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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

 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

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.
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 & 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). Google Scholar [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. doi: 10.1016/S0362-546X(01)00528-4. Google Scholar [3] Centers for Disease Control, Alcohol-related disease impact (ARDI) software,, Atlanta, (2004). Google Scholar [4] Centers for Disease Control and Prevention, Binge Drinking,, , (). Google Scholar [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, (2010). Google Scholar [6] C. Chen and C. Storr, et al., Early-onset drug use and risk for drug dependence problems,, Addictive Behaviors, 34 (2009), 319. doi: 10.1016/j.addbeh.2008.10.021. Google Scholar [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. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar [8] K. R. Fister, S. Lenhart and J. S. Mcnally, Optimizing chemotherapy in an HIV model,, Electronic Journal of Differential Equations, 32 (1998), 1. Google Scholar [9] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control,, No. 1, (1975). Google Scholar [10] M. Glavas and J. Weinberg, Stress, alcohol consumption and the hypothalamic-pituitary-adrenal axis,, in Nutrients, (2005), 165. doi: 10.1385/1-59259-952-4:165. Google Scholar [11] L. Göllmann, Theory and applications of optimal control problems with multiple time-delays,, Journal of Industrial and management Optimization, 10 (2014), 413. Google Scholar [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. Google Scholar [13] A. Halanay and J. A. Yorke, Some new results and problems in the theory of differential-delay equations,, SIAM Rev, 13 (1971), 55. doi: 10.1137/1013004. Google Scholar [14] J. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations,, Springer-Verlag, (1993). doi: 10.1007/978-1-4612-4342-7. Google Scholar [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. Google Scholar [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). doi: 10.5402/2012/215124. Google Scholar [17] K. Hattaf, N. Yousfi and A. Tridane, A Delay virus dynamics model with general incidence rate,, Differ. Equ. Dyn. Syst., 22 (2014), 181. doi: 10.1007/s12591-013-0167-5. Google Scholar [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. doi: 10.1016/j.amc.2013.07.005. Google Scholar [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. doi: 10.1007/s40314-014-0143-x. Google Scholar [20] F. Herbert, Heavy Drinking: The Myth of Alcoholism as a Disease,, University of California Press, (1988). Google Scholar [21] H. W. Hethcote, Qualitative analyses of communicable disease models,, Math. Biosci., 28 (1976), 335. doi: 10.1016/0025-5564(76)90132-2. Google Scholar [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. doi: 10.1016/j.aml.2011.02.007. Google Scholar [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). Google Scholar [24] Y. Kuang, Delay Differential Equations with Application in Population Dynamics,, Academic Press, (1993). Google Scholar [25] E. Kuntsche and R. Knibbe, et al., Why do young people drink? A review of drinking motives,, Clinical Psychology Review, 25 (2005), 841. doi: 10.1016/j.cpr.2005.06.002. Google Scholar [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. doi: 10.1007/s10441-015-9244-1. Google Scholar [27] J. P. LaSalle, The Stability of Dynamical Systems,, Regional Conference Series in Applied Mathematics, (1976). Google Scholar [28] S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models,, Chapman & Hall/CRC Mathematical and Computational Biology Series, (2007). Google Scholar [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). doi: 10.1155/2014/186437. Google Scholar [30] D. L. Lukes, Differential Equations: Classical to Controlled, Mathematics in Science and Engineering,, Academic Press, (1982). Google Scholar [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. doi: 10.2748/tmj/1113247650. Google Scholar [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. doi: 10.1016/j.nonrwa.2008.10.014. Google Scholar [33] C. Connell McCluskey, Global stability of an SIR epidemic model with delay and general nonlinear incidence,, Mathematical Biosciences and Engineering, 7 (2010), 837. doi: 10.3934/mbe.2010.7.837. Google Scholar [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. doi: 10.1142/S0218339011004020. Google Scholar [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. doi: 10.1016/j.seps.2009.02.002. Google Scholar [36] D. Muller and R. Koch, et al., Neurophisiologic findings in chronic alcohol abuse,, Pyschiatr. Neurol. Med. Pyschol., 37 (1985), 129. Google Scholar [37] G. Mulone and B. Straughan, Modeling binge drinking,, International Journal of Biomathematics, 5 (2012). doi: 10.1142/S1793524511001453. Google Scholar [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. doi: 10.1007/s11071-011-9942-4. Google Scholar [39] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes,, Wiley, (1962). Google Scholar [40] R. Room, T. Babor and J. Rehm, Alcohol and public health,, The Lancet, 365 (2005), 519. doi: 10.1016/S0140-6736(05)70276-2. Google Scholar [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, (2007). Google Scholar [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. doi: 10.1016/S0362-546X(99)00138-8. Google Scholar [43] G. Testino, Alcoholic diseases in hepato-gastroenterology: A point of view,, Hepato-Gastroenterology, 55 (2008), 371. Google Scholar [44] H. Walter and K. Gutierrez, et al., Gender specific differences in alcoholism: Implications for treatment,, Arch. Womens Ment. Health, 6 (2003), 253. doi: 10.1007/s00737-003-0014-8. Google Scholar [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). doi: 10.1155/2014/954069. Google Scholar [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. doi: 10.1016/j.nonrwa.2008.10.013. Google Scholar

