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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

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