2013, 10(4): 1135-1157. doi: 10.3934/mbe.2013.10.1135

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

1. 

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

Received  September 2012 Revised  February 2013 Published  June 2013

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.
Citation: Deborah Lacitignola. Saturated treatments and measles resurgence episodes in South Africa: A possible linkage. Mathematical Biosciences & Engineering, 2013, 10 (4) : 1135-1157. doi: 10.3934/mbe.2013.10.1135
References:
[1]

R. M. Anderson and R. M. May, "Infectious Diseases in Humans: Dynamics and Control,", Oxford University Press, (1991).   Google Scholar

[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.  doi: 10.1137/S0036139902413829.  Google Scholar

[3]

F. Brauer, Backward bifurcations in simple vaccination/treatment models,, Journal of Biological Dynamics, 5 (2011), 410.  doi: 10.1080/17513758.2010.510584.  Google Scholar

[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.  doi: 10.1016/j.micinf.2004.12.018.  Google Scholar

[5]

B. Buonomo and D. Lacitignola, On the dynamics of an SEIR epidemic model with a convex incidence rate,, Ric. Mat., 57 (2008), 261.  doi: 10.1007/s11587-008-0039-4.  Google Scholar

[6]

B. Buonomo and D. Lacitignola, Analysis of a tuberculosis model with a case study in Uganda,, J. Biol. Dyn., 4 (2010), 571.  doi: 10.1080/17513750903518441.  Google Scholar

[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.   Google Scholar

[8]

C. Castillo-Chavez and B. Song, Dynamical models of tubercolosis and their applications,, Math. Biosci. Engin., 1 (2004), 361.  doi: 10.3934/mbe.2004.1.361.  Google Scholar

[9]

Centers for Disease Control and Prevention, Atlanta, USA., Progress toward measles elimination - Southern Africa 1996-1998., MMWR, 48 (1999), 585.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.jtbi.2008.05.015.  Google Scholar

[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.  doi: 10.3934/dcdsb.2003.3.299.  Google Scholar

[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).   Google Scholar

[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.  doi: 10.1007/s002850050099.  Google Scholar

[15]

Z. Feng and H. Thieme, Recurrent outbreaks of childhood diseases revisited: The impact of isolation,, Math. Biosci., 128 (1995), 93.  doi: 10.1016/0025-5564(94)00069-C.  Google Scholar

[16]

Z. Feng, C. Castillo-Chavez and A. F. Capurro, A model for tuberculosis with exogenous reinfection,, Theor. Popul. Biol., 57 (2000), 235.  doi: 10.1006/tpbi.2000.1451.  Google Scholar

[17]

P. Glendinning, "Stability, Instability and Chaos: An Introduction to the Theory of Nonlinear Differential Equations,", Cambridge University Press, (1994).   Google Scholar

[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.  doi: 10.1007/s00285-005-0331-9.  Google Scholar

[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.  doi: 10.1007/s00285-008-0206-y.  Google Scholar

[20]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields,", Springer-Verlag, (1983).   Google Scholar

[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.  doi: 10.1016/S0096-3003(02)00372-7.  Google Scholar

[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.  doi: 10.1098/rspb.2004.2800.  Google Scholar

[23]

K. P. Hadeler and P. van den Driesche, Backward bifurcation in epidemic control,, Math. Biosci., 146 (1997), 15.  doi: 10.1016/S0025-5564(97)00027-8.  Google Scholar

[24]

D. L. Heyman, "Control of Communicable Diseases Manual,", American Public Health Association, (2008).  doi: 10.1086/605668.  Google Scholar

[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.  doi: 10.1137/0152047.  Google Scholar

[26]

H. Inaba and H. Sekine, A mathematical model for Chagas disease with infection-age-dependent infectivity,, Math. Biosci., 190 (2004), 39.  doi: 10.1016/j.mbs.2004.02.004.  Google Scholar

[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.  doi: 10.1016/S0264-410X(02)00482-6.  Google Scholar

[28]

Y. Jin, W. Wang and S. Xiao, A SIRS model with a nonlinear incidence,, Chaos Solitons Fractals, 34 (2007), 1482.  doi: 10.1016/j.chaos.2006.04.022.  Google Scholar

[29]

W. O. Kermack and A. G. McKendrick, A Contribution to the mathematical theory of epidemics. I.,, Proc. R. Soc. A, 115 (1927), 700.   Google Scholar

[30]

C. M. Kribs-Zaleta and J. X. Velasco-Hernandez, A simple vaccination model with multiple endemic states,, Math. Biosci., 164 (2000), 183.  doi: 10.1016/S0025-5564(00)00003-1.  Google Scholar

[31]

C. M. Kribs-Zaleta, Center manifolds and normal forms in epidemic models,, Institute for Mathematics and Its Applications, 125 (2000), 269.   Google Scholar

[32]

M. Y. Li and L. Wang, Global stability in some SEIR epidemic models,, in, 126 (2002), 295.  doi: 10.1007/978-1-4613-0065-6_17.  Google Scholar

[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.  doi: 10.1007/BF00276956.  Google Scholar

