Saturated treatments and measles resurgence episodes in South Africa: A possible linkage

Deborah Lacitignola

doi:10.3934/mbe.2013.10.1135

Article Contents

2013, Volume 10, Issue 4: 1135-1157. Doi: 10.3934/mbe.2013.10.1135

This issue Previous Article Regulation of Th1/Th2 cells in asthma development: A mathematical model Next Article Modelling seasonal HFMD with the recessive infection in Shandong, China

Saturated treatments and measles resurgence episodes in South Africa: A possible linkage

Deborah Lacitignola^1,

1.
Department of Electrical and Information Engineering, University of Cassino and Southern Lazio, Via di Biasio 43, I-03043 Cassino

Received: August 31, 2012

Revised: January 31, 2013

Published: May 2013

Abstract Related Papers Cited by

Abstract

We consider the case of measles in South Africa to show that an high vaccination coverage may be not enough - alone - to ensure measles eradication. The occurrence of certain epidemic episodes may in fact be encouraged by delays in the treatments or by not adequately fast clinical case management, which may be related to the backward bifurcation phenomenon as well as to an intriguing spiking dynamics which appears in the system for specific ranges of parameter values.

Keywords:

Mathematics Subject Classification: Primary: 92B05, 34C23, 34C60.

Citation:

Related Papers

Cited by

References

[1]	R. M. Anderson and R. M. May, "Infectious Diseases in Humans: Dynamics and Control," Oxford University Press, Oxford, 1991.
[2]	J. Arino, C. 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.
[3]	F. Brauer, Backward bifurcations in simple vaccination/treatment models, Journal of Biological Dynamics, 5 (2011), 410-418.doi: 10.1080/17513758.2010.510584.
[4]	H. Broutin, N. B. Mantilla-Beniers, F.Simondon, P. Aaby, B. T. Grenfell, J. F. Gugan and P. Rohani, Epidemiological impact of vaccination on the dynamics of two childhood diseases in rural Senegal, Microbes and Infection, 7 (2005), 593-599.doi: 10.1016/j.micinf.2004.12.018.
[5]	B. Buonomo and D. Lacitignola, On the dynamics of an SEIR epidemic model with a convex incidence rate, Ric. Mat., 57 (2008), 261-281.doi: 10.1007/s11587-008-0039-4.
[6]	B. Buonomo and D. Lacitignola, Analysis of a tuberculosis model with a case study in Uganda, J. Biol. Dyn., 4 (2010), 571-593.doi: 10.1080/17513750903518441.
[7]	B. Buonomo and D. Lacitignola, On the backward bifurcation of a vaccination model with nonlinear incidence, Nonlinear Analysis: Modelling and Control, 16 (2011), 30-46.
[8]	C. Castillo-Chavez and B. Song, Dynamical models of tubercolosis and their applications, Math. Biosci. Engin., 1 (2004), 361-404.doi: 10.3934/mbe.2004.1.361.
[9]	Centers for Disease Control and Prevention, Atlanta, USA., Progress toward measles elimination - Southern Africa 1996-1998. MMWR, 48 (1999), 585-589.
[10]	C. Cohen, A. Buys, J. Mc Anerney, L. Mahlaba, M. Mashele, G. Ntshoe, A. Puren, B. Singh and S. Smit, Suspected measles case-based surveillance, South Africa, 2009, Comm. Dis. Surveill. Bull., 8 (2009), 2-3.
[11]	J. Cui, X. Mu and H. Wan, Saturation recovery leads to multiple endemic equilibria and backward bifurcation, Journal of Theoretical Biology, 254 (2008), 275-283.doi: 10.1016/j.jtbi.2008.05.015.
[12]	W. R. Derrick and P. van den Driessche, Homoclinic orbits in a disease transmission model with nonlinear incidence and nonconstant population, Discrete Contin. Dynam. Syst. Ser. B, 3 (2003), 299-309.doi: 10.3934/dcdsb.2003.3.299.
[13]	C. A. de Quadros, H. Izurieta, L. Venczel and P. Carrasco, Measles eradication in the Americas: Progress to date, Journal of Infectious Diseases, 189 (2004), S227-S235.
