Stability analysis of a delayed social epidemics model with general contact rate and its optimal control

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October 2016, 12(4): 1267-1285. doi: 10.3934/jimo.2016.12.1267

Stability analysis of a delayed social epidemics model with general contact rate and its optimal control

Xun-Yang Wang ^1, , Khalid Hattaf ^2, , Hai-Feng Huo ^1, and Hong Xiang ^3,

1.	College of Electrical and Information engineering, Lanzhou University of Technology, Lanzhou, Gansu, 730050, China, China
2.	Department of Mathematics and Computer Science, Faculty of Sciences Ben M'sik, Hassan II University, P.O. Box 7955, Sidi Othman, Casablanca, Morocco
3.	Department of Applied Mathematics, Lanzhou University of Technology, Lanzhou, Gansu, 730050, China

Received May 2014 Revised October 2015 Published January 2016

Abstract
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In this paper, we formulate an alcohol quitting model in which we consider the impact of distributed time delay between contact and infection process by characterizing dynamic nature of alcoholism behaviours, and we generalize the infection rate to the general case, simultaneously, we consider two different control strategies. Next, we discuss the qualities on the model, the existence and boundedness as well as positivity of the equilibrium are involved. Then, under certain proper conditions, we construct appropriate Lyapunov functionals to prove the global stability of alcohol free equilibrium point $E_{0}$ and alcoholism equilibrium $E^{*}$ respectively. Furthermore, the optimal control strategies are derived by proposing an objective functional and using classic Pontryagin's Maximum Principle. Numerical simulations are conducted to support our theoretical results derived in optimal control.

Keywords: general infection rate, stability analysis, Alcoholism dynamics, time delay, delay optimal control..

Mathematics Subject Classification: Primary: 92D2.

Citation: 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

References:

