The Effects of a Backward Bifurcation on a Continuous Time Markov Chain Model for the Transmission Dynamics of Single Strain Dengue Virus

Affiliation(s)

Centre for Advanced Studies in Mathematics, Lahore University of Management Sciences, Lahore, Pakistan.

Centre for Advanced Studies in Mathematics, Lahore University of Management Sciences, Lahore, Pakistan.

ABSTRACT

Global incidence of dengue, a vector-borne tropical disease, has seen a dramatic increase with several major outbreaks in the past few decades. We formulate and analyze a stochastic epidemic model for the transmission dynamics of a single strain of dengue virus. The stochastic model is constructed using a continuous time Markov chain (CTMC) and is based on an existing deterministic model that suggests the existence of a backward bifurcation for some values of the model parameters. The dynamics of the stochastic model are explored through numerical simulations in this region of bistability. The mean of each random variable is numerically estimated and these are compared to the dynamics of the deterministic model. It is observed that the stochastic model also predicts the co-existence of a locally asymptotically stable disease-free equilibrium along with a locally stable endemic equilibrium. This co-existence of equilibria is important from a public health perspective because it implies that dengue can persist in populations even if the value of the basic reproduction number is less than unity.

Cite this paper

A. Khan, M. Hassan and M. Imran, "The Effects of a Backward Bifurcation on a Continuous Time Markov Chain Model for the Transmission Dynamics of Single Strain Dengue Virus,"*Applied Mathematics*, Vol. 4 No. 4, 2013, pp. 663-674. doi: 10.4236/am.2013.44091.

A. Khan, M. Hassan and M. Imran, "The Effects of a Backward Bifurcation on a Continuous Time Markov Chain Model for the Transmission Dynamics of Single Strain Dengue Virus,"

References

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[18] G. Chowell, P. Diaz-Duenas, J. C. Miller, A. Alcazar Velazco, J. M. Hyman, P. W. Fenimore and C. Castillo Chavez, “Estimation of the Reproduction Number of Dengue Fever from Spatial Epidemic Data,” Mathematical Biosciences, Vol. 208, No. 2, 2007, pp. 571-589. doi:10.1016/j.mbs.2006.11.011

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[21] N. Bailey, “A Simple Stochastic Epidemic,” Biometrika, Vol. 37, No. 3-4, 1950, pp. 193-202. doi:10.2307/2332371

[22] L. J. S. Allen, D. A. Flores, R. K. Ratnayake and J. R. Herbold, “Discrete-Time Deterministic and Stochastic Models for the Spread of Rabies,” Applied Mathematics and Computation, Vol. 132, No. 2, 2002, pp. 271-292. doi:10.1016/S0096-3003(01)00192-8

[23] G. H. Weiss and M. Dishon, “On the Asymptotic Behavior of the Stochastic and Deterministic Models of an Epidemic,” Mathematical Biosciences, Vol. 11, No. 3, 1971, pp. 261-265. doi:10.1016/0025-5564(71)90087-3

[24] A. R. Tuite, J. Tien, M. Eisenberg, D. J. Earn, J. Ma and D. N. Fisman, “Cholera Epidemic in Haiti, 2010: Using a Transmission Model to Explain Spatial Spread of Disease and Identify Optimal Control Interventions,” Annals of Internal Medicine, Vol. 154, No. 9, 2011, pp. 593-601. doi:10.7326/0003-4819-154-9-201105030-00334

[25] L. J. S. Allen and P. Driessche, “Stochastic Epidemic Models with a Backward Bifurcation,” Mathematical Bio sciences and Engineering, Vol. 3, No. 3, 2006, pp. 445-458. doi:10.3934/mbe.2006.3.445

[26] D. R. de Souza, T. Tomé, S. T. Pinho, F. R. Barreto and M. J. de Oliveira, “Stochastic Dynamics of Dengue Epidemics,” Physical Review, Vol. 87, No. 1, 2013, Article ID: 012709.

[27] S. Spencer, “Stochastic Epidemic Models for Emerging Diseases,” Ph.D. Dissertation, University of Nottingham, Nottingham, 2008.

[28] L. J. S. Allen, “An Introduction to Stochastic Processes with Applications to Biology,” Chapman and Hall/CRC, Boca Ratton, 2010.