[14]	J. Dushoff, W. Huang and C. Castillo-Chavez, Backwards bifurcations and catastrophe in simple models of fatal diseases, J. Math. Biol., 36 (1998), 227-248.doi: 10.1007/s002850050099.
[15]	Z. Feng and H. 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.
[16]	Z. Feng, C. Castillo-Chavez and A. F. Capurro, A model for tuberculosis with exogenous reinfection, Theor. Popul. Biol., 57 (2000), 235-247.doi: 10.1006/tpbi.2000.1451.
[17]	P. Glendinning, "Stability, Instability and Chaos: An Introduction to the Theory of Nonlinear Differential Equations," Cambridge University Press, 1994.
[18]	M. G. M. Gomes, A. Margheri, G. F. Medley and C. Rebelo, Dynamical behaviour of epidemiological models with sub-optimal immunity and nonlinear incidence, J. Math. Biol., 51 (2005), 414-430.doi: 10.1007/s00285-005-0331-9.
[19]	D. Greenhalgh and M. Griffiths, Backward bifurcation, equilibrium and stability phenomena in a three-stage extended BRSV epidemic model, J. Math. Biol., 59 (2009), 1-36.doi: 10.1007/s00285-008-0206-y.
[20]	J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields," Springer-Verlag, Berlin, 1983.
[21]	A. B. Gumel and S. M. Moghadas, A qualitative study of a vaccination model with non-linear incidence, App. Math. Comput., 143 (2003), 409-419.doi: 10.1016/S0096-3003(02)00372-7.
[22]	A. B. Gumel, S. Ruan, T. Day, J. Watmough, F. Brauer, P. van den Driessche, D. Gabrielson, C. Bowman, M. E. Alexander, S. Ardal, J. Wu and B. M. Sahai, Modeling strategies for controlling SARS outbreaks, Proc. R. Soc. London B, 271 (2004), 2223-2232.doi: 10.1098/rspb.2004.2800.
[23]	K. P. Hadeler and P. van den Driesche, Backward bifurcation in epidemic control, Math. Biosci., 146 (1997), 15-35.doi: 10.1016/S0025-5564(97)00027-8.
[24]	D. L. Heyman, "Control of Communicable Diseases Manual," American Public Health Association, Washington, 2008.doi: 10.1086/605668.
[25]	W. Huang, K. L. Cooke and C. Castillo-Chavez, Stability and bifurcation for a multiple-group model for the dynamics of HIV/AIDS transmission, SIAM J. Appl. Math., 52 (1992), 835-854.doi: 10.1137/0152047.
[26]	H. Inaba and H. Sekine, A mathematical model for Chagas disease with infection-age-dependent infectivity, Math. Biosci., 190 (2004), 39-69.doi: 10.1016/j.mbs.2004.02.004.
[27]	W. Janaszek, N. J. Gay and W. Gut, Measles vaccine efficacy during an epidemic in 1998 in the highly vaccinated population of Poland, Vaccine, 21 (2003), 473-478.doi: 10.1016/S0264-410X(02)00482-6.
[28]	Y. Jin, W. Wang and S. Xiao, A SIRS model with a nonlinear incidence, Chaos Solitons Fractals, 34 (2007), 1482-1497.doi: 10.1016/j.chaos.2006.04.022.
[29]	W. O. Kermack and A. G. McKendrick, A Contribution to the mathematical theory of epidemics. I., Proc. R. Soc. A, 115 (1927), 700-721. (Reprinted with parts II. and III. in Bulletin of Mathematical Biology, 53 (1991), 33-118.)
[30]	C. M. Kribs-Zaleta and J. X. Velasco-Hernandez, A simple vaccination model with multiple endemic states, Math. Biosci., 164 (2000), 183-201.doi: 10.1016/S0025-5564(00)00003-1.
[31]	C. M. Kribs-Zaleta, Center manifolds and normal forms in epidemic models, Institute for Mathematics and Its Applications, 125 (2000), 269-286.
[32]	M. Y. Li and L. Wang, Global stability in some SEIR epidemic models, in "Mathematical Approaches for Emerging and Reemerging Infectious Diseases: Models, Methods, and Theory," IMA Math. Appl., 126 (2002), 295-311.doi: 10.1007/978-1-4613-0065-6_17.
[33]	W. M. Liu, S. A. Levin and Y. Iwasa, Influence of nonlinear incidence rates upon the behavior of SIRS epidemiological models, J. Math. Biol., 23 (1986), 187-204.doi: 10.1007/BF00276956.
