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Impact of discontinuous treatments on disease dynamics in an SIR epidemic model
1. | College of Mathematics and Econometrics, Hunan University, Changsha, Hunan 410082, China, China |
2. | Department of Applied Mathematics, University of Western Ontario, London, Ontario N6A 5B7 |
References:
[1] |
M. E. Alexander, C. Bowman, S. M. Moghadas, R. Summers, A. B. Gumel and B. M. Sahai, A vaccination model for transmission dynamics of influenza, SIAM J. Appl. Dyn. Syst., 3 (2004), 503-524.
doi: 10.1137/030600370. |
[2] |
M. E. Alexander, S. M. Moghadas, P. Rohani and A. R. Summers, Modelling the effect of a booster vaccination on disease epidemiology, J. Math. Biol., 52 (2006), 290-306.
doi: 10.1007/s00285-005-0356-0. |
[3] |
M. E. Alexander, S. M. Moghadas, G. Röst and J. Wu, A delay differential model for pandemic influenza with antiviral treatment, Bull. Math. Biol., 70 (2008), 382-397.
doi: 10.1007/s11538-007-9257-2. |
[4] |
R. M. Anderson and R. M. May, "Infectious Diseases of Humans, Dynamics and Control," Oxford University, Oxford, 1991. |
[5] |
J. Arino, C. McCluskey and P. van den Driessche, Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM J. Appl. Math., 64 (2003), 260-276.
doi: 10.1137/S0036139902413829. |
[6] |
J. Arino, R. Jordan and P. van den Driessche, Quarantine in a multi-species epidemic model with spatial dynamics, Math. Biosci., 206 (2007), 46-60.
doi: 10.1016/j.mbs.2005.09.002. |
[7] |
J.-P. Aubin and A. Cellina, "Differential Inclusions. Set-Valued Maps and Viability Theory," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 264, Springer-Verlag, Berlin, 1984. |
[8] |
A. Baciotti and F. Ceragioli, Stability and stabilization of discontinuous systems and nonsmooth Lyapunov function, ESAIM Control Optim. Calc. Var., 4 (1999), 361-376.
doi: 10.1051/cocv:1999113. |
[9] |
F. Brauer, Backward bifurcations in simple vaccination models, J. Math. Anal. Appl., 298 (2004), 418-431.
doi: 10.1016/j.jmaa.2004.05.045. |
[10] |
F. Brauer, Epidemic models with heterogeneous mixing and treatment, Bull. Math. Biol., 70 (2008), 1869-1885.
doi: 10.1007/s11538-008-9326-1. |
[11] |
F. Brauer, P. van den Driessche and J. Wu, eds., "Mathematical Epidemiology," Lecture Notes in Mathematics, 1945, Mathematical Biosciences Subseries, Springer-Verlag, Berlin, 2008. |
[12] |
C. Castillo-Chavez and Z. Feng, To treat or not to treat: The case of tuberculosis, J. Math. Biol., 35 (1997), 629-656.
doi: 10.1007/s002850050069. |
[13] |
F. Ceragioli, "Discontinuous Ordinary Differential Equations and Stabilization," Universita di Firenze, 2000. |
[14] |
F. H. Clarke, "Optimization and Non-Smooth Analysis," Wiley, New York, 1983. |
[15] |
Z. Feng, Final and peak epidemic sizes for SEIR models with quarantine and isolation, Math. Biosci. Eng., 4 (2007), 675-686.
doi: 10.3934/mbe.2007.4.675. |
[16] |
Z. Feng and H. R. Thieme, Recurrent outbreaks of childhood diseases revisited: The impact of isolation, Math. Biosci., 128 (1995), 93-130.
doi: 10.1016/0025-5564(94)00069-C. |
[17] |
A. F. Filippov, "Differential Equations with Discontinuous Righthand Sides," Translated from the Russian, Mathematics and its Applications (Soviet Series), 18, Kluwer Academic Publishers Group, Dordrecht, 1988. |
[18] |
M. Forti, M. Grazzini, P. Nistri and L. Pancioni, Generalized Lyapunov approach for convergence of neural networks with discontinuous or non-Lipschitz activations, Phys. D, 214 (2006), 88-99.
doi: 10.1016/j.physd.2005.12.006. |
[19] |
J. M. Hyman and J. Li, Modeling the effectiveness of isolation strategies in preventing STD epidemics, SIAM J. Appl. Math., 58 (1998), 912-925.
doi: 10.1137/S003613999630561X. |
[20] |
M. Nuño, Z. Feng, M. Martcheva and C. Castillo-Chavez, Dynamics of two-strain influenza with isolation and partial cross-immunity, SIAM J. Appl. Math., 65 (2005), 964-982.
doi: 10.1137/S003613990343882X. |
[21] |
W. Wang, Backward bifurcation of an epidemic model with treatment, Math. Biosci., 201 (2006), 58-71.
doi: 10.1016/j.mbs.2005.12.022. |
[22] |
L. Wu and Z. Feng, Homoclinic bifurcation in an SIQR model for childhood diseases, J. Differ. Equat., 168 (2000), 150-167. |
[23] |
X. Zhang and X. Liu, Backward bifurcation and global dynamics of an SIS epidemic model with general incidence rate and treatment, Nonl. Anal. RWA, 10 (2009), 565-575.