[1]	J. Adnani, K. Hattaf and N. Yousfi, Stability Analysis of a Stochastic SIR Epidemic Model with Specific Nonlinear Incidence Rate, International Journal of Stochastic Analysis, (2013), Article ID 431257, 4 pages.
[2]	E. Beretta, T. Hara, W. Ma and Y. Takeuchi, Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonlinear Anal., 47 (2001), 4107-4115. doi: 10.1016/S0362-546X(01)00528-4.
[3]	Centers for Disease Control, Alcohol-related disease impact (ARDI) software, Atlanta, GA: CDC, 2004.
[4]	Centers for Disease Control and Prevention, Binge Drinking,, , ().
[5]	C. M. Chen and H. Yi, Trends in Alcohol-related Morbidity Among Short-stay Community Hospital Discharges United States, 1979-2007, US Department of Health and Human Services, Public Health Service, National Institutes of Health, 2010.
[6]	C. Chen and C. Storr, et al., Early-onset drug use and risk for drug dependence problems, Addictive Behaviors, 34 (2009), 319-322. doi: 10.1016/j.addbeh.2008.10.021.
[7]	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. doi: 10.1016/S0025-5564(02)00108-6.
[8]	K. R. Fister, S. Lenhart and J. S. Mcnally, Optimizing chemotherapy in an HIV model, Electronic Journal of Differential Equations, 32 (1998), 1-12.
[9]	W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, No. 1, Springer Verlag, Berlin-New York, 1975.
[10]	M. Glavas and J. Weinberg, Stress, alcohol consumption and the hypothalamic-pituitary-adrenal axis, in Nutrients, Stress, and Medical Disorders, 2005, 165-183. doi: 10.1385/1-59259-952-4:165.
[11]	L. Göllmann, Theory and applications of optimal control problems with multiple time-delays, Journal of Industrial and management Optimization, 10 (2014), 413-441.
[12]	L. Göllmann, D. Kern and H. Maurer, Optimal control problems with delays in state and control variables subject to mixed control-state constraints, Optim. Control Appl. Meth., 32 (2008), 1-25.
[13]	A. Halanay and J. A. Yorke, Some new results and problems in the theory of differential-delay equations, SIAM Rev, 13 (1971), 55-80. doi: 10.1137/1013004.
[14]	J. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.
[15]	K. Hattaf, A. A. Lashari, Y. Louartassi and N. Yousfi, A delayed SIR epidemic model with general incidence rate, Electronic Journal of Qualitative Theory of Differential Equations, 3 (2013), 1-9.
[16]	K. Hattaf and N. Yousfi, Optimal control of a delayed HIV infection model with immune response using an efficient numerical method, ISRN Biomathematics, 2012 (2012), Article ID 215124, 7 pages. doi: 10.5402/2012/215124.
[17]	K. Hattaf, N. Yousfi and A. Tridane, A Delay virus dynamics model with general incidence rate, Differ. Equ. Dyn. Syst., 22 (2014), 181-190. doi: 10.1007/s12591-013-0167-5.
[18]	K. Hattaf, N. Yousfi and A. Tridane, Stability analysis of a virus dynamics model with general incidence rate and two delays, Appl. Math. Comput., 221 (2013), 514-521. doi: 10.1016/j.amc.2013.07.005.
[19]	K. Hattaf and N. Yousfi, Global dynamics of a delay reaction-diffusion model for viral infection with specific functional response, Comp. Appl. Math., 34 (2015), 807-818. doi: 10.1007/s40314-014-0143-x.
[20]	F. Herbert, Heavy Drinking: The Myth of Alcoholism as a Disease, University of California Press, 1988.
[21]	H. W. Hethcote, Qualitative analyses of communicable disease models, Math. Biosci., 28 (1976), 335-356. doi: 10.1016/0025-5564(76)90132-2.
[22]	G. Huang, W. Ma and Y. Takeuchi, Global analysis for delay virus dynamics model with Beddington-DeAngelis functional response, Applied Mathematics Letters, 24 (2011), 1199-1203. doi: 10.1016/j.aml.2011.02.007.
[23]	H. F. Huo and N. N. Song, Global stability for a binge drinking model with two stages, Discrete Dynamics in Nature and Society, (2012), Art. ID 829386, 15pp.
[24]	Y. Kuang, Delay Differential Equations with Application in Population Dynamics, Academic Press, New York, 1993.
