for all would save millions of young people from HIV/AIDS, London, GCE, 2004, http://www.campaignforeducation.org/resources.

[9]

H. W. Hethcote, The mathematics of infectious diseases, SIAM Rev., 42(4) (2000), 599-653.

[10]

H. R. Joshi, S. Lenhart, K. Albright and K. Gipson, Modeling the Effect of Information Campaigns On the HIV Epidemic In Uganda, Mathematical Biosciences and Engineering, 5(4) (2008), 757-770.

[11]

H. R. Joshi, S. Lenhart, S. Hota and F. Agusto, Optimal Control of SIR Model with Changing Behavior through an Education Campaign, Electronic Journal of Differential Equations, 50 (2015), 1-14.

[12]

W. O. Kermack and A. G. McKendrick, Contribution to the mathematical theory of epidemics, Proc. Roy. Soc., A 115 (1927), 700-721.

[13]

W. O. Kermack and A. G. McKendrick, Contributions to the mathematical theory of epidemics II. The problem of endemicity, Proc. Roy. Soc., A 138 (1932), 55-83.

[14]

H. Laarabi, M. Rachik, O. E. Kahlaoui and E. H. Labriji, Optimal Vaccination Strategies of an SIR Epidemic Model with a Saturated Treatment, Universal Journal of Applied Mathematics, 1(3) (2013), 185-191.

[15]

S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, 1st Edition, Chapman and Hall/CRC, 2007.

[16]

D. Low-Beer, R. Stoneburner, T. Barnett and M. Whiteside, Knowledge Diffusion and Personalizing Risk: Key Indicators of Behavior Change in Uganda Compared to South Africa, XIII International AIDS Conference,

[17]

S. Okware, J. Kinsman, S. Onyango, A. Opio, and P. Kaggwa, Revisiting the ABC strategy: HIV prevention in Uganda in the era of antiretroviral therapy, Postgraduate Medical Journal, 81(960) (2005), 625-628.

[18]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience Publishers, 1962.

[19]

P. van den Driessche and J. Watmough, Reproduction numbers and sub-threshold endemic equilibria for compartmental models of disease transmission, Math. Biosci., 180 (2002), 29-48.

[20]

Control of communicable diseases and prevention of epidemics, Environmental health in emergencies and disasters: a practical guide, World Health Organization 2003, http://www.who.int/water_sanitation_health/hygiene/emergencies/em2002chap11.pdf

[1]

Yali Yang, Sanyi Tang, Xiaohong Ren, Huiwen Zhao, Chenping Guo. Global stability and optimal control for a tuberculosis model with vaccination and treatment. Discrete and Continuous Dynamical Systems - B, 2016, 21 (3) : 1009-1022. doi: 10.3934/dcdsb.2016.21.1009
[2]

Xiaomei Feng, Zhidong Teng, Kai Wang, Fengqin Zhang. Backward bifurcation and global stability in an epidemic model with treatment and vaccination. Discrete and Continuous Dynamical Systems - B, 2014, 19 (4) : 999-1025. doi: 10.3934/dcdsb.2014.19.999
[3]

Hongyong Zhao, Peng Wu, Shigui Ruan. Dynamic analysis and optimal control of a three-age-class HIV/AIDS epidemic model in China. Discrete and Continuous Dynamical Systems - B, 2020, 25 (9) : 3491-3521. doi: 10.3934/dcdsb.2020070
[4]

Cristiana J. Silva, Delfim F. M. Torres. A TB-HIV/AIDS coinfection model and optimal control treatment. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4639-4663. doi: 10.3934/dcds.2015.35.4639
[5]

Ellina Grigorieva, Evgenii Khailov, Andrei Korobeinikov. An optimal control problem in HIV treatment. Conference Publications, 2013, 2013 (special) : 311-322. doi: 10.3934/proc.2013.2013.311
[6]

Xun-Yang Wang, Khalid Hattaf, Hai-Feng Huo, Hong Xiang. Stability analysis of a delayed social epidemics model with general contact rate and its optimal control. Journal of Industrial and Management Optimization, 2016, 12 (4) : 1267-1285. doi: 10.3934/jimo.2016.12.1267
[7]