doi: 10.1016/j.nonrwa.2007.10.011. |
show all references
References:
[1] |
M. E. Alexander, C. Bowman, S. M. Moghadas, R. Summers, A. B. Gumel and B. M. Sahai, A vaccination model for transmission dynamics of influenza, SIAM J. Appl. Dyn. Syst., 3 (2004), 503-524.
doi: 10.1137/030600370. |
[2] |
M. E. Alexander, S. M. Moghadas, P. Rohani and A. R. Summers, Modelling the effect of a booster vaccination on disease epidemiology, J. Math. Biol., 52 (2006), 290-306.
doi: 10.1007/s00285-005-0356-0. |
[3] |
M. E. Alexander, S. M. Moghadas, G. Röst and J. Wu, A delay differential model for pandemic influenza with antiviral treatment, Bull. Math. Biol., 70 (2008), 382-397.
doi: 10.1007/s11538-007-9257-2. |
[4] |
R. M. Anderson and R. M. May, "Infectious Diseases of Humans, Dynamics and Control," Oxford University, Oxford, 1991. |
[5] |
J. Arino, C. McCluskey and P. van den Driessche, Global results for an epidemic model with vaccination that exhibits backward bifurcation, SIAM J. Appl. Math., 64 (2003), 260-276.
doi: 10.1137/S0036139902413829. |
[6] |
J. Arino, R. Jordan and P. van den Driessche, Quarantine in a multi-species epidemic model with spatial dynamics, Math. Biosci., 206 (2007), 46-60.
doi: 10.1016/j.mbs.2005.09.002. |
[7] |
J.-P. Aubin and A. Cellina, "Differential Inclusions. Set-Valued Maps and Viability Theory," Grundlehren der Mathematischen Wissenschaften [Fundamental Principles of Mathematical Sciences], 264, Springer-Verlag, Berlin, 1984. |
[8] |
A. Baciotti and F. Ceragioli, Stability and stabilization of discontinuous systems and nonsmooth Lyapunov function, ESAIM Control Optim. Calc. Var., 4 (1999), 361-376.
doi: 10.1051/cocv:1999113. |
[9] |
F. Brauer, Backward bifurcations in simple vaccination models, J. Math. Anal. Appl., 298 (2004), 418-431.
doi: 10.1016/j.jmaa.2004.05.045. |
[10] |
F. Brauer, Epidemic models with heterogeneous mixing and treatment, Bull. Math. Biol., 70 (2008), 1869-1885.
doi: 10.1007/s11538-008-9326-1. |
[11] |
F. Brauer, P. van den Driessche and J. Wu, eds., "Mathematical Epidemiology," Lecture Notes in Mathematics, 1945, Mathematical Biosciences Subseries, Springer-Verlag, Berlin, 2008. |
[12] |
C. Castillo-Chavez and Z. Feng, To treat or not to treat: The case of tuberculosis, J. Math. Biol., 35 (1997), 629-656.
doi: 10.1007/s002850050069. |
[13] |
F. Ceragioli, "Discontinuous Ordinary Differential Equations and Stabilization," Universita di Firenze, 2000. |
[14] |
F. H. Clarke, "Optimization and Non-Smooth Analysis," Wiley, New York, 1983. |
[15] |
Z. Feng, Final and peak epidemic sizes for SEIR models with quarantine and isolation, Math. Biosci. Eng., 4 (2007), 675-686.
doi: 10.3934/mbe.2007.4.675. |
[16] |
Z. Feng and H. R. Thieme, Recurrent outbreaks of childhood diseases revisited: The impact of isolation, Math. Biosci., 128 (1995), 93-130.
doi: 10.1016/0025-5564(94)00069-C. |
[17] |
A. F. Filippov, "Differential Equations with Discontinuous Righthand Sides," Translated from the Russian, Mathematics and its Applications (Soviet Series), 18, Kluwer Academic Publishers Group, Dordrecht, 1988. |
[18] |
M. Forti, M. Grazzini, P. Nistri and L. Pancioni, Generalized Lyapunov approach for convergence of neural networks with discontinuous or non-Lipschitz activations, Phys. D, 214 (2006), 88-99.
doi: 10.1016/j.physd.2005.12.006. |
[19] |
J. M. Hyman and J. Li, Modeling the effectiveness of isolation strategies in preventing STD epidemics, SIAM J. Appl. Math., 58 (1998), 912-925.
doi: 10.1137/S003613999630561X. |
[20] |
M. Nuño, Z. Feng, M. Martcheva and C. Castillo-Chavez, Dynamics of two-strain influenza with isolation and partial cross-immunity, SIAM J. Appl. Math., 65 (2005), 964-982.
doi: 10.1137/S003613990343882X. |
[21] |
W. Wang, Backward bifurcation of an epidemic model with treatment, Math. Biosci., 201 (2006), 58-71.
doi: 10.1016/j.mbs.2005.12.022. |
[22] |
L. Wu and Z. Feng, Homoclinic bifurcation in an SIQR model for childhood diseases, J. Differ. Equat., 168 (2000), 150-167. |
[23] |
X. Zhang and X. Liu, Backward bifurcation and global dynamics of an SIS epidemic model with general incidence rate and treatment, Nonl. Anal. RWA, 10 (2009), 565-575.
doi: 10.1016/j.nonrwa.2007.10.011. |
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