[34]

M. Martcheva and H. R. Thieme, Progression age enhanced backward bifurcation in an epidemic model with superinfection,, J. Math. Biol., 46 (2003), 385.  doi: 10.1007/s00285-002-0181-7.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[38]

J. D. Murray, "Mathematical Biology,", Springer, (1998).  doi: 10.1007/b98869.  Google Scholar

[39]

M. G. Roberts, The pluses and minuses of $R_0$,, J. R. Soc. Interface, 4 (2007), 949.   Google Scholar

[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.  doi: 10.1137/070700966.  Google Scholar

[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.  doi: 10.1007/s00285-006-0028-8.  Google Scholar

[42]

UNICEF., "Unicef Info by Country - South Africa Statistics,", Available on line at: , ().   Google Scholar

[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.  doi: 10.1093/ije/31.5.968.  Google Scholar

[44]

P. van den Driessche and J. Watmough, A simple SIS epidemic model with a backward bifurcation,, J. Math. Biol., 40 (2000), 525.  doi: 10.1007/s002850000032.  Google Scholar

[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.  doi: 10.1016/j.vaccine.2011.12.123.  Google Scholar

[46]

W. Wang, Backward bifurcation of an epidemic model with treatment,, Math. Biosci., 201 (2006), 58.  doi: 10.1016/j.mbs.2005.12.022.  Google Scholar

[47]

Z. Wang, Backward bifurcation in simple SIS model,, Acta Matematicae Applicatae Sinica, 25 (2009), 127.  doi: 10.1007/s10255-006-6160-9.  Google Scholar

[48]

WHO/UNICEF, "Global Plan for Reducing Measles Mortality 2006-2010,", World Health Organization. Available from , ().   Google Scholar

[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.  doi: 10.1016/S0140-6736(07)60107-X.  Google Scholar

[50]

L. Wu and Z. Feng, Homoclinic bifurcation in an SIQR model for childhood diseases,, J. Differential Equations, 168 (2000), 150.  doi: 10.1006/jdeq.2000.3882.  Google Scholar

[51]

X. Zhang and X. Liu, Backward bifurcation of an epidemic model with saturated treatment function,, J. Math. Anal. Appl., 348 (2008), 433.  doi: 10.1016/j.jmaa.2008.07.042.  Google Scholar

[52]

Z. Zhonghua and S. Yaohong, Qualitative analysis of a SIR epidemic model with saturated treatment rate,, J. Appl. Math. Comput., 34 (2010), 177.  doi: 10.1007/s12190-009-0315-9.  Google Scholar

[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.  doi: 10.1016/j.cnsns.2011.03.026.  Google Scholar

show all references

References:
[1]

R. M. Anderson and R. M. May, "Infectious Diseases in Humans: Dynamics and Control,", Oxford University Press, (1991).   Google Scholar

[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.  doi: 10.1137/S0036139902413829.  Google Scholar

[3]

F. Brauer, Backward bifurcations in simple vaccination/treatment models,, Journal of Biological Dynamics, 5 (2011), 410.  doi: 10.1080/17513758.2010.510584.  Google Scholar

[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.  doi: 10.1016/j.micinf.2004.12.018.  Google Scholar

[5]

B. Buonomo and D. Lacitignola, On the dynamics of an SEIR epidemic model with a convex incidence rate,, Ric. Mat., 57 (2008), 261.  doi: 10.1007/s11587-008-0039-4.  Google Scholar

[6]

B. Buonomo and D. Lacitignola, Analysis of a tuberculosis model with a case study in Uganda,, J. Biol. Dyn., 4 (2010), 571.  doi: 10.1080/17513750903518441.  Google Scholar

[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.   Google Scholar

[8]

C. Castillo-Chavez and B. Song, Dynamical models of tubercolosis and their applications,, Math. Biosci. Engin., 1 (2004), 361.  doi: 10.3934/mbe.2004.1.361.  Google Scholar

[9]

Centers for Disease Control and Prevention, Atlanta, USA., Progress toward measles elimination - Southern Africa 1996-1998., MMWR, 48 (1999), 585.   Google Scholar

[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.   Google Scholar

[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.  doi: 10.1016/j.jtbi.2008.05.015.  Google Scholar

[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.  doi: 10.3934/dcdsb.2003.3.299.  Google Scholar

[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).   Google Scholar

[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.  doi: 10.1007/s002850050099.  Google Scholar

[15]

Z. Feng and H. Thieme, Recurrent outbreaks of childhood diseases revisited: The impact of isolation,, Math. Biosci., 128 (1995), 93.  doi: 10.1016/0025-5564(94)00069-C.  Google Scholar

[16]

Z. Feng, C. Castillo-Chavez and A. F. Capurro, A model for tuberculosis with exogenous reinfection,, Theor. Popul. Biol., 57 (2000), 235.  doi: 10.1006/tpbi.2000.1451.  Google Scholar

[17]

P. Glendinning, "Stability, Instability and Chaos: An Introduction to the Theory of Nonlinear Differential Equations,", Cambridge University Press, (1994).   Google Scholar