[25]	E. Kuntsche and R. Knibbe, et al., Why do young people drink? A review of drinking motives, Clinical Psychology Review, 25 (2005), 841-861. doi: 10.1016/j.cpr.2005.06.002.
[26]	H. Laarabi, A. Abta and K. Hattaf, Optimal control of a delayed sirs epidemic model with vaccination and treatment, Acta Biotheor., 63 (2015), 87-97. doi: 10.1007/s10441-015-9244-1.
[27]	J. P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1976.
[28]	S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, Chapman & Hall/CRC Mathematical and Computational Biology Series, Boca Raton, FL, 2007.
[29]	E. M. Lotfi, M. Maziane, K. Hattaf and N. Yousfi, Partial differential equations of an epidemic model with spatial diffusion, International Journal of Partial Differential Equations, 2014 (2014), Article ID 186437, 6pp. doi: 10.1155/2014/186437.
[30]	D. L. Lukes, Differential Equations: Classical to Controlled, Mathematics in Science and Engineering, Academic Press, New York, 1982.
[31]	W. Ma, Y. Takeuchi, T. Hara and E. Beretta, Permanence of an SIR epidemic model with distributed time delays, Tohoku Math. J., 54 (2002), 581-591. doi: 10.2748/tmj/1113247650.
[32]	C. Connell McCluskey, Complete global stability for an SIR epidemic model with delay-Distributed or discrete, Nonlinear Analysis: Real World Applications, 11 (2010), 55-59. doi: 10.1016/j.nonrwa.2008.10.014.
[33]	C. Connell McCluskey, Global stability of an SIR epidemic model with delay and general nonlinear incidence, Mathematical Biosciences and Engineering, 7 (2010), 837-850. doi: 10.3934/mbe.2010.7.837.
[34]	A. K. Misra, A. Sharma and V. Singh, Effect of awareness programs in controlling the prevelence of an epidemic with time delay, Journal of Biological Systems, 19 (2011), 389-402. doi: 10.1142/S0218339011004020.
[35]	A. Mubayi and P. Greenwood, et al., The impact of relative residence times on the distribution of heavy drinkers in highly distinct environments, Socio-Economic Planning Sciences, 44 (2010), 45-56. doi: 10.1016/j.seps.2009.02.002.
[36]	D. Muller and R. Koch, et al., Neurophisiologic findings in chronic alcohol abuse, Pyschiatr. Neurol. Med. Pyschol., 37 (1985), 129-132.
[37]	G. Mulone and B. Straughan, Modeling binge drinking, International Journal of Biomathematics, 5 (2012), 1250005, 14pp. doi: 10.1142/S1793524511001453.
[38]	S. S. Mushayabasa and C. P. Bhunu, Modelling the effects of heavy alcohol consumption on the transmission dynamics of gonorrhea, Nonlinear Dynamics, 66 (2011), 695-706. doi: 10.1007/s11071-011-9942-4.
[39]	L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Wiley, New Jersey, 1962.
[40]	R. Room, T. Babor and J. Rehm, Alcohol and public health, The Lancet, 365 (2005), 519-530. doi: 10.1016/S0140-6736(05)70276-2.
[41]	F. Sanchez and X. Wang, et al., Drinking as an Epidemica Simple Mathematical Model with Recovery and Relapse, Guide to Evidence-Based Relapse Prevention, Elsevier, New York, 2007.
[42]	Y. Takeuchi, W. Ma and E. Beretta, Global asymptotic properties of a delay SIR epidemic model with finite incubation times, Nonlinear Anal., 42 (2000), 931-947. doi: 10.1016/S0362-546X(99)00138-8.
[43]	G. Testino, Alcoholic diseases in hepato-gastroenterology: A point of view, Hepato-Gastroenterology, 55 (2008), 371-377.
[44]	H. Walter and K. Gutierrez, et al., Gender specific differences in alcoholism: Implications for treatment, Arch. Womens Ment. Health, 6 (2003), 253-258. doi: 10.1007/s00737-003-0014-8.
[45]	X.-Y. Wang, H.-F. Huo, Q.-K. Kong and W.-X. Shi, Optimal control strategies in an alcoholism model, Abstract and Applied Analysis, (2014), Article ID 954069, 18pp. doi: 10.1155/2014/954069.
[46]	R. Xu and Z. Ma, Global stability of a SIR epidemic model with nonlinear incidence rate and time delay, Nonlinear Anal. RWA, 10 (2009), 3175-3189. doi: 10.1016/j.nonrwa.2008.10.013.

