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

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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

Abstract
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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. 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. 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. doi: 10.1007/s11538-007-9257-2.
[4]	R. M. Anderson and R. M. May, "Infectious Diseases of Humans, Dynamics and Control,", Oxford University, (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. 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. 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 (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. doi: 10.1051/cocv:1999113.
[9]	F. Brauer, Backward bifurcations in simple vaccination models,, J. Math. Anal. Appl., 298 (2004), 418. doi: 10.1016/j.jmaa.2004.05.045.
[10]	F. Brauer, Epidemic models with heterogeneous mixing and treatment,, Bull. Math. Biol., 70 (2008), 1869. 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 (2008).
[12]	C. Castillo-Chavez and Z. Feng, To treat or not to treat: The case of tuberculosis,, J. Math. Biol., 35 (1997), 629. 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, (1983).
[15]	Z. Feng, Final and peak epidemic sizes for SEIR models with quarantine and isolation,, Math. Biosci. Eng., 4 (2007), 675. 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. doi: 10.1016/0025-5564(94)00069-C.
[17]	A. F. Filippov, "Differential Equations with Discontinuous Righthand Sides,", Translated from the Russian, 18 (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. 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. 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. doi: 10.1137/S003613990343882X.
[21]	W. Wang, Backward bifurcation of an epidemic model with treatment,, Math. Biosci., 201 (2006), 58. 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.
[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. doi: 10.1016/j.nonrwa.2007.10.011.

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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. 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. 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. doi: 10.1007/s11538-007-9257-2.
[4]	R. M. Anderson and R. M. May, "Infectious Diseases of Humans, Dynamics and Control,", Oxford University, (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. 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. 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 (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. doi: 10.1051/cocv:1999113.
[9]	F. Brauer, Backward bifurcations in simple vaccination models,, J. Math. Anal. Appl., 298 (2004), 418. doi: 10.1016/j.jmaa.2004.05.045.
[10]	F. Brauer, Epidemic models with heterogeneous mixing and treatment,, Bull. Math. Biol., 70 (2008), 1869. 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 (2008).
[12]	C. Castillo-Chavez and Z. Feng, To treat or not to treat: The case of tuberculosis,, J. Math. Biol., 35 (1997), 629. 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, (1983).
[15]	Z. Feng, Final and peak epidemic sizes for SEIR models with quarantine and isolation,, Math. Biosci. Eng., 4 (2007), 675. 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. doi: 10.1016/0025-5564(94)00069-C.
[17]	A. F. Filippov, "Differential Equations with Discontinuous Righthand Sides,", Translated from the Russian, 18 (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. 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. 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. doi: 10.1137/S003613990343882X.
[21]	W. Wang, Backward bifurcation of an epidemic model with treatment,, Math. Biosci., 201 (2006), 58. 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.
[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. doi: 10.1016/j.nonrwa.2007.10.011.

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