[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.  doi: 10.1007/s00285-005-0331-9.  Google Scholar

[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.  doi: 10.1007/s00285-008-0206-y.  Google Scholar

[20]

J. Guckenheimer and P. Holmes, "Nonlinear Oscillations, Dynamical Systems and Bifurcations of Vector Fields,", Springer-Verlag, (1983).   Google Scholar

[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.  doi: 10.1016/S0096-3003(02)00372-7.  Google Scholar

[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.  doi: 10.1098/rspb.2004.2800.  Google Scholar

[23]

K. P. Hadeler and P. van den Driesche, Backward bifurcation in epidemic control,, Math. Biosci., 146 (1997), 15.  doi: 10.1016/S0025-5564(97)00027-8.  Google Scholar

[24]

D. L. Heyman, "Control of Communicable Diseases Manual,", American Public Health Association, (2008).  doi: 10.1086/605668.  Google Scholar

[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.  doi: 10.1137/0152047.  Google Scholar

[26]

H. Inaba and H. Sekine, A mathematical model for Chagas disease with infection-age-dependent infectivity,, Math. Biosci., 190 (2004), 39.  doi: 10.1016/j.mbs.2004.02.004.  Google Scholar

[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.  doi: 10.1016/S0264-410X(02)00482-6.  Google Scholar

[28]

Y. Jin, W. Wang and S. Xiao, A SIRS model with a nonlinear incidence,, Chaos Solitons Fractals, 34 (2007), 1482.  doi: 10.1016/j.chaos.2006.04.022.  Google Scholar

[29]

W. O. Kermack and A. G. McKendrick, A Contribution to the mathematical theory of epidemics. I.,, Proc. R. Soc. A, 115 (1927), 700.   Google Scholar

[30]

C. M. Kribs-Zaleta and J. X. Velasco-Hernandez, A simple vaccination model with multiple endemic states,, Math. Biosci., 164 (2000), 183.  doi: 10.1016/S0025-5564(00)00003-1.  Google Scholar

[31]

C. M. Kribs-Zaleta, Center manifolds and normal forms in epidemic models,, Institute for Mathematics and Its Applications, 125 (2000), 269.   Google Scholar

[32]

M. Y. Li and L. Wang, Global stability in some SEIR epidemic models,, in, 126 (2002), 295.  doi: 10.1007/978-1-4613-0065-6_17.  Google Scholar

[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.  doi: 10.1007/BF00276956.  Google Scholar

[34]

M. Martcheva and H. R. Thieme, Progression age enhanced backward bifurcation in an epidemic model with superinfection,, J. Math. Biol., 46 (2003), 385.  doi: 10.1007/s00285-002-0181-7.  Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[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.   Google Scholar

[38]

J. D. Murray, "Mathematical Biology,", Springer, (1998).  doi: 10.1007/b98869.  Google Scholar

[39]

M. G. Roberts, The pluses and minuses of $R_0$,, J. R. Soc. Interface, 4 (2007), 949.   Google Scholar

[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.  doi: 10.1137/070700966.  Google Scholar

[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.  doi: 10.1007/s00285-006-0028-8.  Google Scholar

[42]

UNICEF., "Unicef Info by Country - South Africa Statistics,", Available on line at: , ().   Google Scholar

[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.  doi: 10.1093/ije/31.5.968.  Google Scholar

[44]

P. van den Driessche and J. Watmough, A simple SIS epidemic model with a backward bifurcation,, J. Math. Biol., 40 (2000), 525.  doi: 10.1007/s002850000032.  Google Scholar

[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.  doi: 10.1016/j.vaccine.2011.12.123.  Google Scholar

[46]

W. Wang, Backward bifurcation of an epidemic model with treatment,, Math. Biosci., 201 (2006), 58.  doi: 10.1016/j.mbs.2005.12.022.  Google Scholar

[47]

Z. Wang, Backward bifurcation in simple SIS model,, Acta Matematicae Applicatae Sinica, 25 (2009), 127.  doi: 10.1007/s10255-006-6160-9.  Google Scholar

[48]

WHO/UNICEF, "Global Plan for Reducing Measles Mortality 2006-2010,", World Health Organization. Available from , ().   Google Scholar

[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.  doi: 10.1016/S0140-6736(07)60107-X.  Google Scholar

[50]

L. Wu and Z. Feng, Homoclinic bifurcation in an SIQR model for childhood diseases,, J. Differential Equations, 168 (2000), 150.  doi: 10.1006/jdeq.2000.3882.  Google Scholar

[51]

X. Zhang and X. Liu, Backward bifurcation of an epidemic model with saturated treatment function,, J. Math. Anal. Appl., 348 (2008), 433.  doi: 10.1016/j.jmaa.2008.07.042.  Google Scholar

[52]

Z. Zhonghua and S. Yaohong, Qualitative analysis of a SIR epidemic model with saturated treatment rate,, J. Appl. Math. Comput., 34 (2010), 177.  doi: 10.1007/s12190-009-0315-9.  Google Scholar

[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.  doi: 10.1016/j.cnsns.2011.03.026.  Google Scholar

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