[29] L. J. S. Allen and A. M. Burgin, “Comparison of Deterministic and Stochastic SIS and SIR Models in Discrete Time,” Mathematical Biosciences, Vol. 163, No. 1, 2000, pp. 1-34. doi:10.1016/S0025-5564(99)00047-4

[30] P. van den Driessche and J. Watmough, “Reproduction Numbers and Sub-Threshold Endemic Equilibria for Compartmental Models of Disease Transmission,” Mathematical Biosciences, Vol. 180, No. 1, 2002, pp. 29-48. doi:10.1016/S0025-5564(02)00108-6

[1] S. Ranjit and N. Kissoon, “Dengue Hemorrhagic Fever and Shock Syndromes,” Pediatric Critical Care Medicine, Vol. 12, No. 1, 2011, pp. 90-100. doi:10.1097/PCC.0b013e3181e911a7

[2] World Health Organization, “Dengue and Dengue Hemorrhagic Fever,” 2012. http://www.who.int/mediacentre/factsheets/fs117/en/

[3] D. J. Gubler, “Dengue and Dengue Hemorrhagic Fever,” Clinical Microbiology Reviews, Vol. 11, No. 3, 1998, pp. 480-496.

[4] S. B. Halstead, S. Nimmannitya and S. N. Cohen, “Ob servations Related to Pathogenesis of Dengue Hemorrhagic Fever IV. Relation of Disease Severity to Antibody Response and Virus Recovered,” The Yale Journal of Biology and Medicine, Vol. 42, No. 5, 1970, pp. 311-322.

[5] I. Kautner, M. J. Robinson and U. Kuhnle, “Dengue Virus Infection: Epidemiology, Pathogenesis, Clinical Presentation, Diagnosis, and Prevention,” The Journal of Pediatrics, Vol. 131, No. 4, 1997, pp. 516-524.

[6] C. Shekhar, “Deadly Dengue: New Vaccines Promise to Tackle This Escalating Global Menace,” Chemistry and Biology, Vol. 14, No. 8, 2007, pp. 871-872. doi:10.1016/j.chembiol.2007.08.004

[7] E. C. Holmes and S. S. Twiddy, “The Origin, Emergence and Evolutionary Genetics of Dengue Virus,” Infection, Genetics and Evolution, Vol. 3, No. 1, 2003, pp. 19-28. doi:10.1016/S1567-1348(03)00004-2

[8] J. Whitehorn and J. Farrar, “Dengue,” British Medical Bulletin, Vol. 95, No. 1, 2010, pp. 161-173.

[9] D. J. Gubler and G. Kuno, “Dengue and Dengue Hemorrhagic Fever,” CAB International, London, 1997.

[10] I. Kawaguchi, A. Sasaki and M. Boots, “Why Are Dengue Virus Serotypes so Distantly Related? Enhancement and Limiting Serotype Similarity between Dengue Virus Srains,” Proceedings of the Royal Society of London, Series B: Biological Sciences, Vol. 270, No. 1530, 2003, pp. 2241-2247. doi:10.1098/rspb.2003.2440

[11] S. M. Garba, A. B. Gumel and M. R. Abu Bakar, “Back ward Bifurcations in Dengue Transmission Dynamics,” Mathematical Biosciences, Vol. 215, No. 1, 2008, pp. 11-25. doi:10.1016/j.mbs.2008.05.002

[12] S. M. Garba, A. B. Gumel and M. R. Abu Bakar, “Effect of Cross-Immunity on the Transmission Dynamics of Two Strains of Dengue,” International Journal of Computer Mathematics, Vol. 87, No. 10, 2010, pp. 2361-2384. doi:10.1080/00207160802660608

[13] H. J. Wearing and P. Rohani, “Ecological and Immunological Determinants of Dengue Epidemics,” Proceed ings of the National Academy of Sciences, Vol. 103, No. 31, 2006, pp. 11802-11807. doi:10.1073/pnas.0602960103

[14] L. Esteva and C. Vargas, “Coexistence of Different Serotypes of Dengue Virus,” Journal of Mathematical Biology, Vol. 46, No. 1, 2003, pp. 31-47. doi:10.1007/s00285-002-0168-4