[34]	M. Martcheva and H. R. Thieme, Progression age enhanced backward bifurcation in an epidemic model with superinfection, J. Math. Biol., 46 (2003), 385-424.doi: 10.1007/s00285-002-0181-7.
[35]	J. Mc Anerney, C. Cohen, A. Puren, S. Smit, M. Mashele and J. van den Heever, Measles outbreak, South Africa, 2009. Preliminary data on laboratory-confirmed cases, Comm. Dis. Surveill. Bull., 7 (2009), 15-21.
[36]	M. L. McMorrow, G. Gebremedhin, J. van den Heever, R. Kezaala, B. N. Harris, R. Nandy, P. Strebel, A. Jack and K. L. Cairns, Measles outbreak in South Africa, 2003-2005, S. Afr. Med. J., 99 (2009), 314-319.
[37]	G. F. Medley, N. A. Lindop, W. J. Edmunds and D. J. Nokes, Hepatitis-B virus endemicity: Heterogeneity, catastrophic dynamics and control, Nat. Med., 5 (2001), 619-624.
[38]	J. D. Murray, "Mathematical Biology," Springer, Berlin, 1998.doi: 10.1007/b98869.
[39]	M. G. Roberts, The pluses and minuses of $R_0$, J. R. Soc. Interface, 4 (2007), 949-961.
[40]	Y. Tang, D. Huang, S. Ruan and W. Zhang, Coexistence of limit cycles and homoclinic loops in a SIRS model with a nonlinear incidence rate, SIAM J. Appl. Math., 69 (2008), 621-639.doi: 10.1137/070700966.
[41]	M. Safan, J. A. P. Heesterbeek and K. Dietz, The minimum effort required to eradicate infections in models with backward bifurcation, J. Math. Biol., 53 (2006), 703-718.doi: 10.1007/s00285-006-0028-8.
[42]	UNICEF., "Unicef Info by Country - South Africa Statistics," Available on line at: http://www.unicef.org/infobycountry/southafrica_statistics.html.
[43]	A. Uzicanin, R. Eggers, E. Webb, B. Harris, D. Durrheim, G. Ogunbanjo, V. Isaacs, A. Hawkridge, R. Biellik and P. Strebel, Impact of the 1996-1997 supplementary measles vaccination caimpaigns in South Africa, Int. J. Epidemiol., 31 (2002), 968-976.doi: 10.1093/ije/31.5.968.
[44]	P. van den Driessche and J. Watmough, A simple SIS epidemic model with a backward bifurcation, J. Math. Biol., 40 (2000), 525-540.doi: 10.1007/s002850000032.
[45]	S. Verguet, W. Jassat, C. Hedberg, S. Tollman, D. T. Jamison and K. J. Hofman, Measles control in Sub-Saharan Africa: South Africa as a case study, Vaccine, 30 (2012), 1594-1600.doi: 10.1016/j.vaccine.2011.12.123.
[46]	W. Wang, Backward bifurcation of an epidemic model with treatment, Math. Biosci., 201 (2006), 58-71.doi: 10.1016/j.mbs.2005.12.022.
[47]	Z. Wang, Backward bifurcation in simple SIS model, Acta Matematicae Applicatae Sinica, English Series, 25 (2009), 127-136.doi: 10.1007/s10255-006-6160-9.
[48]	WHO/UNICEF, "Global Plan for Reducing Measles Mortality 2006-2010," World Health Organization. Available from http://www.who.int/immunization_delivery/adc/measles/Measles Global Plan_Eng.pdf.
[49]	L. Wolfson, P. M. Strebel, M. Gagic-Dobo, E. J. Hoekstra, J. W. McFarland and B. S. Hersh, Has the 2005 measles mortality reduction goal been achieved? A natural history modelling study, Lancet, 369 (2007), 191-200.doi: 10.1016/S0140-6736(07)60107-X.
[50]	L. Wu and Z. Feng, Homoclinic bifurcation in an SIQR model for childhood diseases, J. Differential Equations, 168 (2000), 150-167.doi: 10.1006/jdeq.2000.3882.
[51]	X. Zhang and X. Liu, Backward bifurcation of an epidemic model with saturated treatment function, J. Math. Anal. Appl., 348 (2008), 433-443.doi: 10.1016/j.jmaa.2008.07.042.
[52]	Z. Zhonghua and S. Yaohong, Qualitative analysis of a SIR epidemic model with saturated treatment rate, J. Appl. Math. Comput., 34 (2010), 177-194.doi: 10.1007/s12190-009-0315-9.
[53]	X. Zhou and J. Cui, Analysis of stability and bifurcation for an SEIR epidemic model with saturated recovery rate, Commun. Nonlinear Sci. Numer. Simulat., 16 (2011), 4438-4450.doi: 10.1016/j.cnsns.2011.03.026.