2012, 9(4): 915-935. doi: 10.3934/mbe.2012.9.915

Basic stochastic models for viral infection within a host

1. 

Texas Tech University, Department of Mathematics and Statistics, Lubbock, Texas 79409-1042, United States

2. 

Department of Mathematics and Statistics, Texas Tech University, Lubbock, TX 79409-1042, United States

Received  September 2011 Revised  June 2012 Published  October 2012

Stochastic differential equation (SDE) models are formulated for intra-host virus-cell dynamics during the early stages of viral infection, prior to activation of the immune system. The SDE models incorporate more realism into the mechanisms for viral entry and release than ordinary differential equation (ODE) models and show distinct differences from the ODE models. The variability in the SDE models depends on the concentration, with much greater variability for small concentrations than large concentrations. In addition, the SDE models show significant variability in the timing of the viral peak. The viral peak is earlier for viruses that are released from infected cells via bursting rather than via budding from the cell membrane.
Citation: Sukhitha W. Vidurupola, Linda J. S. Allen. Basic stochastic models for viral infection within a host. Mathematical Biosciences & Engineering, 2012, 9 (4) : 915-935. doi: 10.3934/mbe.2012.9.915
References:
[1]

E. Allen, "Modeling with Itô Stochastic Differential Equations," Springer, Dordrecht, The Netherlands, 2007.

[2]

E. J. Allen, L. J. S. Allen, A. Arciniega and P. E. Greenwood, Construction of equivalent stochastic differential equation models, Stochastic Analysis and Applications, 26 (2008), 274-297. doi: 10.1080/07362990701857129.

[3]

L. J. S. Allen, "An Introduction to Stochastic Processes with Applications to Biology," $2^{nd}$ edition, Chapman Hall/CRC Press, Boca Raton, FL, 2010.

[4]

D. Burg, L. Rong, A. U. Neumann and H. Dahari, Mathematical modeling of viral kinetics under immune control during primary HIV-1 infection, Journal of Theoretical Biology, 259 (2009), 751-759. doi: 10.1016/j.jtbi.2009.04.010.

[5]

D. Chao, M. Davenport, S. Forrest and A. Perelson, A stochastic model of cytotoxic T cell responses, Journal of Theoretical Biology, 228 (2004), 227-240. doi: 10.1016/j.jtbi.2003.12.011.

[6]

E. T. Clayson, L. V. Jones Brando and R. W. Compans, Release of simian virus 40 virions from epithelial cells is polarized and occurs without cell lysis, Journal of Virology, 63 (1989), 2278-2288.

[7]

A. J. Ekanayake and L. J. S. Allen, Comparison of Markov chain and stochastic differential equation population models under higher-order moment closure approximations, Stochastic Analysis and Applications, 28 (2010), 907-927. doi: 10.1080/07362990903415882.

[8]

D. T. Gillespie, The chemical Langevin equation, The Journal of Chemical Physics, 113 (2000), 297-306. doi: 10.1063/1.481811.

[9]

J. Heesterbeek and M. G. Roberts, The type-reproduction number T in models for infectious disease control, Mathematical Biosciences, 206 (2007), 3-10. doi: 10.1016/j.mbs.2004.10.013.

[10]

C. B. Jonsson, L. T. M. Figueiredo and O. Vapalahti, A global perspective on hantavirus ecology, epidemiology, and disease, Clinical Microbiology Reviews, 23 (2010), 412-441. doi: 10.1128/CMR.00062-09.

[11]

J. C. Kamgang and G. Sallet, Computation of threshold conditions for epidemiological models and global stability of the disease-free equilibrium (DFE), Mathematical Biosciences, 213 (2008), 1-12. doi: 10.1016/j.mbs.2008.02.005.

[12]

H. Kamina, R. Makuch and H. Zhao, A stochastic modeling of early HIV-1 population dynamics, Mathematical Biosciences, 170 (2001), 187-198. doi: 10.1016/S0025-5564(00)00069-9.

[13]

M. J. Keeling, Metapopulation moments: Coupling, stochasticity and persistence, Journal of Animal Ecology, 69 (2000), 725-736. doi: 10.1046/j.1365-2656.2000.00430.x.

[14]

M. J. Keeling, Multiplicative moments and measure of persistence in ecology, Journal of Theoretical Biology, 205 (2000), 269-281. doi: 10.1006/jtbi.2000.2066.

[15]

N. L. Komarova, Viral reproductive strategies: How can lytic viruses be evolutionarily competitive?, Journal of Theoretical Biology, 249 (2007), 766-784. doi: 10.1016/j.jtbi.2007.09.013.

[16]

I. Krishnarajah, A. Cook, G. Marion and G. Gibson, Novel moment closure approximations in stochastic epidemics, Bulletin of Mathematical Biology, 67 (2005), 855-873. doi: 10.1016/j.bulm.2004.11.002.

[17]

T. G. Kurtz, Strong approximation theorems for density dependent Markov chains, Stochastic Processes and their Applications, 6 (1978), 223-240.

[18]

A. L. Lloyd, Estimating variability in models for recurrent epidemics: Assessing the use of moment closure techniques, Theoretical Population Biology, 65 (2004), 49-65. doi: 10.1016/j.tpb.2003.07.002.

[19]

J. H. Matis and T. Kiffe, "Stochastic Population Models," Springer, New York, Berlin and Heidelberg, 2000.

[20]

M. N. Matrosovich, T. Y. Matrosovich, T. Gray, N. A. Roberts and H. D. Klenk, Human and avian influenza viruses target different cell types in cultures of human airway epithelium, Proceedings of the National Academy of Sciences, 101 (2004), 4620-4624. doi: 10.1073/pnas.0308001101.

[21]

M. A. Nowak and R. M. May, "Virus Dynamics," Oxford Univ. Press, New York, 2000.

[22]

B. Øksendal, "Stochastic Differential Equations: An Introduction with Applications," Springer, Verlag, Berlin, Heidelberg, $5^{th}$ edition, 2000.

[23]

J. E. Pearson, P. Krapivsky and A. S. Perelson, Stochastic theory of early viral infection: continuous versus burst production of virions, PLoS Computational Biology, 7 (2011), 1-17.

[24]

A. S. Perelson, D. E. Kirschner and R. De Boer, Dynamics of HIV infection of CD4+ T cells, Mathematical Biosciences, 114 (1993), 81-125. doi: 10.1016/0025-5564(93)90043-A.

[25]

A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Review, 41 (1999), 3-44. doi: 10.1137/S0036144598335107.

[26]

A. N. Phillips, Reduction of HIV concentration during acute infection: independence from a specific immune response, Science, 271 (1996), 497-499. doi: 10.1126/science.271.5248.497.

[27]

M. G. Roberts and J. Heesterbeek, A new method to estimate the effort required to control an infectious disease, Proceedings of the Royal Society London B, 270 (2003), 1359-1364. doi: 10.1098/rspb.2003.2339.

[28]

A. Singh and J. P. Hespanha, Moment closure techniques for stochastic models in population biology, Proceedings of the 2006 American Control Conference, (2006), 4730-4735.

[29]

W. Y. Tan and H. Wu, Stochastic modeling of the dynamics of CD4+ T-cells infection by HIV and some Monte-Carlo studies, Mathematical Biosciences, 147 (1998), 173-205 doi: 10.1016/S0025-5564(97)00094-1.

[30]

H. Tuckwell and F. Wan, First passage time to detection in stochastic population dynamical models for HIV-1, Applied Mathematics Letters, 13 (2000), 79-83. doi: 10.1016/S0893-9659(00)00037-9.

[31]

H. C. Tuckwell and E. Le Corfec, A stochastic model for early HIV-1 population dynamics, Journal of Theoretical Biology, 195 (1998), 451-463. doi: 10.1006/jtbi.1998.0806.

[32]

P. van den Driesssche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.

[33]

P. van den Driesssche and J. Watmough, Chapter 6: Further notes on the basic reproduction number, in "Mathematical Epidemiology" (eds. F. Brauer, P. van den Driessche and J. Wu), Springer, Verlag, Berlin, Heidelberg, (2008), 159-178.

[34]

F. Verhulst, "Nonlinear Differential Equations and Dynamical Systems," Springer-Verlag, Berlin, Heidelberg, New York, 1985.

[35]

S. W. Vidurupola, "Deterministic and Stochastic Models for Early Viral Infection within a Host," M. S. Thesis, Texas Tech University, Lubbock, Texas, U.S.A., 2010.

[36]

D. Wodarz and M. A. Nowak, Mathematical models of HIV pathogenesis and treatment, BioEssays, 24 (2002), 1178-1187. doi: 10.1002/bies.10196.

[37]

Y. Yuan and L. J. S. Allen, Stochastic models for virus and immune system dynamics, Mathematical Biosciences, 234 (2011), 84-94. doi: 10.1016/j.mbs.2011.08.007.

[38]

S. R. Zaki, P. W. Greer, L. M. Coffield, C. S. Goldsmith, K. B. Nolte, K. Foucar, R. M. Feddersen, R. E. Zumwalt, G. L. Miller, A. S. Khan, P. E. Rollin, T. G. Ksiazek, S. T. Nichol, B. W. J. Mahy and C. J. Peters, Hantavirus pulmonary syndrome: Pathogenesis of an emerging infectious disease, American Journal of Pathology, 146 (1995), 552-578.

show all references

References:
[1]

E. Allen, "Modeling with Itô Stochastic Differential Equations," Springer, Dordrecht, The Netherlands, 2007.

[2]

E. J. Allen, L. J. S. Allen, A. Arciniega and P. E. Greenwood, Construction of equivalent stochastic differential equation models, Stochastic Analysis and Applications, 26 (2008), 274-297. doi: 10.1080/07362990701857129.

[3]

L. J. S. Allen, "An Introduction to Stochastic Processes with Applications to Biology," $2^{nd}$ edition, Chapman Hall/CRC Press, Boca Raton, FL, 2010.

[4]

D. Burg, L. Rong, A. U. Neumann and H. Dahari, Mathematical modeling of viral kinetics under immune control during primary HIV-1 infection, Journal of Theoretical Biology, 259 (2009), 751-759. doi: 10.1016/j.jtbi.2009.04.010.

[5]

D. Chao, M. Davenport, S. Forrest and A. Perelson, A stochastic model of cytotoxic T cell responses, Journal of Theoretical Biology, 228 (2004), 227-240. doi: 10.1016/j.jtbi.2003.12.011.

[6]

E. T. Clayson, L. V. Jones Brando and R. W. Compans, Release of simian virus 40 virions from epithelial cells is polarized and occurs without cell lysis, Journal of Virology, 63 (1989), 2278-2288.

[7]

A. J. Ekanayake and L. J. S. Allen, Comparison of Markov chain and stochastic differential equation population models under higher-order moment closure approximations, Stochastic Analysis and Applications, 28 (2010), 907-927. doi: 10.1080/07362990903415882.

[8]

D. T. Gillespie, The chemical Langevin equation, The Journal of Chemical Physics, 113 (2000), 297-306. doi: 10.1063/1.481811.

[9]

J. Heesterbeek and M. G. Roberts, The type-reproduction number T in models for infectious disease control, Mathematical Biosciences, 206 (2007), 3-10. doi: 10.1016/j.mbs.2004.10.013.

[10]

C. B. Jonsson, L. T. M. Figueiredo and O. Vapalahti, A global perspective on hantavirus ecology, epidemiology, and disease, Clinical Microbiology Reviews, 23 (2010), 412-441. doi: 10.1128/CMR.00062-09.

[11]

J. C. Kamgang and G. Sallet, Computation of threshold conditions for epidemiological models and global stability of the disease-free equilibrium (DFE), Mathematical Biosciences, 213 (2008), 1-12. doi: 10.1016/j.mbs.2008.02.005.

[12]

H. Kamina, R. Makuch and H. Zhao, A stochastic modeling of early HIV-1 population dynamics, Mathematical Biosciences, 170 (2001), 187-198. doi: 10.1016/S0025-5564(00)00069-9.

[13]

M. J. Keeling, Metapopulation moments: Coupling, stochasticity and persistence, Journal of Animal Ecology, 69 (2000), 725-736. doi: 10.1046/j.1365-2656.2000.00430.x.

[14]

M. J. Keeling, Multiplicative moments and measure of persistence in ecology, Journal of Theoretical Biology, 205 (2000), 269-281. doi: 10.1006/jtbi.2000.2066.

[15]

N. L. Komarova, Viral reproductive strategies: How can lytic viruses be evolutionarily competitive?, Journal of Theoretical Biology, 249 (2007), 766-784. doi: 10.1016/j.jtbi.2007.09.013.

[16]

I. Krishnarajah, A. Cook, G. Marion and G. Gibson, Novel moment closure approximations in stochastic epidemics, Bulletin of Mathematical Biology, 67 (2005), 855-873. doi: 10.1016/j.bulm.2004.11.002.

[17]

T. G. Kurtz, Strong approximation theorems for density dependent Markov chains, Stochastic Processes and their Applications, 6 (1978), 223-240.

[18]

A. L. Lloyd, Estimating variability in models for recurrent epidemics: Assessing the use of moment closure techniques, Theoretical Population Biology, 65 (2004), 49-65. doi: 10.1016/j.tpb.2003.07.002.

