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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 |
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).
|
[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. |
[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. |
[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. |
[8] |
K. R. Fister, S. Lenhart and J. S. Mcnally, Optimizing chemotherapy in an HIV model,, Electronic Journal of Differential Equations, 32 (1998), 1.
|
[9] |
W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control,, No. 1, (1975).
|
[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. |
[11] |
L. Göllmann, Theory and applications of optimal control problems with multiple time-delays,, Journal of Industrial and management Optimization, 10 (2014), 413.
|
[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. |
[14] |
J. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations,, Springer-Verlag, (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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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).
|
[24] |
Y. Kuang, Delay Differential Equations with Application in Population Dynamics,, Academic Press, (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.
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.
doi: 10.1007/s10441-015-9244-1. |
[27] |
J. P. LaSalle, The Stability of Dynamical Systems,, Regional Conference Series in Applied Mathematics, (1976).
|
[28] |
S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models,, Chapman & Hall/CRC Mathematical and Computational Biology Series, (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).
doi: 10.1155/2014/186437. |
[30] |
D. L. Lukes, Differential Equations: Classical to Controlled, Mathematics in Science and Engineering,, Academic Press, (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.
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.
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.
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.
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.
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. Google Scholar |
[37] |
G. Mulone and B. Straughan, Modeling binge drinking,, International Journal of Biomathematics, 5 (2012).
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.
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, (1962).
|
[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. |
[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. |
[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. |
[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. |
[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. |
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).
|
[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. |
[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. |
[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. |
[8] |
K. R. Fister, S. Lenhart and J. S. Mcnally, Optimizing chemotherapy in an HIV model,, Electronic Journal of Differential Equations, 32 (1998), 1.
|
[9] |
W. H. Fleming and R. W. Rishel, Deterministic and Stochastic Optimal Control,, No. 1, (1975).
|
[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. |
[11] |
L. Göllmann, Theory and applications of optimal control problems with multiple time-delays,, Journal of Industrial and management Optimization, 10 (2014), 413.
|
[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. |
[14] |
J. Hale and S. M. Verduyn Lunel, Introduction to Functional Differential Equations,, Springer-Verlag, (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.
|
[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. |
[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. |
[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. |
[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. |
[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. |
[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. |
[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).
|
[24] |
Y. Kuang, Delay Differential Equations with Application in Population Dynamics,, Academic Press, (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.
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.
doi: 10.1007/s10441-015-9244-1. |
[27] |
J. P. LaSalle, The Stability of Dynamical Systems,, Regional Conference Series in Applied Mathematics, (1976).
|
[28] |
S. Lenhart and J. T. Workman, Optimal Control Applied to Biological Models,, Chapman & Hall/CRC Mathematical and Computational Biology Series, (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).
doi: 10.1155/2014/186437. |
[30] |
D. L. Lukes, Differential Equations: Classical to Controlled, Mathematics in Science and Engineering,, Academic Press, (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.
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.
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.
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.
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.
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. Google Scholar |
[37] |
G. Mulone and B. Straughan, Modeling binge drinking,, International Journal of Biomathematics, 5 (2012).
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.
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, (1962).
|
[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. |
[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. |
[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. |
[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. |
[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. |
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