American Institute of Mathematical Sciences

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2012, 9(1): 97-110. doi: 10.3934/mbe.2012.9.97

Impact of discontinuous treatments on disease dynamics in an SIR epidemic model

Zhenyuan Guo 1, , Lihong Huang 1, and  Xingfu Zou 2,

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

Received  September 2010 Revised  March 2011 Published  December 2011

We consider an SIR epidemic model with discontinuous treatment strategies. Under some reasonable assumptions on the discontinuous treatment function, we are able to determine the basic reproduction number $\mathcal{R}_0$, confirm the well-posedness of the model, describe the structure of possible equilibria as well as establish the stability/instability of the equilibria. Most interestingly, we find that in the case that an equilibrium is asymptotically stable, the convergence to the equilibrium can actually be achieved in finite time, and we can estimate this time in terms of the model parameters, initial sub-populations and the initial treatment strength. This suggests that from the view point of eliminating the disease from the host population, discontinuous treatment strategies would be superior to continuous ones. The methods we use to obtain the mathematical results are the generalized Lyapunov theory for discontinuous differential equations and some results on non-smooth analysis.
Keywords: discontinuous treatment, Infectious disease, convergence in finite time., SIR model, generalized Lyapunov method, stability.
Mathematics Subject Classification: Primary: 92D30; Secondary:34C2.
Citation: Zhenyuan Guo, Lihong Huang, Xingfu Zou. Impact of discontinuous treatments on disease dynamics in an SIR epidemic model. Mathematical Biosciences & Engineering, 2012, 9 (1) : 97-110. doi: 10.3934/mbe.2012.9.97
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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

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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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

F. Brauer, Epidemic models with heterogeneous mixing and treatment, Bull. Math. Biol., 70 (2008), 1869-1885. doi: 10.1007/s11538-008-9326-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

F. Ceragioli, "Discontinuous Ordinary Differential Equations and Stabilization," Universita di Firenze, 2000. Google Scholar

[14]

F. H. Clarke, "Optimization and Non-Smooth Analysis," Wiley, New York, 1983. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

L. Wu and Z. Feng, Homoclinic bifurcation in an SIQR model for childhood diseases, J. Differ. Equat., 168 (2000), 150-167.  Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[4]

R. M. Anderson and R. M. May, "Infectious Diseases of Humans, Dynamics and Control," Oxford University, Oxford, 1991. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

F. Brauer, Epidemic models with heterogeneous mixing and treatment, Bull. Math. Biol., 70 (2008), 1869-1885. doi: 10.1007/s11538-008-9326-1.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

F. Ceragioli, "Discontinuous Ordinary Differential Equations and Stabilization," Universita di Firenze, 2000. Google Scholar

[14]

F. H. Clarke, "Optimization and Non-Smooth Analysis," Wiley, New York, 1983. Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

L. Wu and Z. Feng, Homoclinic bifurcation in an SIQR model for childhood diseases, J. Differ. Equat., 168 (2000), 150-167.  Google Scholar

[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.  Google Scholar

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