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). Google Scholar [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. doi: 10.1016/S0362-546X(01)00528-4. Google Scholar [3] Centers for Disease Control, Alcohol-related disease impact (ARDI) software,, Atlanta, (2004). Google Scholar [4] Centers for Disease Control and Prevention, Binge Drinking,, , (). Google Scholar [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, (2010). Google Scholar [6] C. Chen and C. Storr, et al., Early-onset drug use and risk for drug dependence problems,, Addictive Behaviors, 34 (2009), 319. doi: 10.1016/j.addbeh.2008.10.021. Google Scholar [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. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar [8] K. R. Fister, S. Lenhart and J. S. Mcnally, Optimizing chemotherapy in an HIV model,, Electronic Journal of Differential Equations, 32 (1998), 1. Google Scholar [9] W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control,, No. 1, (1975). Google Scholar [10] M. Glavas and J. Weinberg, Stress, alcohol consumption and the hypothalamic-pituitary-adrenal axis,, in Nutrients, (2005), 165. doi: 10.1385/1-59259-952-4:165. Google Scholar [11] L. Göllmann, Theory and applications of optimal control problems with multiple time-delays,, Journal of Industrial and management Optimization, 10 (2014), 413. Google Scholar [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. Google Scholar [13] A. Halanay and J. A. Yorke, Some new results and problems in the theory of differential-delay equations,, SIAM Rev, 13 (1971), 55. doi: 10.1137/1013004. Google Scholar [14] J. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations,, Springer-Verlag, (1993). doi: 10.1007/978-1-4612-4342-7. Google Scholar [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. Google Scholar [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). doi: 10.5402/2012/215124. Google Scholar [17] K. Hattaf, N. Yousfi and A. Tridane, A Delay virus dynamics model with general incidence rate,, Differ. Equ. Dyn. Syst., 22 (2014), 181. doi: 10.1007/s12591-013-0167-5. Google Scholar [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. doi: 10.1016/j.amc.2013.07.005. Google Scholar [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. doi: 10.1007/s40314-014-0143-x. Google Scholar [20] F. Herbert, Heavy Drinking: The Myth of Alcoholism as a Disease,, University of California Press, (1988). Google Scholar [21] H. W. Hethcote, Qualitative analyses of communicable disease models,, Math. Biosci., 28 (1976), 335. doi: 10.1016/0025-5564(76)90132-2. Google Scholar [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. doi: 10.1016/j.aml.2011.02.007. Google Scholar [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). Google Scholar [24] Y. Kuang, Delay Differential Equations with Application in Population Dynamics,, Academic Press, (1993). Google Scholar [25] E. Kuntsche and R. Knibbe, et al., Why do young people drink? A review of drinking motives,, Clinical Psychology Review, 25 (2005), 841. doi: 10.1016/j.cpr.2005.06.002. Google Scholar [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. doi: 10.1007/s10441-015-9244-1. Google Scholar [27] J. P. LaSalle, The Stability of Dynamical Systems,, Regional Conference Series in Applied Mathematics, (1976). Google Scholar [28] S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models,, Chapman & Hall/CRC Mathematical and Computational Biology Series, (2007). Google Scholar [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). doi: 10.1155/2014/186437. Google Scholar [30] D. L. Lukes, Differential Equations: Classical to Controlled, Mathematics in Science and Engineering,, Academic Press, (1982). Google Scholar [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. doi: 10.2748/tmj/1113247650. Google Scholar [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. doi: 10.1016/j.nonrwa.2008.10.014. Google Scholar [33] C. Connell McCluskey, Global stability of an SIR epidemic model with delay and general nonlinear incidence,, Mathematical Biosciences and Engineering, 7 (2010), 837. doi: 10.3934/mbe.2010.7.837. Google Scholar [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. doi: 10.1142/S0218339011004020. Google Scholar [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. doi: 10.1016/j.seps.2009.02.002. Google Scholar [36] D. Muller and R. Koch, et al., Neurophisiologic findings in chronic alcohol abuse,, Pyschiatr. Neurol. Med. Pyschol., 37 (1985), 129. Google Scholar [37] G. Mulone and B. Straughan, Modeling binge drinking,, International Journal of Biomathematics, 5 (2012). doi: 10.1142/S1793524511001453. Google Scholar [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. doi: 10.1007/s11071-011-9942-4. Google Scholar [39] L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes,, Wiley, (1962). Google Scholar [40] R. Room, T. Babor and J. Rehm, Alcohol and public health,, The Lancet, 365 (2005), 519. doi: 10.1016/S0140-6736(05)70276-2. Google Scholar [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, (2007). Google Scholar [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. doi: 10.1016/S0362-546X(99)00138-8. Google Scholar [43] G. Testino, Alcoholic diseases in hepato-gastroenterology: A point of view,, Hepato-Gastroenterology, 55 (2008), 371. Google Scholar [44] H. Walter and K. Gutierrez, et al., Gender specific differences in alcoholism: Implications for treatment,, Arch. Womens Ment. Health, 6 (2003), 253. doi: 10.1007/s00737-003-0014-8. Google Scholar [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). doi: 10.1155/2014/954069. Google Scholar [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. doi: 10.1016/j.nonrwa.2008.10.013. Google Scholar
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