show all references

References:

[1]	J. Adnani, K. Hattaf and N. Yousfi, Stability Analysis of a Stochastic SIR Epidemic Model with Specific Nonlinear Incidence Rate, International Journal of Stochastic Analysis, (2013), Article ID 431257, 4 pages.
[2]	E. Beretta, T. Hara, W. Ma and Y. Takeuchi, Global asymptotic stability of an SIR epidemic model with distributed time delay, Nonlinear Anal., 47 (2001), 4107-4115. doi: 10.1016/S0362-546X(01)00528-4.
[3]	Centers for Disease Control, Alcohol-related disease impact (ARDI) software, Atlanta, GA: CDC, 2004.
[4]	Centers for Disease Control and Prevention, Binge Drinking,, , ().
[5]	C. M. Chen and H. Yi, Trends in Alcohol-related Morbidity Among Short-stay Community Hospital Discharges United States, 1979-2007, US Department of Health and Human Services, Public Health Service, National Institutes of Health, 2010.
[6]	C. Chen and C. Storr, et al., Early-onset drug use and risk for drug dependence problems, Addictive Behaviors, 34 (2009), 319-322. doi: 10.1016/j.addbeh.2008.10.021.
[7]	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. doi: 10.1016/S0025-5564(02)00108-6.
[8]	K. R. Fister, S. Lenhart and J. S. Mcnally, Optimizing chemotherapy in an HIV model, Electronic Journal of Differential Equations, 32 (1998), 1-12.
[9]	W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control, No. 1, Springer Verlag, Berlin-New York, 1975.
[10]	M. Glavas and J. Weinberg, Stress, alcohol consumption and the hypothalamic-pituitary-adrenal axis, in Nutrients, Stress, and Medical Disorders, 2005, 165-183. doi: 10.1385/1-59259-952-4:165.
[11]	L. Göllmann, Theory and applications of optimal control problems with multiple time-delays, Journal of Industrial and management Optimization, 10 (2014), 413-441.
[12]	L. Göllmann, D. Kern and H. Maurer, Optimal control problems with delays in state and control variables subject to mixed control-state constraints, Optim. Control Appl. Meth., 32 (2008), 1-25.
[13]	A. Halanay and J. A. Yorke, Some new results and problems in the theory of differential-delay equations, SIAM Rev, 13 (1971), 55-80. doi: 10.1137/1013004.
[14]	J. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations, Springer-Verlag, New York, 1993. doi: 10.1007/978-1-4612-4342-7.
[15]	K. Hattaf, A. A. Lashari, Y. Louartassi and N. Yousfi, A delayed SIR epidemic model with general incidence rate, Electronic Journal of Qualitative Theory of Differential Equations, 3 (2013), 1-9.
[16]	K. Hattaf and N. Yousfi, Optimal control of a delayed HIV infection model with immune response using an efficient numerical method, ISRN Biomathematics, 2012 (2012), Article ID 215124, 7 pages. doi: 10.5402/2012/215124.
[17]	K. Hattaf, N. Yousfi and A. Tridane, A Delay virus dynamics model with general incidence rate, Differ. Equ. Dyn. Syst., 22 (2014), 181-190. doi: 10.1007/s12591-013-0167-5.
[18]	K. Hattaf, N. Yousfi and A. Tridane, Stability analysis of a virus dynamics model with general incidence rate and two delays, Appl. Math. Comput., 221 (2013), 514-521. doi: 10.1016/j.amc.2013.07.005.
[19]	K. Hattaf and N. Yousfi, Global dynamics of a delay reaction-diffusion model for viral infection with specific functional response, Comp. Appl. Math., 34 (2015), 807-818. doi: 10.1007/s40314-014-0143-x.
[20]	F. Herbert, Heavy Drinking: The Myth of Alcoholism as a Disease, University of California Press, 1988.
[21]	H. W. Hethcote, Qualitative analyses of communicable disease models, Math. Biosci., 28 (1976), 335-356. doi: 10.1016/0025-5564(76)90132-2.
[22]	G. Huang, W. Ma and Y. Takeuchi, Global analysis for delay virus dynamics model with Beddington-DeAngelis functional response, Applied Mathematics Letters, 24 (2011), 1199-1203. doi: 10.1016/j.aml.2011.02.007.
[23]	H. F. Huo and N. N. Song, Global stability for a binge drinking model with two stages, Discrete Dynamics in Nature and Society, (2012), Art. ID 829386, 15pp.
[24]	Y. Kuang, Delay Differential Equations with Application in Population Dynamics, Academic Press, New York, 1993.