Junyuan Yang, Yuming Chen, Jiming Liu. Stability analysis of a two-strain epidemic model on complex networks with latency. Discrete and Continuous Dynamical Systems - B, 2016, 21 (8) : 2851-2866. doi: 10.3934/dcdsb.2016076
[8]

Kazuyuki Yagasaki. Optimal control of the SIR epidemic model based on dynamical systems theory. Discrete and Continuous Dynamical Systems - B, 2022, 27 (5) : 2501-2513. doi: 10.3934/dcdsb.2021144
[9]

Hongshan Ren. Stability analysis of a simplified model for the control of testosterone secretion. Discrete and Continuous Dynamical Systems - B, 2004, 4 (3) : 729-738. doi: 10.3934/dcdsb.2004.4.729
[10]

Maciej Leszczyński, Urszula Ledzewicz, Heinz Schättler. Optimal control for a mathematical model for anti-angiogenic treatment with Michaelis-Menten pharmacodynamics. Discrete and Continuous Dynamical Systems - B, 2019, 24 (5) : 2315-2334. doi: 10.3934/dcdsb.2019097
[11]

Andrea Signori. Penalisation of long treatment time and optimal control of a tumour growth model of Cahn–Hilliard type with singular potential. Discrete and Continuous Dynamical Systems, 2021, 41 (6) : 2519-2542. doi: 10.3934/dcds.2020373
[12]

Maria do Rosário de Pinho, Helmut Maurer, Hasnaa Zidani. Optimal control of normalized SIMR models with vaccination and treatment. Discrete and Continuous Dynamical Systems - B, 2018, 23 (1) : 79-99. doi: 10.3934/dcdsb.2018006
[13]

M. Teresa T. Monteiro, Isabel Espírito Santo, Helena Sofia Rodrigues. An optimal control problem applied to a wastewater treatment plant. Discrete and Continuous Dynamical Systems - S, 2022, 15 (3) : 587-601. doi: 10.3934/dcdss.2021153
[14]

Changjie Fang, Weimin Han. Stability analysis and optimal control of a stationary Stokes hemivariational inequality. Evolution Equations and Control Theory, 2020, 9 (4) : 995-1008. doi: 10.3934/eect.2020046
[15]

Marek Bodnar, Monika Joanna Piotrowska, Urszula Foryś. Gompertz model with delays and treatment: Mathematical analysis. Mathematical Biosciences & Engineering, 2013, 10 (3) : 551-563. doi: 10.3934/mbe.2013.10.551
[16]

Tao Feng, Zhipeng Qiu. Global analysis of a stochastic TB model with vaccination and treatment. Discrete and Continuous Dynamical Systems - B, 2019, 24 (6) : 2923-2939. doi: 10.3934/dcdsb.2018292
[17]

Elena Fimmel, Yury S. Semenov, Alexander S. Bratus. On optimal and suboptimal treatment strategies for a mathematical model of leukemia. Mathematical Biosciences & Engineering, 2013, 10 (1) : 151-165. doi: 10.3934/mbe.2013.10.151
[18]

Mudassar Imran, Hal L. Smith. A model of optimal dosing of antibiotic treatment in biofilm. Mathematical Biosciences & Engineering, 2014, 11 (3) : 547-571. doi: 10.3934/mbe.2014.11.547
[19]

Federico Papa, Francesca Binda, Giovanni Felici, Marco Franzetti, Alberto Gandolfi, Carmela Sinisgalli, Claudia Balotta. A simple model of HIV epidemic in Italy: The role of the antiretroviral treatment. Mathematical Biosciences & Engineering, 2018, 15 (1) : 181-207. doi: 10.3934/mbe.2018008
[20]

Nasser Sweilam, Fathalla Rihan, Seham AL-Mekhlafi. A fractional-order delay differential model with optimal control for cancer treatment based on synergy between anti-angiogenic and immune cell therapies. Discrete and Continuous Dynamical Systems - S, 2020, 13 (9) : 2403-2424. doi: 10.3934/dcdss.2020120

 Impact Factor: 

Tools

Metrics

  • PDF downloads (255)
  • HTML views (0)
  • Cited by (0)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy