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, (2007). Google Scholar

[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. doi: 10.1080/07362990701857129. Google Scholar

[3]

L. J. S. Allen, "An Introduction to Stochastic Processes with Applications to Biology,", $2^{nd}$ edition, (2010). Google Scholar

[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. doi: 10.1016/j.jtbi.2009.04.010. Google Scholar

[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. doi: 10.1016/j.jtbi.2003.12.011. Google Scholar

[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. Google Scholar

[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. doi: 10.1080/07362990903415882. Google Scholar

[8]

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

[9]

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

[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. doi: 10.1128/CMR.00062-09. Google Scholar

[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. doi: 10.1016/j.mbs.2008.02.005. Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

J. H. Matis and T. Kiffe, "Stochastic Population Models,", Springer, (2000). Google Scholar

[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. doi: 10.1073/pnas.0308001101. Google Scholar

[21]

M. A. Nowak and R. M. May, "Virus Dynamics,", Oxford Univ. Press, (2000). Google Scholar

[22]

B. Øksendal, "Stochastic Differential Equations: An Introduction with Applications,", Springer, (2000). Google Scholar

[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. Google Scholar

[24]

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

[25]

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

[26]

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

[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. doi: 10.1098/rspb.2003.2339. Google Scholar

[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. Google Scholar

[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. doi: 10.1016/S0025-5564(97)00094-1. Google Scholar

[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. doi: 10.1016/S0893-9659(00)00037-9. Google Scholar

[31]

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

[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. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar

[33]

P. van den Driesssche and J. Watmough, Chapter 6: Further notes on the basic reproduction number,, in, (2008), 159. Google Scholar

[34]

F. Verhulst, "Nonlinear Differential Equations and Dynamical Systems,", Springer-Verlag, (1985). Google Scholar

[35]

S. W. Vidurupola, "Deterministic and Stochastic Models for Early Viral Infection within a Host,", M. S. Thesis, (2010). Google Scholar

[36]

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

[37]

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

[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. Google Scholar

show all references

References:
[1]

E. Allen, "Modeling with Itô Stochastic Differential Equations,", Springer, (2007). Google Scholar

[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. doi: 10.1080/07362990701857129. Google Scholar

[3]

L. J. S. Allen, "An Introduction to Stochastic Processes with Applications to Biology,", $2^{nd}$ edition, (2010). Google Scholar

[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. doi: 10.1016/j.jtbi.2009.04.010. Google Scholar

[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. doi: 10.1016/j.jtbi.2003.12.011. Google Scholar

[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. Google Scholar

[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. doi: 10.1080/07362990903415882. Google Scholar

[8]

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

[9]

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

[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. doi: 10.1128/CMR.00062-09. Google Scholar

[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. doi: 10.1016/j.mbs.2008.02.005. Google Scholar

[12]

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

[13]

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

[14]

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

[15]

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

[16]

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

[17]

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

[18]

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

[19]

J. H. Matis and T. Kiffe, "Stochastic Population Models,", Springer, (2000). Google Scholar

[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. doi: 10.1073/pnas.0308001101. Google Scholar

[21]

M. A. Nowak and R. M. May, "Virus Dynamics,", Oxford Univ. Press, (2000). Google Scholar

[22]

B. Øksendal, "Stochastic Differential Equations: An Introduction with Applications,", Springer, (2000). Google Scholar

[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. Google Scholar

[24]

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

[25]

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

[26]

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

[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. doi: 10.1098/rspb.2003.2339. Google Scholar

[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. Google Scholar

[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. doi: 10.1016/S0025-5564(97)00094-1. Google Scholar

[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. doi: 10.1016/S0893-9659(00)00037-9. Google Scholar

[31]

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

[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. doi: 10.1016/S0025-5564(02)00108-6. Google Scholar

[33]

P. van den Driesssche and J. Watmough, Chapter 6: Further notes on the basic reproduction number,, in, (2008), 159. Google Scholar

[34]

F. Verhulst, "Nonlinear Differential Equations and Dynamical Systems,", Springer-Verlag, (1985). Google Scholar

[35]

S. W. Vidurupola, "Deterministic and Stochastic Models for Early Viral Infection within a Host,", M. S. Thesis, (2010). Google Scholar

[36]

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

[37]

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

[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. Google Scholar

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