[15] N. Ferguson, R. Anderson and S. Gupta, “The Effect of Antibody-Dependent Enhancement on the Transmission Dynamics and Persistence of Multiple-Strain Pathogens,” Proceedings of the National Academy of Sciences, Vol. 96, No. 2, 1999, pp. 790-794. doi:10.1073/pnas.96.2.790

[16] L. Esteva and C. Vargas, “A Model for Dengue Disease with Variable Human Population,” Journal of Mathematical Biology, Vol. 38, No. 3, 1999, pp. 220-240. doi:10.1007/s002850050147

[17] L. Esteva and C. Vargas, “Analysis of a Dengue Disease Transmission Model,” Mathematical Biosciences, Vol. 150, No. 2, 1998, pp. 131-151. doi:10.1016/S0025-5564(98)10003-2

[18] G. Chowell, P. Diaz-Duenas, J. C. Miller, A. Alcazar Velazco, J. M. Hyman, P. W. Fenimore and C. Castillo Chavez, “Estimation of the Reproduction Number of Dengue Fever from Spatial Epidemic Data,” Mathematical Biosciences, Vol. 208, No. 2, 2007, pp. 571-589. doi:10.1016/j.mbs.2006.11.011

[19] L. J. S. Allen, “An Introduction to Stochastic Epidemic Models,” In: F. Brauer, P. Van den Driessche and J. Wu, Eds., Mathematical Epidemiology, Springer-Verlag, Berlin, 2008, pp. 77-128. doi:10.1007/978-3-540-78911-6_3

[20] A. J. Keeling and J. V. Ross, “On Methods for Studying Stochastic Disease Dynamics,” Journal of The Royal Society Interface, Vol. 5, No. 19, 2008, pp. 171-181. doi:10.1098/rsif.2007.1106

[21] N. Bailey, “A Simple Stochastic Epidemic,” Biometrika, Vol. 37, No. 3-4, 1950, pp. 193-202. doi:10.2307/2332371

[22] L. J. S. Allen, D. A. Flores, R. K. Ratnayake and J. R. Herbold, “Discrete-Time Deterministic and Stochastic Models for the Spread of Rabies,” Applied Mathematics and Computation, Vol. 132, No. 2, 2002, pp. 271-292. doi:10.1016/S0096-3003(01)00192-8

[23] G. H. Weiss and M. Dishon, “On the Asymptotic Behavior of the Stochastic and Deterministic Models of an Epidemic,” Mathematical Biosciences, Vol. 11, No. 3, 1971, pp. 261-265. doi:10.1016/0025-5564(71)90087-3

[24] A. R. Tuite, J. Tien, M. Eisenberg, D. J. Earn, J. Ma and D. N. Fisman, “Cholera Epidemic in Haiti, 2010: Using a Transmission Model to Explain Spatial Spread of Disease and Identify Optimal Control Interventions,” Annals of Internal Medicine, Vol. 154, No. 9, 2011, pp. 593-601. doi:10.7326/0003-4819-154-9-201105030-00334

[25] L. J. S. Allen and P. Driessche, “Stochastic Epidemic Models with a Backward Bifurcation,” Mathematical Bio sciences and Engineering, Vol. 3, No. 3, 2006, pp. 445-458. doi:10.3934/mbe.2006.3.445

[26] D. R. de Souza, T. Tomé, S. T. Pinho, F. R. Barreto and M. J. de Oliveira, “Stochastic Dynamics of Dengue Epidemics,” Physical Review, Vol. 87, No. 1, 2013, Article ID: 012709.

[27] S. Spencer, “Stochastic Epidemic Models for Emerging Diseases,” Ph.D. Dissertation, University of Nottingham, Nottingham, 2008.

[28] L. J. S. Allen, “An Introduction to Stochastic Processes with Applications to Biology,” Chapman and Hall/CRC, Boca Ratton, 2010.

[29] L. J. S. Allen and A. M. Burgin, “Comparison of Deterministic and Stochastic SIS and SIR Models in Discrete Time,” Mathematical Biosciences, Vol. 163, No. 1, 2000, pp. 1-34. doi:10.1016/S0025-5564(99)00047-4

[30] P. van den Driessche and J. Watmough, “Reproduction Numbers and Sub-Threshold Endemic Equilibria for Compartmental Models of Disease Transmission,” Mathematical Biosciences, Vol. 180, No. 1, 2002, pp. 29-48. doi:10.1016/S0025-5564(02)00108-6