[25]	E. Kuntsche and R. Knibbe, et al., Why do young people drink? A review of drinking motives, Clinical Psychology Review, 25 (2005), 841-861. doi: 10.1016/j.cpr.2005.06.002.
[26]	H. Laarabi, A. Abta and K. Hattaf, Optimal control of a delayed sirs epidemic model with vaccination and treatment, Acta Biotheor., 63 (2015), 87-97. doi: 10.1007/s10441-015-9244-1.
[27]	J. P. LaSalle, The Stability of Dynamical Systems, Regional Conference Series in Applied Mathematics, SIAM, Philadelphia, 1976.
[28]	S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models, Chapman & Hall/CRC Mathematical and Computational Biology Series, Boca Raton, FL, 2007.
[29]	E. M. Lotfi, M. Maziane, K. Hattaf and N. Yousfi, Partial differential equations of an epidemic model with spatial diffusion, International Journal of Partial Differential Equations, 2014 (2014), Article ID 186437, 6pp. doi: 10.1155/2014/186437.
[30]	D. L. Lukes, Differential Equations: Classical to Controlled, Mathematics in Science and Engineering, Academic Press, New York, 1982.
[31]	W. Ma, Y. Takeuchi, T. Hara and E. Beretta, Permanence of an SIR epidemic model with distributed time delays, Tohoku Math. J., 54 (2002), 581-591. doi: 10.2748/tmj/1113247650.
[32]	C. Connell McCluskey, Complete global stability for an SIR epidemic model with delay-Distributed or discrete, Nonlinear Analysis: Real World Applications, 11 (2010), 55-59. doi: 10.1016/j.nonrwa.2008.10.014.
[33]	C. Connell McCluskey, Global stability of an SIR epidemic model with delay and general nonlinear incidence, Mathematical Biosciences and Engineering, 7 (2010), 837-850. doi: 10.3934/mbe.2010.7.837.
[34]	A. K. Misra, A. Sharma and V. Singh, Effect of awareness programs in controlling the prevelence of an epidemic with time delay, Journal of Biological Systems, 19 (2011), 389-402. doi: 10.1142/S0218339011004020.
[35]	A. Mubayi and P. Greenwood, et al., The impact of relative residence times on the distribution of heavy drinkers in highly distinct environments, Socio-Economic Planning Sciences, 44 (2010), 45-56. doi: 10.1016/j.seps.2009.02.002.
[36]	D. Muller and R. Koch, et al., Neurophisiologic findings in chronic alcohol abuse, Pyschiatr. Neurol. Med. Pyschol., 37 (1985), 129-132.
[37]	G. Mulone and B. Straughan, Modeling binge drinking, International Journal of Biomathematics, 5 (2012), 1250005, 14pp. doi: 10.1142/S1793524511001453.
[38]	S. S. Mushayabasa and C. P. Bhunu, Modelling the effects of heavy alcohol consumption on the transmission dynamics of gonorrhea, Nonlinear Dynamics, 66 (2011), 695-706. doi: 10.1007/s11071-011-9942-4.
[39]	L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Wiley, New Jersey, 1962.
[40]	R. Room, T. Babor and J. Rehm, Alcohol and public health, The Lancet, 365 (2005), 519-530. doi: 10.1016/S0140-6736(05)70276-2.
[41]	F. Sanchez and X. Wang, et al., Drinking as an Epidemica Simple Mathematical Model with Recovery and Relapse, Guide to Evidence-Based Relapse Prevention, Elsevier, New York, 2007.
[42]	Y. Takeuchi, W. Ma and E. Beretta, Global asymptotic properties of a delay SIR epidemic model with finite incubation times, Nonlinear Anal., 42 (2000), 931-947. doi: 10.1016/S0362-546X(99)00138-8.
[43]	G. Testino, Alcoholic diseases in hepato-gastroenterology: A point of view, Hepato-Gastroenterology, 55 (2008), 371-377.
[44]	H. Walter and K. Gutierrez, et al., Gender specific differences in alcoholism: Implications for treatment, Arch. Womens Ment. Health, 6 (2003), 253-258. doi: 10.1007/s00737-003-0014-8.
[45]	X.-Y. Wang, H.-F. Huo, Q.-K. Kong and W.-X. Shi, Optimal control strategies in an alcoholism model, Abstract and Applied Analysis, (2014), Article ID 954069, 18pp. doi: 10.1155/2014/954069.
[46]	R. Xu and Z. Ma, Global stability of a SIR epidemic model with nonlinear incidence rate and time delay, Nonlinear Anal. RWA, 10 (2009), 3175-3189. doi: 10.1016/j.nonrwa.2008.10.013.

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