[19]

J. H. Matis and T. Kiffe, "Stochastic Population Models," Springer, New York, Berlin and Heidelberg, 2000.

[20]

M. N. Matrosovich, T. Y. Matrosovich, T. Gray, N. A. Roberts and H. D. Klenk, Human and avian influenza viruses target different cell types in cultures of human airway epithelium, Proceedings of the National Academy of Sciences, 101 (2004), 4620-4624. doi: 10.1073/pnas.0308001101.

[21]

M. A. Nowak and R. M. May, "Virus Dynamics," Oxford Univ. Press, New York, 2000.

[22]

B. Øksendal, "Stochastic Differential Equations: An Introduction with Applications," Springer, Verlag, Berlin, Heidelberg, $5^{th}$ edition, 2000.

[23]

J. E. Pearson, P. Krapivsky and A. S. Perelson, Stochastic theory of early viral infection: continuous versus burst production of virions, PLoS Computational Biology, 7 (2011), 1-17.

[24]

A. S. Perelson, D. E. Kirschner and R. De Boer, Dynamics of HIV infection of CD4+ T cells, Mathematical Biosciences, 114 (1993), 81-125. doi: 10.1016/0025-5564(93)90043-A.

[25]

A. S. Perelson and P. W. Nelson, Mathematical analysis of HIV-1 dynamics in vivo, SIAM Review, 41 (1999), 3-44. doi: 10.1137/S0036144598335107.

[26]

A. N. Phillips, Reduction of HIV concentration during acute infection: independence from a specific immune response, Science, 271 (1996), 497-499. doi: 10.1126/science.271.5248.497.

[27]

M. G. Roberts and J. Heesterbeek, A new method to estimate the effort required to control an infectious disease, Proceedings of the Royal Society London B, 270 (2003), 1359-1364. doi: 10.1098/rspb.2003.2339.

[28]

A. Singh and J. P. Hespanha, Moment closure techniques for stochastic models in population biology, Proceedings of the 2006 American Control Conference, (2006), 4730-4735.

[29]

W. Y. Tan and H. Wu, Stochastic modeling of the dynamics of CD4+ T-cells infection by HIV and some Monte-Carlo studies, Mathematical Biosciences, 147 (1998), 173-205 doi: 10.1016/S0025-5564(97)00094-1.

[30]

H. Tuckwell and F. Wan, First passage time to detection in stochastic population dynamical models for HIV-1, Applied Mathematics Letters, 13 (2000), 79-83. doi: 10.1016/S0893-9659(00)00037-9.

[31]

H. C. Tuckwell and E. Le Corfec, A stochastic model for early HIV-1 population dynamics, Journal of Theoretical Biology, 195 (1998), 451-463. doi: 10.1006/jtbi.1998.0806.

[32]

P. van den Driesssche and J. Watmough, Reproduction numbers and subthreshold endemic equilibria for compartmental models of disease transmission, Mathematical Biosciences, 180 (2002), 29-48. doi: 10.1016/S0025-5564(02)00108-6.

[33]

P. van den Driesssche and J. Watmough, Chapter 6: Further notes on the basic reproduction number, in "Mathematical Epidemiology" (eds. F. Brauer, P. van den Driessche and J. Wu), Springer, Verlag, Berlin, Heidelberg, (2008), 159-178.

[34]

F. Verhulst, "Nonlinear Differential Equations and Dynamical Systems," Springer-Verlag, Berlin, Heidelberg, New York, 1985.

[35]

S. W. Vidurupola, "Deterministic and Stochastic Models for Early Viral Infection within a Host," M. S. Thesis, Texas Tech University, Lubbock, Texas, U.S.A., 2010.

[36]

D. Wodarz and M. A. Nowak, Mathematical models of HIV pathogenesis and treatment, BioEssays, 24 (2002), 1178-1187. doi: 10.1002/bies.10196.

[37]

Y. Yuan and L. J. S. Allen, Stochastic models for virus and immune system dynamics, Mathematical Biosciences, 234 (2011), 84-94. doi: 10.1016/j.mbs.2011.08.007.

[38]

S. R. Zaki, P. W. Greer, L. M. Coffield, C. S. Goldsmith, K. B. Nolte, K. Foucar, R. M. Feddersen, R. E. Zumwalt, G. L. Miller, A. S. Khan, P. E. Rollin, T. G. Ksiazek, S. T. Nichol, B. W. J. Mahy and C. J. Peters, Hantavirus pulmonary syndrome: Pathogenesis of an emerging infectious disease, American Journal of Pathology, 146 (1995), 552-578.

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